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## **Meet the editor**

Professor Abraão Jessé Capistrano de Souza is a graduate of Physics awarded by Federal University of Pará, Brazil (2004) and obtained his master's degree in Physics (2006) and doctorate in Theoretical Physics from Universidade de Brasília, Brazil (2009). His research focused on problems in mathematical physics, quantum fields, astrophysics, cosmology, and gravitation. Also,

he is performing studies in rocket science and robotics focused on undergraduate engineering projects as well as activities in teaching physics and astronomy to a broader audience through institutional extension projects. He is currently a full professor at Federal University of Latin American Integration and collaborator researcher at Casimiro Montenegro Filho Astronomy Center (Technological Park of Itaipu).

## Contents




#### Chapter 10 **Semi-Analytic Techniques for Solving Quasi-Normal Modes 197** Chun-Hung Chen, Hing-Tong Cho and Alan S. Cornell

## Preface

Chapter 7 **The Impact of Baryons on the Large-Scale Structure of the**

Chapter 8 **Cosmological Consequences of a Quantum Theory of Mass**

Chapter 9 **Deformed Phase Space in Cosmology and Black Holes 177**

Chun-Hung Chen, Hing-Tong Cho and Alan S. Cornell

E.A. Mena-Barboza, L.F. Escamilla-Herrera, J.C. López-Domínguez

Weiguang Cui and Youcai Zhang

**Universe 137**

**VI** Contents

**and Gravity 159** Brian Albert Robson

and J. Torres-Arenas

Chapter 10 **Semi-Analytic Techniques for Solving Quasi-Normal Modes 197**

The modern cosmology has been turned into an outstanding field of active research through the years. Historically, since its origin as a search of understanding the universe came from a mythological, religious, or philosophical point of view, the cosmology depended from es‐ sentially what culture, society, and/or what time we are referring to. It was not considered strictly speaking as a *natural science* rather than a philosophical thinking about the world we are living in. The breakthrough occurred in the sixteenth century with Galileo Galilei and his enhanced version of telescope and the finding of new ways of not only thinking about the natural world but new ways *to see* the world. Accordingly, the scenario radically changed in the beginning of the twentieth century. To get this far, it is important to note that this development has been long in the making. This is a fine example of the scientific *way of thinking* and its *modus faciendi* as a human construction of knowledge. More importantly, any advance comes from a collective work and collaborative spirit since our generation is em‐ bedded in the greatest challenging issues of scientific history. For instance, we cite seminal works from Henrietta Leavitt with the observations of variable stars, Albert Einstein's gen‐ eral relativity, Friedman's metric launching the basics of the standard cosmological model of an expanding universe, Vesto Slipher with the measurements of radial velocities of galaxies and the use of spectroscopy for astronomical observations that led to the finding of galactic redshift, Edwin Hubble's expanding universe, and Fritz Zwicky's anomalies in velocities of the clusters of galaxies that were not predicted by Newtonian theory of gravity. Today, such anomaly is called *the dark matter problem*. This was reinforced by Vera Rubin's observations in spiral galaxies in the 1970s also showing a huge discrepancy of Newton's law to observa‐ tions and further developments in both theoretical and observational methods from genera‐ tions of scientists.

Particularly, in the last two decades, we have an intense growth of scientific interest in cos‐ mology due to the improvement of observational instruments of astronomical science, which motivated theoretical advances from the necessity to deal with the resulting huge amount of scientific data that should be taken into account, analyzed, and properly justified. Accordingly, cosmology as a discipline turned to be a scientific ground-based branch of physics. Today, we have more scientific data in modern cosmology (and also astronomy and astrophysics) than we could get rid of it, which makes the present days an exciting era for scientific knowledge.

In 1998, we have the discovery that our universe is not only expanding but expanding in an accelerated way. The currently simpler explanation to this effect is that the universe is fulfil‐ led with a negative pressure fluid (regarded as fluctuations of the quantum vacuum energy density) producing some sort of *dark energy* that drives the universe to speed up. Of course, this is just one of the explanations for the accelerated regime of the universe, which has been

confirmed again by the recent Planck mission (European space agency) in their last report in 2015. Accordingly, more effort is currently made to understand the physics of the so-called dark sector. Intriguingly enough, roughly 4% of the universe is barely known, and the rest is not properly explained by the standard particle physics or cosmology, and new fresh modes of perception are needed.

More recently, we had an incredible technological advance from the discovery of gravita‐ tional waves in the Laser Interferometer Gravitational-Wave Observatory (LIGO) confirm‐ ing Einstein's prediction. It has given to scientific community a brand new open arena for physics and several areas, particularly on the understanding of black hole physics, another challenging problem on gravitational physics and astrophysics.

The title of this book could be no other. The chapters presented discuss different aspects of cosmology pointed out by some experts in different areas. This book is divided into three sections with related chapters. The first section is devoted to cultural aspects of cosmology in which, as a human construction, it influenced other human activities such as arts, philoso‐ phy, and politics during the human history. The subsequent chapters are intended to dis‐ cuss technical scientific advances in the field with capital themes today as the problems of the physics of dark sector (dark matter and dark energy problems), black holes, galaxies, large structure formation, and particles. In fact, it shows our endless searching for better un‐ derstanding of the universe, and it remembers the human odyssey that is a journey of knowledge as a legacy for next generations. It has been a great opportunity to work with InTech's editorial team and such an honor to read all the chapters proposed by professional researchers in the field.

> **Abraão Jessé Capistrano de Souza,** Federal University of Latin American Integration, Casimiro Montenegro Filho Astronomy Center (Technological Park of Itaipu), Brazil

## **Cosmology and Culture**

confirmed again by the recent Planck mission (European space agency) in their last report in 2015. Accordingly, more effort is currently made to understand the physics of the so-called dark sector. Intriguingly enough, roughly 4% of the universe is barely known, and the rest is not properly explained by the standard particle physics or cosmology, and new fresh modes

More recently, we had an incredible technological advance from the discovery of gravita‐ tional waves in the Laser Interferometer Gravitational-Wave Observatory (LIGO) confirm‐ ing Einstein's prediction. It has given to scientific community a brand new open arena for physics and several areas, particularly on the understanding of black hole physics, another

The title of this book could be no other. The chapters presented discuss different aspects of cosmology pointed out by some experts in different areas. This book is divided into three sections with related chapters. The first section is devoted to cultural aspects of cosmology in which, as a human construction, it influenced other human activities such as arts, philoso‐ phy, and politics during the human history. The subsequent chapters are intended to dis‐ cuss technical scientific advances in the field with capital themes today as the problems of the physics of dark sector (dark matter and dark energy problems), black holes, galaxies, large structure formation, and particles. In fact, it shows our endless searching for better un‐ derstanding of the universe, and it remembers the human odyssey that is a journey of knowledge as a legacy for next generations. It has been a great opportunity to work with InTech's editorial team and such an honor to read all the chapters proposed by professional

**Abraão Jessé Capistrano de Souza,**

(Technological Park of Itaipu),

Brazil

Federal University of Latin American Integration, Casimiro Montenegro Filho Astronomy Center

challenging problem on gravitational physics and astrophysics.

of perception are needed.

VIII Preface

researchers in the field.

## **The Importance of Cosmology in Culture: Contexts and Consequences**

Nicholas Campion

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67976

#### **Abstract**

Scientific cosmology is the study of the universe through astronomy and physics. However, cosmology also has a significant cultural impact. People construct anthropological cosmologies (notions about the way the world works), drawing in scientific theories in order to construct models for activities in disciplines, such as politics and psychology. In addition, the arts (literature, film and painting, for example) comment on cosmological ideas and use them to develop plot lines and content. This chapter illustrates examples of such work, arguing that scientific cosmology should be understood as a significant cultural influence.

**Keywords:** cosmology, culture, politics, psychology, literature, film, space travel

#### **1. Introduction**

Modern scientific cosmology is valuable in itself for what it reveals about the nature of the cosmos we inhabit [1]. It is a demonstration of the power of modern science to transform our understanding of who we are and where we came from. However, most cosmologists focus on scientific questions and are not fully aware of the impact of cosmological theories on culture, including politics and the arts. This chapter introduces this wider context on the basis that both scientists and the public should be aware of the broader importance of their work and its influence on the way we think. Cosmologists often rely on the fascination the subject brings: as Rowe observed in his textbook way back in 1968, 'In the fields of astronomy and cosmology we live in a period of excitement' [2]. Cosmology therefore both impacts culture and is described and represented by it. This chapter explores some ways in which this happens. As Muriel Rukeyser wrote, 'The universe is made of stories, not of atoms' [3]; see also Impey [4].

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

If we select four fundamental causes of changes in our perceptions of the world in the last century, then they would be first relativity, second quantum mechanics, third the expanding universe and fourth, the space programme. The first three date from a fairly narrow time band, if we date special relativity from 1905, general relativity from 1915, that the universe is expanding and is much bigger than previous thought from Edwin Hubble's publications from 1924 to around 1930 and quantum mechanics from Niels Bohr and Werner Heisenberg's formulation of the Copenhagen interpretation in 1925–1927 [5]. This epic revision of scientific knowledge of underlying structures of the universe was therefore concentrated into just a quarter of a century. The dramatic period of the human space programme was concentrated into just over 8 years from the first human space flight in 1961 to the Moon landing in 1969.

All have fundamentally altered the way that we think about life here on Earth. Often these changes are taken for granted. For example, mobile phone technology, dependent as it is on satellite networks, is transforming not only the social lives of teenagers in the west, but also the economic muscle of poor farmers across the third world. Meanwhile, super‐fast quantum computing makes use of phenomena such as entanglement and is driving the development of artificial intelligence, and hence of robotics. The implications for society over the next few decades are potentially enormous. The most important conclusion to be drawn from this combination of revolutionary changes is the role of the observer: as the basis of differing perspectives of time and space in relativity, an influence on the world (at least, at the sub‐atomic level) in quantum mechanics, and the witness for the first time, of the spherical earth, hanging in space, in photographs taken by Apollo astronauts in 1968. Such ideas and experiences have decisively underpinned modern ideas that one person's complete individual experience or perception is as equally valid as anyone else's. Einstein is held particularly responsible for these ideas [6, 7] as a result of popular equations between relativity on the one hand, and cultural relativism (the idea that no one culture is superior or inferior to another) on the other. Moral relativism (the idea that no one culture is morally superior or inferior to another) is controversial and widely rejected, but cultural relativism does have beneficial scholarly consequences. This is especially the case in the new field of cultural anthropology in which academic rigour requires that in order to better understand other cultures, researchers must abandon any idea that one culture is superior or inferior to another.

#### **2. Defining cosmology**

The term cosmology can be traced to the 1730s, although its appearance in a scientific sense dates from only after the Second World War [8]. The *logos*, which is the root of 'logy', means 'an account', so that, as a preliminary working definition, cosmology is simply 'an account of the cosmos'. The primary Latin equivalent of the Greek *Kosmos* is *Universus*, from *Unus verto*, or 'changing into one', thereby suggesting unity. We can divide the definitions of cosmology into two: the scientific and the anthropological. The scientific are perhaps the more familiar, but even here there is variation. Scientific definitions range from the narrow, such as 'the study of the universe' [9], to the broad ('the science, theory or study of the universe as an orderly system, and of the laws that govern it; in particular, a branch of astronomy that deals with the structure and evolution of the universe') [10]. The idea that cosmology is synonymous with astronomy is widespread [2, 11]. Yet the relationship between astronomy and cosmology is not settled, and Hawley and Holcomb [9] argue that cosmology is a separate discipline, which combines features of both astronomy and physics. There are other places where disciplines overlap. For example, astronomy has to include geophysics precisely because the Earth itself is in space [12].

Anthropological cosmologies are based on the proposition that ideas about the cosmos are integral part of human cultural and social systems. For example, the archaeologist Timothy Darvill talks of a cosmology as being, 'The world view and belief system of a community based upon their understanding of order in the universe' [13]. George Gumerman and Miranda Warburton argued that '… to truly comprehend a culture we must have some sense of its cosmology – the group's conception of themselves in relation to the heavens' [14]. And without diminishing the scientific status of modern cosmology, ideas do not come from theory and experiment alone, but can be inspired by wider cultural influences, as Holton [15] illustrates in relation to Einstein's reading and education before he formulated the theory of special relativity. Heisenberg [16] actually argued that science and art are parallel attempts to describe the world, and may both be part of a wider cultural picture. Bell [17] then talks of 'complex subsystems of cosmic exchange' which underpin mundane systems of behaviour, such as socio‐economic exchange, but are designed to reinforce individual and social existence within the cosmos.

#### **3. Defining cosmology**

If we select four fundamental causes of changes in our perceptions of the world in the last century, then they would be first relativity, second quantum mechanics, third the expanding universe and fourth, the space programme. The first three date from a fairly narrow time band, if we date special relativity from 1905, general relativity from 1915, that the universe is expanding and is much bigger than previous thought from Edwin Hubble's publications from 1924 to around 1930 and quantum mechanics from Niels Bohr and Werner Heisenberg's formulation of the Copenhagen interpretation in 1925–1927 [5]. This epic revision of scientific knowledge of underlying structures of the universe was therefore concentrated into just a quarter of a century. The dramatic period of the human space programme was concentrated into just over 8 years from the first human space flight in 1961 to the Moon landing in 1969.

All have fundamentally altered the way that we think about life here on Earth. Often these changes are taken for granted. For example, mobile phone technology, dependent as it is on satellite networks, is transforming not only the social lives of teenagers in the west, but also the economic muscle of poor farmers across the third world. Meanwhile, super‐fast quantum computing makes use of phenomena such as entanglement and is driving the development of artificial intelligence, and hence of robotics. The implications for society over the next few decades are potentially enormous. The most important conclusion to be drawn from this combination of revolutionary changes is the role of the observer: as the basis of differing perspectives of time and space in relativity, an influence on the world (at least, at the sub‐atomic level) in quantum mechanics, and the witness for the first time, of the spherical earth, hanging in space, in photographs taken by Apollo astronauts in 1968. Such ideas and experiences have decisively underpinned modern ideas that one person's complete individual experience or perception is as equally valid as anyone else's. Einstein is held particularly responsible for these ideas [6, 7] as a result of popular equations between relativity on the one hand, and cultural relativism (the idea that no one culture is superior or inferior to another) on the other. Moral relativism (the idea that no one culture is morally superior or inferior to another) is controversial and widely rejected, but cultural relativism does have beneficial scholarly consequences. This is especially the case in the new field of cultural anthropology in which academic rigour requires that in order to better understand other cultures, researchers must

The term cosmology can be traced to the 1730s, although its appearance in a scientific sense dates from only after the Second World War [8]. The *logos*, which is the root of 'logy', means 'an account', so that, as a preliminary working definition, cosmology is simply 'an account of the cosmos'. The primary Latin equivalent of the Greek *Kosmos* is *Universus*, from *Unus verto*, or 'changing into one', thereby suggesting unity. We can divide the definitions of cosmology into two: the scientific and the anthropological. The scientific are perhaps the more familiar, but even here there is variation. Scientific definitions range from the narrow, such as 'the study of the universe' [9], to the broad ('the science, theory or study of the universe as an orderly system, and of the laws that govern it; in particular, a branch of astronomy that deals with the

abandon any idea that one culture is superior or inferior to another.

**2. Defining cosmology**

4 Trends in Modern Cosmology

The philosopher Terry Eagleton concluded, '"Culture" is said to be one of the two or three most complex words in the English language' [18]. A good working definition, though, is offered by the anthropologist Clifford Geertz, who argued that culture is 'an historically transmitted pattern of meanings embodied in symbols, a system of inherited conceptions expressed in symbolic forms by means of which men communicate, perpetuate and develop their knowledge of and attitudes toward life' [19]. The influence of cosmology on culture then becomes a matter of exploring its impact on political and religious ideas, and its use in the arts, perhaps mainly in literature, painting and film. Michael Rowan‐Robinson wrote that 'the intellectual horizon of the human race at any time has always been inextricably bound up with the scale of the universe… there can be little doubt that a people's perceived scale of the universe must play a fundamental role in its culture and consciousness' [20]. He could equally have said that culture and consciousness are bound up with conceptions of the universe as a whole. It was Einstein himself who made the case for the cosmologist intervening in culture. In 1936 he wrote, 'the physicist cannot simply surrender to the philosopher the critical contemplation of the theoretical foundations' of the universe, and 'the critical thinking of the physicist cannot possibly be restricted to examination of his own field. He cannot proceed without considering critically a much more difficult problem, the problem of analysing the nature of everyday thinking' [21]. Mostly the cosmologist does not intervene directly in modern culture. Instead other people interpret and represent cosmological ideas.

### **4. Politics**

Cosmology has had a substantial impact on political theory and continues to do so. We can therefore talk about 'political cosmology' [22]. The earliest recorded societies used the stars or planets to represent their rulers or guide their actions, functions which tended to overlap with religion. The Egyptians and Inca saw their rulers embodied in the Sun, the Chinese in the Pole Star, and in Babylon the chief god, Marduk, was represented by Jupiter. On a practical level, most pre‐modern cultures structured their religious sanctuaries, and sometimes their urban communities, according to their conceptions of the cosmos. Hence we may speak of 'cosmopolises' or 'cosmograms'. A 'cosmic state' is one in which the entire state is organized to embody the structure of the cosmos.

The significant step into modelling politics on the cosmos as an organised system was taken by the Greek philosopher Plato (ca. 428/7?–348/7). Plato's cosmos emanated out of a single consciousness which existed in a realm of unchanging eternity. The material world was then an imperfect representation of the world of Ideas (Plato spoke of the physical world being embedded in the world‐soul) and was governed by time, the mathematically regulated, harmonious rhythms measurable by the motions of the planets. Plato's mathematical universe provides a rationale for scientific speculation to the present day [23, 24].

Plato advocated an education emphasising such subjects as music, mathematics and geometry, and a political system based on rule by philosophers, all designed to harmonise society with the cosmos, for the common good. This concept of the perfect society underpins the entire history of utopianism down to the present day. There were two main consequences of Plato's system, with consequences down to the present day. First, the perfect ruler was envisioned as the Philosopher King, whose right to rule was justified by his wisdom and understanding of cosmic principles. Second, the system tended to be authoritarian because, being founded on cosmic principle, it could brook no opposition. Plato's ideas were revived in Renaissance Europe and were to become extremely influential. The notion of human history as the progressive unfolding of the world soul towards a final, perfect condition was central to the ideas of Georg Friedrich Hegel (1770–1831). Hegel's influence on Karl Marx (1818–1883) led the twentieth century philosopher of science, Popper [25], to see Plato's thought as the foundation of modern totalitarianism: where Plato influenced Marx, via Hegel, was in the notion that history has an inescapable trajectory, founded in the structure of the cosmos itself. For revolutionary Marxists, such as Lenin, Stalin and Mao Tse‐tung, it was then inconceivable that anyone could oppose their rule, for to do so was to oppose the cosmos.

Separate to the platonic strand in European political cosmology, the astronomical discoveries from Copernicus onwards all helped shape western politics. Copernicus' argument that the Sun, not the Earth, is the centre of the universe (or the solar system as we would now say) was attached to the ancient idea that the Sun is associated with kings. It was then used to support claims that, just as the entire universe orbits the Sun, so the whole of society orbits the king [26]. Propaganda in support of absolute monarchy then reached its height in the iconography of the French king Louis XIV. Such authoritarian ideas were directly countered by what I have called Political Newtonianism [22]. This held that, just as Newton had argued that one law governed the whole universe, so the same principle must apply to terrestrial affairs and one law must govern the whole of human society. In principle, then, all people were equal, and there was no justification for monarchy. Such ideas were influential among both the American revolutionaries of 1776 [27] and the French in 1789.

Newtonianism was taken one step further by Auguste Comte (1798–1857), the founder of sociology. Adding Galileo and Kepler to Newton as sources of authority, Comte [28] argued that if the entire universe was a mathematically regulated mechanism, so human society must also be governed by the same principle. If planets moved in mathematically determined patterns, Comte reasoned, so must people. By collecting and analysing data on human behaviour, Comte concluded, the same laws that controlled the wider universe should be discovered in human affairs. And in turn the state, governed by experts who were the modern equivalent of Plato's philosophers, could be managed for the good of all. This remains the foundation of twenty‐first century sociology.

There have been a few attempts, for example, to identify Einsteinian relativity either as a form of political discourse, or to draw political implications from it. As far as the former is concerned, I refer to the French feminist and social theorist Luce Irigaray who has identified the theory of relativity as a political rather than scientific formula [29]. Sokal and Bricmont [28], meanwhile, noted how the notion of relativity in time and space was used by postmodern theorists in order to advocate cultural relativity on the grounds that, if the universe has no single centre, neither does culture.

The anthropologist Falzon [30] has defended multi‐sited ethnography against the charge of lack of depth by arguing that it takes into account shifting perceptions of space and time. The anthropologist Marcus [31] refers to 'space‐time compression', in which the essential difference between space and time contracts in light of the recognition that both are socially produced as a result of what Falzon calls 'a product of interrelations'; Marcus derived his understanding from special relativity, and so directly from his understanding of Einstein. Elsewhere, Einstein's call for humanity in general to take on the implications of the new cosmology has been used to advocate a collaborative global order in which international problems are solved thought global institutions rather than war [32].

#### **5. Psychology**

**4. Politics**

6 Trends in Modern Cosmology

to embody the structure of the cosmos.

Cosmology has had a substantial impact on political theory and continues to do so. We can therefore talk about 'political cosmology' [22]. The earliest recorded societies used the stars or planets to represent their rulers or guide their actions, functions which tended to overlap with religion. The Egyptians and Inca saw their rulers embodied in the Sun, the Chinese in the Pole Star, and in Babylon the chief god, Marduk, was represented by Jupiter. On a practical level, most pre‐modern cultures structured their religious sanctuaries, and sometimes their urban communities, according to their conceptions of the cosmos. Hence we may speak of 'cosmopolises' or 'cosmograms'. A 'cosmic state' is one in which the entire state is organized

The significant step into modelling politics on the cosmos as an organised system was taken by the Greek philosopher Plato (ca. 428/7?–348/7). Plato's cosmos emanated out of a single consciousness which existed in a realm of unchanging eternity. The material world was then an imperfect representation of the world of Ideas (Plato spoke of the physical world being embedded in the world‐soul) and was governed by time, the mathematically regulated, harmonious rhythms measurable by the motions of the planets. Plato's mathematical universe

Plato advocated an education emphasising such subjects as music, mathematics and geometry, and a political system based on rule by philosophers, all designed to harmonise society with the cosmos, for the common good. This concept of the perfect society underpins the entire history of utopianism down to the present day. There were two main consequences of Plato's system, with consequences down to the present day. First, the perfect ruler was envisioned as the Philosopher King, whose right to rule was justified by his wisdom and understanding of cosmic principles. Second, the system tended to be authoritarian because, being founded on cosmic principle, it could brook no opposition. Plato's ideas were revived in Renaissance Europe and were to become extremely influential. The notion of human history as the progressive unfolding of the world soul towards a final, perfect condition was central to the ideas of Georg Friedrich Hegel (1770–1831). Hegel's influence on Karl Marx (1818–1883) led the twentieth century philosopher of science, Popper [25], to see Plato's thought as the foundation of modern totalitarianism: where Plato influenced Marx, via Hegel, was in the notion that history has an inescapable trajectory, founded in the structure of the cosmos itself. For revolutionary Marxists, such as Lenin, Stalin and Mao Tse‐tung, it was then inconceivable

provides a rationale for scientific speculation to the present day [23, 24].

that anyone could oppose their rule, for to do so was to oppose the cosmos.

Separate to the platonic strand in European political cosmology, the astronomical discoveries from Copernicus onwards all helped shape western politics. Copernicus' argument that the Sun, not the Earth, is the centre of the universe (or the solar system as we would now say) was attached to the ancient idea that the Sun is associated with kings. It was then used to support claims that, just as the entire universe orbits the Sun, so the whole of society orbits the king [26]. Propaganda in support of absolute monarchy then reached its height in the iconography of the French king Louis XIV. Such authoritarian ideas were directly countered by what I have called Political Newtonianism [22]. This held that, just as Newton had argued that one law The consequences of Newtonianism (the belief that the entire universe is mathematically regulated) permeate western thought wherever there has been a search for a universal law based on supposedly hard data. Psychology is a prime example. When the word 'psychology' was coined around 1800, it was thought that, since Newtonian science explained everything that exists and occurs in the material world, there could and should be just one science explaining what exists and occurs in the psychological world. As Gilbert Ryle wrote, '"Psychology" was supposed to be the title of the empirical study of "mental phenomena"', a counterpart to Newtonian celestial mechanics [33]. Concepts such as normality and deviation have dominated some of the major schools of western psychology, their roots in Newtonian cosmology's devotion to predictable order unrecognised and forgotten. The measurement and mathematical analysis of the human mind then became the basis of much psychiatry and academic psychology. Even the notion developed by Sigmund Freud (1856–1939), that psychic—or psychological—material can be 'repressed', as if by some kind of downward pressure, is derived from Newtonian mechanics through the influential German physicist Hermann von Helmholtz (1821–1894) [34]. The goal of Freudian psychoanalysis is for the patient to become aware of such repressed material through the so‐called talking cure, the conversations which take place during sessions with the psychoanalyst, thus releasing in downward pressure.

Freud's student C.G. Jung opened a radically different strand of thought in modern psychology which is highly influential in many schools of psychotherapy and counselling practiced in society as a whole, although usually outside the academic system. Jung revived the Platonic theory that everything in the world is a manifestation of an original pure idea or archetype. Jung's system, known as analytical psychology, adheres to a kind of archetypal philosophy in which all psychological types correspond to an archetype, such as the eternal youth (the puer aeternus), anima (female principle) or senex (wise old man), which exist in the collective unconscious, Jung's update of the Platonic world‐soul. The aim of Jung's therapeutic model is for the individual to become truly themselves by recognising the role of the archetypes in their lives and, in effect, understanding their true connections to the cosmos. The idea that one can become one who truly is also relates back to Aristotelian cosmology in which it was thought that the four elements (fire, earth, air and water) all try to find their natural place in the world. This, Aristotle thought, was why flames go up to the sky, where fire belongs, and water falls to the ground, because that is where it finds its natural home. In Aristotelian politics, kings are at the top of society and peasants at the bottom, because that is the natural state of affairs; in Aristotelian psychology every individual then has a natural way of being. Jung, though, was equally concerned with the latest science, and formed a collaboration and friendship with the quantum physicist Wolfgang Pauli (1900–1958) [35]. Together they formulated the concept of synchronicity by which meaningful events are connected because they take place at the same time, without any causal connection [36]. Newtonian psychology—the belief that all mental states can be measured—survives in university departments and psychiatry. But in the wider world, where increasing numbers of people seek counselling and psychotherapy, Einstein is taken as the inspiration for the argument that therapists and analysts must be 'less concerned with the basic nature of time and more with the human experience of it' [37].

#### **6. Literature and film**

One of the major genres of writing in western culture goes under the name 'celestial journey literature', derived from ancient texts on the soul's journey. The soul's journey was secularised in Dante's (1265–1321) 'Divine Comedy' [38]. Inspired by Plato's myth of Er, Dante is guided by the poet Virgil and his love, Beatrice, through the spheres of Hell, Purgatory and Paradise (Hell and Purgatory are structured in spheres analogous to the spheres on which the planets and stars orbit). The last great example of the celestial journey of the soul or a dream world was Johannes Kepler's 'Somnium', an account of lunar astronomy written in 1608 but published in 1634, and sometimes referred to as the first work of science fiction. The genre took a decisive step forward in Francis Godwin's 'Man in the Moon' [39], published in 1638. Like Dante, Godwin used his story to describe the structure of the cosmos, now, after Johannes Kepler and Galileo, rejecting the planetary spheres and challenging the Aristotelian idea that all things have their natural place. Godwin departed from the old idea of a journey of the soul or a dream world. Instead, his hero, Gonzales, flies to the Moon carried by giant geese. From Godwin onwards the celestial journey becomes physical, arriving at the Moon rocket in Jules Verne's 1865 novel, 'From the Earth to the Moon'. Verne's book was one of the inspirations for what may be the first space film, Georges Méliès' 'Le Voyage dans la Lune'—in English 'A Trip to the Moon' after which astronomy and cosmology have a regular presence in film culture [40]. In Méliès' film, the astronomers encounter the inhabitants of the Moon, known as Selenites, in what is clearly a parable for European colonialism: the film was released midway through the so‐called 'Scramble for Africa', the final face of the European take‐over of Africa from the 1880s to 1914. From Méliès on, the major celestial journey novels have often been filmed. Perhaps the most famous is Stanley Kubrick's film of Arthur C. Clarke's '2001: A Space Odyssey'. Clarke's metaphysical story is vividly portrayed first through the transition from ape to human, and then, in the final scene, the transformation of the dying astronaut into the star child. Accompanied by the stirring music of Wagner's 'Also sprach Zarathustra', Kubrick created a vivid evocation of both ancient beliefs in the soul's ascent to the stars and modern ideas that human destiny may take us to realms beyond our current imagination.

devotion to predictable order unrecognised and forgotten. The measurement and mathematical analysis of the human mind then became the basis of much psychiatry and academic psychology. Even the notion developed by Sigmund Freud (1856–1939), that psychic—or psychological—material can be 'repressed', as if by some kind of downward pressure, is derived from Newtonian mechanics through the influential German physicist Hermann von Helmholtz (1821–1894) [34]. The goal of Freudian psychoanalysis is for the patient to become aware of such repressed material through the so‐called talking cure, the conversations which take place during sessions with the psychoanalyst, thus releasing in downward pressure.

Freud's student C.G. Jung opened a radically different strand of thought in modern psychology which is highly influential in many schools of psychotherapy and counselling practiced in society as a whole, although usually outside the academic system. Jung revived the Platonic theory that everything in the world is a manifestation of an original pure idea or archetype. Jung's system, known as analytical psychology, adheres to a kind of archetypal philosophy in which all psychological types correspond to an archetype, such as the eternal youth (the puer aeternus), anima (female principle) or senex (wise old man), which exist in the collective unconscious, Jung's update of the Platonic world‐soul. The aim of Jung's therapeutic model is for the individual to become truly themselves by recognising the role of the archetypes in their lives and, in effect, understanding their true connections to the cosmos. The idea that one can become one who truly is also relates back to Aristotelian cosmology in which it was thought that the four elements (fire, earth, air and water) all try to find their natural place in the world. This, Aristotle thought, was why flames go up to the sky, where fire belongs, and water falls to the ground, because that is where it finds its natural home. In Aristotelian politics, kings are at the top of society and peasants at the bottom, because that is the natural state of affairs; in Aristotelian psychology every individual then has a natural way of being. Jung, though, was equally concerned with the latest science, and formed a collaboration and friendship with the quantum physicist Wolfgang Pauli (1900–1958) [35]. Together they formulated the concept of synchronicity by which meaningful events are connected because they take place at the same time, without any causal connection [36]. Newtonian psychology—the belief that all mental states can be measured—survives in university departments and psychiatry. But in the wider world, where increasing numbers of people seek counselling and psychotherapy, Einstein is taken as the inspiration for the argument that therapists and analysts must be 'less concerned

with the basic nature of time and more with the human experience of it' [37].

One of the major genres of writing in western culture goes under the name 'celestial journey literature', derived from ancient texts on the soul's journey. The soul's journey was secularised in Dante's (1265–1321) 'Divine Comedy' [38]. Inspired by Plato's myth of Er, Dante is guided by the poet Virgil and his love, Beatrice, through the spheres of Hell, Purgatory and Paradise (Hell and Purgatory are structured in spheres analogous to the spheres on which the planets and stars orbit). The last great example of the celestial journey of the soul or a dream world was Johannes Kepler's 'Somnium', an account of lunar astronomy written in 1608 but published in 1634,

**6. Literature and film**

8 Trends in Modern Cosmology

While celestial journey films can be enjoyed as simple adventures, they often contain deeper meanings. The 1951 movie 'The Day the Earth Stood Still' exploited the current public interest in Flying Saucers. Released at a time when the Cold War was reaching its height with the conflict in Korea, the story featured a wise alien who arrived from space in order to reveal to humanity the error of its ways. The rejection of the alien's words of wisdom presented a gloomy view of humanity as incapable of solving its problems. Cosmology, through film, then becomes a means of commenting on societal change. 'Star Trek', the biggest celestial journey TV franchise of them all, was launched in 1966. 'Star Trek' was altogether more sophisticated than 'Lost in Space' and was entirely more optimistic. It is set in a utopian future in which there is one world government, collaborating with other worlds through United Federation of Planets, and money has been abolished. In the crew's adventures, the European voyages of the fifteenth and sixteenth centuries are replayed in a universe of an infinite number of galaxies, except that now alien cultures are to be respected and preserved rather than conquered and destroyed. The values espoused by the Federation were American: freedom from tyranny, freedom of expression and respect for minorities. Compared to 'Star Trek', the blockbuster film franchise, 'Star Wars', launched in 1976 and still going strong forty years later, projects into space a simpler version of the endless struggle for freedom against an evil empire, very much an update of anti‐Nazi war films. In all such cases the cosmos is seen as a blank slate, a tabula rasa, on which human concerns are imposed.

There is a constant strand of literary comment on cosmology in the nineteenth century. Edgar Allan Poe (1809–1849), often known as the author of the first detective novel, wrote a remarkable work which he called 'Eureka' [41], deliberately suggesting an imaginative breakthrough in the understanding of the cosmos. Poe realised that in a Newtonian universe the stars are likely to collapse in on each other, and that therefore the universe must be evolving [42]. Thomas Hardy (1840–1928) was as fascinated by cosmology as Poe, but unlike him lived to see the publication of Darwin's 'Origin of Species' in 1959 (when he was just 19). For Hardy evolution was a reality. He combined his encyclopaedic knowledge of myth, astronomy and cosmology into a 'moral astrophysics' [43] which provided the background for the individual conflicts and tragedies in novels such as 'Far from the Madding Crowd' (1874) and 'Tess of the d'Urbervilles' (1891/1892). Virginia Woolf (1882–1941) used Edwin Hubble's discovery of the expanding universe in the 1920s as a metaphor for personal and political insecurity in the 1930s. In Woolf's view, as we all live together on the same delicate, vulnerable planet in a vastness of space, it is incumbent upon us to live together rather than fight [44]. The Marxist playwright Bertolt Brecht (1898–1956) looked back to an earlier cosmology but equally wanted to illustrate a modern point in his 1938 play, 'Galileo'. His portrayal of Galileo as the heroic intellectual defending Copernicus, struggling against an obscurantist Inquisition (inaccurate because many in the senior Catholic hierarchy were Copernicans), was an allegory of the revolutionary struggles of the 1930s.

One of the other familiar tropes derived from modern cosmology is time travel, a topic rarely dealt with in 'Star Trek', in spite of the regular use of faster‐than‐light travel. There is now a considerable literature which draws on Gerald Feinberg's 1967 paper 'Possibility of faster‐ than‐light particles', which proposed the existence of the tachyon [45]. This hypothetical particle, the tachyon, might as Martin Rees [46], says alter the order of events, if a signal from a tachyon arrived before it was sent. The genre's earliest notable example was Wells' novel 'The Time Machine' (1895) [47]. Wells coined the phrase time machine to describe a time‐travelling vehicle which moved because the fourth dimension was of time rather than space. Wells was a utopian socialist and his main preoccupation was to explore varieties of human society, considering whether progress inevitably resulted in human improvement: it's clear that he didn't think that this was the case, and that ignorance and superstition could easily flourish in the future. Neither does 'Dr Who', the most successful TV time travel franchise, explain how it is possible to travel to the distant past or future. The time machine, the TARDIS (short for Time and Relative Dimensions in Space), is bigger inside than outside and references Einstein by referencing relativity in its name. The Doctor himself is increasingly represented as a lonely figure, destined to exist in perpetual sadness caused by the death or departure of his companions.

The concept of alienation is developed by Alan Lightman in his 'Einstein's Dreams' [48], a journal set in 1905—the 'annus mirabilis' when Einstein developed the theory of Special Relativity. Lightman's character experiences the alienation of a world in which any particular point in space‐time is delicate, temporary and liable to vanish, an 'exile in time' [48]. In a later entry, Lightman's diarist writes 'There is a place where time stands still. Raindrops hang motionless in air. Pendulums of clocks float mid‐swing' [49]. At the centre of space‐time nothing moves. The concept that all time exists simultaneously actually has a long lineage. It is central to Plato's cosmology, occurs in the Bible (Ecclesiastes 3.15), and was elaborated by St. Augustine (V.9) in the fifth century [50]. He described a universal paradox whereby even if a future event in our individual lives already exists, it depends on an act of our free‐will in order to take place. T.S. Elliot, impressed by Einstein, combined the lessons of relativity with Plato, Ecclesiastes and Augustine. Following Einstein's English visit in 1921, the year he won the Nobel Prize, Elliot wrote, 'Einstein the Great has visited England (and) has taken his place in the newspapers with the comet, the sunspots, the poisonous xxx‐jellyfish and octopus at Margate, and other natural phenomena' [51].

In two poems composed in the 1930s, and published in 1941–1942, Elliot considered the conundrum of time for the human condition. In 'Burnt Norton' he wrote that 'All time is unredeemable', for if the past exists in the future and the future exists in the past, all possibilities are eternally present, and in 'East Coker' he wrote the famous line 'In my beginning is my end' [52]. Elliot's speculations on time were shared by Priestly in such metaphysical plays as 'Time and the Conways' (1937) and, perhaps his most famous work, 'An Inspector Calls' (1947), in which a detective from the future extracts confessions of guilt for a poor girl's suicide from a comfortable middle class family. Priestley's immediate inspiration was Dunne's [53] work on time, which drew on Einstein (a cautiously supportive note was included by Arthur Eddington in the appendix to the third edition) in order to explain why the future could be predicted by precognition.

The popular end of such speculation is best represented by the collected works of Philip K. Dick (1928–1982). Like Elliot, Dick was inspired by ancient philosophy and modern science, especially the conclusions of quantum mechanics as expressed in Heisenberg's uncertainty principle and Erwin Schrȍdinger's famous thought‐experiment with the cat (1935), ideas responsible for modern multi‐verse theory. If one cannot tell both where a particle is and where it is travelling to, whether it is even a particle at all (or a wave), and how far the act of observing it has altered its state, how can one ever trust what appears to be real. For example, in the 'The Cosmic Puppets' (1957), an ordinary suburban couple return to their hometown after a gap of several years to find that everyone and everything has changed, and nobody recognises them. The novel then shifts into a traditional religious mode, located in the Zoroastrian (Persian) struggle between the good god Ormazd and his evil rival Ahriman. Eventually Ormazd triumphs, the illusory world created by Ahriman is removed, and reality returns. In Dick's award‐winning counter‐factual history, 'The Man in the High Castle' (1962), the ability of the observer to act on—and change—the material world is described via the lead characters' use of the Chinese oracle, the I Ching, to alter the future.

Dick's intensity is absent from the most whimsical of recent cosmological fiction, that of Italo Calvino (1923–1985). Calvino took cosmological ideas and exaggerated them until they were absurd. His short story, 'The Form of Space' (2002), points out that if one fell in curved space, one would logically fall for ever, while 'the Distance of the Moon' imagines a time in the distant past when the Moon was closer to the Earth, close enough for people to jump up to it and gather such delicacies as Moon‐milk.

#### **7. The visual arts**

likely to collapse in on each other, and that therefore the universe must be evolving [42]. Thomas Hardy (1840–1928) was as fascinated by cosmology as Poe, but unlike him lived to see the publication of Darwin's 'Origin of Species' in 1959 (when he was just 19). For Hardy evolution was a reality. He combined his encyclopaedic knowledge of myth, astronomy and cosmology into a 'moral astrophysics' [43] which provided the background for the individual conflicts and tragedies in novels such as 'Far from the Madding Crowd' (1874) and 'Tess of the d'Urbervilles' (1891/1892). Virginia Woolf (1882–1941) used Edwin Hubble's discovery of the expanding universe in the 1920s as a metaphor for personal and political insecurity in the 1930s. In Woolf's view, as we all live together on the same delicate, vulnerable planet in a vastness of space, it is incumbent upon us to live together rather than fight [44]. The Marxist playwright Bertolt Brecht (1898–1956) looked back to an earlier cosmology but equally wanted to illustrate a modern point in his 1938 play, 'Galileo'. His portrayal of Galileo as the heroic intellectual defending Copernicus, struggling against an obscurantist Inquisition (inaccurate because many in the senior Catholic hierarchy were Copernicans), was an allegory of the

One of the other familiar tropes derived from modern cosmology is time travel, a topic rarely dealt with in 'Star Trek', in spite of the regular use of faster‐than‐light travel. There is now a considerable literature which draws on Gerald Feinberg's 1967 paper 'Possibility of faster‐ than‐light particles', which proposed the existence of the tachyon [45]. This hypothetical particle, the tachyon, might as Martin Rees [46], says alter the order of events, if a signal from a tachyon arrived before it was sent. The genre's earliest notable example was Wells' novel 'The Time Machine' (1895) [47]. Wells coined the phrase time machine to describe a time‐travelling vehicle which moved because the fourth dimension was of time rather than space. Wells was a utopian socialist and his main preoccupation was to explore varieties of human society, considering whether progress inevitably resulted in human improvement: it's clear that he didn't think that this was the case, and that ignorance and superstition could easily flourish in the future. Neither does 'Dr Who', the most successful TV time travel franchise, explain how it is possible to travel to the distant past or future. The time machine, the TARDIS (short for Time and Relative Dimensions in Space), is bigger inside than outside and references Einstein by referencing relativity in its name. The Doctor himself is increasingly represented as a lonely figure, destined to exist in perpetual sadness caused by the death or departure of

The concept of alienation is developed by Alan Lightman in his 'Einstein's Dreams' [48], a journal set in 1905—the 'annus mirabilis' when Einstein developed the theory of Special Relativity. Lightman's character experiences the alienation of a world in which any particular point in space‐time is delicate, temporary and liable to vanish, an 'exile in time' [48]. In a later entry, Lightman's diarist writes 'There is a place where time stands still. Raindrops hang motionless in air. Pendulums of clocks float mid‐swing' [49]. At the centre of space‐time nothing moves. The concept that all time exists simultaneously actually has a long lineage. It is central to Plato's cosmology, occurs in the Bible (Ecclesiastes 3.15), and was elaborated by St. Augustine (V.9) in the fifth century [50]. He described a universal paradox whereby even if a future event in our individual lives already exists, it depends on an act of our free‐will in order to take place. T.S. Elliot, impressed by Einstein, combined the lessons of relativity with

revolutionary struggles of the 1930s.

10 Trends in Modern Cosmology

his companions.

Representations of the sky, stars or cosmos in visual form date back to the Stone Age and are familiar throughout the ancient world. They may be symbolic, as in Egyptian astronomical‐ceilings, or take on human form, as in Roman images of planetary deities. Later they might be decorative, as in Renaissance star maps, or attempt accuracy, as in modern star maps, or be entirely abstract, as in twentieth century surrealism. The Sun and Moon make regular appearances in western painting, as one would expect. The cosmological statements, though, are often simple. Often the Sun and Moon are poetic additions, symbolising time or heaven in medieval and Renaissance art, casting light or embodying the power of nature, and even serve as political satire in the nineteenth century [54]. Cosmological changes encouraged the new spirit: the philosopher Berlin [55] argued that the astronomical revolution's abandonment of the old crystalline planetary spheres in favour of a universe without boundaries encouraged the emergence of the chaos and adventurism of Romantic art. Painting began the shift towards the abstract with the extraordinary work of Joseph Turner, who drew on both esoteric wisdom and the latest science in his portrayal of light [56, 57]. Perhaps the most famous example of nineteenth century astronomical art is van Gogh's 1889 masterpiece 'The Starry Night', a painting partly inspired by the pre‐dawn rising of Venus, but easily interpreted as representing the swirling chaos of van Gogh's inner world.

The relationship between modern art movements and science is complicated by many artists' multiple affiliations. For example, many notable early twentieth century painters were followers of Theosophy, a spiritual teaching highly indebted to Plato, Renaissance alchemy and Freudian psychoanalysis, all of which could deal with unseen realities and the interdependence of all things in the cosmos. It is therefore not easy to distinguish scientific influences on twentieth art from mystical or magic ones and it is up to art historians to interpret [58]. However, it is clear that the new physics encouraged the move towards radical, abstract forms of expression. Heisenberg's uncertainty principle appears to have supported the playful, apparently chaotic, practices of Dada, in which nothing is quite as it seems. When Marcel Duchamp (1887–1968) displayed a urinal in 1917 he was making a radical statement that, if art is not what one imagines it is, neither is anything else.

André Breton (1896–1966), the poet and author of the Surrealist Manifesto, singled out Einstein (along with Freud) as significant in 1922. J.W. Dunne's adaptation of Einstein to precognition and psychology was popular with the surrealists, as it as with Priestley. That the observer stands at the centre of time and space, as popular conceptions of relativity and quantum mechanics assume, underpins the playful juxtaposition of images and ideas, sense and nonsense, which runs through the entire history of modern abstract and conceptual art. When Joan Miro paints a picture such as 'Dog barking at the Moon' (1926), he is alluding to ideas that the Moon makes one mad— lunatic—as well as departing significantly from naturalism, but also raising a smile.

A separate strand of painting drew on mathematical conceptions of the universe, as did Duchamp's earlier painting 'Nude Descending a Staircase' (1912), or Man Ray's (1890–1976) geometrical models, such as 'Polyèdres' (1934–1936). The distinctive angular lines of Picasso's painting were also decisively influenced by the idea of multi‐dimensional mathematics as explained by Miller [59]. By contrast, the whirling lines in Max Ernst's '…sur le plan de la Physique' (1943) evoke the spinning of atoms. And for Klee [60], writing in 1920, movement in painting was essential because everything in the universe is characterised by motion. Perhaps the most famous portrayals of the new physics are Salvador Dali's (1904–1989) paintings of bent clocks as representations of distorted space‐time, as in 'The Persistence of the Disintegration of Memory' (1952).

#### **8. Space flight**

astronomical‐ceilings, or take on human form, as in Roman images of planetary deities. Later they might be decorative, as in Renaissance star maps, or attempt accuracy, as in modern star maps, or be entirely abstract, as in twentieth century surrealism. The Sun and Moon make regular appearances in western painting, as one would expect. The cosmological statements, though, are often simple. Often the Sun and Moon are poetic additions, symbolising time or heaven in medieval and Renaissance art, casting light or embodying the power of nature, and even serve as political satire in the nineteenth century [54]. Cosmological changes encouraged the new spirit: the philosopher Berlin [55] argued that the astronomical revolution's abandonment of the old crystalline planetary spheres in favour of a universe without boundaries encouraged the emergence of the chaos and adventurism of Romantic art. Painting began the shift towards the abstract with the extraordinary work of Joseph Turner, who drew on both esoteric wisdom and the latest science in his portrayal of light [56, 57]. Perhaps the most famous example of nineteenth century astronomical art is van Gogh's 1889 masterpiece 'The Starry Night', a painting partly inspired by the pre‐dawn rising of Venus, but easily interpreted as representing the

The relationship between modern art movements and science is complicated by many artists' multiple affiliations. For example, many notable early twentieth century painters were followers of Theosophy, a spiritual teaching highly indebted to Plato, Renaissance alchemy and Freudian psychoanalysis, all of which could deal with unseen realities and the interdependence of all things in the cosmos. It is therefore not easy to distinguish scientific influences on twentieth art from mystical or magic ones and it is up to art historians to interpret [58]. However, it is clear that the new physics encouraged the move towards radical, abstract forms of expression. Heisenberg's uncertainty principle appears to have supported the playful, apparently chaotic, practices of Dada, in which nothing is quite as it seems. When Marcel Duchamp (1887–1968) displayed a urinal in 1917 he was making a radical statement that, if art is not what one imagines it is, neither is anything else.

André Breton (1896–1966), the poet and author of the Surrealist Manifesto, singled out Einstein (along with Freud) as significant in 1922. J.W. Dunne's adaptation of Einstein to precognition and psychology was popular with the surrealists, as it as with Priestley. That the observer stands at the centre of time and space, as popular conceptions of relativity and quantum mechanics assume, underpins the playful juxtaposition of images and ideas, sense and nonsense, which runs through the entire history of modern abstract and conceptual art. When Joan Miro paints a picture such as 'Dog barking at the Moon' (1926), he is alluding to ideas that the Moon makes one mad— lunatic—as well as departing significantly from naturalism,

A separate strand of painting drew on mathematical conceptions of the universe, as did Duchamp's earlier painting 'Nude Descending a Staircase' (1912), or Man Ray's (1890–1976) geometrical models, such as 'Polyèdres' (1934–1936). The distinctive angular lines of Picasso's painting were also decisively influenced by the idea of multi‐dimensional mathematics as explained by Miller [59]. By contrast, the whirling lines in Max Ernst's '…sur le plan de la Physique' (1943) evoke the spinning of atoms. And for Klee [60], writing in 1920, movement in painting was essential because everything in the universe is characterised by motion.

swirling chaos of van Gogh's inner world.

12 Trends in Modern Cosmology

but also raising a smile.

It is well understood that the photographs of the whole Earth taken by the Apollo astronauts encouraged concepts of the global village, a world devoid of racial divisions, religious schism and political boundaries [61, 62]. The first major use of the Apollo photographs was on the cover of the first Whole Earth Catalogue in 1968, placed there by the editor, political activist Stewart Brand. Subsequently, the photographs became an inspiration for the emerging environmental movement.

Since 1968/1969 we have been able to look down on our sky from space. The euphoric consequences of this experience, still enjoyed by a few hundred people, was named the 'Overview Effect' by Frank White in 1987 [63]. Interviewed on BBC Radio 4's iPM programme on 25 May 2013, the astronaut Geoff Hoffman described his own experience of the effect [64]. He recalled the strange sensation of looking down at the Earth, watching the terrestrial sky from above instead of from below, witnessing the flash of lightning storms and streaks of light as meteors plunged into the atmosphere. He saw the world as one, drawing salutary ecological lessons from the visible deforestation of tropical areas. Inspired by the ethereal nature of the Earth's halo, Hoffman hesitated to use the word 'spiritual', put to him by his interviewer in a leading question, but was happy to describe the condition he experienced on his mission as being a 'state of grace', words which he said had been suggested to him by a Jesuit priest. Shamans, Pharaohs and Platonic souls may have seen the Earth in their imaginations, but astronauts experience it physically.

The 'Overview Effect' has been institutionalised in the Overview Institute, whose purpose is to utilise the Effect for the common good. The Institute's apocalyptic and utopian agenda draws a direct connection between the experience of space travel and the need to save the Earth: 'We live at a critical moment in human history. The challenges of climate change, food, water and energy shortages as well as the increasing disparity between the developed and developing nations are testing our will to unite, while differences in religions, cultures, and politics continue to keep us apart. The creation of a 'global village' through satellite TV and the Internet is still struggling to connect the world into one community. At this critical moment, our greatest need is for a global vision of planetary unity and purpose for humanity as a whole' [65]. In this sense, the institute completes the earlier visions of Virginia Woolf and Stewart Brand.

#### **9. Conclusion**

Modern scientific cosmology needs to be valued not just for what it tells us about the universe, but for how what it tells us informs the ways that people think and behave in wider culture. A number of themes emerge, including the vastness of space as a metaphor for loneliness and insecurity, and the new physics as a source of freedom and adventure. Scientific cosmology's wider significance needs to be more widely acknowledged, for modern society still benefits from 'complex subsystems of cosmic exchange' between scientists —cosmologists—and the general public.

#### **Author details**

Nicholas Campion

Address all correspondence to: n.campion@tsd.uwtsd.ac.uk

University of Wales Trinity Saint David, Swansea, Wales, UK

#### **References**


number of themes emerge, including the vastness of space as a metaphor for loneliness and insecurity, and the new physics as a source of freedom and adventure. Scientific cosmology's wider significance needs to be more widely acknowledged, for modern society still benefits from 'complex subsystems of cosmic exchange' between scientists —cosmologists—and the general public.

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**Dark Matter Phenomenology**

## **Constraining the Parameters of a Model for Cold Dark Matter**

### Abdessamad Abada and Salah Nasri

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69044

#### Abstract

This chapter aims at reviewing how modeling cold dark matter as weakly interacting massive particles (WIMPs) gets increasingly constrained as models have to face stringent cosmological and phenomenological experimental results as well as internal theoretical requirements like those coming from a renormalization-group analysis. The review is based on the work done on a two-singlet extension of the Standard Model of elementary particles. We conclude that the model stays viable in physically meaningful regions that soon will be probed by direct-detection experiments.

PACS numbers: 95.35.þd; 98.80.�k; 12.15.�y; 11.30.Qc

Keywords: cold dark matter, light WIMP, extension of Standard Model, rare decays, RGE

#### 1. Introduction

Dark matter accounts for about 26.5% of the total mass-energy density of the Universe [1], but we still do not know what it is. It is called dark because it is not accounted by the visible matter, the conventional baryons and leptons, which take about 4.9% of the total mass-energy density [1]. As it clearly interacts through gravity, some argue that it could still be baryonic, in the form of massive astrophysical compact halo objects (MACHOs) which emit dim or no light [2] or some sort of huge gravitational objects like galaxy-sized black holes. Indeed, such high concentrations of matter would bend passing light, the so-called gravitational lensing phenomenon, including microlensing, in ways we can detect. But the amount of dark matter we know of would produce gravitational lensing with a significantly higher number of occurrences than what observation accounts for.

Neutrinos have long been thought of composing the dark matter around us. However, Standard Model neutrinos are light, and so too fast-moving (hot) to compose the (cold) dark matter

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

structures we see. But sterile neutrinos, non-Standard Model particles, can be heavier, and so could be dark matter candidates. This possibility has been reignited with the recent detection of an X-ray emission line at an energy of 3:55 keV coming from galaxy clusters, the Andromeda galaxy, the Galactic Center and the Draco dwarf spheroidal galaxy. This line is consistent with the decay of a 7:1 keV sterile neutrino [3].

In fact, there is by now quasi-consensus that dark matter ought to be understood outside the realm of conventional matter. One other scenario is that of (pseudo)scalar particles of tiny mass � <sup>10</sup>�<sup>22</sup> eV, the so-called ultralight axions that could account for the dark matter content of the Universe. This is supported by high-resolution cosmological simulations [4]. Axions originated in quantum chromodynamics, the theory of quarks and gluons, in relation to the axial anomaly in this theory and the strong Charge Conjugation Parity Symmetry Violation (CP violation) problem. But like anything else related to dark matter, they elude detection. The Axion Dark Matter Experiment (ADMX) may bring in answers in the near future [5].

But maybe the most popular candidate for dark matter is an electrically neutral and colorless weakly interacting massive particle (WIMP). Such a particle originated in supersymmetric (SUSY) extensions of the StandardModel. The most obvious such a candidate is the neutralino, a neutralRodd supersymmetric particle. Indeed, neutralinos are only produced or destroyed in pairs, thus constituting the lightest SUSYparticles. However, alas, as rich, attractive and beautiful as SUSY can be, supersymmetric particles continue to elude detection at the Large Hadron Collider (LHC), at least in Run 1 experiments with a center-of-mass energy ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 8 TeV [6]. Run 2 experiments with ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV are currently under way, targeting a final luminosity of about 100 fb�<sup>1</sup> , and so are tested in more involved and less stringent formulations of supersymmetry [7].

It must be stressed that until now, we have not detected dark matter, at least not in a conclusive manner. Indeed, we know dark matter is there only because of its gravitational interactions, and this is why and how we believe it contributes about a quarter of the mass energy of the known Universe. But we still do not know whether dark matter really interacts with ordinary matter. We believe it does, even if very weakly. We believe these interactions can yield signals with enough strength so that we can detect dark matter or produce it in collisions of Standard Model particles [8].

We must also understand that a detection process relies primarily on a theory or a model. A theory like supersymmetry, which originated in the realm of elementary particle physics, is devised as an extension to the Standard Model that is based on a yet-to-be-detected symmetry between fermionic and bosonic states [9]. Its DM connection came only later. In fact, in the rather long period between the Higgs mechanism proposal [10] and the detection of the Higgs particle [11], various extensions of the Standard Model were proposed in order to alleviate some of its shortcomings, the so-called "Beyond the Standard Model" (BSM) Physics [12]. A number of these BSM models bear in them extra fields, meaning extra particles with specific properties. Until today, such particles have never been detected. With time and change in focus, the most stable of these hypothetical particles have then been proposed as candidates for dark matter, many in the form of WIMPs. The advantage of such a paradigm is clear: the calculational techniques that built strength in the realm of particle physics were ready at the service of dark matter search with little extra effort in development. But the experimental framework was also ready. Such a state of affairs could partly explain the popularity of WIMP physics, compared to other possible scenarios for dark matter.

structures we see. But sterile neutrinos, non-Standard Model particles, can be heavier, and so could be dark matter candidates. This possibility has been reignited with the recent detection of an X-ray emission line at an energy of 3:55 keV coming from galaxy clusters, the Andromeda galaxy, the Galactic Center and the Draco dwarf spheroidal galaxy. This line is consistent

In fact, there is by now quasi-consensus that dark matter ought to be understood outside the realm of conventional matter. One other scenario is that of (pseudo)scalar particles of tiny mass � <sup>10</sup>�<sup>22</sup> eV, the so-called ultralight axions that could account for the dark matter content of the Universe. This is supported by high-resolution cosmological simulations [4]. Axions originated in quantum chromodynamics, the theory of quarks and gluons, in relation to the axial anomaly in this theory and the strong Charge Conjugation Parity Symmetry Violation (CP violation) problem. But like anything else related to dark matter, they elude detection. The Axion Dark

But maybe the most popular candidate for dark matter is an electrically neutral and colorless weakly interacting massive particle (WIMP). Such a particle originated in supersymmetric (SUSY) extensions of the StandardModel. The most obvious such a candidate is the neutralino, a neutralRodd supersymmetric particle. Indeed, neutralinos are only produced or destroyed in pairs, thus constituting the lightest SUSYparticles. However, alas, as rich, attractive and beautiful as SUSY can be, supersymmetric particles continue to elude detection at the Large Hadron Collider (LHC), at

It must be stressed that until now, we have not detected dark matter, at least not in a conclusive manner. Indeed, we know dark matter is there only because of its gravitational interactions, and this is why and how we believe it contributes about a quarter of the mass energy of the known Universe. But we still do not know whether dark matter really interacts with ordinary matter. We believe it does, even if very weakly. We believe these interactions can yield signals with enough strength so that we can detect dark matter or produce it in collisions of Standard

We must also understand that a detection process relies primarily on a theory or a model. A theory like supersymmetry, which originated in the realm of elementary particle physics, is devised as an extension to the Standard Model that is based on a yet-to-be-detected symmetry between fermionic and bosonic states [9]. Its DM connection came only later. In fact, in the rather long period between the Higgs mechanism proposal [10] and the detection of the Higgs particle [11], various extensions of the Standard Model were proposed in order to alleviate some of its shortcomings, the so-called "Beyond the Standard Model" (BSM) Physics [12]. A number of these BSM models bear in them extra fields, meaning extra particles with specific properties. Until today, such particles have never been detected. With time and change in focus, the most stable of these hypothetical particles have then been proposed as candidates for dark matter, many in the form of WIMPs. The advantage of such a paradigm is clear: the calculational techniques that built strength in the realm of particle physics were ready at the

<sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV are currently under way, targeting a final luminosity of about 100 fb�<sup>1</sup>

tested in more involved and less stringent formulations of supersymmetry [7].

<sup>s</sup> <sup>p</sup> <sup>¼</sup> 8 TeV [6]. Run 2 experiments with

, and so are

Matter Experiment (ADMX) may bring in answers in the near future [5].

least in Run 1 experiments with a center-of-mass energy ffiffi

ffiffi

Model particles [8].

with the decay of a 7:1 keV sterile neutrino [3].

22 Trends in Modern Cosmology

Accordingly, many experiments have been devised specifically to detect dark matter. Each, of course, must be based on a specific scheme that is based on a specific scenario. There are experiments that try to detect dark matter directly, through missing energy momentum after a WIMP collides directly with an ordinary nucleus. The low-background DAMA (NaI) and then DAMA/LIBRA (NaI[Ti]) experiments at Gran Sasso in Italy [13] add a twist to this by trying to detect dark matter in the galactic halo via its suggested model-independent flux annual modulation [14]. The CoGeNT experiment [15] in Soudan (Minnesota, USA) also tries to detect this annual modulation, but in the region where the WIMP mass is ≲10 GeV. The CDMS I (Stanford, USA) [16], then CDMS II (Soudan, USA) [17], and now the superCDMS (Soudan, USA, then SNOLAB, Sudbury, Canada) [18] perform direct detection, measuring ionization and phonon signals resulting from a WIMP-nucleus collision, sensitive in the lowmass region. The XENON10 [19], then XENON100 [20], then the coming XENON1t [21], all in Gran Sasso, Italy, use liquid Xenon as a detecting medium for WIMP-nucleon and WIMPelectron collisions. There is also the Large Underground Xenon (LUX) experiment (South Dakota, USA) [22], as a direct-detection experiment, and its more sensitive successor LZ experiment [23]. The CRESST experiment [24], followed by CRESST II [25], both at Gran Sasso, Italy, also try to detect dark matter directly with low mass. We also have the series of EDEL-WEISS experiments [26] (Modane, France), which target low-mass WIMPs. The list is exhaustive, and could not be accounted here due to space constraints.

The above experiments are terrestrial, with instruments buried underground to reduce noise. But there are other experiments which are space borne that carry out indirect detection in cosmic rays. There is the Fermi Gamma-Ray Space Telescope (Fermi-LAT), which has found excess of gamma rays in the galactic center that cannot be explained by conventional sources and which is compatible with the presence of dark matter [27]. Fermi-LAT uses what we call indirect methods, namely, collecting gamma-ray signals and removing from these those emitted by all possible known sources. Another space-borne experiment is the Alpha Magnetic Spectrometer (AMS) experiment at the international space station [28], collecting and analyzing signals from cosmic rays. In addition, the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA) experiment [29] is a particle identifier that uses a permanent magnet spectrometer for space cosmic-ray direct measurements.

A third prong in the dark matter search enterprise is to produce it in particle colliders like the LHC [8]. There is an added difficulty here, which is that we do not know in which mass range we should look into. It could well be that the present center-of-mass energy that is available, 13 TeV, may not be sufficient. Nevertheless, the search for dark matter at the LHC is intense. One reason is that, experimentally, this is feasible now: small amounts of missing energy and transverse momentum can be detected now. Note that the present detectors are not built to detect dark matter directly. Rather, the latter would appear as a missing energy or missing momentum. For example, we now look at events in which a Z boson and a missing transverse momentum are produced in a proton-proton collision at ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV. The <sup>Z</sup> boson decays into two charged leptons, a recognizable signature, and a possible missing transverse momentum, which would indicate the production of dark matter in the process. A similar search, conducted previously by the CMS Collaboration and based on data collected with ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 8 TeV (Run 1), found no evidence of new physics and hence set limits on dark matter production. A recent search performed by the ATLAS Collaboration with ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV with an integrated luminosity of 3:2 fb�<sup>1</sup> also reported no evidence [30].

What should be clear by now is that interpreting signals as dark matter necessitates modeling. On the other hand, any model needs experimental results to restrict the range of its free parameters, to fine-tune these parameters, and, ultimately, in many cases, to be eliminated. The aim of this chapter is to shed light on the main steps a phenomenologist takes when building a model for dark matter, then testing the model against experimental results. It is an attempt to look into the modeling process itself, from the "cradle to the grave," so to speak. The discussion is based on a model proposed in Ref. [31] for cold dark matter, exposed to particle-physics phenomenology in [32], and further restricted by internal consistency in Ref. [33]. We will see how gradually the parameters of the model are constrained, and how the region of viability is reached. To carry out the discussion smoothly, we have chosen a model which is simple enough to avoid confusion created by the often involved details of the calculations and could-be-complexity of the model itself, but at the same time rich enough to be able to accommodate a vast range of experimental results. The material presented in this chapter is drawn from the works just cited.

This chapter is organized as follows. After this Introduction, Section 2 motivates and then presents the model based on WIMP physics, namely, a two-singlet extension of the Standard Model of elementary particles. We will try to avoid lengthy arguments and focus on the essentials. Section 3 shows how the measured amount of dark matter relic-density constrains the value of the dark matter annihilation cross-section, a constraint any model has to satisfy. We then discuss how the two-singlet extension fits into this, and add to it a perturbativity ingredient. Section 4 takes the two-singlet model into the arena of particle phenomenology and sees how it copes with rare meson decays. Section 5 goes back to the fundamentals and runs a renormalization-group analysis to inquire into the sustainability of the model. Section 6 puts all these constraints together and determines the regions of viability of the model. Section 7 is left for concluding remarks.

#### 2. A model for dark matter: motivation and parametrization

As mentioned in the Introduction, the most popular candidate for dark matter is an electrically neutral colorless weakly interacting massive particle (WIMP), and the neutralino, the lightest supersymmetric particle, is a robust fit for this role. However, as explained in Ref. [31] and references therein, it is hard to argue in favor of a neutralino when it comes to light cold dark matter, say, a WIMP mass of up to 10 GeV. In addition, up to now, we have not detected supersymmetric signatures at the LHC [34].

Therefore, with no prior hints as to what the internal structure of the WIMP might be, one adopts a bottom-up approach, in which one extends the Standard Model by adding to it the simplest of fields, one real spinless scalar, which will be the WIMP. This field must be a Standard Model gauge singlet so that we avoid any "direct contact" with any of the Standard Model particles. It is allowed to interact with visible particles only via the Higgs field. It is made stable against annihilation by enforcing upon it the simplest of symmetries, a discrete Z<sup>2</sup> symmetry that does not break spontaneously. This construction is called the minimal extension to the Standard Model. In view of its cosmological implication, the minimal extension has first been proposed in Ref. [35] and has been extensively studied and explored in Ref. [36]. However, this model is shown in Ref. [37] to be inadequate if we want the WIMP to be light.

into two charged leptons, a recognizable signature, and a possible missing transverse momentum, which would indicate the production of dark matter in the process. A similar search,

(Run 1), found no evidence of new physics and hence set limits on dark matter production. A

What should be clear by now is that interpreting signals as dark matter necessitates modeling. On the other hand, any model needs experimental results to restrict the range of its free parameters, to fine-tune these parameters, and, ultimately, in many cases, to be eliminated. The aim of this chapter is to shed light on the main steps a phenomenologist takes when building a model for dark matter, then testing the model against experimental results. It is an attempt to look into the modeling process itself, from the "cradle to the grave," so to speak. The discussion is based on a model proposed in Ref. [31] for cold dark matter, exposed to particle-physics phenomenology in [32], and further restricted by internal consistency in Ref. [33]. We will see how gradually the parameters of the model are constrained, and how the region of viability is reached. To carry out the discussion smoothly, we have chosen a model which is simple enough to avoid confusion created by the often involved details of the calculations and could-be-complexity of the model itself, but at the same time rich enough to be able to accommodate a vast range of experimental results. The material presented in this

This chapter is organized as follows. After this Introduction, Section 2 motivates and then presents the model based on WIMP physics, namely, a two-singlet extension of the Standard Model of elementary particles. We will try to avoid lengthy arguments and focus on the essentials. Section 3 shows how the measured amount of dark matter relic-density constrains the value of the dark matter annihilation cross-section, a constraint any model has to satisfy. We then discuss how the two-singlet extension fits into this, and add to it a perturbativity ingredient. Section 4 takes the two-singlet model into the arena of particle phenomenology and sees how it copes with rare meson decays. Section 5 goes back to the fundamentals and runs a renormalization-group analysis to inquire into the sustainability of the model. Section 6 puts all these constraints together and determines the regions of viability of the model. Section 7 is left for concluding remarks.

As mentioned in the Introduction, the most popular candidate for dark matter is an electrically neutral colorless weakly interacting massive particle (WIMP), and the neutralino, the lightest supersymmetric particle, is a robust fit for this role. However, as explained in Ref. [31] and references therein, it is hard to argue in favor of a neutralino when it comes to light cold dark matter, say, a WIMP mass of up to 10 GeV. In addition, up to now, we have not detected

Therefore, with no prior hints as to what the internal structure of the WIMP might be, one adopts a bottom-up approach, in which one extends the Standard Model by adding to it the

2. A model for dark matter: motivation and parametrization

<sup>s</sup> <sup>p</sup> <sup>¼</sup> 8 TeV

<sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV with an integrated

conducted previously by the CMS Collaboration and based on data collected with ffiffi

recent search performed by the ATLAS Collaboration with ffiffi

luminosity of 3:2 fb�<sup>1</sup> also reported no evidence [30].

24 Trends in Modern Cosmology

chapter is drawn from the works just cited.

supersymmetric signatures at the LHC [34].

In the logic of this bottom-up approach, adding another real scalar seems the natural step forward. This field will also be endowed with a Z<sup>2</sup> symmetry, but this one we will break spontaneously, and the reason is to open new channels for dark matter annihilation, which implies an increase in the corresponding annihilation cross-section, which in turn would allow smaller WIMP masses, something we want to achieve. Needless to say that this auxiliary field must also be a Standard Model gauge singlet.

Therefore, we extend the Standard Model by adding two real, spinless and Z2-symmetric fields: the dark matter field S<sup>0</sup> for which the Z<sup>2</sup> symmetry is unbroken and an auxiliary field for which it is spontaneously broken. Both fields are Standard Model gauge singlets and hence can interact with "visible" particles only via the Higgs doublet, taken in the unitary gauge. We must also assume all processes calculable in perturbation theory. The details of the spontaneous breaking of the electroweak gauge symmetry and the additional auxiliary Z<sup>2</sup> symmetry are left aside [31].

The potential function that involves the physical scalar Higgs field h, the dark matter field S0, and the physical auxiliary scalar field S<sup>1</sup> is as follows:

$$\begin{split} \mathcal{U} &= \frac{1}{2} m\_0^2 S\_0^2 + \frac{1}{2} m\_h^2 h^2 + \frac{1}{2} m\_1^2 S\_1^2 \\ &+ \frac{\lambda\_0^{(3)}}{2} S\_0^2 h + \frac{\eta\_{01}^{(3)}}{2} S\_0^2 S\_1 + \frac{\lambda^{(3)}}{6} h^3 + \frac{\eta\_1^{(3)}}{6} S\_1^3 + \frac{\lambda\_1^{(3)}}{2} h^2 S\_1 + \frac{\lambda\_2^{(3)}}{2} h S\_1^2 \\ &+ \frac{\eta\_0}{24} S\_0^4 + \frac{\lambda^{(4)}}{24} h^4 + \frac{\eta\_1^{(4)}}{24} S\_1^4 + \frac{\lambda\_0^{(4)}}{4} S\_0^2 h^2 + \frac{\eta\_{01}^{(4)}}{4} S\_0^2 S\_1^2 + \frac{\lambda\_{01}^{(4)}}{2} S\_0^2 h S\_1 \\ &+ \frac{\lambda\_1^{(4)}}{6} h^3 S\_1 + \frac{\lambda\_2^{(4)}}{4} h^2 S\_1^2 + \frac{\lambda\_3^{(4)}}{6} h S\_1^3. \end{split} (1)$$

The quantities m0, mh, and m<sup>1</sup> are the masses of the corresponding fields S0, h, and S1, respectively, and all the other parameters are real coupling constants. Also, the part of the Standard Model Lagrangian that is relevant to Dark matter annihilation is given in terms of the physical fields h and S<sup>1</sup> by the following potential function:

$$\begin{split} \mathcal{U}\_{\rm SM} &= \sum\_{f} \left( \lambda\_{\rm l\overline{\mu}} h \overline{\!\!\!f} + \lambda\_{\rm l\overline{\tau}} \mathcal{S}\_{\rm l} \overline{\!\!\!f} \right) + \lambda\_{\rm l\overline{\mu}}^{(3)} h \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} + \lambda\_{\rm l\overline{\mu}}^{(3)} \mathcal{S}\_{1} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} \\ &+ \lambda\_{\rm l\overline{\mu}}^{(3)} h \left( \mathcal{Z}\_{\mu} \right)^{2} + \lambda\_{\rm l\overline{\tau}}^{(3)} \mathcal{S}\_{1} \left( \mathcal{Z}\_{\mu} \right)^{2} + \lambda\_{\rm l\overline{\mu}}^{(4)} h^{2} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} + \lambda\_{\rm l\overline{\mu}}^{(4)} \mathcal{S}\_{1}^{2} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} \\ &+ \lambda\_{\rm l\overline{\mu}} h \mathcal{S}\_{1} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} + \lambda\_{\rm l\overline{\mu}}^{(4)} h^{2} \left( \mathcal{Z}\_{\mu} \right)^{2} + \lambda\_{\rm l\overline{\mu}}^{(4)} \mathcal{S}\_{1}^{2} \left( \mathcal{Z}\_{\mu} \right)^{2} + \lambda\_{\rm l\overline{\mu}} h \mathcal{S}\_{1} \left( \mathcal{Z}\_{\mu} \right)^{2}. \end{split} \tag{2}$$

The coupling constants in the above expression are given by the following relations, in which the quantities mf , mw, and mz are the masses of the fermion f , the W, and the Z gauge bosons, respectively:

$$\begin{aligned} \lambda\_{\text{hf}} &= -\frac{m\_f}{v} \cos \Theta; \quad \lambda\_{1f} = \frac{m\_f}{v} \sin \Theta; \\ \lambda\_{\text{hz}}^{(3)} &= 2 \frac{m\_w^2}{v} \cos \Theta; \quad \lambda\_{1w}^{(3)} = -2 \frac{m\_w^2}{v} \sin \Theta; \\ \lambda\_{\text{hz}}^{(3)} &= \frac{m\_z^2}{v} \cos \Theta; \quad \lambda\_{1z}^{(3)} = -\frac{m\_z^2}{v} \sin \Theta; \\ \lambda\_{\text{hw}}^{(4)} &= \frac{m\_w^2}{v^2} \cos^2 \Theta; \quad \lambda\_{1w}^{(4)} = \frac{m\_w^2}{v^2} \sin^2 \Theta; \quad \lambda\_{h1w} = -\frac{m\_w^2}{v^2} \sin 2\Theta; \\ \lambda\_{\text{hz}}^{(4)} &= \frac{m\_z^2}{2v^2} \cos^2 \Theta; \quad \lambda\_{1z}^{(4)} = \frac{m\_z^2}{2v^2} \sin^2 \Theta; \quad \lambda\_{h1z} = -\frac{m\_z^2}{2v^2} \sin 2\Theta. \end{aligned} \tag{3}$$

The angle θ is the mixing angle between the fields h and S<sup>1</sup> [31]. The quantities v and v1, both positive, are the vacuum expectation values of the Higgs and auxiliary fields, respectively.

This model has nine free parameters to start with, three mass parameters and six coupling constants [31]. As already mentioned, perturbativity is assumed, which means all the original coupling constants are small. The dark matter self-coupling constant η<sup>0</sup> in Eq. (1) will not enter the lowest-order calculations we will consider, and so this parameter stays free for the time being and we are left with eight parameters. The spontaneous breaking of the electroweak and Z<sup>2</sup> symmetries for the Higgs and auxiliary fields, respectively, introduces the two vacuum expectation values v and v1. The value of v is fixed experimentally to be 246 GeV [38] and for the present discussion, we fix the value of v<sup>1</sup> at the order of the electroweak scale, say, 100 GeV. In addition, the Higgs mass is now known [11], mh ¼ 125 GeV. Hence, five free parameters remain. Three of these are chosen to be the two physical masses m<sup>0</sup> (dark matter) and m<sup>1</sup> (S<sup>1</sup> field), plus the mixing angle θ between S<sup>1</sup> and h. The two last parameters we choose are the two physical mutual coupling constants λð Þ<sup>4</sup> <sup>0</sup> (dark matter—Higgs) and <sup>η</sup>ð Þ<sup>4</sup> <sup>01</sup> (dark matter—S<sup>1</sup> particle), see Eq. (1).

#### 3. Constraints from cosmology and perturbativity

Any model of dark matter has to comply with astrophysical observations. Indeed, dark matter is believed to have been produced in the early Universe. A most popular paradigm for this production is the so-called "freeze-out scenario" by which dark matter, thought of as a set of elementary particles, interacts with ordinary matter, weakly but with enough strength to generate common thermal equilibrium at high temperature. However, as the cosmos is cooling down, at some temperature Tf , the rate of expansion of the Universe becomes higher than the rate of dark matter particle annihilation, which forces dark matter to decouple from ordinary matter, and hence a "freeze-out"—Tf is thus called the freeze-out temperature. The DM relic density ΩDM is essentially the one we measure today [1]:

$$
\Omega\_{\rm DM} \overline{h}^2 = 0.1199 \pm 0.0022 \approx 0.12,\tag{4}
$$

where <sup>h</sup> is Hubble constant in units of 100 km � <sup>s</sup>�<sup>1</sup> � Mpc�<sup>1</sup> .

The coupling constants in the above expression are given by the following relations, in which the quantities mf , mw, and mz are the masses of the fermion f , the W, and the Z gauge bosons,

sin θ;

m2 w v

sin θ;

sin θ;

<sup>θ</sup>; <sup>λ</sup>h1<sup>w</sup> ¼ � <sup>m</sup><sup>2</sup>

<sup>θ</sup>; <sup>λ</sup>h1<sup>z</sup> ¼ � <sup>m</sup><sup>2</sup>

<sup>0</sup> (dark matter—Higgs) and <sup>η</sup>ð Þ<sup>4</sup>

w <sup>v</sup><sup>2</sup> sin 2θ; ð3Þ

<sup>01</sup> (dark matter—S<sup>1</sup>

z <sup>2</sup>v<sup>2</sup> sin 2θ:

v

<sup>1</sup><sup>w</sup> ¼ �2

<sup>1</sup><sup>z</sup> ¼ � <sup>m</sup><sup>2</sup> z v

> <sup>1</sup><sup>w</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup> w <sup>v</sup><sup>2</sup> sin <sup>2</sup>

<sup>1</sup><sup>z</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup> z <sup>2</sup>v<sup>2</sup> sin <sup>2</sup>

The angle θ is the mixing angle between the fields h and S<sup>1</sup> [31]. The quantities v and v1, both positive, are the vacuum expectation values of the Higgs and auxiliary fields, respectively.

This model has nine free parameters to start with, three mass parameters and six coupling constants [31]. As already mentioned, perturbativity is assumed, which means all the original coupling constants are small. The dark matter self-coupling constant η<sup>0</sup> in Eq. (1) will not enter the lowest-order calculations we will consider, and so this parameter stays free for the time being and we are left with eight parameters. The spontaneous breaking of the electroweak and Z<sup>2</sup> symmetries for the Higgs and auxiliary fields, respectively, introduces the two vacuum expectation values v and v1. The value of v is fixed experimentally to be 246 GeV [38] and for the present discussion, we fix the value of v<sup>1</sup> at the order of the electroweak scale, say, 100 GeV. In addition, the Higgs mass is now known [11], mh ¼ 125 GeV. Hence, five free parameters remain. Three of these are chosen to be the two physical masses m<sup>0</sup> (dark matter) and m<sup>1</sup> (S<sup>1</sup> field), plus the mixing angle θ between S<sup>1</sup> and h. The two last parameters we choose are the

Any model of dark matter has to comply with astrophysical observations. Indeed, dark matter is believed to have been produced in the early Universe. A most popular paradigm for this production is the so-called "freeze-out scenario" by which dark matter, thought of as a set of elementary particles, interacts with ordinary matter, weakly but with enough strength to generate common thermal equilibrium at high temperature. However, as the cosmos is cooling down, at some temperature Tf , the rate of expansion of the Universe becomes higher than the rate of dark matter particle annihilation, which forces dark matter to decouple from ordinary matter, and hence a "freeze-out"—Tf is thus called the freeze-out temperature. The DM relic

cos <sup>θ</sup>; <sup>λ</sup>1<sup>f</sup> <sup>¼</sup> mf

cos θ; λð Þ<sup>3</sup>

θ; λð Þ<sup>4</sup>

θ; λð Þ<sup>4</sup>

cos θ; λð Þ<sup>3</sup>

respectively:

26 Trends in Modern Cosmology

<sup>λ</sup>hf ¼ � mf v

> m2 w v

λð Þ<sup>3</sup> hw ¼ 2

λð Þ<sup>3</sup> hz <sup>¼</sup> <sup>m</sup><sup>2</sup> z v

λð Þ<sup>4</sup> hw <sup>¼</sup> <sup>m</sup><sup>2</sup> w <sup>v</sup><sup>2</sup> cos <sup>2</sup>

λð Þ<sup>4</sup> hz <sup>¼</sup> <sup>m</sup><sup>2</sup> z <sup>2</sup>v<sup>2</sup> cos <sup>2</sup>

two physical mutual coupling constants λð Þ<sup>4</sup>

3. Constraints from cosmology and perturbativity

density ΩDM is essentially the one we measure today [1]:

particle), see Eq. (1).

In a model where dark matter is seen as WIMPs that can annihilate into ordinary elementary particles, the relic density ΩDM can be related to the annihilation DM cross-section σann. Indeed, in the framework of the standard cosmological model, one can derive the following relation [39]:

$$
\Omega\_{\rm DM} \overline{h}^2 \simeq \frac{1.07 \times 10^9 \text{x}\_{\circ}}{\sqrt{\mathcal{g}\_\*} m\_{\rm Pl} \langle \sigma\_{\rm ann} v \rangle \,\text{GeV}}; \quad \mathbf{x}\_{\circ} \simeq \ln \frac{0.0038 m\_{\rm Pl} m\_0 \langle \sigma\_{\rm ann} v \rangle}{\sqrt{\mathcal{g}\_\* \overline{\chi}\_f}}. \tag{5}
$$

The quantity <sup>m</sup>Pl <sup>¼</sup> <sup>1</sup>:<sup>22</sup> � 1019GeV is the Planck mass, <sup>m</sup><sup>0</sup> is the dark matter mass, xf <sup>¼</sup> <sup>m</sup>0=Tf , and g<sup>∗</sup> is the number of relativistic degrees of freedom with a mass less than Tf . The quantity 〈σannv〉 is the thermally averaged annihilation cross-section of a pair of two dark matter particles multiplied by their relative speed in their center-of-mass reference frame. Solving (4) with the current value (5) for ΩDM with xf between 19.2 and 21.6 [40], we obtain the following constraint on the annihilation cross-section:

$$
\langle \sigma\_{\rm ann} v \rangle \simeq 2 \times 10^{-9} \text{GeV}.\tag{6}
$$

This is one major constraint any WIMP model like the one we discuss here has to satisfy. Indeed, the quantity 〈σannv〉 is calculable in perturbation theory, and so, the implementation of (6) will induce an admittedly complicated but important relation between the free parameters of the model, hence reducing their space of freedom, reducing their number by one. Also, the constraint induced by (6) can be used to examine aspects of the theory like perturbativity. To implement perturbativity in the present two-singlet model, we use (6) to obtain the mutual coupling constant ηð Þ<sup>4</sup> <sup>01</sup> (coupling between the DM field S<sup>0</sup> and auxiliary field S1) in terms of the dark matter mass m<sup>0</sup> for given values of λð Þ<sup>4</sup> <sup>0</sup> (coupling between S<sup>0</sup> and Higgs) and study its behavior to tell which dark matter mass regions are consistent with perturbativity. It should be mentioned that once the two mutual coupling constants λð Þ<sup>4</sup> <sup>0</sup> and <sup>η</sup>ð Þ<sup>4</sup> <sup>01</sup> are small, all the other physical coupling constants will be small.

The quantity 〈σannv〉 is calculated in perturbation theory using all possible annihilation channels the model allows for [31]. As the model has many parameters, the behavior of the mutual coupling constant ηð Þ<sup>4</sup> <sup>01</sup> is bound to be rich. Sampling is therefore necessary. In this review, we briefly comment on the behavior of ηð Þ<sup>4</sup> <sup>01</sup> for two sets of the parameters (θ, m1, <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ). A more substantial discussion can be found in Ref. [31].

The first set of parameters is a small mixing angle <sup>θ</sup><sup>¼</sup> 10o, a weak mutual <sup>S</sup>0-Higgs coupling constant λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> <sup>¼</sup> <sup>0</sup>:01, and a <sup>S</sup>1-mass <sup>m</sup><sup>1</sup> <sup>¼</sup> 10 GeV. The corresponding behavior of <sup>η</sup><sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> versus m<sup>0</sup> is shown in Figure 1. The range of m<sup>0</sup> displayed is from 0:1 to 200 GeV. In this regime, the first feature we see is that the relic-density constraint on dark matter annihilation forbids WIMP

Figure 1. η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> versus m<sup>1</sup> for very light S1, small mixing, and very small WIMP-Higgs coupling.

masses m0≲1:3 GeV. Furthermore, just about m0≃1:3 GeV, the c-quark threshold, the S<sup>0</sup> � S<sup>1</sup> mutual coupling constant η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> starts at about 0:8, a value, while perturbative, that is roughly 80-fold larger than the mutual S<sup>0</sup> Higgs coupling constant λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> . Then as the DM mass increases, η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> decreases, steeply first, more slowly as we cross the τ mass toward the b mass. Just before m1=2, the coupling η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> hops onto another solution branch that is just emerging from negative territory, gets back to the first one at precisely m1=2 as this latter carries now smaller values, and then jumps up again onto the second branch as the first crosses the m<sup>0</sup> axis down. It goes up this branch with a moderate slope until m<sup>0</sup> becomes equal to m1, a value at which the S<sup>1</sup> annihilation channel opens. Just beyond m1, there is a sudden fall to a value η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> ≃0:0046 that is about half the value of λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> , and <sup>η</sup><sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> stays flat till m0≃45 GeV where it starts increasing, sharply after 60 GeV. In the mass interval m<sup>0</sup> ≃ 66–79 GeV, there is a "desert" with no positive real solutions to the relic-density constraint, hence no viable dark matter candidate exists. Beyond m0≃79 GeV, the mutual coupling constant η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> keeps increasing monotonously, with a small notch at the W mass and a less noticeable one at the Z mass. As it increases, its values remain perturbative.

The second set of parameters we feature is still a small Higgs <sup>S</sup><sup>1</sup> mixing angle <sup>θ</sup><sup>¼</sup> 10o, an increased <sup>S</sup>0-Higgs mutual coupling constant <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ¼ 0:2, and a moderate S<sup>1</sup> mass m<sup>1</sup> ¼ 20 GeV. The behavior of the <sup>S</sup><sup>0</sup> � <sup>S</sup><sup>1</sup> mutual coupling constant <sup>η</sup><sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> versus the DM mass m<sup>0</sup> is displayed in Figure 2. Here too, no viable DM masses exist below roughly 1:4 GeV, at which value η<sup>ð</sup>4<sup>Þ</sup> 01 starts at 1:95. It decreases with a sharp change of slope at the b-quark threshold, then makes a sudden dive at about 5 GeV, a change of branch at m1=2 down till about 12 GeV where it jumps up back onto the previous branch just before going to cross into negative territory. It drops

Figure 2. η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> versus m<sup>0</sup> for small mixing, moderate m1, and WIMP-Higgs coupling.

sharply at m<sup>0</sup> ¼ m<sup>1</sup> and then increases slowly until m<sup>0</sup> ≃43:3 GeV. Then, no viable WIMP masses exist, a desert. As we see, for this set of parameters (θ, λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> , m1), the model constrains the dark matter mass inside the interval 1:6 GeV ≲ m<sup>0</sup> ≲ 43:3 GeV, with perturbative coupling constants.

With the same mixing angle <sup>θ</sup> <sup>¼</sup> 10o and mutual coupling constant <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ¼ 0:2, larger masses m<sup>1</sup> yield roughly the same behavior, but with values of η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> that could be nonperturbative. For example, when <sup>m</sup><sup>1</sup> <sup>¼</sup> 60 GeV, the mutual coupling <sup>η</sup><sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> starts very high (≃85) at m<sup>0</sup> ≃ 1:5 GeV, and then decreases rapidly. There is a usual change of branches and a desert starting at about 49 GeV, a behavior that is peculiar in a way because the desert starts at a mass m<sup>0</sup> < m1, that is, before the opening of the S<sup>1</sup> annihilation channel. In other words, the dark matter is annihilating into the light fermions only and the model is perturbatively viable in the range of 20–49 GeV.

#### 4. Constraints from direct detection

masses m0≲1:3 GeV. Furthermore, just about m0≃1:3 GeV, the c-quark threshold, the S<sup>0</sup> � S<sup>1</sup>

<sup>01</sup> versus m<sup>1</sup> for very light S1, small mixing, and very small WIMP-Higgs coupling.

<sup>01</sup> decreases, steeply first, more slowly as we cross the τ mass toward the b mass. Just before

territory, gets back to the first one at precisely m1=2 as this latter carries now smaller values, and then jumps up again onto the second branch as the first crosses the m<sup>0</sup> axis down. It goes up this branch with a moderate slope until m<sup>0</sup> becomes equal to m1, a value at which the S<sup>1</sup>

after 60 GeV. In the mass interval m<sup>0</sup> ≃ 66–79 GeV, there is a "desert" with no positive real solutions to the relic-density constraint, hence no viable dark matter candidate exists. Beyond

notch at the W mass and a less noticeable one at the Z mass. As it increases, its values remain

The second set of parameters we feature is still a small Higgs <sup>S</sup><sup>1</sup> mixing angle <sup>θ</sup><sup>¼</sup> 10o, an

in Figure 2. Here too, no viable DM masses exist below roughly 1:4 GeV, at which value η<sup>ð</sup>4<sup>Þ</sup>

starts at 1:95. It decreases with a sharp change of slope at the b-quark threshold, then makes a sudden dive at about 5 GeV, a change of branch at m1=2 down till about 12 GeV where it jumps up back onto the previous branch just before going to cross into negative territory. It drops

annihilation channel opens. Just beyond m1, there is a sudden fall to a value η<sup>ð</sup>4<sup>Þ</sup>

<sup>01</sup> starts at about 0:8, a value, while perturbative, that is roughly

<sup>01</sup> stays flat till m0≃45 GeV where it starts increasing, sharply

<sup>01</sup> keeps increasing monotonously, with a small

<sup>0</sup> ¼ 0:2, and a moderate S<sup>1</sup> mass m<sup>1</sup> ¼ 20 GeV.

<sup>01</sup> versus the DM mass m<sup>0</sup> is displayed

<sup>01</sup> hops onto another solution branch that is just emerging from negative

<sup>0</sup> . Then as the DM mass increases,

<sup>01</sup> ≃0:0046 that is

01

mutual coupling constant η<sup>ð</sup>4<sup>Þ</sup>

m1=2, the coupling η<sup>ð</sup>4<sup>Þ</sup>

about half the value of λ<sup>ð</sup>4<sup>Þ</sup>

perturbative.

η<sup>ð</sup>4<sup>Þ</sup>

Figure 1. η<sup>ð</sup>4<sup>Þ</sup>

28 Trends in Modern Cosmology

80-fold larger than the mutual S<sup>0</sup> Higgs coupling constant λ<sup>ð</sup>4<sup>Þ</sup>

<sup>0</sup> , and <sup>η</sup><sup>ð</sup>4<sup>Þ</sup>

m0≃79 GeV, the mutual coupling constant η<sup>ð</sup>4<sup>Þ</sup>

increased <sup>S</sup>0-Higgs mutual coupling constant <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup>

The behavior of the <sup>S</sup><sup>0</sup> � <sup>S</sup><sup>1</sup> mutual coupling constant <sup>η</sup><sup>ð</sup>4<sup>Þ</sup>

Perhaps the most known constraints on a WIMP model are those coming from direct-detection experiments like the many we have cited in the introductory section. In such experiments, the signal sought for would typically come from the elastic scattering of a WIMP off a nonrelativistic nucleon target. However, as mentioned in the Introduction, until now, none of these direct-detection experiments have yielded an unambiguous dark matter signal. Rather, with increasing precision from one generation to the next, these experiments put increasingly stringent exclusion bounds on the dark matter-nucleon elastic-scattering total cross-section σdet in terms of the dark matter mass m0, and because of these constraints, many models can get excluded.

Therefore, a theoretical dark matter model like the two-singlet extension we discuss here has to satisfy these bounds to remain viable. For this purpose, we calculate σdet as a function of m<sup>0</sup> for different values of the parameters (θ, λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> , m1Þ and compare its behavior against the experimental bounds. The calculation is carried out with sufficient details in Ref. [31], and the total cross-section for non-relativistic S0-nucleon elastic scattering is given by

$$\sigma\_{\rm det} = \frac{m\_N^2 \left( m\_N - \frac{7}{9} m\_B \right)^2}{4 \pi (m\_N + m\_0)^2 v^2} \left[ \frac{\lambda\_0^{(3)} \cos \theta}{m\_h^2} - \frac{\eta\_{01}^{(3)} \sin \theta}{m\_1^2} \right]^2. \tag{7}$$

In this relation, mN is the nucleon mass and mB is the baryon mass in the chiral limit. The mutual coupling constants λ<sup>ð</sup>3<sup>Þ</sup> <sup>0</sup> and <sup>η</sup><sup>ð</sup>3<sup>Þ</sup> <sup>01</sup> are defined in Eq. (1). The relic-density constraint on the dark matter annihilation cross-section (6) has to be imposed throughout. In addition, we require now that the coupling constants be perturbative, and we do this by imposing the additional requirement 0 ≤ η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> ≤ 1.

Generically, as m<sup>0</sup> increases, the detection cross-section σdet starts from high values, slopes down to minima that depend on the parameters, and then picks up moderately. There are features and action at the usual mass thresholds, with varying sizes and shapes. Regions coming from the relic-density constraint and new ones originating from the additional perturbativity requirement are excluded.

For the purpose of illustration, we choose three indicative sets of values for the parameters (θ, λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> , m1). We start first with a Higgs-S<sup>1</sup> mixing angle <sup>θ</sup><sup>¼</sup> <sup>10</sup>o, a weak mutual <sup>S</sup>0-Higgs coupling λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ¼ 0:01, and an S<sup>1</sup> mass m<sup>1</sup> ¼ 20 GeV. The behavior of σdet versus m<sup>0</sup> is shown in Figure 3. There, we see that for the two mass intervals 20–65 GeV and 75–100 GeV, plus an almost singled-out dip at m<sup>0</sup> ¼ m1=2, the elastic scattering cross-section is below the sensitivity of SuperCDMS. However, XENON1T should probe all these masses, except m<sup>0</sup> ≃58 and 85 GeV.

Increasing m<sup>1</sup> has the effect of closing possibilities for very light dark matter and thinning the intervals as it drives the predicted masses to larger values. Indeed, in Figure 4, where

Figure 3. Elastic N � S<sup>0</sup> scattering cross-section as a function of m<sup>0</sup> for moderate m1, small mixing, and small WIMP-Higgs coupling.

σdet ¼

mutual coupling constants λ<sup>ð</sup>3<sup>Þ</sup>

30 Trends in Modern Cosmology

additional requirement 0 ≤ η<sup>ð</sup>4<sup>Þ</sup>

(θ, λ<sup>ð</sup>4<sup>Þ</sup>

85 GeV.

Higgs coupling.

coupling λ<sup>ð</sup>4<sup>Þ</sup>

perturbativity requirement are excluded.

m2

<sup>N</sup> mN � <sup>7</sup>

4πð Þ mN þ m<sup>0</sup>

<sup>0</sup> and <sup>η</sup><sup>ð</sup>3<sup>Þ</sup>

<sup>01</sup> ≤ 1.

9 mB � �<sup>2</sup>

> 2 v2

In this relation, mN is the nucleon mass and mB is the baryon mass in the chiral limit. The

the dark matter annihilation cross-section (6) has to be imposed throughout. In addition, we require now that the coupling constants be perturbative, and we do this by imposing the

Generically, as m<sup>0</sup> increases, the detection cross-section σdet starts from high values, slopes down to minima that depend on the parameters, and then picks up moderately. There are features and action at the usual mass thresholds, with varying sizes and shapes. Regions coming from the relic-density constraint and new ones originating from the additional

For the purpose of illustration, we choose three indicative sets of values for the parameters

Figure 3. There, we see that for the two mass intervals 20–65 GeV and 75–100 GeV, plus an almost singled-out dip at m<sup>0</sup> ¼ m1=2, the elastic scattering cross-section is below the sensitivity of SuperCDMS. However, XENON1T should probe all these masses, except m<sup>0</sup> ≃58 and

Increasing m<sup>1</sup> has the effect of closing possibilities for very light dark matter and thinning the intervals as it drives the predicted masses to larger values. Indeed, in Figure 4, where

Figure 3. Elastic N � S<sup>0</sup> scattering cross-section as a function of m<sup>0</sup> for moderate m1, small mixing, and small WIMP-

<sup>0</sup> , m1). We start first with a Higgs-S<sup>1</sup> mixing angle <sup>θ</sup><sup>¼</sup> <sup>10</sup>o, a weak mutual <sup>S</sup>0-Higgs

<sup>0</sup> ¼ 0:01, and an S<sup>1</sup> mass m<sup>1</sup> ¼ 20 GeV. The behavior of σdet versus m<sup>0</sup> is shown in

λð Þ<sup>3</sup> <sup>0</sup> cos θ m<sup>2</sup> h

� <sup>η</sup>ð Þ<sup>3</sup> <sup>01</sup> sin θ m<sup>2</sup> 1

<sup>01</sup> are defined in Eq. (1). The relic-density constraint on

: ð7Þ

" #<sup>2</sup>

Figure 4. Elastic N � S<sup>0</sup> scattering cross-section as a function of m<sup>0</sup> for moderate m1, small mixing, and small WIMP-Higgs coupling.

m<sup>1</sup> ¼ 40 GeV, in addition to the dip at m1=2 that crosses SuperCDMS but not XENON1T, we see acceptable masses in the ranges of 40–65 GeV and 78 GeV up. The intervals narrow as we descend, surviving XENON1T only as spiked dips at 62 GeV and around 95 GeV.

On the other hand, a larger mutual coupling constant λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> has the general effect of squeezing the acceptable intervals of m<sup>0</sup> by pushing the values of σdet up, and it may even happen that at some point, the model has no predictability. This case is shown in Figure 5, where <sup>θ</sup><sup>¼</sup> 10o, <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ¼ 0:4, and <sup>m</sup><sup>1</sup> <sup>¼</sup> 60 GeV. In this example, the effects of increasing the values of both <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> and m1. As we see, the model cannot even escape Cryogenic Dark Matter Search II (CDMSII).

Figure 5. Elastic cross-section σdet versus m<sup>0</sup> for heavy S1, small mixing, and relatively large WIMP-Higgs coupling.

#### 5. Constraints from particle phenomenology

If a dark matter model based on WIMP physics is not killed already by the constraints coming from cosmology, perturbativity, and direct detection, it has to undergo the tests of particle phenomenology. To see how this works, we discuss here the constraints on our two-singlet model that come from a small selection of low-energy processes, namely, the rare decays of ϒ mesons. The forthcoming discussion is based on work done in Ref. [32]. There, the interested reader will find a fuller account of this study, together with relation to Higgs phenomenology. Note that the dark matter relic-density constraint in Eq. (6) and the perturbativity requirement 0 < ηð Þ<sup>4</sup> <sup>01</sup> < 1 are implemented systematically. Also, as in Ref. [32], we will restrict the discussion to light cold dark matter.

We therefore look at the constraints that come from the decay of the meson ϒ in the state nS (n ¼ 1; 3) into one photon γ and one particle S1. For m1≲8 GeV, the branching ratio for this process is given by the relation:

$$\mathrm{Br}(\mathbf{Y}\_{n\mathbb{S}} \rightarrow \boldsymbol{\mathcal{Y}} + \boldsymbol{\mathcal{S}}\_{1}) = \frac{G\boldsymbol{\pi}m\_{b}^{2}\sin^{2}\theta}{\sqrt{2}\pi\alpha} \,\mathrm{x}\_{\boldsymbol{n}}\left(1 - \frac{4\alpha\_{\mathrm{s}}}{3\pi}f(\mathbf{x}\_{\boldsymbol{n}})\right) \mathrm{Br}^{(\mu)}\,\boldsymbol{\Theta}(\boldsymbol{m}\_{\mathbf{Y}\_{\boldsymbol{n}\mathbb{S}}} - \boldsymbol{m}\_{1}).\tag{8}$$

In the above expression, xn � <sup>1</sup> � <sup>m</sup><sup>2</sup> 1=m<sup>2</sup> <sup>ϒ</sup>ns � � with the mass of <sup>ϒ</sup><sup>1</sup>ð3Þ<sup>S</sup> given by <sup>m</sup><sup>ϒ</sup>1ð3Þ<sup>S</sup> <sup>¼</sup> <sup>9</sup>:46ð10:355<sup>Þ</sup> GeV, the branching ratio Brðμ<sup>Þ</sup> � Br <sup>ϒ</sup><sup>1</sup>ð3Þ<sup>S</sup> ! <sup>μ</sup>þμ� � � <sup>¼</sup> <sup>2</sup>:48ð2:18Þ � <sup>10</sup>�<sup>2</sup> [41], <sup>α</sup> is the QCD coupling constant, α<sup>s</sup> ¼ 0:184 the QCD coupling constant at the scale m<sup>ϒ</sup>nS , the quantity GF is the Fermi coupling constant, and mb is the b-quark mass [38]. The function fðxÞ incorporates the effect of QCD radiative corrections given in [42] and the step function is denoted by Θð Þx . However, a rough estimate of the lifetime of S<sup>1</sup> indicates that the latter is likely to decay inside a typical particle detector, which means we should take into account its most dominant decay products. We first have a process by which S<sup>1</sup> decays into a pair of pions, with the following decay rate:

$$\begin{split} \Gamma(S\_1 \to \pi\pi) &\simeq \frac{G\_F m\_1}{4\sqrt{2}\pi} \sin^2 \theta \left[ \frac{m\_1^2}{27} \left( 1 + \frac{11m\_n^2}{2m\_1^2} \right)^2 \right. \\ &\quad \times \left( 1 - \frac{4m\_n^2}{m\_1^2} \right)^{\frac{1}{2}} \Theta(m\_1 - 2m\_n)(2m\_K - m\_1) \right] \\ &\quad + 3 \left( M\_u^2 + M\_d^2 \right) \left( 1 - \frac{4m\_n^2}{m\_1^2} \right)^{\frac{3}{2}} \Theta(m\_1 - 2m\_K) \, . \end{split} \tag{9}$$

Here, m<sup>π</sup> is the pion mass and mK is the kaon mass. Also, chiral perturbation theory is used below the kaon pair production threshold [43, 44], and the spectator-quark model above up to roughly 3 GeV, with the dressed u and d quark masses Mu ¼ Md ≃ 0:05 GeV. Note that this rate includes all pions, charged and neutral. Above the 2mK threshold, there is the production of both a pair of kaons and η particles. The decay rate for K production is

Constraining the Parameters of a Model for Cold Dark Matter http://dx.doi.org/10.5772/intechopen.69044 33

$$\Gamma(\mathbb{S}\_1 \to \mathbb{K} \mathbb{K}) \simeq \frac{9}{13} \frac{\mathfrak{Z} G\_{\mathbb{F}} M\_s^2 m\_1}{4\sqrt{2}\pi} \sin^2 \theta \left(1 - \frac{4m\_K^2}{m\_1^2}\right)^{\frac{3}{2}} \Theta(m\_1 - 2m\_\mathbb{K}).\tag{10}$$

In the above rate, Ms ≃ 0:45 GeV is the s-quark mass in the spectator-quark model [45, 46]. For η production, replace mK by m<sup>η</sup> and 9=13 by 4=13.

5. Constraints from particle phenomenology

0 < ηð Þ<sup>4</sup>

to light cold dark matter.

32 Trends in Modern Cosmology

process is given by the relation:

BrðϒnS ! <sup>γ</sup> <sup>þ</sup> <sup>S</sup>1Þ ¼ GFm<sup>2</sup>

In the above expression, xn � <sup>1</sup> � <sup>m</sup><sup>2</sup>

pions, with the following decay rate:

If a dark matter model based on WIMP physics is not killed already by the constraints coming from cosmology, perturbativity, and direct detection, it has to undergo the tests of particle phenomenology. To see how this works, we discuss here the constraints on our two-singlet model that come from a small selection of low-energy processes, namely, the rare decays of ϒ mesons. The forthcoming discussion is based on work done in Ref. [32]. There, the interested reader will find a fuller account of this study, together with relation to Higgs phenomenology. Note that the dark matter relic-density constraint in Eq. (6) and the perturbativity requirement

<sup>01</sup> < 1 are implemented systematically. Also, as in Ref. [32], we will restrict the discussion

xn <sup>1</sup> � <sup>4</sup>α<sup>s</sup>

<sup>3</sup><sup>π</sup> <sup>f</sup>ðxn<sup>Þ</sup> � �

Brðμ<sup>Þ</sup> <sup>Θ</sup>ð Þ <sup>m</sup><sup>ϒ</sup>nS � <sup>m</sup><sup>1</sup> : <sup>ð</sup>8<sup>Þ</sup>

#

ð9Þ

Θð Þ� m<sup>1</sup> � 2mK :

with the mass of ϒ<sup>1</sup>ð3Þ<sup>S</sup> given by m<sup>ϒ</sup>1ð3Þ<sup>S</sup> ¼

We therefore look at the constraints that come from the decay of the meson ϒ in the state nS (n ¼ 1; 3) into one photon γ and one particle S1. For m1≲8 GeV, the branching ratio for this

<sup>9</sup>:46ð10:355<sup>Þ</sup> GeV, the branching ratio Brðμ<sup>Þ</sup> � Br <sup>ϒ</sup><sup>1</sup>ð3Þ<sup>S</sup> ! <sup>μ</sup>þμ� � � <sup>¼</sup> <sup>2</sup>:48ð2:18Þ � <sup>10</sup>�<sup>2</sup> [41], <sup>α</sup> is the QCD coupling constant, α<sup>s</sup> ¼ 0:184 the QCD coupling constant at the scale m<sup>ϒ</sup>nS , the quantity GF is the Fermi coupling constant, and mb is the b-quark mass [38]. The function fðxÞ incorporates the effect of QCD radiative corrections given in [42] and the step function is denoted by Θð Þx . However, a rough estimate of the lifetime of S<sup>1</sup> indicates that the latter is likely to decay inside a typical particle detector, which means we should take into account its most dominant decay products. We first have a process by which S<sup>1</sup> decays into a pair of

sin <sup>2</sup>

π m2 1

<sup>u</sup> <sup>þ</sup> <sup>M</sup><sup>2</sup> d � � <sup>1</sup> � <sup>4</sup>m<sup>2</sup>

Here, m<sup>π</sup> is the pion mass and mK is the kaon mass. Also, chiral perturbation theory is used below the kaon pair production threshold [43, 44], and the spectator-quark model above up to roughly 3 GeV, with the dressed u and d quark masses Mu ¼ Md ≃ 0:05 GeV. Note that this rate includes all pions, charged and neutral. Above the 2mK threshold, there is the production of

<sup>θ</sup> <sup>m</sup><sup>2</sup> 1 27

2

1 þ

� �<sup>2</sup> "

11m<sup>2</sup> π 2m<sup>2</sup> 1

Θð Þ m<sup>1</sup> � 2m<sup>π</sup> ð Þ 2mK � m<sup>1</sup>

2

π m<sup>2</sup> 1

� �<sup>3</sup>

<sup>b</sup> sin <sup>2</sup> θ

� �

1=m<sup>2</sup> ϒns

ffiffiffi 2 <sup>p</sup> πα

<sup>Γ</sup>ð Þ <sup>S</sup><sup>1</sup> ! ππ <sup>≃</sup> GFm<sup>1</sup>

both a pair of kaons and η particles. The decay rate for K production is

4 ffiffiffi 2 <sup>p</sup> <sup>π</sup>

<sup>þ</sup> <sup>3</sup> <sup>M</sup><sup>2</sup>

� <sup>1</sup> � <sup>4</sup>m<sup>2</sup>

� �<sup>1</sup>

The particle S<sup>1</sup> also decays into c and b quarks (mainly c). Including the radiative QCD corrections, the corresponding decay rates are given by

$$\Gamma(\mathbb{S}\_1 \to q\overline{q}) \simeq \frac{3G\_F \overline{m}^{\theta^2} m\_1}{4\sqrt{2}\pi} \sin^2 \theta \left(\frac{1 - 4\overline{m}\_q^2}{m\_h^2}\right)^{\frac{3}{2}} \left(1 + 5.67 \frac{\overline{a}\_s}{\pi}\right) \Theta\left(m\_1 - 2\overline{m}\_q\right). \tag{11}$$

The dressed quark mass mq � mqðm1Þ and the running strong coupling constant α<sup>s</sup> � αsðm1Þ are defined at the energy scale m<sup>1</sup> [47]. There is also a decay into a pair of gluons, with the rate

$$\Gamma(\mathbb{S}\_1 \to \mathbb{g}\mathbb{g}) \simeq \frac{G\_\mathbb{F} m\_1^3 \sin^2 \theta}{12\sqrt{2}\pi} \left(\frac{\alpha'\_s}{\pi}\right)^2 \left[6 - 2\left(1 - \frac{4m\_n^2}{m\_1^2}\right)^{\frac{3}{2}} - \left(1 - \frac{4m\_K^2}{m\_1^2}\right)^{\frac{3}{2}}\right] \Theta(m\_1 - 2m\_K). \tag{12}$$

Here, α<sup>0</sup> <sup>s</sup> ¼ 0:47 is the QCD coupling constant at the spectator-quark model scale, between roughly 1 and 3 GeV.

We then have the decay of S<sup>1</sup> into leptons, the corresponding rate given by

$$\Gamma\left(\mathbb{S}\_1 \to \ell^+ \ell^-\right) = \frac{G\_F m\_\ell^2 m\_1}{4\sqrt{2}\pi} \sin^2 \theta \left(1 - \frac{4m\_\ell^2}{m\_1^2}\right)^{\frac{2}{2}} \Theta(m\_1 - 2m\_\ell) \tag{13}$$

where m<sup>ℓ</sup> is the lepton mass. Finally, S<sup>1</sup> can decay into a pair of dark matter particles, with a decay rate:

$$
\Gamma(\mathcal{S}\_1 \to \mathcal{S}\_0 \mathcal{S}\_0) = \frac{\left(\eta\_{01}^{(3)}\right)^2}{32\pi m\_1} \sqrt{1 - \frac{4m\_0^2}{m\_1^2}} \Theta(m\_1 - 2m\_0). \tag{14}
$$

The coupling constant η<sup>ð</sup>3<sup>Þ</sup> <sup>01</sup> is given in Eq. (1). The branching ratio for ϒnS decaying via S<sup>1</sup> into a photon plus X, where X represents any kinematically allowed final state, will be

$$\operatorname{Br}(\Upsilon\_{n\mathcal{S}}\to\mathcal{Y}+X) = \operatorname{Br}(\Upsilon\_{n\mathcal{S}}\to\mathcal{Y}+\mathcal{S}\_1)\times\operatorname{Br}(\mathcal{S}\_1\to X).\tag{15}$$

In particular, X � S0S<sup>0</sup> corresponds to a decay into invisible particles.

The best available experimental upper bounds on 1S-state branching ratios are (i) Brð Þ <sup>ϒ</sup>1<sup>S</sup> ! <sup>γ</sup> <sup>þ</sup> ττ <sup>&</sup>lt; <sup>5</sup> � <sup>10</sup>�<sup>5</sup> for 3:5 GeV <sup>&</sup>lt; <sup>m</sup><sup>1</sup> <sup>&</sup>lt; <sup>9</sup>:2 GeV [48]; (ii) Br <sup>ϒ</sup>1<sup>S</sup> ! <sup>γ</sup> <sup>þ</sup> <sup>π</sup>þπ� ð Þ <sup>&</sup>lt; <sup>6</sup>:<sup>3</sup> � <sup>10</sup>�<sup>5</sup> for 1 GeV <sup>&</sup>lt; <sup>m</sup><sup>1</sup> [49]; (iii) Br <sup>ϒ</sup>1<sup>S</sup> ! <sup>γ</sup> <sup>þ</sup> <sup>K</sup>þK� ð Þ <sup>&</sup>lt; <sup>1</sup>:<sup>14</sup> � <sup>10</sup>�<sup>5</sup> for 2 GeV <sup>&</sup>lt; <sup>m</sup><sup>1</sup> <sup>&</sup>lt; 3 GeV [50]. Figure 6 displays the corresponding branching ratios of ϒ1<sup>S</sup> decays via S<sup>1</sup> as

Figure 6. Typical branching ratios of ϒ1<sup>S</sup> decaying into τ's, charged pions, and charged kaons as functions of m1. The corresponding experimental upper bounds are shown.

Figure 7. Typical branching ratios of ϒ3<sup>S</sup> decaying into muons and dark matter as functions of m1. The corresponding experimental upper bounds are shown.

functions of m1, together with these upper bounds. Also, the best available experimental upper bounds on <sup>ϒ</sup>3<sup>S</sup> branching ratios are: (i) Br <sup>ϒ</sup>3<sup>S</sup> ! <sup>γ</sup> <sup>þ</sup> μμ <sup>&</sup>lt; <sup>3</sup> � <sup>10</sup>�<sup>6</sup> for 1 GeV <sup>&</sup>lt; <sup>m</sup><sup>1</sup> <sup>&</sup>lt; 10 GeV; (ii) Brð Þ <sup>ϒ</sup>3<sup>S</sup> ! <sup>γ</sup> <sup>þ</sup> invisible <sup>&</sup>lt; <sup>3</sup> � <sup>10</sup>�<sup>6</sup> for 1 GeV <sup>&</sup>lt; <sup>m</sup><sup>1</sup> <sup>&</sup>lt; <sup>7</sup>:8 GeV [51]. Typical corresponding branching ratios are shown in Figure 7.

If we perform a systematic scan of the parameter space, we find that the main effect of the Higgs-dark matter coupling constant λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> and the dark matter mass m<sup>0</sup> is to exclude, via the relic-density and perturbativity constraints, regions of applicability of the model. This is shown in Figures 6 and 7, where the region m<sup>1</sup> ≲ 1:4 GeV is excluded. Otherwise, these two parameters have little effect on the shapes of the branching ratios themselves. The onset of the S0S<sup>0</sup> channel for m<sup>1</sup> ≥ 2m<sup>0</sup> abates sharply the other channels, and this one becomes dominant by far. The effect of the mixing angle θ is to enhance all branching ratios as it increases, due to the factor sin <sup>2</sup> θ. The dark matter decay channel reaches the invisible upper bound already for θ≃15o, for fairly small m0, say, 0.5 GeV. The other channels find it hard to get to their respective experimental upper bounds, even for large values of θ. There are further constraints that come from particle phenomenology tests. The interested reader may refer to [32] for further details.

#### 6. Internal constraints

functions of m1, together with these upper bounds. Also, the best available experimental upper bounds on <sup>ϒ</sup>3<sup>S</sup> branching ratios are: (i) Br <sup>ϒ</sup>3<sup>S</sup> ! <sup>γ</sup> <sup>þ</sup> μμ <sup>&</sup>lt; <sup>3</sup> � <sup>10</sup>�<sup>6</sup> for 1 GeV <sup>&</sup>lt; <sup>m</sup><sup>1</sup> <sup>&</sup>lt; 10 GeV; (ii) Brð Þ <sup>ϒ</sup>3<sup>S</sup> ! <sup>γ</sup> <sup>þ</sup> invisible <sup>&</sup>lt; <sup>3</sup> � <sup>10</sup>�<sup>6</sup> for 1 GeV <sup>&</sup>lt; <sup>m</sup><sup>1</sup> <sup>&</sup>lt; <sup>7</sup>:8 GeV [51]. Typical corresponding

Figure 7. Typical branching ratios of ϒ3<sup>S</sup> decaying into muons and dark matter as functions of m1. The corresponding

Figure 6. Typical branching ratios of ϒ1<sup>S</sup> decaying into τ's, charged pions, and charged kaons as functions of m1. The

If we perform a systematic scan of the parameter space, we find that the main effect of the

relic-density and perturbativity constraints, regions of applicability of the model. This is shown in Figures 6 and 7, where the region m<sup>1</sup> ≲ 1:4 GeV is excluded. Otherwise, these two

<sup>0</sup> and the dark matter mass m<sup>0</sup> is to exclude, via the

branching ratios are shown in Figure 7.

experimental upper bounds are shown.

corresponding experimental upper bounds are shown.

34 Trends in Modern Cosmology

Higgs-dark matter coupling constant λ<sup>ð</sup>4<sup>Þ</sup>

Further constraints on a field-theory dark matter model come from internal consistencies. Indeed, one must ask how high in the energy scale the model is computationally reliable. To answer this question, one investigates the running of the coupling constants as a function of the scale Λ via the renormalization-group equations (RGE). One-loop calculations are amply sufficient. A detailed study of the RGE for our two-singlet model was carried out in Ref. [33]. The brief subsequent discussion is drawn from there, and the reader is referred to that article for more details.

In an RGE study, there are two standard issues to monitor, namely, the perturbativity of the scalar coupling constants and the vacuum stability of the theory. Imposing these two latter as conditions on the model will indicate at what scale Λ<sup>m</sup> it is valid. As mentioned in the Introduction, it has been anticipated that new physics, such as supersymmetry would appear at the LHC at the scale Λ � 1 TeV. Present results from ATLAS and CMS indicate no such signs yet. One consequence of this is that the cutoff scale Λ<sup>m</sup> may be higher. In this model, the RGE study suggests that it can be � 40 TeV. As ever, the DM relic-density constraint is systematically imposed, together with the somewhat less stringent perturbativity restriction 0 ≤ η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> <sup>≤</sup> ffiffiffiffiffiffi <sup>4</sup><sup>π</sup> <sup>p</sup> .

Remember that the model is obtained by extending the Standard Model with two real, spinless, and Z2-symmetric SM-gauge-singlet fields. The potential function of the scalar sector after spontaneous breaking of the gauge and one of the Z<sup>2</sup> symmetries is given in Eq. (1). The potential function before symmetry breaking is the one we need in this section. It is given in Eq. [31]:

$$\begin{split} \mathcal{U} &= \frac{\tilde{m}\_0^2}{2} \mathcal{S}\_0^2 - \mu^2 H^\dagger H - \frac{\mu\_1^2}{2} \chi\_1^2 \\ &+ \frac{\eta\_0}{24} \mathcal{S}\_0^4 + \frac{\lambda}{6} \left( H^\dagger H \right)^2 + \frac{\eta\_1}{24} \chi\_1^4 + \frac{\lambda\_0}{2} S\_0^2 H^\dagger H + \frac{\eta\_{01}}{4} S\_0^2 \chi\_1^2 + \frac{\lambda\_1}{2} H^\dagger H \chi\_1^2. \end{split} \tag{16}$$

The field S<sup>0</sup> is still the WIMP with unbroken Z<sup>2</sup> symmetry, and χ<sup>1</sup> is the auxiliary field before spontaneously breaking its Z<sup>2</sup> symmetry. Both fields interact with the SM particles via the Higgs doublet H. The masses m~ <sup>2</sup> <sup>0</sup>, μ<sup>2</sup>, and μ<sup>2</sup> <sup>1</sup> as well as all the coupling constants are real positive numbers.<sup>1</sup>

<sup>1</sup> The mutual couplings can be negative as discussed below, see (21).

A one-loop renormalization-group calculation yields the following β-functions for the above scalar coupling constants [33]:

βη0 <sup>¼</sup> <sup>3</sup> <sup>16</sup>π<sup>2</sup> <sup>η</sup><sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup> <sup>01</sup> <sup>þ</sup> <sup>4</sup>λ<sup>2</sup> 0 � �; βη1 <sup>¼</sup> <sup>3</sup> <sup>16</sup>π<sup>2</sup> <sup>η</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup> <sup>01</sup> <sup>þ</sup> <sup>4</sup>λ<sup>2</sup> 1 � �; βλ <sup>¼</sup> <sup>3</sup> 16π<sup>2</sup> 4 3 <sup>λ</sup><sup>2</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> <sup>1</sup> � <sup>48</sup>λ<sup>4</sup> <sup>t</sup> <sup>þ</sup> <sup>8</sup>λλ<sup>2</sup> <sup>t</sup> � <sup>3</sup>λg<sup>2</sup> � <sup>λ</sup><sup>g</sup> 0 2 þ 3 2 g2 g 0 2 þ 9 4 g4 � �; βη01 <sup>¼</sup> <sup>1</sup> <sup>16</sup>π<sup>2</sup> <sup>4</sup>η<sup>2</sup> <sup>01</sup> þ η0η<sup>01</sup> þ η1η<sup>01</sup> þ 4λ0λ<sup>1</sup> � �; βλ<sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>16</sup>π<sup>2</sup> <sup>4</sup>λ<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>λ</sup>0η<sup>0</sup> <sup>þ</sup> <sup>2</sup>λ0<sup>λ</sup> <sup>þ</sup> <sup>η</sup>01λ<sup>1</sup> <sup>þ</sup> <sup>12</sup>λ0λ<sup>2</sup> <sup>t</sup> � <sup>9</sup> 2 <sup>λ</sup>0g<sup>2</sup> � <sup>3</sup> 2 λ0g 0 2 � �; βλ<sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>16</sup>π<sup>2</sup> <sup>4</sup>λ<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>λ</sup>1η<sup>1</sup> <sup>þ</sup> <sup>2</sup>λ1<sup>λ</sup> <sup>þ</sup> <sup>η</sup>01λ<sup>0</sup> <sup>þ</sup> <sup>12</sup>λ1λ<sup>2</sup> <sup>t</sup> � <sup>9</sup> 2 <sup>λ</sup>1g<sup>2</sup> � <sup>3</sup> 2 λ1g 0 2 � �: ð17Þ

As usual, by definition β<sup>g</sup> � dg=dlnΛ, where Λ is the running mass scale, starting from Λ<sup>0</sup> ¼ 100 GeV. Note that the DM self-coupling constant η<sup>0</sup> has so far been decoupled from the other coupling constants, but not anymore in view of Eq. (17) now that the running is the focus. However, its initial value η0ð Þ Λ<sup>0</sup> is arbitrary and its β-function is always positive. This means η0ð Þ Λ will only increase as Λ increases, quickly if starting from a rather large initial value, slowly if not. Therefore, without losing generality in the subsequent discussion, we fix <sup>η</sup>0ð Þ¼ <sup>Λ</sup><sup>0</sup> 1. Hence, here too we still effectively have four free parameters: <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> , θ, m0, and m1.

Furthermore, the constants g, g 0 , and gs are the SM and strong gauge couplings, known [52] and given to one-loop order by the expression:

$$G(\Lambda) = \frac{G(\Lambda\_0)}{\sqrt{1 - 2a\_G G^2(\Lambda\_0) \ln\left(\frac{\Lambda}{\Lambda\_0}\right)}},\tag{18}$$

where aG <sup>¼</sup> �<sup>19</sup> <sup>96</sup>π<sup>2</sup> , <sup>41</sup> <sup>96</sup>π<sup>2</sup> , �<sup>7</sup> <sup>16</sup>π<sup>2</sup> and Gð Þ¼ Λ<sup>0</sup> 0:65, 0:36; 1:2 for G ¼ g, g<sup>0</sup> , gs, respectively. The coupling constant λ<sup>t</sup> is that between the Higgs field and the top quark. To one-loop order, it runs according to Ref. [52] the following expression:

$$\beta\_{\lambda\_t} = \frac{\lambda\_t}{16\pi^2} \left( 9\lambda\_t^2 - 8g\_s^2 - \frac{9}{4}g^2 - \frac{17}{12}g'^2 \right),\tag{19}$$

with λtð Þ¼ Λ<sup>0</sup> mtð Þ Λ<sup>0</sup> <sup>v</sup> ¼ 0:7, where v is the Higgs vacuum expectation value and mt is the top mass. Note that we are taking into consideration the fact that the top-quark contribution is dominant over that of the other fermions of the Standard Model.

After the two spontaneous breakings of symmetry, we end up with the two vacuum expectation values: v ¼ 246GeV for the Higgs field h, and v<sup>1</sup> for the auxiliary field S1. In this section, we take v<sup>1</sup> ¼ 150GeV. Above v, the fields and parameters of the theory are those of (16). Below v1, the fields and parameters are those of Eq. (1). We take the values of the physical parameters at the mass scale Λ<sup>0</sup> ¼ 100GeV. The initial conditions for the coupling constants in (16) in terms of these physical free parameters are as follows:

$$\begin{aligned} \eta\_1(\Lambda\_0) &= \frac{3}{2v\_1^2} \left[ m\_1^2 + m\_h^2 + |m\_1^2 - m\_h^2| \left( \cos \left( 2\theta \right) + \frac{v}{2v\_1} \sin \left( 2\theta \right) \right) \right]; \\ \lambda(\Lambda\_0) &= \frac{3}{2v^2} \left[ m\_1^2 + m\_h^2 - |m\_1^2 - m\_h^2| \left( \cos \left( 2\theta \right) - \frac{v\_1}{2v} \sin \left( 2\theta \right) \right) \right]; \\ \lambda\_1(\Lambda\_0) &= \frac{\sin \left( 2\theta \right)}{2vv\_1} |m\_1^2 - m\_h^2|; \\ \eta\_{01}(\Lambda\_0) &= \frac{1}{\cos \left( 2\theta \right)} \left[ \eta\_{01}^{(4)} \cos^2 \theta - \lambda\_0^{(4)} \sin^2 \theta \right]; \\ \lambda\_0(\Lambda\_0) &= \frac{1}{\cos \left( 2\theta \right)} \left[ \lambda\_0^{(4)} \cos^2 \theta - \eta\_{01}^{(4)} \sin^2 \theta \right]. \end{aligned} \tag{20}$$

Note that, normally, as we go down the mass scale, we should seam quantities in steps: at v, v1, and <sup>Λ</sup>0. However, the corrections to (20) are of one-loop order times ln <sup>v</sup> <sup>v</sup><sup>1</sup> or ln <sup>v</sup><sup>1</sup> Λ<sup>0</sup> , small enough for our present purposes to neglect. The perturbativity constraint we impose on all dimensionless scalar coupling constants is <sup>G</sup>ð Þ <sup>Λ</sup> <sup>≤</sup> ffiffiffiffiffiffi <sup>4</sup><sup>π</sup> <sup>p</sup> . Also, vacuum stability means that <sup>G</sup>ð Þ <sup>Λ</sup> <sup>≥</sup> 0 for the self-coupling constants η0, λ, and η1, and the conditions:

$$-\frac{1}{6}\sqrt{\eta\_0\lambda} \le \lambda\_0 \le \sqrt{4\pi}; \qquad -\frac{1}{6}\sqrt{\eta\_0\eta\_1} \le \eta\_{01} \le \sqrt{4\pi}; \qquad -\frac{1}{6}\sqrt{\eta\_1\lambda} \le \lambda\_1 \le \sqrt{4\pi} \tag{21}$$

for the mutual couplings λ0, η01, and λ1.

A one-loop renormalization-group calculation yields the following β-functions for the above

<sup>t</sup> <sup>þ</sup> <sup>8</sup>λλ<sup>2</sup>

� �

� �

� �

As usual, by definition β<sup>g</sup> � dg=dlnΛ, where Λ is the running mass scale, starting from Λ<sup>0</sup> ¼ 100 GeV. Note that the DM self-coupling constant η<sup>0</sup> has so far been decoupled from the other coupling constants, but not anymore in view of Eq. (17) now that the running is the focus. However, its initial value η0ð Þ Λ<sup>0</sup> is arbitrary and its β-function is always positive. This means η0ð Þ Λ will only increase as Λ increases, quickly if starting from a rather large initial value, slowly if not. Therefore, without losing generality in the subsequent discussion, we fix

> <sup>G</sup>ð Þ¼ <sup>Λ</sup> <sup>G</sup>ð Þ <sup>Λ</sup><sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>2</sup>aG <sup>G</sup><sup>2</sup>

<sup>16</sup>π<sup>2</sup> and Gð Þ¼ Λ<sup>0</sup> 0:65, 0:36; 1:2 for G ¼ g, g<sup>0</sup>

pling constant λ<sup>t</sup> is that between the Higgs field and the top quark. To one-loop order, it runs

<sup>t</sup> � <sup>8</sup>g<sup>2</sup>

mass. Note that we are taking into consideration the fact that the top-quark contribution is

After the two spontaneous breakings of symmetry, we end up with the two vacuum expectation values: v ¼ 246GeV for the Higgs field h, and v<sup>1</sup> for the auxiliary field S1. In this section,

<sup>s</sup> � <sup>9</sup> 4

� �

<sup>t</sup> � <sup>3</sup>λg<sup>2</sup> � <sup>λ</sup><sup>g</sup>

<sup>t</sup> � <sup>9</sup> 2

<sup>t</sup> � <sup>9</sup> 2

, and gs are the SM and strong gauge couplings, known [52]

<sup>s</sup> � � , <sup>ð</sup>18<sup>Þ</sup>

ð Þ <sup>Λ</sup><sup>0</sup> ln <sup>Λ</sup>

<sup>g</sup><sup>2</sup> � <sup>17</sup> <sup>12</sup> <sup>g</sup>0<sup>2</sup>

<sup>v</sup> ¼ 0:7, where v is the Higgs vacuum expectation value and mt is the top

Λ<sup>0</sup>

0 2 þ 3 2 g2 g 0 2 þ 9 4 g4

<sup>λ</sup>0g<sup>2</sup> � <sup>3</sup> 2 λ0g 0 2

<sup>λ</sup>1g<sup>2</sup> � <sup>3</sup> 2 λ1g 0 2 ;

ð17Þ

;

:

<sup>0</sup> , θ, m0, and m1.

, gs, respectively. The cou-

, ð19Þ

scalar coupling constants [33]:

<sup>¼</sup> <sup>3</sup> <sup>16</sup>π<sup>2</sup> <sup>η</sup><sup>2</sup>

<sup>¼</sup> <sup>3</sup> <sup>16</sup>π<sup>2</sup> <sup>η</sup><sup>2</sup>

<sup>¼</sup> <sup>1</sup> <sup>16</sup>π<sup>2</sup> <sup>4</sup>η<sup>2</sup>

βλ <sup>¼</sup> <sup>3</sup> 16π<sup>2</sup>

βλ<sup>0</sup> <sup>¼</sup> <sup>1</sup>

βλ<sup>1</sup> <sup>¼</sup> <sup>1</sup>

Furthermore, the constants g, g

<sup>96</sup>π<sup>2</sup> , <sup>41</sup>

mtð Þ Λ<sup>0</sup>

<sup>96</sup>π<sup>2</sup> , �<sup>7</sup>

according to Ref. [52] the following expression:

where aG <sup>¼</sup> �<sup>19</sup>

with λtð Þ¼ Λ<sup>0</sup>

and given to one-loop order by the expression:

<sup>0</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup>

<sup>λ</sup><sup>2</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

4 3

<sup>16</sup>π<sup>2</sup> <sup>4</sup>λ<sup>2</sup>

<sup>16</sup>π<sup>2</sup> <sup>4</sup>λ<sup>2</sup>

<sup>01</sup> <sup>þ</sup> <sup>4</sup>λ<sup>2</sup> 0

<sup>01</sup> <sup>þ</sup> <sup>4</sup>λ<sup>2</sup> 1

<sup>0</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

<sup>1</sup> � <sup>48</sup>λ<sup>4</sup>

<sup>0</sup> <sup>þ</sup> <sup>λ</sup>0η<sup>0</sup> <sup>þ</sup> <sup>2</sup>λ0<sup>λ</sup> <sup>þ</sup> <sup>η</sup>01λ<sup>1</sup> <sup>þ</sup> <sup>12</sup>λ0λ<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>λ</sup>1η<sup>1</sup> <sup>þ</sup> <sup>2</sup>λ1<sup>λ</sup> <sup>þ</sup> <sup>η</sup>01λ<sup>0</sup> <sup>þ</sup> <sup>12</sup>λ1λ<sup>2</sup>

<sup>01</sup> þ η0η<sup>01</sup> þ η1η<sup>01</sup> þ 4λ0λ<sup>1</sup> � �;

<sup>η</sup>0ð Þ¼ <sup>Λ</sup><sup>0</sup> 1. Hence, here too we still effectively have four free parameters: <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup>

0

βλ<sup>t</sup> <sup>¼</sup> <sup>λ</sup><sup>t</sup>

dominant over that of the other fermions of the Standard Model.

<sup>16</sup>π<sup>2</sup> <sup>9</sup>λ<sup>2</sup>

� �;

� �;

βη0

36 Trends in Modern Cosmology

βη1

βη01

Figure 8 displays the behavior of the self-couplings under RGE for <sup>θ</sup><sup>¼</sup> 10o, <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ¼ 0:01;m<sup>1</sup> ¼ 110 GeV, and m<sup>0</sup> ¼ 55 GeV. The dramatic effect is on the Higgs self-coupling constant λ which quickly gets into negative territory, at about 15 TeV, thus rendering the theory unstable beyond this mass scale. This is better displayed in Figure 9, where the Renormalization Group (RG) behavior of λ is shown by itself. Such a negative slope forλ is expected, given the negative contributions to βλ in (17). The coupling constant η<sup>1</sup> is dominant over the other couplings and controls perturbativity, leaving its region much later, at about 1600TeV. This seems to be a somewhat general trend: the non-Higgs SM particles seem to flatten the runnings of the scalar couplings.

The runnings of the mutual coupling constants for the same set of parameters' values are displayed in Figure 10. They also get flattened by the other SM particles, but they stay positive. They dwell well below the self-couplings. Increasing m<sup>0</sup> and m<sup>1</sup> will raise the mutual coupling η<sup>01</sup> and not the two others, higher than η<sup>1</sup> in some regions.

Raising λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> will also make the self-couplings η<sup>1</sup> and η<sup>0</sup> run faster while affecting very little λ. It will also make the mutual coupling η<sup>01</sup> starts higher, and so demarked from λ<sup>0</sup> and λ1.

Figure 8. Running of the self-couplings. η<sup>1</sup> controls perturbativity and the Higgs coupling λ becomes negative quickly.

Figure 9. The running of the Higgs self-coupling λ. It becomes negative at about 15 TeV for this set of parameter values.

By contrast, the effect of θ is not very dramatic: the self-couplings are not much affected and the mutuals only evolve differently, without any particular boosting of η01. Details and further comments are found in [33].

Figure 10. Running of the mutual couplings. The inclusion of the other SM particles flattens the runnings.

#### 7. All constraints together: viability regions

The above RGE analysis taught us two lessons: (i) The two couplings η<sup>1</sup> and η<sup>01</sup> control perturbativity. (ii) The change of sign of λ controls vacuum stability. Equipped with these indicators, we can try to systematically locate the regions in parameter space in which the model is viable. We have by now a number of tools at our disposal. First, the DM relic-density constraint (6), which has been and will continue to be applied throughout. We have the RGE analysis of the previous section. We will require both <sup>η</sup>1ð Þ <sup>Λ</sup> and <sup>η</sup>01ð Þ <sup>Λ</sup> to be smaller than ffiffiffiffiffiffi <sup>4</sup><sup>π</sup> <sup>p</sup> , and λð Þ Λ to be positive. From the phenomenological implications we deduced in Section 5, we will retain only two: the mixing angle θ and the physical self-coupling λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> are to be chosen small. Last, we want the model to comply with the experimental direct-detection upper bounds. The condition we impose is that σdet of Eq. (7) be within the XENON 100 upper bounds [20]. We will vary λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> and θ and track the viability regions in the ð Þ m0, m<sup>1</sup> plane. The relevant mass range for m<sup>0</sup> and m<sup>1</sup> is 1–160 GeV. This is because there are no reliable data to discuss below the GeV and beyond 160 GeV takes us outside the perturbativity region.2

One important issue must be addressed before we proceed: How far do we want the model to be perturbatively predictive and stable? The maximum value Λ<sup>m</sup> for the mass scale Λ should not be very high. One reason, more conceptual, is that we want to allow the model to be intermediary between the current Standard Model and some possible higher structure at higher energies. Another one, more practical, is that a too high Λ<sup>m</sup> is too restrictive for the

By contrast, the effect of θ is not very dramatic: the self-couplings are not much affected and the mutuals only evolve differently, without any particular boosting of η01. Details and further

Figure 9. The running of the Higgs self-coupling λ. It becomes negative at about 15 TeV for this set of parameter values.

Figure 8. Running of the self-couplings. η<sup>1</sup> controls perturbativity and the Higgs coupling λ becomes negative quickly.

comments are found in [33].

38 Trends in Modern Cosmology

<sup>2</sup> In practice, m<sup>0</sup> is taken up to 200 GeV, but there are no additional features to report.

parameters themselves. From the results of the RGE analysis [33], a reasonable compromise is to set Λ<sup>m</sup> ≃ 40 TeV.

With all this in mind, Figure 11 displays the regions (blue) for which the model is viable when λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> <sup>¼</sup> <sup>0</sup>:01 and <sup>θ</sup><sup>¼</sup> 1o. The mass <sup>m</sup><sup>1</sup> is confined to the interval 116–138 GeV while the DM mass is confined mainly to the region above 118 GeV, the left boundary of which having a positive slope as m<sup>1</sup> increases. In addition, m<sup>0</sup> has a small showing in the narrow interval 57–68 GeV. The effect of increasing the mixing angle θ is to enrich the existing regions without relocating them. This is displayed in Figure 12 for which θ is increased to 15o. As θ increases, the region between the narrow band and the larger one to the right gets populated. This means more viable DM masses above 60 GeV, but m<sup>1</sup> stays in the same interval.

By contrast, increasing the Higgs-DM mutual coupling λð Þ<sup>4</sup> <sup>0</sup> has the opposite effect, that of shrinking existing viability regions. To see this, compare Figure 13, for which λð Þ<sup>4</sup> <sup>0</sup> ¼ 0:1 and <sup>θ</sup><sup>¼</sup> <sup>15</sup><sup>o</sup>, with Figure 12. We see indeed shrunk regions, pushed downward by a few GeVs, which is not a substantial relocation. This effect should be expected because increasing λð Þ<sup>4</sup> 0 raises <sup>η</sup>01ð Þ <sup>Λ</sup><sup>0</sup> , well enough above 1 so that we leave perturbativity sooner. Increasing <sup>λ</sup>ð Þ<sup>4</sup> <sup>0</sup> is also caught up by the relic-density constraint, which tends to shut down such larger values of λð Þ<sup>4</sup> <sup>0</sup> when m<sup>0</sup> is large. The direct-detection constraint has also a similar effect. Further comments can be found in [33].

Figure 11. Regions of viability of the two-singlet model (in dark grey). Physical Higgs self-coupling λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> and mixing angle θ very small.

Constraining the Parameters of a Model for Cold Dark Matter http://dx.doi.org/10.5772/intechopen.69044 41

Figure 12. The region of viability (dark grey) is even richer for a larger mixing angle θ.

parameters themselves. From the results of the RGE analysis [33], a reasonable compromise is

With all this in mind, Figure 11 displays the regions (blue) for which the model is viable when

<sup>0</sup> <sup>¼</sup> <sup>0</sup>:01 and <sup>θ</sup><sup>¼</sup> 1o. The mass <sup>m</sup><sup>1</sup> is confined to the interval 116–138 GeV while the DM mass is confined mainly to the region above 118 GeV, the left boundary of which having a positive slope as m<sup>1</sup> increases. In addition, m<sup>0</sup> has a small showing in the narrow interval 57–68 GeV. The effect of increasing the mixing angle θ is to enrich the existing regions without relocating them. This is displayed in Figure 12 for which θ is increased to 15o. As θ increases, the region between the narrow band and the larger one to the right gets populated. This means more

<sup>0</sup> has the opposite effect, that of

<sup>0</sup> ¼ 0:1 and

<sup>0</sup> and mixing angle

0

<sup>0</sup> is

viable DM masses above 60 GeV, but m<sup>1</sup> stays in the same interval.

shrinking existing viability regions. To see this, compare Figure 13, for which λð Þ<sup>4</sup>

Figure 11. Regions of viability of the two-singlet model (in dark grey). Physical Higgs self-coupling λ<sup>ð</sup>4<sup>Þ</sup>

<sup>θ</sup><sup>¼</sup> <sup>15</sup><sup>o</sup>, with Figure 12. We see indeed shrunk regions, pushed downward by a few GeVs, which is not a substantial relocation. This effect should be expected because increasing λð Þ<sup>4</sup>

raises <sup>η</sup>01ð Þ <sup>Λ</sup><sup>0</sup> , well enough above 1 so that we leave perturbativity sooner. Increasing <sup>λ</sup>ð Þ<sup>4</sup>

also caught up by the relic-density constraint, which tends to shut down such larger values of

<sup>0</sup> when m<sup>0</sup> is large. The direct-detection constraint has also a similar effect. Further com-

By contrast, increasing the Higgs-DM mutual coupling λð Þ<sup>4</sup>

to set Λ<sup>m</sup> ≃ 40 TeV.

40 Trends in Modern Cosmology

λ<sup>ð</sup>4<sup>Þ</sup>

λð Þ<sup>4</sup>

θ very small.

ments can be found in [33].

Figure 13. The physical Higgs self-coupling λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> shrinks the viability region (dark grey) as it increases.

#### 8. Concluding remarks

The purpose of this chapter was to help the reader understand how modeling cold dark matter evolves from motivating the model itself to constraining the space of its parameters. We took as prototype a two-singlet extension to the Standard Model of elementary particles within the paradigm of weakly interacting massive particles.

The first set of constraints the model had to undergo came from cosmology and perturbativity. The model had to reproduce the known relic density of cold dark matter while being consistent with perturbation theory. The second set of tests came from direct detection, in the form of the total elastic cross-section of a WIMP scattering off a non-relativistic nucleon that had to satisfy bounds set by several direct-detection experiments. We have seen that the model is capable of satisfying all the existing bounds and will soon be probed by the coming XENON1t experiment. The third set of constraints came from particle phenomenology. We have seen how ϒ rare decays constrain the predictions of the model for light cold dark matter. The fourth set of constraints came from internal consistency of the model, in the form of viability and stability under running coupling constants via a renormalization-group analysis. We have concluded that the model can still make sound predictions in important and useful physical regions. We then have investigated the regions in the space of parameters in which the model is viable when all these four sets of constraints are applied together with a maximum cutoff Λm≃40 TeV, a scale at which heavy degrees of freedom may start to be relevant. We have deduced that for small λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> and θ, the auxiliary field mass m<sup>1</sup> is confined to the interval 116–138 GeV, while the DM mass m<sup>0</sup> is confined mainly to the region above 118 GeV, with a small showing in the narrow interval 57– 68 GeV. Increasing θ enriches the existing viability regions without relocating them, while increasing λð Þ<sup>4</sup> <sup>0</sup> has the opposite effect, that of shrinking them without substantial relocation.

There is one aspect of the study we have not touched upon in this review, and that is the connection with and consequences from Higgs physics. This has been analyzed in Refs. [32, 33]. This aspect is important, of course, too important maybe to be just touched upon in this limited space. Such an analysis also needs to be reactualized in view of the many advances made in Higgs physics [53].

Despite all our efforts, dark matter stays elusive. Many models that tried to understand it have failed. The fate of the two-singlet model may not be different. But this will not be a source of disappointment. On the contrary, failure will only fuel motivation to try and explore new ideas.

### Author details

Abdessamad Abada1,2\* and Salah Nasri3

\*Address all correspondence to: a.abada@uaeu.ac.ae

1 Physics Department, United Arab Emirates University, Al Ain, United Arab Emirates

2 Laboratoire de Physique des Particules et Physique Statistique, Ecole Normale Supérieure, Alger, Algeria

3 United Arab Emirates University, United Arab Emirates

#### References

8. Concluding remarks

42 Trends in Modern Cosmology

increasing λð Þ<sup>4</sup>

Author details

Alger, Algeria

Abdessamad Abada1,2\* and Salah Nasri3

\*Address all correspondence to: a.abada@uaeu.ac.ae

3 United Arab Emirates University, United Arab Emirates

paradigm of weakly interacting massive particles.

The purpose of this chapter was to help the reader understand how modeling cold dark matter evolves from motivating the model itself to constraining the space of its parameters. We took as prototype a two-singlet extension to the Standard Model of elementary particles within the

The first set of constraints the model had to undergo came from cosmology and perturbativity. The model had to reproduce the known relic density of cold dark matter while being consistent with perturbation theory. The second set of tests came from direct detection, in the form of the total elastic cross-section of a WIMP scattering off a non-relativistic nucleon that had to satisfy bounds set by several direct-detection experiments. We have seen that the model is capable of satisfying all the existing bounds and will soon be probed by the coming XENON1t experiment. The third set of constraints came from particle phenomenology. We have seen how ϒ rare decays constrain the predictions of the model for light cold dark matter. The fourth set of constraints came from internal consistency of the model, in the form of viability and stability under running coupling constants via a renormalization-group analysis. We have concluded that the model can still make sound predictions in important and useful physical regions. We then have investigated the regions in the space of parameters in which the model is viable when all these four sets of constraints are applied together with a maximum cutoff Λm≃40 TeV, a scale at which heavy

degrees of freedom may start to be relevant. We have deduced that for small λ<sup>ð</sup>4<sup>Þ</sup>

auxiliary field mass m<sup>1</sup> is confined to the interval 116–138 GeV, while the DM mass m<sup>0</sup> is confined mainly to the region above 118 GeV, with a small showing in the narrow interval 57– 68 GeV. Increasing θ enriches the existing viability regions without relocating them, while

There is one aspect of the study we have not touched upon in this review, and that is the connection with and consequences from Higgs physics. This has been analyzed in Refs. [32, 33]. This aspect is important, of course, too important maybe to be just touched upon in this limited space. Such an analysis also needs to be reactualized in view of the many advances made in Higgs physics [53]. Despite all our efforts, dark matter stays elusive. Many models that tried to understand it have failed. The fate of the two-singlet model may not be different. But this will not be a source of disappointment. On the contrary, failure will only fuel motivation to try and explore new ideas.

1 Physics Department, United Arab Emirates University, Al Ain, United Arab Emirates

2 Laboratoire de Physique des Particules et Physique Statistique, Ecole Normale Supérieure,

<sup>0</sup> has the opposite effect, that of shrinking them without substantial relocation.

<sup>0</sup> and θ, the


[16] CDMS Collaboration. Physical Review Letters. 2000;84:5699. arXiv:astro-ph/0002471

[18] Super CDMS Collaboration. Physical Review Letters. 2016;116:071301. arXiv:1509.02448

[20] XENON100 Collaboration. Physical Review D. 2016;94:122001. arXiv:1609.06154; Science. 2015;349:851. arXiv:1507.07747; EPJ Web of Conferences. 2016;121:06006. arXiv:1501.03492

[21] The XENON Collaboration. Journal of Cosmology and Astroparticle Physics. 2016;04:

[24] Jochum J, et al. [CRESST Collaboration], Physics of Atomic Nuclei. 2000;63:1242;

[25] Angloher G, et al. [CRESST II Collaboration], The European Physical Journal C. 2014;74:3184. arXiv:1407.3146; The European Physical Journal C. 2016;76:25. arXiv:

[26] Armengaud E, et al. Journal of Cosmology and Astroparticle Physics. 2016;05:019. arxiv:1603.

[27] Ackermann M, et al. [Fermi-LAT], Physical Review Letters. 2015;115:231301. arXiv: 1503.02641; Fermi-LAT Collaboration, The Astrophysical Journal. 2016;819:44. arXiv:

[28] Aguilar M, et al. [AMS Collaboration]. Physical Review Letters. 2016;110:141102; Aguilar

[29] Adriani O, et al. [PAMELA Collaboration], Nature. 2009;458:607. arXiv:0810.4995; PAMELA Collaboration, Physical Review Letters. 2016;111:081102. arXiv:1308.0133;

M, et al. [AMS-02 Collaboration], Physical Review Letters. 2016;117:091103

Adriani O, et al. [PAMELA collaboration]. Physics Reports. 2014;544:323

[31] Abada A, Ghafor D, Nasri S. Physical Review D. 2011;83:095021. arXiv:1101.0365

[34] Autermann C. Progress in Particle and Nuclear Physics. 2016;90:125. arXiv:1609.01686

[36] McDonald J. Physical Review D. 1994;50:3637; Burgess CP, Pospelov M, ter Veldhuis T ter. Nuclear Physics B. 2001;619:709; Barger V, et al. Physical Review D. 2008;77:035005.

arXiv:0706.4311; Gonderinger M, et al. JHEP. 2010;053:1001. arXiv:0910.3167

[30] ATLAS Collaboration. Physics Letters B. 2016;763:251. arXiv:1608.02372

[32] Abada A, Nasri S. Physical Review D. 2012;85:075009. arXiv:1201.1413 [33] Abada A, Nasri S. Physical Review D. 2013;88:016006. arXiv:1201.1413

[22] Akerib DS, et al. Physical Review Letters. 2016;116:161301. arXiv:1512.03506

1509.01515; Physical Review Letters. 2016;117:021303. arXiv:1601.04447

[17] CDMS Collaboration. Physical Review Letters. 2013;111:251301. arXiv:1304.4279

[19] Aprile E, et al. Astroparticle Physics. 2011;34:679. arXiv:1001.2834

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44 Trends in Modern Cosmology

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## **Neutrino Interactions with Nuclei and Dark Matter**

Paraskevi C. Divari

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68196

#### Abstract

Even though the combined laboratory, astrophysical and cosmological evidence implies that neutrinos have masses, neutrinos provide only a small cosmic dark matter component. The study of solar neutrinos provides important information on nuclear processes inside the Sun as well as on matter densities. Moreover, supernova neutrinos provide sensitive probes for studying supernova explosions, neutrino properties and stellar collapse mechanisms. Neutrino-nucleus reactions at energies below 100 MeV play essential roles in core-collapse supernovae, explosive and r-process nucleosynthesis, as well as observation of solar and supernova neutrinos by earthbound detectors. On the other hand, recent experimental data of high-energy extragalactic neutrinos at 1 PeV open a new window to probe non-standard neutrino properties, such as resonant effects in the oscillation probability.

Keywords: neutrino physics, neutrino oscillations, charge current neutrino-nucleus scattering, dark matter, sterile neutrino

#### 1. Introduction

Neutrinos play a fundamental role in cosmology and astrophysics, two rapidly progressing fields. The origin of neutrino masses and the nature of dark matter (DM) are two most pressing open questions in modern astro-particle physics. We know from the observation of neutrino oscillations that neutrinos have masses [1, 2]. The smallness of neutrino masses relative to those of the standard model (SM) charged fermions remains a puzzle. The effect of small neutrino masses may be probed in precision cosmic microwave background (CMB) radiation observations [3–10] as well as large-scale galaxy surveys [11–13]. The absolute scale of neutrino mass may also affect the long-standing issue of cosmic structure formation. Furthermore, neutrinos govern big-bang nucleosynthesis so that neutrino properties can be inferred from the observed light-element abundances [14–16]. Massive neutrinos may also be responsible to account for the mystery of the matter to anti-matter asymmetry in the Universe. Finally, corecollapse supernovae are powerful 'laboratories' to probe neutrino properties if in the future

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

one were to observe a high-statistics neutrino signal. For example, a stellar core collapse in the Milky Way satellite galaxies may produce an enormous burst of neutrinos 'visible' by terrestrial detectors. Such an effect will carry important information in astrophysics, cosmology and particle physics [17, 18]. In addition, dedicated experiments are now planned involving intense accelerator-produced neutrino beams to study neutrino properties over long baselines. These will traverse the mantle or/and core of the Earth [19] so that the interpretation of the results will require geophysical details [20, 21].

Neutrinos could be key particles to unravel the nature of the DM in the Universe. The dark matter problem has been a long-standing one in physics [22, 23]. Even though we know that it must exist [24–27], we do not know much about its true nature. It is clear, though, that massive neutrinos and dark matter are both part of nature and should be incorporated in models of physics beyond the standard model. It may be that they are related to each other [28] and that, in addition, both originate from new physics at the TeV scale.

Several studies have noted that the existence of light sterile neutrinos would have important consequences for dark matter searches [29]. Moreover, MSW-enhanced transitions between active and sterile neutrinos would have a substantial impact on searches for neutrinos from dark matter annihilation in the Sun [30, 31]. Furthermore, if sterile neutrinos in addition to theirmixing with the active neutrinos possess some new gauge interactions, they could lead to signals which appear to favour a dark matter interpretation. These can be used to investigate sterile states and may also generate strong signals in DM detectors [32–34]. Couplings between neutrino, either active or sterile, and dark matter have been studied in many different contexts [35–46].

Understanding the explosion of supernovae or the physics of the early universe, where neutrinos play an essential role, requires a solid theoretical background in astrophysics and cosmology and reliable input from nuclear physics. Neutrino-nucleus scattering at energies below 100 MeV plays an essential role in core-collapse supernova simulations in various interactions of neutrinos with the supernova environment. Based on the improved supernova simulations, it is found that inelastic neutrino-nucleus reactions will also allow for an additional mode of energy deposition to the matter ahead of the shock wave in the post-shock explosion phase, supporting the shock propagation.

A call for reliable neutrino-nucleus cross sections has also been made in the context of explosive nucleosynthesis, occurring when the shock wave passes through the exploding star and leads to fast nuclear reactions. It has also been pointed out that neutrino-induced reactions in the outer layers of the star can actually be the major source for the production of certain nuclides in nature. This is the so-called ν-process. Such ν-process is sensitive to those neutrinos, which are detectable at the new generation of supernova neutrino detectors. The latter can distinguish the incoming neutrino types and hence will probe the supernova neutrino distributions. An analysis of the events observed by these detectors requires detailed calculations of the interaction of neutrinos with the detector material. Such accurate determination of the neutrino-nucleus cross sections for nuclei, like 12C (KamLAND, Borexino) [47–49], 16O (SNO, Super-Kamiokande) [50, 51], 40Ar (ICARUS) [52], 208Pb (OMNIS) [53, 54], 56Fe (MINOS) [55], 114,116Cd (COBRA) [56, 57], 132Xe (XENON) [58], will be especially useful for present or near future experiments.

Recently, the IceCube Collaboration has reported the detection of ultra-high energy (UHE) neutrino events coming from extraterrestrial sources, that is, neutrinos with energies in the range TeV–PeV [59–61]. The most plausible sources that these events are connected are from unique high-energy cosmic ray accelerators like semi-relativistic hypernova remnants (HNRs) [62–64], and remnants from gamma ray bursts in star-burst galaxies, which can produce primary cosmic rays with the required energies and abundance [65]. Neutrino interactions with DM could have strong implications at cosmological scales, such as reduction of the relic neutrino density, modification of the CMB spectra [37] or even a connection between the smallness of neutrino mass and a MeV-mass scalar field DM [66]. Many DM candidates have been proposed in this context: heavy neutrinos as dark matter, lightest supersymmetric particles (LSP) and MeV-mass scalar field. Furthermore, sterile neutrinos appear in models attempting to explain the dark matter problem either as the main component for the dark matter content or as an additional subleading component of a multiparticle dark matter model. Those particles interact with matter through mixing with the active neutrino states. If there is a mixing between active and sterile neutrinos, UHE neutrinos interacting with dark matter may experience an enhancement in the oscillation probability when they propagate in a DM medium. This is a mechanism that could be tested from future UHE experimental data.

one were to observe a high-statistics neutrino signal. For example, a stellar core collapse in the Milky Way satellite galaxies may produce an enormous burst of neutrinos 'visible' by terrestrial detectors. Such an effect will carry important information in astrophysics, cosmology and particle physics [17, 18]. In addition, dedicated experiments are now planned involving intense accelerator-produced neutrino beams to study neutrino properties over long baselines. These will traverse the mantle or/and core of the Earth [19] so that the interpretation of the results

Neutrinos could be key particles to unravel the nature of the DM in the Universe. The dark matter problem has been a long-standing one in physics [22, 23]. Even though we know that it must exist [24–27], we do not know much about its true nature. It is clear, though, that massive neutrinos and dark matter are both part of nature and should be incorporated in models of physics beyond the standard model. It may be that they are related to each other [28] and that,

Several studies have noted that the existence of light sterile neutrinos would have important consequences for dark matter searches [29]. Moreover, MSW-enhanced transitions between active and sterile neutrinos would have a substantial impact on searches for neutrinos from dark matter annihilation in the Sun [30, 31]. Furthermore, if sterile neutrinos in addition to theirmixing with the active neutrinos possess some new gauge interactions, they could lead to signals which appear to favour a dark matter interpretation. These can be used to investigate sterile states and may also generate strong signals in DM detectors [32–34]. Couplings between neutrino, either active or

Understanding the explosion of supernovae or the physics of the early universe, where neutrinos play an essential role, requires a solid theoretical background in astrophysics and cosmology and reliable input from nuclear physics. Neutrino-nucleus scattering at energies below 100 MeV plays an essential role in core-collapse supernova simulations in various interactions of neutrinos with the supernova environment. Based on the improved supernova simulations, it is found that inelastic neutrino-nucleus reactions will also allow for an additional mode of energy deposition to the matter ahead of the shock wave in the post-shock explosion phase,

A call for reliable neutrino-nucleus cross sections has also been made in the context of explosive nucleosynthesis, occurring when the shock wave passes through the exploding star and leads to fast nuclear reactions. It has also been pointed out that neutrino-induced reactions in the outer layers of the star can actually be the major source for the production of certain nuclides in nature. This is the so-called ν-process. Such ν-process is sensitive to those neutrinos, which are detectable at the new generation of supernova neutrino detectors. The latter can distinguish the incoming neutrino types and hence will probe the supernova neutrino distributions. An analysis of the events observed by these detectors requires detailed calculations of the interaction of neutrinos with the detector material. Such accurate determination of the neutrino-nucleus cross sections for nuclei, like 12C (KamLAND, Borexino) [47–49], 16O (SNO, Super-Kamiokande) [50, 51], 40Ar (ICARUS) [52], 208Pb (OMNIS) [53, 54], 56Fe (MINOS) [55], 114,116Cd (COBRA) [56, 57], 132Xe (XENON) [58], will be especially useful for present or near

will require geophysical details [20, 21].

48 Trends in Modern Cosmology

supporting the shock propagation.

future experiments.

in addition, both originate from new physics at the TeV scale.

sterile, and dark matter have been studied in many different contexts [35–46].

The chapter has been organized as follows. In Section 2, we outline the basic formalism used in the evaluation of neutrino-nucleus cross sections. Section 3 presents original cross section calculations for charged current (CC) neutrino and antineutrino scattering off targets from 12C to 208Pb, at energies below 100 MeV. Illustrative test calculations are performed for CC (anti)neutrino reactions on 56Fe and 40Ar, and the results are compared with other previous theoretical studies. Such cross section calculations provide us with significant information regarding the range of efficiency of these isotopes in low-energy neutrino searches. The event estimates are made by convolving the calculated cross sections with two different distributions: the Fermi-Dirac (FD) flux and the Livermore one. In Section 4, results are presented concerning the interaction potential of extragalactic neutrinos, at ultra-high energies, with dark matter, which might induce resonant effects in the oscillation survival probability. Finally, in Section 5, the main conclusions extracted from the present work are summarized.

#### 2. Charge current neutrino-nucleus cross-section formalism

Let us consider a neutrino-nucleus interaction in which a low or intermediate energy neutrino (or antineutrino) is scattered inelastically from a nucleus (A, Z) being in its ground state.

The standard model effective Hamiltonian in a charge current interaction can be written as:

$$\mathcal{H} = \frac{G\cos\theta\_c}{\sqrt{2}} j\_\mu(\mathbf{x}) l^\mu(\mathbf{x}),\tag{1}$$

Here, <sup>G</sup> <sup>¼</sup> <sup>1</sup>:<sup>1664</sup> · <sup>10</sup>�<sup>5</sup> GeV�<sup>2</sup> denotes the Fermi weak coupling constant and θ<sup>c</sup> ≃13<sup>o</sup> is the Cabibbo angle. According to V-A theory, the leptonic current takes the form:

$$j\_{\mu} = \overline{\psi}\_{\nu\_{\ell}}(\mathbf{x}) \gamma\_{\mu}(\mathbf{1} - \gamma\_{5}) \psi\_{\nu\_{\ell}}(\mathbf{x})\,,\tag{2}$$

where, Ψν<sup>ℓ</sup> are the neutrino/antineutrino spinors. The hadronic current of vector, axial-vector and pseudo-scalar components is written as:

$$J\_{\mu} = \overline{\psi}\_{N} \left[ F\_{1}(q^{2}) \underline{\boldsymbol{\gamma}}\_{\mu} + F\_{2}(q^{2}) \frac{i \boldsymbol{\sigma}\_{\mu\nu} q^{\nu}}{2 \mathbf{M}\_{N}} + F\_{A}(q^{2}) \underline{\boldsymbol{\gamma}}\_{\mu} \underline{\boldsymbol{\gamma}}\_{5} + F\_{P}(q^{2}) \frac{1}{2 \mathbf{M}\_{N}} q\_{\mu} \underline{\boldsymbol{\gamma}}\_{5} \right] \boldsymbol{\psi}\_{N} \tag{3}$$

(MN stands for the nucleon mass and ψ<sup>N</sup> denotes the nucleon spinors). By the conservation of the vector current (CVC), the vector form factors <sup>F</sup><sup>1</sup>;<sup>2</sup>ðq<sup>2</sup><sup>Þ</sup> can be written in terms of the proton and neutron electromagnetic form factors [67]. The axial-vector form factor FAðq<sup>2</sup><sup>Þ</sup> is assumed to be of dipole form [68]:

$$F\_A = -g\_A (1 - q^2 / M\_A^2)^{-2} \,, \tag{4}$$

where MA ¼ 1:05 GeV is the axial cut-off mass and gA is the static value (at q ¼ 0) of the axial form factor. Recently, it has been shown in modelling the GT<sup>+</sup> and GT� transition strengths that in both channels the quenching factor 0.8 in the axial vector coupling constant is necessary to describe the experimentally measured GT strengths. Therefore, in our work, the effective quenched static value gA = 1.0 is employed [69]. Moreover, the pseudoscalar form factor FPðq<sup>2</sup><sup>Þ</sup> is obtained from the Goldberger-Treiman relation [70]:

$$F\_P(q^2) = \frac{2M\_N F\_A(q^2)}{m\_\pi^2 - q^2} \tag{5}$$

where m<sup>π</sup> ¼ 139:57 MeV represents the mass of the charged pion. The strangeness contributions are not taken into account since the energy region considered here is below the quasielastic region where the contributions from strangeness can be neglected [71].

In the convention we used in the present work q2, the square of the four-momentum transfer q � ðq0;qÞ is written as:

$$
\eta^2 = \eta^\mu \eta\_\mu = \omega^2 - \mathbf{q}^2 = (\varepsilon\_f - \varepsilon\_i)^2 - (\mathbf{p}\_f - \mathbf{p}\_i)^2 < 0,\tag{6}
$$

where ω ¼ �q<sup>0</sup> ¼ ε<sup>i</sup> � ε<sup>f</sup> is the excitation energy of the nucleus. ε<sup>i</sup> denotes the energy of the incoming lepton and ε<sup>f</sup> that of the outgoing lepton. p<sup>i</sup> and p<sup>f</sup> are the corresponding 3-momenta of the incoming and outgoing leptons, respectively.

The neutrino/antineutrino-nucleus differential cross section, after applying a multipole analysis of the weak hadronic current, is written as:

$$\sigma(\varepsilon i) = \frac{2G^2 \cos^2 \theta\_c}{2J\_i + 1} \sum\_f |\mathbf{p}\_f| \varepsilon\_f \int\_{-1}^1 d(\cos \theta) F(\varepsilon\_f, \mathbf{Z}\_f) \left(\sum\_{f=0}^m \sigma\_{\rm CL}^f + \sum\_{f=1}^m \sigma\_{\rm T}^f\right) \tag{7}$$

θ denotes the lepton scattering angle. The summations in Eq. (7) contain the contributions σ<sup>J</sup> CL, for the Coulomb <sup>M</sup>c<sup>J</sup> and longitudinal <sup>L</sup>bJ, and <sup>σ</sup><sup>J</sup> <sup>T</sup>, for the transverse electric bT el <sup>J</sup> and magnetic

#### <sup>b</sup><sup>T</sup> mag <sup>J</sup> multipole operators [72]. These operators include both polar-vector and axial-vector weak interaction components.

The contributions of σ<sup>J</sup> CL and <sup>σ</sup><sup>J</sup> <sup>T</sup> are written as:

j <sup>μ</sup> ¼ ψν<sup>ℓ</sup>

<sup>Þ</sup>γμ <sup>þ</sup> <sup>F</sup>2ðq<sup>2</sup>

FPðq<sup>2</sup><sup>Þ</sup> is obtained from the Goldberger-Treiman relation [70]:

of the incoming and outgoing leptons, respectively.

cos<sup>2</sup>θ<sup>c</sup> 2Ji þ 1

X f

jp<sup>f</sup> jε<sup>f</sup>

Z <sup>1</sup> �1

sis of the weak hadronic current, is written as:

for the Coulomb <sup>M</sup>c<sup>J</sup> and longitudinal <sup>L</sup>bJ, and <sup>σ</sup><sup>J</sup>

<sup>σ</sup>ðεiÞ ¼ <sup>2</sup>G<sup>2</sup>

Þ iσμνq<sup>ν</sup> 2MN

and pseudo-scalar components is written as:

<sup>J</sup><sup>μ</sup> <sup>¼</sup> <sup>ψ</sup><sup>N</sup> <sup>F</sup>1ðq<sup>2</sup>

to be of dipole form [68]:

50 Trends in Modern Cosmology

q � ðq0;qÞ is written as:

ðxÞγμð1 � γ5Þψν<sup>ℓ</sup>

<sup>þ</sup> FAðq<sup>2</sup>

=M<sup>2</sup> AÞ

� �

<sup>Þ</sup>γμγ<sup>5</sup> <sup>þ</sup> FPðq<sup>2</sup>

Þ 1 2MN

where, Ψν<sup>ℓ</sup> are the neutrino/antineutrino spinors. The hadronic current of vector, axial-vector

(MN stands for the nucleon mass and ψ<sup>N</sup> denotes the nucleon spinors). By the conservation of the vector current (CVC), the vector form factors <sup>F</sup><sup>1</sup>;<sup>2</sup>ðq<sup>2</sup><sup>Þ</sup> can be written in terms of the proton and neutron electromagnetic form factors [67]. The axial-vector form factor FAðq<sup>2</sup><sup>Þ</sup> is assumed

where MA ¼ 1:05 GeV is the axial cut-off mass and gA is the static value (at q ¼ 0) of the axial form factor. Recently, it has been shown in modelling the GT<sup>+</sup> and GT� transition strengths that in both channels the quenching factor 0.8 in the axial vector coupling constant is necessary to describe the experimentally measured GT strengths. Therefore, in our work, the effective quenched static value gA = 1.0 is employed [69]. Moreover, the pseudoscalar form factor

> Þ ¼ <sup>2</sup>MNFAðq<sup>2</sup><sup>Þ</sup> m<sup>2</sup>

where m<sup>π</sup> ¼ 139:57 MeV represents the mass of the charged pion. The strangeness contributions are not taken into account since the energy region considered here is below the quasi-

In the convention we used in the present work q2, the square of the four-momentum transfer

where ω ¼ �q<sup>0</sup> ¼ ε<sup>i</sup> � ε<sup>f</sup> is the excitation energy of the nucleus. ε<sup>i</sup> denotes the energy of the incoming lepton and ε<sup>f</sup> that of the outgoing lepton. p<sup>i</sup> and p<sup>f</sup> are the corresponding 3-momenta

The neutrino/antineutrino-nucleus differential cross section, after applying a multipole analy-

θ denotes the lepton scattering angle. The summations in Eq. (7) contain the contributions σ<sup>J</sup>

<sup>2</sup> � ðp<sup>f</sup> � <sup>p</sup><sup>i</sup>

<sup>d</sup><sup>ð</sup> cos <sup>θ</sup>ÞFðε<sup>f</sup> ;Zf<sup>Þ</sup> <sup>X</sup><sup>∞</sup>

Þ

J¼0 σJ CL <sup>þ</sup>X<sup>∞</sup> J¼1 σJ T

<sup>T</sup>, for the transverse electric bT

!

FA ¼ �gAð<sup>1</sup> � <sup>q</sup><sup>2</sup>

FPðq<sup>2</sup>

elastic region where the contributions from strangeness can be neglected [71].

<sup>q</sup><sup>2</sup> <sup>¼</sup> <sup>q</sup><sup>μ</sup>q<sup>μ</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> � <sup>q</sup><sup>2</sup> ¼ ðε<sup>f</sup> � <sup>ε</sup>i<sup>Þ</sup>

ðxÞ ; ð2Þ

qμγ<sup>5</sup>

�<sup>2</sup> ; <sup>ð</sup>4<sup>Þ</sup>

<sup>π</sup> � <sup>q</sup><sup>2</sup> <sup>ð</sup>5<sup>Þ</sup>

<sup>2</sup> <sup>&</sup>lt; <sup>0</sup> ; <sup>ð</sup>6<sup>Þ</sup>

el

ð7Þ

CL,

<sup>J</sup> and magnetic

ψ<sup>N</sup> ð3Þ

$$\begin{aligned} \sigma\_{\mathcal{CL}}^{\vec{I}} &= (1 + a \cos \theta) \left| \langle f\_{\vec{I}} || \hat{M}\_{\vec{I}} || \hat{I}\_{\vec{i}} \rangle \right|^2 + (1 + a \cos \theta - 2b \sin^2 \theta) \left| \langle f\_{\vec{I}} || \hat{L}\_{\vec{I}} || \hat{I}\_{\vec{i}} \rangle \right|^2 \\ &+ \left[ \frac{\varepsilon\_{\hat{I}} - \varepsilon\_{\hat{I}}}{q} (1 + a \cos \theta) + c \right] 2 \, \Re(\hat{J}\_{\vec{I}} || \hat{L}\_{\vec{I}} || \hat{J}\_{\vec{I}} || \hat{M}\_{\vec{I}} || \hat{J}\_{\vec{I}})^\* \\\\ \sigma\_{\mathcal{T}}^{\vec{I}} &= (1 - a \cos \theta + b \sin^2 \theta) \left[ \left| \langle f\_{\vec{I}} || \hat{T}\_{\vec{I}}^{\text{mag}} || \hat{I}\_{\vec{I}} \rangle \right|^2 + \left| \langle f\_{\vec{I}} || \hat{T}\_{\vec{I}}^{\text{eff}} || \hat{J}\_{\vec{I}} \rangle \right|^2 \right] \\ &\mp \left[ \frac{(\varepsilon\_{\hat{I}} + \varepsilon\_{\hat{I}})}{q} (1 - a \cos \theta) - c \right] 2 \, \Re \left< \mathcal{J}\_{\vec{I}} | \hat{T}\_{\vec{I}}^{\text{mag}} || \hat{J}\_{\vec{I}} \rangle \langle f\_{\vec{I}} || \hat{T}\_{\vec{I}}^{\text{eff}} || \hat{J}\_{\vec{I}} \rangle \right]^\* \end{aligned} (9)$$

where <sup>b</sup> <sup>¼</sup> <sup>ε</sup>iε<sup>f</sup> <sup>a</sup><sup>2</sup>=jq<sup>j</sup> 2 , <sup>a</sup> ¼ jp<sup>f</sup> <sup>j</sup>=ε<sup>f</sup> and <sup>c</sup> ¼ ðmf <sup>c</sup><sup>2</sup><sup>Þ</sup> 2 =ðjqjεfÞ. In Eq. (9), the (�) sign corresponds to neutrino scattering and the (+) sign to antineutrino. The absolute value of the three-momentum transfer is given by:

$$|\mathbf{q}| = \sqrt{a^2 + 2\varepsilon\_f \varepsilon\_i (1 - a \cos \theta) - \left(m\_f c^2\right)^2} \tag{10}$$

For charge current (CC) reactions, the cross section of Eq. (7) must be corrected for the distortion of the outgoing lepton wave function by the Coulomb field of the daughter nucleus [73] and references therein.

#### 3. Original cross sections

Development of large mass detectors for low energy neutrinos and dark matter may allow supernova detection via neutrino-nucleus scattering (elastic or inelastic). An analysis of the events observed by these detectors requires a detailed calculation of the interaction cross sections of neutrinos with the detector material. Especially interesting is modelling the reaction cross sections of neutrinos scattering on nuclei that can be used as targets for SN neutrino detectors. The target materials include a range of isotopes from <sup>4</sup> He to 208Pb. In this chapter, we report results concerning the cross sections of charge current (CC) (anti)neutrino-nucleus reactions for some isotopes of astrophysical interest. The results refer to the target isotopes 12C, 16O, 18O, 40Ar, 56Fe, 114Cd, 116Cd, 132Xe and 208Pb. The nuclear matrix elements entering in Eqs. (8) and (9) have been calculated in the framework of pnQRPA [73–75]. The respective cross sections are listed in Tables 1 and 2 for various incoming (anti)neutrino energies E<sup>ν</sup> below 100 MeV. Cross-section results for 208Pb are taken from Ref. [76]. The reliability of our calculations is justified from the comparison of the CC neutrino-nucleus cross sections with other calculations. In Figure 1, we compare our calculated cross sections for the reactions <sup>ν</sup>e=νe�<sup>56</sup>Fe and <sup>ν</sup>e=νe�<sup>40</sup>Ar with those of Refs. [76] and [77], respectively.


The cross sections are given in units of 10�<sup>42</sup> cm<sup>2</sup> , exponents are given in parentheses.

Table 1. Total cross sections σtot for the indicated neutrino-nucleus charge current reactions as a function of the incoming neutrino energy.

Charge current interactions proceed through interaction of ν<sup>e</sup> and ν<sup>e</sup> with neutrons and protons, respectively, in nuclei ν<sup>e</sup> þ ðN;ZÞ!ðN � 1;Z þ 1Þ þ e� and ν<sup>e</sup> þ ðN;ZÞ!ðN þ 1;Z � 1Þ þ eþ. The kinematic threshold is <sup>E</sup>thres <sup>¼</sup> <sup>M</sup><sup>2</sup> <sup>f</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> <sup>e</sup> <sup>þ</sup> <sup>2</sup>Mf me � <sup>M</sup><sup>2</sup> i <sup>2</sup>Mi <sup>e</sup> Mf � Mi <sup>þ</sup> me, where Mf and Mi are the initial- and final-state nuclear masses and me is the electron mass. The corresponding thresholds for CC reactions on the above target isotopes are given in Table 3. Note that at supernova energies, νμ and ντ are below the CC interaction threshold and thus are kinematically unable to produce their partner leptons.

An important application of microscopic models of neutrino-nucleus reactions is the calculation of cross sections for supernova neutrinos. Thus, in order to estimate the response of a nucleus to a specific source of neutrinos, the calculated cross sections given in Tables 1 and 2 must be folded with a specific supernova neutrino energy distribution. The neutrino spectrum of a core-collapse supernova is believed to be similar to a Fermi-Dirac (FD) spectrum, with temperatures in the range 3–8 MeV [78]. The FD energy distribution is given by:

$$\eta\_{FD} = \frac{N\_2(a)}{T^3} \frac{E\_v^2}{1 + \exp[(E\_v/T) - a]} \tag{11}$$

where T is the neutrino temperature and α being a degeneracy parameter. N2(α) denotes the normalization factor depending on α given from


The cross sections are given in units of 10�<sup>42</sup> cm<sup>2</sup> , exponents are given in parentheses

Charge current interactions proceed through interaction of ν<sup>e</sup> and ν<sup>e</sup> with neutrons and protons, respectively, in nuclei ν<sup>e</sup> þ ðN;ZÞ!ðN � 1;Z þ 1Þ þ e� and ν<sup>e</sup> þ ðN;ZÞ!ðN þ 1;Z � 1Þ þ eþ.

Table 1. Total cross sections σtot for the indicated neutrino-nucleus charge current reactions as a function of the incoming

E<sup>ν</sup><sup>e</sup> (MeV) 12C 16O 18O 40Ar 56Fe 114Cd 116Cd 132Xe 208Pb 7.5 2.72(0) 1.29(0) 3.34(�1) 1.73(+1) 3.40(+1) 1.07(0) 2.47(�4) 10.0 5.72(0) 4.59(0) 2.10(0) 6.24(+1) 9.93(+1) 2.09(+1) 8.49(0) 5.0 1.75(+1) 2.25(+1) 2.03(+1) 2.55(+2) 3.32(+2) 1.97(+2) 1.75(+2) 20.0 4.80(�1) 4.48(�2) 3.86(+1) 5.90(+1) 6.23(+1) 5.58(+2) 6.73(+2) 6.27(+2) 8.53(+2) 25.0 2.02(0) 2.95(�1) 7.10(+1) 1.17(+2) 1.28(+2) 9.45(+2) 1.10(+3) 1.30(+3) 2.86(+3) 30.0 5.8(0) 8.91(�1) 1.17(+2) 1.98(+2) 2.18(+2) 1.29(+3) 1.41(+3) 1.82(+3) 4.90(+3) 40.0 2.78(+1) 8.20(0) 2.60(+2) 4.42(+2) 4.74(+2) 1.92(+3) 2.07(+3) 2.76(+3) 7.13(+3) 50.0 7.89(+1) 3.97(+1) 4.88(+2) 8.07(+2) 8.25(+2) 2.65(+3) 2.83(+3) 3.74(+3 1.13(+4) 60.0 1.71(+2) 1.19(+2) 8.29(+2) 1.30(+3) 1.27(+3) 3.43(+3) 3.65(+3) 4.76(+3) 1.63(+4) 70.0 3.07(+2) 2.74(+2) 1.30(+3) 1.89(+3) 1.81(+3) 4.21(+3) 4.46(+3) 5.75(+3) 2.20(+4) 80.0 4.87(+2) 5.33(+2) 1.91(+3) 2.55(+3) 2.42(+3) 4.94(+3) 5.22(+3) 6.63(+3) 2.83(+4) 90.0 7.06(+2) 9.17(+2) 2.65(+3) 3.27(+3) 3.07(+3) 5.65(+3) 5.95(+3) 7.32(+3) 3.50(+4) 100.0 9.95(+2) 1.43(+3) 3.51(+3) 4.03(+3) 3.75(+3) 6.33(+3) 6.65(+3) 7.78(+3) 4.16(+4)

<sup>e</sup> <sup>þ</sup> <sup>2</sup>Mf me � <sup>M</sup><sup>2</sup>

initial- and final-state nuclear masses and me is the electron mass. The corresponding thresholds for CC reactions on the above target isotopes are given in Table 3. Note that at supernova energies, νμ and ντ are below the CC interaction threshold and thus are kinematically unable to

An important application of microscopic models of neutrino-nucleus reactions is the calculation of cross sections for supernova neutrinos. Thus, in order to estimate the response of a nucleus to a specific source of neutrinos, the calculated cross sections given in Tables 1 and 2 must be folded with a specific supernova neutrino energy distribution. The neutrino spectrum of a core-collapse supernova is believed to be similar to a Fermi-Dirac (FD) spectrum, with

where T is the neutrino temperature and α being a degeneracy parameter. N2(α) denotes the

i

, exponents are given in parentheses.

E2 ν

<sup>2</sup>Mi <sup>e</sup> Mf � Mi <sup>þ</sup> me, where Mf and Mi are the

<sup>1</sup> <sup>þ</sup> exp½ðEν=TÞ � <sup>α</sup>� <sup>ð</sup>11<sup>Þ</sup>

<sup>f</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup>

temperatures in the range 3–8 MeV [78]. The FD energy distribution is given by:

<sup>η</sup>FD <sup>¼</sup> <sup>N</sup>2ðα<sup>Þ</sup> T3

normalization factor depending on α given from

The kinematic threshold is <sup>E</sup>thres <sup>¼</sup> <sup>M</sup><sup>2</sup>

The cross sections are given in units of 10�<sup>42</sup> cm<sup>2</sup>

σtot(10�<sup>42</sup> cm<sup>2</sup>

52 Trends in Modern Cosmology

)

produce their partner leptons.

neutrino energy.

Table 2. Total cross sections σ for the indicated antineutrino-nucleus charge current reactions as a function of the incoming neutrino energy.

$$N\_k(\alpha) = \left(\int\_0^\infty \frac{x^k}{1 + e^{x-\alpha}} d\mathbf{x}\right)^{-1} \tag{12}$$

for k = 2. The degeneracy parameter α is called the chemical potential parameter. Characteristic of the FD energy distribution is that the peak shifts to higher neutrino energies and the width increases as the neutrino temperature increases (Figure 2).

Following Ref. [79], the average neutrino energy 〈Eν〉 can be written in terms of the functions of Eq. (12) as:

$$
\langle E\_v \rangle = \frac{N\_2(\alpha)}{N\_3(\alpha)} T \tag{13}
$$

Some characteristic values of 〈Eν〉 are listed in Table 4.

For a connection of the present theoretical results with the neutrino experiments and the neutrino sources, we carry out the folding (convolution) of the calculated cross sections given in Tables 1 and 2 with the distribution ηFD and estimate the response of the given isotopes to the corresponding spectrum. These responses (signals to the detector) are evaluated by:

Figure 1. Cross sections for relevant neutrino (antineutrino) reactions on isotopes 56Fe and 40Ar. The results denoted by square symbols are taken from Refs. [76] and [77].

$$
\langle \sigma \rangle = \int\_0^\infty \sigma(E\_\nu) \eta\_{FD}(E\_\nu) dE\_\nu \tag{14}
$$

In Figure 3, we compare the respective neutrino flux-averaged cross sections for some of target nuclei given in Table 1.

We close this subsection by exploiting our predictions of total cross sections to estimate the number of expected electron (anti)neutrino events in a detector. For current detectors [80], typical event yields are a few hundred events per kt of detector material for a core-collapse event at D = 10 kpc (3.1 · 10<sup>22</sup> cm) away from the Earth. A supernova radiates via neutrinos, an amount of total energy 3 · 1053 erg in about 10s. Assuming an equal partition of energy among neutrinos, the supernova radiates <sup>N</sup><sup>ν</sup><sup>e</sup> <sup>¼</sup> <sup>3</sup>:<sup>0</sup> · 1057 electron neutrinos and <sup>N</sup>ν<sup>e</sup> <sup>¼</sup> <sup>2</sup>:<sup>1</sup> · 1057 electron antineutrinos. The neutrino fluence ΦðEνÞ for electron neutrinos/antineutrinos is given by the relation:


Table 3. Thresholds values Ethres (in MeV) for charge current antineutrino-nucleus interactions.

〈σ〉 ¼

**20 40 60 80 100**

<sup>ν</sup>*<sup>e</sup> <sup>−</sup>*<sup>40</sup> Ar

**E**ν  **(MeV)**

square symbols are taken from Refs. [76] and [77].

**0 20 40 60 80 100**

ν e − 56Fe

 **(MeV)**

**E**ν

**10−4**

**10−10**

**10−5**

**100**

σ**(10−40 cm2**

**)**

**105**

**10−2**

**100**

σ**(10−40 cm2**

**)**

**102**

54 Trends in Modern Cosmology

nuclei given in Table 1.

the relation:

Z <sup>∞</sup> 0

In Figure 3, we compare the respective neutrino flux-averaged cross sections for some of target

Figure 1. Cross sections for relevant neutrino (antineutrino) reactions on isotopes 56Fe and 40Ar. The results denoted by

**10−4**

**10−4**

**10−2**

**100**

σ**(10−40 cm2**

**)**

**102**

**10−2**

**100**

σ**(10−40 cm2**

a)

**)**

**102**

We close this subsection by exploiting our predictions of total cross sections to estimate the number of expected electron (anti)neutrino events in a detector. For current detectors [80], typical event yields are a few hundred events per kt of detector material for a core-collapse event at D = 10 kpc (3.1 · 10<sup>22</sup> cm) away from the Earth. A supernova radiates via neutrinos, an amount of total energy 3 · 1053 erg in about 10s. Assuming an equal partition of energy among neutrinos, the supernova radiates <sup>N</sup><sup>ν</sup><sup>e</sup> <sup>¼</sup> <sup>3</sup>:<sup>0</sup> · 1057 electron neutrinos and <sup>N</sup>ν<sup>e</sup> <sup>¼</sup> <sup>2</sup>:<sup>1</sup> · 1057 electron antineutrinos. The neutrino fluence ΦðEνÞ for electron neutrinos/antineutrinos is given by

σðEνÞηFDðEνÞdE<sup>ν</sup> ð14Þ

**E**ν  **(MeV)**

c) d)

**20 40 60 80 100**

<sup>ν</sup>¯*<sup>e</sup> <sup>−</sup>*<sup>40</sup> Ar

**0 20 40 60 80 100**

 **(MeV)**

b)

<sup>ν</sup>¯*<sup>e</sup> <sup>−</sup>*<sup>56</sup> F e

**E**ν

$$\Phi(E\_{\nu}) = \frac{N\_{\nu\_{\nu}/\overline{\mathcal{V}\_{\epsilon}}}}{4\pi D^{2}} \eta\_{FD}(E\_{\nu}) \tag{15}$$

Two examples of supernova models are used to predict the neutrino flux: (i) the model based on the FD distribution with a single temperature (3.5 MeV for neutrinos and 5 MeV for antineutrinos) and zero chemical parameter (α = 0) and (ii) the numerical simulation of supernova neutrino emission model called Livermore [81], which assumes the FD spectra with α = 0 and with the average energies indicated as a function of time integrated from 0 to 14 seconds after the core collapse. The Livermore energy spectrum for the ν<sup>e</sup> and ν<sup>e</sup> flavour components is shown in Figure 4. The nature of the neutrino spectra and their time evolution depend on mass, oscillation parameters, such as θ<sup>13</sup> and the mass hierarchy. Furthermore, the chance that the supernova neutrinos will traverse Earth matter on their way to a detector is not negligible [82] and oscillations in the Earth modulate the supernova neutrino spectrum for either ν<sup>e</sup> or ν<sup>e</sup> [83–85]. In a single detector, an Earth matter-induced spectral modulation may give information about oscillations, involving probably sterile neutrino states (e.g., [19, 86]).

Figure 2. The normalized unity Fermi-Dirac spectrum for α = 0.


Table 4. The average supernova neutrino energies as a function of the parameters α and T.

If the mass of the target material is mt, corresponding to Nat atoms, then the number of expected events per energy are:

$$\frac{dN\_{\text{events}}}{dE\_{\text{v}}} = N\_{dt} \Phi(E\_{\text{v}}) \sigma\_{\text{tot}}(E\_{\text{v}}) \tag{16}$$

where σtotðEνÞ is the total cross section (see Tables 1 and 2). Figure 5 shows the event rates in 1 kt of the target material for the Livermore model. The total number of events per kiloton for each of the two neutrino fluxes are listed in Table 5. The actual detected number of events may

Figure 3. Temperature dependence of the energy-weighted cross section for charge current neutrino-nucleus reactions, whose neutrino spectra obey the Fermi-Dirac distribution with chemical parameter α = 0.

If the mass of the target material is mt, corresponding to Nat atoms, then the number of expected

**0 20 40 60 80 100**

**(MeV)**

**T=3.5MeV T=5MeV T=8MeV**

**E** ν

α T = 3.5MeV T = 5MeV T = 8MeV 0 11.03 15.76 25.21 0.76 11.46 16.37 26.19 1.52 12.10 17.28 27.65 2.28 12.96 18.52 29.63 3.04 14.03 20.05 32.08 4.56 16.66 23.80 38.09 5.76 19.06 27.23 43.58

where σtotðEνÞ is the total cross section (see Tables 1 and 2). Figure 5 shows the event rates in 1 kt of the target material for the Livermore model. The total number of events per kiloton for each of the two neutrino fluxes are listed in Table 5. The actual detected number of events may

¼ NatΦðEνÞσtotðEνÞ ð16Þ

dNevents dE<sup>ν</sup>

Table 4. The average supernova neutrino energies as a function of the parameters α and T.

events per energy are:

**0**

Figure 2. The normalized unity Fermi-Dirac spectrum for α = 0.

< Eν > (MeV)

**0.02**

**0.04**

η**FD(E**

**)**

56 Trends in Modern Cosmology

ν

**0.06**

**0.08**

Figure 4. Livermore fluence for ν<sup>e</sup> and ν<sup>e</sup> flavour components, assuming Fermi-Dirac spectra with the average energies indicated as a function of time and zero chemical potential, integrated from core collapse till about 14 seconds later.

Figure 5. Event rates in 1 kt of the target isotope for the Livermore model. The event rates of <sup>ν</sup>e�<sup>16</sup><sup>O</sup> and <sup>ν</sup>e�<sup>208</sup>Pb, which are less than 10�<sup>2</sup> , are not shown.


We consider the Fermi-Dirac distribution with temperature T = 3.5 MeV for neutrinos and T = 5 MeV for antineutrinos (second column) and the Livermore numerical simulation for supernova neutrino emission (third column). No detector efficiency (detector threshold, energy response, background effects, etc.) is taken into account.

Table 5. Number of supernova events per kt at 10 kpc away from the Earth, on different targets relevant for existing detector types.

be significantly fewer, if the detector energy threshold, detector efficiency and other background contamination effects coming from radioactive isotopes are taken into account. The results show that there is no considerable variation in the total antineutrino events between the two supernova models used in the calculation.

#### 4. Interaction of neutrinos with dark matter

Dark matter particles (hereafter generically denoted by χ) may interact with ordinary matter through Z boson exchanges. Therefore, they have to be heavy, or else they would have been pair-produced in Z decays. A light dark matter candidate should have no significant direct coupling to the Z boson, but it could still interact with ordinary matter through the exchanges of other spin-1 gauge bosons or of spin-0 Higgs bosons.

If there is a mixing between active and sterile neutrinos, high-energy neutrinos interacting with dark matter may suffer a kind of MSW effect when they propagate in a dark matter medium. In a simplified model, which includes ordinary and dark matter potentials, the evolution equation with one sterile ν<sup>s</sup> and an active one να is written as:

$$i\frac{\mathbf{d}}{\mathbf{d}t} \begin{pmatrix} \nu\_a \\ \nu\_s \end{pmatrix} = (\mathcal{U}H\_0\mathcal{U}^\dagger + V) \begin{pmatrix} \nu\_a \\ \nu\_s \end{pmatrix},\tag{17}$$

with

**0 20 40 60 80 100**

<sup>ν</sup>*<sup>e</sup> <sup>−</sup>*<sup>12</sup> <sup>C</sup> <sup>ν</sup>*<sup>e</sup> <sup>−</sup>*<sup>40</sup> Ar <sup>ν</sup>*<sup>e</sup> <sup>−</sup>*<sup>56</sup> F e <sup>ν</sup>*<sup>e</sup> <sup>−</sup>*<sup>114</sup> Cd <sup>ν</sup>*<sup>e</sup> <sup>−</sup>*<sup>132</sup> Xe <sup>ν</sup>*<sup>e</sup> <sup>−</sup>*<sup>208</sup> P b

Channel Nevents Nevents

<sup>ν</sup>eþ12C ! <sup>e</sup>�þ12Nð�Þ 3 2 <sup>ν</sup>eþ12C ! <sup>e</sup>þþ12Bð�Þ <sup>15</sup> <sup>13</sup> <sup>ν</sup>eþ16O ! <sup>e</sup>�þ16Fð�Þ 1 1 <sup>ν</sup>eþ16O ! <sup>e</sup>þþ16Nð�Þ 4 4 <sup>ν</sup>eþ40A<sup>r</sup> ! <sup>e</sup>�þ40K� <sup>71</sup> <sup>28</sup> <sup>ν</sup>eþ40A<sup>r</sup> ! <sup>e</sup>þþ40Cl� <sup>40</sup> <sup>26</sup> <sup>ν</sup>eþ56F<sup>e</sup> ! <sup>e</sup>�þ56Co� <sup>48</sup> <sup>20</sup> <sup>ν</sup>eþ56F<sup>e</sup> ! <sup>e</sup>þþ56Mn� <sup>58</sup> <sup>31</sup> <sup>ν</sup>eþ114C<sup>d</sup> ! <sup>e</sup>�þ114In� <sup>229</sup> <sup>88</sup> <sup>ν</sup>eþ114C<sup>d</sup> ! <sup>e</sup>þþ114Ag� <sup>35</sup> <sup>20</sup> <sup>ν</sup>eþ132X<sup>e</sup> ! <sup>e</sup>�þ132Cs� <sup>198</sup> <sup>78</sup>

� 12 8

We consider the Fermi-Dirac distribution with temperature T = 3.5 MeV for neutrinos and T = 5 MeV for antineutrinos (second column) and the Livermore numerical simulation for supernova neutrino emission (third column). No detector

Table 5. Number of supernova events per kt at 10 kpc away from the Earth, on different targets relevant for existing

<sup>ν</sup>eþ208P<sup>b</sup> ! <sup>e</sup>�þ208Bi� <sup>219</sup> <sup>89</sup> <sup>ν</sup>eþ208P<sup>b</sup> ! <sup>e</sup>þþ208Tl� 0 0

efficiency (detector threshold, energy response, background effects, etc.) is taken into account.

a)

Figure 5. Event rates in 1 kt of the target isotope for the Livermore model. The event rates of <sup>ν</sup>e�<sup>16</sup><sup>O</sup> and <sup>ν</sup>e�<sup>208</sup>Pb, which

**10−2**

Fermi-Dirac Livermore

**10−1**

**100**

**Events per MeV per kt**

**101**

**0 20 40 60 80 100**

<sup>ν</sup>¯*<sup>e</sup> <sup>−</sup>*<sup>12</sup> <sup>C</sup> <sup>ν</sup>¯*<sup>e</sup> <sup>−</sup>*<sup>16</sup> <sup>O</sup> <sup>ν</sup>¯*<sup>e</sup> <sup>−</sup>*<sup>40</sup> Ar <sup>ν</sup>¯*<sup>e</sup> <sup>−</sup>*<sup>56</sup> F e <sup>ν</sup>¯*<sup>e</sup> <sup>−</sup>*<sup>114</sup> Cd <sup>ν</sup>¯*<sup>e</sup> <sup>−</sup>*<sup>132</sup> Xe

b)

**E**ν **(MeV)**

**E**ν **(MeV)**

, are not shown.

**10−1**

are less than 10�<sup>2</sup>

<sup>ν</sup>eþ132X<sup>e</sup> ! <sup>e</sup>þþ132I

detector types.

**100**

**Events per MeV per kt**

**101**

**102**

58 Trends in Modern Cosmology

$$H\_0 = \frac{1}{2E} \text{diag} \left\{ 0, \Delta m\_{\alpha 4}^2 \right\} \tag{18}$$

$$\mathcal{V} = \text{diag}\left\{ V\_{\mathbf{v}\_d f} + V\_{\mathbf{v}\_d \chi}, V\_{\mathbf{v}\_\ell \chi} \right\} \tag{19}$$

and

$$
\mathcal{U} = \begin{pmatrix}
\cos \theta\_0 & -\sin \theta\_0 \\
\sin \theta\_0 & \cos \theta\_0
\end{pmatrix},
\tag{20}
$$

where E is the neutrino energy, Δm<sup>2</sup> <sup>α</sup><sup>4</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup> <sup>4</sup> � <sup>m</sup><sup>2</sup> <sup>α</sup> is the mass-squared splitting and θ<sup>0</sup> is the vacuum mixing angle between the sterile and the active neutrino. The matter potentials are defined as:

$$\mathcal{V}\_{\nu\_{\rm af}} = \sqrt{2} \mathcal{G}\_{\rm F} (\mathcal{N}\_{\alpha} - \mathcal{N}\_{n}/2) \; ; \tag{21}$$

$$\boldsymbol{V}\_{\boldsymbol{\nu}\_{\mathcal{K}}\boldsymbol{\chi}} = \boldsymbol{\varepsilon}\_{\boldsymbol{\nu}\_{\mathcal{K}}\boldsymbol{\chi}} \boldsymbol{G}\_{\mathcal{F}} \mathbf{N}\_{\boldsymbol{\chi}} \; ; \tag{22}$$

$$\mathcal{V}\_{\nu\_{\mathcal{X}}} = \varepsilon\_{\nu\_{\mathcal{X}}} \mathcal{G} \mathsf{r} \mathcal{N}\_{\mathcal{X}} \; , \tag{23}$$

where Nα, Nn and N<sup>χ</sup> are, respectively, the number density of leptons, neutrons and dark matter particles interacting with neutrinos. The parameters ενα;s<sup>χ</sup> account for the coupling strength in terms of Fermi constant GF <sup>¼</sup> <sup>1</sup>:<sup>166</sup> · <sup>10</sup>�<sup>5</sup> GeV�<sup>2</sup> . A list of values is given in [87]. Considering an astrophysical environment where Ne ≈ Nn=2 and N<sup>μ</sup> ≈ N<sup>τ</sup> ≈ 0, the contribution, V<sup>ν</sup>e;μ;τ<sup>f</sup> , from electron, tau and muon neutrinos is negligible in comparison with the neutrino and dark matter interactions Vναχ and V<sup>ν</sup>s<sup>χ</sup>. The dark matter number density, Nχ, can be written as N<sup>χ</sup> ¼ ρχ=mχ, where m<sup>χ</sup> being the dark matter particle mass and ρχ the dark matter density. Around our galactic halo, it is expected that ρχ <sup>¼</sup> <sup>0</sup>:3 GeV � cm�<sup>3</sup> [88]. Even though there exists firm indirect evidence for a halo of dark matter in galaxies from the observed rotational curves, see for example the review [89], it is essential to directly detect such matter. The possibility of such detection, however, depends on the nature of the dark matter constituents and their interactions. There are quite a few dark matter candidates such as WIMPs (weakly interacting massive particles), superWIMPs, light gravitinos, hidden dark matter, sterile neutrinos, Kaluza-Klein particles and axions. We will pay special attention to WIMPs. WIMPs have masses m<sup>χ</sup> in the range of few GeV to few TeV [90–94]. In this context, we take m<sup>χ</sup> = 20 GeV.

It is interesting to compute the survival probability Pðνα ! ναÞ for active neutrinos for various values of sin <sup>2</sup> ð2θ0Þ. Figure 6 depicts P(να ! να) as a function of neutrino energy E<sup>ν</sup> with a coupling <sup>j</sup>εχj¼jεναχ � ενs<sup>χ</sup>j ¼ <sup>3</sup> · 1011. As it is seen, a resonant effect happens at the energy around 0.4 PeV which corresponds to an oscillation length <sup>L</sup> <sup>¼</sup> <sup>4</sup>π<sup>E</sup> sin <sup>ð</sup>2θ0ÞΔm<sup>2</sup> <sup>e</sup><sup>1018</sup> Km in accordance with the expected dark matter halo dimension. This suggests that the high-energy spectrum of extragalactic neutrinos could be affected by the existence of sterile neutrino and its interaction with dark matter. If the various experiments such as IceCube [59, 60, 95–97]

Figure 6. Survival probability Pðνα ! ναÞ as a function of the neutrino energy Eν, considering the galactic halo average dark matter density. The (black) dashed line corresponds to sin <sup>2</sup>ð2θ0Þ ¼ <sup>0</sup>:05, the (red) dotted line to 0.15 while the (blue) solid line to sin <sup>2</sup>ð2θ0Þ ¼ <sup>0</sup>:25. The neutrino squared mass difference is taken <sup>Δ</sup>m<sup>2</sup> <sup>¼</sup> <sup>7</sup> · <sup>10</sup>�<sup>13</sup> eV<sup>2</sup> .

collect in future sufficient data, it might be possible to observe the MSW mechanism for dark matter as a distortion in the UHE neutrino spectrum. Resonance enhancement in the oscillation probability can also be found considering a more realistic halo density profile of the form:

$$\rho(r) = \frac{\rho\_0}{(r/\mathcal{R})^\delta [1 + (r/\mathcal{R})^a]^{(\beta - \delta)/a}} \,, \tag{24}$$

where the parameters α, β, δ and R (in kpc) depend on the specific model to be considered. A list of parameters is given in Table 6 for various model density profiles [98–101]. The left panel of Figure 7 illustrates the four different density profiles, whereas the right one depicts the corresponding survival probability as a function of neutrino energy for constant density and as an example of the survival probability corresponding to the density profile [101].



Figure 7. Left panel: Dark matter density profiles, Ref. [98] (red) dot-dashed line, Ref. [99] (black) solid line, Ref. [100] (blue) dashed line, and Ref. [101] (green) dotted line. Right panel: Survival probabilities for constant ρ (blue) solid line and for the density profile [101] (green) dotted line.

#### 5. Conclusions

astrophysical environment where Ne ≈ Nn=2 and N<sup>μ</sup> ≈ N<sup>τ</sup> ≈ 0, the contribution, V<sup>ν</sup>e;μ;τ<sup>f</sup> , from electron, tau and muon neutrinos is negligible in comparison with the neutrino and dark matter interactions Vναχ and V<sup>ν</sup>s<sup>χ</sup>. The dark matter number density, Nχ, can be written as N<sup>χ</sup> ¼ ρχ=mχ, where m<sup>χ</sup> being the dark matter particle mass and ρχ the dark matter density. Around our galactic halo, it is expected that ρχ <sup>¼</sup> <sup>0</sup>:3 GeV � cm�<sup>3</sup> [88]. Even though there exists firm indirect evidence for a halo of dark matter in galaxies from the observed rotational curves, see for example the review [89], it is essential to directly detect such matter. The possibility of such detection, however, depends on the nature of the dark matter constituents and their interactions. There are quite a few dark matter candidates such as WIMPs (weakly interacting massive particles), superWIMPs, light gravitinos, hidden dark matter, sterile neutrinos, Kaluza-Klein particles and axions. We will pay special attention to WIMPs. WIMPs have masses m<sup>χ</sup> in the

It is interesting to compute the survival probability Pðνα ! ναÞ for active neutrinos for various

coupling <sup>j</sup>εχj¼jεναχ � ενs<sup>χ</sup>j ¼ <sup>3</sup> · 1011. As it is seen, a resonant effect happens at the energy

dance with the expected dark matter halo dimension. This suggests that the high-energy spectrum of extragalactic neutrinos could be affected by the existence of sterile neutrino and its interaction with dark matter. If the various experiments such as IceCube [59, 60, 95–97]

**<sup>100</sup> <sup>101</sup> <sup>0</sup>**

**(PeV)**

**E** ν

Figure 6. Survival probability Pðνα ! ναÞ as a function of the neutrino energy Eν, considering the galactic halo average dark matter density. The (black) dashed line corresponds to sin <sup>2</sup>ð2θ0Þ ¼ <sup>0</sup>:05, the (red) dotted line to 0.15 while the (blue)

solid line to sin <sup>2</sup>ð2θ0Þ ¼ <sup>0</sup>:25. The neutrino squared mass difference is taken <sup>Δ</sup>m<sup>2</sup> <sup>¼</sup> <sup>7</sup> · <sup>10</sup>�<sup>13</sup> eV<sup>2</sup>

ð2θ0Þ. Figure 6 depicts P(να ! να) as a function of neutrino energy E<sup>ν</sup> with a

sin <sup>ð</sup>2θ0ÞΔm<sup>2</sup> <sup>e</sup><sup>1018</sup> Km in accor-

.

range of few GeV to few TeV [90–94]. In this context, we take m<sup>χ</sup> = 20 GeV.

around 0.4 PeV which corresponds to an oscillation length <sup>L</sup> <sup>¼</sup> <sup>4</sup>π<sup>E</sup>

values of sin <sup>2</sup>

60 Trends in Modern Cosmology

**0.2**

**P(**

ν

 **−−−−>** 

ν

**)**

α

α

**0.4**

**0.6**

**0.8**

**1**

The study of neutrino scattering with nuclei provides the most attractive mechanism to detect or distinguish neutrinos of different flavour and to investigate the basic structure of weak interactions. Further studies involving neutrino-induced transitions between discrete nuclear states may help us to explore the structure of the weak hadronic currents and also constitute good sources of explanation for neutrino properties.

Neutrino-induced reactions are of particular significance in view of studies on modern detectors, based on neutrino scattering on various isotopes. So far, experimental neutrino cross sections are not available for modest energies below 100 MeV, with the exception of 12C and, with large uncertainty, 56Fe. These are rather important for astrophysical and cosmological applications and must be calculated. In this chapter, we have presented neutrino(antineutrino)-nucleus reactions via charge current related to a range of targets from 12C to 208Pb. The calculated cross sections are tabulated for a set of neutrino energies which are relevant for supernova neutrinos. The rather low neutrino energies involved introduce, however, some sensitivity to nuclear structure effects and, in particular, for neutrinos with energies lower than 20 MeV, where state-of-the-art nuclear models must be employed which describe the manybody correlations in the nucleus accurately. The model of choice is the pnQRPA yielding to reasonable cross sections in a wide range of nuclear isotopes. The nuclear responses of these isotopes (used in common detector materials) to supernova neutrinos have been studied for two neutrino flux models. The two-parameter Fermi-Dirac neutrino energy model with zero chemical potential and the Livermore model assuming Fermi-Dirac spectra with average neutrino energies indicated as a function of time integrated over 14 seconds burst. The expected number of events per kt are predicted for supernova-detector distance 10 kpc. The results show that there is no considerable variation in the total antineutrino events between the two supernova models used in the calculation.

Furthermore, we also present results concerning the interaction potential of extragalactic neutrinos, at high energies 1 PeV, with DM in the presence of sterile neutrino state. Highenergy neutrinos interacting with DM may suffer a kind of MSW effect when propagating in DM medium. The resonance effect happens at around 0.4–0.6 PeV for various density DM profiles. The existence of light sterile neutrinos can impact existing and future dark matter searches. The mechanism of MSW effect in the UHE neutrino survival probability may be tested in future experimental searches using experimental data, for instance, as those taken from IceCube operation.

#### Author details

Paraskevi C. Divari

Address all correspondence to: pdivari@gmail.com

Department of Physical Sciences and Applications, Hellenic Army Academy, Attica, Greece

#### References

[1] D. Forero, M. Tortola and J. Valle, Phys. Rev. D 86, 073012 (2012).

[2] F. Capozzi et al., Phys. Rev. D 89, 093018 (2014).

states may help us to explore the structure of the weak hadronic currents and also constitute

Neutrino-induced reactions are of particular significance in view of studies on modern detectors, based on neutrino scattering on various isotopes. So far, experimental neutrino cross sections are not available for modest energies below 100 MeV, with the exception of 12C and, with large uncertainty, 56Fe. These are rather important for astrophysical and cosmological applications and must be calculated. In this chapter, we have presented neutrino(antineutrino)-nucleus reactions via charge current related to a range of targets from 12C to 208Pb. The calculated cross sections are tabulated for a set of neutrino energies which are relevant for supernova neutrinos. The rather low neutrino energies involved introduce, however, some sensitivity to nuclear structure effects and, in particular, for neutrinos with energies lower than 20 MeV, where state-of-the-art nuclear models must be employed which describe the manybody correlations in the nucleus accurately. The model of choice is the pnQRPA yielding to reasonable cross sections in a wide range of nuclear isotopes. The nuclear responses of these isotopes (used in common detector materials) to supernova neutrinos have been studied for two neutrino flux models. The two-parameter Fermi-Dirac neutrino energy model with zero chemical potential and the Livermore model assuming Fermi-Dirac spectra with average neutrino energies indicated as a function of time integrated over 14 seconds burst. The expected number of events per kt are predicted for supernova-detector distance 10 kpc. The results show that there is no considerable variation in the total antineutrino events between the

Furthermore, we also present results concerning the interaction potential of extragalactic neutrinos, at high energies 1 PeV, with DM in the presence of sterile neutrino state. Highenergy neutrinos interacting with DM may suffer a kind of MSW effect when propagating in DM medium. The resonance effect happens at around 0.4–0.6 PeV for various density DM profiles. The existence of light sterile neutrinos can impact existing and future dark matter searches. The mechanism of MSW effect in the UHE neutrino survival probability may be tested in future experimental searches using experimental data, for instance, as those taken

Department of Physical Sciences and Applications, Hellenic Army Academy, Attica, Greece

[1] D. Forero, M. Tortola and J. Valle, Phys. Rev. D 86, 073012 (2012).

good sources of explanation for neutrino properties.

62 Trends in Modern Cosmology

two supernova models used in the calculation.

Address all correspondence to: pdivari@gmail.com

from IceCube operation.

Author details

Paraskevi C. Divari

References


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## **Relativistic Celestial Metrology: Dark Matter as an Inertial Gauge Effect**

Luca Lusanna and Ruggero Stanga

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68115

#### Abstract

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In canonical tetrad gravity, it is possible to identify the gauge variables, describing relativistic inertial effects, in Einstein general relativity. One of these is the York time, the trace of the extrinsic curvature of the instantaneous non-Euclidean 3-spaces (global Euclidean 3 spaces are forbidden by the equivalence principle). The extrinsic curvature depends both on gauge variables and on dynamical ones like the gravitational waves after linearization. The fixation of these gauge variables is done by relativistic metrology with its identification of time and space. Till now, the International Celestial Reference Frame ICRF uses Euclidean 3-spaces outside the Solar System. It is shown that York time and non-Euclidean 3-spaces may explain the main signatures of dark matter in ordinary space-time before using cosmology. Also dark energy may be connected to these inertial gauge effects, because both red-shift and luminosity distance depend on them.

Keywords: dark matter

#### 1. Introduction

An extremely important, till now not explicitly clarified, point in Einstein general relativity (GR) (and in every generally covariant theory of gravity), whose gauge group is the group of diffeomorphisms of the Lorentzian 4-dimensional space-time,<sup>1</sup> is that the fixation of the gauge freedom is nothing else than the establishment of conventions for relativistic metrology, an operation performed from atomic physicists, NASA engineers and astronomers [2] with the introduction of a notion of clock synchronization and with a definition of the axes for the 4 coordinates in each point, that is, with the identification of a non-inertial frame of the spacetime (global inertial frames are forbidden by the equivalence principle). See Ref. [3, 4] for a review of the existing conventions in the Solar System.

<sup>1</sup> See Ref. [1] for theoretical considerations concerning the nature of space and time in GR.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

According with the International Astronomic Union IAU inside the Solar System, the choice of the 4-coordinates is solved at the experimental level by the choice of a convention for the description of matter based on special post-Newtonian (PN) solutions of linearized Einstein equations in a fixed given harmonic gauge [2–4]: (a) for satellites near the Earth (like the GPS ones) one uses NASA 4-coordinates compatible with the reference frames of the International Terrestrial Reference System ITRS20032 and of the Geocentric Celestial Reference System GCRS IAU2000; (b) for planets in the Solar System one uses the frame of the Barycentric Celestial Reference System BCRS-IAU2000.

These frames are compatible with the usual interpretation as quasi-inertial frames in Minkowski space-time and are metrology choices like the choice of a certain atomic clock as standard of time. However, already in the Solar System, the instantaneous 3-spaces are not Euclidean in the selected solutions, but the existing technology is not yet able to show it, being a property of order <sup>O</sup>ð1=c<sup>2</sup><sup>Þ</sup> 3

In astronomy, data like luminosity, light spectrum and angles are used to determine the positions of stars and galaxies and their temporal evolution in a 4-dimensional nearly Galilei space-time with the International Celestial Reference System ICRS [2, 3], a frame considered as a "quasi-inertial frame" and with all galactic dynamics described by PN gravity.

This is in accord with the smallness of the intrinsic 3-curvature of the 3-spaces as implied by the CMB data, a property included in the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) Λ CDM cosmological model with its isotropy and homogeneity symmetries. However, to reconcile all the existing data with this 4-dimensional description, one must postulate the existence of dark matter and dark energy as the dominant components of the classical universe [6–8] after the recombination 3-surface (before it quantum mechanics is entering in the description and there is no acceptable description for the transition from quantum to classical astrophysics) already within galaxies before making the transition to cosmology and the replacement of ordinary space-time with the standard cosmological FLWR one, whose points describe a mean over a volume of 100 Mega-parsecs of the ordinary space-time. The attempts to avoid the appearance of "darkness" have led to many proposals of modifications of GR like MOND [9], fðRÞ gravity [10–12] and the ones analyzed in Refs. [6–8].

After a description of ICRS and of the measurements in ordinary astrophysics (not cosmology) of quantities like luminosity distance, rotation curves of galaxies, gravitational lensing,…. implying "darkness," we will study canonical ADM tetrad gravity and its gauge freedom after a suitable but arbitrary 3 þ 1 splitting of the space-time in a family of Einstein space-times able to include the extension of the models of particle physics to GR. We will identify which are the gauge variables to be fixed with astrophysical metrology and how the interpretation of "dark

<sup>2</sup> A relativistic version of ITRS is not yet existing, so that one cannot yet connect the time of the atomic clocks in different laboratories to the clock on the Space Station with a suitable Lorentz transformations.

<sup>3</sup> See however the LATOR proposal [5] of measuring the deviation from 2π of the sum of the three angles of a triangle formed by the Space Station and two spacecrafts behind the Sun. When this non-Euclidean nature will be measured, one will have to redefine the standard of length measurements [2].

matter" and probably also of "dark energy" depends on the fixation of these gauge variables in a family of gauge-fixings different from the harmonic ones used in the IAU conventions. Therefore, our suggestion is that "darkness" may be interpreted as a relativistic inertial effect and that ICRS should be reformulated in a suitable relativistic way.

#### 2. Astrophysical metrology

According with the International Astronomic Union IAU inside the Solar System, the choice of the 4-coordinates is solved at the experimental level by the choice of a convention for the description of matter based on special post-Newtonian (PN) solutions of linearized Einstein equations in a fixed given harmonic gauge [2–4]: (a) for satellites near the Earth (like the GPS ones) one uses NASA 4-coordinates compatible with the reference frames of the International Terrestrial Reference System ITRS20032 and of the Geocentric Celestial Reference System GCRS IAU2000; (b) for planets in the Solar System one uses the frame of the Barycentric Celestial Reference

These frames are compatible with the usual interpretation as quasi-inertial frames in Minkowski space-time and are metrology choices like the choice of a certain atomic clock as standard of time. However, already in the Solar System, the instantaneous 3-spaces are not Euclidean in the selected solutions, but the existing technology is not yet able to show it, being a property of

In astronomy, data like luminosity, light spectrum and angles are used to determine the positions of stars and galaxies and their temporal evolution in a 4-dimensional nearly Galilei space-time with the International Celestial Reference System ICRS [2, 3], a frame considered as

This is in accord with the smallness of the intrinsic 3-curvature of the 3-spaces as implied by the CMB data, a property included in the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) Λ CDM cosmological model with its isotropy and homogeneity symmetries. However, to reconcile all the existing data with this 4-dimensional description, one must postulate the existence of dark matter and dark energy as the dominant components of the classical universe [6–8] after the recombination 3-surface (before it quantum mechanics is entering in the description and there is no acceptable description for the transition from quantum to classical astrophysics) already within galaxies before making the transition to cosmology and the replacement of ordinary space-time with the standard cosmological FLWR one, whose points describe a mean over a volume of 100 Mega-parsecs of the ordinary space-time. The attempts to avoid the appearance of "darkness" have led to many proposals of modifications

After a description of ICRS and of the measurements in ordinary astrophysics (not cosmology) of quantities like luminosity distance, rotation curves of galaxies, gravitational lensing,…. implying "darkness," we will study canonical ADM tetrad gravity and its gauge freedom after a suitable but arbitrary 3 þ 1 splitting of the space-time in a family of Einstein space-times able to include the extension of the models of particle physics to GR. We will identify which are the gauge variables to be fixed with astrophysical metrology and how the interpretation of "dark

A relativistic version of ITRS is not yet existing, so that one cannot yet connect the time of the atomic clocks in different

See however the LATOR proposal [5] of measuring the deviation from 2π of the sum of the three angles of a triangle formed by the Space Station and two spacecrafts behind the Sun. When this non-Euclidean nature will be measured, one

a "quasi-inertial frame" and with all galactic dynamics described by PN gravity.

of GR like MOND [9], fðRÞ gravity [10–12] and the ones analyzed in Refs. [6–8].

laboratories to the clock on the Space Station with a suitable Lorentz transformations.

will have to redefine the standard of length measurements [2].

System BCRS-IAU2000.

68 Trends in Modern Cosmology

3

order <sup>O</sup>ð1=c<sup>2</sup><sup>Þ</sup>

2

3

Reference data for positional astronomy, such as the data in astrometric star catalogs, are specified in the International Celestial Reference System ICRS [2, 3] with origin in the solar system barycenter and with kinematically non-rotating spatial axes fixed with respect to space according to the IAU conventions [2, 3]. It is based on the position of extragalactic radio sources that are distant enough to be considered stationary, in the limit of today's capabilities, and whose position is known with a precision of 0.001arcsec, thanks to the Very Long Baseline Interferometry technique [13]. These sources are assumed to have no observable intrinsic angular momentum. The International Celestial Reference Frame ICRF is a realization of ICRS obtained by supposing that the origin is a quasi-inertial observer and that we have a quasiinertial (essentially non-relativistic) reference frame with rectangular 3-coordinates in a nearly Galilean space-time whose 3-spaces are Euclidean.

However, a number of different categories of astronomical observations are explained in the usual Euclidean 3-space only in terms of so far undetermined dark matter and dark energy: rotational curves of galaxies [14–17], gravitational lensing [18–20], application of the virial theorem to galaxy clusters [21–23] and the acceleration of the expansion of the universe [24–29]. This already happens before the transition from the ordinary space-time to the cosmological one, the FLWR space-time which is not a Galilean space-time but has nearly internally flat 3-spaces and uses a theoretical cosmic time. What is still not explored is the possibility that in Einstein GR one can use non Euclidean 3-spaces with small internal 3-curvature, but with an extrinsic curvature (as 3-submanifolds of the space-time) depending on the gauge variables, namely on the metrology conventions.

In all the astronomical observations, the distance of the objects needs to be known. Measuring distances in astronomy is a difficult task, especially when dealing with extragalactic objects. Different methods must be applied at increasing distances, which need to be inter-calibrated appropriately. To get relevant quantities like distances and absolute luminosity of stars from the directly measured quantities, that is, apparent luminosity, angles and red-shift, it is important to know the geometry of the 3-spaces crossed by the propagating rays of light on null 4 geodesics of the space-time.

The most important methods rely on the absolute intrinsic luminosity L of a standard candle compared to the apparent brightness F as measured on Earth. In terms of these quantities, one defines the luminosity distance [6, 18–20] of a luminous object dL <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L=4 π F p , which is the proper distance of an object at rest with respect to the observer in a Euclidean stationary universe. In an expanding universe, the luminosity distance is dependent on the red-shift z of the light arriving on the Earth from the object and to the comoving distance r1. If ao is the scale factor, Ho is the Hubble constant, H\_ <sup>o</sup> its time derivative, and if one keeps only the first-order terms in the expansion, one has

$$d\_L = a\_o \, r\_1 \, (1 + z) = \frac{z}{H\_o} \left[ 1 + \left( 1 + \frac{\dot{H}\_o}{2H\_o^2} \right) z \right],\tag{1}$$

Also used is the angular diameter distance dA ¼ D=θ, where θ is the angular diameter of the source as measured by the observer and D is the diameter of well-known close galaxies. Also the angular diameter distance depends on the red-shift: dA <sup>¼</sup> ao <sup>r</sup><sup>1</sup> <sup>1</sup>þ<sup>z</sup> <sup>¼</sup> <sup>z</sup> Ho <sup>1</sup> � ð<sup>1</sup> � <sup>H</sup>\_ <sup>o</sup> 2 H<sup>2</sup> o Þ z h i.

In both luminosity distance and angular diameter distance, the terms which depart from Euclidean geometry enter only at higher orders, which depend on the rate of expansion of the universe and on the curvature parameter. For the galaxies with the most reliable rotation curves that are within a range of a few tens of Mega-parsec, they can be neglected, and we can consider the 3-space to be Euclidean. Higher order terms need instead to be considered when the objects have a distance of hundreds of Mega-parsec or more.

For larger z, one has to take into account a model of cosmology: In a FLWR metric, one has F ¼ L=½4 π ðao r1Þ <sup>2</sup> <sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>z</sup><sup>Þ</sup> 2 � with ao ¼ 1=ð1 þ zÞ and with r<sup>1</sup> depending also on z.

Assuming that all supernovae (SN) Ia have the same intrinsic luminosity, it was found [26–29] that the SN1a's at z ≤ 0:5 are about 10 per cent fainter than expected, and this has been interpreted as evidence of an accelerated expansion of the universe and dark energy has been invoked to take care of the accelerated expansion.

#### 3. Einstein general relativity

We shall use the formulation of Einstein GR in a 4-dimensional Lorentzian space-time (the one used in classical astrophysics, not in cosmology, after the recombination surface for the propagation of light) with the Lagrangian description implied by the ADM action principle [30, 31], because it allows to make the transition to the canonical formalism and to use Dirac theory of constraints [32], in particular to use the Shanmugadhasan canonical transformation [33, 34] to find canonical bases adapted to the constraints (see Ref. [35] for reviews). Light and visible stars and galaxies constitute the matter.

We will restrict ourselves to globally hyperbolic, topologically trivial and asymptotically Minkowskian space-times, in the absence of Killing symmetries (see Ref. [36] for their inclusion as Dirac constraints) and with the asymptotic SPI symmetries at spatial infinity of Ref. [37] restricted to the asymptotic ADM Poincaré group [38] by eliminating the super-translations with suitable boundary conditions on the 4-metric. This framework is defined in Refs. [39–43], where the matter consists of electrically charged positive-energy scalar point particles plus the electromagnetic field. In the limit of vanishing Newton constant (G ¼ 0), the asymptotic Poincare' group becomes the Poincare' group of particle physics, where elementary particles are always considered as irreducible representations of this group.

While in the family of spatially compact without spatial boundary space-times<sup>4</sup> , considered in loop quantum gravity [44, 45], the Dirac Hamiltonian is a combination of constraints because the canonical Hamiltonian vanishes, in our space-times there is not a frozen picture, because the canonical Hamiltonian is the weak ADM energy E^ADM <sup>5</sup> plus a combination of constraints. In the absence of matter, Christodoulou-Klainermann space-times [46] are compatible with this description.

factor, Ho is the Hubble constant, H\_ <sup>o</sup> its time derivative, and if one keeps only the first-order

Ho

Also used is the angular diameter distance dA ¼ D=θ, where θ is the angular diameter of the source as measured by the observer and D is the diameter of well-known close galaxies. Also

In both luminosity distance and angular diameter distance, the terms which depart from Euclidean geometry enter only at higher orders, which depend on the rate of expansion of the universe and on the curvature parameter. For the galaxies with the most reliable rotation curves that are within a range of a few tens of Mega-parsec, they can be neglected, and we can consider the 3-space to be Euclidean. Higher order terms need instead to be considered

For larger z, one has to take into account a model of cosmology: In a FLWR metric, one has

Assuming that all supernovae (SN) Ia have the same intrinsic luminosity, it was found [26–29] that the SN1a's at z ≤ 0:5 are about 10 per cent fainter than expected, and this has been interpreted as evidence of an accelerated expansion of the universe and dark energy has been

We shall use the formulation of Einstein GR in a 4-dimensional Lorentzian space-time (the one used in classical astrophysics, not in cosmology, after the recombination surface for the propagation of light) with the Lagrangian description implied by the ADM action principle [30, 31], because it allows to make the transition to the canonical formalism and to use Dirac theory of constraints [32], in particular to use the Shanmugadhasan canonical transformation [33, 34] to find canonical bases adapted to the constraints (see Ref. [35] for reviews). Light and visible

We will restrict ourselves to globally hyperbolic, topologically trivial and asymptotically Minkowskian space-times, in the absence of Killing symmetries (see Ref. [36] for their inclusion as Dirac constraints) and with the asymptotic SPI symmetries at spatial infinity of Ref. [37] restricted to the asymptotic ADM Poincaré group [38] by eliminating the super-translations with suitable boundary conditions on the 4-metric. This framework is defined in Refs. [39–43], where the matter consists of electrically charged positive-energy scalar point particles plus the electromagnetic field. In the limit of vanishing Newton constant (G ¼ 0), the asymptotic Poincare' group becomes the Poincare' group of particle physics, where elementary particles are always consid-

� with ao ¼ 1=ð1 þ zÞ and with r<sup>1</sup> depending also on z.

½1 þ ð1 þ

H\_ o 2 H<sup>2</sup> o

<sup>1</sup>þ<sup>z</sup> <sup>¼</sup> <sup>z</sup>

Þ z�; ð1Þ

h i

2 H<sup>2</sup> o Þ z .

Ho <sup>1</sup> � ð<sup>1</sup> � <sup>H</sup>\_ <sup>o</sup>

dL <sup>¼</sup> ao <sup>r</sup><sup>1</sup> <sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>z</sup>Þ ¼ <sup>z</sup>

the angular diameter distance depends on the red-shift: dA <sup>¼</sup> ao <sup>r</sup><sup>1</sup>

when the objects have a distance of hundreds of Mega-parsec or more.

terms in the expansion, one has

70 Trends in Modern Cosmology

F ¼ L=½4 π ðao r1Þ

<sup>2</sup> <sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>z</sup><sup>Þ</sup> 2

3. Einstein general relativity

stars and galaxies constitute the matter.

ered as irreducible representations of this group.

invoked to take care of the accelerated expansion.

In the ADM Lagrangian, the basic variable is the 4-metric <sup>4</sup>gμνðx<sup>Þ</sup> of the space-time (x<sup>μ</sup> are local 4-coordinates with an arbitrary origin): it determines the dynamical chrono-geometrical structure of space-time by means of the line element ds<sup>2</sup> <sup>¼</sup> <sup>4</sup>gμνðx<sup>Þ</sup> dx<sup>μ</sup> dx<sup>ν</sup>, and it teaches to massless particles which are the allowed trajectories in each point.

However, to include the coupling of gravity to the spin of fermions, we must use ADM tetrad gravity: the 10 components of the 4-metric appearing in the ADM Lagrangian are decomposed on a set of cotetrads [31] E<sup>ð</sup>α<sup>Þ</sup> <sup>μ</sup> <sup>ð</sup>xÞ, <sup>4</sup>gμνðxÞ ¼ <sup>E</sup><sup>ð</sup>α<sup>Þ</sup> <sup>μ</sup> <sup>ð</sup>x<sup>Þ</sup> <sup>η</sup>ðαÞðβ<sup>Þ</sup> <sup>E</sup><sup>ð</sup>β<sup>Þ</sup> <sup>ν</sup> ðxÞ. <sup>6</sup> This leads to an interpretation of gravity based on a congruence of time-like observers endowed with orthonormal tetrads Eμ ðαÞ <sup>ð</sup>x<sup>Þ</sup> (i.e., the inverse of the cotetrads <sup>E</sup><sup>μ</sup> ðαÞ <sup>ð</sup>x<sup>Þ</sup> <sup>E</sup><sup>ð</sup>β<sup>Þ</sup> <sup>μ</sup> ðxÞ ¼ δ ðβÞ ðαÞ ): in each point of space-time, the time-like axis is the unit 4-velocity of a time-like observer, whereas the spatial axes are a (gauge) convention for the three gyroscopes of the observer.

#### A. Metrology as the Fixation of the Gauge Freedom of General Relativity

While the ADM action for metric gravity is invariant under space-time diffeomorphisms, the decomposition of the 4-metric on the cotetrads gives an ADM action [30] invariant not only under the space-time diffeomorphisms but also on a local O(3,1) Lorentz group describing the freedom in the orientation and transport of the gyroscopes along the time-like world lines of observers. Let us remark that the same gauge freedoms are present in all the generally covariant formulations of GR proposed as modifications of Einstein GR.

In electromagnetism and in Yang-Mills theories, the Lagrangian description in terms of potentials implies the presence of a gauge group acting on an internal space and implying the gauge nature of certain scalar and longitudinal components of the potentials: the gauge fixings imply the description of physics in terms of electric and magnetic fields or of their non-abelian analogues. Instead, in the metric formulation of GR, the gauge freedom is connected with the freedom in the choice of the metrology conventions, described in the previous section, for the definitions of clocks (i.e., time) and 3-space in each point of the space-time. As we shall see a metrology convention implies the fixation of 8 of the 10 components of the 4-metric, so that the remaining two components describe the physical degrees of freedom of the gravitational field

<sup>4</sup> Therefore, it is not possible to define a Poincare' group and to find a connection with particle physics.

<sup>5</sup> It is a volume integral over 3-space of a coordinate-dependent energy density. It is weakly equal to the strong ADM energy, which is a flux through a 2-surface at spatial infinity.

<sup>6</sup> <sup>ð</sup>α<sup>Þ</sup> are flat indices and <sup>η</sup>ðαÞðβ<sup>Þ</sup> is the flat 4-metric of Minkowski space-time. The signature of the 4-metrics is <sup>E</sup> ¼ � so that <sup>η</sup>ðαÞðβ<sup>Þ</sup> <sup>¼</sup> <sup>E</sup>ð1; � <sup>1</sup>; � <sup>1</sup>; � <sup>1</sup>Þ. <sup>E</sup> <sup>¼</sup> 1 is the convention of particle physics, whereas <sup>E</sup> ¼ �1 is the convention usually used in GR

(the gravitational waves (GW) of its linearization in the case of weak fields). In tetrad gravity, we have 16 fields, but the extra 6 fields are fixed by metrology conventions on the orientation of three gyroscopes and on their transport along time-like world lines in each point of the space-time.

In special relativity, the metrology conventions amount to the choice of a standard atomic clock and of the instantaneous Euclidean 3-spaces of a global inertial frame, whose extension to global non-inertial frames was done in Ref. [47] with an application to relativistic atomic physics described in Ref. [48].

In GR, due to the equivalence principle forbidding the existence of global inertial frames, one has to use the cited theory of global non-inertial frames in the form of the so-called <sup>3</sup>þ1 point of view<sup>7</sup> : one gives the world line of a time-like observer and a nice foliation of the space-time whose leaves are the instantaneous 3-spaces. Instead of standard local 4-coordinates x<sup>μ</sup> centered in a point of the observer world line, one uses 4-scalar observer-dependent radar 4-coordinates<sup>8</sup> <sup>σ</sup><sup>A</sup> ¼ ðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup>, where <sup>τ</sup> is an arbitrary increasing function of the observer proper time and <sup>σ</sup><sup>r</sup> is curvilinear 3-coordinates on the 3-spaces Στ (diffeomorphic to R<sup>3</sup> ) with the observer as origin.

The inverse transformation <sup>σ</sup><sup>A</sup>↦x<sup>μ</sup> <sup>¼</sup> <sup>z</sup><sup>μ</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ defines the embeddings of the 3-spaces Στ into the space-time and the induced 4-metric is gAB½zðτ; <sup>σ</sup><sup>r</sup> Þ� ¼ z μ <sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ zν <sup>B</sup>ðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>g</sup>μνðzðτ; <sup>σ</sup><sup>r</sup> ÞÞ, where z μ <sup>A</sup> <sup>¼</sup> <sup>∂</sup> <sup>z</sup><sup>μ</sup>=<sup>∂</sup> <sup>σ</sup><sup>A</sup>, while the cotetrads take the form <sup>E</sup><sup>ð</sup>α<sup>Þ</sup> <sup>A</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ ¼ z μ <sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>E</sup><sup>ð</sup>α<sup>Þ</sup> <sup>μ</sup> <sup>ð</sup>zðτ; <sup>σ</sup><sup>r</sup> ÞÞ. As shown explicitly in Ref. [51], the use of the 4-scalar radar 4-coordinates implies that the ten components <sup>4</sup>gABðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> and the sixteen components E<sup>ð</sup>α<sup>Þ</sup> <sup>A</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ are 4-scalars of the space-time. Also, all the components of radar tensors (i.e., tensors expressed in radar 4-coordinates) are 4-scalars of the space-time.

While the 4-vectors z μ <sup>r</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>u</sup><sup>Þ</sup> are tangent to Στ, so that in each point of the 3-space, the unit normal l <sup>μ</sup>ðτ; <sup>σ</sup><sup>u</sup><sup>Þ</sup> is proportional to <sup>E</sup><sup>μ</sup> αβγ z<sup>α</sup> <sup>1</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ z β <sup>2</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ z γ <sup>3</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>u</sup>Þ, we have <sup>z</sup> μ <sup>τ</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ ¼ <sup>N</sup>ðτ; <sup>σ</sup><sup>r</sup> Þl <sup>μ</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ þ <sup>N</sup><sup>r</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ z μ <sup>r</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup>, where <sup>N</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ ¼ E z μ <sup>τ</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup>lμðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> and Nrðτ; <sup>σ</sup><sup>r</sup> ޼�Egτ<sup>r</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ are the lapse and shift functions of canonical GR.

In the chosen family of space-times, the foliation needed for the 3þ1 splitting is nice and admissible if the lapse function satisfies <sup>N</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ > 0 in every point of Στ, <sup>9</sup> if <sup>E</sup><sup>4</sup>gττðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>&</sup>gt; 010 and if the positive-definite 3-metric <sup>3</sup>grsðτ; <sup>σ</sup><sup>u</sup>޼�<sup>E</sup> <sup>4</sup>grsðτ; <sup>σ</sup><sup>u</sup><sup>Þ</sup> has three positive eigenvalues. These are the Møller conditions [52, 53].

Moreover, all the 3-spaces Στ must tend to the same space-like hyperplane at spatial infinity. Due to the imposed absence of super-translations [39, 40], the non-Euclidean 3-spaces are orthogonal to the conserved ADM 4-momentum at spatial infinity; therefore, each 3-space is a

<sup>7</sup> Instead the usually used 1þ3 point of view using the world line of a time-like observer leads only to local coordinate systems like the Riemann and Fermi ones valid only in a neighborhood of a time-like world line, because locally the 3 spaces are identified with the tangent spaces orthogonal to the observer 4-velocity so that they intersect each other. 8 They were introduced by Bondi in Ref. [49, 50].

<sup>9</sup> Therefore, the 3-spaces never intersect, avoiding the coordinate singularity of Fermi coordinates.

<sup>10</sup>This property avoids the coordinate singularity of the rotating disk.

non-inertial rest frame [39–41] of the 3-universe with vanishing ADM 3-momentum, and there are asymptotic inertial observers with spatial axes identified by means of the fixed stars of star catalogues. In each 3-space Στ, there are cotriads <sup>3</sup>eðaÞ<sup>r</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ ¼ <sup>X</sup> <sup>b</sup> <sup>R</sup>ðaÞð<sup>b</sup>Þðαð<sup>c</sup>Þðτ; <sup>σ</sup><sup>r</sup> ÞÞ <sup>3</sup>eðaÞ<sup>r</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ defined modulo rotations (RðaÞðb<sup>Þ</sup> are rotation matrices and <sup>α</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ are angles).

(the gravitational waves (GW) of its linearization in the case of weak fields). In tetrad gravity, we have 16 fields, but the extra 6 fields are fixed by metrology conventions on the orientation of three gyroscopes and on their transport along time-like world lines in each point of the

In special relativity, the metrology conventions amount to the choice of a standard atomic clock and of the instantaneous Euclidean 3-spaces of a global inertial frame, whose extension to global non-inertial frames was done in Ref. [47] with an application to relativistic atomic physics

In GR, due to the equivalence principle forbidding the existence of global inertial frames, one has to use the cited theory of global non-inertial frames in the form of the so-called <sup>3</sup>þ1 point of view<sup>7</sup>

one gives the world line of a time-like observer and a nice foliation of the space-time whose leaves are the instantaneous 3-spaces. Instead of standard local 4-coordinates x<sup>μ</sup> centered in a point of the observer world line, one uses 4-scalar observer-dependent radar 4-coordinates<sup>8</sup>

explicitly in Ref. [51], the use of the 4-scalar radar 4-coordinates implies that the ten components

components of radar tensors (i.e., tensors expressed in radar 4-coordinates) are 4-scalars of the

In the chosen family of space-times, the foliation needed for the 3þ1 splitting is nice and

and if the positive-definite 3-metric <sup>3</sup>grsðτ; <sup>σ</sup><sup>u</sup>޼�<sup>E</sup> <sup>4</sup>grsðτ; <sup>σ</sup><sup>u</sup><sup>Þ</sup> has three positive eigenvalues.

Moreover, all the 3-spaces Στ must tend to the same space-like hyperplane at spatial infinity. Due to the imposed absence of super-translations [39, 40], the non-Euclidean 3-spaces are orthogonal to the conserved ADM 4-momentum at spatial infinity; therefore, each 3-space is a

Instead the usually used 1þ3 point of view using the world line of a time-like observer leads only to local coordinate systems like the Riemann and Fermi ones valid only in a neighborhood of a time-like world line, because locally the 3 spaces are identified with the tangent spaces orthogonal to the observer 4-velocity so that they intersect each other.

Therefore, the 3-spaces never intersect, avoiding the coordinate singularity of Fermi coordinates.

<sup>A</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup>

αβγ z<sup>α</sup> <sup>1</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ z β <sup>2</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ z γ

<sup>Þ</sup>, where <sup>N</sup>ðτ; <sup>σ</sup><sup>r</sup>

curvilinear 3-coordinates on the 3-spaces Στ (diffeomorphic to R<sup>3</sup>

The inverse transformation <sup>σ</sup><sup>A</sup>↦x<sup>μ</sup> <sup>¼</sup> <sup>z</sup><sup>μ</sup>ðτ; <sup>σ</sup><sup>r</sup>

μ

Þ þ <sup>N</sup><sup>r</sup>

the space-time and the induced 4-metric is gAB½zðτ; <sup>σ</sup><sup>r</sup>

<sup>A</sup> <sup>¼</sup> <sup>∂</sup> <sup>z</sup><sup>μ</sup>=<sup>∂</sup> <sup>σ</sup><sup>A</sup>, while the cotetrads take the form <sup>E</sup><sup>ð</sup>α<sup>Þ</sup>

<sup>Þ</sup> and the sixteen components E<sup>ð</sup>α<sup>Þ</sup>

<sup>μ</sup>ðτ; <sup>σ</sup><sup>u</sup><sup>Þ</sup> is proportional to <sup>E</sup><sup>μ</sup>

Þ are the lapse and shift functions of canonical GR.

<sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ z μ <sup>r</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup>

admissible if the lapse function satisfies <sup>N</sup>ðτ; <sup>σ</sup><sup>r</sup>

These are the Møller conditions [52, 53].

They were introduced by Bondi in Ref. [49, 50].

10This property avoids the coordinate singularity of the rotating disk.

<sup>Þ</sup>, where <sup>τ</sup> is an arbitrary increasing function of the observer proper time and <sup>σ</sup><sup>r</sup> is

Þ� ¼ z μ <sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ zν <sup>B</sup>ðτ; <sup>σ</sup><sup>r</sup>

<sup>r</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>u</sup><sup>Þ</sup> are tangent to Στ, so that in each point of the 3-space, the unit

Þ > 0 in every point of Στ,

Þ ¼ E z μ <sup>τ</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup>

Þ ¼ z μ <sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup>

<sup>A</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup>

:

ÞÞ, where

ÞÞ. As shown

μ <sup>τ</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ ¼

޼�Egτ<sup>r</sup>

<sup>Þ</sup> <sup>&</sup>gt; 010

) with the observer as origin.

<sup>Þ</sup> <sup>g</sup>μνðzðτ; <sup>σ</sup><sup>r</sup>

Þ defines the embeddings of the 3-spaces Στ into

<sup>Þ</sup> <sup>E</sup><sup>ð</sup>α<sup>Þ</sup> <sup>μ</sup> <sup>ð</sup>zðτ; <sup>σ</sup><sup>r</sup>

Þ are 4-scalars of the space-time. Also, all the

<sup>3</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>u</sup>Þ, we have <sup>z</sup>

<sup>Þ</sup> and Nrðτ; <sup>σ</sup><sup>r</sup>

<sup>9</sup> if <sup>E</sup><sup>4</sup>gττðτ; <sup>σ</sup><sup>r</sup>

<sup>Þ</sup>lμðτ; <sup>σ</sup><sup>r</sup>

space-time.

72 Trends in Modern Cosmology

<sup>σ</sup><sup>A</sup> ¼ ðτ; <sup>σ</sup><sup>r</sup>

z μ

<sup>4</sup>gABðτ; <sup>σ</sup><sup>r</sup>

space-time.

normal l

<sup>N</sup>ðτ; <sup>σ</sup><sup>r</sup> Þl <sup>μ</sup>ðτ; <sup>σ</sup><sup>r</sup>

<sup>ð</sup>τ; <sup>σ</sup><sup>r</sup>

7

8

9

While the 4-vectors z

described in Ref. [48].

In Refs. [39–43], there is a parametrization of tetrads, cotetrads and 4-metric in the framework of the 3þ1 splitting of space-time. The basic configuration variables, that is, the cotetrads, are connected to cotetrads adapted to the 3þ1 splitting of space-time (so that the adapted time-like tetrad is the unit normal to the 3-space Στ) by standard Wigner boosts Lðα<sup>Þ</sup> <sup>ð</sup>β<sup>Þ</sup> for time-like vectors depending upon boost parameters <sup>ϕ</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup>: <sup>4</sup>E<sup>ð</sup>α<sup>Þ</sup> <sup>A</sup> <sup>¼</sup> <sup>L</sup>ðα<sup>Þ</sup> <sup>ð</sup><sup>β</sup>Þðϕð<sup>a</sup>ÞÞ <sup>4</sup>E<sup>o</sup> <sup>A</sup>ðβÞ. The adapted tetrads and cotetrads have the expression11, 12

<sup>4</sup>E<sup>A</sup> <sup>ð</sup>α<sup>Þ</sup> ¼ def <sup>4</sup> <sup>E</sup> ∘ A ðβÞ Lðβ<sup>Þ</sup> <sup>ð</sup><sup>α</sup>Þðϕð<sup>a</sup>ÞÞ ¼def <sup>E</sup> ∘ A <sup>ð</sup>o<sup>Þ</sup> <sup>L</sup>ðo<sup>Þ</sup> <sup>ð</sup><sup>α</sup>Þðϕð<sup>c</sup>ÞÞþ þ X ab 4 E ∘ A <sup>ð</sup>b<sup>Þ</sup> RT <sup>ð</sup>bÞð<sup>a</sup>Þðαð<sup>c</sup>ÞÞ <sup>L</sup>ða<sup>Þ</sup> <sup>ð</sup><sup>α</sup>Þðϕð<sup>c</sup>ÞÞ; 4 gAB <sup>¼</sup> <sup>E</sup><sup>ð</sup>α<sup>Þ</sup> A 4 <sup>η</sup>ðαÞðβ<sup>Þ</sup> <sup>E</sup><sup>ð</sup>β<sup>Þ</sup> <sup>B</sup> ¼ ¼4 E ∘ ðαÞ A 4 <sup>η</sup>ðαÞðβ<sup>Þ</sup> 4 E ∘ ðβÞ <sup>B</sup> <sup>¼</sup> <sup>4</sup> E ∘ ðαÞ A 4 <sup>η</sup>ðαÞðβ<sup>Þ</sup> 4 E ∘ ðβÞ <sup>B</sup> ; 4 E ∘ A <sup>ð</sup>o<sup>Þ</sup> <sup>¼</sup> <sup>4</sup> E ∘ A <sup>ð</sup>o<sup>Þ</sup> <sup>¼</sup> <sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>n</sup> <sup>ð</sup>1; �<sup>X</sup> a nða<sup>Þ</sup> 3 e r <sup>ð</sup><sup>a</sup>ÞÞ ¼ <sup>l</sup> <sup>A</sup>; <sup>4</sup> E ∘ A ð<sup>a</sup>Þ¼<sup>ð</sup>0;<sup>3</sup> e r <sup>ð</sup><sup>a</sup>ÞÞ; 4 E ∘ ðoÞ <sup>A</sup> <sup>¼</sup> <sup>4</sup> E ∘ ðoÞ <sup>A</sup> ¼ ð1 þ nÞ ð1; 0 ! Þ ¼ <sup>E</sup>lA; <sup>4</sup> E ∘ ðaÞ <sup>A</sup> ¼ ðnðaÞ; 3 eðaÞ<sup>r</sup>Þ; 4 E ∘ ðaÞ <sup>A</sup> <sup>¼</sup> <sup>X</sup> b RðaÞðb<sup>Þ</sup> ðnðbÞ; 3 eðbÞ<sup>r</sup>Þ; 4 gττ ¼ E½ð1 þ nÞ <sup>2</sup> �<sup>X</sup> a n2 <sup>ð</sup><sup>a</sup>Þ�; <sup>4</sup> gτ<sup>r</sup> ¼ �E nr ¼ �E X a nða<sup>Þ</sup> 3 eðaÞ<sup>r</sup>; 4 grs ¼ �<sup>E</sup> <sup>3</sup> grs ¼ �E X a 3 eðaÞ<sup>r</sup> 3 <sup>e</sup>ðaÞ<sup>s</sup>; ffiffiffiffiffiffi �<sup>g</sup> <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffi j 4 gj q ¼ ffiffiffiffiffi <sup>3</sup>g p ffiffiffiffiffiffiffiffiffiffiffi <sup>E</sup> <sup>4</sup>gττ <sup>p</sup> <sup>¼</sup> ffiffiffi <sup>γ</sup> <sup>p</sup> <sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup>Þ: ð2Þ

From Eq. (5.5) of the third paper in Ref. [43], we assume the following (direction-independent, so to kill super-translations) boundary conditions at spatial infinity (r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X <sup>r</sup> <sup>ð</sup> <sup>σ</sup><sup>r</sup><sup>Þ</sup> <sup>2</sup> q ; E > 0; <sup>M</sup> <sup>¼</sup> const:): <sup>n</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ!<sup>r</sup>! <sup>∞</sup> <sup>O</sup>ðr�ð2þE<sup>Þ</sup> <sup>Þ</sup>, <sup>π</sup>nðτ; <sup>σ</sup><sup>r</sup> Þ!<sup>r</sup>!<sup>∞</sup> <sup>O</sup>ðr�<sup>3</sup>Þ, <sup>n</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ!<sup>r</sup>!<sup>∞</sup> Oðr�<sup>E</sup> <sup>Þ</sup>, <sup>π</sup><sup>n</sup>ða<sup>Þ</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ !<sup>r</sup>!<sup>∞</sup> <sup>O</sup>ðr�<sup>3</sup>Þ, <sup>ϕ</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ!<sup>r</sup>!<sup>∞</sup> <sup>O</sup>ðr�ð1þE<sup>Þ</sup> Þ, πϕða<sup>Þ</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ!<sup>r</sup>!<sup>∞</sup> <sup>O</sup>ðr�<sup>2</sup>Þ, <sup>3</sup>eðaÞ<sup>r</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ!<sup>r</sup> !<sup>∞</sup> <sup>1</sup> <sup>þ</sup> <sup>M</sup> 2r � � <sup>δ</sup>ar <sup>þ</sup> <sup>O</sup>ðr�3=<sup>2</sup>Þ, <sup>3</sup>π<sup>r</sup> ðaÞ <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ!<sup>r</sup>!<sup>∞</sup> <sup>O</sup>ðr�5=<sup>2</sup>Þ.

<sup>11</sup>Nðτ; <sup>σ</sup><sup>r</sup> Þ ¼ <sup>1</sup> <sup>þ</sup> <sup>n</sup>ðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> and <sup>n</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ¼ðN<sup>r</sup> <sup>3</sup>e<sup>r</sup> ð<sup>a</sup>ÞÞðτ; <sup>σ</sup><sup>r</sup> Þ ¼ <sup>X</sup> <sup>b</sup> <sup>R</sup>ðaÞð<sup>b</sup>Þðαð<sup>c</sup>Þðτ; <sup>σ</sup><sup>r</sup> ÞÞ <sup>n</sup>ð<sup>b</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ are the lapse and shift functions respectively.

<sup>124</sup> E ∘ A <sup>ð</sup>β<sup>Þ</sup> and <sup>4</sup> <sup>E</sup> ∘ A <sup>ð</sup>o<sup>Þ</sup> are tetrads adapted to the 3þ1 splitting.

As shown in Refs. [[39–43], due to the existence of the asymptotic ADM Poincare' group, the isolated system gravitational field plus matter, namely the 3-universe, has the mass given by the ADM weak energy and the spin by the ADM angular momentum. Therefore, at each time, the 3-universe can be described as a decoupled non-covariant non-observable external pseudoparticle (the center of mass of the 3-universe) carrying a pole (the mass)-dipole (the spin) structure. Since the ADM 3-momentum vanishes due to the rest-frame condition, the conjugate non-observable internal center of mass of the 3-universe may be eliminated from the observable variables by imposing the vanishing of the ADM Lorentz boosts.

As a conclusion to fix the gauge in GR with a metrology convention, so to visualize the associated gauge-dependent inertial effects, we need to separate the gauge variables from the dynamical ones, the so-called Dirac observables (DO), and only the Hamiltonian formalism has the tools to face this problem. The usual criticism that this can be done only in a noncovariant coordinate-dependent way is avoided due to the use of the radar coordinates implying the existence of 4-scalar tensors.

#### B. Canonical ADM Tetrad Gravity and Its Gauge Variables

The parametrization of cotetrads given in the previous subsection for ADM tetrad gravity implies [40] that the ADM action may be considered function of the 16 configurational variables <sup>ϕ</sup>ðaÞ, 1 <sup>þ</sup> <sup>n</sup>, <sup>n</sup>ðaÞ, <sup>3</sup>eðaÞ<sup>r</sup>. At the Hamiltonian level, there is a phase space spanned by these 16 configuration variables and their conjugated 16 momenta, and there are 14 first class constraints. Ten of them are primary constraints (the vanishing of the 7 momenta of boosts, lapse and shift variables plus three constraints describing the gauge freedom in the rotation on the flat indices ðaÞ of the cotriads), whereas four are secondary ones (the super-Hamiltonian and super-momentum constraints). Therefore, there are 14 gauge variables describing inertial effects and 2 canonical pairs of physical degrees of freedom describing the tidal effects of the gravitational field (namely GW in the weak field limit).

The basis of canonical variables for this formulation of tetrad gravity, naturally adapted to 7 of the 14 first-class constraints, is (only the momenta <sup>3</sup>π<sup>r</sup> <sup>ð</sup>a<sup>Þ</sup> conjugated to the cotriads are not vanishing)

$$
\begin{array}{ccccc}
\mathfrak{o}\_{(a)} & \mathfrak{n} & \overline{\mathfrak{n}}\_{(a)} & \mathfrak{d}\_{(a)r} \\
\mathfrak{m}\_{\psi\_{(a)}} \simeq \mathbf{0} & \pi\_{\mathfrak{n}} \simeq \mathbf{0} & \pi\_{\mathfrak{n}\_{(a)}} \simeq \mathbf{0} & \left| \,^{3}\pi\_{\mathfrak{n}\_{(a)}}^{r} \right.
\end{array} \tag{3}
$$

In Ref. [42], a York canonical basis, adapted to 10 first-class constraints (not to the super-Hamiltonian and super-momentum ones, whose solution is unknown), was identified by means of a Shanmugadhasan canonical transformation [33, 34]; this allows for the first time to get the explicit identification of the inertial and tidal variables. It implements the York map of Ref. [54] and diagonalizes the York-Lichnerowicz approach [55]. Its final form is13

<sup>13</sup>G is Newton constant. The set of numerical parameters <sup>γ</sup>aa satisfies <sup>X</sup> <sup>u</sup> <sup>γ</sup>au <sup>¼</sup> 0, <sup>X</sup> <sup>u</sup> <sup>γ</sup>au <sup>γ</sup>bu <sup>¼</sup> <sup>δ</sup>ab, <sup>X</sup><sup>a</sup> <sup>γ</sup>au <sup>γ</sup>av <sup>¼</sup> <sup>δ</sup>uv � <sup>1</sup> 3. Each solution of these equations defines a different York canonical basis.

Relativistic Celestial Metrology: Dark Matter as an Inertial Gauge Effect http://dx.doi.org/10.5772/68115 75

$$
\pi\_{\vartheta\_{(a)}} \simeq 0 \quad \pi\_{(a)}^{(a)} \simeq 0 \quad \pi\_{\pi} \simeq 0 \quad \pi\_{\pi} \simeq 0 \quad \pi\_{\pi\_{\vartheta}} \simeq 0 \quad \pi\_{\vartheta}^{(0)} \quad \pi \ddot{\phi} = \frac{c^{3}}{12 \pi G} \, ^{3} \mathrm{K} \quad \Pi\_{\overline{a}} \tag{4}
$$

$$
\begin{aligned}
\varphi\_{(a)r} &= \sum\_{b} R\_{(a)(b)}(\alpha\_{(c)}) \, V\_{rb}(\theta^{\dagger}) \, \dot{\phi} 1/3 \, e^{\frac{\pi}{4}} \quad & \tag{5} \\
\,^{4}g\_{rr} &= \epsilon[(1+n)^{2} - \sum\_{a} \overline{n}\_{(a)}^{2}], \\
\end{aligned}
$$

$$
\begin{aligned}
\varphi\_{rr} &= -\epsilon \, \overline{n}\_{(a)} \, V\_{m}(\theta^{\dagger}) \, \dot{\phi} 1/3 \, e^{\frac{1.2}{\overline{a}}} \quad & \tag{5} \\
\,^{4}g\_{rs} &= -\epsilon \, \overline{n}\_{(a)} \, V\_{m}(\theta^{\dagger}) \, \dot{\phi} 1/3 \, e^{\frac{1.2}{\overline{a}}} \quad & \tag{5} \\
\,^{4}g\_{rs} &= -\epsilon \, \dot{\phi} 2/3 \sum\_{a} V\_{m}(\theta^{\dagger}) \, V\_{sr}(\theta^{\dagger}) \, e^{\frac{1.2}{\overline{a}}} \quad & \}
\end{aligned}
$$

In this York canonical basis, the inertial effects are described by the arbitrary gauge variables14 <sup>α</sup>ðaÞ, <sup>ϕ</sup>ðaÞ, 1 <sup>þ</sup> <sup>n</sup>, <sup>n</sup>ðaÞ, <sup>θ</sup><sup>i</sup> , <sup>3</sup>K, whereas the tidal effects, that is, the physical degrees of freedom of the gravitational field (the two polarizations of GW in the linearized theory), by the two canonical pairs Ra and Πa, a ¼ 1; 2 (Ra are eigenvalues of the 3-metric with determinant one).

As shown in Refs. [[39–43], due to the existence of the asymptotic ADM Poincare' group, the isolated system gravitational field plus matter, namely the 3-universe, has the mass given by the ADM weak energy and the spin by the ADM angular momentum. Therefore, at each time, the 3-universe can be described as a decoupled non-covariant non-observable external pseudoparticle (the center of mass of the 3-universe) carrying a pole (the mass)-dipole (the spin) structure. Since the ADM 3-momentum vanishes due to the rest-frame condition, the conjugate non-observable internal center of mass of the 3-universe may be eliminated from the observ-

As a conclusion to fix the gauge in GR with a metrology convention, so to visualize the associated gauge-dependent inertial effects, we need to separate the gauge variables from the dynamical ones, the so-called Dirac observables (DO), and only the Hamiltonian formalism has the tools to face this problem. The usual criticism that this can be done only in a noncovariant coordinate-dependent way is avoided due to the use of the radar coordinates imply-

The parametrization of cotetrads given in the previous subsection for ADM tetrad gravity implies [40] that the ADM action may be considered function of the 16 configurational variables <sup>ϕ</sup>ðaÞ, 1 <sup>þ</sup> <sup>n</sup>, <sup>n</sup>ðaÞ, <sup>3</sup>eðaÞ<sup>r</sup>. At the Hamiltonian level, there is a phase space spanned by these 16 configuration variables and their conjugated 16 momenta, and there are 14 first class constraints. Ten of them are primary constraints (the vanishing of the 7 momenta of boosts, lapse and shift variables plus three constraints describing the gauge freedom in the rotation on the flat indices ðaÞ of the cotriads), whereas four are secondary ones (the super-Hamiltonian and super-momentum constraints). Therefore, there are 14 gauge variables describing inertial effects and 2 canonical pairs of physical degrees of freedom describing the tidal effects of the gravita-

The basis of canonical variables for this formulation of tetrad gravity, naturally adapted to 7 of the

<sup>ϕ</sup>ða<sup>Þ</sup> <sup>n</sup> <sup>n</sup>ða<sup>Þ</sup>

[54] and diagonalizes the York-Lichnerowicz approach [55]. Its final form is13

πϕða<sup>Þ</sup> <sup>≈</sup> <sup>0</sup> <sup>π</sup><sup>n</sup> <sup>≈</sup> <sup>0</sup> <sup>π</sup><sup>n</sup>ða<sup>Þ</sup> <sup>≈</sup> <sup>0</sup> <sup>j</sup>

In Ref. [42], a York canonical basis, adapted to 10 first-class constraints (not to the super-Hamiltonian and super-momentum ones, whose solution is unknown), was identified by means of a Shanmugadhasan canonical transformation [33, 34]; this allows for the first time to get the explicit identification of the inertial and tidal variables. It implements the York map of Ref.

<sup>ð</sup>a<sup>Þ</sup> conjugated to the cotriads are not vanishing)

<sup>u</sup> <sup>γ</sup>au <sup>γ</sup>bu <sup>¼</sup> <sup>δ</sup>ab,

ð3Þ

3.

<sup>X</sup><sup>a</sup> <sup>γ</sup>au <sup>γ</sup>av <sup>¼</sup> <sup>δ</sup>uv � <sup>1</sup>

<sup>3</sup>eðaÞ<sup>r</sup>

<sup>u</sup> <sup>γ</sup>au <sup>¼</sup> 0, <sup>X</sup>

3 πr ðaÞ

able variables by imposing the vanishing of the ADM Lorentz boosts.

B. Canonical ADM Tetrad Gravity and Its Gauge Variables

tional field (namely GW in the weak field limit).

14 first-class constraints, is (only the momenta <sup>3</sup>π<sup>r</sup>

13G is Newton constant. The set of numerical parameters <sup>γ</sup>aa satisfies <sup>X</sup>

Each solution of these equations defines a different York canonical basis.

ing the existence of 4-scalar tensors.

74 Trends in Modern Cosmology

The momenta <sup>π</sup><sup>ð</sup>θ<sup>Þ</sup> <sup>r</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> and the 3-volume element <sup>φ</sup><sup>~</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det <sup>3</sup>grsðτ; <sup>σ</sup><sup>u</sup><sup>Þ</sup> <sup>p</sup> have to be found as solutions of the super-momentum (Hð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ ≈ 0) and super-Hamiltonian (i.e., the Lichnerowicz equation [55] <sup>H</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ ≈ 0) constraints, respectively.

Instead, the DO's (gauge invariant under the Hamiltonian gauge transformations generated by all the first class constraints; see Ref. [51]) of the gravitational field are not known15; they would be the two pairs of 4-scalar tidal variables in a Shanmugadhasan canonical basis adapted to all the 14 first class constraints.

The extra O(3,1) gauge freedom of the tetrads<sup>16</sup> is described by the gauge variables <sup>α</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ, <sup>ϕ</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup>. In the Schwinger time gauges, one imposes the gauge fixings <sup>ϕ</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ ≈ 0, <sup>α</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ ≈ 0 so that the time-like tetrad coincides with the unit normal to the 3-space and the space-like ones became tangent to it (namely the tetrads become adapted to the 3þ1 splitting).

<sup>14</sup>αðaÞ, <sup>ϕ</sup>ðaÞ, <sup>θ</sup><sup>i</sup> and <sup>3</sup><sup>K</sup> are the primary gauge variables, whereas <sup>n</sup> and <sup>n</sup>ða<sup>Þ</sup> are the secondary ones, which are determined as a consequence of the gauge fixing of the primary ones.

<sup>15</sup>Ra, Π<sup>a</sup> are not gauge invariant under the Hamiltonian gauge transformations generated by the super-Hamiltonian and super-momentum constraints.

<sup>16</sup>The gauge freedom for each observer to choose three gyroscopes as spatial axes and to choose the law for their transport along the time-like world line.

The gauge angles θ<sup>i</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ <sup>17</sup> describe the freedom in the choice of the axes for the 3-coordinates <sup>σ</sup><sup>r</sup> on each 3-space: their fixation implies the determination of the shift gauge variables <sup>n</sup>ðaÞ, namely the appearances of gravitomagnetism in the chosen 3-coordinate system [55]. The 3-orthogonal gauges are defined by the gauge fixings θ<sup>i</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ ≈ 0: in them, the 3-metric

$$\epsilon^3 \mathcal{g}\_{rs}(\tau, \sigma^\mu) = -\epsilon^4 \mathcal{g}\_{rs}(\tau, \sigma^\mu) = \delta\_{rs} \tilde{\phi} 2/3 \,\epsilon^2 \sum \overline{\pi}^{1,2} \mathcal{\eta}\_{\tilde{\pi}r} \mathcal{R}\_{\tilde{\pi}} \text{ is diagonal.}$$

Only one momentum is a gauge variable (a reflection of the Lorentz signature): the York time [56, 57], that is, the trace <sup>3</sup>Kðτ; <sup>σ</sup><sup>r</sup> Þ of the extrinsic curvature of the non-Euclidean 3-spaces as 3 submanifolds of space-time.<sup>18</sup> This inertial effect describes the GR version of the specialrelativistic gauge freedom in clock synchronization [47, 48] when one has to describe physics in non-inertial frames. Its fixation determines the lapse function.

The Dirac Hamiltonian is HD <sup>¼</sup> <sup>1</sup> <sup>c</sup> <sup>E</sup>^ADM <sup>þ</sup> ð d3 <sup>σ</sup> <sup>½</sup><sup>n</sup> <sup>H</sup> � <sup>n</sup>ða<sup>Þ</sup> <sup>H</sup>ð<sup>a</sup>Þ�ðτ; <sup>σ</sup><sup>u</sup>Þ þ <sup>ð</sup> d3 σ ½λ<sup>n</sup> πnþ λ<sup>n</sup>ða<sup>Þ</sup> π<sup>n</sup>ða<sup>Þ</sup> þ

λϕða<sup>Þ</sup> πϕða<sup>Þ</sup> <sup>þ</sup> λαða<sup>Þ</sup> <sup>π</sup><sup>ð</sup>α<sup>Þ</sup> <sup>ð</sup>a<sup>Þ</sup> �ðτ; <sup>σ</sup><sup>u</sup>Þ, where the weak ADM energy is an explicit function of all the variables, and the λ's are arbitrary Dirac multipliers (to be determined as a consequence of the gauge fixings).

In the family of Schwinger time gauges, the fixation of the primary gauge variables <sup>3</sup>Kðτ; <sup>σ</sup><sup>r</sup> Þ, θi <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ implies elliptic equations on the instantaneous 3-space Στ for the determination of the lapse and shift functions (the secondary gauge variables) and then of their Dirac multipliers λ's. Instead in the usually used harmonic gauges, one imposes the primary gauge fixing <sup>χ</sup><sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ ¼ ∂<sup>τ</sup> � <sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup>ðτ; <sup>σ</sup><sup>r</sup> ÞÞ <sup>3</sup>eðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>4</sup>g<sup>τ</sup><sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ � ≈ 0, whose stability in time, that is, <sup>∂</sup><sup>τ</sup> <sup>χ</sup><sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ ≈ 0, implies hyperbolic equations for the lapse and shift functions, namely the necessity of Cauchy conditions in the past for these metrology gauge variables.

This parametrization of canonical tetrad gravity clarifies the meaning of the metrology conventions.

The fixation of the York time determines the sequence of instantaneous non-Euclidean 3-spaces Στ of the 3þ1 splitting of space-time centered on an observer either on the Earth or on the Space Station19: all the clocks on each 3-space are synchronized with the atomic clock (τ is its proper time) of the observer at the intersection of the 3-space with the observer world line. This time metrology convention implies also the determination of the lapse function, which describes how the unit of time of the atomic clock changes when one goes from a 3-space to an infinitesimally near successive one. The metrology conventions on the choice of the three space coordinates σ<sup>r</sup> also imply the determination of the shift functions, which say in which point of the infinitesimally near next 3-space there are the same 3-coordinates of the chosen point on the original 3-space.

<sup>17</sup>They identify the direction cosines of the tangents to the three coordinate lines in each point of the 3-space Στ.

<sup>18</sup>It is absent in the Galilean space-time of Newtonian gravity with its absolute notions of time and Euclidean 3-space.

<sup>19</sup>The detailed structure of these non-Euclidean 3-spaces depends on the extrinsic curvature 3-tensor <sup>3</sup>Krs, which depends not only from all the gauge variables but also on the tidal variables, so that it is determined by the chosen solution of Einstein equations.

#### C. Einstein Hamilton Equations of Tetrad Gravity and their Linearization

In the York canonical basis, the Hamilton equations generated by the Dirac Hamiltonian HD <sup>¼</sup> <sup>E</sup>^ADM þ ðconstraints<sup>Þ</sup> are divided into four groups after the fixation of the O(3,1) gauge variables with the Schwinger time gauges:


The gauge angles θ<sup>i</sup>

76 Trends in Modern Cosmology

<sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> Þ

<sup>3</sup>grsðτ; <sup>σ</sup><sup>u</sup>޼�<sup>E</sup> <sup>4</sup>grsðτ; <sup>σ</sup><sup>u</sup>Þ ¼ <sup>δ</sup>rs <sup>φ</sup><sup>~</sup> <sup>2</sup>=<sup>3</sup> <sup>e</sup>

[56, 57], that is, the trace <sup>3</sup>Kðτ; <sup>σ</sup><sup>r</sup>

The Dirac Hamiltonian is HD <sup>¼</sup> <sup>1</sup>

λϕða<sup>Þ</sup> πϕða<sup>Þ</sup> <sup>þ</sup> λαða<sup>Þ</sup> <sup>π</sup><sup>ð</sup>α<sup>Þ</sup>

the gauge fixings).

Þ ¼ ∂<sup>τ</sup> �

point on the original 3-space.

Einstein equations.

<sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup>ðτ; <sup>σ</sup><sup>r</sup>

θi <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup>

<sup>χ</sup><sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup>

<sup>∂</sup><sup>τ</sup> <sup>χ</sup><sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup>

ventions.

3-orthogonal gauges are defined by the gauge fixings θ<sup>i</sup>

in non-inertial frames. Its fixation determines the lapse function.

ÞÞ <sup>3</sup>eðτ; <sup>σ</sup><sup>r</sup>

<sup>c</sup> <sup>E</sup>^ADM <sup>þ</sup>

ð d3

<sup>Þ</sup> <sup>4</sup>g<sup>τ</sup><sup>A</sup>ðτ; <sup>σ</sup><sup>r</sup>

necessity of Cauchy conditions in the past for these metrology gauge variables.

2 Xa 1;2

<sup>17</sup> describe the freedom in the choice of the axes for the 3-coordinates

γar Ra is diagonal.

<sup>σ</sup> <sup>½</sup><sup>n</sup> <sup>H</sup> � <sup>n</sup>ða<sup>Þ</sup> <sup>H</sup>ð<sup>a</sup>Þ�ðτ; <sup>σ</sup><sup>u</sup>Þ þ

<sup>ð</sup>a<sup>Þ</sup> �ðτ; <sup>σ</sup><sup>u</sup>Þ, where the weak ADM energy is an explicit function of all the

<sup>ð</sup>τ; <sup>σ</sup><sup>r</sup>

Þ of the extrinsic curvature of the non-Euclidean 3-spaces as 3-

ð d3

≈ 0, whose stability in time, that is,

Þ ≈ 0: in them, the 3-metric

σ ½λ<sup>n</sup> πnþ λ<sup>n</sup>ða<sup>Þ</sup> π<sup>n</sup>ða<sup>Þ</sup>

þ

Þ,

<sup>σ</sup><sup>r</sup> on each 3-space: their fixation implies the determination of the shift gauge variables <sup>n</sup>ðaÞ, namely the appearances of gravitomagnetism in the chosen 3-coordinate system [55]. The

Only one momentum is a gauge variable (a reflection of the Lorentz signature): the York time

submanifolds of space-time.<sup>18</sup> This inertial effect describes the GR version of the specialrelativistic gauge freedom in clock synchronization [47, 48] when one has to describe physics

variables, and the λ's are arbitrary Dirac multipliers (to be determined as a consequence of

In the family of Schwinger time gauges, the fixation of the primary gauge variables <sup>3</sup>Kðτ; <sup>σ</sup><sup>r</sup>

Þ �

This parametrization of canonical tetrad gravity clarifies the meaning of the metrology con-

The fixation of the York time determines the sequence of instantaneous non-Euclidean 3-spaces Στ of the 3þ1 splitting of space-time centered on an observer either on the Earth or on the Space Station19: all the clocks on each 3-space are synchronized with the atomic clock (τ is its proper time) of the observer at the intersection of the 3-space with the observer world line. This time metrology convention implies also the determination of the lapse function, which describes how the unit of time of the atomic clock changes when one goes from a 3-space to an infinitesimally near successive one. The metrology conventions on the choice of the three space coordinates σ<sup>r</sup> also imply the determination of the shift functions, which say in which point of the infinitesimally near next 3-space there are the same 3-coordinates of the chosen

17They identify the direction cosines of the tangents to the three coordinate lines in each point of the 3-space Στ. 18It is absent in the Galilean space-time of Newtonian gravity with its absolute notions of time and Euclidean 3-space. 19The detailed structure of these non-Euclidean 3-spaces depends on the extrinsic curvature 3-tensor <sup>3</sup>Krs, which depends not only from all the gauge variables but also on the tidal variables, so that it is determined by the chosen solution of

Þ implies elliptic equations on the instantaneous 3-space Στ for the determination of the lapse and shift functions (the secondary gauge variables) and then of their Dirac multipliers λ's. Instead in the usually used harmonic gauges, one imposes the primary gauge fixing

Þ ≈ 0, implies hyperbolic equations for the lapse and shift functions, namely the

The Hamilton equations become completely deterministic after a fixation of the gauge freedom. In the York canonical basis, it is convenient to use a family of non-harmonic 3-orthogonal Schwinger time gauges <sup>α</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>≈</sup> 0, <sup>ϕ</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>≈</sup> 0, <sup>θ</sup><sup>i</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>≈</sup> 0, <sup>3</sup>Kðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>≈</sup> <sup>F</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ parametrized by the numerical values <sup>F</sup>ðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> of the York time <sup>3</sup>Kðτ; <sup>σ</sup><sup>r</sup> Þ and having the 3-metric in the 3 spaces diagonal and well determined lapse and shift functions. In these gauges, given a solution of the super-momentum and super-Hamiltonian constraints, one can find a solution of Einstein's equations in radar 4-coordinates adapted to a time-like observer giving the Cauchy data on an initial 3-space only for the tidal variables. This happens in the associated 3þ1 splitting of space-time with dynamically selected instantaneous 3-spaces in accord with Ref. [1]. Then, one can pass to adapted world 4-coordinates (x<sup>μ</sup> <sup>¼</sup> <sup>z</sup><sup>μ</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ ¼ x μ <sup>o</sup> þ E μ <sup>A</sup> σ<sup>A</sup>) and can describe the solution in every 4-coordinate system by means of 4-diffeomorphisms.

In Ref. [43], this class of asymptotically Minkowskian space-times without super-translations is used to study the coupling of N charged scalar point particles (with the inertial and gravitational masses equal as required by the equivalence principle) plus the electromagnetic field to ADM tetrad gravity. The use of Grassmann-valued electric charges and the signs of the energy of the particles allows to regularize the self-energies. The theory can be reformulated in terms of transverse electromagnetic fields by using the non-covariant radiation gauge; this allows to extract the generalization of the Coulomb interaction among the particles in the Riemannian instantaneous 3-spaces of global non-inertial frames.

From the Hamilton equations in the York canonical basis [43], followed by a Hamiltonian Post-Minkowskian (HPM) linearization (disregarding terms of order <sup>O</sup>ðG<sup>2</sup> Þ in the Newton constant and using an ultra-violet cutoff for matter) with the asymptotic flat Minkowski 4-metric at spatial infinity as background, it has been possible to develop a theory of GW's with asymptotic background propagating in the non-Euclidean 3-spaces Στ of a family of non-harmonic 3 orthogonal Schwinger time gauges <sup>α</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>≈</sup> 0, <sup>ϕ</sup>ð<sup>a</sup>Þðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>≈</sup> 0, <sup>θ</sup><sup>i</sup> <sup>ð</sup>τ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>≈</sup> 0, <sup>3</sup>Kðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> <sup>≈</sup> <sup>F</sup>ðτ; <sup>σ</sup><sup>r</sup> Þ parametrized by the numerical values <sup>F</sup>ðτ; <sup>σ</sup><sup>r</sup> <sup>Þ</sup> of the York time <sup>3</sup>Kðτ; <sup>σ</sup><sup>r</sup> Þ and having the 3 metric in the 3-spaces diagonal and well-determined lapse and shift functions.

Since the celestial reference frame ICRS has diagonal 3-metric, our 3-orthogonal Schwinger time gauges are a good choice for celestial metrology.

The open problem is that the GCRS and BCRS conventions in the Solar System are using the special harmonic gauge of IAU [2, 3], in which the lapse function satisfies a hyperbolic equation like the tidal variables and needs initial data in the past, differently from what happens in the 3-orthogonal Schwinger time gauges. See Subsection 3.3 of the third paper in Ref. [43] for the comparison of the IAU harmonic gauge for BCRS with the 3-orthogonal gauges and Subsection 3.3 of the second paper in Ref. [43] for the equations identifying the 4 coordinate transformation from the 3-orthogonal gauges to the harmonic ones after the linearization, which have to be solved to get the reformulation of IAU conventions in our gauges.

#### 4. Dark matter as a relativistic inertial effect

The linearized HPM Hamilton equations for point particles of mass mi, <sup>i</sup> <sup>¼</sup> <sup>1</sup>; ::;N20, whose world lines x μ <sup>i</sup> <sup>ð</sup>τÞ ¼ <sup>z</sup><sup>μ</sup>ðτ;η<sup>r</sup> <sup>i</sup>ðτÞÞ are identified by radar 3-coordinates <sup>η</sup><sup>r</sup> <sup>i</sup>ðτÞ due to the 3þ1 splitting, and for the electromagnetic field coupled to tetrad gravity have been written explicitly in Refs. [43]: among the forces acting on matter, there are both the inertial potentials and the GW's.

In the third paper of Ref. [43], electro-magnetism is eliminated and there is a detailed studied of the HPM equations of motion of the particles. Then, the PN expansion of these regularized HPM equations of motion for the particles was studied, and it was shown that the particle 3 coordinates η<sup>r</sup> <sup>i</sup>ð<sup>τ</sup> <sup>¼</sup> ctÞ ¼ <sup>η</sup>~<sup>r</sup> <sup>i</sup>ðtÞ (coinciding with the Newtonian coordinates of the world lines at this level of approximation) satisfy the equation of motion

$$\begin{split} &\frac{d}{dt}\Big[m\_i\Big(1+\frac{1}{c}\frac{d}{dt}\,^3\tilde{\mathcal{K}}\_{(1)}(t,\overrightarrow{\tilde{\eta}\_i}^{\cdot}(t))\Big)\frac{d\,\tilde{\eta}\_i^{\prime}(t)}{dt}\Big] \doteq \\ & -G\frac{\partial}{\partial\,\tilde{\eta}\_i^{\prime}}\sum\_{j\neq i}\eta\_j\frac{m\_i m\_j}{|\overrightarrow{\tilde{\eta}\_i}^{\cdot}(t)-\overrightarrow{\tilde{\eta}\_j}^{\cdot}(t)|} + \mathcal{O}(\mathcal{G}^2). \end{split} \tag{6}$$

where at the lowest order, there is the standard Newton gravitational force

$$\overrightarrow{F}\_{i(\text{New} \text{to})}(t) = -m\_i \text{G} \frac{\partial}{\partial \, \tilde{\eta}\_i^{\tau}} \sum\_{j \neq i} \frac{m\_j}{|\overrightarrow{\tilde{\eta}\_i}^{\tau}(t) - \overrightarrow{\tilde{\eta}\_j}^{\tau}(t)|} = -m\_i \frac{\partial \Phi(t, \overrightarrow{\tilde{\eta}}\_i(t))}{\partial \, \tilde{\eta}\_i^{\tau}}.\tag{7}$$

Since Eqs. (4) imply

$$\kappa^4 \mathcal{g}\_{\rm rr}(\tau = ct, \sigma') - 1 = 2\,\text{n}(\tau = ct, \sigma') + \mathcal{O}(G^2) = 2\,\frac{\Phi(t, \sigma')}{c^2} - \frac{2}{c} \frac{\partial}{\partial t} \, ^3 \hat{\mathcal{K}}\_{(1)}(\tau = ct, \sigma') + \mathcal{O}(G^2),\tag{8}$$

<sup>20</sup>mi is both the inertial and the gravitational mass, since they coincide in Einstein GR due to the equivalence principle.

there is a 0.5 PN inertial effect (hidden in the lapse function) not existing in the Newton theory where the Euclidean 3-space is an absolute notion like the Newtonian time. It does not depend on the York time <sup>3</sup>Kð1<sup>Þ</sup> but on the non-local York time (<sup>Δ</sup> is the Laplacian associated to the asymptotic Minkowski 4-metric)

$$\,^3\tilde{\mathcal{K}}\_{(1)}(\tau,\sigma^r) = \left(\frac{1}{\Delta}\,^3\mathcal{K}\_{(1)}\right)(\tau,\sigma^r). \tag{9}$$

If we put <sup>3</sup>Kð1<sup>Þ</sup> <sup>¼</sup> 0, the standard results about binaries are reproduced.

Since the celestial reference frame ICRS has diagonal 3-metric, our 3-orthogonal Schwinger

The open problem is that the GCRS and BCRS conventions in the Solar System are using the special harmonic gauge of IAU [2, 3], in which the lapse function satisfies a hyperbolic equation like the tidal variables and needs initial data in the past, differently from what happens in the 3-orthogonal Schwinger time gauges. See Subsection 3.3 of the third paper in Ref. [43] for the comparison of the IAU harmonic gauge for BCRS with the 3-orthogonal gauges and Subsection 3.3 of the second paper in Ref. [43] for the equations identifying the 4 coordinate transformation from the 3-orthogonal gauges to the harmonic ones after the linearization, which have to be solved to get the reformulation of IAU conventions in our gauges.

The linearized HPM Hamilton equations for point particles of mass mi, <sup>i</sup> <sup>¼</sup> <sup>1</sup>; ::;N20, whose world

for the electromagnetic field coupled to tetrad gravity have been written explicitly in Refs. [43]:

In the third paper of Ref. [43], electro-magnetism is eliminated and there is a detailed studied of the HPM equations of motion of the particles. Then, the PN expansion of these regularized HPM equations of motion for the particles was studied, and it was shown that the particle 3-

� � d η~<sup>r</sup>

mi mj

ðtÞ � η~<sup>j</sup> ! ðtÞj

mj

ðtÞ � η~<sup>j</sup> ! ðtÞj

Þ ¼ <sup>2</sup> <sup>Φ</sup>ðt; <sup>σ</sup><sup>r</sup>

Þ <sup>c</sup><sup>2</sup> � <sup>2</sup> c ∂ ∂ t 3

� �

jη~i !

> jη~i !

<sup>20</sup>mi is both the inertial and the gravitational mass, since they coincide in Einstein GR due to the equivalence principle.

<sup>i</sup>ðtÞ (coinciding with the Newtonian coordinates of the world lines at

<sup>i</sup>ðtÞ dt

<sup>þ</sup> <sup>O</sup>ðG<sup>2</sup> Þ:

¼ �mi

¼ ∘

> ∂ Φðt; η~ ! i ðtÞÞ

∂ η~<sup>r</sup> i

<sup>K</sup><sup>~</sup> <sup>ð</sup><sup>1</sup>Þð<sup>τ</sup> <sup>¼</sup> ct; <sup>σ</sup><sup>r</sup>

<sup>i</sup>ðτÞ due to the 3þ1 splitting, and

ð6Þ

: ð7Þ

Þ þ <sup>O</sup>ðG<sup>2</sup>

Þ; ð8Þ

<sup>i</sup>ðτÞÞ are identified by radar 3-coordinates <sup>η</sup><sup>r</sup>

among the forces acting on matter, there are both the inertial potentials and the GW's.

time gauges are a good choice for celestial metrology.

4. Dark matter as a relativistic inertial effect

this level of approximation) satisfy the equation of motion d

dt mi <sup>1</sup> <sup>þ</sup>

�<sup>G</sup> <sup>∂</sup> ∂ η~<sup>r</sup> i X j=¼i ηj

<sup>i</sup>ðNewtonÞðt޼�mi <sup>G</sup> <sup>∂</sup>

Þ � <sup>1</sup> <sup>¼</sup> <sup>2</sup> <sup>n</sup>ð<sup>τ</sup> <sup>¼</sup> ct; <sup>σ</sup><sup>r</sup>

1 c d dt 3 <sup>K</sup><sup>~</sup> <sup>ð</sup><sup>1</sup>Þðt;η~<sup>i</sup> ! ðtÞÞ

where at the lowest order, there is the standard Newton gravitational force

∂ η~<sup>r</sup> i X j=¼i

Þ þ <sup>O</sup>ðG<sup>2</sup>

lines x μ

coordinates η<sup>r</sup>

<sup>i</sup> <sup>ð</sup>τÞ ¼ <sup>z</sup><sup>μ</sup>ðτ;η<sup>r</sup>

78 Trends in Modern Cosmology

<sup>i</sup>ð<sup>τ</sup> <sup>¼</sup> ctÞ ¼ <sup>η</sup>~<sup>r</sup>

F !

Since Eqs. (4) imply

<sup>g</sup>ττð<sup>τ</sup> <sup>¼</sup> ct; <sup>σ</sup><sup>r</sup>

E 4 The term in the non-local York time can be interpreted as the introduction of an effective (time-, velocity- and position-dependent) inertial mass term for the kinetic energy of each particle:

$$
\hat{\lambda}m\_i \mapsto m\_i \left(1 + \frac{1}{c} \left. \frac{d}{dt} \right.\\ \left. \left. \hat{\mathcal{K}}\_{(1)}(t, \overrightarrow{\tilde{\eta}}\_i(t)) \right. \right) \right. \tag{10}
$$

in each instantaneous 3-space. Since, in the Newton potential, there are the gravitational masses mi of the particles, the effect is due to a modification of the effective inertial mass in each non-Euclidean 3-space depending on its shape as a 3-submanifold of space-time. Therefore, we find it is the equality of the inertial and gravitational masses of Newtonian gravity to be violated in a gauge-dependent way in Einstein GR!

In the two-body case, one gets that for Keplerian circular orbits of radius r the modulus of the relative 3-velocity can be written in the form ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G ðmþΔ mðrÞÞ r q with <sup>Δ</sup> <sup>m</sup>ðr<sup>Þ</sup> function only of <sup>3</sup>K<sup>~</sup> <sup>ð</sup>1Þ.

The data on the rotation curves of spiral galaxies [14–17] imply that the relative 3-velocity goes to constant for large r instead of vanishing like in Kepler theory. As shown in Subsection 6.4 of the third paper in Ref. [43], this result can be simulated by fitting Δ mðrÞ (i.e., the non-local York time) to the experimental data with Δ mðrÞ interpreted as a dark matter halo around the galaxy.

Therefore, this dark matter can be explained as a relativistic inertial gauge effect consequence of the non-trivial shape of the non-Euclidean 3-space as a 3-submanifold of space-time. There is the concrete possibility to explain the rotation curves of galaxies [14–17] as a relativistic inertial effect inside Einstein GR (choice of a non-local York time compatible with observations) without modifications: (a) of Newton gravity like in MOND [9]; (b) of GR like in fðRÞ theories [10–12]; (c) of particle physics with the introduction of WIMPS [58].

A similar interpretation (see Subsections 6.2 and 6.3 of the third paper in Ref. [43]) can be given for the other two main signatures of the existence of dark matter in the observed masses of galaxies and clusters of galaxies, namely the mass determination with weak and strong gravitational lensing<sup>21</sup> [18–20] and the mass determination with the virial theorem [21–23].

<sup>21</sup>In the case of gravitational lensing Einstein's deflection angle, <sup>α</sup> <sup>¼</sup> <sup>4</sup> G M=c<sup>2</sup> <sup>ξ</sup> (<sup>ξ</sup> is the impact parameter of the ray of light deflected at the position of the mass M) has M ¼ Mbaryon þ MDM with the dark matter term given by G MDM ¼ �<sup>2</sup> <sup>c</sup><sup>2</sup> <sup>j</sup> <sup>σ</sup> ! <sup>j</sup> <sup>∂</sup><sup>τ</sup> <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðτ;σ<sup>u</sup>Þ.

Therefore, there is the possibility of describing part (or maybe all) dark matter as a relativistic inertial effect.

The quoted three main experimental signatures of dark matter are well-defined functional of the time and space derivatives of the non-local York time<sup>22</sup> the inertial gauge variable describing the general relativistic remnant of the gauge freedom in clock synchronization.

Since the time evolution of the signatures of dark matter is not known, at best from the data, we can extract information only on a mean value in time of the time- and space derivatives of the non-local York time. Since from Eq. (7), we see that �∂<sup>τ</sup> <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ is a modification of the Newton potential, we can assume that in Einstein GR the gauge variable non-local York time can be equated to the time-independent potentials <sup>V</sup><sup>ð</sup> <sup>σ</sup><sup>r</sup> Þ used either in phenomenology or in modified theories of GR to describe dark matter in either galaxies or cluster of galaxies. Then, we can make the ansatz <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðτ; <sup>σ</sup><sup>r</sup> ޼�<sup>τ</sup> <sup>V</sup><sup>ð</sup> <sup>σ</sup><sup>r</sup> <sup>Þ</sup> and find the local York time <sup>3</sup>Kð<sup>1</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ ¼ <sup>Δ</sup><sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðτ; <sup>σ</sup><sup>r</sup> Þ connected with the dark matter of the chosen either galaxy or cluster of galaxies.

Since there is no indication of dark matter in the voids existing among the clusters of galaxies, we can get an idea on the form of the local York time in the 3-space Στ (i.e., the whole 3 universe) by summing its value for all the known galaxies and clusters of galaxies. This would produce an indication of which could be a metrology convention on the inertial gauge variable describing the general relativistic gauge freedom in clock synchronization in the Einstein space-time outside the Solar System. One expects that, with this metrology convention, the resulting 3-spaces (each one with all the clocks synchronized) are nearly Euclidean except where there is need of introducing dark matter.

In Ref. [59], there is a first attempt to fit some data of dark matter by using a Yukawa-like ansatz on the non-local York time of a galaxy. In each galaxy, the Yukawa-like potential of fðRÞ theories [10–12] is put equal to a contribution to the extra potential depending on the non-local York time present in the lapse function appearing in Eq. (8); in this way, the good fits of the rotation curves of galaxies obtainable with fðRÞ theories can be reproduced inside Einstein's GR as an inertial gauge effect.

#### 5. Metrology against darkness

In conclusion, a suitable metrology convention on the inertial gauge variable York time could reduce or maybe eliminate the necessity of introducing dark matter in the classical universe and in its extension to classical cosmology after the recombination surface.

A needed natural proposal is now to define a Post-Minkowskian ICRS with non-Euclidean 3 spaces, whose intrinsic 3-curvature (due essentially to GW and matter) is small, in such a way

<sup>22</sup>∂<sup>τ</sup> <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðτ;σ<sup>r</sup> <sup>Þ</sup> in the gravitational lensing case, <sup>d</sup> dt <sup>3</sup>K<sup>~</sup> ð<sup>1</sup>Þðc t;η<sup>~</sup> ! i <sup>ð</sup>tÞÞ ¼ ð <sup>∂</sup> <sup>∂</sup> <sup>t</sup> <sup>þ</sup> \_ η~ ! i <sup>ð</sup>tÞ � <sup>∂</sup> ∂ η~ ! i <sup>Þ</sup> <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðc t; <sup>η</sup>~ðtÞÞ ! in the rotation curve case and <sup>d</sup><sup>2</sup> dt<sup>2</sup> <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðc t;η~<sup>i</sup> ! ðtÞÞ in the virial theorem case.

that the York time be (at least partially) fitted to the observational data implying the presence of dark matter. As a consequence, BCRS would be its quasi-Minkowskian approximation for the Solar System.23 Let us remark that the 3-spaces can be quasi-Euclidean (i.e. with a small internal 3-curvature tensor), as required by CMB data in the astrophysical context, even when their shape as 3-submanifolds of space-time is not trivial and is described by a not-small York time.

Therefore, there is the possibility of describing part (or maybe all) dark matter as a relativistic

The quoted three main experimental signatures of dark matter are well-defined functional of the time and space derivatives of the non-local York time<sup>22</sup> the inertial gauge variable describ-

Since the time evolution of the signatures of dark matter is not known, at best from the data, we can extract information only on a mean value in time of the time- and space derivatives of the

Newton potential, we can assume that in Einstein GR the gauge variable non-local York time

modified theories of GR to describe dark matter in either galaxies or cluster of galaxies. Then,

Since there is no indication of dark matter in the voids existing among the clusters of galaxies, we can get an idea on the form of the local York time in the 3-space Στ (i.e., the whole 3 universe) by summing its value for all the known galaxies and clusters of galaxies. This would produce an indication of which could be a metrology convention on the inertial gauge variable describing the general relativistic gauge freedom in clock synchronization in the Einstein space-time outside the Solar System. One expects that, with this metrology convention, the resulting 3-spaces (each one with all the clocks synchronized) are nearly Euclidean except

In Ref. [59], there is a first attempt to fit some data of dark matter by using a Yukawa-like ansatz on the non-local York time of a galaxy. In each galaxy, the Yukawa-like potential of fðRÞ theories [10–12] is put equal to a contribution to the extra potential depending on the non-local York time present in the lapse function appearing in Eq. (8); in this way, the good fits of the rotation curves of galaxies obtainable with fðRÞ theories can be reproduced inside Einstein's

In conclusion, a suitable metrology convention on the inertial gauge variable York time could reduce or maybe eliminate the necessity of introducing dark matter in the classical universe

A needed natural proposal is now to define a Post-Minkowskian ICRS with non-Euclidean 3 spaces, whose intrinsic 3-curvature (due essentially to GW and matter) is small, in such a way

> <sup>3</sup>K<sup>~</sup> ð<sup>1</sup>Þðc t;η<sup>~</sup> ! i <sup>ð</sup>tÞÞ ¼ ð <sup>∂</sup> <sup>∂</sup> <sup>t</sup> <sup>þ</sup> \_ η~ ! i <sup>ð</sup>tÞ � <sup>∂</sup> ∂ η~ ! i

and in its extension to classical cosmology after the recombination surface.

dt

Þ connected with the dark matter of the chosen either galaxy or cluster of galaxies.

޼�<sup>τ</sup> <sup>V</sup><sup>ð</sup> <sup>σ</sup><sup>r</sup>

Þ is a modification of the

Þ ¼

Þ used either in phenomenology or in

<sup>Þ</sup> and find the local York time <sup>3</sup>Kð<sup>1</sup>Þðτ; <sup>σ</sup><sup>r</sup>

<sup>Þ</sup> <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðc t; <sup>η</sup>~ðtÞÞ !

in the rotation curve

ing the general relativistic remnant of the gauge freedom in clock synchronization.

non-local York time. Since from Eq. (7), we see that �∂<sup>τ</sup> <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðτ; <sup>σ</sup><sup>r</sup>

can be equated to the time-independent potentials <sup>V</sup><sup>ð</sup> <sup>σ</sup><sup>r</sup>

we can make the ansatz <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðτ; <sup>σ</sup><sup>r</sup>

where there is need of introducing dark matter.

GR as an inertial gauge effect.

<sup>22</sup>∂<sup>τ</sup> <sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðτ;σ<sup>r</sup>

<sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðc t;η~<sup>i</sup> !

case and <sup>d</sup><sup>2</sup> dt<sup>2</sup>

5. Metrology against darkness

<sup>Þ</sup> in the gravitational lensing case, <sup>d</sup>

ðtÞÞ in the virial theorem case.

inertial effect.

80 Trends in Modern Cosmology

<sup>Δ</sup><sup>3</sup>K<sup>~</sup> <sup>ð</sup><sup>1</sup>Þðτ; <sup>σ</sup><sup>r</sup>

In this way, one would get a solution to the gauge problem for the PM space-times of GR: one chooses a reference system of 4-coordinates in a 3-orthogonal gauge selected by the observational conventions for matter. A PM definition of ICRS will be also useful for the ESA-GAIA mission [60] (cartography of the Milky Way) and for the possible anomalies (different from the already explained Pioneer one) inside the Solar System [5].

Regarding dark energy in cosmology [24–29], we can remark that in the FLRW cosmological solution, the Killing symmetries connected with homogeneity and isotropy imply (τ is the cosmic time, <sup>a</sup>ðτ<sup>Þ</sup> the scale factor) <sup>3</sup>Kðτ޼� <sup>a</sup>\_ðτ<sup>Þ</sup> <sup>a</sup>ðτ<sup>Þ</sup> ¼ �H, namely the York time is no more a gauge variable but coincides with the Hubble constant. However, in cosmological perturbation theory, we have <sup>3</sup><sup>K</sup> ¼ �Hþ<sup>3</sup>Kð1<sup>Þ</sup> at the first order with <sup>3</sup>Kð1<sup>Þ</sup> being again an inertial gauge variable.

Let us also remark that in Szekeres space-times [61–63], that is, in inhomogeneous space-times without Killing symmetries, the York time remains an inertial gauge variable.

As said in Section2, the red-shift and luminosity distance of SNIa is a signal of dark energy. In Section3 of the third paper in Ref. [43], there is the evaluation of the dependence on the nonlocal York time of the PM time-like geodesics, whereas in Section4 of that paper, there is evaluated the dependence on it of the PM null geodesics, of the PM red-shift, of the PM geodesics deviation equation, of the PM luminosity distance and of the Hubble old red-shift distance relation (becoming the Hubble law if cosmology in introduced in the description). Like in the case of dark matter, one has a dependence on the second derivatives ∂<sup>2</sup> <sup>τ</sup>, ∂<sup>τ</sup> ∂<sup>r</sup> and ∂<sup>r</sup> ∂<sup>s</sup> of the non-local York time now concentrated along the either time- or null geodesics. Therefore, also, this indication of dark energy is metrology dependent!

Let us also remark that in the back-reaction approach [64–69], in which to take into account the inhomogeneity of the observed universe when trying to get a cosmological description of it, one considers spatial mean values on large scales, dark energy in cosmology is a byproduct of the nonlinearities of GR. In this approach, one gets that the spatial average of the 4-scalar gauge variable York time gives the effective Hubble constant of this approach.

Finally, as shown in Eq. (10) of the last paper in Ref. [35], it can be shown that the York time is responsible for the negative terms in the kinetic energy term in the ADM energy, whose existence was known but whose explicit form could be given only in the York canonical basis. It is therefore possible that the connected Landau-Lifschitz energy-momentum pseudo-tensor [70] of GR could be reformulated as the energy-momentum tensor of a viscous pseudo-fluid,

<sup>23</sup>To test this possibility, one has to study the transition from harmonic gauges to 3-orthogonal ones in linearized Einstein GR.

which could have a negative pressure for certain choices of the York time like the dark energy fluid in FLWR cosmology.

In conclusion, the York time has a central position in all the cases where darkness is required to fit the data!

#### Author details

Luca Lusanna<sup>1</sup> \* and Ruggero Stanga<sup>2</sup>

\*Address all correspondence to: lusanna@fi.infn.it

1 Retired Research Director of Sezione INFN di Firenze, Sesto Fiorentino (FI), Italy

2 Dipartimento di Fisica e Astronomia, Firenze Univ., and Sezione INFN di Firenze, Sesto Fiorentino (FI), Italy

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which could have a negative pressure for certain choices of the York time like the dark energy

In conclusion, the York time has a central position in all the cases where darkness is required to

1 Retired Research Director of Sezione INFN di Firenze, Sesto Fiorentino (FI), Italy

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2 Dipartimento di Fisica e Astronomia, Firenze Univ., and Sezione INFN di Firenze, Sesto

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[2] M. Soffel, S.A. Klioner, G. Petit, P. Wolf, S.M. Kopeikin, P. Bretagnon, V.A. Brumberg, N. Capitaine, T. Damour, T. Fukushima, B. Guinot, T. Huang, L. Lindegren, C. Ma, K. Nordtvedt, J. Ries, P.K. Seidelmann, D. Vokroulicky', C. Will and Ch. Xu, The IAU 2000 Resolutions for Astrometry, Celestial Mechanics and Metrology in the Relativistic Framework:

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[5] S.G. Turishev, M. Shao and K.L. Nordtvedt, Experimental Design of the LATOR Mission, arXiv gr-qc/0410044 (2004). S.G. Turyshev and V.T. Toth, The Pioneer Anomaly Living Rev.

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\* and Ruggero Stanga<sup>2</sup>

\*Address all correspondence to: lusanna@fi.infn.it

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Author details

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Fiorentino (FI), Italy

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## **Superfluid Quantum Space and Evolution of the Universe**

Valeriy I. Sbitnev and Marco Fedi

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68113

#### Abstract

We assume that dark energy and dark matter filling up the whole cosmic space behave as a special superfluid, here named "superfluid quantum space." We analyze the relationship between intrinsic pressure of SQS (dark energy's repulsive force) and gravity, described as an inflow of dark energy into massive particles, causing a negative pressure gradient around massive bodies. Since no superfluid has exact zero viscosity, we analyze the consequences of SQS's viscosity on light propagation, and we show that a static Universe could be possible, by solving a modified Navier-Stokes equation. Indeed, Hubble's law may actually refer to tired light, though described as energy loss due to SQS's nonzero viscosity instead of Compton scattering, bypassing known historical problems concerning tired light. We see that SQS's viscosity may also account for the Pioneer anomaly. Our evaluation gives a magnitude of the anomalous acceleration aP <sup>=</sup> <sup>H</sup>Λ<sup>c</sup> <sup>=</sup> 8.785<sup>10</sup><sup>10</sup> ms<sup>2</sup> . Here, H<sup>Λ</sup> is the Hubble parameter loaded by the cosmological constant Λ. Furthermore, the vorticity equation stemming from the modified Navier-Stokes equation gives a solution for flat profile of the orbital speed of spiral galaxies and discloses what one might call a breathing of galaxies due to energy exchange between the galactic vortex and dark energy.

Keywords: gravity, dark energy, Hubble's law, tired light, Pioneer anomaly, flat profile

#### 1. Introduction

A recent view of the evolution of the Universe suggests that it pre-existed the Big Bang. What we now observe seems, however, to be the result of such event. The Universe apparently continues to expand at an accelerated pace, as evidenced by the Doppler redshift of light coming from distant sources. To explain this accelerated expansion, scientists resort to dark energy. In addition, it turns out that spiral galaxies demonstrate a flat profile of orbital speeds.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Dark matter is used to explain this riddle [1]. Current evaluations of the presence of dark energy and dark matter in the cosmos say that the former is of about 69.1% and the latter about 25.9%. In total, they are about 95% of the whole energy matter in the Universe. The residue of 5% corresponds to baryon matter, which is the constituent material of all observed galaxies, stars, planets, etc. At present, space as a mere container of matter is therefore being revised. It is not an empty vessel: On the contrary, it may act as a quantum superfluid, named by us "superfluid quantum space" (SQS) [2]. It consists of dark energy and dark matter whose hydrodynamics generates perennially fluctuating particle-antiparticle pairs, which annihilate and newly arise, forming a dark fluid whose features are similar to a Bose-Einstein condensate [3–7]. Within this concept, the repulsive action of dark energy may be simply explained as the internal pressure of the SQS. It should be noted that there are scientists [8–12] who do not agree with a concept of Universe based on Big Bang, inflation and Doppler redshift. They explain its evolution without calling into play any "Deus ex machina" as cosmic inflation. On the contrary, they believe that light loses energy as a function of the traveled distance. We assert that this happens because of a nonzero viscosity of the SQS, in perfect agreement with the empirical Hubble's law. This could be interpreted as a revised phenomenon of tired light, different from that proposed in 1929 by F. Zwicky. In effect, while Zwicky's hypothesis based on light scattering [13] may be disproved, for example, by the absent blurring of distant cosmic objects, tired light due to SQS's viscosity is a more robust concept, which seems not to conflict with the current observations. In addition, while a viscosity-related tired light would let us observe a Doppler-alike redshift, pressure phenomena of opposite sign, that is, repulsion caused by SQS's internal pressure and gravity as an inflow of dark energy into massive particles [14], could balance and permit a not expanding Universe.

It is interesting to note the critical opinion of a greatest theorist of our time, of Roger Penrose. In his recently published book "Fashion, Faith, and Fantasy in the New Physics of the Universe" [15], he argues that most of the current imaginary ideas about the origins of the Universe could be not true. We agree with Penrose, being unsatisfied with the current mainstream. In this key, we speculate and analyze a different framework.

At the beginning, in Section 2, we present as much detail as possible on our idea of space as superfluid quantum space, including a short historical overview about the concept of ether, vacuum, and physical space. In Section 3, we introduce a general relativistic hydrodynamic equation, and we analyze the corresponding equation in non-relativistic limit, as a modified Navier-Stokes equation. Here, we discuss the issue of tired light, and we evaluate the Pioneer anomaly according to the nonzero viscosity of SQS. Section 4 deals with solutions of the vorticity equation derived from the modified Navier-Stokes equation. We obtain exact formulas for the flat profile of orbital speeds of spiral galaxies. Section 5 gives concluding remarks and a look on the overall issue of a superfluid Universe.

#### 2. Space, vacuum and ether: toward a Superfluid Quantum Space

The issue concerning the concepts of space, time, motion and the existence, or not, of a real vacuum has accompanied the human knowledge all along [16]. The most distinct form of representation about space and time has developed in the form of two dialectically opposite ideas, later known as the conceptions of Democritus-Newton and Aristotle-Leibniz. According to Democritus everything is formed of "atoms," each of them is considered indivisible. Between atoms, we have empty space. Philosophical views of Sir Isaac Newton were focused on the idea that all material bodies move in Absolute Space and Absolute Time. Such a philosophy is extremely convenient in the analysis of motion based on Newton's mechanics [17]. Huygens championed a different concept, according to which, the whole space is filled with a special substance, the ether [18]. In his view, each point in space was a virtual source of light waves. This implied the homogeneity of space, a feature which is important also in modern quantum field theory (QFT), where wave functions propagate along all available paths.

Dark matter is used to explain this riddle [1]. Current evaluations of the presence of dark energy and dark matter in the cosmos say that the former is of about 69.1% and the latter about 25.9%. In total, they are about 95% of the whole energy matter in the Universe. The residue of 5% corresponds to baryon matter, which is the constituent material of all observed galaxies, stars, planets, etc. At present, space as a mere container of matter is therefore being revised. It is not an empty vessel: On the contrary, it may act as a quantum superfluid, named by us "superfluid quantum space" (SQS) [2]. It consists of dark energy and dark matter whose hydrodynamics generates perennially fluctuating particle-antiparticle pairs, which annihilate and newly arise, forming a dark fluid whose features are similar to a Bose-Einstein condensate [3–7]. Within this concept, the repulsive action of dark energy may be simply explained as the internal pressure of the SQS. It should be noted that there are scientists [8–12] who do not agree with a concept of Universe based on Big Bang, inflation and Doppler redshift. They explain its evolution without calling into play any "Deus ex machina" as cosmic inflation. On the contrary, they believe that light loses energy as a function of the traveled distance. We assert that this happens because of a nonzero viscosity of the SQS, in perfect agreement with the empirical Hubble's law. This could be interpreted as a revised phenomenon of tired light, different from that proposed in 1929 by F. Zwicky. In effect, while Zwicky's hypothesis based on light scattering [13] may be disproved, for example, by the absent blurring of distant cosmic objects, tired light due to SQS's viscosity is a more robust concept, which seems not to conflict with the current observations. In addition, while a viscosity-related tired light would let us observe a Doppler-alike redshift, pressure phenomena of opposite sign, that is, repulsion caused by SQS's internal pressure and gravity as an inflow of dark energy into massive

It is interesting to note the critical opinion of a greatest theorist of our time, of Roger Penrose. In his recently published book "Fashion, Faith, and Fantasy in the New Physics of the Universe" [15], he argues that most of the current imaginary ideas about the origins of the Universe could be not true. We agree with Penrose, being unsatisfied with the current main-

At the beginning, in Section 2, we present as much detail as possible on our idea of space as superfluid quantum space, including a short historical overview about the concept of ether, vacuum, and physical space. In Section 3, we introduce a general relativistic hydrodynamic equation, and we analyze the corresponding equation in non-relativistic limit, as a modified Navier-Stokes equation. Here, we discuss the issue of tired light, and we evaluate the Pioneer anomaly according to the nonzero viscosity of SQS. Section 4 deals with solutions of the vorticity equation derived from the modified Navier-Stokes equation. We obtain exact formulas for the flat profile of orbital speeds of spiral galaxies. Section 5 gives concluding remarks

2. Space, vacuum and ether: toward a Superfluid Quantum Space

The issue concerning the concepts of space, time, motion and the existence, or not, of a real vacuum has accompanied the human knowledge all along [16]. The most distinct form of

particles [14], could balance and permit a not expanding Universe.

90 Trends in Modern Cosmology

stream. In this key, we speculate and analyze a different framework.

and a look on the overall issue of a superfluid Universe.

Exactly QFT has triggered the current concept of a not inert vacuum, seen as the scene of continuous, frantic physical events. What John Wheeler named quantum foam [19]. A sea of particle-antiparticle pairs which arise and annihilate according to Heisenberg's principle of uncertainty, in a vacuum where energy can't be always and surely zero. These pairs perform an endless dance by infinitely arising and annihilating. In Dirac's opinion, the new theory of electrodynamics, which implies a vacuum filled with virtual particles, forces us to take into account the existence of an ether. In 1951, he stated [20]: "If one examines the question in the light of present-day knowledge, one finds that the aether is no longer ruled out by relativity, and good reasons can now be advanced for postulating an aether." His new ether model was based on a stochastic covariant distribution of subquantum motion, which generates a vacuum dominated by fluctuations and randomness.

De Broglie stated that: "any particle, even isolated, has to be imagined as in continuous energetic contact with a hidden medium" [21]. The hydrodynamics of this medium could explain the outcome of the double slit experiment using electron beams, where the leptons interfere as waves, probably driven through the aether by pressure waves, generated by their motion, exactly as pressure waves forming the same patterns are involved in the case of sound propagating through a double slit. De Broglie-Bohm's pilot-waves could be then explained as aether waves, which guide the electrons and show analogies with Faraday waves guiding a bouncing droplet along a surface of silicon oil [22]. Petroni and Vigier stated that: "one can deduce the De Broglie waves as real collective Markov processes on the top of Dirac's aether" [23].

Robert Betts Laughlin, Nobel Laureate for the fractional quantum Hall effect, in his work [24], writes: "Studies with large particle accelerators have now led us to understand that space is more like a piece of window glass than ideal Newtonian emptiness. It is filled with "stuff" that is normally transparent but can be made visible by hitting it sufficiently hard to knock out a part. The modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic ether. But we do not call it this because it is taboo." Laughlin also tells us that this false vacuum can be treated with the laws of fluid dynamics: "About the time relativity was becoming accepted, studies of radioactivity began showing that the empty space had spectroscopic structure similar to that of ordinary quantum solids and fluids."

In summary, it seems that any phenomenon occurring in quantum mechanics needs to interact with the vacuum, which consequently possesses a quantum physical structure, rather than being real empty (zero-energy) space. Thus, a single unbound particle is always and anyway connected to its environment. We believe that this fact might also facilitate the explanation to quantum entanglement, in which quantum information would be transmitted from a particle to the other through, and thanks to, the quantum structure of space. Petroni and Vigier debate that: "the quantum potential associated with this ether's modification, by the presence of EPR photon pairs, yields a relativistic causal action at a distance which interprets the superluminal correlations recently established by Aspect et al." [23].

In our opinion, this is also the case of gravitational waves, for which the asserted space-time deformation could be actually interpreted as a negative pressure wave traveling through a superfluid quantum space (SQS) from the source up to a measuring point due to a mechanism that we call superfluid quantum gravity (SQG) [14], a quantum fluid dynamic explanation of gravity. In a few words, gravitational waves could be a hydrodynamic phenomenon in a SQS instead of a deformation of space. After all, it is unlikely that a deformation occurs in a non-solid substance. If space is not solid, we can then only observe fluid dynamic events, which can, indeed, fully replace and better justify any effect of SR and GR [14]. SQS also shares interesting analogies with Higgs field, being an ubiquitous fundamental scalar field with non-zero viscosity, which gives mass to particles. In our case, thanks to quantum fluid dynamic perturbations of the field, with formation of superfluid quantum vortices, akin to what happens in superfluid He-4.

This suggests to even reconsider the pre-existence of a quantum space (as quantized dark energy) even before the Big Bang. By assuming that what we know to be the ubiquitous dark energy is a quantum superfluid, it could exactly correspond to our idea of SQS in a state of rest. Since dark energy still pervades the cosmos, corresponding to 69.1% of its energy, thinking that the Big Bang has rather been a perturbation event occurred in a previously quiet sea of dark energy, seems to be reasonable. From then on, cascade perturbations at Planck scale would have generated any existing particle as superfluid vortices or as pulses. Since no fluid or superfluid has real zero-viscosity, vortex-particles could attract the surrounding quanta, causing gravity as a fluid dynamic phenomenon. If the attracted space's quanta were packed and re-emitted as virtual photons, stable particles could exist, and the link gravity-electromagnetism would be clear [2]. In such a view, quantum gravity is an apparent force which does not accelerate bodies by directly acting on them thanks to gravitons, they are rather dragged by the superfluid quantum space in which they are immersed that flows toward the site where greater absorption is exerted, i.e. toward the greater mass of a gravitational system, according to Newton's law of universal gravitation. Cahill came to a similar conclusion in 2003, describing gravity as an inflow of quantum foam [25], though we consider more likely absorption of quantized dark energy.

Compared to QFT's quantum vacuum, SQS would be at the lowest level (we believe that it is the very fundamental scalar field in nature) made up of dark energy's quanta, whose hydrodynamic perturbation produces the continuous fluctuations which allow the formation and annihilation of particle-antiparticle pairs. In addition, Bohm and Vigier, moving from Dirac's ether model, introduced in 1954 the idea of a sub-quantum medium, a hidden medium which all particles of the microphysical level constantly interact with [26]. The surface level of SQS, that is, the currently defined quantum foam or quantum vacuum, has to possess superfluid features as well and may act as a special Bose-Einstein condensate. The historical problem of vacuum's infinite energy is solved by the infinite extent of the SQS.

As far as the Michelson-Morley test run in 1887 is concerned, we could wonder whether light interacts with the SQS, if it really exists, that is, if light interacts with dark energy. That test hypothesized a static ether and took into account Earth's motion. However, if we change the premise, by supposing that the Earth absorbs the ether, since massive particles absorb dark energy, we deal with a radial ether wind, independent of Earth's motion through the space, an ether wind which transports any object pointing toward the center of the Earth. In the hypothesis of fluid quantum gravity, this vertical ether wind exactly corresponds to the gravitational field [14]. This view would explain all the relativistic effects due to curved space-time, for example, the gravitational lensing and the Lense-Thirring precession. The correspondence between ether wind and gravitational field seems to be confirmed in a test run in 2009 by Martin Grusenick, who used a vertically placed Michelson's interferometer [27]. Maxwell's idea of an electromagnetic ether should be then revisited since, if a SQS exists, light could be a mechanical wave which propagates through an ether and its speed would merely correspond to the speed of sound through that specific fluid medium (i.e. of a pulse through dark energy) analogously to the case of sound through the air and for any other mechanical wave. In the case of light, this pulse would spin1 and its velocity would arise from SQS's parameters such as density and compressibility [2, 14]. In short, a photon would be a spinning phonon through superfluid dark energy, whose mechanical interaction with dark energy's quanta would excite them, producing the photon's electromagnetic field. By starting from the formula which indicates the speed of a mechanical wave through a fluid, <sup>a</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffi <sup>K</sup>=<sup>ρ</sup> <sup>p</sup> , in which <sup>K</sup> <sup>¼</sup> VdP=dV is the bulk modulus, calculated by dividing the pressure increment, dP, by relative increment of the volume, dV/V, and ρ is the mass density and by putting <sup>β</sup><sup>S</sup> <sup>¼</sup> <sup>1</sup>=<sup>K</sup> as isentropic compressibility, we have <sup>a</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffiffiffiffiffi <sup>β</sup>S<sup>ρ</sup> <sup>p</sup> . If we consider <sup>β</sup><sup>S</sup> <sup>¼</sup> <sup>β</sup><sup>0</sup> as SQS'<sup>s</sup> compressibility and ρ<sup>0</sup> as its mass density, we get

$$a = \frac{1}{\sqrt{\beta\_0 \rho\_0}},\tag{1}$$

expressing the speed of a photon as a phonon through the SQS, mathematically analogous to

$$\mathcal{L} = \frac{1}{\sqrt{\varepsilon\_0 \mu\_0}},\tag{2}$$

as resulting from Maxwell's equations. The nonzero viscosity of the superfluid medium (SQS) would compel light to undergo redshift over very large distances: the more distant a galaxy the more stronger the observed redshift. This is fully compatible with Hubble's law, letting us doubt that an accelerated expansion of the Universe is really occurring.

#### 3. Hydrodynamics of SQS

being real empty (zero-energy) space. Thus, a single unbound particle is always and anyway connected to its environment. We believe that this fact might also facilitate the explanation to quantum entanglement, in which quantum information would be transmitted from a particle to the other through, and thanks to, the quantum structure of space. Petroni and Vigier debate that: "the quantum potential associated with this ether's modification, by the presence of EPR photon pairs, yields a relativistic causal action at a distance which interprets the superluminal

In our opinion, this is also the case of gravitational waves, for which the asserted space-time deformation could be actually interpreted as a negative pressure wave traveling through a superfluid quantum space (SQS) from the source up to a measuring point due to a mechanism that we call superfluid quantum gravity (SQG) [14], a quantum fluid dynamic explanation of gravity. In a few words, gravitational waves could be a hydrodynamic phenomenon in a SQS instead of a deformation of space. After all, it is unlikely that a deformation occurs in a non-solid substance. If space is not solid, we can then only observe fluid dynamic events, which can, indeed, fully replace and better justify any effect of SR and GR [14]. SQS also shares interesting analogies with Higgs field, being an ubiquitous fundamental scalar field with non-zero viscosity, which gives mass to particles. In our case, thanks to quantum fluid dynamic perturbations of the field,

with formation of superfluid quantum vortices, akin to what happens in superfluid He-4.

This suggests to even reconsider the pre-existence of a quantum space (as quantized dark energy) even before the Big Bang. By assuming that what we know to be the ubiquitous dark energy is a quantum superfluid, it could exactly correspond to our idea of SQS in a state of rest. Since dark energy still pervades the cosmos, corresponding to 69.1% of its energy, thinking that the Big Bang has rather been a perturbation event occurred in a previously quiet sea of dark energy, seems to be reasonable. From then on, cascade perturbations at Planck scale would have generated any existing particle as superfluid vortices or as pulses. Since no fluid or superfluid has real zero-viscosity, vortex-particles could attract the surrounding quanta, causing gravity as a fluid dynamic phenomenon. If the attracted space's quanta were packed and re-emitted as virtual photons, stable particles could exist, and the link gravity-electromagnetism would be clear [2]. In such a view, quantum gravity is an apparent force which does not accelerate bodies by directly acting on them thanks to gravitons, they are rather dragged by the superfluid quantum space in which they are immersed that flows toward the site where greater absorption is exerted, i.e. toward the greater mass of a gravitational system, according to Newton's law of universal gravitation. Cahill came to a similar conclusion in 2003, describing gravity as an inflow of quantum foam [25], though we consider more likely absorption of quantized dark energy.

Compared to QFT's quantum vacuum, SQS would be at the lowest level (we believe that it is the very fundamental scalar field in nature) made up of dark energy's quanta, whose hydrodynamic perturbation produces the continuous fluctuations which allow the formation and annihilation of particle-antiparticle pairs. In addition, Bohm and Vigier, moving from Dirac's ether model, introduced in 1954 the idea of a sub-quantum medium, a hidden medium which all particles of the microphysical level constantly interact with [26]. The surface level of SQS, that is, the currently defined quantum foam or quantum vacuum, has to possess superfluid features as well and may act as a special Bose-Einstein condensate. The historical problem of

vacuum's infinite energy is solved by the infinite extent of the SQS.

correlations recently established by Aspect et al." [23].

92 Trends in Modern Cosmology

General relativity describes the Universe with a curved space-time metric due to presence of mass and energy. Observations show that the Universe, nevertheless, is flat at large distances

<sup>1</sup> Spinning sound waves have already been demonstrated [57], and thus, we can think of a photon as a spinning phonon through a superfluid medium.

and long times [28]. It means that the curvature tensor in the Einstein's field equations has to be omitted. In fact, as discussed below, we believe that what is supposed to be the curvature of space-time is rather a pressure force acting in a fluid, flat space, whose effect is compatible with that of general relativity's differential geometry. We come then to the general relativistic hydrodynamic equations [29, 30] containing the local conservation laws of the stress-energy tensor (the Bianchi identities) and of matter current density (the continuity equation) [5]:

$$
\partial\_{\mu} T^{\mu\nu} = 0,\tag{3}
$$

$$
\partial\_{\mu}l^{\mu} = 0.\tag{4}
$$

Here, ∂<sup>μ</sup> is the covariant derivative associated with the four-dimensional space-time metric ημν having the signature ð� þ þþÞ. The density current is given by J <sup>μ</sup> <sup>¼</sup> <sup>ρ</sup>mu<sup>μ</sup>, where <sup>u</sup><sup>μ</sup> is the fluid 4-velocity and ρ<sup>m</sup> is the rest-mass density in a locally inertial reference frame:

$$
\rho\_m = m\rho = m\frac{N\_B}{\Delta V} = \frac{M}{\Delta V}.\tag{5}
$$

Here, ρ is the density distribution of N particles within the unit volume ΔV, where each of them has mass m. So, M ¼ mNB represents the bulk mass of the fluid occupying this volume.

The stress-energy tensor, T<sup>μ</sup><sup>v</sup> , is expressed in units of pressure, whereas we need it in units of energy. Indeed, we further adopt the expression Tμν=ρ in order to have the possibility of getting the quantum potential Q ¼ PQ=ρ, where PQ is the internal quantum pressure arising in SQS under influence of the external environment. We consider an incompressible, viscous fluid along with the gravitational potential ϕ. So, Eq. (3) reads [5]:

$$
\partial\_{\mu} \left( \frac{T^{\mu \nu}}{\rho} \right) = \partial\_{\mu} \left( \frac{\varepsilon + p}{\rho} \gamma \mu^{\mu} \mu^{\nu} \right) + \partial^{\nu} Q - \partial^{\nu} \phi + \partial\_{\mu} \left( \mu(t) / \rho \right) \pi^{\mu \nu} = 0. \tag{6}
$$

Here, ε and p are functions per unit volume. Divided by ρ, the sum ε þ p has the dimension of energy. The term <sup>μ</sup>ðt<sup>Þ</sup> is the dynamic viscosity coefficient having the dimension of [N s/m<sup>2</sup> ]. Divided by ρm, this function represents the kinetic viscosity coefficient νðtÞ, having the dimension of [m<sup>2</sup> /s]. In our case, viscosity is a fluctuating-about-zero function of time. We suppose that its expectation vanishes in time but the variance is not zero. That is, we suppose the following average quantities:

$$
\langle \mu(\mathbf{t}) \rangle = \varepsilon \to 0\_{+},\\
\langle \mu(t)\mu(\mathbf{0}) \rangle > 0. \tag{7}
$$

Here, 0+ is an arbitrarily close-to-zero positive value, which describes the energy exchange with the zero-point energy of the SQS. The term πμν reads:

$$
\pi^{\mu\nu} = \mathfrak{c}(\partial^{\mu}u^{\nu} + \partial^{\nu}u^{\mu}) - \mathfrak{c}\frac{2}{3}\partial^{\mu}u\_{\mu}\eta^{\mu\nu}.\tag{8}
$$

In order to bring Eq. (6) to the relativistic Navier-Stokes equation, we shall repeat the computations of van Holten [31]. These calculations are reproduced also in Ref. [5].

We shall further consider only the non-relativistic limit, since the orbital speeds of galaxies and of many intergalactic bodies are predominantly much lower than the speed of light [32]. The factor γ in Eq. (6) is a sign of relativistic/non-relativistic limit. When it tends to infinity, we have the relativistic limit. In this case, also the mass m tends to infinity. On the other hand, when γ converges to unit, it denotes a non-relativistic limit. In this case, the mass m becomes the rest mass. For the sake of simplicity, we further take into account only shear viscosity. Given the quantum and granular nature of the SQS, a dilatant behavior under a linear, great increase of shear stress would be plausible and that would help to explain the upper limit to the acceleration of a body in the Universe [14]: The more acceleration is supplied the much more resistance is encountered, following Lorentz factor. In the non-relativistic limit, the viscosity term can be cut up to � μðtÞ=ρ � ∂i∂<sup>i</sup> cvi ! � μðtÞ=ρ � ∇<sup>2</sup> v !. As a result, we come to the following non-relativistic modified Navier-Stokes equation:

and long times [28]. It means that the curvature tensor in the Einstein's field equations has to be omitted. In fact, as discussed below, we believe that what is supposed to be the curvature of space-time is rather a pressure force acting in a fluid, flat space, whose effect is compatible with that of general relativity's differential geometry. We come then to the general relativistic hydrodynamic equations [29, 30] containing the local conservation laws of the stress-energy tensor (the Bianchi identities) and of matter current density (the continuity equation) [5]:

∂μJ

having the signature ð� þ þþÞ. The density current is given by J

fluid along with the gravitational potential ϕ. So, Eq. (3) reads [5]:

ε þ p <sup>ρ</sup> <sup>γ</sup>u<sup>μ</sup>u<sup>ν</sup> 

The stress-energy tensor, T<sup>μ</sup><sup>v</sup>

94 Trends in Modern Cosmology

∂μ

following average quantities:

sion of [m<sup>2</sup>

Tμν ρ 

¼ ∂<sup>μ</sup>

with the zero-point energy of the SQS. The term πμν reads:

<sup>π</sup>μν <sup>¼</sup> <sup>c</sup>ð∂<sup>μ</sup>u<sup>ν</sup> <sup>þ</sup> <sup>∂</sup><sup>ν</sup>

tations of van Holten [31]. These calculations are reproduced also in Ref. [5].

4-velocity and ρ<sup>m</sup> is the rest-mass density in a locally inertial reference frame:

ρ<sup>m</sup> ¼ mρ ¼ m

Here, ∂<sup>μ</sup> is the covariant derivative associated with the four-dimensional space-time metric ημν

Here, ρ is the density distribution of N particles within the unit volume ΔV, where each of them has mass m. So, M ¼ mNB represents the bulk mass of the fluid occupying this volume.

energy. Indeed, we further adopt the expression Tμν=ρ in order to have the possibility of getting the quantum potential Q ¼ PQ=ρ, where PQ is the internal quantum pressure arising in SQS under influence of the external environment. We consider an incompressible, viscous

<sup>þ</sup> <sup>∂</sup><sup>ν</sup>

Here, ε and p are functions per unit volume. Divided by ρ, the sum ε þ p has the dimension of energy. The term <sup>μ</sup>ðt<sup>Þ</sup> is the dynamic viscosity coefficient having the dimension of [N s/m<sup>2</sup>

Divided by ρm, this function represents the kinetic viscosity coefficient νðtÞ, having the dimen-

that its expectation vanishes in time but the variance is not zero. That is, we suppose the

Here, 0+ is an arbitrarily close-to-zero positive value, which describes the energy exchange

In order to bring Eq. (6) to the relativistic Navier-Stokes equation, we shall repeat the compu-

<sup>u</sup><sup>μ</sup>Þ � <sup>c</sup> 2 3

<sup>Q</sup> � <sup>∂</sup><sup>ν</sup>

/s]. In our case, viscosity is a fluctuating-about-zero function of time. We suppose

ϕ þ ∂<sup>μ</sup> μðtÞ=ρ 

〈μðtÞ〉 ¼ ε ! 0þ,〈μðtÞμð0Þ〉 > 0: ð7Þ

<sup>∂</sup><sup>μ</sup>uμημν: <sup>ð</sup>8<sup>Þ</sup>

NB <sup>Δ</sup><sup>V</sup> <sup>¼</sup> <sup>M</sup>

<sup>∂</sup>μTμν <sup>¼</sup> <sup>0</sup>, <sup>ð</sup>3<sup>Þ</sup>

, is expressed in units of pressure, whereas we need it in units of

<sup>μ</sup> <sup>¼</sup> <sup>0</sup>: <sup>ð</sup>4<sup>Þ</sup>

<sup>μ</sup> <sup>¼</sup> <sup>ρ</sup>mu<sup>μ</sup>, where <sup>u</sup><sup>μ</sup> is the fluid

πμν <sup>¼</sup> <sup>0</sup>: <sup>ð</sup>6<sup>Þ</sup>

].

<sup>Δ</sup><sup>V</sup> : <sup>ð</sup>5<sup>Þ</sup>

$$
\rho\_m \frac{d\overrightarrow{v}}{dt} = -\rho\_m \nabla \phi + \rho \nabla \Sigma Q + \mu(t) \nabla^2 \overrightarrow{v}.\tag{9}
$$

Here, ΣQ calculates the contributions of the quantum potential within SQS. The gravitational potential, ϕ, is a function coming from a continuous mass distribution ρ<sup>m</sup> [33]. Gravity by itself described as an inflow of SQS obeys Gauss's law for gravity (gravity as an incoming flux), which in differential form is ð∇ � g !޼�<sup>4</sup>πGρm. In our view, the classical gravitational potential ϕ ¼ GM=r of the absorbing body, associated to the radial field in each point, that is,g ! ¼ �∇ϕ, can be interpreted [14] as a quantum potential expressed as the ratio of the pressure PG of the incoming flux to mass density, ρm:

$$\varphi = -G \int\_{V} \frac{\rho\_m(\overrightarrow{r})}{r} dV = \frac{P\_G}{\rho\_m}. \tag{10}$$

Here, G is the Newtonian gravitational constant, and r is the distance from the volume element dV to a point in the field, and the integration is performed in the entire volume of the body, creating a field. We note that the gravitational constant in the rightmost part of Eq. (10) is absent, and we cleanly look at gravity as a quantum phenomenon driven by the ratio pressure/density. The continuous mass distribution ρmðr !Þ can be expressed using the Laplace operator, <sup>Δ</sup>:

$$
\rho\_m(\overrightarrow{r}) = \frac{1}{4\pi G} \Delta \varphi. \tag{11}
$$

We note that the mass density, in addition, submits to the continuity equation

$$\frac{\partial \rho\_m}{\partial t} + (\nabla \cdot \overrightarrow{v})\rho\_m = 0.\tag{12}$$

We can express from Eqs. (10) and (11) the gravitational pressure PG as a function of the gravitational potential ϕ:

$$P\_G = \frac{\varrho \cdot \Delta \varrho}{4\pi G} = \frac{\varrho^2}{4\pi G} ((\nabla \ln \varrho)^2 + \nabla^2 \ln \varrho). \tag{13}$$

We now see that in Eq. (9) two opposite quantum potentials act:

$$
\rho\_m \frac{d\overrightarrow{v}}{dt} = -\rho\_m \nabla \left(\frac{P\_G}{\rho\_m}\right) + \rho \nabla \Sigma \left(\frac{P\_Q}{\rho}\right) + \mu(t) \nabla^2 \overrightarrow{v} \tag{14}
$$

The Navier-Stokes equation is written above in the modified form [34]. The modification is due to (a) presence of the quantum potentials Q ¼ PQ=ρ and Q<sup>ϕ</sup> ¼ Mϕ ¼ MPG=ρ<sup>m</sup> ¼ PG=ρ (M is the mass of Universe, about 10<sup>53</sup> kg) and (b) existence of the dynamic viscosity coefficient μ(t) that fluctuates about zero. In other words, we accept that there is an energy exchange between baryon matter and the SQS. The pair of Eqs. (12) and (14) represents a full set of equations, sufficient for describing the motion of baryon matter through SQS in the non-relativistic limit of the Euclidean geometry.

Referring to Eq. (14), we believe that baryon matter is reciprocally attracted due to the gravitational quantum potential Q<sup>ϕ</sup> ¼ PG=ρ that we suppose justified by a hydrodynamic interaction occurring between SQS and baryon matter (attraction of dark energy's quanta toward vortexparticles, causing decrease of pressure and a consequent apparent attractive force [14]). On the contrary, the quantum potential Q ¼ PQ=ρ existing in SQS causes reciprocal repulsion of the baryon matter on large distance.

By omitting from consideration the viscous term in Eq. (14), we assume μ ¼ 0 and we obtain Newton's second law describing variations of the acceleration a ! <sup>¼</sup> d v!=dt under the action of the two opposite quantum forces described above, ∇Q<sup>ϕ</sup> and ∇ΣQ:

$$\begin{split} \rho\_m \frac{d\overrightarrow{\boldsymbol{\varphi}}}{dt} &= -\rho\_m \nabla \left(\frac{P\_G}{\rho\_m}\right) + \rho \nabla \Sigma \left(\frac{P\_Q}{\rho}\right) = -\rho \nabla Q\_\psi + \rho \nabla \Sigma Q \\ &= -\nabla P\_G + \nabla P\_Q + \left(P\_G \nabla \ln(\rho) - P\_Q \nabla \ln(\rho)\right) = f\_{QG} .\end{split} \tag{15}$$

We generally consider the volume of the whole visible Universe, ΔV ! V, so the rest mass density <sup>ρ</sup><sup>m</sup> <sup>¼</sup> mNB=<sup>V</sup> <sup>¼</sup> <sup>M</sup>=V, where <sup>M</sup> <sup>¼</sup> mNB is the total mass of the Universe (about 1053 kg). Here, f GQ is the force density. It arises from the superposition of two forces within the considered volume, which are expressed through the gradient of the gravitational potential and that of the intrinsic quantum potential of SQS. They are represented by a negative pressure gradient around baryonic bodies, ∇PG (Superfluid Quantum Gravity) [14], and the quantum pressure gradient acting on SQS, ∇PQ.

The acceleration, a ! <sup>¼</sup> d v!=dt, vanishes if both potentials, <sup>Q</sup><sup>ϕ</sup> and <sup>Q</sup>, are uniformly distributed across the space. The uniformity of the potentials can be justified according to the reports of the Planck Observatory [28]. So that E ¼ �Qϕ=m þ ΣQ=m reads

$$\mathcal{E} = G \int\_{V} \frac{\rho\_m(\overrightarrow{r'})}{r} dV + N\_D D^2 \left( \frac{\nabla^2 \rho\_m}{\rho\_m} - \frac{1}{2} \frac{(\nabla \rho\_m)^2}{\rho\_m^2} \right). \tag{16}$$

It should be constant at least within the visible Universe. The first term follows from Eq. (10) that is ϕ ¼ PG=ρ<sup>m</sup> ¼ Qϕ=M and the second term is the intrinsic quantum potential of SQS [5, 34] divided by mass. The integer multiplier ND is equal to the sum of all the quantum potentials, which arise from the contribution of all dark energy and dark matter in SQS. This value is calculated from the fact that about 95% of mass-energy in the Universe accounts for this dark fluid, respectively, 69.1% dark energy and 25.9% dark matter. So, from here, we find ND ¼ 95=ð100 � 95Þ � NB ¼ 19 � M=m.The number NB ¼ M=m follows from Eq. (5).

As for the mass density distribution ρ<sup>m</sup> under the integral, we permit the existence of a static spherically symmetric Gaussian density of baryon matter

$$\rho\_m(r) = \frac{M}{\left(\sigma\sqrt{2\pi}\right)^3} \exp\left\{-\frac{r^2}{2\sigma^2}\right\}.\tag{17}$$

By accepting this result, Eq. (16) gives the following solution:

PG <sup>¼</sup> <sup>ϕ</sup> � <sup>Δ</sup><sup>ϕ</sup>

dt ¼ �ρm<sup>∇</sup>

Newton's second law describing variations of the acceleration a

the two opposite quantum forces described above, ∇Q<sup>ϕ</sup> and ∇ΣQ:

PG ρm � �

¼ �∇PG þ ∇PQ þ

We now see that in Eq. (9) two opposite quantum potentials act:

ρm d v !

of the Euclidean geometry.

96 Trends in Modern Cosmology

baryon matter on large distance.

<sup>ρ</sup><sup>m</sup> d v ! dt ¼ �ρm∇

pressure gradient acting on SQS, ∇PQ.

! <sup>¼</sup> d v

Planck Observatory [28]. So that E ¼ �Qϕ=m þ ΣQ=m reads

V

ρmðr !Þ r

E ¼ G ð

The acceleration, a

<sup>4</sup>π<sup>G</sup> <sup>¼</sup> <sup>ϕ</sup><sup>2</sup>

PG ρm � �

<sup>4</sup>π<sup>G</sup> ðð∇lnϕ<sup>Þ</sup>

The Navier-Stokes equation is written above in the modified form [34]. The modification is due to (a) presence of the quantum potentials Q ¼ PQ=ρ and Q<sup>ϕ</sup> ¼ Mϕ ¼ MPG=ρ<sup>m</sup> ¼ PG=ρ (M is the mass of Universe, about 10<sup>53</sup> kg) and (b) existence of the dynamic viscosity coefficient μ(t) that fluctuates about zero. In other words, we accept that there is an energy exchange between baryon matter and the SQS. The pair of Eqs. (12) and (14) represents a full set of equations, sufficient for describing the motion of baryon matter through SQS in the non-relativistic limit

Referring to Eq. (14), we believe that baryon matter is reciprocally attracted due to the gravitational quantum potential Q<sup>ϕ</sup> ¼ PG=ρ that we suppose justified by a hydrodynamic interaction occurring between SQS and baryon matter (attraction of dark energy's quanta toward vortexparticles, causing decrease of pressure and a consequent apparent attractive force [14]). On the contrary, the quantum potential Q ¼ PQ=ρ existing in SQS causes reciprocal repulsion of the

By omitting from consideration the viscous term in Eq. (14), we assume μ ¼ 0 and we obtain

<sup>þ</sup> <sup>ρ</sup>∇<sup>Σ</sup> PQ ρ � �

We generally consider the volume of the whole visible Universe, ΔV ! V, so the rest mass density <sup>ρ</sup><sup>m</sup> <sup>¼</sup> mNB=<sup>V</sup> <sup>¼</sup> <sup>M</sup>=V, where <sup>M</sup> <sup>¼</sup> mNB is the total mass of the Universe (about 1053 kg). Here, f GQ is the force density. It arises from the superposition of two forces within the considered volume, which are expressed through the gradient of the gravitational potential and that of the intrinsic quantum potential of SQS. They are represented by a negative pressure gradient around baryonic bodies, ∇PG (Superfluid Quantum Gravity) [14], and the quantum

across the space. The uniformity of the potentials can be justified according to the reports of the

dV <sup>þ</sup> NDD<sup>2</sup> <sup>∇</sup><sup>2</sup>ρ<sup>m</sup>

ρm

�

<sup>þ</sup> <sup>ρ</sup>∇<sup>Σ</sup> PQ ρ � �

<sup>2</sup> <sup>þ</sup> <sup>∇</sup><sup>2</sup>

<sup>þ</sup> <sup>μ</sup>ðtÞ∇<sup>2</sup> <sup>v</sup>

! <sup>¼</sup> d v

�

¼ f QG:

¼ �ρ∇Q<sup>ϕ</sup> þ ρ∇ΣQ

PG∇lnðρÞ � PQ∇lnðρÞ

!=dt, vanishes if both potentials, Q<sup>ϕ</sup> and Q, are uniformly distributed

� 1 2

!

ð∇ρmÞ 2

> ρ2 m

lnϕÞ: ð13Þ

! <sup>ð</sup>14<sup>Þ</sup>

!=dt under the action of

: ð16Þ

ð15Þ

$$\varepsilon = \frac{GM}{r} \text{erf} \left\{ \frac{r}{\sigma \sqrt{2}} \right\} + N\_D D^2 \left( \frac{r^2}{2\sigma^4} - \frac{1}{\sigma^2} \right) \tag{18}$$

The expression of ε reduced to dimensionless form by multiplying by c �<sup>2</sup> (c is the speed of light) is shown as a function of r in Figure 1. We see that there is a flat potential plateau of baryon matter ranging in the radius of the visible Universe <sup>r</sup> <sup>&</sup>lt; <sup>σ</sup> <sup>≈</sup> <sup>4</sup>:<sup>5</sup> � 1026m <sup>≈</sup> <sup>14</sup>:6 Gpc. The negative pressure arising among the baryon bodies determines the attraction.

On the other hand, the repulsion is due to the quantum vacuum fluctuations in SQS. This repulsion is conditioned by the quantum potential Q represented by the second term In Eq. (18). In this case, the diffusion coefficient D reads

$$D = \frac{\hbar}{2m} \tag{19}$$

In the case of the proton mass, <sup>m</sup>, that is, about 1:<sup>67</sup> � <sup>10</sup>�27kg, we have <sup>D</sup> <sup>≈</sup> <sup>4</sup>:<sup>6</sup> � <sup>10</sup>�<sup>8</sup> <sup>m</sup><sup>2</sup> � <sup>s</sup>�1. The term NDD<sup>2</sup> in Eq. (18), however, reaches the enormous value of about 1065m4 � <sup>s</sup>�2.

Figure 1. Function ε reduced to dimensionless form by multiplying by c �<sup>2</sup> (c is the speed of light) as a function of r, where σ is the radius of the visible Universe. A flat plateau ranging from 0 to about 10 Gpc tells us that the expansion of the Universe is almost absent. Dotted curve outside the chart shows divergence due to the quadratic term r 2 /2σ<sup>4</sup> in the quantum potential.

Therefore, the quantum potential outside the visible horizon gives divergence because of the quadratic term r<sup>2</sup>=2σ<sup>4</sup> in its representation. In Figure 1, this divergence is shown by a dotted curve.

#### 3.1. Viscosity of SQS: tired-light and the Pioneer anomaly

Let us return to the relativistic hydrodynamic equation [5] by considering the Klein-Gordon equation, loaded by the viscosity term. The kinetic energy of a relativistic particle, in this case, can be written as follows:

$$E = E\_0 - 2m\nu(t)\frac{d\ln(\rho\_m)}{dt} = E\_0 - E\_0\frac{2}{c^2}\nu(t)\frac{d\ln(\rho\_m)}{dt}.\tag{20}$$

Here, <sup>ν</sup>ðtÞ ¼ <sup>μ</sup>ðtÞ=ρ<sup>m</sup> is the kinematic viscosity coefficient. Its dimension is m<sup>2</sup> � <sup>s</sup>�1. The second term here describes energy exchange with vacuum fluctuations during a particle's motion through SQS. Here, we took into account that <sup>E</sup><sup>0</sup> <sup>¼</sup> mc2. By adopting <sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>ℏ</sup>ω0, we can write a suitable wave function for a photon coming from a distant source:

$$\Psi(t) = \exp\left\{-i\omega\_0 t \left(1 - \frac{2}{t} \Big| \frac{\nu(\tau)}{c^2} \frac{d \ln(\rho\_m)}{dt} d\tau\right)\right\}.\tag{21}$$

We let the integral under the exponent be linked with the expanded Hubble parameter HΛ, as follows:

$$H\_{\Lambda} = \frac{2}{t^2} \left[ \frac{\nu(\tau)}{c^2} \frac{d \ln(\rho\_m)}{dt} d\tau = \frac{\dot{a}}{a} = \sqrt{\frac{8\pi G \rho\_m}{3} - k\frac{c^2}{a^2} + \Lambda} \frac{c^2}{3} \right. \tag{22}$$

The rightmost terms under root in (22) result from the first Friedmann equation. Here, Λ is the cosmological constant (which refers to dark energy, i.e., to the SQS itself, being Λ ¼ κρsqs, where <sup>κ</sup> ¼ ð8πGÞ=c<sup>2</sup> is Einstein's constant), <sup>a</sup> the dimensionless scale factor, and <sup>k</sup> its Gaussian curvature. We further consider the case of a flat Universe, k = 0:

$$H\_{\Lambda} = \sqrt{H\_0^2 + \frac{\Lambda c^2}{3}}, \qquad H\_0 = \sqrt{\frac{8\pi G \rho\_m}{3}}. \tag{23}$$

Being Λ omitted, the parameter H<sup>Λ</sup> degenerates to H0. We may evaluate H<sup>0</sup> at the known critical densityρ<sup>c</sup> <sup>¼</sup> <sup>ρ</sup><sup>m</sup> <sup>≈</sup> <sup>10</sup>�<sup>26</sup> kg∙m–<sup>3</sup> and knowing <sup>G</sup> [33]. We find <sup>H</sup><sup>0</sup> <sup>≈</sup> <sup>2</sup>:<sup>36</sup> � <sup>10</sup>�18s�<sup>1</sup> in SI unit, while, in units adopted in astrophysics, it is about 73 km � Mpc�<sup>1</sup> � <sup>s</sup>�1. <sup>H</sup><sup>0</sup> fits well within the confidence interval estimated by Friedmann and others in [35] (see Figure 2).

We know that, to justify a ratio of the actual density to the critical density corresponding to a flat Universe, that is, <sup>Ω</sup> <sup>¼</sup> <sup>ρ</sup>m=ρ<sup>c</sup> <sup>¼</sup> 1, we have to solve the flatness problem (j<sup>Ω</sup> � <sup>1</sup><sup>j</sup> <sup>&</sup>lt; <sup>10</sup>�<sup>62</sup> at the Planck era [36]). Its solution, as well known, is given by the theory of cosmic inflation, to

Therefore, the quantum potential outside the visible horizon gives divergence because of the quadratic term r<sup>2</sup>=2σ<sup>4</sup> in its representation. In Figure 1, this divergence is shown by a dotted

Let us return to the relativistic hydrodynamic equation [5] by considering the Klein-Gordon equation, loaded by the viscosity term. The kinetic energy of a relativistic particle, in this case,

dt <sup>¼</sup> <sup>E</sup><sup>0</sup> � <sup>E</sup><sup>0</sup>

t ðt

a ¼

, H<sup>0</sup> ¼

The rightmost terms under root in (22) result from the first Friedmann equation. Here, Λ is the cosmological constant (which refers to dark energy, i.e., to the SQS itself, being Λ ¼ κρsqs, where <sup>κ</sup> ¼ ð8πGÞ=c<sup>2</sup> is Einstein's constant), <sup>a</sup> the dimensionless scale factor, and <sup>k</sup> its Gaussian

Being Λ omitted, the parameter H<sup>Λ</sup> degenerates to H0. We may evaluate H<sup>0</sup> at the known critical densityρ<sup>c</sup> <sup>¼</sup> <sup>ρ</sup><sup>m</sup> <sup>≈</sup> <sup>10</sup>�<sup>26</sup> kg∙m–<sup>3</sup> and knowing <sup>G</sup> [33]. We find <sup>H</sup><sup>0</sup> <sup>≈</sup> <sup>2</sup>:<sup>36</sup> � <sup>10</sup>�18s�<sup>1</sup> in SI unit, while, in units adopted in astrophysics, it is about 73 km � Mpc�<sup>1</sup> � <sup>s</sup>�1. <sup>H</sup><sup>0</sup> fits well within

We know that, to justify a ratio of the actual density to the critical density corresponding to a flat Universe, that is, <sup>Ω</sup> <sup>¼</sup> <sup>ρ</sup>m=ρ<sup>c</sup> <sup>¼</sup> 1, we have to solve the flatness problem (j<sup>Ω</sup> � <sup>1</sup><sup>j</sup> <sup>&</sup>lt; <sup>10</sup>�<sup>62</sup> at the Planck era [36]). Its solution, as well known, is given by the theory of cosmic inflation, to

r

We let the integral under the exponent be linked with the expanded Hubble parameter HΛ, as

0

νðτÞ c2

dlnðρmÞ dt <sup>d</sup><sup>τ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8πGρ<sup>m</sup> 3

r

c2 <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>Λ</sup> <sup>c</sup><sup>2</sup> 3

8πGρ<sup>m</sup> <sup>3</sup> � <sup>k</sup>

Here, <sup>ν</sup>ðtÞ ¼ <sup>μ</sup>ðtÞ=ρ<sup>m</sup> is the kinematic viscosity coefficient. Its dimension is m<sup>2</sup> � <sup>s</sup>�1. The second term here describes energy exchange with vacuum fluctuations during a particle's motion through SQS. Here, we took into account that <sup>E</sup><sup>0</sup> <sup>¼</sup> mc2. By adopting <sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>ℏ</sup>ω0, we can write

2 <sup>c</sup><sup>2</sup> <sup>ν</sup>ðt<sup>Þ</sup> dlnðρmÞ

1 A

9 =

dt : <sup>ð</sup>20<sup>Þ</sup>

;: <sup>ð</sup>21<sup>Þ</sup>

: ð22Þ

: ð23Þ

dlnðρmÞ

0 @

3.1. Viscosity of SQS: tired-light and the Pioneer anomaly

E ¼ E<sup>0</sup> � 2mνðtÞ

<sup>H</sup><sup>Λ</sup> <sup>¼</sup> <sup>2</sup> t 2 ðt

0

νðτÞ c2

curvature. We further consider the case of a flat Universe, k = 0:

r

H<sup>Λ</sup> ¼

a suitable wave function for a photon coming from a distant source:

<sup>Ψ</sup>ðtÞ ¼ exp �iω0<sup>t</sup> <sup>1</sup> � <sup>2</sup>

dlnðρmÞ

dt <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>a</sup>\_

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2 <sup>0</sup> þ Λc<sup>2</sup> 3

the confidence interval estimated by Friedmann and others in [35] (see Figure 2).

8 < :

curve.

98 Trends in Modern Cosmology

follows:

can be written as follows:

Figure 2. A Hubble diagram of distance versus velocity for secondary distance indicators, calibrated to Cepheids. Figure is taken from Ref. [35] and modified by adding the parameter HΛ. Note that H0 belongs to the confidence interval, whereas H<sup>Λ</sup> lies beyond it.

which many of the scientific community resort also to solve the magnetic-monopole problem and the homogeneity problem. In accordance with what above, we believe that no accelerating Universe exists, and consequently, the Doppler effect does not influence the observed redshift. We identify the cause of the redshift in the phenomenon of tired light, which in our case is due to the weak viscosity of SQS that leads to Eq. (21). The frequency ω shifted with respect to the initial frequency ω<sup>0</sup> after the time t will be [37]:

$$
\omega = \omega\_0 e^{-H\_0 t} \tag{24}
$$

Cosmic inflation appears to us as a deus ex machina, which could actually hide the effect of viscosity on photons traveling through the SQS. As follows from Eq. (22), tired light occurs due to the existence of a tiny viscosity of SQS, in which photons are subject to by traveling through the cosmos. Fluctuations of the space-time metric (fluid dynamic fluctuations of dark energy, in our case) at the Planck scale [38] give a crucial contribution to the viscosity effect.

From Eq. (22), we can evaluate <sup>H</sup><sup>Λ</sup> <sup>≈</sup> <sup>2</sup>:<sup>93</sup> � <sup>10</sup>�18s�<sup>1</sup> (about 90 km � Mpc�<sup>1</sup> � <sup>s</sup>�1) at the adopted value of <sup>Λ</sup> <sup>¼</sup> <sup>10</sup>�<sup>52</sup>m�2. We observe that this parameter lies far outside the confidence interval, marked in Figure 2. It means that the parameter HΛ, most possibly, plays another role different from the Hubble constant H0. Let us compute the acceleration of an object, a ! <sup>¼</sup> d v!=dt traveling through the Universe. It can be found from Eq. (9), by setting ∇ðmϕ þ ΣQÞ ¼ 0:

$$
\overrightarrow{a} = \nu(t)\nabla^2 \overrightarrow{v} = -\nu(t)\nabla(d\ln(\rho\_m)/dt). \tag{25}
$$

The term ∇v ! ¼ �<sup>d</sup> lnðρmÞ=dt comes from the continuity Eq. (12). Now, by multiplying Eq. (22) by t <sup>2</sup>=2 and differentiating it with respect to t, we gain:

$$\frac{\nu(t)}{c^2} \frac{d \ln(\rho\_m)}{dt} = \frac{1}{2} \frac{d}{dt} t^2 \sqrt{\frac{8\pi G\rho\_m}{3} + \Lambda} \frac{c^2}{3} = tH\_\Lambda + \frac{t^2}{4H\_\Lambda} \left(\frac{8\pi G\rho\_m}{3}\right) \frac{d \ln(\rho\_m)}{dt}.\tag{26}$$

Then, by multiplying by c <sup>2</sup> and by applying the operator ∇, we get

$$a = -\frac{d}{d\ell}t c^2 H\_\Lambda - \frac{d}{d\ell} \frac{t^2 c^2}{4H\_\Lambda} (H\_0^2) \frac{d \ln(\rho\_m)}{dt}. \tag{27}$$

Here, the operator <sup>∇</sup> <sup>¼</sup> <sup>d</sup>=d<sup>ℓ</sup> calculates a gradient along the increment <sup>d</sup>ℓ. Let us suppose that dℓ=dt represents an updated rate for the cosmic microwave background (CMB) fluctuations at the frequency <sup>Ω</sup>CMB <sup>¼</sup> <sup>ω</sup>0, that is, the speed of light <sup>d</sup>ℓ=dt <sup>¼</sup> <sup>λ</sup>CMBΩCMB <sup>¼</sup> <sup>c</sup>. By substituting <sup>c</sup> into Eq. (27) instead of dℓ=dt, we get:

$$a = -H\_{\Lambda}c \left( 1 + \frac{H\_0^2}{H\_{\Lambda}^2} \cdot \left( \frac{1}{4} \frac{t}{\rho\_m} \frac{d\rho\_m}{dt} \right) \right). \tag{28}$$

We observe that the first term, <sup>H</sup>Λc, is equal to 8:<sup>785</sup> � <sup>10</sup>�10m � <sup>s</sup>�2. This indicates a good agreement with the acceleration aP ¼ ð8:<sup>74</sup> � <sup>1</sup>:33Þ � <sup>10</sup>�10m � <sup>s</sup>�<sup>2</sup> which became known as the Pioneer anomaly [39–41]. From this, we have the fact that the term in second brackets vanishes, namely, dρm=dt ¼ 0, as follows from the continuity Eq. (12). It means that ρ<sup>m</sup> ¼ ρ<sup>c</sup> ¼ const: Eq. (28) suggests that the Pioneer anomaly is due to the presence of non-zero energy density (dark energy) of the vacuum, as reflected in the cosmological constant Λ ≈ 10�52m�<sup>2</sup> in metric units.

The Hubble parameters, H<sup>0</sup> and HΛ, concern different manifestations of SQS. The first parameter is due to presence of the tiny non-zero, positive viscosity of the SQS, whereby light undergoes loss of energy (redshift) proportional to the traveled distance. The Hubble diagram in Figure 2 shows in fact the relationship with distance expressed by our hypothesis, in which the role of the recessional velocity in causing the cosmological redshift has to be however substituted by that of energy dissipation. Since in our analysis (see Ch.2 and [14]) photons are phonons through the SQS, i.e. waves carrying a momentum, in agreement with the concept of photon, they lose energy while traveling huge distances, as no superfluid has perfectly zero viscosity.

As for the parameter HΛ, it results from the trigonometric shear of the Hubble parameter H<sup>0</sup> by adding the contribution of the cosmological constant Λ, see Eq. (23). The calculated acceleration (28), excellently close to the acceleration aP of the Pioneer apparatus, is due to the contribution of the SQS (i.e., of dark energy and of its hydrodynamic perturbations) expressed by the cosmological constant [42, 43]. Indeed, we know that the relationship between Λ and the energy density of free space is <sup>Λ</sup> <sup>¼</sup> κρ0, where <sup>κ</sup> <sup>¼</sup> <sup>8</sup>πGc�<sup>2</sup> is Einstein's constant and <sup>ρ</sup><sup>0</sup> is vacuum's (i.e., SQS's) energy density. Thus, by considering the SQS as a ubiquitous sea of quantized, perturbed dark energy, partially condensed as dark matter (25.9%), we see that its mass is mdarkenergy <sup>¼</sup> <sup>ρ</sup>0ðtÞVðtÞ ¼ Edarkenergyc�<sup>2</sup> <sup>¼</sup> <sup>β</sup>0ρ0Edarkenergy. Where V(t) is the volume of the Universe at an instant t (even in an expanding/shrinking Universe, we have ρ0V ¼ const, so at any moment dark energy is neither created nor annihilated) and <sup>c</sup>�<sup>2</sup> <sup>¼</sup> <sup>β</sup>0ρ<sup>0</sup> from Eq. (1), where β<sup>0</sup> and ρ<sup>0</sup> are physical parameters of dark energy, responsible for a small non-zero, positive viscosity of free space and, consequently, for the investigated anomalous deceleration.

marked in Figure 2. It means that the parameter HΛ, most possibly, plays another role different

! ¼ �<sup>d</sup> lnðρmÞ=dt comes from the continuity Eq. (12). Now, by multiplying Eq. (22)

¼ tH<sup>Λ</sup> þ

t 2 4H<sup>Λ</sup>

dlnðρmÞ

dρ<sup>m</sup> dt

! <sup>¼</sup> d v

dt : <sup>ð</sup>27<sup>Þ</sup>

: ð28Þ

! ¼ �νðtÞ∇ðdlnðρmÞ=dtÞ: <sup>ð</sup>25<sup>Þ</sup>

8πGρ<sup>m</sup> 3

� � <sup>d</sup>lnðρm<sup>Þ</sup>

!=dt traveling

dt : <sup>ð</sup>26<sup>Þ</sup>

from the Hubble constant H0. Let us compute the acceleration of an object, a

! <sup>¼</sup> <sup>ν</sup>ðtÞ∇<sup>2</sup> <sup>v</sup>

r

<sup>a</sup> ¼ � <sup>d</sup> dℓ tc2

a ¼ �HΛc 1 þ

a

<sup>2</sup>=2 and differentiating it with respect to t, we gain:

The term ∇v

100 Trends in Modern Cosmology

νðtÞ c2

Then, by multiplying by c

dlnðρmÞ dt <sup>¼</sup> <sup>1</sup> 2 d dt t 2

into Eq. (27) instead of dℓ=dt, we get:

by t

through the Universe. It can be found from Eq. (9), by setting ∇ðmϕ þ ΣQÞ ¼ 0:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8πGρ<sup>m</sup>

<sup>3</sup> <sup>þ</sup> <sup>Λ</sup> <sup>c</sup><sup>2</sup> 3

> <sup>H</sup><sup>Λ</sup> � <sup>d</sup> dℓ t <sup>2</sup>c<sup>2</sup> 4H<sup>Λ</sup> <sup>ð</sup>H<sup>2</sup> 0Þ

<sup>2</sup> and by applying the operator ∇, we get

Here, the operator <sup>∇</sup> <sup>¼</sup> <sup>d</sup>=d<sup>ℓ</sup> calculates a gradient along the increment <sup>d</sup>ℓ. Let us suppose that dℓ=dt represents an updated rate for the cosmic microwave background (CMB) fluctuations at the frequency <sup>Ω</sup>CMB <sup>¼</sup> <sup>ω</sup>0, that is, the speed of light <sup>d</sup>ℓ=dt <sup>¼</sup> <sup>λ</sup>CMBΩCMB <sup>¼</sup> <sup>c</sup>. By substituting <sup>c</sup>

> H2 0 H2 Λ � 1 4 t ρm

We observe that the first term, <sup>H</sup>Λc, is equal to 8:<sup>785</sup> � <sup>10</sup>�10m � <sup>s</sup>�2. This indicates a good agreement with the acceleration aP ¼ ð8:<sup>74</sup> � <sup>1</sup>:33Þ � <sup>10</sup>�10m � <sup>s</sup>�<sup>2</sup> which became known as the Pioneer anomaly [39–41]. From this, we have the fact that the term in second brackets vanishes, namely, dρm=dt ¼ 0, as follows from the continuity Eq. (12). It means that ρ<sup>m</sup> ¼ ρ<sup>c</sup> ¼ const: Eq. (28) suggests that the Pioneer anomaly is due to the presence of non-zero energy density (dark energy)

The Hubble parameters, H<sup>0</sup> and HΛ, concern different manifestations of SQS. The first parameter is due to presence of the tiny non-zero, positive viscosity of the SQS, whereby light undergoes loss of energy (redshift) proportional to the traveled distance. The Hubble diagram in Figure 2 shows in fact the relationship with distance expressed by our hypothesis, in which the role of the recessional velocity in causing the cosmological redshift has to be however substituted by that of energy dissipation. Since in our analysis (see Ch.2 and [14]) photons are phonons through the SQS, i.e. waves carrying a momentum, in agreement with the concept of photon, they lose

As for the parameter HΛ, it results from the trigonometric shear of the Hubble parameter H<sup>0</sup> by adding the contribution of the cosmological constant Λ, see Eq. (23). The calculated acceleration (28), excellently close to the acceleration aP of the Pioneer apparatus, is due to the contribution of the SQS (i.e., of dark energy and of its hydrodynamic perturbations) expressed by the

of the vacuum, as reflected in the cosmological constant Λ ≈ 10�52m�<sup>2</sup> in metric units.

energy while traveling huge distances, as no superfluid has perfectly zero viscosity.

! � �

#### 4. Vorticity equation and solutions for orbital speeds of spiral galaxies

The modified Navier-Stokes Eqs. (9) and (14) can exhibit a manifestation of long-lived vortices in SQS. The last term in this equation is dissipative due to the presence of a weak viscosity of the medium fluctuating about zero. If the viscosity coefficient μ is a fluctuating function of time, we can assume (see Eq. (7)) that (a) time-averaged, the viscosity coefficient vanishes; (b) its variance is not zero. Therefore, the viscosity coefficient is a function fluctuating about zero. We suppose that such fluctuations determine energy exchange between the existing baryon matter and the zero-point fluctuations of the superfluid physical vacuum [44].

Note first that the total derivative of v ! with respect to t in the Navier-Stokes equation (9), rewritten through the partial derivatives reads:

$$\frac{d\overrightarrow{\boldsymbol{v}}}{dt} = \frac{\partial\overrightarrow{\boldsymbol{v}}}{\partial t} + (\overrightarrow{\boldsymbol{v}} \cdot \nabla)\overrightarrow{\boldsymbol{v}}.\tag{29}$$

Let us apply now the curl operator to the Navier-Stokes equation. We come to the equation for vorticity ω ! ¼ ½<sup>∇</sup> � <sup>v</sup> !� [45]:

$$\frac{\partial \overrightarrow{\boldsymbol{\omega}}}{\partial t} + (\overrightarrow{\boldsymbol{v}} \cdot \nabla) \overrightarrow{\boldsymbol{\omega}} = \nu(t) \nabla^2 \overrightarrow{\boldsymbol{\omega}}.\tag{30}$$

Here, νðtÞ ¼ μðtÞ=ρ<sup>m</sup> is the kinematic viscosity coefficient. The vector ω ! is directed along the rotation axis. In order to simplify this task, let us move to the coordinate system in which the rotation occurs in the plane (x, y) and the z-axis lies along the vorticity, Figure 3.

Under this transformation, the vorticity equation takes a particularly simple form:

$$\frac{\partial \omega}{\partial t} = \nu(t) \left( \frac{\partial^2 \omega}{\partial r^2} + \frac{1}{r} \frac{\partial \omega}{\partial r} \right). \tag{31}$$

A general solution of this equation has the following view [5, 46]:

$$\omega(r,t) = \frac{\Gamma}{4\Sigma(\nu, t, \sigma)} \exp\left\{-\frac{r^2}{4\Sigma(\nu, t, \sigma)}\right\},\tag{32}$$

Figure 3. A simulation of a rotating spiral galaxy: the orbital velocity v ! lies in the plane (x, y). The vorticity vector ω ! is oriented perpendicular to this plane.

$$w(r,t) = \frac{1}{r} \left[ \omega(r',t)r'dr' = \frac{\Gamma}{2r} \left(1 - \exp\left\{-\frac{r^2}{4\Sigma(\mathbf{v},t,\sigma)}\right\} \right) . \tag{33}$$

The first function is vorticity; the second is the orbital speed. We do not mark the arrows above the letters ν and ω since the orbital velocity lies in the (x, y) plane and vorticity lies on z-axis. The denominator Σðν, t, σÞ in these formulas reads:

$$\Sigma(\nu, t, \sigma) = \int\_0^t \nu(\tau)d\tau + \sigma^2. \tag{34}$$

Here, σ is an arbitrary constant such that the denominator is always positive.

Taking into account 〈νðtÞ〉 ¼ 0þ, see Eq. (7), we can see that the integral in Eq. (34) tends to zero and solutions of (32) and (33) in the limit of t ! ∞ reduce to

$$
\omega\_{\rm Gxc}(r,t) = \frac{\Gamma}{4\sigma^2} \exp\left\{-\frac{r^2}{4\sigma^2}\right\},\tag{35}
$$

$$v\_{\rm Grcc}(r,t) = \frac{\Gamma}{2r} \left(1 - \exp\left\{-\frac{r^2}{4\sigma^2}\right\}\right). \tag{36}$$

That is, vorticity and angular speed are permanent in time. Here, the circulation Γ and the average radius σ are initially existing. The extra parameter σ comes from the Gaussian coherent vortex cloud [47]. The subscript Gcvc indicates the Gaussian coherent vortex cloud. The vortex cloud represents localized concentration of vorticity energy with a lifetime tending to infinity [48]. It does not significantly interact with any form of matter and exists in itself as long as possible.

#### 4.1. Flat profile of the orbital speed (evaluations)

Solution (36) gives no flat profile. The function monotonically decreases with r ! ∞. This velocity is shown by curve 1 in Figure 4.

Let us begin to search for a solution of Eq. (31) by perturbing the solution (32) through a function gðrÞ not equal to one, that is, ωðr, tÞ � gðrÞ. When we substitute this function into Eq. (31), we get:

$$g\frac{\partial\omega}{\partial t} = g\nu(t)\left(\frac{\partial^2\omega}{\partial r^2} + \frac{1}{r}\frac{\partial\omega}{\partial r}\right) + \nu(t)\left(\omega\frac{\partial^2g}{\partial r^2} + \left(2\frac{\partial\omega}{\partial r} + \frac{1}{r}\omega\right)\frac{\partial g}{\partial r}\right). \tag{37}$$

Here, we obtain two independent differential equations. The first one is for the function ωðr, tÞ. We return to the same solution (32). While the second equation for the function gðrÞ becomes equal to zero. In this case, we introduce an auxiliary function ϕ ¼ ∂g=∂r for which this equation takes the form:

$$
\frac{\partial \phi}{\partial r} + \left(\frac{2}{\omega} \frac{\partial \omega}{\partial r} + \frac{1}{r}\right) \phi = \frac{\partial \phi}{\partial r} + \left(-\frac{r}{\Sigma} + \frac{1}{r}\right) \phi = 0. \tag{38}
$$

In the second part instead of ð2=ωÞ � ∂ω=∂r, we put its solution –r=Σ. For the sake of simplicity, we write Σ instead of Σðν, t, σÞ. The function gðrÞ stemming from the solution of Eq. (38) reads:

$$g(r) = \int\_0^r \frac{1}{\xi} \exp\left\{\frac{\xi^2}{2\Sigma}\right\} d\xi. \tag{39}$$

Next, we find the orbital speed

<sup>v</sup>ðr, tÞ ¼ <sup>1</sup> r ðr

oriented perpendicular to this plane.

102 Trends in Modern Cosmology

The denominator Σðν, t, σÞ in these formulas reads:

0 ωðr 0 , tÞr 0 dr<sup>0</sup> <sup>¼</sup> <sup>Γ</sup> 2r

Figure 3. A simulation of a rotating spiral galaxy: the orbital velocity v

and solutions of (32) and (33) in the limit of t ! ∞ reduce to

<sup>1</sup> � exp � <sup>r</sup><sup>2</sup>

The first function is vorticity; the second is the orbital speed. We do not mark the arrows above the letters ν and ω since the orbital velocity lies in the (x, y) plane and vorticity lies on z-axis.

ðt

<sup>ν</sup>ðτÞd<sup>τ</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

<sup>4</sup>σ<sup>2</sup> exp � <sup>r</sup><sup>2</sup>

<sup>1</sup> � exp � <sup>r</sup><sup>2</sup>

� � � �

4σ<sup>2</sup> � �

4σ<sup>2</sup>

0

Taking into account 〈νðtÞ〉 ¼ 0þ, see Eq. (7), we can see that the integral in Eq. (34) tends to zero

Σðν, t, σÞ ¼

Here, σ is an arbitrary constant such that the denominator is always positive.

<sup>ω</sup>Gcvcðr, tÞ ¼ <sup>Γ</sup>

2r

vGcvcðr, tÞ ¼ <sup>Γ</sup>

� � � �

4Σðν, t, σÞ

: ð33Þ

! is

: ð34Þ

! lies in the plane (x, y). The vorticity vector ω

, ð35Þ

: ð36Þ

Figure 4. (1, 2, 3) are monotonically decreasing profiles with r ! ∞; 4 is an example of flat profile for large r, but when r tends to infinity, the curve vanishes.

$$
\sigma'(r,t) = \frac{1}{r} \left[ \omega(r',t) \cdot \mathbf{g}(r') \cdot r' dr'. \tag{40}
$$

This speed is shown as curve 2 in Figure 4. We have to observe, however, that the weight function gðrÞ can be approximated by the continued fraction [49]

$$E\_1(x) = \frac{e^{-x}}{x + \cfrac{1}{1 + \cfrac{1}{x + \cfrac{2}{1 + \cfrac{2}{1 + \cfrac{3}{1 + \cfrac{4}{x + \cfrac{4}{1 + \cfrac{4}{...}}}}}}}}} \tag{41}$$

In this way, we find an approximated function of the orbital speed

$$\sigma''(r,t) = \frac{1}{r} \left[ \omega(r',t) \cdot E\_1 \left(\frac{r'}{2\Sigma}\right) \cdot r' dr'. \tag{42}$$

This speed is shown as curve 3 in Figure 4. One can see that all curves, 1, 2, and 3, accurate to the scaling, show good accordance with each other.

As for the curve 4 in this figure, it follows from the function

$$\upsilon'''(r,t) = \frac{\Gamma}{r} \sum\_{n=1}^{N} \frac{1}{4\sigma\_n^2} \left[ E\_1\left(\left(\frac{r'}{2\sigma\_n}\right)^2\right) \cdot r' dr'.\tag{43}$$

This function is drawn with linear growth of σ<sup>n</sup> when n goes on, σ<sup>n</sup> ¼ 10 � n. For n large enough, it shows a good outcome for the flat profile at r ≫ 1.

#### 4.2. Flat profile of the orbital speed (a general case)

A rich gallery of galactic rotation curves showing output on a flat profile is presented in [50].

These flat profiles of the orbital speeds are here rearranged, and they are shown in Figure 5. The curves draw approximations of these profiles.

Equation (43) gives a hint for getting flat profiles of orbital speeds, which are typical for spiral galaxies. In this section, we present formulas which show the formation of flat profiles evolving in time. First, we hypothesize that the above-mentioned Gaussian coherent vortex clouds (see Eqs. (35) and (36)) have a long-term memory, and they can therefore manifest themselves as dark matter. As shown in Figure 4 by the curve 4, the clouds can support flat profiles for a long time through their superposition (see Eq. (43)). For the sake of demonstration, let us set [46]

Superfluid Quantum Space and Evolution of the Universe http://dx.doi.org/10.5772/68113 105

v0

104 Trends in Modern Cosmology

function gðrÞ can be approximated by the continued fraction [49]

In this way, we find an approximated function of the orbital speed

the scaling, show good accordance with each other.

As for the curve 4 in this figure, it follows from the function

enough, it shows a good outcome for the flat profile at r ≫ 1.

4.2. Flat profile of the orbital speed (a general case)

The curves draw approximations of these profiles.

<sup>v</sup>000ðr, tÞ ¼ <sup>Γ</sup>

r X N

n¼1

1 4σ<sup>2</sup> n ðr

0 E1

This function is drawn with linear growth of σ<sup>n</sup> when n goes on, σ<sup>n</sup> ¼ 10 � n. For n large

A rich gallery of galactic rotation curves showing output on a flat profile is presented in [50]. These flat profiles of the orbital speeds are here rearranged, and they are shown in Figure 5.

Equation (43) gives a hint for getting flat profiles of orbital speeds, which are typical for spiral galaxies. In this section, we present formulas which show the formation of flat profiles evolving in time. First, we hypothesize that the above-mentioned Gaussian coherent vortex clouds (see Eqs. (35) and (36)) have a long-term memory, and they can therefore manifest themselves as dark matter. As shown in Figure 4 by the curve 4, the clouds can support flat profiles for a long time

through their superposition (see Eq. (43)). For the sake of demonstration, let us set [46]

<sup>v</sup>00ðr, tÞ ¼ <sup>1</sup>

r ðr

0 ωðr 0 , tÞ � E<sup>1</sup>

This speed is shown as curve 3 in Figure 4. One can see that all curves, 1, 2, and 3, accurate to

<sup>ð</sup>r, tÞ ¼ <sup>1</sup> r ðr

<sup>E</sup>1ðxÞ ¼ <sup>e</sup>�<sup>x</sup>

1 þ

x þ

x þ

0 ωðr 0 , tÞ � gðr 0 Þ � r 0 dr<sup>0</sup>

This speed is shown as curve 2 in Figure 4. We have to observe, however, that the weight

1

1 þ 2 x þ

1

1þ

r0 2Σ � �

r0 2σ<sup>n</sup> � �2 !

2

3

x þ

� r 0 dr<sup>0</sup>

> � r 0 dr<sup>0</sup>

3

1 þ 4 ⋯

4

:

: ð40Þ

: ð42Þ

: ð43Þ

ð41Þ

Figure 5. Families F#, U#, and ESO# of the flat profiles of the orbital speeds taken from [50]. The curves approximate these profiles marked by the black points.

$$\nu(t) = \nu \frac{e^{i\Omega t} + e^{-i\Omega t}}{2} = \nu \cos\left(\Omega t\right). \tag{44}$$

In this case <sup>Σ</sup>ðν, t, <sup>σ</sup>Þ¼ðν=Ω<sup>Þ</sup> sin <sup>ð</sup>ΩtÞ þ <sup>σ</sup><sup>2</sup> ¼ ðν=Ω<sup>Þ</sup> sin ðΩtÞ þ ζ , <sup>σ</sup><sup>2</sup> ¼ ðν=ΩÞ � <sup>ζ</sup> and <sup>ζ</sup> <sup>&</sup>gt; 1. Note first that the Gaussian coherent vortex clouds show self-similarity.

From this view, let us assume that the fluctuating viscosity reads as follows [5]:

$$\nu\_n(t) = \frac{c^2}{\Omega\_n} \cos \left(\Omega\_n t\right). \tag{45}$$

The kinetic viscosity coefficient c<sup>2</sup>=Ω<sup>n</sup> has dimension [m<sup>2</sup> ∙s -1]. Here, c is the speed of light, and Ω<sup>n</sup> is the angular frequency of a vacuum oscillation. That is, there is a periodic exchange of energy <sup>ℏ</sup>Ω<sup>n</sup> <sup>¼</sup> <sup>n</sup>�<sup>1</sup> with SQS. The energy tends to zero at <sup>n</sup> going to infinity, whereas the viscosity goes to infinity. It can mean that SQS acquires a high viscosity on very small frequencies of the vortex energy exchange. Note, however, that 〈νnðtÞ〉 ¼ 0<sup>þ</sup> for any n.

Let us compute the flat profile for the orbital speed of a spiral galaxy guided by the rule formulated above. To see its formation, we perform computations of sets collected from modes (45), n ¼ 1, 2, ⋯, N. Let us substitute the expression (45) into the integral (34). After computing it, we get the following view of the denominator Σðν, t, σÞ:

$$
\Sigma\_n(t) = \frac{c^2}{\Omega\_n^2} \left( \sin \left( \Omega\_n t \right) + \zeta \right). \tag{46}
$$

Since <sup>Ω</sup><sup>n</sup> <sup>¼</sup> <sup>n</sup>�1, the coefficient <sup>Σ</sup><sup>n</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup>=Ω<sup>2</sup> <sup>n</sup> tends to infinity as Ω<sup>n</sup> goes to zero while n increases. From here, it follows that the expression 1 � exp { � <sup>r</sup><sup>2</sup>=4Σn} in Eq. (33) reaches 1 the more slowly with increasing r, the larger is Σn. As a result, the set of coefficients Σ<sup>n</sup> for n ¼ 1, 2, ⋯ can give output to the flat profile of the orbital speed. Let us, therefore, compute a sum of possible orbital speeds of galaxies for all entangled modes for which baryon matter is allowed to exchange energy with the SQS. Our statistical sum reads as follows:

$$V(r,t) = \frac{\Gamma}{2r} \sum\_{n=1}^{N} \left(1 - \exp\left\{-\frac{r^2}{4\Sigma\_n(t)}\right\}\right). \tag{47}$$

The orbital speed Vðr, tÞ versus r and t is shown in Figure 6. Here, for the evaluated calculations, we used <sup>Γ</sup> <sup>¼</sup> <sup>3</sup> � <sup>10</sup>25m2 � <sup>s</sup>�<sup>1</sup> and the angular frequency <sup>Ω</sup><sup>n</sup> ranges from 10�<sup>11</sup> <sup>s</sup> -1 to 1.667�10�<sup>13</sup> s �<sup>1</sup> as n runs from 1 to 60. The angular frequencies are extremely small, while the wavelengths, λ<sup>n</sup> ¼ c=Ωn, are in the range from 0.97 to 58.3 kpc. These oscillating modes cover areas from the galactic core up to the size of the galaxy itself.

Figure 6. Orbital speed V is a function of the radius r from the galactic center (in kiloparsec) and time t (in light years). Variations of the orbital speed in time are evident.

Figure 6 shows that the orbital speed experiences small fluctuations in time, resembling the breathing of the galaxy. This trembling of galaxies within the 1=f spectrum is caused by the exchange of vortex energy with the SQS on the ultra-low frequencies Ωn.

De Broglie wavelength, λ<sup>n</sup> ¼ c=Ωn, by changing in the range from about 20 kpc to 2 Mpc covers all galactic scales. One can evaluate the mass of axion-like particles [51],

$$
\hbar m = \frac{\hbar \Omega\_n}{c^2}.\tag{48}
$$

It ranges from about 10�<sup>62</sup> kg to 5�10�<sup>66</sup> kg. They are in the range shown in Ref. [52]. These particles may correspond to dark energy's quanta and be responsible for exchange phenomena among baryon objects in the frequency range from <sup>Ω</sup><sup>n</sup> <sup>¼</sup> <sup>10</sup>�11s�<sup>1</sup> to <sup>Ω</sup><sup>n</sup> <sup>≈</sup> <sup>5</sup> � <sup>10</sup>�15s�1. We note that the frequency <sup>Ω</sup><sup>c</sup> is 2:<sup>2</sup> � <sup>10</sup>�18s�1. We obtain <sup>c</sup>=Ω<sup>c</sup> <sup>≈</sup> <sup>1</sup>:<sup>36</sup> � 1026m, which is close enough to the Compton wavelength evaluated for the visible Universe in [53]. This corresponds to the radius of the Hubble sphere rHS <sup>¼</sup> <sup>c</sup>=H0, which is about 4 � 103 Mpc (at the Hubble constant H<sup>0</sup> ¼ 73 km � ðs � MpcÞ �1 ). On these cosmological scales, we can evaluate the mass of a graviton (or more likely of a quantum of dark energy, since we don't need gravitons in superfluid quantum gravity [2, 14]), by resorting to a wavelength that is commensurable with the radius of the Universe stated above. An extreme mass of the axion-like particle for the observable Universe is mg <sup>¼</sup> <sup>ℏ</sup>Ω=c<sup>2</sup> <sup>≈</sup> <sup>2</sup>:<sup>6</sup> � <sup>10</sup>�<sup>69</sup> kg. This value finds a good agreement with the evaluation that comes from the holographic screen model to be the boundary of the visible Universe [52]. This evaluation is also in agreement with the graviton mass given in Ref. [54], here interpreted as a quantum of dark energy. We finally add that ultra-light dark matter particles produced in the vacuum have been predicted in Ref. [10].

We can continue the calculation of the orbital speed (44) up to the point <sup>Ω</sup><sup>c</sup> <sup>¼</sup> <sup>2</sup>:<sup>2</sup> � <sup>10</sup>�18s�1. This would allow us to affirm that the observable Universe rotates about some center with an orbital speed, which has a flat profile through enormous distances. Excepting a central region where the orbital speed grows from zero to the maximal value corresponding to the profile level. This rotation possibly takes place around the richest Super Cluster in the Sloan Great Wall, SCl~126, and especially around its core, resembling a very rich filament [55].

#### 5. Conclusion

(45), n ¼ 1, 2, ⋯, N. Let us substitute the expression (45) into the integral (34). After computing

increases. From here, it follows that the expression 1 � exp { � <sup>r</sup><sup>2</sup>=4Σn} in Eq. (33) reaches 1 the more slowly with increasing r, the larger is Σn. As a result, the set of coefficients Σ<sup>n</sup> for n ¼ 1, 2, ⋯ can give output to the flat profile of the orbital speed. Let us, therefore, compute a sum of possible orbital speeds of galaxies for all entangled modes for which baryon matter is

The orbital speed Vðr, tÞ versus r and t is shown in Figure 6. Here, for the evaluated calculations, we used <sup>Γ</sup> <sup>¼</sup> <sup>3</sup> � <sup>10</sup>25m2 � <sup>s</sup>�<sup>1</sup> and the angular frequency <sup>Ω</sup><sup>n</sup> ranges from 10�<sup>11</sup> <sup>s</sup>

wavelengths, λ<sup>n</sup> ¼ c=Ωn, are in the range from 0.97 to 58.3 kpc. These oscillating modes cover

Figure 6. Orbital speed V is a function of the radius r from the galactic center (in kiloparsec) and time t (in light years).

sin ðΩntÞ þ ζ

<sup>1</sup> � exp � <sup>r</sup><sup>2</sup>

� � � �

�<sup>1</sup> as n runs from 1 to 60. The angular frequencies are extremely small, while the

�

4ΣnðtÞ

: ð46Þ

: ð47Þ


<sup>n</sup> tends to infinity as Ω<sup>n</sup> goes to zero while n

it, we get the following view of the denominator Σðν, t, σÞ:

Since <sup>Ω</sup><sup>n</sup> <sup>¼</sup> <sup>n</sup>�1, the coefficient <sup>Σ</sup><sup>n</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup>=Ω<sup>2</sup>

1.667�10�<sup>13</sup> s

106 Trends in Modern Cosmology

<sup>Σ</sup>nðtÞ ¼ <sup>c</sup><sup>2</sup> Ω2 n �

allowed to exchange energy with the SQS. Our statistical sum reads as follows:

n¼1

2r X N

<sup>V</sup>ðr, tÞ ¼ <sup>Γ</sup>

areas from the galactic core up to the size of the galaxy itself.

Variations of the orbital speed in time are evident.

We have shown that the fluid dynamics of SQS could explain the astrophysical observations without resorting to far-fetched auxiliary concepts, such as cosmic inflation and accelerated expansion.

In general, the fluid dynamics of SQS is described by the conservation equations of energy, momentum, orbital momentum, etc. In the non-relativistic limit, these equations are reduced to the modified Navier-Stokes equation and to the continuity equation of mass density. The modification leads to the emergence of a quantum potential, Q(t), and reduces the viscosity coefficient, μðtÞ, to a weak term fluctuating about zero, 〈μðtÞ〉 ¼ 0þ, 〈μðtÞμð0Þ〉 > 0. Because of that, this term acquires an absolutely different physical meaning. Firstly, an active exchange of energy between cosmological structures and SQS takes place and extends their lifetime.

We applied the modified Navier-Stokes equation to describe a balance within the visible Universe between the gravitational potential, ϕ, expressed as the quantum potential Q<sup>ϕ</sup> ¼ PG=ρ<sup>m</sup> and the intrinsic quantum potential, Q, of SQS. Outside this range, strong repelling forces act (see dotted curve in Figure 1), probably due to osmotic expansion of dark energy in a really empty space. Figuratively speaking, baryon matter in the Universe is similar to a hydrophobic droplet floating in a hydrophilic medium filling the vast space. However, there is a difference between a "droplet model" and the Universe, since the latter consists of numerous clumps of baryonic matter separated by vast voids. These baryonic clumps are concentrated on vortex filaments that permeate the whole Universe and form an intricate cosmic web [56] with galaxies strung on these filaments.

Since 〈μðtÞ〉 ¼ 0<sup>þ</sup> (it differs from zero to a tiny value), light coming from distant stars shows a frequency shift due to a loss of energy when traveling through the SQS. We therefore introduce an updated concept of tired light without resorting to Compton scattering and overcoming in this way the known objections to the classical concept of tired light.

The Pioneer anomaly has a lot in common with the revised tired light effect. The same loss of energy due to motion through the SQS most likely led to a deceleration of the space apparatus. An essential contribution to the deceleration comes from a non-zero small correction of the Hubble parameter thanks to the cosmological constant, which refers to dark energy, i.e., to SQS itself). This correction gives a value of the negative acceleration of the cosmic apparatus a ¼ �HΛc ¼ �c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2 <sup>0</sup> þ Λc<sup>2</sup>=3 q <sup>≈</sup> � <sup>8</sup>:<sup>785</sup> � <sup>10</sup>�10m � <sup>s</sup>�2, which acceptably falls within the measured anomalous acceleration of the Pioneer probes 10 and 11.

Eventually, the considered superfluid dark medium is capable of explaining the flat profile of the orbital speed of spiral galaxies, due to their interactions with the SQS. We can observe flat profile solutions by putting (46) as denominator in Eqs. (30) and (31), with σ<sup>0</sup> > 1, where the set of coefficients Σ<sup>n</sup> for n ¼ 1, 2, ⋯ shows the flat profile.

#### Author details

Valeriy I. Sbitnev1,2\* and Marco Fedi3

\*Address all correspondence to: valery.sbitnev@gmail.com

1 Petersburg B. P. Konstantinov Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, Russia

2 Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA

3 Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR), Italy

#### References

of that, this term acquires an absolutely different physical meaning. Firstly, an active exchange of energy between cosmological structures and SQS takes place and extends their lifetime.

We applied the modified Navier-Stokes equation to describe a balance within the visible Universe between the gravitational potential, ϕ, expressed as the quantum potential Q<sup>ϕ</sup> ¼ PG=ρ<sup>m</sup> and the intrinsic quantum potential, Q, of SQS. Outside this range, strong repelling forces act (see dotted curve in Figure 1), probably due to osmotic expansion of dark energy in a really empty space. Figuratively speaking, baryon matter in the Universe is similar to a hydrophobic droplet floating in a hydrophilic medium filling the vast space. However, there is a difference between a "droplet model" and the Universe, since the latter consists of numerous clumps of baryonic matter separated by vast voids. These baryonic clumps are concentrated on vortex filaments that permeate the whole Universe and form an intricate cosmic web [56] with galaxies strung on these filaments. Since 〈μðtÞ〉 ¼ 0<sup>þ</sup> (it differs from zero to a tiny value), light coming from distant stars shows a frequency shift due to a loss of energy when traveling through the SQS. We therefore introduce an updated concept of tired light without resorting to Compton scattering and overcoming in

The Pioneer anomaly has a lot in common with the revised tired light effect. The same loss of energy due to motion through the SQS most likely led to a deceleration of the space apparatus. An essential contribution to the deceleration comes from a non-zero small correction of the Hubble parameter thanks to the cosmological constant, which refers to dark energy, i.e., to SQS itself). This correction gives a value of the negative acceleration of the cosmic apparatus

Eventually, the considered superfluid dark medium is capable of explaining the flat profile of the orbital speed of spiral galaxies, due to their interactions with the SQS. We can observe flat profile solutions by putting (46) as denominator in Eqs. (30) and (31), with σ<sup>0</sup> > 1, where the

1 Petersburg B. P. Konstantinov Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina,

2 Department of Electrical Engineering and Computer Sciences, University of California,

3 Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR), Italy

<sup>≈</sup> � <sup>8</sup>:<sup>785</sup> � <sup>10</sup>�10m � <sup>s</sup>�2, which acceptably falls within the mea-

this way the known objections to the classical concept of tired light.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2

q

Valeriy I. Sbitnev1,2\* and Marco Fedi3

<sup>0</sup> þ Λc<sup>2</sup>=3

sured anomalous acceleration of the Pioneer probes 10 and 11.

set of coefficients Σ<sup>n</sup> for n ¼ 1, 2, ⋯ shows the flat profile.

\*Address all correspondence to: valery.sbitnev@gmail.com

a ¼ �HΛc ¼ �c

108 Trends in Modern Cosmology

Author details

Berkeley, CA, USA

Russia


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Astrophys. J. 2011; 736: 25 p. doi:10.1088/0004-637X/736/1/51

## **Modified Gravity Theories: Distinguishing from ΛCDM Model**

Koichi Hirano

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68281

#### Abstract

The method and probability of distinguishing between the Λ cold dark matter (ΛCDM) model and modified gravity are studied from future observations for the growth rate of cosmic structure (Euclid redshift survey). We compare the mock observational data to the theoretical cosmic growth rate by modified gravity models, including the extended Dvali–Gabadadze–Porrati (DGP) model, kinetic gravity braiding model, and Galileon model. In the original DGP model, the growth rate f σ<sup>8</sup> is suppressed in comparison with that in the ΛCDM model in the setting of the same value of the today's energy density of matter Ω<sup>m</sup>;0, due to suppression of the effective gravitational constant. In the case of the kinetic gravity braiding model and the Galileon model, the growth rate f σ<sup>8</sup> is enhanced in comparison with the ΛCDM model in the same value of Ω<sup>m</sup>;0, due to enhancement of the effective gravitational constant. For the cosmic growth rate data from the future observation (Euclid), the compatible value of Ω<sup>m</sup>;<sup>0</sup> differs according to the model. Furthermore, Ω<sup>m</sup>;<sup>0</sup> can be stringently constrained. Thus, we find the ΛCDM model is distinguishable from modified gravity by combining the growth rate data of Euclid with other observations.

Keywords: accelerated expansion, gravitational theory, dark energy, observational test, cosmic growth rate

#### 1. Introduction

Cosmological observations, including type Ia supernovae (SNIa) [1, 2], cosmic microwave background (CMB) anisotropies, and baryon acoustic oscillations (BAO), indicate that the universe is undergoing an accelerated phase of expansion. This late-time acceleration is one of the biggest mysteries in current cosmology. The standard explanation is that this acceleration is

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caused by dark energy [3–6]. This would mean that a large part of components in the universe is unknown. The cosmological constant is a candidate of dark energy. To explain the late-time accelerated expansion of the universe, the cosmological constant must be a very small value. However, its value is not compatible with a prediction from particle physics, and it has finetuning and coincidence problems.

An alternative explanation for the current acceleration of the universe is to modify general relativity to be a more general theory of gravity at a long-distance scale. Several modified gravity theories have been studied, such as f(R) gravity (for reviews, see, e.g., [7]), scalar-tensor theories [8–10], and the Dvali–Gabadadze–Porrati (DGP) braneworld model [11–13].

Furthermore, as an alternative to general relativity, Galileon gravity models have been proposed [14–22]. These models are built by introducing a scalar field with a self-interaction whose Lagrangian, which is invariant under Galileon symmetry ∂μφ ! ∂μφ þ bμ, keeps the field equation of motion as a second-order differential equation. This avoids presenting a new degree of freedom, and perturbation of the theory is free from ghost or instability problems. The simplest term of the self-interaction is □φð∇φ<sup>Þ</sup> 2 , which induces decoupling of the Galileon field from gravity at small scales via the Vainshtein mechanism [23]. Therefore, the Galileon theory recovers general relativity at scales around the high-density region, as is not inconsistent with solar system experiments.

Galileon theory has been covariantized and studied in curved backgrounds [24, 25]. Although Galileon symmetry cannot be maintained in the case that the theory is covariantized, it is possible to preserve the equation of motion at second order, which means that the theory does not raise ghost-like instabilities. Galileon gravity induces self-accelerated expansion of the current universe. Thus, inflation models inspired by the Galileon gravity theory have been studied [26–28]. In Ref. [29], the parameters of the generalized Galileon cosmology were constrained from the observational data of SNIa, CMB, and BAO. The evolution of matter density perturbations for Galileon cosmology has also been investigated [16–18, 30, 31].

Almost 40 years ago, Horndeski derived the action of most general scalar-tensor theories with second-order equations of motion [32]. His theory received much attention as an extension of covariant Galileons [14, 24, 25, 33]. One can show that the four-dimensional action of generalized Galileons derived by Deffayet et al. [34] is equivalent to Horndeski's action under field redefinition [35]. Because Horndeski's theory contains all modified gravity models and singlefield inflation models with one scalar degree of freedom as specific cases, considerable attention has been paid to various aspects of Horndeski's theory and its importance in cosmology.

Recently, more general modified gravity theories have been studied, including Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories [36, 37] and eXtended Galileon with 3-space covariance (XG3) [38].

In this chapter, the probability of distinguishing between the Λ cold dark matter (ΛCDM) model and modified gravity is studied by using future observations for the growth rate of cosmic structure (e.g., Euclid redshift survey [39]). We computed the growth rate of matter density perturbations in modified gravity and compared it with mock observational data. Whereas the background expansion history in modified gravity is almost identical to that of dark energy models, the evolution of matter density perturbations of modified gravity is different from that of dark energy models. Thus, it is important to study the growth history of perturbations to distinguish modified gravity from models based on the cosmological constant or dark energy.

Although past observations of the growth rate of matter density perturbations have been used to study modified gravity [40], we focus on future observations of the growth rate by Euclid. We adopt the extended DGP model [41], kinetic gravity braiding model [30], and Galileon model [16, 17] as modified gravity models. The kinetic gravity braiding model and the Galileon model are specific aspects of Horndeski's theory.

This chapter is organized as follows. In the next section, we present the background evolution and the effective gravitational constant in modified gravity models. In Section 3, we describe the theoretical computations and the mock observational data of the growth rate of matter density perturbations. In Section 4, we study the probability of distinguishing between the ΛCDM model and modified gravity by comparing the predicted cosmic growth rate by models to the mock observational data. Finally, conclusions are given in Section 5.

#### 2. Modified gravity models

#### 2.1. Extended DGP model

In the DGP model [11], it is assumed that we live on a 4D brane embedded in a 5D Minkowski bulk. Matter is trapped on the 4D brane, and only gravity experiences the 5D Minkowski bulk.

The action is

caused by dark energy [3–6]. This would mean that a large part of components in the universe is unknown. The cosmological constant is a candidate of dark energy. To explain the late-time accelerated expansion of the universe, the cosmological constant must be a very small value. However, its value is not compatible with a prediction from particle physics, and it has fine-

An alternative explanation for the current acceleration of the universe is to modify general relativity to be a more general theory of gravity at a long-distance scale. Several modified gravity theories have been studied, such as f(R) gravity (for reviews, see, e.g., [7]), scalar-tensor

Furthermore, as an alternative to general relativity, Galileon gravity models have been proposed [14–22]. These models are built by introducing a scalar field with a self-interaction whose Lagrangian, which is invariant under Galileon symmetry ∂μφ ! ∂μφ þ bμ, keeps the field equation of motion as a second-order differential equation. This avoids presenting a new degree of freedom, and perturbation of the theory is free from ghost or instability problems.

field from gravity at small scales via the Vainshtein mechanism [23]. Therefore, the Galileon theory recovers general relativity at scales around the high-density region, as is not inconsis-

Galileon theory has been covariantized and studied in curved backgrounds [24, 25]. Although Galileon symmetry cannot be maintained in the case that the theory is covariantized, it is possible to preserve the equation of motion at second order, which means that the theory does not raise ghost-like instabilities. Galileon gravity induces self-accelerated expansion of the current universe. Thus, inflation models inspired by the Galileon gravity theory have been studied [26–28]. In Ref. [29], the parameters of the generalized Galileon cosmology were constrained from the observational data of SNIa, CMB, and BAO. The evolution of matter density perturbations for Galileon cosmology has also been investigated [16–18, 30, 31].

Almost 40 years ago, Horndeski derived the action of most general scalar-tensor theories with second-order equations of motion [32]. His theory received much attention as an extension of covariant Galileons [14, 24, 25, 33]. One can show that the four-dimensional action of generalized Galileons derived by Deffayet et al. [34] is equivalent to Horndeski's action under field redefinition [35]. Because Horndeski's theory contains all modified gravity models and singlefield inflation models with one scalar degree of freedom as specific cases, considerable attention has been paid to various aspects of Horndeski's theory and its importance in cosmology. Recently, more general modified gravity theories have been studied, including Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories [36, 37] and eXtended Galileon with 3-space covari-

In this chapter, the probability of distinguishing between the Λ cold dark matter (ΛCDM) model and modified gravity is studied by using future observations for the growth rate of cosmic structure (e.g., Euclid redshift survey [39]). We computed the growth rate of matter density perturbations in modified gravity and compared it with mock observational data. Whereas the background expansion history in modified gravity is almost identical to that of dark energy

2

, which induces decoupling of the Galileon

theories [8–10], and the Dvali–Gabadadze–Porrati (DGP) braneworld model [11–13].

tuning and coincidence problems.

116 Trends in Modern Cosmology

The simplest term of the self-interaction is □φð∇φ<sup>Þ</sup>

tent with solar system experiments.

ance (XG3) [38].

$$S = \frac{1}{16\pi} M\_{(5)}^3 \int\_{\text{bulk}} d^5 \mathbf{x} \sqrt{-\mathcal{G}\_{(5)}} \mathcal{R}\_{(5)} + \frac{1}{16\pi} M\_{(4)}^2 \int\_{\text{brane}} d^4 \mathbf{x} \sqrt{-\mathcal{G}\_{(4)}} (\mathcal{R}\_{(4)} + L\_{\text{m}})\_{\text{\textquotedblleft}} \tag{1}$$

where quantities of the 4D brane and the 5D Minkowski bulk are represented with subscripts (4) and (5), respectively. M is the Planck mass, and Lm is the Lagrangian of matter confined on the 4D brane. The transition between 4D and 5D gravity occurs at the crossover scale rc.

$$r\_c = \frac{M\_{(4)}^2}{2M\_{(5)}^3}.\tag{2}$$

At scales larger than rc, gravity appears in 5D. At scales smaller than rc, gravity is effectively bound to the brane, and 4D Newtonian dynamics is recovered to a good approximation. rc is a parameter in this model, which has the unit of length [42].

Under spatial homogeneity and isotropy, a Friedmann-like equation is obtained on the brane [43, 44]:

$$H^2 = \frac{8\pi G}{3}\rho + \varepsilon \frac{H}{r\_c},\tag{3}$$

where ρ represents the total fluid energy density on the 4D brane. The DGP model has two

branches (ε = �1). The choice of ε = þ1 is called the self-accelerating branch. In this branch, the accelerated expansion of the universe is induced without dark energy, since the Hubble parameter comes close to a constant H = 1/rc as time passes. By contrast, ε = �1 is the normal branch. In this case, the expansion cannot accelerate without a dark energy component. Therefore, in the following, we adopt the self-accelerating branch (ε = þ1).

The original DGP model, however, is plagued by the ghost problem [45] and is incompatible with cosmological observations [46].

Dvali and Turner [41] phenomenologically extended the Friedmann-like equation of the DGP model (Eq. (3)). This model interpolates between the original DGP model and the ΛCDM model by adding the parameter α. The modified Friedmann-like equation is

$$H^2 = \frac{8\pi G}{3}\rho + \frac{H^a}{r\_c^{2-a}}.\tag{4}$$

For α = 1, this is equivalent to the original DGP Friedmann-like equation, whereas α = 0 leads to an expansion history identical to ΛCDM cosmology. This is important for distinguishing the ΛCDM model from the original DGP model between α = 0 and 1. In the extended DGP model, the crossover scale rc can be expressed as follows:

$$r\_c = (1 - \Omega\_{m,0})^{\frac{1}{n-2}} H\_0^{-1} \,. \tag{5}$$

Thus, the independent parameters of the cosmological model are α and today's energy density parameter of matter Ω<sup>m</sup>;0. The effective gravitational constant of the extended DGP model is given in order to interpolate between ΛCDM and the original DGP model. The effective gravitational constant is as follows.

$$\frac{G\_{\rm eff}}{G} = 1 + \frac{1}{3\beta},\tag{6}$$

where

$$\beta \equiv 1 - \frac{2\left(r\_{\epsilon}H\right)^{2-\alpha}}{\alpha} \left[1 + \frac{1}{3}\frac{(2-\alpha)\dot{H}}{H^2}\right].\tag{7}$$

Geff / G is the effective gravitational constant normalized to Newton's gravitational constant, and an overdot represents differentiation with respect to cosmic time t.

#### 2.2. Kinetic gravity braiding model

The kinetic gravity braiding model [30] is proposed as an alternative to the dark energy model. One can say that the kinetic gravity braiding model is a specific aspect of Horndeski's theory [32].

The most general four-dimensional scalar-tensor theories keeping the field equations of motion at second order are described by the Lagrangian [32–35, 47]

Modified Gravity Theories: Distinguishing from ΛCDM Model http://dx.doi.org/10.5772/intechopen.68281 119

$$
\mathcal{L} = \sum\_{i=2}^{5} \mathcal{L}\_i,\tag{8}
$$

where

branches (ε = �1). The choice of ε = þ1 is called the self-accelerating branch. In this branch, the accelerated expansion of the universe is induced without dark energy, since the Hubble parameter comes close to a constant H = 1/rc as time passes. By contrast, ε = �1 is the normal branch. In this case, the expansion cannot accelerate without a dark energy component.

The original DGP model, however, is plagued by the ghost problem [45] and is incompatible

Dvali and Turner [41] phenomenologically extended the Friedmann-like equation of the DGP model (Eq. (3)). This model interpolates between the original DGP model and the ΛCDM

<sup>3</sup> <sup>ρ</sup> <sup>þ</sup>

For α = 1, this is equivalent to the original DGP Friedmann-like equation, whereas α = 0 leads to an expansion history identical to ΛCDM cosmology. This is important for distinguishing the ΛCDM model from the original DGP model between α = 0 and 1. In the extended DGP model,

Thus, the independent parameters of the cosmological model are α and today's energy density parameter of matter Ω<sup>m</sup>;0. The effective gravitational constant of the extended DGP model is given in order to interpolate between ΛCDM and the original DGP model. The effective

H<sup>α</sup> rc

1 <sup>α</sup>�<sup>2</sup>H�<sup>1</sup>

1 3β

<sup>ð</sup><sup>2</sup> � <sup>α</sup>ÞH\_ H2

" #

<sup>2</sup>�<sup>α</sup> : <sup>ð</sup>4<sup>Þ</sup>

<sup>0</sup> : ð5Þ

; ð6Þ

: ð7Þ

Therefore, in the following, we adopt the self-accelerating branch (ε = þ1).

model by adding the parameter α. The modified Friedmann-like equation is

<sup>H</sup><sup>2</sup> <sup>¼</sup> <sup>8</sup>π<sup>G</sup>

rc ¼ ð1 � Ω<sup>m</sup>;<sup>0</sup>Þ

Geff <sup>G</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

α

2�α

Geff / G is the effective gravitational constant normalized to Newton's gravitational constant,

The kinetic gravity braiding model [30] is proposed as an alternative to the dark energy model. One can say that the kinetic gravity braiding model is a specific aspect of Horndeski's theory [32]. The most general four-dimensional scalar-tensor theories keeping the field equations of motion

<sup>β</sup> � <sup>1</sup> � <sup>2</sup>ðrcH<sup>Þ</sup>

and an overdot represents differentiation with respect to cosmic time t.

at second order are described by the Lagrangian [32–35, 47]

with cosmological observations [46].

118 Trends in Modern Cosmology

the crossover scale rc can be expressed as follows:

gravitational constant is as follows.

2.2. Kinetic gravity braiding model

where

$$\mathcal{L}\_2 = \mathcal{K}(\phi, X), \tag{9}$$

$$\mathcal{L}\_3 = -\mathcal{G}\_3(\phi, X)\Box\phi,\tag{10}$$

$$\mathcal{L}\_4 = \mathsf{G}\_4(\phi, X)\mathsf{R} + \mathsf{G}\_{4,X}[(\Box \phi)^2 - (\nabla\_\mu \nabla\_\nu \phi)(\nabla^\mu \nabla^\nu \phi)],\tag{11}$$

$$\begin{split} \mathcal{L}\_5 &= \mathsf{G}\_5(\phi, \mathsf{X}) \mathsf{G}\_{\mu\nu} (\nabla^{\mu} \nabla^{\nu} \phi) - \frac{1}{6} \mathsf{G}\_{5, \mathsf{X}} [(\Box \phi)^3 - \mathsf{3} (\Box \phi) (\nabla\_{\mu} \nabla\_{\nu} \phi) (\nabla^{\mu} \nabla^{\nu} \phi) \\ &+ 2 (\nabla^{\mu} \nabla\_{a} \phi) (\nabla^{a} \nabla\_{\beta} \phi) (\nabla^{\beta} \nabla\_{\mu} \phi)]. \end{split} \tag{12}$$

Here, K and Gi (i = 3, 4, 5) are functions of the scalar field φ and its kinetic energy <sup>X</sup> ¼ �∂μφ∂μφ=2 with the partial derivatives Gi;<sup>X</sup> � <sup>∂</sup>Gi=∂X. <sup>R</sup> is the Ricci scalar, and <sup>G</sup>μν is the Einstein tensor. The above Lagrangian was first derived by Horndeski in a different form Ref. [32]. This Lagrangian (Eqs. (8)–(12)) is equivalent to that derived by Horndeski [35]. The total action is then given by

$$S = \int d^4x \sqrt{-g} (\mathcal{L} + \mathcal{L}\_m) \,\prime \,\tag{13}$$

where g represents a determinant of the metric gμν, and L<sup>m</sup> is the Lagrangian of non-relativistic matter.

Variation with respect to the metric produces the gravity equations, and variation with respect to the scalar field φ yields the equation of motion. By using the notation K � Kðφ;XÞ, G � G3ðφ;XÞ, F � <sup>2</sup> M<sup>2</sup> pl G4ðφ;XÞ, and assuming G5ðφ;XÞ ¼ 0 for Friedmann–Robertson–Walker spacetime, the gravity equations give

$$\Im \mathcal{M}\_{\text{pl}}^2 F H^2 = \rho\_m + \rho\_r - \Im \mathcal{M}\_{\text{pl}}^2 H \dot{F} - K + 2\mathcal{X} \mathcal{K}\_{,\text{X}} + 6H \dot{\phi} \mathcal{X} \mathcal{G}\_{,\text{X}} - 2\mathcal{X} \mathcal{G}\_{,\phi},\tag{14}$$

$$-M\_{\rm pl}^2 F(3H^2 + 2\dot{H}) = p\_r + 2M\_{\rm pl}^2 H \dot{F} + M\_{\rm pl}^2 \ddot{F} + K - 2XG\_{,\dot{X}}\ddot{\phi} - 2XG\_{,\phi},\tag{15}$$

and the equation of motion for the scalar field gives

$$\begin{aligned} &(\mathbf{K}\_{,X} + 2\mathbf{X}\mathbf{K}\_{,XX} + 6\mathbf{H}\dot{\phi}\mathbf{G}\_{,X} + 6\mathbf{H}\dot{\phi}\mathbf{X}\mathbf{G}\_{,XX} - 2\mathbf{X}\mathbf{G}\_{,\phi X} - 2\mathbf{G}\_{,\phi})\ddot{\phi} \\ &+ (\mathbf{3}H\mathbf{K}\_{,X} + \dot{\phi}\mathbf{K}\_{,\phi X} + 9\mathbf{H}^{2}\dot{\phi}\mathbf{G}\_{,X} + 3\dot{\mathbf{H}}\dot{\phi}\mathbf{G}\_{,X} + 6\mathbf{H}\mathbf{X}\mathbf{G}\_{,\phi X} - 6\mathbf{H}\mathbf{G}\_{,\phi} - \mathbf{G}\_{,\phi\phi}\dot{\phi})\dot{\phi} \\ &- \mathbf{K}\_{,\phi} - 6\mathbf{M}\_{\mathrm{pl}}^{2}\mathbf{H}^{2}F\_{,\phi} - 3\mathbf{M}\_{\mathrm{pl}}^{2}\dot{\mathbf{H}}F\_{,\phi} = 0. \end{aligned} \tag{16}$$

Here, an overdot denotes differentiation with respect to cosmic time t, and H ¼ a\_=a is the Hubble expansion rate. Note that we use the partial derivative notation K;<sup>X</sup> � ∂K=∂X and <sup>K</sup>;XX � <sup>∂</sup><sup>2</sup>K=∂X<sup>2</sup> and a similar notation for other variables. Also, <sup>ρ</sup><sup>m</sup> and <sup>ρ</sup><sup>r</sup> are the energy densities of matter and radiation, respectively, and pr is the pressure of the radiation.

In the kinetic gravity braiding model [30], the functions in Horndeski's theory are given as follows:

$$K(\phi, X) = -X,\tag{17}$$

$$\mathcal{G}\_{\rm 3}(\phi, X) = M\_{\rm pl} \left( \frac{r\_c^2}{M\_{\rm pl}^2} X \right)^n,\tag{18}$$

$$\mathcal{G}\_4(\phi, \mathcal{X}) = \frac{M\_{\text{pl}}^2}{2},\tag{19}$$

$$G\_{\mathbb{S}}(\phi, X) = 0.\tag{20}$$

<sup>M</sup>pl is the reduced Planck mass related to Newton's gravitational constant by <sup>M</sup>pl <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffiffiffiffiffiffiffi <sup>8</sup>π<sup>G</sup> <sup>p</sup> , and rc is called the crossover scale in the DGP model [42]. The kinetic braiding model we study is characterized by parameter n in Eq. (18). For n = 1, this corresponds to Deffayet's Galileon cosmological model [22]. For n equal to infinity, the background expansion of the universe of the kinetic braiding model approaches that of the ΛCDM model. This helps distinguish the kinetic braiding model from the ΛCDM model.

In the case of the kinetic braiding model using the Hubble parameter as the present epoch H0, the crossover scale rc is given by

$$r\_c = \left(\frac{2^{n-1}}{3n}\right)^{1/2n} \left[\frac{1}{6(1-\Omega\_{m,0}-\Omega\_{r,0})}\right]^{(2n-1)/4n} H\_0^{-1},\tag{21}$$

where Ωr,0 is the density parameter of the radiation at the present time. Thus, the independent parameters of the cosmological model are n and Ωm,0. The effective gravitational constant normalized to Newton's gravitational constant Geff / G of the kinetic braiding model is given by

$$\frac{G\_{\rm eff}}{G} = \frac{2n + 3n\Omega\_m - \Omega\_m}{\Omega\_m(5n - \Omega\_m)},\tag{22}$$

where <sup>Ω</sup><sup>m</sup> is the matter energy density parameter defined as <sup>Ω</sup><sup>m</sup> <sup>¼</sup> <sup>ρ</sup>m=3M<sup>2</sup> plH<sup>2</sup> . Here, we used the attractor condition. Although the background evolution for large n approaches the ΛCDM model, the growth history of matter density perturbations is different due to the timedependent effective gravitational constant.

#### 2.3. Galileon model

The Galileon gravity model is proposed as an alternative to the dark energy model. It is thought that the Galileon model studied in Refs. [16, 17] is a specific aspect of Horndeski's theory [32].

In the Galileon model [16, 17], the functions in the Lagrangian (Eqs. (8)–(12)) of the Horndeski's theory are given as follows:

$$K(\phi, X) = 2\frac{\omega}{\phi}X,\tag{23}$$

$$\mathcal{G}\_3(\phi, \mathcal{X}) = 2\xi(\phi)\mathcal{X},\tag{24}$$

$$\mathcal{G}\_4(\phi, X) = \phi,\tag{25}$$

$$G\_5(\phi, X) = 0,\tag{26}$$

where ω is the Brans–Dicke parameter and ξðφÞ is a function of φ.

<sup>K</sup>;XX � <sup>∂</sup><sup>2</sup>K=∂X<sup>2</sup> and a similar notation for other variables. Also, <sup>ρ</sup><sup>m</sup> and <sup>ρ</sup><sup>r</sup> are the energy

In the kinetic gravity braiding model [30], the functions in Horndeski's theory are given as

r2 c M<sup>2</sup> pl X !<sup>n</sup>

> M<sup>2</sup> pl

Kðφ;X޼�X; ð17Þ

G5ðφ;XÞ ¼ 0: ð20Þ

H�<sup>1</sup>

<sup>Ω</sup>mð5<sup>n</sup> � <sup>Ω</sup>m<sup>Þ</sup> ; <sup>ð</sup>22<sup>Þ</sup>

plH<sup>2</sup>

. Here, we used

<sup>0</sup> ; ð21Þ

; ð18Þ

<sup>8</sup>π<sup>G</sup> <sup>p</sup> ,

<sup>2</sup> ; <sup>ð</sup>19<sup>Þ</sup>

densities of matter and radiation, respectively, and pr is the pressure of the radiation.

G3ðφ;XÞ ¼ Mpl

kinetic braiding model from the ΛCDM model.

dependent effective gravitational constant.

2.3. Galileon model

theory [32].

rc <sup>¼</sup> <sup>2</sup><sup>n</sup>�<sup>1</sup> 3n

the crossover scale rc is given by

G4ðφ;XÞ ¼

<sup>M</sup>pl is the reduced Planck mass related to Newton's gravitational constant by <sup>M</sup>pl <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffiffiffiffiffiffiffi

and rc is called the crossover scale in the DGP model [42]. The kinetic braiding model we study is characterized by parameter n in Eq. (18). For n = 1, this corresponds to Deffayet's Galileon cosmological model [22]. For n equal to infinity, the background expansion of the universe of the kinetic braiding model approaches that of the ΛCDM model. This helps distinguish the

In the case of the kinetic braiding model using the Hubble parameter as the present epoch H0,

6ð1 � Ω<sup>m</sup>;<sup>0</sup> � Ω<sup>r</sup>;<sup>0</sup>Þ

where Ωr,0 is the density parameter of the radiation at the present time. Thus, the independent parameters of the cosmological model are n and Ωm,0. The effective gravitational constant normalized to Newton's gravitational constant Geff / G of the kinetic braiding model is given by

<sup>G</sup> <sup>¼</sup> <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>3</sup>nΩ<sup>m</sup> � <sup>Ω</sup><sup>m</sup>

the attractor condition. Although the background evolution for large n approaches the ΛCDM model, the growth history of matter density perturbations is different due to the time-

The Galileon gravity model is proposed as an alternative to the dark energy model. It is thought that the Galileon model studied in Refs. [16, 17] is a specific aspect of Horndeski's

� �ð2n�1Þ=4<sup>n</sup>

� �<sup>1</sup>=2<sup>n</sup> 1

Geff

where <sup>Ω</sup><sup>m</sup> is the matter energy density parameter defined as <sup>Ω</sup><sup>m</sup> <sup>¼</sup> <sup>ρ</sup>m=3M<sup>2</sup>

follows:

120 Trends in Modern Cosmology

In this case, the Friedmann-like equations, Eqs. (14) and (15), can be written in the following forms, respectively:

$$\mathcal{G}H^2 = \frac{1}{M\_{\text{pl}}^2} (\rho\_m + \rho\_r + \rho\_\phi)\_\prime \tag{27}$$

$$-3H^2 - 2\dot{H} = \frac{1}{M\_{\text{pl}}^2} (p\_r + p\_\phi)\_\prime \tag{28}$$

where the effective dark energy density ρφ is defined as

$$\rho\_{\phi} = 2\phi \left[ -3H\frac{\dot{\phi}}{\phi} + \frac{\omega}{2} \left( \frac{\dot{\phi}}{\phi} \right)^2 + \phi^2 \xi(\phi) \left\{ 3H + \frac{\dot{\phi}}{\phi} \right\} \left( \frac{\dot{\phi}}{\phi} \right)^3 \right] + 3H^2 (M\_{\text{pl}}^2 - 2\phi), \tag{29}$$

and the effective pressure of dark energy p<sup>φ</sup> is

$$p\_{\phi} = 2\phi \left[ \frac{\ddot{\phi}}{\phi} + 2H \frac{\dot{\phi}}{\phi} + \frac{\omega}{2} \left( \frac{\dot{\phi}}{\phi} \right)^2 - \phi^2 \xi(\phi) \left\{ \frac{\ddot{\phi}}{\phi} - \left( \frac{\dot{\phi}}{\phi} \right)^2 \right\} \left( \frac{\dot{\phi}}{\phi} \right)^2 \right] - (3H^2 + 2\dot{H})(M\_{\text{pl}}^2 - 2\phi) . \tag{30}$$

The equation of motion for the scalar field is given by Eq. (16).

For the numerical analysis, we adopt a specific model in which

$$
\xi(\phi) = \frac{r\_c^2}{\phi^2},
\tag{31}
$$

where rc is the crossover scale [15]. This Galileon model extends the Brans–Dicke theory by adding the self-interaction term, ξðφÞð∇φÞ 2 □ φ. Thus, ω of this model is not exactly the same as the original Brans–Dicke parameter. The evolution of matter density perturbations of this model was computed in Refs. [16, 17].

At early times, we set the initial condition φ≃M<sup>2</sup> pl=2 to recover general relativity. Using these initial conditions reduces the Friedmann equations (Eqs. (27) and (28)) to their usual forms: <sup>3</sup>H<sup>2</sup> ¼ ðρ<sup>m</sup> <sup>þ</sup> <sup>ρ</sup>rÞ=M<sup>2</sup> pl and �3H<sup>2</sup> � <sup>2</sup>H\_ <sup>¼</sup> pr=M<sup>2</sup> pl. This is the cosmological version of the Vainshtein effect [23], which is a method to recover general relativity below a certain scale. At present, to induce cosmic acceleration, the value of rc must be fine tuned.

The energy density parameter of matter at present in this model is defined as <sup>Ω</sup><sup>m</sup>;<sup>0</sup> <sup>¼</sup> <sup>ρ</sup><sup>m</sup>;<sup>0</sup>=3H<sup>2</sup> <sup>0</sup>φ0. Therefore, in the numerical analysis, the value of rc is fine tuned so that Ωm,0 becomes an assumed value. Thus, the independent parameters of the cosmological model are ω and Ωm,0. For the Galileon model specified by Eqs. (23)–(26) and (31), the effective gravitational constant is given by

$$G\_{\rm eff} = \frac{1}{16\pi\phi} \left[ 1 + \frac{(1 + \xi(\phi)\dot{\phi}^2)^2}{J} \right],\tag{32}$$

where

$$J \equiv 3 + 2\omega + \phi^2 \xi(\phi) \left[ 4\frac{\ddot{\phi}}{\phi} - 2\frac{\dot{\phi}^2}{\phi^2} + 8H\frac{\dot{\phi}}{\phi} - \phi^2 \xi(\phi) \frac{\dot{\phi}^4}{\phi^4} \right]. \tag{33}$$

The effective gravitational constant Geff is close to Newton's constant G at early times, but increases at later times.

#### 3. Cosmic growth rate

#### 3.1. Density perturbations

Under the quasistatic approximation on sub-horizon scales, the evolution equation for cold dark matter over-density δ in linear theory is given by

$$
\ddot{\delta} + 2H\dot{\delta} - 4\pi G\_{\text{eff}}\rho\delta \simeq 0,\tag{34}
$$

where Geff represents the effective gravitational constant of the modified gravity models described in the previous section.

We set the same initial conditions as in the conventional ΛCDM case (δ ≈ a and \_ δ ≈ a\_) because we trace the difference between the evolution of the matter perturbations in modified gravity and the evolution in the ΛCDM model. From the evolution equation, we numerically obtain the growth factor δ / a for modified gravity models. The linear growth rate is written as

$$f = \frac{d\ln\delta}{d\ln a}.\tag{35}$$

where δ is the matter density fluctuations and a is the scale factor. The growth rate can be parameterized by the growth index γ, defined by

$$f = \Omega\_m^\vee. \tag{36}$$

Refs. [48, 49] showed that the growth rate f in the Galileon model specified by Eqs. (23)–(26) and (31) is enhanced compared with the ΛCDM model for the same value of Ωm,0 due to enhancement of the effective gravitational constant.

#### 3.2. Euclid

At early times, we set the initial condition φ≃M<sup>2</sup>

pl and �3H<sup>2</sup> � <sup>2</sup>H\_ <sup>¼</sup> pr=M<sup>2</sup>

present, to induce cosmic acceleration, the value of rc must be fine tuned.

<sup>G</sup>eff <sup>¼</sup> <sup>1</sup>

ξðφÞ 4

€<sup>δ</sup> <sup>þ</sup> <sup>2</sup><sup>H</sup> \_

We set the same initial conditions as in the conventional ΛCDM case (δ ≈ a and \_

φ€ <sup>φ</sup> � <sup>2</sup>

The effective gravitational constant Geff is close to Newton's constant G at early times, but

Under the quasistatic approximation on sub-horizon scales, the evolution equation for cold

where Geff represents the effective gravitational constant of the modified gravity models

we trace the difference between the evolution of the matter perturbations in modified gravity and the evolution in the ΛCDM model. From the evolution equation, we numerically obtain the growth factor δ / a for modified gravity models. The linear growth rate is written as

> <sup>f</sup> <sup>¼</sup> <sup>d</sup> ln<sup>δ</sup> d lna

<sup>J</sup> � <sup>3</sup> <sup>þ</sup> <sup>2</sup><sup>ω</sup> <sup>þ</sup> <sup>φ</sup><sup>2</sup>

dark matter over-density δ in linear theory is given by

<sup>3</sup>H<sup>2</sup> ¼ ðρ<sup>m</sup> <sup>þ</sup> <sup>ρ</sup>rÞ=M<sup>2</sup>

122 Trends in Modern Cosmology

where

increases at later times.

3. Cosmic growth rate

3.1. Density perturbations

described in the previous section.

initial conditions reduces the Friedmann equations (Eqs. (27) and (28)) to their usual forms:

Vainshtein effect [23], which is a method to recover general relativity below a certain scale. At

The energy density parameter of matter at present in this model is defined as <sup>Ω</sup><sup>m</sup>;<sup>0</sup> <sup>¼</sup> <sup>ρ</sup><sup>m</sup>;<sup>0</sup>=3H<sup>2</sup>

Therefore, in the numerical analysis, the value of rc is fine tuned so that Ωm,0 becomes an assumed value. Thus, the independent parameters of the cosmological model are ω and Ωm,0. For the Galileon model specified by Eqs. (23)–(26) and (31), the effective gravitational constant is given by

<sup>16</sup>πφ <sup>1</sup> <sup>þ</sup> <sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>ξ</sup>ðφÞφ\_ <sup>2</sup>

φ\_ 2 <sup>φ</sup><sup>2</sup> <sup>þ</sup> <sup>8</sup><sup>H</sup> <sup>φ</sup>\_ Þ 2

<sup>φ</sup> � <sup>φ</sup><sup>2</sup>

ξðφÞ φ\_ 4 φ4

δ � 4πGeffρδ≃0; ð34Þ

: ð35Þ

J " #

" #

pl=2 to recover general relativity. Using these

pl. This is the cosmological version of the

; ð32Þ

: ð33Þ

δ ≈ a\_) because

<sup>0</sup>φ0.

Euclid [39] is a European Space Agency mission that is prepared for a launch at the end of 2020. The aim of Euclid is to study the origin of the accelerated expansion of the universe. Euclid will investigate the distance-redshift relationship and the evolution of cosmic structures by measuring shapes and redshifts of galaxies and the distribution of clusters of galaxies over a large part of the sky. Its main subject of research is the nature of dark energy. However, Euclid will cover topics including cosmology, galaxy evolution, and planetary research.

In this study, Euclid parameters are adopted as the growth rate observations. The growth rate can be parameterized by using the growth index <sup>γ</sup>, defined by <sup>f</sup> <sup>¼</sup> <sup>Ω</sup>m<sup>γ</sup> . Mock data of the cosmic growth rate are built based on the 1σ marginalized errors of the growth rate by Euclid. These data are listed in Table 4 in the paper by Amendola et al. [39]. Table 1 shows the 1σ marginalized errors for the cosmic growth rates with respect to each redshift in accordance with Table 4 in [39]. In Figure 1, the mock data of the cosmic growth rate used in this study are plotted.

The mock data are used to compute the statistical χ<sup>2</sup> function. The χ<sup>2</sup> function for the growth rate is defined as

$$\chi^2\_f = \sum\_{i=1}^{14} \frac{\left(f\_{\text{theory}}(z\_i) - f\_{\text{obs}}(z\_i)\right)^2}{\sigma\_{f\_\text{s}}(z\_i)^2} \tag{37}$$

where f obsðziÞ is the future observational (mock) data of the growth rate. The theoretical growth rate f theoryðziÞ is computed as Eq. (35). In Ref. [50], constraints on neutrino masses are estimated based on future observations of the growth rate of the cosmic structure from the Euclid redshift survey.

The estimated errors from the observational technology of Euclid are known, but the center value of future observations is not known. Therefore, the purpose of this study is not to validate the ΛCDM model or modified gravity but to find ways and probabilities to distinguish between the ΛCDM model and modified gravity.


Here, z represents the redshift and σ<sup>f</sup> <sup>g</sup> represents the 1σ marginalized errors of the growth rates.

Table 1. 1σ marginalized errors for the growth rates in each redshift bin based on Table 4 in the study by Amendola et al. [39].

Figure 1. Plot of mock data of the cosmic growth rate.

#### 4. Comparison with observations

#### 4.1. Extended DGP model

Experiment z σ<sup>f</sup> <sup>g</sup> (ref.) Euclid [39] 0.7 0.011

Here, z represents the redshift and σ<sup>f</sup> <sup>g</sup> represents the 1σ marginalized errors of the growth rates.

0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10

Figure 1. Plot of mock data of the cosmic growth rate.

*f*

124 Trends in Modern Cosmology

Table 1. 1σ marginalized errors for the growth rates in each redshift bin based on Table 4 in the study by Amendola et al. [39].

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1

*z*

0.8 0.010 0.9 0.009 1.0 0.009 1.1 0.009 1.2 0.009 1.3 0.010 1.4 0.010 1.5 0.011 1.6 0.012 1.7 0.014 1.8 0.014 1.9 0.017 2.0 0.023

In Figure 2, we plot the probability contours in the (α, σ8)-plane in the extended DGP model from the observational (mock) data of the cosmic growth rate by Euclid. The blue (dark) and light blue (light) contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits, respectively. Part of α = 0 for the horizontal axis corresponds to the ΛCDM model, and part of α = 1 corresponds to the original DGP model. σ<sup>8</sup> is the root mean square (rms) amplitude of over-density at the comoving 8 h�<sup>1</sup> Mpc scale (where <sup>h</sup> is the normalized Hubble parameter <sup>H</sup><sup>0</sup> <sup>¼</sup> <sup>100</sup> <sup>h</sup>kmsec�1Mpc�<sup>1</sup> ). In Figures 3–5, we demonstrate why σ<sup>8</sup> is stringently constrained.

We plot fσ<sup>8</sup> (the product of growth rate and σ8) in the extended DGP model as a function of redshift z for various values of the energy density parameter of matter at the present Ωm,0 in Figures 3–5. In Figure 3, the parameters are fixed by α = 1, σ<sup>8</sup> = 0.6. For the various values of Ωm,0, the theoretical curves seem to revolve around the dashed circle. Hence, the value of σ<sup>8</sup> = 0.6 is incompatible with the observational (mock) data.

In Figure 4, the parameters are fixed by α = 1, σ<sup>8</sup> = 1.0. For the various values of Ωm,0, the theoretical curves seem to revolve around the dashed circle. Hence, the value of σ<sup>8</sup> = 1.0 is incompatible with the observational (mock) data.

In Figure 5, the parameters are fixed by α = 1, σ<sup>8</sup> = 0.855. For the various values of Ωm,0, although the theoretical curves seem to revolve around the dashed circle, some theoretical curves are comparatively close to the observational (mock) data. Hence, the value of σ<sup>8</sup> = 0.855 is compatible with the observational (mock) data in the original DGP model (α = 1).

Figure 2. Probability contours in the (α, σ8)-plane for the extended DGP model from the observational (mock) data of the cosmic growth rate by Euclid. The contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits.

Figure 3. fσ<sup>8</sup> (the product of growth rate and σ8) in the extended DGP model as a function of redshift z for various values of Ωm,0. The parameters are fixed by α = 1, σ<sup>8</sup> = 0.6.

Figure 4. fσ<sup>8</sup> in the extended DGP model as a function of redshift z for various values of Ωm,0. The parameters are fixed by α = 1, σ<sup>8</sup> = 1.0.

Figure 5. fσ<sup>8</sup> in the extended DGP model as a function of redshift z for various values of Ωm,0. The parameters are fixed by α = 1, σ<sup>8</sup> = 0.855.

In Figure 6, we plot the probability contours in the (α, Ωm,0)-plane in the extended DGP model from the observational (mock) data of the cosmic growth rate by Euclid. The red (dark) and pink (light) contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits, respectively. We demonstrate why Ωm,0 is positively correlated with α in Figure 7.

We plot fσ<sup>8</sup> in the extended DGP model as a function of redshift z in Figure 7. The red (solid) line is the theoretical curve for the best-fit parameter in the ΛCDM model (α = 0, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803). In the case of changing only α =1, the growth rate fσ<sup>8</sup> is suppressed due to suppression of the effective gravitational constant (green (dashed) line: α = 1, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803). For α = 1, by tuning the value of Ωm,0 and σ8, the theoretical curve is compatible with the observational (mock) data again (blue (dotted) line: α = 1, Ωm,0 = 0.395, σ<sup>8</sup> = 0.855).

In Figure 8, we add constraints on Ωm,0 for the extended DGP model from the combination of CMB, BAO, and SNIa data (black solid lines) [46] to the probability contours in the (α, Ωm,0) plane by the growth rate (mock) data by Euclid of Figure 6.

Because Ωm,0 is stringently constrained by the cosmic growth rate data from Euclid, we find the ΛCDM model is distinguishable from the original DGP model by combining the growth rate data of Euclid with other observations.

#### 4.2. Kinetic gravity braiding model

0.20

of Ωm,0. The parameters are fixed by α = 1, σ<sup>8</sup> = 0.6.

0.20

0.25

0.30

Ωm,0=0.1 Ωm,0=0.3 Ωm,0=0.5 Ωm,0=0.7 Ωm,0=0.9

0.35

*f* 

by α = 1, σ<sup>8</sup> = 1.0.

σ8

0.40

0.45

0.50

0.5 1.0 1.5 2.0

0.5 1.0 1.5 2.0

Figure 4. fσ<sup>8</sup> in the extended DGP model as a function of redshift z for various values of Ωm,0. The parameters are fixed

*z*

Figure 3. fσ<sup>8</sup> (the product of growth rate and σ8) in the extended DGP model as a function of redshift z for various values

α= 1*,* σ<sup>8</sup> = 1.0

α= 1*,* σ<sup>8</sup> = 0.6

Ωm,0=0.1 Ωm,0=0.3 Ωm,0=0.5 Ωm,0=0.7 Ωm,0=0.9

*z*

0.25

0.30

0.35

*f* 

σ8

126 Trends in Modern Cosmology

0.40

0.45

0.50

In Figure 9, we plot the probability contours in the (n, Ωm,0)-plane in the kinetic gravity braiding model from the observational (mock) data of the cosmic growth rate by Euclid. The red (dark) and pink (light) contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits, respectively.

Figure 6. Probability contours in the (α, Ωm,0)-plane for the extended DGP model from the observational (mock) data of the cosmic growth rate by Euclid. The contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits.

Figure 7. fσ<sup>8</sup> in the extended DGP model as a function of redshift z. The values of the parameters are as follows. Red (solid) line: α = 0, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803. Green (dashed) line: α = 1, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803. Blue (dotted) line: α = 1, Ωm,0 = 0.395, σ<sup>8</sup> = 0.855.

Figure 8. Addition of constraints on Ωm,0 for the extended DGP model from CMB, BAO and SNIa data [46] to the data in Figure 6.

0.00

0.20

0.25

0.30

0.35

*f* 

0.395, σ<sup>8</sup> = 0.855.

σ<sup>8</sup>

0.40

0.45

0.50

0.0 0.2 0.4 0.6 0.8 1.0

α

0.5 1.0 1.5 2.0

α =0, Ωm,0=0.257, σ8=0.803 α =1, Ωm,0=0.257, σ8=0.803 α =1, Ωm,0=0.395, σ8=0.855

Figure 7. fσ<sup>8</sup> in the extended DGP model as a function of redshift z. The values of the parameters are as follows. Red (solid) line: α = 0, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803. Green (dashed) line: α = 1, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803. Blue (dotted) line: α = 1, Ωm,0 =

*z*

Figure 6. Probability contours in the (α, Ωm,0)-plane for the extended DGP model from the observational (mock) data of

the cosmic growth rate by Euclid. The contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits.

0.10

0.20

0.30

Ω*m,*0

128 Trends in Modern Cosmology

0.40

0.50

Figure 9. Probability contours in the (n, Ωm,0)-plane for the kinetic gravity braiding model from the observational (mock) data of the cosmic growth rate by Euclid. The contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits.

We plot fσ<sup>8</sup> in the kinetic gravity braiding model as a function of redshift z in Figure 10. The red (solid) line is the theoretical curve for the best-fit parameter in the ΛCDM model (Ωm,0 = 0.257, σ<sup>8</sup> = 0.803). In the case of the kinetic gravity braiding model for n = 1, the growth rate fσ<sup>8</sup> is enhanced due to enhancement of the effective gravitational constant (Green (dashed) line: n = 1, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803). For n = 100, by tuning the value of Ωm,0 and σ8, the theoretical curve is compatible with the observational (mock) data again (blue (dotted) line: n = 100, Ωm,0 = 0.196, σ<sup>8</sup> = 0.820).

In Figure 11, we add constraints on Ωm,0 for the kinetic gravity braiding model from CMB (black dashed lines) and from SNIa (black solid lines) [30], respectively, to the probability contours in the (n, Ωm,0)-plane by the growth rate (mock) data by Euclid of Figure 9.

In the kinetic gravity braiding model, the allowed parameter region obtained by using only the growth rate data does not overlap with the allowed parameter region obtained from CMB or from SNIa data.

#### 4.3. Galileon model

In Figure 12, we plot the probability contours in the (ω, Ωm,0)-plane in the Galileon model from the observational (mock) data of the cosmic growth rate by Euclid. The red (dark) and pink (light) contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits, respectively. We also plot the constraints on Ωm,0 for the Galileon model from the combination of CMB, BAO, and SNIa (black solid lines) [49].

Figure 10. fσ<sup>8</sup> in kinetic gravity braiding model as a function of redshift z. The values of the parameters are as follows. Red (solid) line: ΛCDM, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803. Green (dashed) line: n = 1, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803. Blue (dotted) line: n = 100, Ωm,0 = 0.196, σ<sup>8</sup> = 0.820.

We plot fσ<sup>8</sup> in the kinetic gravity braiding model as a function of redshift z in Figure 10. The red (solid) line is the theoretical curve for the best-fit parameter in the ΛCDM model (Ωm,0 = 0.257, σ<sup>8</sup> = 0.803). In the case of the kinetic gravity braiding model for n = 1, the growth rate fσ<sup>8</sup> is enhanced due to enhancement of the effective gravitational constant (Green (dashed) line: n = 1, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803). For n = 100, by tuning the value of Ωm,0 and σ8, the theoretical curve is compatible with the observational (mock) data again (blue (dotted) line: n = 100, Ωm,0 = 0.196,

In Figure 11, we add constraints on Ωm,0 for the kinetic gravity braiding model from CMB (black dashed lines) and from SNIa (black solid lines) [30], respectively, to the probability

In the kinetic gravity braiding model, the allowed parameter region obtained by using only the growth rate data does not overlap with the allowed parameter region obtained from CMB or

In Figure 12, we plot the probability contours in the (ω, Ωm,0)-plane in the Galileon model from the observational (mock) data of the cosmic growth rate by Euclid. The red (dark) and pink (light) contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits, respectively. We also plot the constraints on Ωm,0 for the Galileon model from the combination of CMB, BAO, and SNIa

0.5 1 1.5 2

Figure 10. fσ<sup>8</sup> in kinetic gravity braiding model as a function of redshift z. The values of the parameters are as follows. Red (solid) line: ΛCDM, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803. Green (dashed) line: n = 1, Ωm,0 = 0.257, σ<sup>8</sup> = 0.803. Blue (dotted) line: n =

*z*

ΛCDM, Ωm,0=0.257, σ8=0.803 n=1, Ωm,0=0.257, σ8=0.803 n=100, Ωm,0=0.196, σ8=0.820

contours in the (n, Ωm,0)-plane by the growth rate (mock) data by Euclid of Figure 9.

σ<sup>8</sup> = 0.820).

130 Trends in Modern Cosmology

from SNIa data.

4.3. Galileon model

(black solid lines) [49].

0.25

0.30

0.35

0.40

*f* 

100, Ωm,0 = 0.196, σ<sup>8</sup> = 0.820.

σ<sup>8</sup>

0.45

0.50

0.55

Figure 11. Addition of constraints on Ωm,0 for the kinetic gravity braiding model from CMB (black dashed lines) and from SNIa (black solid lines), respectively, [30] to Figure 9.

Figure 12. Probability contours in the (ω, Ωm,0)-plane for the Galileon model from the observational (mock) data of the cosmic growth rate by Euclid. The contours show the 1σ (68.3%) and 2σ (95.0%) confidence limits. The added constraints on Ωm,0 for the Galileon model are from the combination of CMB, BAO, and SNIa (black solid lines) [49].

In the Galileon model, the allowed parameter region obtained by using only the growth rate data do not overlap at all with the allowed parameter region obtained from the combination of CMB, BAO, and SNIa data.

#### 5. Conclusions

The growth rate fσ<sup>8</sup> in the original DGP model is suppressed in comparison with that in the ΛCDM case in the setting of the same value of Ωm,0 due to suppression of the effective gravitational constant. In the case of the kinetic gravity braiding model and the Galileon model, the growth rate fσ<sup>8</sup> is enhanced in comparison with the ΛCDM case in the same value of Ωm,0 due to enhancement of the effective gravitational constant. For the cosmic growth rate data from the future observation, compatible values of Ωm,0 differ according to the model. Furthermore, values of Ωm,0 can be stringently constrained. Thus, we find the ΛCDM model is distinguishable from modified gravity by combining the growth rate data of Euclid with other observations.

The estimated errors from the observational technology of Euclid are known, but the center value of future observations is not known. If the center value of the cosmic growth rate of future observations is different from that of this chapter, the valid model can differ from that of this chapter. However, the methods in this chapter are useful for distinguishing between the ΛCDM model and modified gravity.

In this chapter, assuming the function G5ðφ; XÞ in Horndeski's theory G5ðφ; XÞ ¼ 0, we compute the linear matter density perturbations for the growth rate. In future work, we will study the model having non-zero function G5ðφ; XÞ in Horndeski's theory and investigate the nonlinear effect.

#### Acknowledgements

This work was supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (Grant Number 25400264).

#### Author details

Koichi Hirano

Address all correspondence to: k\_hirano@tsuru.ac.jp

Department of Primary Education, Tsuru University, Yamanashi, Japan

#### References

In the Galileon model, the allowed parameter region obtained by using only the growth rate data do not overlap at all with the allowed parameter region obtained from the combination of

The growth rate fσ<sup>8</sup> in the original DGP model is suppressed in comparison with that in the ΛCDM case in the setting of the same value of Ωm,0 due to suppression of the effective gravitational constant. In the case of the kinetic gravity braiding model and the Galileon model, the growth rate fσ<sup>8</sup> is enhanced in comparison with the ΛCDM case in the same value of Ωm,0 due to enhancement of the effective gravitational constant. For the cosmic growth rate data from the future observation, compatible values of Ωm,0 differ according to the model. Furthermore, values of Ωm,0 can be stringently constrained. Thus, we find the ΛCDM model is distinguishable from modified gravity by combining the growth rate data of Euclid with other

The estimated errors from the observational technology of Euclid are known, but the center value of future observations is not known. If the center value of the cosmic growth rate of future observations is different from that of this chapter, the valid model can differ from that of this chapter. However, the methods in this chapter are useful for distinguishing between the

In this chapter, assuming the function G5ðφ; XÞ in Horndeski's theory G5ðφ; XÞ ¼ 0, we compute the linear matter density perturbations for the growth rate. In future work, we will study the model having non-zero function G5ðφ; XÞ in Horndeski's theory and investigate the

This work was supported by a Grant-in-Aid for Scientific Research from the Japan Society for

CMB, BAO, and SNIa data.

5. Conclusions

132 Trends in Modern Cosmology

observations.

nonlinear effect.

Author details

Koichi Hirano

Acknowledgements

ΛCDM model and modified gravity.

the Promotion of Science (Grant Number 25400264).

Address all correspondence to: k\_hirano@tsuru.ac.jp

Department of Primary Education, Tsuru University, Yamanashi, Japan


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134 Trends in Modern Cosmology


## **The Impact of Baryons on the Large-Scale Structure of the Universe**

Weiguang Cui and Youcai Zhang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68116

#### **Abstract**

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and Gravitation. 2016;48:138. DOI: 10.1007/s10714-016-2129-z

PTP.127.1041

136 Trends in Modern Cosmology

Numerical simulations play an important role in current astronomy researches. Previous dark-matter-only simulations have represented the large-scale structure of the Universe. However, nowadays, hydro-dynamical simulations with baryonic models, which can directly present realistic galaxies, may twist these results from dark-matter-only simulations. In this chapter, we mainly focus on these three statistical methods: power spectrum, two-point correlation function and halo mass function, which are normally used to characterize the large-scale structure of the Universe. We review how these baryon processes influence the cosmology structures from very large scale to quasi-linear and nonlinear scales by comparing dark-matter-only simulations with their hydro-dynamical counterparts. At last, we make a brief discussion on the impacts coming from different baryon models and simulation codes.

**Keywords:** large-scale structure, simulation, statistical methods, hydro-dynamical simulation, baryonic models

#### **1. Introduction**

The core of current research foci in cosmology is to interpret the distribution and properties of observed galaxies in the sky and to understand their formation and evolution. The current standard cosmology model—lambda-cold-dark-matter (ΛCDM) paradigm—provides a general explanation for the galaxy formation and evolution: matter is dominated by the dark matter, which only subjects to gravitational interactions; inside dark matter halo that acts as a gravitational potential well, baryonic matters go through a series of physical processes, such as gas cooling, star forming and death with Supernova feedback.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

From the cosmic micro background (CMB) observation, such as WMAP [1] and Plank [2], matters occupy roughly one-fourth of total energy of the Universe. The rest comes from dark energy. Dark matter is about 20% of the total energy, while baryons only occupy 5% (see more accurate fractions from [3]). At the CMB time (z ~ 1100), matters are distributed nearly 'homogeneous' in the Universe with little fluctuations at small scales. Started from that time, dark matter and baryons are assembled by the gravitational force. They follow a pattern of hierarchical structure formation, where the smaller structures form first, then merge to build massive ones. At very large scale, this structure formation process can be roughly described by the Zel'dovich approximation [4]. However, the formation of structures with gravity is a nonlinear process, which cannot be fully described analytically, especially at small scales. Therefore, building these structures and tracing their evolutions require numerically solving the gravitational equation.

Combining with modern computers, this problem can be solved with numerical methods— N-body simulations, which boosts a new area of research in astronomy. Initially, different numerical methods are developed to simulate only dark matter component, such as particle-mesh (PM), particle-particle/particle-mesh (P3M) and tree-PM algorithms. Dark matter is described numerically by data points/particles that trace a mass element corresponding to a volume element of the early 'homogeneous' universe. Those methods successfully describe the formation of structures by implementing gravitational interactions. Thus, over decades, such simulations have been widely used with little variation in term of physics. Combined with ever decreasing limitations of computer resources and vast improvement in terms of implementations, larger volumes can be explored with increasing resolution to reserve the small-scale information. The properties of cosmology structures (such as cosmic web, voids), halos and even subhaloes are well understood.

#### *However, these simulations cannot give any information of galaxies, which are resident inside dark matter halos*.

To connect these theoretical investigation results with observed galaxies, numerous methods are developed. They can be roughly separated into these three approaches:


extracted from N-body simulations (e.g. [11–17]). Interested readers are encouraging to find the differences between these models (including HOD models) in the nIFTy cosmology comparison project [18] and their following works. As the formation of halos can be traced in the form of halo merger trees, both the formation and interaction of galaxies can be explored within the time frame and the mass resolution explored by the simulation. Although these methods can provide more physical views of galaxy formation, they are still lacking the consistency of co-evolving between baryon and dark matter.

From the cosmic micro background (CMB) observation, such as WMAP [1] and Plank [2], matters occupy roughly one-fourth of total energy of the Universe. The rest comes from dark energy. Dark matter is about 20% of the total energy, while baryons only occupy 5% (see more accurate fractions from [3]). At the CMB time (z ~ 1100), matters are distributed nearly 'homogeneous' in the Universe with little fluctuations at small scales. Started from that time, dark matter and baryons are assembled by the gravitational force. They follow a pattern of hierarchical structure formation, where the smaller structures form first, then merge to build massive ones. At very large scale, this structure formation process can be roughly described by the Zel'dovich approximation [4]. However, the formation of structures with gravity is a nonlinear process, which cannot be fully described analytically, especially at small scales. Therefore, building these structures and tracing their evolutions require numerically solving

Combining with modern computers, this problem can be solved with numerical methods— N-body simulations, which boosts a new area of research in astronomy. Initially, different numerical methods are developed to simulate only dark matter component, such as particle-mesh (PM), particle-particle/particle-mesh (P3M) and tree-PM algorithms. Dark matter is described numerically by data points/particles that trace a mass element corresponding to a volume element of the early 'homogeneous' universe. Those methods successfully describe the formation of structures by implementing gravitational interactions. Thus, over decades, such simulations have been widely used with little variation in term of physics. Combined with ever decreasing limitations of computer resources and vast improvement in terms of implementations, larger volumes can be explored with increasing resolution to reserve the small-scale information. The properties of cosmology structures (such as cosmic

*However, these simulations cannot give any information of galaxies, which are resident inside dark* 

To connect these theoretical investigation results with observed galaxies, numerous methods

i. Much simpler approaches are halo occupation distribution (HOD) models, where observed galaxies are assigned to halos by matching both the halo mass and stellar mass functions (e.g. [5–10]). Such methods are tuned to directly link the luminosity functions with halo mass functions. Thus, they are successful in defining the stellar mass halo mass (SMHM) relation. However, this method cannot provide useful individual galaxy information. Furthermore, the scatter in this relation still remains uncertain and difficult to interpret. It can be constrained by comparing specific galaxies, their environments, the

ii. Other less computationally intensive methods involve applying sub-grid models on the scale of dark matter halos, starting from the accretion of gas by the potential well, following recipes of gas cooling, star forming, supernova (SN) and active galactic nuclei (AGN) feedbacks, at last galaxies are formed and evolved under the halo merger tree. These semi-analytical models (SAMs) have been successfully applied to halo catalogues

are developed. They can be roughly separated into these three approaches:

inter galactic medium (IGM) and their full formation history.

web, voids), halos and even subhaloes are well understood.

the gravitational equation.

138 Trends in Modern Cosmology

*matter halos*.

iii. Hydro-dynamical simulations are the only way to overcome the problem faced by SAMs. They can directly solve the physical processes of the baryonic component on top of the dark matter one, which can provide consistent co-evolution with the same gravitational force. These hydro-simulations require complex implementations of baryonic models with gas described either as (a) numerical data points with associated density (smooth particles hydrodynamics (SPH): [19–22], etc.), (b) grid cells fixed in the volume (cells are refined and unrefined as required to explore highest gas density while neglecting low-density regions with nested mesh or adaptive mesh refinement (AMR): [23–25], etc.) or (c) moving mesh (the gas element is associated with a numerical point within a volume defined from the distribution of nearby mesh point through Voronoi tessellation) [26]. The key aspect is the description of the physical processes within these gas elements. These recipes from SAM can be implanted in hydro-dynamical simulations with moderate modifications. However, hydro-dynamical simulation is suffered from its time-consuming computation, with which the numerous free parameters from these sub-grid baryonic models cannot be easily tuned to represent these observational relations as they are in SAM.

Although hydro-dynamical simulations are the heaviest and most time-consuming tool for connecting the dark part with the luminous part in the Universe, they are irreplaceable in investigating/understanding galaxy formations in a full picture. Those HOD and SAM models, which are used to create mock galaxy catalogues, have been quite successfully in reproducing the observational statistical features, such as the two-point correlation functions, luminosity functions, colour distributions and star formation rates. Nevertheless, they are based on the assumption that baryon processes are independent of dark matter halo formation, which is apparently not true [27–29]. As both observation and simulation are becoming more and more accurate, the back reaction of baryons to dark matter cannot be ignored. Thanks to the Morse's law, more and more efforts are being put in these areas in recent years, for example, [30, 31], the OWLS project [32], the EAGLE project [33], the Illustris project [34] and the Horizon-AGN simulation [35] for these cosmological simulations; the NIHAO project [36] and the FIRE project [37] for these zoom-in simulations. Interested readers refer to the Aquila project [38], the AGORA project [39] and the nIFTy cluster comparison project [40–44] for the comparison of different hydro-dynamical simulation codes. A number of studies based on cosmological hydrodynamic simulations have been recently carried out to analyse in detail the effect of baryonic processes on different properties of the total mass distribution, such as the power spectrum of matter density fluctuations (e.g. [45–48]), the halo correlation functions (e.g. [49, 50]), the halo density profiles (e.g. [29, 51–53]), concentration (e.g. [54, 55]) and shape (e.g. [56, 57]) and the halo mass function (e.g. [30, 31, 58–61]).

In this chapter, we will focus on the impacts of baryons through these comparisons between hydro-dynamical simulations with dark-matter-only simulations and summarize the results in these three aspects: power spectrum, two-point correlation function (2-PCF) and halo mass function (HMF).

#### **2. Chapter**

In the last decade, dark-matter-only simulations have been vastly used to theoretically investigate the large-scale structure of the Universe. Through different statistical methods, such as power spectrum, two-point correlation function, halo mass function and so on, the formation and evolution of the large-scale structures have been clearly characterized by those cosmological dark-matter-only simulations. However, the observed Universe can only show the distribution of baryonic matters at such scales. To connect these theoretical understanding with observations of the large-scale structure of the Universe, we need hydro-dynamical simulations, which can provide a consistent evolution driving by the gravitational force for both dark matter and baryons. With these hydro-simulations, we can directly compare simulations with observations through mock techniques (e.g. [62–64]); explore the galaxy formation process in details; correct and improve our understanding of these baryon models, and so on. In this chapter, we only concentrate on one simple question: How do the baryon processes react on dark matter? This is a question, which these simplified analytical models such as HOD and SAM with ad hoc parameters lack the ability to deal with. As baryons occupy only a small fraction of total matter, we are expecting a very weak effect on the dark matter structures. Nevertheless, baryons dominate at small scales such as in galaxies, where the effect cannot be ignored anymore. Thus, we will address this question with different statistical quantities at different scales, which are listed in Sections 2.1, 2.2 and 2.3.

#### **2.1. Power spectrum**

The power spectrum P(k) (here k is the comoving wavenumber corresponding to a comoving spatial scale *λ* = 2 \_\_ *π <sup>k</sup>* ) is one of the most powerful and basic statistical measurements that describes the distribution of mass in the Universe, and one of the most thoroughly investigated quantities in modelling the structure formation process. Due to the large amount of data from both observation and simulation, the power spectra are measured mostly using the fast Fourier transform (FFT) technique. Lots of methods are used to improve the accuracy of the measurement for power spectrum especially at nonlinear scale, for example [65, 66]. However, such algorithm improvements cannot deal with the power spectrum changes caused by the physical models.

Using the OWSL simulations, [46] studied the influence of baryonic models on matter power spectrum through a comparison between a dark-matter-only (DMONLY) one and hydro-dynamical simulations (REF and AGN). Starting from the same initial condition, these simulations from various models are listed in **Table 1**.


In this chapter, we will focus on the impacts of baryons through these comparisons between hydro-dynamical simulations with dark-matter-only simulations and summarize the results in these three aspects: power spectrum, two-point correlation function (2-PCF) and halo mass

In the last decade, dark-matter-only simulations have been vastly used to theoretically investigate the large-scale structure of the Universe. Through different statistical methods, such as power spectrum, two-point correlation function, halo mass function and so on, the formation and evolution of the large-scale structures have been clearly characterized by those cosmological dark-matter-only simulations. However, the observed Universe can only show the distribution of baryonic matters at such scales. To connect these theoretical understanding with observations of the large-scale structure of the Universe, we need hydro-dynamical simulations, which can provide a consistent evolution driving by the gravitational force for both dark matter and baryons. With these hydro-simulations, we can directly compare simulations with observations through mock techniques (e.g. [62–64]); explore the galaxy formation process in details; correct and improve our understanding of these baryon models, and so on. In this chapter, we only concentrate on one simple question: How do the baryon processes react on dark matter? This is a question, which these simplified analytical models such as HOD and SAM with ad hoc parameters lack the ability to deal with. As baryons occupy only a small fraction of total matter, we are expecting a very weak effect on the dark matter structures. Nevertheless, baryons dominate at small scales such as in galaxies, where the effect cannot be ignored anymore. Thus, we will address this question with different statistical quantities at different scales, which are listed in Sections

The power spectrum P(k) (here k is the comoving wavenumber corresponding to a comoving

describes the distribution of mass in the Universe, and one of the most thoroughly investigated quantities in modelling the structure formation process. Due to the large amount of data from both observation and simulation, the power spectra are measured mostly using the fast Fourier transform (FFT) technique. Lots of methods are used to improve the accuracy of the measurement for power spectrum especially at nonlinear scale, for example [65, 66]. However, such algorithm improvements cannot deal with the power spectrum changes caused by the

Using the OWSL simulations, [46] studied the influence of baryonic models on matter power spectrum through a comparison between a dark-matter-only (DMONLY) one and hydro-dynamical simulations (REF and AGN). Starting from the same initial condition, these

*<sup>k</sup>* ) is one of the most powerful and basic statistical measurements that

function (HMF).

140 Trends in Modern Cosmology

**2. Chapter**

2.1, 2.2 and 2.3.

**2.1. Power spectrum**

spatial scale *λ* = 2 \_\_

physical models.

*π*

simulations from various models are listed in **Table 1**.

**Table 1.** Different variations on the reference simulation that are compared in the chapter. Unless noted otherwise, all simulations use a set of cosmological parameters derived from the WMAP3 results and use identical initial conditions.

In **Figure 1**, they showed the dimensionless matter power spectrum *Δ*<sup>2</sup>(*k*) <sup>=</sup> *<sup>k</sup>* <sup>3</sup> *<sup>P</sup>*(*<sup>k</sup>* ) /<sup>2</sup> *<sup>π</sup>*<sup>2</sup> on the upper panel and the relative difference to the DMONLY run on the lower panel. It is clear that the contribution of the baryons is significant: they decrease the power by more than 1% for k ≈ 0.8–5 h Mpc−1 by comparing the DMONLY simulation with the REF simulation; the power is greatly increased at smaller scales < 1 h−1 Mpc (k ≥ 6 h Mpc−1). The decreased power is caused by the gas pressure, which smooths the density field relative to that expected from dark matter alone. While, the increased power in the REF simulation is because radiative cooling enables gas to cluster on smaller scales than the dark matter. These results confirm the findings of previous studies, at least qualitatively (e.g. [45, 67, 68]). However, with the AGN feedback, which is required to match observations of groups and clusters, its effect on the power spectrum is enormous: the power is reduced by ≥10% for k ≥ 1 h Mpc−1. This could be caused by that large amounts of gas are moved to large radii due to the AGN feedback (see also [43]). Because the AGN normally reside in massive and thus strongly clustered objects, the power is suppressed out to scales, where the removed gas can reach.

In **Figure 2**, they showed power spectra from the REF (left panel) and AGN (right panel) simulations at z = 0. As indicated on the top left of each panel, different components are shown by different colour lines. The power spectrum for DMONLY (dashed black lines) is shown as a reference. The power spectra on top row is calculated with δ<sup>i</sup> <sup>≡</sup>(ρ<sup>i</sup> <sup>−</sup> ¯ <sup>ρ</sup>ı)/¯ ρı . This definition guarantees that all power spectra from component i converge on large scales, thus enabling a straightforward comparison of their shapes. The bottom row, on the other hand, shows the

**Figure 1.** Upper panel: the total matter power spectra of REF (top solid line with highest value at k ~ 500), AGN (middle solid line) and DMONLY (bottom solid line), at redshift z = 0. Lower panel: the power spectrum difference between the two hydro runs and the DMONLY one; solid (dashed) curves indicate that the power is higher (lower) than for DMONLY. The dotted, horizontal line indicates the 1% level. This figure is from Ref. [46]. (note: Please refer to the online publication for a colorful figure).

power spectra of δ<sup>i</sup> ≡(ρ<sup>i</sup> − ¯ ρtot ) /¯ <sup>ρ</sup>tot, which allows one to estimate the contributions of different components to the total matter power spectrum. From the top-left panel, the baryonic components trace the dark matter well at the largest scales. However, significant differences exist for λ ≤ 10 h−1 Mpc. At scales of several hundred kpc and smaller, the difference between the CDM component (also the total component) of the reference simulation and DMONLY exceeds the change between the latter and the analytic models. This is caused by the back-reaction of the baryons on the dark matter. On the bottom left panel of **Figure 2**, it is clear that CDM dominates the power spectrum on large scales. While the contribution of baryons is significant for λ ≤ 0.1 h−1 kpc and dominates below 0.06 h−1 Mpc. The strong small-scale baryonic clustering is a direct consequence of gas cooling and galaxy formation. For the baryonic component, the baryonic power spectrum is dominated by gas component on large scales, which has a flatter power for λ ≤ 1 h−1 Mpc (corresponding to the virial radii of groups of galaxies) and a slightly

The Impact of Baryons on the Large-Scale Structure of the Universe http://dx.doi.org/10.5772/68116 143

**Figure 2.** Decomposing the z = 0 total power spectra into the contributions from different components. The left- and right-hand columns show results for REF and AGN. For reference, the power spectrum for DMONLY is shown with dashed black lines. This figure is from Ref. [46].

steeper power again for λ ≤ 0.1 h−1 Mpc (galaxy scales). While the stellar power spectrum takes control for λ ≤ 1 h−1 Mpc. The inclusion of AGN feedback greatly impacts the matter power spectrum on a wide range of scales. Comparing the top panels of **Figure 2**, the power in both the gas and stellar components is decreased by AGN feedback for λ ≤ 1 h−1 Mpc. Through comparing the two bottom panels, the stellar power spectrum is reduced the most: about an order of magnitude on the largest scales; more than two orders of magnitude on the smallest scales. This is an expected result of the AGN feedback, which suppresses star formation, as required to solve the overcooling problem. The gas power spectrum is also dramatically dropped as a consequence of the AGN feedback. The suppression of baryonic structure by AGN feedback also makes the dominant dark matter component of the power spectrum on small scales down.

In addition, different baryonic models investigated in [46] (see more details in their **Figure 3**) showed significant changes of power spectrum at non-linear scale. It means that these baryonic models need very subtle tuning of their parameters to represent the observational results.

#### **2.2. Two-point correlation function**

power spectra of δ<sup>i</sup>

142 Trends in Modern Cosmology

publication for a colorful figure).

≡(ρ<sup>i</sup> − ¯ ρtot ) /¯

<sup>ρ</sup>tot, which allows one to estimate the contributions of different

components to the total matter power spectrum. From the top-left panel, the baryonic components trace the dark matter well at the largest scales. However, significant differences exist for λ ≤ 10 h−1 Mpc. At scales of several hundred kpc and smaller, the difference between the CDM component (also the total component) of the reference simulation and DMONLY exceeds the change between the latter and the analytic models. This is caused by the back-reaction of the baryons on the dark matter. On the bottom left panel of **Figure 2**, it is clear that CDM dominates the power spectrum on large scales. While the contribution of baryons is significant for λ ≤ 0.1 h−1 kpc and dominates below 0.06 h−1 Mpc. The strong small-scale baryonic clustering is a direct consequence of gas cooling and galaxy formation. For the baryonic component, the baryonic power spectrum is dominated by gas component on large scales, which has a flatter power for λ ≤ 1 h−1 Mpc (corresponding to the virial radii of groups of galaxies) and a slightly

**Figure 1.** Upper panel: the total matter power spectra of REF (top solid line with highest value at k ~ 500), AGN (middle solid line) and DMONLY (bottom solid line), at redshift z = 0. Lower panel: the power spectrum difference between the two hydro runs and the DMONLY one; solid (dashed) curves indicate that the power is higher (lower) than for DMONLY. The dotted, horizontal line indicates the 1% level. This figure is from Ref. [46]. (note: Please refer to the online

> The correlation function, ξ(r), through the calculation of the excess probability to a random distribution to find the possibility of two objects at a given separation r. It is a very useful

**Figure 3.** Upper panel: the subhalo autocorrelation function for the three simulations: DMONLY (solid), REF (dashed) and AGN (dot-dashed lines). Different total subhalo masses results are shown with different colours, and the number of objects in each bin is indicated in the legend. The median rvir of the subhaloes are indicated by vertical dotted lines. Middle panel: the relative difference of subhalo clustering between REF and DMONLY. For radii, may biased due to subhalo non-detections, the curves are shown in grey. Bottom: similar to middle panel but for AGN and DMONLY. Both REF and AGN show increased clustering with a stronger effect on smaller scales. This figure is from Ref. [50].

measure of the clustering of these objects as a function of scale. Comparing power spectrum, correlation can provide different views of cosmological structures. Using galaxy as a tracer, it can be used to investigate the clustering of dark matter halo, for example, [69, 70].

Following their work on power spectrum [46], they studied the baryon effect on two-point correlation functions in [50] with the OWLS simualtions. A parallelized brute force approach is used to calculate the correlation function. Through simple pair counts, ξ(r) can be easily expressed as:

$$
\varepsilon\_{\chi\chi}(r) = \frac{D \, D\_{\chi\chi}(r)}{R \, R\_{\chi\chi}(r)} - 1. \tag{1}
$$

Here, X and Y denote two (not necessarily distinct) sets of objects (e.g. subhaloes and particles or haloes and haloes), DDXY (r) is the number of unique pairs consisting of an object from set X and an object from set Y separated by a distance r, and RRXY (r) is the expected number of pairs at this separation if the positions of the objects in these sets were random.

Subhalos from their simulations are identified by the SUBFIND algorithm [71, 72] inside Friends-of-Friends haloes. Interested readers refer to Ref. [73] for the comparison of different subhalo finding codes, as well as the effect from the included baryonic models. Top panel of **Figure 3** shows the subhalo autocorrelation function, ξss(r), for three different simulations: DMONLY, REF and AGN. Different colours indicate different subsamples, selected by the total mass of the subhaloes, Msh,tot. The median virial radii of subhaloes in each mass bin are indicated by vertical dotted lines. These radii are similar to the scales at which the subhalo correlation functions for DMONLY turn over. It is clear that subhalo clustering in the darkmatter-only simulation behaves quite differently from that in the baryonic models, especially on small scales (r ≤ 1 h−1 Mpc).

The middle and bottom panels show the relative 2-PCF difference between REF (middle)/AGN (bottom) and DMONLY simulation. All subhaloes in the baryonic simulations are typically ~10% more strongly clustered on large scales than their dark-matter-only counterparts. This difference is due to the reduction of subhalo mass caused by baryonic processes. For the larger subhaloes, 10<sup>13</sup> < Msh,tot/[M⊙/h] < 10<sup>14</sup>, this offset is somewhat larger when AGN feedback is included, because supernova feedback alone cannot change the subhalo mass by as much as it can for lower halo masses [74]. The differences between the baryonic and dark-matter-only simulations increase rapidly for r < 2rvir, at least for Msh,tot < 1014 h−1M⊙. Subhaloes from the REF simulation show significant larger clustering signal on small scales than from the AGN simulation. This seems to contradict to the results from the previous section. This is because at fixed mass range, subhaloes from the AGN simulation are less compact compared with these from the REF simulation. Due to the additional form of feedback in the AGN run, more material from the centre are pushed into outer radii, which results in a lower concentration. Similar to the subhalo 2-PCF, the galaxy 2-PCFs (ξgg(r)) are very similar between REF and AGN at smaller galaxy mass bins. However, it is worth to note that there is a significant difference at the largest halo mass bin, which is shown in [50].

**Figure 4** shows the subhalo-mass 2-PCF, ξsm(r) on the upper panel; the fractional difference between ξREF sm(r) and ξDMONLY sm(r) on the middle panel; the fractional difference between the ξAGN sm(r) and ξDMONLY sm(r) on the bottom panel. Again, subhaloes are generally more strongly clustered with matter in the REF and AGN than in DMONLY for scales r > rvir. There is also a constant ~5% difference in favour of both REF and AGN simulations on large scales, regardless of subhalo mass. The largest differences can be up to 40% (20% for AGN) higher on intermediate scales for the lowest mass subhaloes. If sufficiently small scales are considered, this difference can be much higher for any subhalo mass. The AGN run does show a stronger

measure of the clustering of these objects as a function of scale. Comparing power spectrum, correlation can provide different views of cosmological structures. Using galaxy as a tracer, it

**Figure 3.** Upper panel: the subhalo autocorrelation function for the three simulations: DMONLY (solid), REF (dashed) and AGN (dot-dashed lines). Different total subhalo masses results are shown with different colours, and the number of objects in each bin is indicated in the legend. The median rvir of the subhaloes are indicated by vertical dotted lines. Middle panel: the relative difference of subhalo clustering between REF and DMONLY. For radii, may biased due to subhalo non-detections, the curves are shown in grey. Bottom: similar to middle panel but for AGN and DMONLY. Both REF and AGN show increased clustering with a stronger effect on smaller scales. This figure is from Ref. [50].

Following their work on power spectrum [46], they studied the baryon effect on two-point correlation functions in [50] with the OWLS simualtions. A parallelized brute force approach

can be used to investigate the clustering of dark matter halo, for example, [69, 70].

144 Trends in Modern Cosmology

**Figure 4.** As Figure 3, but now for the subhalo-mass cross-correlation function, ξsm(r). This figure is from Ref. [50].

decrease in clustering up to scales r ~ 0.1 h−1 Mpc. While the ξsm(r) at smaller mass bins from REF also show similar decrease. It is worth to note the strongly non-monotonic changes of the subhalo-mass 2-PCF between the two baryonic runs and the DMONLY one. This can be caused by the interplay between the changes in both the total subhalo mass and its mass profile. On the one hand, the lowered halo masses in the baryonic simulations tend to increase clustering at fixed mass on all scales. On the other hand, galaxy formation dissipates smoothed gas component and causes the inner halo profile to steepen (increasing clustering on small scales); the associated feedback causes the outer layers of the halo to expand (decreasing clustering on intermediate scales). These conclusions are proved in Ref. [50] through the 2-PCF ξsm(r) that have been linked between a baryonic simulation and DMONLY which are selected based on their mass in the latter. This procedure removes the effects of changes in the subhalo masses, leaving only the effect on the mass profiles and the changes in the positions of the subhaloes.

Through these comparisons, the major reason for the increased clustering in the hydrodynamical simulations is the lowering of the mass of objects due to galaxy formation with strong feedback. However, secondary effects, such as the resulting changes in the dynamics and density profiles of haloes, are also expected to be significant. Interestingly, Despali and Vegetti [75] find that the presence of baryons reduces the number of subhaloes, especially at the low mass end, by different amounts depending on the model. The variations in the subhalo mass function are strongly dependent on those in the halo mass function, which is shifted by the effect of stellar and AGN feedback. We will investigate these effects on the halo mass function in Section 2.3.

#### **2.3. Halo mass function**

Different to the power spectrum and 2-PCF, HMF shows another interesting statistic of the large-scale structure. Located on the central structure which connects theory with observation, HMF provides the statistical view of the halo abundance. The two most common methods used for halo identification in simulations are the FoF algorithm (e.g. [76]) and the spherical overdensity (SO) algorithm (e.g. [77]). Interested readers refer to Ref. [78] for the comparison of different halo finding codes.

A series of three versions of cosmological simulations are used in Ref. [31] for their study. Starting from the same initial condition, these simulations share the same number of dark matter particles (1024<sup>3</sup> ) and gas particles (1024<sup>3</sup> ) within a simulation box size of 410 h−1 Mpc. A first hydro-dynamical simulation includes radiative cooling, star formation and kinetic SN feedback (CSF hereafter), while the second one also includes the effect of AGN feedback (AGN<sup>1</sup> hereafter). As for the DM simulation, it simply replaces the gas particle by collisionless particles, so as to have the same description of the initial density and velocity fields as in the hydro-dynamical simulations.

FoF HMFs are compared on the top panel of **Figure 5** between the three different versions of simulations. While the bottom panel shows the halo number ratio in a mass bin respected to the DM simulation. The baryonic effect from the CSF with respect to the DM case has clear redshift evolution as well as halo mass dependence. From higher redshift to lower redshift, the HMF ratios between the CSF and DM runs decrease from ∼1.6 to ∼1.1, with a weak increasing trend along halo mass changes. Quite remarkably, including AGN feedback in the baryonic model reduces the difference with respect to the DM-only case: the HMF ratio drops to about

decrease in clustering up to scales r ~ 0.1 h−1 Mpc. While the ξsm(r) at smaller mass bins from REF also show similar decrease. It is worth to note the strongly non-monotonic changes of the subhalo-mass 2-PCF between the two baryonic runs and the DMONLY one. This can be caused by the interplay between the changes in both the total subhalo mass and its mass profile. On the one hand, the lowered halo masses in the baryonic simulations tend to increase clustering at fixed mass on all scales. On the other hand, galaxy formation dissipates smoothed gas component and causes the inner halo profile to steepen (increasing clustering on small scales);

**Figure 4.** As Figure 3, but now for the subhalo-mass cross-correlation function, ξsm(r). This figure is from Ref. [50].

146 Trends in Modern Cosmology

<sup>1</sup> Note that this simulation with AGN feedback is also named AGN. It is different in simulation box, resolution and models from the OWLS AGN simulation shown before.

<sup>2</sup> https://github.com/ilaudy/PIAO.

**Figure 5.** Upper panel: FoF HMFs from DM, CSF and AGN. The HMFs from CSF are always higher than the results from AGN and DM. Different redshifts are shown with different line styles (see the lower left legend for details). Lower panel: relative difference between the HMFs from the hydro-dynamic simulations and from the DM simulation from all four redshifts. This figure is from Ref. [31]. (Please refer to the online publication for a colorful figure).

unity for massive haloes with MFoF ≈ 1014 h−1 M⊙, while at smaller halo mass it decreases to ∼0.9 for MFoF ≈ 1013 h−1 M⊙. Different to the CSF case, there is no clear redshift evolution in these ratios from z = 1 to 0. At the highest redshift, z = 2.2, this HMF ratio keeps fluctuating around 1. This could be a consequence of the limited statistics of haloes due to the finite box size.

Using the PIAO<sup>2</sup> code [79], the SO haloes are identified with three overdensities Δ<sup>c</sup> = 2500, 500 and 200. These HMFs are shown in **Figure 6** from left to right top panels, respectively. While the HMF ratios from the CSF and AGN simulations with respect to the one from DM run are shown in lower panels. Baryons show a larger impact on the HMF at the higher overdensity. With Δ<sup>c</sup> = 2500, the ratio between the CSF and DM HMFs shows a redshift evolution ranging from ∼1.4 at z = 0 to ∼2.5 at z = 2.2, but with no significant dependence on the halo mass. At lower overdensities, the redshift evolution becomes weaker and the differences with respect to the DM case are also reduced. When AGN feedback is included in the hydro-dynamical simulation, the corresponding HMF drops below the HMF from the DM simulation, by an amount that decreases for lower Δ<sup>c</sup> values, with no evidence for redshift dependence on the HMF difference. Generally speaking, the baryonic effect on the HMF goes in the same direction, qualitatively independent of whether FoF or SO halo finders are used. However, as expected, quantitative differences between FoF and SO results are found, especially for the AGN case. This is rooted in intrinsic algorithm difference of these halo finder methods (we refer interested readers to Ref. [78] for details).

The three simulations share the same dark matter particles, which have the same progressive identification number (ID). Therefore, we can use the halo from the DM simulation as the reference. The halo in the CSF or AGN simulation is defined as the counterpart of the DM halo, if it includes the largest number of DM particles belonging to the latter. In their paper,

**Figure 6.** Similar to Figure 5 but for the HMFs of SO haloes. Three different overdensities Δ<sup>c</sup> = 2500 (left-hand panel), 500 (middle panel) and 200 (right-hand panel) are used to identify SO haloes. Again, top panels show the SO HMFs, while bottom panels show the HMF difference. This figure is from Ref. [31].

unity for massive haloes with MFoF ≈ 1014 h−1 M⊙, while at smaller halo mass it decreases to ∼0.9 for MFoF ≈ 1013 h−1 M⊙. Different to the CSF case, there is no clear redshift evolution in these ratios from z = 1 to 0. At the highest redshift, z = 2.2, this HMF ratio keeps fluctuating around 1. This could be a consequence of the limited statistics of haloes due to the finite box size.

**Figure 5.** Upper panel: FoF HMFs from DM, CSF and AGN. The HMFs from CSF are always higher than the results from AGN and DM. Different redshifts are shown with different line styles (see the lower left legend for details). Lower panel: relative difference between the HMFs from the hydro-dynamic simulations and from the DM simulation from all four

redshifts. This figure is from Ref. [31]. (Please refer to the online publication for a colorful figure).

148 Trends in Modern Cosmology

a matching rate is defined as the ratio of matched to total number of dark matter particles in the DM halo. To avoid multiple-to-1 matching from CSF/AGN simulation to the DM one, only haloes with matching rate larger than 0.5 are selected.

**Figure 7** shows the halo mass ratios between these matched haloes. Red points indicate the halo pairs, which are coming from CSF and DM simulations, while green points are for the pairs from AGN and DM simulations. The thick lines show the mean value of these data points computed within each mass bin (magenta for CSF and blue for AGN, respectively). For the CSF-DM halo pairs, the increased halo mass is almost independent of redshift. At each redshift, the ratio shows a weak decrease with halo mass, from ~ 1.1 at M500 = 1012.5 h-1 M⊙ to ~ 1.05 at M500 > 1013.5 h-1 M⊙, then becoming constant. However, for the AGN-DM pairs, the strong AGN feedback makes the ratio go in the opposite direction (decreased halo masses). Thereby, this will result in a decreased HMF, which has been shown in **Figure 6**. There also shows no evidence of redshift evolution for the halo mass ratio, at least below z = 1.0. However, this ratio

**Figure 7.** The ratio of masses of matched SO haloes as function of MDM computed for Δ<sup>c</sup> = 500 at four different redshifts. Each data point indicates a halo mass ratio between the matched CSF or AGN halo to its corresponding DM one. These misty data points above the horizon dashed lines are normally from CSF run, while lower ones are coming from AGN run. The mean values of these ratios within each mass bin are shown by thick magenta (CSF) and blue lines (AGN), respectively. The solid black lines are the best-fitting results for the mass correction, which are used for the HMF correction in Ref. [31]. This figure is from Ref. [31]. (Please refer to the online publication for a colorful figure).

shows a strong dependence on the halo mass, which increases from ~ 0.8 at M500 = 1012.5 h-1 M<sup>⊙</sup> to ~ 1 for the most massive haloes found in their simulation box.

#### **2.4. Summary**

a matching rate is defined as the ratio of matched to total number of dark matter particles in the DM halo. To avoid multiple-to-1 matching from CSF/AGN simulation to the DM one, only

**Figure 7** shows the halo mass ratios between these matched haloes. Red points indicate the halo pairs, which are coming from CSF and DM simulations, while green points are for the pairs from AGN and DM simulations. The thick lines show the mean value of these data points computed within each mass bin (magenta for CSF and blue for AGN, respectively). For the CSF-DM halo pairs, the increased halo mass is almost independent of redshift. At each redshift, the ratio shows a weak decrease with halo mass, from ~ 1.1 at M500 = 1012.5 h-1 M⊙ to ~ 1.05 at M500 > 1013.5 h-1 M⊙, then becoming constant. However, for the AGN-DM pairs, the strong AGN feedback makes the ratio go in the opposite direction (decreased halo masses). Thereby, this will result in a decreased HMF, which has been shown in **Figure 6**. There also shows no evidence of redshift evolution for the halo mass ratio, at least below z = 1.0. However, this ratio

haloes with matching rate larger than 0.5 are selected.

150 Trends in Modern Cosmology

**Figure 7.** The ratio of masses of matched SO haloes as function of MDM computed for Δ<sup>c</sup>

Each data point indicates a halo mass ratio between the matched CSF or AGN halo to its corresponding DM one. These misty data points above the horizon dashed lines are normally from CSF run, while lower ones are coming from AGN run. The mean values of these ratios within each mass bin are shown by thick magenta (CSF) and blue lines (AGN), respectively. The solid black lines are the best-fitting results for the mass correction, which are used for the HMF correction in Ref. [31]. This figure is from Ref. [31]. (Please refer to the online publication for a colorful figure).

= 500 at four different redshifts.

Stepping from dark-matter-only to hydro-dynamical simulations allows us to view the galaxy formation and evolution in the Universe in a self-consistent and realistic way. Hydro-dynamical simulations estimate tight connections between theoretical and observational researches, therefore providing a perfect test lab for examining theories. Through these comparisons between state-of-the-art hydro-dynamical simulations and dark-matter-only ones, we summarized the recent findings of baryonic effects on the large-scale structure of the Universe by showing the changes on power spectrum, two-point correlation function and halo mass functions:


Besides these statistics methods investigated in upper paragraphs, baryonic processes can also leave an impact on cosmological structures, such as cosmic webs, sheets and voids. Using the EAGLE simulation, Paillas et al. [80] studied the effect of baryons on void statistics. They found that the dark-matter-only simulation produces 24% more voids than the hydrodynamical one, but this difference comes mainly from voids with radii smaller than 5 Mpc. They claimed that there are no significant differences in the density profiles between voids in hydro-dynamical and its dark-matter-only counterpart.

However, as we already see, all these results strongly depend on the included baryonic models and simulation codes. There is no guarantee of a perfect model yet, especially that most of the implanted baryonic models are based on observational relations. Starting from the same initial condition of a galaxy cluster, the recent nIFTy project [40–44] has made vast comparisons between different simulation codes as well as baryonic models included in them. There is a good agreement between these simulation codes for the dark-matter-only runs. A larger disagreement is shown between the classic SPH codes and mesh/modern SPH/moving mesh codes for the non-radiative hydro-dynamical runs. In the full physics runs, the largest difference is lying between the runs with AGN feedback and the ones without AGN feedback. However, even inside both families, there are a lot of variances between different simulation codes.

To simulate the observed Universe, more efforts are needed to understand the sub-grid baryonic models, such as their parameter choice, resolution and method dependence. To understand and pin down these sub-grid models, we need direct and detailed comparisons between the simulation results and observational ones. Thus, a one-to-one comparison is much helpful than the statistical relations. These constrained simulation projects aiming to represent the observed Universe, the ELUCID project [81–83], the CLUES project [84], [85] and the APOSTLE project [86], point to the direction of future simulation studies.

#### **Acknowledgements**

The authors are particularly grateful to Dr Marcel van Daalen for freely using the figures from his papers [46] and [50]. W.C. would like to thank his wife Ms Yufang Liu for her kind help. Y.Z. acknowledges the support from the 973 Program (No. 2015CB857002).

### **Author details**

Weiguang Cui<sup>1</sup> \* and Youcai Zhang<sup>2</sup>

\*Address all correspondence to: cuiweiguang@gmail.com

1 Dover Court, Western Australia, Australia

2 ShangHai Astronomical Observatory, Shanghai, China

#### **References**


[4] Y. B. Zel'dovich, "Gravitational instability: an approximate theory for large density perturbations," *Astron. Astrophys*., Vol. 5, pp. 84–89, 1970.

implanted baryonic models are based on observational relations. Starting from the same initial condition of a galaxy cluster, the recent nIFTy project [40–44] has made vast comparisons between different simulation codes as well as baryonic models included in them. There is a good agreement between these simulation codes for the dark-matter-only runs. A larger disagreement is shown between the classic SPH codes and mesh/modern SPH/moving mesh codes for the non-radiative hydro-dynamical runs. In the full physics runs, the largest difference is lying between the runs with AGN feedback and the ones without AGN feedback. However, even inside both families, there are a lot of variances between different simulation codes.

To simulate the observed Universe, more efforts are needed to understand the sub-grid baryonic models, such as their parameter choice, resolution and method dependence. To understand and pin down these sub-grid models, we need direct and detailed comparisons between the simulation results and observational ones. Thus, a one-to-one comparison is much helpful than the statistical relations. These constrained simulation projects aiming to represent the observed Universe, the ELUCID project [81–83], the CLUES project [84], [85]

The authors are particularly grateful to Dr Marcel van Daalen for freely using the figures from his papers [46] and [50]. W.C. would like to thank his wife Ms Yufang Liu for her kind help.

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1 Dover Court, Western Australia, Australia

\*Address all correspondence to: cuiweiguang@gmail.com

2 ShangHai Astronomical Observatory, Shanghai, China

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## **Cosmological Consequences of a Quantum Theory of Mass and Gravity**

Brian Albert Robson

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68410

#### **Abstract**

The understanding of several cosmological problems that has been obtained from the development of the Generation Model (GM) of particle physics is presented. The GM is presented as a viable simpler alternative to the Standard Model (SM). The GM considers the elementary particles of the SM to be composite particles and this substructure leads to new paradigms for both mass and gravity, which in turn lead to an understanding of several cosmological problems: the matter-antimatter asymmetry of the universe, dark matter and dark energy. The GM provides a unified origin of mass and the composite nature of the leptons and quarks of the GM leads to a solution of the cosmological matterantimatter asymmetry problem. The GM also provides a new universal quantum theory of gravity in terms of a residual interaction of a strong color-like interaction, analogous to quantum chromodynamics (QCD). This very weak residual interaction has two important properties: antiscreening and finite range, that provide an understanding of dark matter and dark energy, respectively, in the universe.

**Keywords:** generation model, gravity, dark matter, MOND theory, dark energy, antimatter, big bang

#### **1. Introduction**

The main purpose of this chapter is to present the contributions to an understanding of several cosmological problems that have been obtained from the development of an alternative to the Standard Model (SM) of particle physics [1]. This alternative model, named the Generation Model (GM) [2], not only describes all the transition probabilities for interactions involving all the elementary particles of the SM but also provides new paradigms, including the origin of both mass [3] and gravity [4].

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The GM considers the elementary particles, the six leptons, six quarks and three weak bosons, of the SM to be composite particles. Their constituents are called rishons and antirishons. It will be demonstrated that this substructure leads to new paradigms for both mass and gravity, which in turn lead to an understanding of several cosmological problems: the matterantimatter asymmetry of the universe, dark matter and dark energy.

It will be shown that the GM provides a unified origin of mass in which the mass of a particle is described in terms of the energy content of its constituents [5], while in the SM, the elementary leptons, quarks and weak bosons, are described in terms of the Higgs mechanism [6, 7]. It will be demonstrated that the composite nature of the leptons and quarks of the GM leads to a solution of the cosmological matter-antimatter asymmetry problem.

In particular, it will be shown that the GM provides a new universal theory of gravity in terms of a residual interaction of a strong color-like interaction [4], analogous to quantum chromodynamics (QCD) [8], the theory of strong interactions, in the SM. This very weak residual interaction has two important properties: asymptotic freedom (antiscreening) [9, 10] and finite range [11], that provide an understanding of dark matter [12] and dark energy [13], respectively, in the universe.

Section 2 discusses the *incompleteness* of the SM and examines some basic assumptions upon which the SM has been built. Section 3 introduces the GM which replaces several *dubious assumptions* within the SM by different and simpler assumptions, without destroying any agreement with experiment. These lead to the notion that the elementary particles, the six leptons, six quarks and three weak bosons of the SM are *composite particles*. Section 4 describes the development of a viable composite GM in which the elementary particles of the SM are composites of elementary spin-1/2 particles called rishons and antirishons. Section 5 discusses a *new paradigm* provided by the composite GM for *all* mass, and indicates a qualitative understanding of the mass hierarchy of leptons and quarks.

Section 6 presents a solution to the *cosmological matter-antimatter asymmetry problem* in terms of the composite GM. The composite GM also leads to a *new paradigm* for *gravity*. This is discussed in Section 7 in which a *quantum theory of gravity* is described. This new law of universal gravitation is shown to provide an understanding of both *dark matter* (Section 8) and *dark energy* (Section 9).

Section 10 discusses the possibility that the photon may be considered to be the singlet state of the corresponding QCD color octet binding together the constituents of the leptons and quarks. Finally, Section 11 gives a brief summary and conclusion.

#### **2. Standard model of particle physics**

The current formulation of the Standard Model (SM) of particle physics [1] was essentially finalized in the mid-1970s following the experimental confirmation of the existence of quarks [14, 15]. However, the model is regarded by most physicists as *incomplete* since it provides little understanding of several empirical observations. First, it does not explain the occurrence of three "generations" of the elementary particles of the SM [16]: the first generation comprising the up and down quarks, the electron and its neutrino, the second generation comprising the charmed and strange quarks, the muon and its neutrino and the third generation comprising the top and bottom quarks, the tau and its neutrino. Each generation behaves similarly except for mass. Second, it does not provide a unified description of the origin of mass nor describe the mass hierarchy of leptons and quarks. The SM also fails to describe the nature of gravity, dark matter, dark energy or the cosmological matter-antimatter asymmetry problem.

Because of the incompleteness of the SM, the basic assumptions upon which the SM has been erected have been examined [17]. There are three basic assumptions, which are considered to be *dubious* and also present major stumbling blocks preventing progress beyond the SM. These are (i) the assumption of a diverse complicated scheme of additive quantum numbers to classify its elementary particles; (ii) the assumption of weak isospin doublets in the quark sector to accommodate the universality of the charge-changing weak interactions and (iii) the assumption that the weak interactions are fundamental interactions described by a local gauge theory.

The elementary particles of the SM are six leptons: electron (*e*<sup>−</sup> ), electron neutrino (*ν<sup>e</sup>* ), muon (*μ*<sup>−</sup> ), muon neutrino (*νμ*), tau (*τ*<sup>−</sup> ), tau neutrino (*ντ* ) and six quarks: up (*u*), down (*d*), charmed (*c*), strange (*s*), top (*t*) and bottom (*b*). These twelve particles all have spin-1/2 and are allotted several additive quantum numbers: charge *Q*, lepton number *L*, muon lepton number *Lμ*, tau lepton number *L<sup>τ</sup>* , baryon number *A*, strangeness *S*, charm *C*, bottomness *B* and topness *T* (see **Table 1**). For each particle additive quantum number *N*, the corresponding antiparticle has the additive quantum number − *N*.

**Table 1** demonstrates that this classification of the elementary particles in the SM is *nonunified*, since, except for charge, the leptons and quarks are allotted different kinds of additive quantum numbers. This diverse complicated scheme of additive quantum numbers for its elementary particles constitutes a basic problem for the SM, especially if the leptons and quarks are *not* elementary particles (see Section 4).

A second problem with the SM involves the method it uses to accommodate the universality of the charge-changing (CC) weak interactions, mediated by the *W*<sup>+</sup> and *W*<sup>−</sup> bosons.

In the SM, the mass eigenstate leptons have weak isospin 1/2, whose third component is related to both charge and lepton number. Restricting the discussion in this chapter to only the first two generations for simplicity means that the two neutrinos interact with their corresponding charged leptons with the full strength of the CC weak interaction and do not interact at all with the other charged lepton. This is guaranteed by the conservation of lepton numbers.

On the other hand the universality of the CC weak interactions in the quark sector is treated differently. It is assumed that the *u* and *c* quarks form weak isospin doublets with so-called weak eigenstate quarks *d*′ and *s*′, respectively, where again for simplicity, the small mixing with the third generation quark *b* is neglected:

$$d' = d\cos\theta \star s\sin\theta,\tag{1}$$

and

The GM considers the elementary particles, the six leptons, six quarks and three weak bosons, of the SM to be composite particles. Their constituents are called rishons and antirishons. It will be demonstrated that this substructure leads to new paradigms for both mass and gravity, which in turn lead to an understanding of several cosmological problems: the matter-

It will be shown that the GM provides a unified origin of mass in which the mass of a particle is described in terms of the energy content of its constituents [5], while in the SM, the elementary leptons, quarks and weak bosons, are described in terms of the Higgs mechanism [6, 7]. It will be demonstrated that the composite nature of the leptons and quarks of the GM leads

In particular, it will be shown that the GM provides a new universal theory of gravity in terms of a residual interaction of a strong color-like interaction [4], analogous to quantum chromodynamics (QCD) [8], the theory of strong interactions, in the SM. This very weak residual interaction has two important properties: asymptotic freedom (antiscreening) [9, 10] and finite range [11], that provide an understanding of dark matter [12] and dark energy [13],

Section 2 discusses the *incompleteness* of the SM and examines some basic assumptions upon which the SM has been built. Section 3 introduces the GM which replaces several *dubious assumptions* within the SM by different and simpler assumptions, without destroying any agreement with experiment. These lead to the notion that the elementary particles, the six leptons, six quarks and three weak bosons of the SM are *composite particles*. Section 4 describes the development of a viable composite GM in which the elementary particles of the SM are composites of elementary spin-1/2 particles called rishons and antirishons. Section 5 discusses a *new paradigm* provided by the composite GM for *all* mass, and indicates a qualitative under-

Section 6 presents a solution to the *cosmological matter-antimatter asymmetry problem* in terms of the composite GM. The composite GM also leads to a *new paradigm* for *gravity*. This is discussed in Section 7 in which a *quantum theory of gravity* is described. This new law of universal gravitation is shown to provide an understanding of both *dark matter* (Section 8) and *dark* 

Section 10 discusses the possibility that the photon may be considered to be the singlet state of the corresponding QCD color octet binding together the constituents of the leptons and

The current formulation of the Standard Model (SM) of particle physics [1] was essentially finalized in the mid-1970s following the experimental confirmation of the existence of quarks [14, 15]. However, the model is regarded by most physicists as *incomplete* since it provides little understanding of several empirical observations. First, it does not explain the occurrence of three "generations" of the elementary particles of the SM [16]: the first generation comprising

antimatter asymmetry of the universe, dark matter and dark energy.

to a solution of the cosmological matter-antimatter asymmetry problem.

respectively, in the universe.

160 Trends in Modern Cosmology

*energy* (Section 9).

standing of the mass hierarchy of leptons and quarks.

**2. Standard model of particle physics**

quarks. Finally, Section 11 gives a brief summary and conclusion.

$$s^\cdot = -d\sin\theta + s\cos\theta\_\prime \tag{2}$$


**Table 1.** SM additive quantum numbers for leptons and quarks.

and *θ* is a mixing angle introduced by Cabibbo [18] into the transition amplitudes prior to the development of the quark model in 1964. In the quark case the third component of weak isospin is related to both charge and baryon number.

The *u* and *c* quarks are assumed to interact with the weak eigenstate quarks *d*′ and *s*′, respectively, with the full strength of the CC weak interaction. In addition the *u* and *c* quarks are assumed to not interact at all with the weak eigenstate quarks *s*′ and *d*′, respectively. This assumption is dubious, since there are no conserved quantum numbers to support this assumption.

A third problem with the SM concerns the origin of mass. In the SM, the masses of hadrons arise mainly from the energy content of their constituent quarks and gluons, in agreement with Einstein's conclusion [5]. On the other hand the masses of the elementary particles, the leptons, the quarks and the *W* and *Z* bosons are interpreted differently, arising from the existence of the so-called Higgs field [6, 7]. The Higgs field was introduced mathematically to spontaneously break the *U*(1) × *SU*(2) local gauge symmetry of the electroweak interaction to generate the masses of the *W* and *Z* bosons. The Higgs field also cured the associated fermion mass problem: by coupling, with appropriate strength, originally massless fermions to the scalar Higgs field, it is possible to produce the observed fermion masses and to maintain local gauge invariance [19].

There are several problems with the SM's interpretation of the origin of mass. First, there is no clear evidence for the existence of the hypothetical Higgs field. Second, the model provides no *unified* origin of mass. Third, the fermion-Higgs coupling strength is dependent upon the mass of the fermion so that a new parameter is introduced into the SM for each fermion mass. In fact fourteen new parameters are required, if one includes two more parameters to describe the masses of the *W* boson and the Higgs particle. Fourth, the Higgs mechanism does *not provide any physical explanation* for the origin of the masses of the elementary particles.

The assumption that the weak interactions are fundamental interactions arising from a local gauge theory, unlike both the electromagnetic and strong colour interactions, is at variance with the experimental facts: both the *W* and *Z* particles, mediating the weak interactions, are massive, and this conflicts with the requirement of a local gauge theory that the mediating particles should be massless in order to guarantee the gauge invariance. This assumption is very dubious, especially since it leads to more problems than it solves. It also leaves several questions unanswered: *How does the spontaneous symmetry breaking mechanism occur within the electroweak theory? What is the principle that determines the large range of fermion masses exhibited by the leptons and quarks?*

### **3. Generation model of particle physics**

and *θ* is a mixing angle introduced by Cabibbo [18] into the transition amplitudes prior to the development of the quark model in 1964. In the quark case the third component of weak

**Particle** *Q L Lμ L<sup>τ</sup> A S C B T e*<sup>−</sup> − 1 1 0 0 0 0 0 0 0 *ν<sup>e</sup>* 0 1 0 0 0 0 0 0 0 *μ*<sup>−</sup> − 1 1 1 0 0 0 0 0 0 *ν<sup>μ</sup>* 0 1 1 0 0 0 0 0 0 *τ*<sup>−</sup> − 1 1 0 1 0 0 0 0 0 *ντ* 0 1 0 1 0 0 0 0 0

3

3

3

3

3

3

0 0 0 0

0 0 0 0

0 1 0 0

− 1 0 0 0

0 0 0 1

0 0 − 1 0

0 0 0 \_\_1

0 0 0 \_\_1

0 0 0 \_\_1

0 0 0 \_\_1

0 0 0 \_\_1

0 0 0 \_\_1

The *u* and *c* quarks are assumed to interact with the weak eigenstate quarks *d*′ and *s*′, respectively, with the full strength of the CC weak interaction. In addition the *u* and *c* quarks are assumed to not interact at all with the weak eigenstate quarks *s*′ and *d*′, respectively. This assumption is dubious, since there are no conserved quantum numbers to support this

A third problem with the SM concerns the origin of mass. In the SM, the masses of hadrons arise mainly from the energy content of their constituent quarks and gluons, in agreement with Einstein's conclusion [5]. On the other hand the masses of the elementary particles, the leptons, the quarks and the *W* and *Z* bosons are interpreted differently, arising from the existence of the so-called Higgs field [6, 7]. The Higgs field was introduced mathematically to spontaneously break the *U*(1) × *SU*(2) local gauge symmetry of the electroweak interaction to generate the masses of the *W* and *Z* bosons. The Higgs field also cured the associated fermion mass problem: by coupling, with appropriate strength, originally massless fermions to the scalar Higgs field, it is possible to produce the observed fermion masses and to maintain local

isospin is related to both charge and baryon number.

**Table 1.** SM additive quantum numbers for leptons and quarks.

assumption.

*<sup>u</sup>* +\_\_2

162 Trends in Modern Cosmology

*<sup>d</sup>* <sup>−</sup>\_\_1

*<sup>c</sup>* +\_\_2

*<sup>s</sup>* <sup>−</sup>\_\_1

*<sup>t</sup>* +\_\_2

*<sup>b</sup>* <sup>−</sup>\_\_1

3

3

3

3

3

3

gauge invariance [19].

The GM of particle physics [2, 17] overcomes many of the problems inherent in the SM. In the GM the three dubious assumptions of the SM discussed previously are replaced by three different and simpler assumptions. These are (i) the assumption of a *simpler unified* classification of leptons and quarks; (ii) the assumption that the *mass eigenstate quarks* form weak isospin doublets and that hadrons are composed of *weak eigenstate quarks* and (iii) the assumption that the weak interactions are *not fundamental interactions*.

**Table 2** shows the additive quantum numbers allotted to both leptons and quarks in the GM. This is a much simpler and unified classification scheme involving only three additive quantum numbers: charge *Q*, particle number *p* and generation quantum number *g*. All three quantum numbers are conserved in all interactions. In particular the generation quantum number *g* is strictly conserved in weak interactions unlike some of the quantum numbers, e.g. strangeness *S*, of the SM. As for **Table 1** the corresponding antiparticles have the opposite sign for each particle additive quantum number.

The conservation of the generation quantum number in weak interactions was only achieved by making two postulates, which means that the GM differs fundamentally from the SM in two more ways [20, 21]. First the GM postulates that it is the mass eigenstate quarks of the same generation, which form weak isospin doublets: (*u*, *d*) and (*c*, *s*). Thus the GM assumes, in the two generation approximation, that the *u* and *c* quarks interact with *d* and *s*, respectively, with the *full* strength of the CC weak interaction and that the *u* and *c* quarks do *not* interact at all with *s* and *d*, respectively. This result is a consequence of the conservation of the generation quantum number. Second, while the SM assumes that hadrons are composed of mass eigenstate quarks such as *d* and *s*, the GM postulates that hadrons are composed of the corresponding weak eigenstate quarks *d*′ and *s*′. These two postulates overcome the second dubious assumption of the SM.


**Table 2.** GM additive quantum numbers for leptons and quarks.

The classification scheme given in **Table 2** provides a simpler and unified description of leptons and quarks. It also suggests that leptons and quarks are related and in particular allows the development of a composite model of leptons and quarks [4, 22].

The GM assumes that the leptons, quarks and the *W* and *Z* bosons are *composites*. Consequently, the weak interactions are *not* fundamental interactions arising from an *SU*(2) local gauge theory. They are residual interactions of the strong color interaction binding the constituents of the leptons, quarks and the *W* and *Z* bosons together. This strong color interaction is completely analogous to that of QCD in the SM. The composite nature of leptons, quarks and the *W* bosons overcomes the dubious assumption of the SM that the weak interactions are fundamental. It should be noted that the treatment of the electromagnetic and weak interactions in the SM in terms of a *U*(1) × *SU*(2) local gauge theory is replaced in the GM by a *U*(1) × *SU*(2) global gauge theory [23].

The GM has only the charge *Q* additive quantum number in common with the SM (see **Tables 1** and **2**). The second additive quantum number of the GM, particle number *p*, replaces both baryon number *A* and lepton number *L* of the SM, while the third additive quantum number of the GM, generation quantum number *g*, effectively replaces the remaining additive quantum numbers, *Lμ*, *L<sup>τ</sup>* , *S*, *C*, *B* and *T* of the SM.

#### **4. Composite generation model**

Although there is no direct experimental evidence that leptons and quarks have a substructure, there is considerable indirect evidence that leptons and quarks are actually *composites*.

First, the equal magnitude of the electric charges of the electron and proton, indicates that the charges of the up and down quarks are related to that of the electron.

Second, the leptons and quarks are considered [16] to form three families or generations, containing particles which have similar properties, except for mass. This feature is analogous to Mendeleev's periodic table of the elements, based essentially on elements having similar properties, except for mass. The existence of three repeating patterns suggests strongly that the members of each generation are composites, analogous to the elements of the periodic table.

The nonunified classification scheme of the SM provides a major stumbling block for the development of a composite model of leptons and quarks. On the other hand, the unified classification scheme of the GM (**Table 2**) makes feasible a composite version of the GM (CGM) [22]. Here we shall present the current version [4].

The composite GM is based on the unified classification scheme and also on 1979 composite models of Harari [24] and Shupe [25]. The current composite GM was proposed in 2011 and is described in detail in Chapter 1 [2] of the book *Particle Physics* published by InTech and in a review paper [17] published in *Advances in High Energy Physics*.

Both the models of Harari and Shupe are very similar and treat leptons and quarks as composites of two kinds of spin-1/2 particles, which Harari named "rishons" from the Hebrew word for primary. The CGM adopts this name for the constituents of both leptons and quarks and for consistency the same three additive quantum numbers are assigned to the constituents as were previously allotted in the GM to leptons and quarks (see **Table 2**).

In the Harari-Shupe Model (HSM), two kinds of rishons labeled *T* with charge *Q* = + \_\_1 <sup>3</sup> and *V* with *Q* = 0 and their corresponding antiparticles labeled *T*‾‾ with charge *<sup>Q</sup>* <sup>=</sup> <sup>−</sup>\_\_1 <sup>3</sup> and *V*‾‾ with *Q* = 0, are employed to construct the leptons, quarks and their antiparticles. In the HSM, each spin-1/2 lepton or quark is composed of three rishons or three antirishons.


**Table 3** shows the proposed HSM structures of the first generation of leptons and quarks. Basically the HSM describes only the charge structure of the first generation of leptons and quarks and does not provide a satisfactory understanding of the second and third generations.

**Table 3.** HSM of first generation of leptons and quarks.

The classification scheme given in **Table 2** provides a simpler and unified description of leptons and quarks. It also suggests that leptons and quarks are related and in particular allows

**Particle** *Q p g* **Particle** *Q p g*

<sup>3</sup> \_\_1

<sup>3</sup> \_\_1

<sup>3</sup> \_\_1

3

3

3

3

\_\_1 3

3

\_\_1 3

3

\_\_1 3 0

0

± 1

± 1

0, ± 2

0, ± 2

*<sup>ν</sup><sup>e</sup>* <sup>0</sup> <sup>−</sup> <sup>1</sup> <sup>0</sup> *<sup>u</sup>* +\_\_2

*<sup>e</sup>*<sup>−</sup> <sup>−</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>0</sup> *<sup>d</sup>* <sup>−</sup>\_\_1

*<sup>ν</sup><sup>μ</sup>* <sup>0</sup> <sup>−</sup> <sup>1</sup> <sup>±</sup> <sup>1</sup> *<sup>c</sup>* +\_\_2

*<sup>μ</sup>*<sup>−</sup> <sup>−</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>±</sup> <sup>1</sup> *<sup>s</sup>* <sup>−</sup>\_\_1

*ντ* <sup>0</sup> <sup>−</sup> <sup>1</sup> 0, <sup>±</sup> <sup>2</sup> *<sup>t</sup>* +\_\_2

*<sup>τ</sup>*<sup>−</sup> <sup>−</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup> 0, <sup>±</sup> <sup>2</sup> *<sup>b</sup>* <sup>−</sup>\_\_1

**Table 2.** GM additive quantum numbers for leptons and quarks.

The GM assumes that the leptons, quarks and the *W* and *Z* bosons are *composites*. Consequently, the weak interactions are *not* fundamental interactions arising from an *SU*(2) local gauge theory. They are residual interactions of the strong color interaction binding the constituents of the leptons, quarks and the *W* and *Z* bosons together. This strong color interaction is completely analogous to that of QCD in the SM. The composite nature of leptons, quarks and the *W* bosons overcomes the dubious assumption of the SM that the weak interactions are fundamental. It should be noted that the treatment of the electromagnetic and weak interactions in the SM in terms of a *U*(1) × *SU*(2) local gauge theory is replaced in the GM by a *U*(1) × *SU*(2)

The GM has only the charge *Q* additive quantum number in common with the SM (see **Tables 1** and **2**). The second additive quantum number of the GM, particle number *p*, replaces both baryon number *A* and lepton number *L* of the SM, while the third additive quantum number of the GM, generation quantum number *g*, effectively replaces the remaining additive

Although there is no direct experimental evidence that leptons and quarks have a substructure, there is considerable indirect evidence that leptons and quarks are actually *composites*.

First, the equal magnitude of the electric charges of the electron and proton, indicates that the

, *S*, *C*, *B* and *T* of the SM.

charges of the up and down quarks are related to that of the electron.

the development of a composite model of leptons and quarks [4, 22].

global gauge theory [23].

164 Trends in Modern Cosmology

quantum numbers, *Lμ*, *L<sup>τ</sup>*

**4. Composite generation model**

To overcome some of the deficiencies of the simple HSM, the two-rishon model was extended [4, 22], within the framework of the CGM, by the introduction of a *third* kind of rishon labeled *U* and the allocation of all three additive quantum numbers, *Q*, *p* and *g* to each kind of rishon (see **Table 4**). It should be noted that each rishon additive quantum number *N*, the corresponding antirishon has the additive quantum number − *N*.

In the CGM, the substructure of the leptons and quarks is described in terms of massless rishons and/or antirishons. Each rishon carries a color charge, red, green or blue, while each antirishon carries an anticolor charge, antired, antigreen or antiblue. The constituents of leptons and quarks are bound together by a strong color-type interaction, corresponding to a local gauged *SU*(3) symmetry (analogous to QCD in the SM) mediated by massless *hypergluons* (analogous to gluons in the SM).

**Table 5** gives the structures of the first generation of leptons and quarks in the CGM. The *u*-quark has *p* = + \_\_1 <sup>3</sup> since it contains two *T*-rishons and one *V*‾‾-antirishon. It is essential that the *u*-quark should contain an *V*‾‾-antirishon rather than a *V*-rishon as in the HSM, since its particle number is required to agree with its baryon number *A* = + \_\_1 <sup>3</sup>. It should be noted that leptons




**Table 5.** CGM of first generation of leptons and quarks.

are composed of three rishons, while quarks are composed of one rishon and one rishonantirishon pair.

Each lepton of the first generation is *colorless*, composed of three rishons carrying different colors. Each quark of the first generation is *colored*, composed of one rishon and one colorless rishon-antirishon pair. The first generation of particles are all built out of *T* and *V* rishons and their antiparticles so that each particle has *g* = 0. The second and third generations are identical to the first generation plus one and two *colorless* rishon-antirishon pair(s): *ŪV* or *V*‾‾*U* with *Q* = *p* = 0 but *g* = ± 1 so that the second and third generations have *g* = ± 1 and *g* = 0, ± 2, respectively. This gives *three repeating patterns* [2].

It should be noted that if each of the three kinds of rishons is *conserved* then each of the three additive quantum numbers, *Q*, *p* and *g* is also conserved in all interactions [22].

#### **5. Mass**

To overcome some of the deficiencies of the simple HSM, the two-rishon model was extended [4, 22], within the framework of the CGM, by the introduction of a *third* kind of rishon labeled *U* and the allocation of all three additive quantum numbers, *Q*, *p* and *g* to each kind of rishon (see **Table 4**). It should be noted that each rishon additive quantum number *N*, the corre-

In the CGM, the substructure of the leptons and quarks is described in terms of massless rishons and/or antirishons. Each rishon carries a color charge, red, green or blue, while each antirishon carries an anticolor charge, antired, antigreen or antiblue. The constituents of leptons and quarks are bound together by a strong color-type interaction, corresponding to a local gauged *SU*(3) symmetry (analogous to QCD in the SM) mediated by massless *hyperglu-*

**Table 5** gives the structures of the first generation of leptons and quarks in the CGM. The

*u*-quark should contain an *V*‾‾-antirishon rather than a *V*-rishon as in the HSM, since its particle

3

3

3

<sup>3</sup> +\_\_1

<sup>3</sup> <sup>−</sup>\_\_1

<sup>3</sup> <sup>+</sup>\_\_1

<sup>3</sup> <sup>−</sup>\_\_1

3

3

3

3

<sup>3</sup> since it contains two *T*-rishons and one *V*‾‾-antirishon. It is essential that the

\_\_1

<sup>3</sup>. It should be noted that leptons

0

0

− 1

0

0

0

0

sponding antirishon has the additive quantum number − *N*.

number is required to agree with its baryon number *A* = +

*<sup>V</sup>* <sup>0</sup> +\_\_1

*<sup>U</sup>* <sup>0</sup> +\_\_1

**Table 4.** CGM additive quantum numbers for rishons.

*<sup>u</sup> TTV*‾‾ +\_\_2

*<sup>d</sup>*‾‾ *TV*‾‾*V*‾‾ +\_\_1

*<sup>d</sup> <sup>T</sup>*‾‾*VV* <sup>−</sup>\_\_1

*<sup>ū</sup> <sup>T</sup>*‾‾*T*‾‾*<sup>V</sup>* <sup>−</sup>\_\_2

**Table 5.** CGM of first generation of leptons and quarks.

*ν*

**Rishon** *Q p g*

<sup>3</sup> +\_\_1

**Particle Structure** *Q p g e*<sup>+</sup> *TTT* + 1 + 1 0

*ν<sup>e</sup> V*‾‾*V*‾‾*V*‾‾ 0 − 1 0

‾‾*<sup>e</sup> VVV* 0 + 1 0

*e*<sup>−</sup> *T*‾‾*T*‾‾*T*‾‾ − 1 − 1 0

*ons* (analogous to gluons in the SM).

\_\_1

*u*-quark has *p* = +

166 Trends in Modern Cosmology

*<sup>T</sup>* +\_\_1

Since the mass of a hadron arises mainly from the energy of its constituents [26–28], the CGM suggests [3] that the mass of a lepton, quark or weak boson arises from a characteristic energy *E* associated with its constituent rishons and hypergluons, according to *m* = *E*/*c*<sup>2</sup> . Thus the CGM provides a *new paradigm* and a *unified* description for the origin of *all* mass: the mass of a body arises from the energy content *E* of its constituents. The mass is given by *m* = *E*/*c*<sup>2</sup> in agreement with Einstein's conclusion [5], so that there is no need for the existence of a Higgs field with its accompanying problems. A corollary of this idea is: *If a particle has mass, then it is composite*.

The CGM suggests that the mass hierarchy of the three generations arises from the substructures of the leptons and quarks. The mass of each composite particle is expected to be *greater* if the constituents are on average more widely spaced: this is a consequence of the nature of the strong color interactions, which are assumed to possess the property of "asymptotic freedom" [9, 10], whereby the color interactions become stronger for larger separations of the color charges. Particles with two or more like charged rishons will have larger structures due to electric repulsion.

Qualitatively, for the same generation, one expects that (i) a charged lepton will have a greater mass than the corresponding neutral lepton; (ii) a *Q* = + \_\_2 <sup>3</sup> quark will have a greater mass than the corresponding *<sup>Q</sup>* <sup>=</sup> <sup>−</sup>\_\_1 <sup>3</sup> quark. These are both generally true: (i) the electron has a larger mass than its corresponding neutrino, and (ii) the top quark mass (173 GeV) is > the bottom quark mass (4.2 GeV), the charmed quark mass (1.3 GeV) is > the strange quark mass (95 MeV), although the up quark mass (2 MeV) is < the down quark mass (5 MeV). The first generation quarks seem to present an anomaly since the proton consists of two up quarks and one down quark while the neutron consists of two down quarks and one up quark so that the proton is only stable if the down quark (*<sup>Q</sup>* <sup>=</sup> <sup>−</sup>\_\_1 <sup>3</sup>) is more massive than the up quark (*<sup>Q</sup>* <sup>=</sup> <sup>+</sup> \_\_2 <sup>3</sup>). In the CGM, this anomaly is accounted for by the constituents of hadrons being weak-eigenstate quarks rather than mass-eigenstate quarks. The proton is stable since the weak eigenstate quark *d*′ has a larger mass than the up quark, containing about 5% of the strange quark mass, since sin2 *θ* ≈ 0.05 [29].

#### **6. Matter-antimatter asymmetry problem**

According to the prevailing cosmological model [30] the universe was created in the so-called "Big Bang" from pure energy, and is currently composed of about 5% ordinary matter, 27% dark matter and 68% dark energy. It is generally assumed that the Big Bang and its aftermath produced *equal numbers* of particles and antiparticles, although the universe today appears to consist almost entirely of matter (particles) rather than antimatter (antiparticles). This constitutes the matter-antimatter asymmetry problem: *Where have all the antiparticles gone?* Currently there is no acceptable understanding of this asymmetry problem.

An understanding of the matter-antimatter asymmetry requires a precise definition of both matter and antimatter and knowledge of the physical nature of the Big Bang but unfortunately this knowledge is currently far from complete. The prevailing model of the Big Bang is based upon the theory of general relativity [31]: extrapolation of the expansion of the universe backwards in time yields an infinite density and temperature at a finite time in the past (approximately 13.8 billion years ago). Thus the "birth" of the universe seems to be associated with a "singularity", which not only signals a breakdown of general relativity but also all the laws of physics. This leads to serious impediments to understanding the physical nature of the Big Bang and consequently the development of the matter-antimatter asymmetry in the aftermath of the Big Bang.

Since the physical nature of the Big Bang is still not understood, it is not possible to discuss the matter-antimatter asymmetry problem from the initial singularity. Consequently, the matterantimatter (i.e. particle-antiparticle) asymmetry problem will be discussed in terms of the observed nature of the universe, ignoring the singularity.

In the SM the universe is made essentially of matter comprising three kinds of elementary particles: electrons, up quarks and down quarks. The matter described by the SM refers to the 5% ordinary matter, which prior to the nucleosynthesis, i.e. the fusion into heavier elements, consisted of about 92% hydrogen atoms and 8% helium atoms [32], so that the ordinary matter of the universe was, and still is, essentially electrically neutral and colorless.

For many decades now the SM has been unable to provide an acceptable understanding of the matter-antimatter asymmetry problem. The main reason seems to be that the SM assumes that the leptons and quarks are elementary particles. Since this implies no relationship between leptons and quarks, as indicated by the different sets of additive quantum numbers employed by the SM to describe their interactions, this allows the matter/antimatter nature of leptons and quarks to be decided by pure convention. In the SM both leptons and quarks are assumed to be matter particles.

In the SM, neglecting the singularity, it is assumed that the Big Bang initially produces numerous elementary particle-antiparticle pairs such as electron-positron pairs and quark-antiquark pairs by converting energy into mass according to *m* = *E*/*c*<sup>2</sup> . Thus the early universe consisted of a soup of particle-antiparticle pairs continually being created and annihilated. Later, as the universe cooled following an inflationary period, the quarks and antiquarks would form protons, neutrons, antiprotons, antineutrons, etc., and eventually atoms of hydrogen, antihydrogen, helium and antihelium. These would later annihilate in pairs until only atoms of hydrogen and helium prevailed. In this scenario, it seems unlikely that either electrons or positrons would prevail so that neither hydrogen atoms nor antihydrogen atoms would prevail. This simply reflects that the creation and annihilation of electron-positron pairs constitute a unique process, since in the SM both electrons and positrons are elementary particles.

**6. Matter-antimatter asymmetry problem**

aftermath of the Big Bang.

168 Trends in Modern Cosmology

to be matter particles.

there is no acceptable understanding of this asymmetry problem.

observed nature of the universe, ignoring the singularity.

pairs by converting energy into mass according to *m* = *E*/*c*<sup>2</sup>

According to the prevailing cosmological model [30] the universe was created in the so-called "Big Bang" from pure energy, and is currently composed of about 5% ordinary matter, 27% dark matter and 68% dark energy. It is generally assumed that the Big Bang and its aftermath produced *equal numbers* of particles and antiparticles, although the universe today appears to consist almost entirely of matter (particles) rather than antimatter (antiparticles). This constitutes the matter-antimatter asymmetry problem: *Where have all the antiparticles gone?* Currently

An understanding of the matter-antimatter asymmetry requires a precise definition of both matter and antimatter and knowledge of the physical nature of the Big Bang but unfortunately this knowledge is currently far from complete. The prevailing model of the Big Bang is based upon the theory of general relativity [31]: extrapolation of the expansion of the universe backwards in time yields an infinite density and temperature at a finite time in the past (approximately 13.8 billion years ago). Thus the "birth" of the universe seems to be associated with a "singularity", which not only signals a breakdown of general relativity but also all the laws of physics. This leads to serious impediments to understanding the physical nature of the Big Bang and consequently the development of the matter-antimatter asymmetry in the

Since the physical nature of the Big Bang is still not understood, it is not possible to discuss the matter-antimatter asymmetry problem from the initial singularity. Consequently, the matterantimatter (i.e. particle-antiparticle) asymmetry problem will be discussed in terms of the

In the SM the universe is made essentially of matter comprising three kinds of elementary particles: electrons, up quarks and down quarks. The matter described by the SM refers to the 5% ordinary matter, which prior to the nucleosynthesis, i.e. the fusion into heavier elements, consisted of about 92% hydrogen atoms and 8% helium atoms [32], so that the ordinary mat-

For many decades now the SM has been unable to provide an acceptable understanding of the matter-antimatter asymmetry problem. The main reason seems to be that the SM assumes that the leptons and quarks are elementary particles. Since this implies no relationship between leptons and quarks, as indicated by the different sets of additive quantum numbers employed by the SM to describe their interactions, this allows the matter/antimatter nature of leptons and quarks to be decided by pure convention. In the SM both leptons and quarks are assumed

In the SM, neglecting the singularity, it is assumed that the Big Bang initially produces numerous elementary particle-antiparticle pairs such as electron-positron pairs and quark-antiquark

a soup of particle-antiparticle pairs continually being created and annihilated. Later, as the universe cooled following an inflationary period, the quarks and antiquarks would form protons, neutrons, antiprotons, antineutrons, etc., and eventually atoms of hydrogen, antihydrogen,

. Thus the early universe consisted of

ter of the universe was, and still is, essentially electrically neutral and colorless.

In the GM, neglecting the singularity, it is expected that the Big Bang would initially produce numerous elementary rishon-antirishon pairs. Then as the universe cooled following an inflationary period, the rishons and antirishons would form leptons, quarks and their antiparticles and eventually atoms of hydrogen, antihydrogen, helium and antihelium. These would later annihilate in pairs until only atoms of hydrogen and helium prevailed.

In order to understand this matter-antimatter asymmetry, it is necessary to define the matter/ antimatter nature of composite particles. Historically, the term "particle" defines matter that is naturally occurring, i.e. electrons, protons, hydrogen atoms, etc. This is consistent within the SM in which the electron and the up and down quarks are elementary particles. However, it is not consistent within the GM in which the electron and the up and down quarks are composite particles consisting of elementary particles (rishons) and/or antiparticles (antirishons). In the GM, rishons are considered to be matter (particles) while antirishons are considered to be antimatter (antiparticles), since rishon-antirishon pairs are considered to be created/annihilated in the standard manner, e.g. in the Big Bang.

In the GM the elementary rishons and antirishons have particle number *p* = + \_\_1 <sup>3</sup> and *<sup>p</sup>* <sup>=</sup> <sup>−</sup>\_\_1 3, respectively. Thus the particle number *p* allotted to a composite lepton or quark reflects its degree of matter or antimatter nature. In the GM the quarks are composed of both rishons and antirishons so that they have both a matter and an antimatter nature, although an electron is composed of three *T*‾‾-rishons so that it has *p* = − 1 and is pure antimatter.

The solution of the matter-antimatter asymmetry problem involves the particle number additive quantum number *p* of the GM: in particular the values of *p* corresponding to a weak eigenstate up quark (*p* = + \_\_1 <sup>3</sup>), a weak eigenstate down quark (*<sup>p</sup>* <sup>=</sup> <sup>+</sup> \_\_1 <sup>3</sup>) and an electron (*p* = − 1). The values of *p* = + \_\_1 <sup>3</sup> of the quarks, correspond to the values of their baryon number in the SM, while the value of *p* = − 1 of the electron, corresponds to minus the value of the lepton number of the electron in the SM. In the GM, the electron consists entirely of antirishons, i.e. antiparticles, while in the SM it is assumed to be a particle, although as we have indicated earlier, there is no a priori reason for this assumption. It should be noted that the matter/antimatter nature of an electron in the GM is not merely a revised definition of the term "matter" but is a requirement for consistency of the nature of the constituents of the electron and the initial particle-antiparticle nature of the universe in the Big Bang: the elementary particles in the SM (leptons and quarks) and in the GM (rishons) are different.

In the GM the proton is assumed to consist of three weak eigenstate quarks, two up quarks and one down quark, so that the proton has particle number *p* = + 1. Consequently, a hydrogen atom, consisting of one proton and one electron has particle number *p* = 0: the hydrogen atom in the GM consists basically of an equal number of rishons and antirishons, so that there is no asymmetry of matter and antimatter there.

In the GM the neutron consists of three weak eigenstate quarks, one up quark and two down quarks, so that the neutron also has particle number *p* = + 1. Consequently, a helium atom, consisting of two protons, two neutrons and two electrons has particle number *p* = + 2: the helium atom in the GM consists of six more rishons than antirishons, i.e. more matter than antimatter. In the GM it is assumed that during the formation of helium in the aftermath of the Big Bang that an equivalent surplus of antimatter was formed as neutrinos, which have *p* = − 1, so that overall equal numbers of rishons and antirishons prevailed. This assumption is a consequence of the conservation of *p* in all interactions in the GM.

Thus the ordinary matter present in the universe, prior to the fusion process into heavier elements, has particle number *p* = 0. Since the additive quantum number *p* is conserved in all interactions, this implies that the overall particle number of the universe will remain as *p* = 0, i.e. symmetric in particle and antiparticle matter.

Indeed it should be noted that if the Big Bang produced equal numbers of particles and antiparticles so that the initial state of the universe had particle number *p* = 0, then the GM *predicts* that the present state of the universe should also have *p* = 0, since particle number *p* is conserved in all interactions.

To summarize: the ordinary matter present in the universe has an overall particle number of *p* = 0, so that it contains equal numbers of both rishons and antirishons. This implies that the original antimatter created in the Big Bang is now contained within the stable composite leptons, i.e. electrons and neutrinos, and the stable composite quarks, i.e. the weak eigenstate up and down quarks, which comprise the protons and neutrons. The hydrogen, helium and heavier atoms all consist of electrons, protons and neutrons. This explains where all the antiparticles have gone. Thus there is no matter-antimatter asymmetry in the present universe.

However, the above does not explain why the present universe consists primarily of hydrogen atoms and not antihydrogen atoms. In the SM this is considered to be the matter-antimatter asymmetry problem, since each of the elementary particles, the electron and both the up and down quarks are defined to be matter (particles). In the GM it has been demonstrated that this is not so: both hydrogen atoms and antihydrogen atoms have *p* = 0, consisting of six rishons and six antirishons. Indeed an antihydrogen atom consists of the same rishons and antirishons as does a hydrogen atom, although the rishons and antirishons are differently arranged in the two systems. A hydrogen atom consists of a proton (six rishons and three antirishons) and an electron (three antirishons), while an antihydrogen atom consists of an antiproton (three rishons and six antirishons) and a positron (three rishons).

This implies that both hydrogen atoms and antihydrogen atoms should be formed during the aftermath of the Big Bang with about the same probability. In fact, estimates from the cosmic microwave background data suggest [11] that for every billion hydrogen-antihydrogen pairs there was just one extra hydrogen atom. It is suggested that this extremely small difference, one extra hydrogen atom in 109 hydrogen-antihydrogen pairs, may arise from statistical fluctuations associated with the complex many-body processes involved in the formation of either a hydrogen atom or an antihydrogen atom. The uniformity of the universe [11], in particular the lack of antihydrogen throughout the universe, indicates that the above statistical fluctuations took place prior to the "inflationary period" [33, 34] associated with the Big Bang scenario.

#### **7. Gravity**

In the GM the neutron consists of three weak eigenstate quarks, one up quark and two down quarks, so that the neutron also has particle number *p* = + 1. Consequently, a helium atom, consisting of two protons, two neutrons and two electrons has particle number *p* = + 2: the helium atom in the GM consists of six more rishons than antirishons, i.e. more matter than antimatter. In the GM it is assumed that during the formation of helium in the aftermath of the Big Bang that an equivalent surplus of antimatter was formed as neutrinos, which have *p* = − 1, so that overall equal numbers of rishons and antirishons prevailed. This assumption is

Thus the ordinary matter present in the universe, prior to the fusion process into heavier elements, has particle number *p* = 0. Since the additive quantum number *p* is conserved in all interactions, this implies that the overall particle number of the universe will remain as *p* = 0,

Indeed it should be noted that if the Big Bang produced equal numbers of particles and antiparticles so that the initial state of the universe had particle number *p* = 0, then the GM *predicts* that the present state of the universe should also have *p* = 0, since particle number *p* is con-

To summarize: the ordinary matter present in the universe has an overall particle number of *p* = 0, so that it contains equal numbers of both rishons and antirishons. This implies that the original antimatter created in the Big Bang is now contained within the stable composite leptons, i.e. electrons and neutrinos, and the stable composite quarks, i.e. the weak eigenstate up and down quarks, which comprise the protons and neutrons. The hydrogen, helium and heavier atoms all consist of electrons, protons and neutrons. This explains where all the antiparticles have gone. Thus there is no matter-antimatter asymmetry in the

However, the above does not explain why the present universe consists primarily of hydrogen atoms and not antihydrogen atoms. In the SM this is considered to be the matter-antimatter asymmetry problem, since each of the elementary particles, the electron and both the up and down quarks are defined to be matter (particles). In the GM it has been demonstrated that this is not so: both hydrogen atoms and antihydrogen atoms have *p* = 0, consisting of six rishons and six antirishons. Indeed an antihydrogen atom consists of the same rishons and antirishons as does a hydrogen atom, although the rishons and antirishons are differently arranged in the two systems. A hydrogen atom consists of a proton (six rishons and three antirishons) and an electron (three antirishons), while an antihydrogen atom consists of an antiproton

This implies that both hydrogen atoms and antihydrogen atoms should be formed during the aftermath of the Big Bang with about the same probability. In fact, estimates from the cosmic microwave background data suggest [11] that for every billion hydrogen-antihydrogen pairs there was just one extra hydrogen atom. It is suggested that this extremely small difference,

tuations associated with the complex many-body processes involved in the formation of either a hydrogen atom or an antihydrogen atom. The uniformity of the universe [11], in particular

hydrogen-antihydrogen pairs, may arise from statistical fluc-

a consequence of the conservation of *p* in all interactions in the GM.

(three rishons and six antirishons) and a positron (three rishons).

i.e. symmetric in particle and antiparticle matter.

served in all interactions.

170 Trends in Modern Cosmology

present universe.

one extra hydrogen atom in 109

Let us now consider the nature of gravity within the framework of the GM. It is considered that the rishons of each colorless lepton, e.g. an electron, are very strongly localized since to date there is no direct experimental evidence for any substructure of these particles. It is expected that the product wave function describing the distribution of the constituent rishons is significant for only an extremely small volume of space so that the color fields almost cancel. It should be noted that the color fields would only cancel completely if each of the rishons occupied the same position, but quantum mechanics prevents this. This raises a question: *What is the residual interaction arising from the incomplete cancellation of the strong interactions?*

Between any two colorless leptons (e.g. electrons) there will be a very weak residual interaction, arising from the color interactions acting between the color charges of one lepton and the color charges of the other lepton. There will be a similar residual interaction between any two colorless hadrons such as neutrons and protons, each containing three differently colored quarks.

In two papers [3, 4] it was suggested that this residual interaction, arising from these "interfermion" color interactions, gives rise to the usual gravitational interaction.

The mass of a body of ordinary matter is essentially the total mass of its constituent colorless electrons, neutrons and protons. Each of these three composite particles is in a three-color antisymmetric state so that its behavior with respect to the color interactions is basically the same. This suggests [13] that the residual color interactions between electrons, neutrons and protons have several properties associated with the usual gravitational interaction: universality, very weak strength and attraction.

In the GM gravity essentially arises from the residual color forces between all electrons, neutrons and protons. This leads [13] to a *new law of gravity*: the residual color interactions between any two bodies of masses *m*<sup>1</sup> and *m*<sup>2</sup> , separated by a distance *r*, leads to a universal law of gravitation, which closely resembles Newton's original law given by:

$$F = H(r) \, m\_1 \, m\_2 \vert \, r^2 \,, \tag{3}$$

where Newton's gravitational constant (*G*) is replaced by a function of *r*, *H*(*r*).

The new gravitational interaction of the GM is based upon the residual color interactions acting between electrons, neutrons and protons. The GM assumes that the color interactions acting between rishons have the same characteristics as the color interactions acting between quarks in the SM. These color interactions have two important properties that differ from the Newtonian interaction: (i) asymptotic freedom [9, 10] and (ii) color confinement [11]. These determine the nature of the function *H*(*r*).

In Sections 8 and 9, we shall indicate how these two additional properties of the interfermion color interactions provide an understanding of dark matter and dark energy, respectively.

#### **8. Galaxy rotation problem and dark matter**

For galaxies there is a major gravitational problem [12], which has been around for about 40 years. It was found that the rotation curves for galaxies disagreed with Newton's gravitational law for large *r*: the stars and gas were rotating much faster than expected from Newton's law and their orbital velocities were roughly constant. These observations implied that either Newton's law was incorrect at large distances or some considerable mass was missing.

The rotation curve for a galaxy is the dependence of the orbital velocity of the visible matter in the galaxy on its radial distance from the center of the galaxy. What the observations showed was that the rotation curves were essentially "flat" at the extremities of the visible matter, i.e. at large distances. This implies gross disagreement with Newton's universal law of gravitation, which predicts a fall-off as 1/ <sup>√</sup> \_ *r*, as in the solar system.

Two solutions, which have been very successful, are first the *dark matter hypothesis*, which proposes that a galaxy is embedded within a giant halo of dark matter. This matter is considered to be nonatomic but otherwise its nature is unknown and so far has not been detected. The second solution is the *Modified Newtonian Dynamics* (MOND) hypothesis: Milgrom [35] proposed that gravity varies from Newton's law for low accelerations. This was an empirical hypothesis without physical understanding.

It was found that the *new law of universal gravitation* given by Eq. (3) is essentially equivalent to the MOND hypothesis so that the GM gravitational interaction provides a physical basis for the MOND hypothesis. The continuing success [36, 37] of the MOND hypothesis is a strong argument against the existence of undetected dark matter haloes, consisting of unknown matter embedding galaxies.

Asymptotic freedom is rather a misnomer. A better term is "antiscreening" as used by Wilczek in his 2004 Nobel lecture. The flat galactic rotation curves are described by the property of antiscreening provided by the self-interactions of the hypergluons mediating the residual color interactions. These antiscreening effects lead to an increase in the strength of the residual color interactions so that *H*(*r*) becomes an increasing function of *r*. The flat rotation curves observed for galaxies indicate that:

$$H(r) = G(1+kr),\tag{4}$$

where *k* is a factor representing the relative strengths of the modified and Newtonian gravitational fields.

In the GM *H*(*r*) = *G*(1 + *kr*) arises from the self-interactions of the hypergluons mediating the gravitational interaction and explains the dark matter problem of the galaxy rotational curves: for small *r*, *H*(*r*) is approximately *G* and gravity is approximately Newtonian; for large *r*, *H*(*r*) is approximately *Gkr* and gravity is approximately 1/*r* rather than 1/*r*<sup>2</sup> , and the 1/*r* dependence gives the flat rotation curves observed.

#### **9. Dark energy**

In Sections 8 and 9, we shall indicate how these two additional properties of the interfermion color interactions provide an understanding of dark matter and dark energy, respectively.

For galaxies there is a major gravitational problem [12], which has been around for about 40 years. It was found that the rotation curves for galaxies disagreed with Newton's gravitational law for large *r*: the stars and gas were rotating much faster than expected from Newton's law and their orbital velocities were roughly constant. These observations implied that either Newton's law was incorrect at large distances or some considerable mass was missing.

The rotation curve for a galaxy is the dependence of the orbital velocity of the visible matter in the galaxy on its radial distance from the center of the galaxy. What the observations showed was that the rotation curves were essentially "flat" at the extremities of the visible matter, i.e. at large distances. This implies gross disagreement with Newton's universal law of gravita-

*r*, as in the solar system. Two solutions, which have been very successful, are first the *dark matter hypothesis*, which proposes that a galaxy is embedded within a giant halo of dark matter. This matter is considered to be nonatomic but otherwise its nature is unknown and so far has not been detected. The second solution is the *Modified Newtonian Dynamics* (MOND) hypothesis: Milgrom [35] proposed that gravity varies from Newton's law for low accelerations. This was an empirical

It was found that the *new law of universal gravitation* given by Eq. (3) is essentially equivalent to the MOND hypothesis so that the GM gravitational interaction provides a physical basis for the MOND hypothesis. The continuing success [36, 37] of the MOND hypothesis is a strong argument against the existence of undetected dark matter haloes, consisting of unknown mat-

Asymptotic freedom is rather a misnomer. A better term is "antiscreening" as used by Wilczek in his 2004 Nobel lecture. The flat galactic rotation curves are described by the property of antiscreening provided by the self-interactions of the hypergluons mediating the residual color interactions. These antiscreening effects lead to an increase in the strength of the residual color interactions so that *H*(*r*) becomes an increasing function of *r*. The flat rotation curves

*H*(*r*) = *G*(1 + *kr*), (4)

where *k* is a factor representing the relative strengths of the modified and Newtonian gravi-

In the GM *H*(*r*) = *G*(1 + *kr*) arises from the self-interactions of the hypergluons mediating the gravitational interaction and explains the dark matter problem of the galaxy rotational curves: for small *r*, *H*(*r*) is approximately *G* and gravity is approximately Newtonian; for large *r*, *H*(*r*)

, and the 1/*r* dependence

is approximately *Gkr* and gravity is approximately 1/*r* rather than 1/*r*<sup>2</sup>

\_

**8. Galaxy rotation problem and dark matter**

tion, which predicts a fall-off as 1/ <sup>√</sup>

172 Trends in Modern Cosmology

ter embedding galaxies.

tational fields.

observed for galaxies indicate that:

gives the flat rotation curves observed.

hypothesis without physical understanding.

Color confinement is the phenomenon that color charged particles (e.g. quarks in the SM, rishons in the GM) cannot be isolated and consequently form colorless composite particles (e.g. mesons and baryons in the SM and also leptons in the GM). Color confinement leads to another phenomenon analogous to the "hadronization process" [11], i.e. the formation of hadrons out of quarks and gluons in the SM and implies *H*(*r*) = 0 for sufficiently large *r* in the GM [13].

In the GM, *H*(*r*) = 0 arises if the gravitational field energy is sufficient that it is energetically favorable to produce the mass of a particle-antiparticle colorless pair rather than the color field to extend further. This implies that gravity ceases to exist for sufficiently large cosmological distances.

The strong color interaction is known to have a finite range of approximately 10− <sup>15</sup> m. Gravity is about 10<sup>−</sup> 41 times weaker at 10− <sup>15</sup> m [11] than the strong color interaction. This suggests that the "hadronization process" for gravity occurs at about 10<sup>26</sup> m, i.e. roughly ten billion light years.

The new law of gravity implies that gravity ceases to exist for cosmological distances exceeding several billion light years, resulting in less slowing down of galaxies than expected from Newton's law. This result agrees well with observations [38, 39] of distant Type Ia supernovae, which indicate the onset of an accelerating expansion of the universe at about six billion light years. Thus the new law of gravity suggests that dark energy like dark matter may be understood as an effect arising from the nature of the gravitational interaction.

#### **10. Photon**

The nature of the photon within the framework of the GM is worthy of consideration. The photon has several properties that may be accommodated in the GM: (i) massless; (ii) neutral; (iii) colorless; (iv) spin-1; (v) *U*(1) symmetry and (vi) interacts with matter gravitationally.

It is proposed that the photon may be the *singlet state* of the corresponding QCD color octet binding together the rishons and antirishons of the leptons and quarks. If the photon is the standard singlet state of QCD: [(*rr* ‾‾) <sup>+</sup> (*gg* ‾‾) <sup>+</sup> (*bb* ‾‾) ]/ <sup>√</sup> \_\_ 3, containing all three color charges, red (*r*), green (*g*) and blue (*b*) and all three anticolor charges, antired (*r* ‾‾), antigreen (*<sup>g</sup>* ‾‾) and antiblue (*<sup>b</sup>* ‾‾), then it has each of the above six properties possessed by the photon.

In the GM the singlet state is a hypergluon, which is massless, electrically neutral, colorless and has spin-1 and *U*(1) symmetry. Furthermore, in the GM it interacts with matter via the new gravitational interaction based upon the residual color interactions. In particular the GM predicts that it leads to *twice* the deflection of Newton's universal theory of gravity in agreement with Einstein's theory of general relativity but without any warping of spacetime. In this way the electromagnetic and strong color interactions are *unified* within a *U*(3) = *U*(1) × *SU*(3) symmetry.

#### **11. Summary and conclusion**

The GM considers that the elementary particles of the SM, the leptons, quarks and weak bosons are all composite particles. This substructure leads to new paradigms for both mass and gravity, which in turn lead to an understanding of several cosmological problems: the matter-antimatter asymmetry of the universe, dark matter and dark energy.

The GM provides a *unified* origin of mass and the composite nature of the leptons and quarks leads to a solution of the cosmological matter-antimatter asymmetry problem.

The GM also provides a new universal quantum theory of gravity. In the GM, gravity is a very weak residual interaction of a strong color-like interaction, which possesses two important properties: antiscreening and finite range that provide an understanding of dark matter and dark energy, respectively, in terms of gravitational effects.

#### **Author details**

Brian Albert Robson

Address all correspondence to: brian.robson@anu.edu.au

The Australian National University, Canberra, ACT, Australia

#### **References**


[8] Halzen F, Martin AD. Quarks and Leptons: An Introductory Course in Modern Particle Physics. New York: Wiley; 1984. 396p

**11. Summary and conclusion**

174 Trends in Modern Cosmology

**Author details**

Brian Albert Robson

**References**

The GM considers that the elementary particles of the SM, the leptons, quarks and weak bosons are all composite particles. This substructure leads to new paradigms for both mass and gravity, which in turn lead to an understanding of several cosmological problems: the

The GM provides a *unified* origin of mass and the composite nature of the leptons and quarks

The GM also provides a new universal quantum theory of gravity. In the GM, gravity is a very weak residual interaction of a strong color-like interaction, which possesses two important properties: antiscreening and finite range that provide an understanding of dark matter and

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## **Deformed Phase Space in Cosmology and Black Holes**

E.A. Mena-Barboza, L.F. Escamilla-Herrera,

J.C. López-Domínguez and J. Torres-Arenas

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68282

#### Abstract

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ID 10, 159pp.

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It is well known that one way to study canonical quantum cosmology is through the Wheeler DeWitt (WDW) equation where the quantization is performed on the minisuperspace variables. The original ideas of a deformed minisuperspace were done in connection with noncommutative cosmology, by introducing a deformation into the minisuperspace in order to incorporate an effective noncommutativity. Therefore, studying solutions to Cosmological models through the WDW equation with deformed phase space could be interpreted as studying quantum effects to Cosmology. In this chapter, we make an analysis of scalar field cosmology and conclude that under a phase space transformation and imposed restriction, the effective cosmological constant is positive. On the other hand, obtaining the wave equation for the noncommutativity Kantowski-Sachs model, we are able to derive a modified noncommutative version of the entropy. To that purpose, the Feynman-Hibbs procedure is considered in order to calculate the partition function of the system.

Keywords: noncommutativity, quantum cosmology, thermodynamics of black holes

#### 1. Introduction

Since the initial use of the Hamiltonian formulation to cosmology, different issues have been studied. In particular, thermodynamic properties of black holes, classical and quantum cosmology, dynamics of cosmological scalar fields, and the problem of cosmological constant among others. In this chapter, we present some results in deforming the phase space variables, discussing recent advances on this special topic by presenting three models. In the first model (Section 2), we analyze the effects of the phase space deformations over different scenarios, we start with the noncommutative on Λ cosmological and comment on the possibility that the

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

origin of the cosmological constant in the (4 + 1) Kaluza-Klein universe is related to the deformation parameter associated to the four-dimensional scale factor and the compact extra dimensions. In Section 3, we study the effects of phase space deformations in late time cosmology. To introduce the deformation, we use the approach given in Refs. [1]. We conclude that for this model an effective cosmological constant Λeff appears.

In Section 4, the thermodynamic formalism for rotating black holes, characterized by noncommutative and quantum corrections, is constructed. From a fundamental thermodynamic relation, the equations of state are explicitly given, and the effect of noncommutativity and quantum correction is discussed; in this sense, the goal of this section is to explore how these considerations introduced in Bekenstein-Hawking (BH) entropy change the thermodynamic information contained in this new fundamental relation. Under these considerations, Section 4 examines the different thermodynamic equations of state and their behavior when considering the aforementioned modifications to entropy.

In this chapter, we mainly pretend to indulge in recollections of different studies on the noncommutative proposal that has been put forward in the literature by the authors of this chapter [2–4]; in this sense, our guideline has been to concentrate on resent results that still seem likely to be of general interest to those researchers that are interested in this noncommutative subject.

#### 2. Model 1: Kaluza-Klein cosmology with Λ

Let us begin by introducing the model in a classical scenario which is an empty (4+1) theory of gravity with cosmological constant Λ as shown in Eq. (1). In this setup, the action takes the form:

$$I = \int \sqrt{-g}(R - \Lambda)dt d^3r d\rho,\tag{1}$$

where <sup>f</sup>t;ri g are the coordinates of the 4-dimensional spacetime and ρ represents the coordinate of the fifth dimension. We are interested in Kaluza-Klein cosmology, so a Friedmann-Robertson-Walker (FRW)-type metric is assumed, which is of the form

$$ds^2 = -dt^2 + \frac{a^2(t)dr^i dr^i}{\left(1 + \frac{\kappa r^2}{4}\right)^2} + \phi^2(t)d\rho^2,\tag{2}$$

where κ ¼ 0; � 1 and a(t), φ(t) are the scale factors of the universe and the compact dimension, respectively. Substituting this metric into the action Eq. (1) and integrating over the spatial dimensions, we obtain an effective Lagrangian that only depends on (a, φ):

$$L = \frac{1}{2} \left( a\phi \dot{a}^2 + a^2 \dot{a}\phi - \kappa a \phi + \frac{1}{3} \Lambda a^3 \phi \right). \tag{3}$$

For the purposes of simplicity and calculations, we can rewrite this Lagrangian in a more convenient way:

Deformed Phase Space in Cosmology and Black Holes http://dx.doi.org/10.5772/intechopen.68282 179

$$L = \frac{1}{2} \left[ \left( \dot{\mathbf{x}}^2 - \omega^2 \mathbf{x}^2 \right) - \left( \dot{\mathbf{y}}^2 - \omega^2 \mathbf{y}^2 \right) \right],\tag{4}$$

where the new variables were defined as

origin of the cosmological constant in the (4 + 1) Kaluza-Klein universe is related to the deformation parameter associated to the four-dimensional scale factor and the compact extra dimensions. In Section 3, we study the effects of phase space deformations in late time cosmology. To introduce the deformation, we use the approach given in Refs. [1]. We conclude that for

In Section 4, the thermodynamic formalism for rotating black holes, characterized by noncommutative and quantum corrections, is constructed. From a fundamental thermodynamic relation, the equations of state are explicitly given, and the effect of noncommutativity and quantum correction is discussed; in this sense, the goal of this section is to explore how these considerations introduced in Bekenstein-Hawking (BH) entropy change the thermodynamic information contained in this new fundamental relation. Under these considerations, Section 4 examines the different thermodynamic equations of state and their behavior when considering

In this chapter, we mainly pretend to indulge in recollections of different studies on the noncommutative proposal that has been put forward in the literature by the authors of this chapter [2–4]; in this sense, our guideline has been to concentrate on resent results that still seem likely to be

Let us begin by introducing the model in a classical scenario which is an empty (4+1) theory of gravity with cosmological constant Λ as shown in Eq. (1). In this setup, the action takes the form:

�<sup>g</sup> <sup>p</sup> <sup>ð</sup><sup>R</sup> � <sup>Λ</sup>Þdtd<sup>3</sup>

coordinate of the fifth dimension. We are interested in Kaluza-Klein cosmology, so a

<sup>a</sup><sup>2</sup>ðtÞdr<sup>i</sup> dri

<sup>1</sup> <sup>þ</sup> <sup>κ</sup>r<sup>2</sup> 4 � �<sup>2</sup> <sup>þ</sup> <sup>φ</sup><sup>2</sup>

where κ ¼ 0; � 1 and a(t), φ(t) are the scale factors of the universe and the compact dimension, respectively. Substituting this metric into the action Eq. (1) and integrating over the spatial

For the purposes of simplicity and calculations, we can rewrite this Lagrangian in a more

a\_φ � κaφ þ

� �

1 3 Λa<sup>3</sup> φ

g are the coordinates of the 4-dimensional spacetime and ρ represents the

<sup>ð</sup>tÞdρ<sup>2</sup>

rdρ; ð1Þ

; ð2Þ

: ð3Þ

of general interest to those researchers that are interested in this noncommutative subject.

this model an effective cosmological constant Λeff appears.

the aforementioned modifications to entropy.

178 Trends in Modern Cosmology

where <sup>f</sup>t;ri

convenient way:

2. Model 1: Kaluza-Klein cosmology with Λ

I ¼

ds<sup>2</sup> ¼ �dt<sup>2</sup> <sup>þ</sup>

dimensions, we obtain an effective Lagrangian that only depends on (a, φ):

<sup>2</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

<sup>2</sup> <sup>a</sup>φa\_

<sup>L</sup> <sup>¼</sup> <sup>1</sup>

<sup>ð</sup> ffiffiffiffiffiffi

Friedmann-Robertson-Walker (FRW)-type metric is assumed, which is of the form

$$\chi = \frac{1}{\sqrt{8}} \left( a^2 + a\phi - \frac{3\kappa}{\Lambda} \right), \\ y = \frac{1}{\sqrt{8}} \left( a^2 - a\phi - \frac{3\kappa}{\Lambda} \right), \tag{5}$$

and <sup>ω</sup> ¼ � <sup>2</sup><sup>Λ</sup> <sup>3</sup> . The Hamiltonian for the model is calculated as usual and reads

$$H = \left[ \left( p\_x^2 + \omega^2 \mathbf{x}^2 \right) - \left( p\_y^2 + \omega^2 y^2 \right) \right],\tag{6}$$

which describes an isotropic oscillator-ghost-oscillator system. A full analysis of the quantum behavior of this model is presented in Ref. [1].

#### 2.1. Noncommutative model

As is well known, there are different approaches to introduce noncommutativity in gravity [5]. In particular, to study noncommutative cosmology [6, 7], there exist a well-explored path to introduce noncommutativity into a cosmological setting [6]. In this setup, the noncommutativity is realized in the minisuperspace variables. The deformation of the phase space structure is achieved through the Moyal brackets, which are based on the Moyal product. However, a more appropriate way to introduce the deformation is by means of the Poisson brackets rather than the Moyal ones.

The most conventional way to understand the noncommutativity between the phase space variables (minisuperspace variables) is by replacing the usual product of two arbitrary functions with the Moyal product (or star product) as

$$(f \star g)(\mathbf{x}) = \exp\left[\frac{1}{2}\alpha^{ab}\partial\_a^{(1)}\partial\_b^{(2)}\right]f(\mathbf{x}\_1)g(\mathbf{x}\_2)|\_{\mathbf{x}\_1=\mathbf{x}\_2=\mathbf{x}},\tag{7}$$

such that

$$
\alpha\_{ab} = \begin{pmatrix}
\theta\_{i\dot{\jmath}} & \delta\_{i\dot{\jmath}} + \sigma\_{i\dot{\jmath}} \\
\end{pmatrix},
\tag{8}
$$

where the θ and β are 2 · 2 antisymmetric matrices and represent the noncommutativity in the coordinates and momenta, respectively, and σ ¼ θβ=4. With this product law, a straightforward calculation gives

$$\{\mathbf{x}\_i, \mathbf{x}\_j\} = \theta\_{\vec{\eta}}, \quad \{\mathbf{x}\_i, p\_j\} = \delta\_{\vec{\eta}} + \sigma\_{\vec{\eta}}, \quad \{p\_i, p\_j\} = \beta\_{\vec{\eta}}.\tag{9}$$

The noncommutative deformation has been applied to the minisuperspace variables as well as to the corresponding canonical momenta; this type of noncommutativity can be motivated by string theory correction to gravity [6, 8]. In the rest of this model, we use for the noncommutative parameters θij ¼ �θεij and βij ¼ βεij.

If we consider the following change of variables in the classical phase space {x; y; px; py}

$$\begin{aligned} \hat{y} &= y - \frac{\theta}{2} p\_x, \,\hat{\mathbf{x}} = \mathbf{x} - \frac{\theta}{2} p\_y \\ \hat{p}\_y &= p\_y + \frac{\beta}{2} \mathbf{x}, \,\hat{p}\_x = p\_x - \frac{\beta}{2} y, \end{aligned} \tag{10}$$

it can be verified that if {x;y;px;py} obeys the usual Poisson algebra, then

$$\{\hat{y}, \hat{x}\} = \theta, \quad \{\hat{x}, \hat{p}\_x\} = \left\{\hat{y}, \hat{p}\_y\right\} = 1 + \sigma, \quad \left\{\hat{p}\_y, \hat{p}\_x\right\} = \beta. \tag{11}$$

Now that we have defined the deformed phase space, we can see the effects on the proposed cosmological model. From the action Eq. (4), we can obtain the Hamiltonian constraint Eq. (6); inserting relations Eq. (11), a Wheeler DeWitt (WDW) equation can be constructed as:

$$\begin{split} H\Psi(\hat{\mathbf{x}},\hat{\mathbf{y}}) &= \left\{ \left( \hat{p}\_x - \frac{2(\beta - \theta\omega^2)}{4 - \omega^2\theta^2} \hat{\mathbf{y}} \right)^2 - \left( \hat{p}\_y + \frac{2(\beta - \theta\omega^2)}{4 - \omega^2\theta^2} \hat{\mathbf{x}} \right)^2 \\ &+ \left( \frac{4(\beta - \theta\omega^2)^2}{(4 - \omega^2\theta^2)^2} + \frac{4(\omega^2 - \beta^2/4)}{4 - \omega^2\theta^2} \right) \hat{\mathbf{x}}^2 - \left( \frac{4(\beta - \theta\omega^2)^2}{(4 - \omega^2\theta^2)^2} + \frac{4(\omega^2 - \beta^2/4)}{4 - \omega^2\theta^2} \right) \hat{\mathbf{y}}^2 \right\} \Psi(\hat{\mathbf{x}},\hat{\mathbf{y}}) = 0, \end{split} (12)$$

By a closer inspection of the equation, it is convenient to make the following definitions:

$$\begin{aligned} \omega^{\prime 2} \equiv \frac{4(\beta - \theta \omega^2)^2}{(4 - \omega^2 \theta^2)^2} + \frac{4(\omega^2 - \beta^2/4)}{4 - \omega^2 \theta^2}, \\\ A\_{\hat{X}} \equiv \frac{-2(\beta - \theta \omega^2)}{4 - \omega^2 \theta^2} \hat{y}, \quad A\_{\hat{Y}} \equiv \frac{2(\beta - \theta \omega^2)}{4 - \omega^2 \theta^2} \hat{x}, \end{aligned} \tag{13}$$

With these definitions, we can rewrite Eq. (12) in a much simpler and suggestive form:

$$H = \left\{ \left[ \left( \hat{p}\_x - A\_{\hat{X}} \right)^2 + \omega'^2 \hat{x}^2 \right] - \left[ \left( \hat{p}\_y - A\_{\hat{Y}} \right)^2 + \omega'^2 \hat{y}^2 \right] \right\},\tag{14}$$

which is a two-dimensional anisotropic ghost-oscillator [1]. From Eq. (14), we can see that the terms ðpi � AiÞ can be associated to a minimal coupling term as is done in electromagnetic theory. From this vector potential, we find that <sup>B</sup> <sup>¼</sup> <sup>4</sup>ðβ�ω2θ<sup>Þ</sup> <sup>4</sup>�ω2θ<sup>2</sup> and the vector potential <sup>A</sup> can be rewritten as Ax^ ¼ � <sup>B</sup> <sup>2</sup> <sup>y</sup>^ and Ay^ <sup>¼</sup> <sup>B</sup> <sup>2</sup> x^. On the other hand, we already know from Eq. (11) that p^y; p^<sup>x</sup> n o <sup>¼</sup> <sup>β</sup> and if we set <sup>θ</sup> = 0 in the above equation for <sup>B</sup>, we can conclude that the deformation of the momentum plays a role analogous to a magnetic field.

#### 2.2. Discussion

string theory correction to gravity [6, 8]. In the rest of this model, we use for the noncommutative

<sup>2</sup> px; <sup>x</sup>^ <sup>¼</sup> <sup>x</sup> � <sup>θ</sup>

<sup>x</sup>; <sup>p</sup>^<sup>x</sup> <sup>¼</sup> px � <sup>β</sup>

<sup>2</sup>ð<sup>β</sup> � θω<sup>2</sup><sup>Þ</sup> <sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup> <sup>x</sup>^ � �<sup>2</sup>

2

!

Þ <sup>2</sup> þ

<sup>4</sup>ðω<sup>2</sup> � <sup>β</sup><sup>2</sup>=4<sup>Þ</sup> <sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup> ;

� <sup>p</sup>^<sup>y</sup> � Ay^ � �<sup>2</sup>

<sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup> <sup>y</sup>^; Ay^ � <sup>2</sup>ð<sup>β</sup> � θω<sup>2</sup><sup>Þ</sup>

� � � �

<sup>2</sup> <sup>y</sup>^ and Ay^ <sup>¼</sup> <sup>B</sup>

which is a two-dimensional anisotropic ghost-oscillator [1]. From Eq. (14), we can see that the terms ðpi � AiÞ can be associated to a minimal coupling term as is done in

can conclude that the deformation of the momentum plays a role analogous to a mag-

2 py

ð10Þ

ð12Þ

ð13Þ

¼ β: ð11Þ

Ψðx^; y^Þ ¼ 0;

; ð14Þ

<sup>4</sup>�ω2θ<sup>2</sup> and the vector

<sup>2</sup> x^. On the other hand, we already

2 y;

¼ 1 þ σ; p^y; p^<sup>x</sup>

<sup>4</sup>ðω<sup>2</sup> � <sup>β</sup><sup>2</sup>

<sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup>

<sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup> <sup>x</sup>^;

<sup>þ</sup> <sup>ω</sup>0<sup>2</sup> y^2

¼ β and if we set θ = 0 in the above equation for B, we

=4Þ

y^2 )

n o

If we consider the following change of variables in the classical phase space {x; y; px; py}

2

n o

Now that we have defined the deformed phase space, we can see the effects on the proposed cosmological model. From the action Eq. (4), we can obtain the Hamiltonian constraint Eq. (6); inserting relations Eq. (11), a Wheeler DeWitt (WDW) equation can be constructed

� p^<sup>y</sup> þ

<sup>x</sup>^<sup>2</sup> � <sup>4</sup>ð<sup>β</sup> � θω<sup>2</sup><sup>Þ</sup>

By a closer inspection of the equation, it is convenient to make the following definitions:

2

Þ <sup>2</sup> þ

With these definitions, we can rewrite Eq. (12) in a much simpler and suggestive form:

<sup>þ</sup> <sup>ω</sup>0<sup>2</sup> x^2

electromagnetic theory. From this vector potential, we find that <sup>B</sup> <sup>¼</sup> <sup>4</sup>ðβ�ω2θ<sup>Þ</sup>

<sup>ð</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup>

<sup>y</sup>^ <sup>¼</sup> <sup>y</sup> � <sup>θ</sup>

<sup>p</sup>^<sup>y</sup> <sup>¼</sup> py <sup>þ</sup> <sup>β</sup>

� � <sup>¼</sup> <sup>y</sup>^; <sup>p</sup>^<sup>y</sup>

it can be verified that if {x;y;px;py} obeys the usual Poisson algebra, then

f g y^; x^ ¼ θ; x^; p^<sup>x</sup>

<sup>p</sup>^<sup>x</sup> � <sup>2</sup>ð<sup>β</sup> � θω<sup>2</sup><sup>Þ</sup> <sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup> <sup>y</sup>^ � �<sup>2</sup>

<sup>4</sup>ðω<sup>2</sup> � <sup>β</sup><sup>2</sup>

<sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup>

H ¼ p^<sup>x</sup> � Ax^

potential <sup>A</sup> can be rewritten as Ax^ ¼ � <sup>B</sup>

know from Eq. (11) that p^y; p^<sup>x</sup>

netic field.

=4Þ

<sup>ω</sup>0<sup>2</sup> � <sup>4</sup>ð<sup>β</sup> � θω<sup>2</sup><sup>Þ</sup>

Ax^ � �2ð<sup>β</sup> � θω<sup>2</sup><sup>Þ</sup>

� �

� �<sup>2</sup>

n o

<sup>ð</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup>

parameters θij ¼ �θεij and βij ¼ βεij.

180 Trends in Modern Cosmology

as:

þ

HΨðx^; y^Þ ¼

<sup>4</sup>ð<sup>β</sup> � θω<sup>2</sup><sup>Þ</sup>

<sup>ð</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup>

(

2

!

Þ <sup>2</sup> þ We found that ω is defined in terms of the cosmological constant, then modifications to the oscillator frequency will imply modifications to the effective cosmological constant. Here, we have done a deformation of the phase space of the theory by introducing a modification to the momenta and to the minisuperspace coordinates, this gives two new fundamental constants θ and β. As expected, we obtain a different functional dependence for the frequency ω and the magnetic B field as functions of β and θ. With this in mind, we can construct a new frequency ω~ in terms of ω0<sup>2</sup> and the cyclotron term B<sup>2</sup> =4:

$$
\tilde{\omega}^2 = \omega^{\prime 2} - \frac{B^2}{4} = \frac{4(\omega^2 - \beta^2/4)}{4 - \omega^2 \theta^2}. \tag{15}
$$

This ω~ was obtained by a definition of the effective cosmological constant Λ ˜ ef f ¼ � <sup>3</sup> <sup>2</sup> <sup>ω</sup><sup>~</sup> <sup>2</sup> as was done in Section 2, to finally get a redefinition of the effective cosmological constant due to noncommutative parameters:

$$\bar{\Lambda}\_{\it eff} = \frac{4(\Lambda\_{\it eff} + \frac{3}{8}\beta^2)}{4 - \frac{2}{3}\Theta^2|\Lambda\_{\it eff}|}. \tag{16}$$

Now if we choose the case β = 0, this should be equivalent to the noncommutative minisuperspace model, hence we get an effective cosmological constant given by:

$$\dot{\Lambda}\_{\sharp f} = \frac{4\Lambda\_{\sharp f}}{(4 - \frac{2}{3}\Theta^2|\Lambda\_{\sharp f}|)}\tag{17}$$

We can see from Eq. (17) that the noncommutative parameter θ cannot take the place of the cosmological constant, but depending on the value of θ, the effective cosmological constant Λ ˜ ef f is modified. Equation 17 is in agreement with the results given in Refs. [9, 10].

#### 3. Model 2: Scalar field cosmology

Let us start with a homogeneous and isotropic universe with a flat Friedmann-Robertson-Walker (FRW) metric:

$$ds^2 = -N^2(t)dt^2 + a^2(t)\left[dr^2 + r^2d\Omega\right] \tag{18}$$

As usual, a(t) is the scale factor andN(t) is the lapse function.We use the Einstein-Hilbert action and a scalar field φ as the matter content for the model. In units 8πG ¼ 1, the action takes the form:

$$S = \left\{ dt \left\{ -\frac{3a\dot{a}^2}{N} + a^3 \left( \frac{\dot{\phi}^2}{2N} - N\Lambda \right) \right\} \right\} \tag{19}$$

Now, we make the following change of variables:

$$\mathbf{x} = m^{-1} a^{3/2} \sinh(m\phi), \\ \mathbf{y} = m^{-1} a^{3/2} \cosh(m\phi). \tag{20}$$

where <sup>m</sup>�<sup>1</sup> <sup>¼</sup> <sup>2</sup> ffiffiffiffiffiffiffi 2=3 p . Then the Hamiltonian is

$$H\_c = N\left(\frac{1}{2}P\_x^2 + \frac{\omega^2}{2}\mathbf{x}^2\right) - N\left(\frac{1}{2}P\_y^2 + \frac{\omega^2}{2}\mathbf{y}^2\right),\tag{21}$$

with <sup>ω</sup><sup>2</sup> ¼ � <sup>3</sup> <sup>4</sup> Λ. To find the dynamics, we solve the equations of motion; for this model, it can easily be integrated [9].

To construct the deformed model, we usually follow the canonical quantum cosmology approach, where after canonical quantization [11], one formally obtains the WDW equation. In the deformed phase space approach, the deformation is introduced by the Moyal brackets to get a deformed Poisson algebra. To construct a deformed Poisson algebra, we use the approach given in Refs. [1, 9]. We start with the same transformation on the classical phase space variables {x;y;Px;Py} that satisfy the usual Poisson algebra as shown in Section 2.1, Eqs. (10) and (11). With this deformed theory in mind, we first calculate the Hamiltonian which is formally analogous to Eq. (21) but constructed with the variables that obey the modified algebra Eq. (11)

$$H = \frac{1}{2} \left[ \left( P\_x^2 - P\_y^2 \right) - \omega\_1^2 \left( \mathbf{x} P\_y + y P\_x \right) + \omega\_2^2 \left( \mathbf{x}^2 - y^2 \right) \right]. \tag{22}$$

where we have used the change of variables Eq. (10) and the following definitions:

$$
\omega\_1^2 = \frac{\beta - \omega^2 \theta}{1 - \omega^2 \theta^2 / 4}, \quad \omega\_2^2 = \frac{\omega^2 - \beta^2 / 4}{1 - \omega^2 \theta^2 / 4}. \tag{23}
$$

Written in terms of the original variables, the Hamiltonian explicitly has the effects of the phase space deformation. These effects are encoded by the parameters θ and β. In Ref. [9], the late time behavior of this model was studied. From this formulation, two different physical theories arise, one that considers the variables x and y and a different theory based on x^ and y^. The first theory is interpreted as a "commutative" theory with a modified interaction, and this theory is referred as being realized in the commutative frame "(C-frame)" [12]. The second theory, which privileges the variables x^ and y^, is a theory with "noncommutative" variables but with the standard interaction and is referred to as realized in the noncommutative frame "(NC-frame)." In the "C-frame," our deformed model has a very nice interpretation that of a ghost-oscillator in the presence of constant magnetic field and allows us to write the effects of the noncommutative deformation as minimal coupling on the Hamiltonian and write the Hamiltonian in terms of the magnetic B-field [9].

To obtain the dynamics for the model, we derive the equations of motion from the Hamiltonian Eq. (22). The solutions for the variables x(t) and y(t) in the "C-frame" are:

#### Deformed Phase Space in Cosmology and Black Holes http://dx.doi.org/10.5772/intechopen.68282 183

$$\begin{cases} x(t) = \eta\_0 \, e^{\frac{-\omega\_1^2}{2}} \cosh(\omega' t + \delta\_1) - \zeta\_0 \, e^{\frac{\omega\_1^2}{2}} \cosh(\omega' t + \delta\_2), \\ y(t) = \eta\_0 \, e^{\frac{-\omega\_1^2}{2}} \cosh(\omega' t + \delta\_1) + \zeta\_0 \, e^{\frac{\omega\_1^2}{2}} \cosh(\omega' t + \delta\_2), \end{cases} \tag{24}$$

where ω<sup>0</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffi <sup>β</sup>2�4ω<sup>2</sup> <sup>4</sup>�ω2θ<sup>2</sup> q . For ω0<sup>2</sup> < 0, the hyperbolic functions are replaced by harmonic functions. There is a different solution for β ¼ 2ω, the solutions in the "C-frame" are:

$$\begin{cases} x(t) = (a+bt)e^{\frac{-\omega\_1^2}{2}t} + (c+dt)e^{\frac{\omega\_1^2}{2}t}, \\ y(t) = (a+bt)e^{\frac{-\omega\_1^2}{2}t} - (c+dt)e^{\frac{\omega\_1^2}{2}t}. \end{cases} \tag{25}$$

To compute the volume of the universe in the "C-frame," we use Eqs. (24) and (20).

$$
\sigma^3(t) = V\_0 \cosh^2(\omega' t),
\tag{26}
$$

where we have taken <sup>δ</sup><sup>1</sup> <sup>¼</sup> <sup>δ</sup><sup>2</sup> <sup>¼</sup> 0. For the case <sup>ω</sup>0<sup>2</sup> <sup>&</sup>lt; 0, the hyperbolic function is replaced by a harmonic function. For the case β ¼ 2ω, the volume is given by

$$a^3(t) = V\_0 + At + Bt^2,\tag{27}$$

where V0;A and B are constructed from the integration constants. To develop the dynamics in the "NC-frame," we start from the "C-frame" solutions and use Eq. (10), we get for the volume

$$\hat{a}^3(t) = \begin{cases} \hat{V}\_0 \left[ \cosh^2(\omega' t) - \frac{\omega'^2 \theta^2}{\left(2 - \omega\_1^2 \theta\right)^2} \sinh^2(\omega' t) \right] \text{ for } \omega'^2 > 0;\\ \hat{V}\_0 + Bt + Ct^2 & \text{for } \omega'^2 = 0;\\ \hat{V}\_0 \left[ \cos^2(|\omega'|t) - \frac{|\omega'|^2 \theta^2}{\left(2 - \omega\_1^2 \theta\right)^2} \sin^2(|\omega'|t) \right] \text{ for } \omega'^2 < 0, \end{cases} \tag{28}$$

where V^ <sup>0</sup> is the initial volume in the "NC-frame." We can see that for θ = 0, the descriptions in the two frames are the same.

#### 3.1. Discussion

<sup>x</sup> <sup>¼</sup> <sup>m</sup>�<sup>1</sup>

2=3 p . Then the Hamiltonian is

Hc <sup>¼</sup> <sup>N</sup> <sup>1</sup>

where <sup>m</sup>�<sup>1</sup> <sup>¼</sup> <sup>2</sup> ffiffiffiffiffiffiffi

182 Trends in Modern Cosmology

easily be integrated [9].

modified algebra Eq. (11)

B-field [9].

<sup>H</sup> <sup>¼</sup> <sup>1</sup> 2

P2 <sup>x</sup> � <sup>P</sup><sup>2</sup> y � �

ω2

� <sup>ω</sup><sup>2</sup>

where we have used the change of variables Eq. (10) and the following definitions:

=4 ; ω<sup>2</sup>

Written in terms of the original variables, the Hamiltonian explicitly has the effects of the phase space deformation. These effects are encoded by the parameters θ and β. In Ref. [9], the late time behavior of this model was studied. From this formulation, two different physical theories arise, one that considers the variables x and y and a different theory based on x^ and y^. The first theory is interpreted as a "commutative" theory with a modified interaction, and this theory is referred as being realized in the commutative frame "(C-frame)" [12]. The second theory, which privileges the variables x^ and y^, is a theory with "noncommutative" variables but with the standard interaction and is referred to as realized in the noncommutative frame "(NC-frame)." In the "C-frame," our deformed model has a very nice interpretation that of a ghost-oscillator in the presence of constant magnetic field and allows us to write the effects of the noncommutative deformation as minimal coupling on the Hamiltonian and write the Hamiltonian in terms of the magnetic

To obtain the dynamics for the model, we derive the equations of motion from the Hamilto-

nian Eq. (22). The solutions for the variables x(t) and y(t) in the "C-frame" are:

<sup>1</sup> <sup>¼</sup> <sup>β</sup> � <sup>ω</sup><sup>2</sup><sup>θ</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup>

<sup>1</sup> xPy þ yPx � � <sup>þ</sup> <sup>ω</sup><sup>2</sup> <sup>2</sup> <sup>x</sup><sup>2</sup> � <sup>y</sup><sup>2</sup> � � h i

> <sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> � <sup>β</sup><sup>2</sup>=<sup>4</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup>θ<sup>2</sup>

=4

with <sup>ω</sup><sup>2</sup> ¼ � <sup>3</sup>

a<sup>3</sup>=<sup>2</sup>

2 P2 <sup>x</sup> þ ω2 2 x2 � �

sinhðmφÞ, y <sup>¼</sup> <sup>m</sup>�<sup>1</sup>

To construct the deformed model, we usually follow the canonical quantum cosmology approach, where after canonical quantization [11], one formally obtains the WDW equation. In the deformed phase space approach, the deformation is introduced by the Moyal brackets to get a deformed Poisson algebra. To construct a deformed Poisson algebra, we use the approach given in Refs. [1, 9]. We start with the same transformation on the classical phase space variables {x;y;Px;Py} that satisfy the usual Poisson algebra as shown in Section 2.1, Eqs. (10) and (11). With this deformed theory in mind, we first calculate the Hamiltonian which is formally analogous to Eq. (21) but constructed with the variables that obey the

a<sup>3</sup>=<sup>2</sup>

� <sup>N</sup> <sup>1</sup> 2 P2 <sup>y</sup> þ ω2 2 y2 � �

<sup>4</sup> Λ. To find the dynamics, we solve the equations of motion; for this model, it can

coshðmφÞ: ð20Þ

; ð21Þ

: ð22Þ

: ð23Þ

As already discussed, phase space deformation gives two physical descriptions. If we say that both descriptions should be equal, then comparing the late time behavior for the two frames with the scale factor of de Sitter cosmology, an effective positive cosmological constant exists and is given by

$$
\Lambda\_{\it f\prime} = \frac{1}{3} \left( \frac{\beta^2 + 3\Lambda}{1 + \frac{3}{16}\Lambda \theta^2} \right). \tag{29}
$$

This result is the same as the one obtained from the WDW formalism of Kaluza-Klein cosmology. Therefore, one can start taking seriously the possibility that noncommutativity can shed light on the cosmological constant problem.

### 4. Model 3: Thermodynamics of noncommutative quantum Kerr black hole

Thermodynamics of black holes has a long history, focusing mainly on the problem of thermodynamic stability. It is known for a long time that this problem can be extended beyond the asymptotically flat spacetimes [13]. For example, in de Sitter spacetimes, thermodynamic information of black holes exhibit important differences with the previous case [14, 15]. Gibbons and Hawking found that, in analogy with the asymptotically flat space case, such black holes emit radiation with a perfect blackbody spectrum and its temperature is determined by their surface gravity. However, a feature of de Sitter space is that exists a cosmological event horizon, emitting particles with a temperature which is proportional to its surface gravity. The only way to achieve thermal equilibrium is when both surface gravities are equal, which corresponds to a degenerate case [16, 17].

Regarding AdS manifolds, it was shown that thermodynamic stability of black holes in this spacetime can be achieved [18]. In this manifold, gravitational potential produces a confinement for particles with nonzero mass, which acts as an effective cavity of finite volume, containing the black hole. An important feature of black holes in AdS manifolds is that their heat capacity is positive, opposite to the asymptotically flat case; additionally, this positiveness allows a canonical description of the system.

It is also known that thermodynamic stability of black holes is related with dynamical stability of those systems, which brings an additional motivation to study it. For example, in the asymptotically flat spacetime case, it is well known that Schwarzschild black holes are thermodynamically unstable, although they are dynamically stable [19]. For AdS spacetimes, however, it is known that both thermodynamic and dynamical stability are closely related [20, 21].

In this study, we study black holes in asymptotically flat spacetime, whereby it seems very legitimate to ask whether corrections like the above discussed noncommutativity or even semiclassical ones can modify thermodynamic properties of black holes in order to have thermodynamic stable systems.

In a number of studies [22–24], black hole entropy proposed by Bekenstein and Hawking is postulated to be the fundamental thermodynamic relation for black holes, which contains all thermodynamic information of the system. Under this assumption, corresponding classical thermodynamic formalism is constructed, finding that its thermodynamic structure resembles ordinary magnetic systems instead of fluids.

#### 4.1. Schwarzschild and Kerr black holes

As previously discussed, it is well known that for an asymptotically flat spacetime, temperature of black holes is proportional to its surface gravity κ, as T ¼ κħ=2πkBc, which is commonly known as Hawking temperature [25]; this semiclassical result, along with Bekenstein bound for entropy, leads to the Bekenstein-Hawking entropy,

This result is the same as the one obtained from the WDW formalism of Kaluza-Klein cosmology. Therefore, one can start taking seriously the possibility that noncommutativity can shed light on

4. Model 3: Thermodynamics of noncommutative quantum Kerr black hole

Thermodynamics of black holes has a long history, focusing mainly on the problem of thermodynamic stability. It is known for a long time that this problem can be extended beyond the asymptotically flat spacetimes [13]. For example, in de Sitter spacetimes, thermodynamic information of black holes exhibit important differences with the previous case [14, 15]. Gibbons and Hawking found that, in analogy with the asymptotically flat space case, such black holes emit radiation with a perfect blackbody spectrum and its temperature is determined by their surface gravity. However, a feature of de Sitter space is that exists a cosmological event horizon, emitting particles with a temperature which is proportional to its surface gravity. The only way to achieve thermal equilibrium is when both surface gravities are equal, which

Regarding AdS manifolds, it was shown that thermodynamic stability of black holes in this spacetime can be achieved [18]. In this manifold, gravitational potential produces a confinement for particles with nonzero mass, which acts as an effective cavity of finite volume, containing the black hole. An important feature of black holes in AdS manifolds is that their heat capacity is positive, opposite to the asymptotically flat case; additionally, this positiveness

It is also known that thermodynamic stability of black holes is related with dynamical stability of those systems, which brings an additional motivation to study it. For example, in the asymptotically flat spacetime case, it is well known that Schwarzschild black holes are thermodynamically unstable, although they are dynamically stable [19]. For AdS spacetimes, however, it is known

In this study, we study black holes in asymptotically flat spacetime, whereby it seems very legitimate to ask whether corrections like the above discussed noncommutativity or even semiclassical ones can modify thermodynamic properties of black holes in order to have

In a number of studies [22–24], black hole entropy proposed by Bekenstein and Hawking is postulated to be the fundamental thermodynamic relation for black holes, which contains all thermodynamic information of the system. Under this assumption, corresponding classical thermodynamic formalism is constructed, finding that its thermodynamic structure resembles

As previously discussed, it is well known that for an asymptotically flat spacetime, temperature of black holes is proportional to its surface gravity κ, as T ¼ κħ=2πkBc, which is commonly

that both thermodynamic and dynamical stability are closely related [20, 21].

the cosmological constant problem.

184 Trends in Modern Cosmology

corresponds to a degenerate case [16, 17].

allows a canonical description of the system.

ordinary magnetic systems instead of fluids.

4.1. Schwarzschild and Kerr black holes

thermodynamic stable systems.

$$\mathcal{S}\_{BH} = \frac{c^3}{4G\hbar}A.\tag{30}$$

Where A stands for the area of the event horizon of the black hole. The Kerr metric, which describes a rotating black hole, can be written as:

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right)dt^2 - \frac{4Mra\sin^2\theta}{\Sigma}dt d\theta + \frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2 + \frac{B\sin^2\theta}{\Sigma}d\phi^2;\tag{31}$$

where, <sup>Σ</sup> <sup>¼</sup> <sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> cos <sup>2</sup>θ, <sup>Δ</sup> <sup>¼</sup> <sup>r</sup><sup>2</sup> � <sup>2</sup>Mr <sup>þ</sup> <sup>a</sup>2, <sup>B</sup> ¼ ðr<sup>2</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup><sup>Þ</sup> <sup>2</sup> � <sup>a</sup><sup>2</sup>Δsin <sup>2</sup> θ and a ¼ J=Mc. The area of the event horizon of a black hole is given by A ¼ ð s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detjgμνj q ds. Applying for the elements of the metric tensor given in Eq. (31), the resulting area is:

$$A = 8\pi G^2 M^2 c^{-4} \left[ 1 + \sqrt{1 - \frac{c^2 f^2}{G^2 M^4}} \right]. \tag{32}$$

Assumed thermodynamic fundamental relation for Kerr black holes is found substituting the above result in Eq. (30); where U = Mc<sup>2</sup> is the internal energy of the system and J is its angular momentum. This relation can be written as [22]:

$$\mathcal{S}\_{BH}(\mathcal{U}, \mathcal{J}) = \frac{2\pi k\_B}{\hbar c} \left( \frac{\mathcal{G}\mathcal{U}^2}{c^4} + \sqrt{\frac{\mathcal{G}^2 \mathcal{U}^4}{c^8} - c^2 \mathcal{J}^2} \right);\tag{33}$$

where the following constants appear: G is the universal gravitational constant, c is the speed of light, ħ is the reduced Planck constant, and kB is the Boltzmann constant. In recent years, in the search of suitable candidates of quantum gravity, that is, in the quest to understand microscopic states of black holes [26, 27], a number of quantum corrections to Bekenstein-Hawking (BH) entropy SBH have arisen. We are interested not only in the possible thermodynamic implications of quantum corrections to this entropy but also in the consequences of introducing noncommutativity as proposed by Obregon et al. [28], considering that coordinates of minisuperspace are noncommutative. From a variety of approaches that have emerged in recent years to correct SBH, logarithmic ones are a popular choice among those. These corrections arise from quantum corrections to the string theory partition function [29] and are related to infrared or low-energy properties of gravity. They are also independent of high-energy or ultraviolet properties of the theory [26, 29–31]. We will denote the selected expression for quantum and noncommutative corrected entropy as S\* , which is obtained by following the ideas presented in [28]. The starting point is the diffeomorphism between the Kantowski-Sachs cosmological model, describing a homogeneous but anisotropic universe [32], and the Schwarzschild interior solution, whose line element for r < 2M is given by:

$$ds^2 = -\left(\frac{2M}{t} - 1\right)^{-1}dt^2 + \left(\frac{2M}{t} - 1\right)dr^2 + t^2(d\theta^2 + \sin^2\theta d\phi^2);\tag{34}$$

where the role of temporal t and the spatial r coordinates is swapped, that is, transformation t \$ r is performed, leading to a change on the causal structure of spacetime; considering the Misner parametrization of the Kantowski-Sachs metric it follows:

$$ds^2 = -N^2dt^2 + e^{(2\sqrt{3}\gamma)}dr^2 + e^{(-2\sqrt{3}\gamma)}e^{(-2\sqrt{3}\lambda)}(d\theta^2 + \sin^2\theta d\phi^2). \tag{35}$$

Parameters λ and γ play the role of the cartesian coordinates in the Kantowski-Sachs minisuperspace. If Eqs. (34) and (35) are compared, it is straightforward to notice correspondence between components of the metric tensor, which allows us to identify the functions N, γ, and λ as:

$$N^2 = \left(\frac{2M}{t} - 1\right)^{-1}, \quad e^{(-2\sqrt{3}\gamma)} = \frac{2M}{t} - 1, \quad e^{(-2\sqrt{3}\gamma)}e^{(-2\sqrt{3}\lambda)} = t^2.$$

Next, the Wheeler DeWitt (WDW) equation for Kantowski-Sachs metric with the above parametrization of the Schwarzschild interior solution is found, along with the corresponding Hamiltonian of the system H through the Arnowitt-Deser-Misner (ADM) formalism. This Hamiltonian is introduced into the quantum wave equation HΨ ¼ 0, where Ψðγ;λÞ is the wave function. This process leads to the WDW equation whose solution can be found by separation of variables.

However, we are not interested in the usual case, rather our point of interest is the solution that can be found when the symplectic structure of minisuperspace is modified by the inclusion of a noncommutativity parameter between the coordinates λ and γ, that is, the following commutation relation is obeyed: ½λ;γ� ¼ iθ, where θ is the noncommutative parameter; this relation strongly resembles noncommutative quantum mechanics. It is also possible to introduce the aforementioned deformation in terms of a Moyal product [7], which modifies the original phase space, similarly to noncommutative quantum mechanics [33]:

These modifications allow us to redefine the coordinates of minisuperspace in order to obtain a noncommutative version of the WDW equation:

$$
\left[\frac{\partial^2}{\partial \boldsymbol{\gamma}^2} - \frac{\partial^2}{\partial \boldsymbol{\lambda}^2} + 48e^{(-2\sqrt{3}\boldsymbol{\lambda} + \sqrt{3}\boldsymbol{\theta}\boldsymbol{\theta}\_{\boldsymbol{\gamma}})}\right] \Psi(\boldsymbol{\lambda}, \boldsymbol{\gamma}) = 0;\tag{36}
$$

where P<sup>γ</sup> is the momentum on coordinate γ. The above equation can be solved by separation of variables to obtain the corresponding wave function [6]:

$$\Psi(\lambda,\gamma) = e^{i\sqrt{3}\nu\gamma} K\_{i\nu}[4e^{(-\sqrt{3}(\lambda+\sqrt{3}\nu\theta/2)}];\tag{37}$$

where ν is the separation constant and Ki<sup>ν</sup> are the modified Bessel functions. We can see in Eq. (37) that the wave function has the form <sup>Ψ</sup>ðλ;γÞ ¼ <sup>e</sup><sup>i</sup> ffiffi 3 <sup>p</sup> νγΦðλÞ; therefore, dependence on the coordinate γ is the one of a plane wave. It is worth mentioning that this contribution vanishes when thermodynamic observables are calculated.

With the wave function presented in Eq. (37) for the noncommutative Kantowski-Sachs cosmological model, a modified noncommutative version of the entropy can be obtained. In order to calculate the partition function of the system, the Feynman-Hibbs procedure is considered [34]. Starting with the separated differential equation for λ:

ds<sup>2</sup> ¼ � <sup>2</sup><sup>M</sup>

186 Trends in Modern Cosmology

ds<sup>2</sup> ¼ �N<sup>2</sup>

<sup>N</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup><sup>M</sup>

separation of variables.

<sup>t</sup> � <sup>1</sup> � ��<sup>1</sup>

<sup>t</sup> � <sup>1</sup> � ��<sup>1</sup>

Misner parametrization of the Kantowski-Sachs metric it follows:

<sup>ð</sup><sup>2</sup> ffiffi 3 <sup>p</sup> <sup>γ</sup><sup>Þ</sup>

dr<sup>2</sup> <sup>þ</sup> <sup>e</sup>

nents of the metric tensor, which allows us to identify the functions N, γ, and λ as:

; e ð�<sup>2</sup> ffiffi 3 <sup>p</sup> <sup>γ</sup><sup>Þ</sup> <sup>¼</sup> <sup>2</sup><sup>M</sup>

phase space, similarly to noncommutative quantum mechanics [33]:

<sup>∂</sup>λ<sup>2</sup> <sup>þ</sup> <sup>48</sup><sup>e</sup>

Ψðλ;γÞ ¼ e

� �

<sup>i</sup> ffiffi 3 <sup>p</sup> νγKiν½4<sup>e</sup>

noncommutative version of the WDW equation:

∂2 <sup>∂</sup>γ<sup>2</sup> � <sup>∂</sup><sup>2</sup>

of variables to obtain the corresponding wave function [6]:

that the wave function has the form <sup>Ψ</sup>ðλ;γÞ ¼ <sup>e</sup><sup>i</sup> ffiffi

thermodynamic observables are calculated.

dt<sup>2</sup> <sup>þ</sup> <sup>e</sup>

dt<sup>2</sup> <sup>þ</sup>

2M <sup>t</sup> � <sup>1</sup> � �

where the role of temporal t and the spatial r coordinates is swapped, that is, transformation t \$ r is performed, leading to a change on the causal structure of spacetime; considering the

> ð�<sup>2</sup> ffiffi 3 <sup>p</sup> <sup>γ</sup><sup>Þ</sup> e ð�<sup>2</sup> ffiffi 3 <sup>p</sup> <sup>λ</sup><sup>Þ</sup>

Parameters λ and γ play the role of the cartesian coordinates in the Kantowski-Sachs minisuperspace. If Eqs. (34) and (35) are compared, it is straightforward to notice correspondence between compo-

Next, the Wheeler DeWitt (WDW) equation for Kantowski-Sachs metric with the above parametrization of the Schwarzschild interior solution is found, along with the corresponding Hamiltonian of the system H through the Arnowitt-Deser-Misner (ADM) formalism. This Hamiltonian is introduced into the quantum wave equation HΨ ¼ 0, where Ψðγ;λÞ is the wave function. This process leads to the WDW equation whose solution can be found by

However, we are not interested in the usual case, rather our point of interest is the solution that can be found when the symplectic structure of minisuperspace is modified by the inclusion of a noncommutativity parameter between the coordinates λ and γ, that is, the following commutation relation is obeyed: ½λ;γ� ¼ iθ, where θ is the noncommutative parameter; this relation strongly resembles noncommutative quantum mechanics. It is also possible to introduce the aforementioned deformation in terms of a Moyal product [7], which modifies the original

These modifications allow us to redefine the coordinates of minisuperspace in order to obtain a

where P<sup>γ</sup> is the momentum on coordinate γ. The above equation can be solved by separation

where ν is the separation constant and Ki<sup>ν</sup> are the modified Bessel functions. We can see in Eq. (37)

nate γ is the one of a plane wave. It is worth mentioning that this contribution vanishes when

3

ð� ffiffi 3 <sup>p</sup> <sup>ð</sup>λ<sup>þ</sup> ffiffi 3 <sup>p</sup> νθ=2<sup>Þ</sup>

ð�<sup>2</sup> ffiffi 3 <sup>p</sup> <sup>λ</sup><sup>þ</sup> ffiffi 3 <sup>p</sup> <sup>θ</sup>Pγ<sup>Þ</sup>

dr<sup>2</sup> <sup>þ</sup> <sup>t</sup> 2

<sup>t</sup> � <sup>1</sup>; <sup>e</sup>

<sup>ð</sup>dθ<sup>2</sup> <sup>þ</sup> sin <sup>2</sup>

<sup>ð</sup>dθ<sup>2</sup> <sup>þ</sup> sin <sup>2</sup>

ð�<sup>2</sup> ffiffi 3 <sup>p</sup> <sup>γ</sup><sup>Þ</sup> e ð�<sup>2</sup> ffiffi 3 <sup>p</sup> <sup>λ</sup><sup>Þ</sup> <sup>¼</sup> <sup>t</sup> 2 :

θdφ<sup>2</sup>

θdφ<sup>2</sup>

Ψðλ;γÞ ¼ 0; ð36Þ

<sup>p</sup> νγΦðλÞ; therefore, dependence on the coordi-

�; ð37Þ

Þ; ð34Þ

Þ: ð35Þ

$$\left[-\frac{d^2}{d\lambda^2} + 48e^{-2\sqrt{3}\lambda + 3\nu\theta}\right]\Phi(\lambda) = 3\nu^2\Phi(\lambda);\tag{38}$$

In this equation, the exponential in the potential term <sup>V</sup>ðλÞ ¼ 48 exp ½�<sup>2</sup> ffiffiffi 3 <sup>p</sup> <sup>λ</sup> <sup>þ</sup> <sup>3</sup>νθ� is expanded up to second order in λ and if a change of variables is considered, resulting differential equation can be compared with a one-dimensional quantum harmonic oscillator, which is a non-degenerate quantum system. In the Feynman-Hibbs procedure, the potential under study is modified by quantum effects, for the harmonic oscillator is given by:

$$\mathcal{U}(\mathfrak{x}) = V(\overline{\mathfrak{x}}) + \frac{\beta \mathfrak{h}^2}{24m} V'(\overline{\mathfrak{x}});$$

where x is the mean value of x and V00 ðxÞ stands for the second derivative of the potential. For the considered change of variables, the noncommutative quantum-corrected potential can be written as:

$$\mathcal{U}I(\mathbf{x}) = \frac{3}{4\pi} \frac{E\_p}{l\_p^2} e^{3\nu\theta} \left[ \mathbf{x}^2 + \frac{\beta l\_p^2 E\_p}{12} \right]. \tag{39}$$

The above potential allows us to calculate the canonical partition function of the system:

$$Z(\beta) = \mathbb{C} \int\_{-\infty}^{\infty} e^{-\beta l I(x)} dx;\tag{40}$$

where <sup>β</sup>�<sup>1</sup> is proportional to the Bekenstein-Hawking temperature and <sup>C</sup> <sup>¼</sup> <sup>2</sup>π<sup>l</sup> 2 <sup>p</sup>Epβ h i�1=<sup>2</sup> is a constant. Substituting U(x) into Eq. (40) and performing the integral over x, the partition function is given by:

$$Z(\beta) = \sqrt{\frac{2\pi}{3}} \frac{e^{3\upsilon\theta/2}}{E\_p \beta} \exp\left[-\frac{\beta^2 E\_p^2}{16\pi} e^{3\upsilon\theta}\right];\tag{41}$$

This partition function allows us to calculate any desired thermodynamic observable by means of the thermodynamic connection of the Helmholtz free energy A ¼ �kBTlnZðβÞ, with the internal energy and the Legendre transformation:

$$
\langle E \rangle = -\frac{\partial}{\partial \beta} \ln Z(\beta); \quad \frac{S}{k\_B} = \ln Z(\beta) + \beta \langle E \rangle.
$$

With this equation for 〈E〉, the value of β can be determined as a function of the Hawking temperature <sup>β</sup><sup>H</sup> <sup>¼</sup> <sup>8</sup>πMc<sup>2</sup>=Ep, obtaining:

$$\beta = \beta\_H e^{-3v\theta} \left[ 1 - \frac{1}{\beta\_H e^{-3v\theta}} \frac{1}{Mc^2} \right];\tag{42}$$

With the aid of this relation and the Legendre transformation for Helmholtz free energy presented above, an expression for the noncommutative quantum-corrected black hole entropy can be found:

$$S^{\star} = \mathcal{S}\_{BH}e^{-3\upsilon\theta} - \frac{1}{2}k\_{\text{B}}\ln\left[\frac{\mathcal{S}\_{BH}}{k\_{\text{B}}}e^{-3\upsilon\theta}\right] + \mathcal{O}(\mathcal{S}\_{BH}^{-1}e^{-3\upsilon\theta}).\tag{43}$$

Functional form of S\* is basically the same than quantum-corrected commutative case, besides the addition of multiplicative factor e�3νθ to Bekenstein-Hawking entropy. From now on, we will denote the noncommutative term in this expression, for the sake of simplicity, as:

$$
\Gamma = \exp\left[-\Im \nu \theta\right].
$$

Likewise, natural units, G ¼ ħ ¼ kB ¼ c ¼ 1, will be considered through the rest of this chapter. In this section, the previous result found in Eq. (43) for the Schwarzschild noncommutative black hole is extended to the rotating case, that is, the Kerr black hole. This is not straightforward as an analog expression for the noncommutative entropy of the rotating black hole is required, implying the application of a similar procedure to the one presented above: A diffeomorphism between the Kerr metric and some appropriated cosmological model and the procedure is presented in Ref. [28]. To our knowledge, the implementation of this procedure has not been yet reported. However, we are interested to have an expression to study not only the static case but also the effect of angular momentum over the physical properties of the system. Our proposal to have an approximated relation for the extended Kerr black hole entropy starts with the assumption that for entropy found in Eq. (43), Bekenstein-Hawking entropy for Schwarzschild in this relation SBH can be also substituted for its Kerr counterpart given in Eq. (33). As the noncommutative relation for quantum Schwarzschild black hole entropy is correct, it is clear that our proposal to the quantum noncommutative Kerr black hole entropy will be a good approximation for small values of J when compared to the values of U<sup>2</sup> , whatever be the exact expression for the rotating case. For our proposal, in the vicinity of small values of angular momentum, λ and γ, the coordinates of the minisuperspace are the same than in the Schwarzschild case. Therefore, the corrected entropy that will be analyzed is:

$$S^\star = 2\pi \,\Gamma \left( \mathcal{U}^2 + \sqrt{\mathcal{U}^4 - f^2} \right) - \frac{1}{2} \ln \left[ 2\pi \,\Gamma \left( \mathcal{U}^2 + \sqrt{\mathcal{U}^4 - f^2} \right) \right]. \tag{44}$$

A clarification must be made that Eq. (44) is not a unique valid generalization for the quantumcorrected noncommutative entropy of a rotating black hole in the neighborhood of small J. However, we claim that this is the most natural extension from the Schwarzschild case to the Kerr one. Although, to our knowledge, there is no general argument to support that Eq. (43) remains valid for any other black hole besides the Schwarzschild one. However, there is some evidence that for the case of charged black holes, the functional form of Eq. (43) is maintained, at least partially [35].

Through the rest of this section, all thermodynamic expressions with superindex ⋆ will stand for noncommutative quantum-corrected quantities derived from Eq. (44), meanwhile, all thermodynamic functions without subindexes or superindexes will represent the corresponding noncommutative Bekenstein-Hawking counterparts. It is known that noncommutativity parameter θ in spacetime is small, from observational evidence [36, 37]; although in this study, noncommutativity on the coordinates of minisuperspace is considered instead, it is expected such parameter to be also small [38]; nonetheless, its actual bounds are not well known yet. We will consider that parameter Γ is bounded in the interval 0 < Γ ≤ 1. As previously mentioned for the non-corrected Kerr black hole, Eq. (44) is now assumed to be a fundamental thermodynamic relation for the rotating black hole, when noncommutative and quantum corrections are considered. It is well known from classical thermodynamics that fundamental equations contain all the thermodynamic information of the considered system [39], and, as a consequence, modifications introduced by corrections to entropy (which imply modifications to thermodynamic information) are carried through all thermodynamic quantities.

In Figure 1, plots for both Bekenstein-Hawking entropy and its quantum-corrected counterpart are presented for <sup>Γ</sup> <sup>¼</sup> 1. Figure 1a shows plots for <sup>S</sup> <sup>¼</sup> <sup>S</sup>ðU<sup>Þ</sup> and <sup>S</sup><sup>⋆</sup> <sup>¼</sup> <sup>S</sup><sup>⋆</sup>ðUÞ; Bekenstein-Hawking entropy is above the quantum-corrected one, in all its dominion, even in the region of low masses, where entropy is thermodynamically stable [22, 24]. Figure 1b presents the same curves as function of angular momentum instead, for U ¼ 1; a similar behavior can be noticed in this case. If this analysis is performed over the noncommutative relation, it is found that for small values of θ, differences between both SBH and S\* are negligible.

#### 4.2. Equations of state

β ¼ βHe

�3νθ � <sup>1</sup> 2

<sup>S</sup><sup>⋆</sup> <sup>¼</sup> SBHe

188 Trends in Modern Cosmology

<sup>S</sup><sup>⋆</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>Γ</sup> <sup>U</sup><sup>2</sup> <sup>þ</sup>

at least partially [35].

�3νθ <sup>1</sup> � <sup>1</sup>

With the aid of this relation and the Legendre transformation for Helmholtz free energy presented above, an expression for the noncommutative quantum-corrected black hole entropy can be found:

Functional form of S\* is basically the same than quantum-corrected commutative case, besides the addition of multiplicative factor e�3νθ to Bekenstein-Hawking entropy. From now on, we

Γ ¼ exp ½�3νθ�:

Likewise, natural units, G ¼ ħ ¼ kB ¼ c ¼ 1, will be considered through the rest of this chapter. In this section, the previous result found in Eq. (43) for the Schwarzschild noncommutative black hole is extended to the rotating case, that is, the Kerr black hole. This is not straightforward as an analog expression for the noncommutative entropy of the rotating black hole is required, implying the application of a similar procedure to the one presented above: A diffeomorphism between the Kerr metric and some appropriated cosmological model and the procedure is presented in Ref. [28]. To our knowledge, the implementation of this procedure has not been yet reported. However, we are interested to have an expression to study not only the static case but also the effect of angular momentum over the physical properties of the system. Our proposal to have an approximated relation for the extended Kerr black hole entropy starts with the assumption that for entropy found in Eq. (43), Bekenstein-Hawking entropy for Schwarzschild in this relation SBH can be also substituted for its Kerr counterpart given in Eq. (33). As the noncommutative relation for quantum Schwarzschild black hole entropy is correct, it is clear that our proposal to the quantum noncommutative Kerr black hole entropy will be a good approximation for small values of J when compared to the values of U<sup>2</sup>

whatever be the exact expression for the rotating case. For our proposal, in the vicinity of small values of angular momentum, λ and γ, the coordinates of the minisuperspace are the same than in the Schwarzschild case. Therefore, the corrected entropy that will be analyzed is:

> � 1 2

A clarification must be made that Eq. (44) is not a unique valid generalization for the quantumcorrected noncommutative entropy of a rotating black hole in the neighborhood of small J. However, we claim that this is the most natural extension from the Schwarzschild case to the Kerr one. Although, to our knowledge, there is no general argument to support that Eq. (43) remains valid for any other black hole besides the Schwarzschild one. However, there is some evidence that for the case of charged black holes, the functional form of Eq. (43) is maintained,

ln 2<sup>π</sup> <sup>Γ</sup> <sup>U</sup><sup>2</sup> <sup>þ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

� � � � q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

� � q

kBln SBH kB e �3νθ � �

will denote the noncommutative term in this expression, for the sake of simplicity, as:

βHe�3νθ

� �

1 Mc<sup>2</sup>

> <sup>þ</sup> <sup>O</sup>ðS�<sup>1</sup> BHe

; ð42Þ

�3νθÞ: <sup>ð</sup>43<sup>Þ</sup>

,

: ð44Þ

Working in entropic representation, fundamental Bekenstein-Hawking thermodynamic relation for a Kerr black hole has the form SBH ¼ SBHðU;JÞ. For these systems, partial derivatives of SBH T � ð∂SUÞ<sup>J</sup> and Ω � ð∂JUÞ<sup>S</sup> play the role of thermodynamic equations of state; here, T

Figure 1. A comparison between Bekenstein-Hawking entropy (solid line) and its quantum-corrected counterpart (dashdot line) is presented; both relations exhibit a region where entropy is a concave function, implying the existence of metastable states. (a) Entropy as a function of internal energy, J = 1. (b) Entropy as a function of angular momentum for U = 1, S ¼ Sð1;JÞ.

stands for Hawking temperature and Ω is the angular velocity. In entropic representation, equations of state are defined by:

$$\frac{1}{T} \equiv \left(\frac{\partial S\_{BH}}{\partial \mathcal{U}}\right)\_{\mathcal{J}}; \quad \frac{\Omega}{T} \equiv -\left(\frac{\partial S\_{BH}}{\partial \mathcal{J}}\right)\_{\mathcal{U}}.\tag{45}$$

For the entropy of the quantum-corrected entropy S\* , the above relations remain valid. In entropic representation, T and Ω for the noncommutative quantum-corrected entropy are given by:

$$\frac{1}{T^\star} = \frac{\mathcal{U}\left(4\pi\,\Gamma\sqrt{\mathcal{U}^4 - f^2} + 4\pi\,\Gamma\mathcal{U}^2 - 1\right)}{\sqrt{\mathcal{U}^4 - f^2}},\tag{46a}$$

$$\frac{\Omega^{\star}}{T^{\star}} = \frac{1}{2} \frac{\int \left( 4\pi \,\Gamma \sqrt{\mathcal{U}^{4} - \mathcal{J}^{2}} + 4\pi \,\Gamma \mathcal{U}^{2} - 1 \right)}{\sqrt{\mathcal{U}^{4} - \mathcal{J}^{2}} \left( \mathcal{U}^{2} + \sqrt{\mathcal{U}^{4} - \mathcal{J}^{2}} \right)} \,. \tag{46b}$$

The same relations for noncommutative Bekenstein-Hawking entropy are calculated as:

$$\frac{1}{T} = \frac{4\pi\,\Gamma\mathcal{U}(\mathcal{U}^2 + \sqrt{\mathcal{U}^4 - \mathcal{J}^2})^2}{\sqrt{\mathcal{U}^4 - \mathcal{J}^2}},\tag{47a}$$

$$\frac{\Omega}{T} = \frac{2\pi \,\Gamma \mathfrak{J}}{\sqrt{\mathcal{U}^4 - \mathcal{J}^2}}. \tag{47b}$$

When the overall effect over T and T\* of noncommutativity was analyzed, different values of parameter Γ were tested, including Γ ¼ 1 (commutative case). The corresponding curves present a noticeable effect by the presence of Γ; nonetheless, functional behavior either of T or T\* is not modified. A comparison of the plots of both temperature is presented in Figure 2 for Γ = 1, in order to illustrate how quantum corrections introduced in entropy affect thermodynamic properties of black holes. Resulting curves of T and T\* are very similar, although the latter one is slightly higher than TðU;JÞ, an opposite result to the one obtained when entropy was studied; it indicates that for a given change in its internal energy, variations of entropy are greater for quantum-corrected entropy when compared to the Bekenstein-Hawking one.

As previously mentioned, when values in the vicinity of Γ = 1 are considered, temperature is minimally affected by noncommutativity. We also tested smaller values of noncommutativity parameter, it was found that the maximum values that T and T\* are able to reach are noticeably increased. However, the shape of both curves is not modified by changing the value of Γ.

An interesting result is obtained for angular velocity Ω, this property seems to be independent of both quantum and noncommutative corrections to entropy, namely:

stands for Hawking temperature and Ω is the angular velocity. In entropic representation,

entropic representation, T and Ω for the noncommutative quantum-corrected entropy are

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

� �

<sup>U</sup><sup>2</sup> <sup>þ</sup>

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

<sup>T</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>Γ</sup><sup>J</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

When the overall effect over T and T\* of noncommutativity was analyzed, different values of parameter Γ were tested, including Γ ¼ 1 (commutative case). The corresponding curves present a noticeable effect by the presence of Γ; nonetheless, functional behavior either of T or T\* is not modified. A comparison of the plots of both temperature is presented in Figure 2 for Γ = 1, in order to illustrate how quantum corrections introduced in entropy affect thermodynamic properties of black holes. Resulting curves of T and T\* are very similar, although the latter one is slightly higher than TðU;JÞ, an opposite result to the one obtained when entropy was studied; it indicates that for a given change in its internal energy, variations of entropy are greater for quantum-corrected entropy when compared to the Bekenstein-

As previously mentioned, when values in the vicinity of Γ = 1 are considered, temperature is minimally affected by noncommutativity. We also tested smaller values of noncommutativity parameter, it was found that the maximum values that T and T\* are able to reach are noticeably increased. However, the shape of both curves is not modified by changing the

<sup>T</sup> � � <sup>∂</sup>SBH ∂J � �

<sup>þ</sup> <sup>4</sup><sup>π</sup> <sup>Γ</sup>U<sup>2</sup> � <sup>1</sup>

<sup>þ</sup> <sup>4</sup><sup>π</sup> <sup>Γ</sup>U<sup>2</sup> � <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

> Þ 2

U

: ð45Þ

, the above relations remain valid. In

<sup>q</sup> ; <sup>ð</sup>46a<sup>Þ</sup>

� � <sup>q</sup> : <sup>ð</sup>46b<sup>Þ</sup>

<sup>q</sup> ; <sup>ð</sup>47a<sup>Þ</sup>

<sup>q</sup> : <sup>ð</sup>47b<sup>Þ</sup>

J ; Ω

q

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>U</sup><sup>4</sup> � <sup>J</sup> 2

The same relations for noncommutative Bekenstein-Hawking entropy are calculated as:

<sup>T</sup> <sup>¼</sup> <sup>4</sup><sup>π</sup> <sup>Γ</sup>UðU<sup>2</sup> <sup>þ</sup>

Ω

equations of state are defined by:

190 Trends in Modern Cosmology

given by:

Hawking one.

value of Γ.

1

For the entropy of the quantum-corrected entropy S\*

1 T<sup>⋆</sup> ¼

Ω<sup>⋆</sup> <sup>T</sup><sup>⋆</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>T</sup> � <sup>∂</sup>SBH ∂U � �

U 4π Γ

J 4π Γ

q

1

$$
\Omega = \Omega^\star = \frac{I}{2\mathcal{U}\left(\mathcal{U}^2 + \sqrt{\mathcal{U}^4 - I^2}\right)}.\tag{48}
$$

In Figure 3, plots for angular velocity are presented. As this equation of state is not modified by any of the considered corrections, only one curve per graphic appears; first, in Figure 3a, Ω as a function of the black hole internal energy is presented, as can be noticed, angular velocity steadily decreases as black hole mass is increased, asymptotically going to zero. Figure 3b considers instead the case where the black hole mass is fixed at U ¼ 10, for which Ω grows until it reaches a maximum value determined by the square root that appears in the denominator of Eq. (48), beyond this value angular velocity becomes complex.

Figure 2. Temperature in the commutative case Γ = 1 for Bekenstein-Hawking usual entropy and its quantum-corrected counterpart. (a) Plots of <sup>T</sup>ðU; <sup>1</sup><sup>Þ</sup> (solid line) versus <sup>T</sup><sup>⋆</sup>ðU; <sup>1</sup><sup>Þ</sup> (dash-dot line) as a function of internal energy for a fixed value of angular momentum J = 1. (b) The same curves, considering instead for variations in J at a fixed U ¼ 1.

Figure 3. Angular velocity for Bekenstein-Hawking entropy and the quantum-corrected version are presented in Eq. (44). (a) Ω as a function of internal energy considering a fixed value of angular momentum (J = 1). (b) Angular velocity as a function of angular momentum considering U ¼ 10.

#### 5. Conclusions

In section 2, if we turn our attention to the case where there is no deformation on the coordinates. Taking the noncommutative parameter θ = 0, we have that the frequency and the effective cosmological constant are given by:

$$
\tilde{\omega}^2 = \omega^2 - \frac{\beta^2}{4}, \qquad \text{and} \qquad \dot{\Lambda}\_{\text{eff}} = \Lambda\_{\text{eff}} + \frac{3\beta^2}{8}. \tag{49}
$$

From the last equation, we get the most interesting result of this section. We can see that noncommutative parameter β and Λef f compete to give the effective cosmological constant Λ ˜ ef f . If we consider the case of a flat universe with a vanishing Λef f , we see that Λ ˜ ef f <sup>¼</sup> <sup>3</sup>β<sup>2</sup> <sup>8</sup> . This shows the relationship between the cosmological constant and the deformed parameter. Recently, some evidence on the possibility that the effects of the phase space deformation could be related to the late time acceleration of the universe as well as to the cosmological constant were presented [8]. Interestingly, in the particular case of <sup>β</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>θ, we find that frequency reduces to <sup>ω</sup><sup>~</sup> <sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> and we have that <sup>Λ</sup> ˜ ef f ¼ Λef f . In this case, even as we have done a deformation on the minisuperspace of the theory, the effects cancel out and the resulting theory behaves as in the commutative theory. The results are similar for model 2 (Section 3), where under a totally classical regime, we find the same functional relationship between the cosmological constant and the deformation parameter β. Therefore, we conclude that noncommutative phase space deformations can hold the answer to the cosmological constant problem.

Then, in Section 4, an analysis on the thermodynamic properties of noncommutative quantumcorrected Kerr black holes using an approximate relation was presented. Although the resulting expressions are mathematically more complicated, the thermodynamic properties still retain the same functional behavior with respect to those calculated through Bekenstein-Hawking entropy. It can be proved that Kerr black holes do not pass through a first-order phase transition [4]; since the local criteria to find the critical point are not fulfilled for any value in the domain, corresponding isotherms do not exhibit van der Waals loops, and the Maxwell construction cannot be obtained; all of which are characteristic of this kind of transition. Regarding the effective noncommutativity incorporated in the coordinates of minisuperspace, outside the vicinity where Γ ≈ 1, changes introduced by this parameter over the thermodynamic information of the system are relevant. For a complete analysis using this phase deformations, for example, thermodynamic response functions, thermodynamic stability, and phase transitions for Kerr black holes, see Ref. [4].

#### Acknowledgements

Eri Atahualpa Mena Barboza thanks the financial support from C.U.CI., U. de G. project Desarrollo de la investigación y fortalecimiento del posgrado 235506.

#### Author details

5. Conclusions

192 Trends in Modern Cosmology

Λ ˜

problem.

cosmological constant are given by:

<sup>ω</sup><sup>~</sup> <sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> � <sup>β</sup><sup>2</sup>

frequency reduces to <sup>ω</sup><sup>~</sup> <sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> and we have that <sup>Λ</sup>

ity, and phase transitions for Kerr black holes, see Ref. [4].

Desarrollo de la investigación y fortalecimiento del posgrado 235506.

Acknowledgements

In section 2, if we turn our attention to the case where there is no deformation on the coordinates. Taking the noncommutative parameter θ = 0, we have that the frequency and the effective

˜

ef f ¼ Λef f þ

3β<sup>2</sup>

<sup>8</sup> : <sup>ð</sup>49<sup>Þ</sup>

˜ ef f <sup>¼</sup> <sup>3</sup>β<sup>2</sup>

ef f ¼ Λef f . In this case, even as we have done

<sup>8</sup> . This

<sup>4</sup> ; and <sup>Λ</sup>

ef f . If we consider the case of a flat universe with a vanishing Λef f , we see that Λ

From the last equation, we get the most interesting result of this section. We can see that noncommutative parameter β and Λef f compete to give the effective cosmological constant

shows the relationship between the cosmological constant and the deformed parameter. Recently, some evidence on the possibility that the effects of the phase space deformation could be related to the late time acceleration of the universe as well as to the cosmological constant were presented [8]. Interestingly, in the particular case of <sup>β</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>θ, we find that

˜

a deformation on the minisuperspace of the theory, the effects cancel out and the resulting theory behaves as in the commutative theory. The results are similar for model 2 (Section 3), where under a totally classical regime, we find the same functional relationship between the cosmological constant and the deformation parameter β. Therefore, we conclude that noncommutative phase space deformations can hold the answer to the cosmological constant

Then, in Section 4, an analysis on the thermodynamic properties of noncommutative quantumcorrected Kerr black holes using an approximate relation was presented. Although the resulting expressions are mathematically more complicated, the thermodynamic properties still retain the same functional behavior with respect to those calculated through Bekenstein-Hawking entropy. It can be proved that Kerr black holes do not pass through a first-order phase transition [4]; since the local criteria to find the critical point are not fulfilled for any value in the domain, corresponding isotherms do not exhibit van der Waals loops, and the Maxwell construction cannot be obtained; all of which are characteristic of this kind of transition. Regarding the effective noncommutativity incorporated in the coordinates of minisuperspace, outside the vicinity where Γ ≈ 1, changes introduced by this parameter over the thermodynamic information of the system are relevant. For a complete analysis using this phase deformations, for example, thermodynamic response functions, thermodynamic stabil-

Eri Atahualpa Mena Barboza thanks the financial support from C.U.CI., U. de G. project

E.A. Mena-Barboza<sup>1</sup> \*, L.F. Escamilla-Herrera<sup>2</sup> , J.C. López-Domínguez<sup>3</sup> and J. Torres-Arenas<sup>4</sup>

\*Address all correspondence to: emena@cuci.udg.mx

1 Centro Universitario de la Ciénega, Universidad de Guadalajara Ave. Universidad, Ocotlán, Jalisco, México

2 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad de, México

3 Unidad Académica de Física, Universidad Autónoma de Zacatecas, Calzada Solidaridad, Zacatecas, México

4 Departamento de Física de la Universidad de Guanajuato, León, Guanajuato, México

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Theoretical Physics. 2007;46:2928; Ortiz C, Mena-Barboza E, Sabido M, Socorro J. International Journal of Theoretical Physics. 2008;47:1240; Guzman W, Ortiz C, Sabido M, Socorro J, Aguero MA. International Journal of Modern Physics D. 2007;16:1625; Obregon O, Quiros I. Physical Review. 2011;D84:044005; Guzman W, Sabido M, Socorro J, Physics Letters. 2011;B697:271-274.


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[8] Vakili B, Pedram P, Jalalzadeh S. Physics Letters B. 2010;687:119-123

[10] Sabido M, Yee-Romero C. Physics Letters B. 2016;757:57

[12] Barbosa GD. Physical Review D. 2005;71:063511

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1977;353:499

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1971;26:1344

Mathematical Physics. 2013;321:629-680

strophic holography. Physical Review D. 1999;60:064018

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for a Kerr black hole. Revista Mexicana de Fisica. 2014;60:59-68

[9] Pérez-Payán S, Sabido M, Yee-Romero C. Physical Review D. 2013;88:027503

[11] Cordero R, Garcia-Compean H, Turrubiates FJ. Physical Review D. 2011;83:125030

[13] Gibbons GW, Hawking SW. Cosmological event horizons, thermodynamics, and particle

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[16] Nariai H. On some static solutions of Einstein's gravitational field equations in a spherically symmetric case. Science Reports of the Tohoku Imperial University. 1950;34:160-167

[17] Nariai H. On a new cosmological solution of Einstein's field equations of gravitation.

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[19] Price RH. Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and

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Letters. 2011;B697:271-274.

194 Trends in Modern Cosmology


## **Semi-Analytic Techniques for Solving Quasi-Normal Modes**

Chun-Hung Chen, Hing-Tong Cho and

Alan S. Cornell

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68114

#### Abstract

In this chapter, we discuss an approach to obtaining black hole quasi-normal modes known as the asymptotic iteration method, which was initially developed in mathematics as a new way to solve for eigenvalues in differential equations. Furthermore, we demonstrate that the asymptotic iteration method allows one to also solve for the radial quasi-normal modes on a variety of black hole spacetimes for a variety of perturbing fields. A specific example for Dirac fields in a general dimensional Schwarzschild black hole spacetime is given, as well as for spin-3/2 field quasi-normal modes.

Keywords: extra-dimensions, quasi-normal modes, quantum fields in curved space, supergravity, blackholes

#### 1. Introduction

Quasi-normal modes (QNMs) are one of the most important theoretical results in modern cosmology, especially for studying the perturbations from various fields on black hole spacetimes. In this theory, the behaviour of a particle around a black hole is dominated by the radial equation, and the evolution of QNMs behaves like damped harmonic oscillators with specific frequencies. The frequencies are constructed by complex modes, where the real part is the actual frequency and the imaginary part represents the damping rate due to the gravitational emission. In lay terms, the QNMs are the characteristic sounds of the black hole.

With the recent ground breaking progress into the detection of gravitational wave data, where it is believed that the last part of the gravitational wave emission, called the ring down phase,

is dominated by the QNMs, it is exciting that perturbation theory in curved spacetimes can now be possibly tested in a real experimental system, and that a large number of scientists from all over the world are involved in the data analysis. Related issues of the QNMs within the range of research into cosmology include studying the stability of black holes and probing the dimensionality of spacetime. It is, however, the observable gravitational wave data from the collisions of binary black hole systems, which indicates how the background spacetime will finally become a Kerr black hole spacetime through gravitational wave emission. Perturbation theory on a Kerr black hole spacetime still includes some difficulties in the higher dimensional cases, which will be a challenge for the theoretical community for some time to come.

Methods that are used to obtain QNMs can be both semi-analytic and numerical methods and were introduced by Cho et al. [1], the most famous of these is the WKB approximation methods [2]. Note that, the WKB approximation has been extended to sixth order [3] and is powerful in many cases, but like all methods have several limitations. A new method has been developed in recent years called the asymptotic iteration method (AIM), which is more efficient in some cases. This method was used to solve eigenvalue problems for the second-order homogeneous linear differential equations [4, 5] and also successfully used in calculating QNMs [6]. Reviewing this AIM and providing the tools "in detail" for studying QNMs in the higher dimensional spacetimes are the key focus of this chapter.

As such, this chapter is organised as follows: In the next section, we shall review the recent progress on perturbation theory in curved spacetimes. More precisely, we shall present a comparison of the spin-3/2 field in general dimensional Schwarzschild spacetimes with other spin fields, including the spherical harmonics and the radial equations. In Section 3, we shall review the AIM and present an exercise detailing how the QNMs of Dirac fields are obtained in general dimensional Schwarzschild spacetimes. Furthermore, we can also compare to spin-3/2 QNMs results. We shall conclude with a brief summary.

#### 2. Perturbation theory in a general dimensional Schwarzschild spacetime

#### 2.1. Eigenvalue problem on spheres

For perturbation theory in curved spacetimes, separability can always simplify the equations of motion and plays an important role. For the maximally symmetric spacetime cases, the eigenmodes on spheres allow us to separate the angular part for various spin fields and simplify the equations of motion from the general form into a "radial-time" presentation. For the case of bosonic fields, an earlier study by Rubin and Ordóñez presented a systematic study [7, 8] as well as in a later work by Higuchi [9]. For the case of fermionic fields, Camporesi and Higuchi presented the eigenmodes for spinor fields on arbitrary dimensional spheres [10], and in a recent work by the authors [11], the spinor-vector eigenmodes on arbitrary dimensional spheres were derived using a similar approach to Camporesi and Higuchi's methods. In this section, we review the structure of these eigenmodes, especially for the case of spinor-vector fields, which shall be presented with the characteristics of both spinor and vector fields.

The metric of the N-sphere is given by

$$d\Omega\_N^2 = d\Theta\_N^2 + \sin^2\Theta\_N d\Omega\_{N-1'}^2 \tag{1}$$

where, in this metric, we restrict the sphere to radius r = 1. When we consider the eigenvalue problem in this spacetime, the bosonic field and fermionic field shall be studied with different operators. In the case of bosonic fields, the operator for the eigenvalue equation will be the Laplacian operator ∇<sup>μ</sup>∇μ, whereas, in the case of fermionic fields, it will be the Dirac operator γ<sup>μ</sup>∇μ, where γμ is the Dirac gamma matrices. In Table 1, we present the structure of the eigenmodes with various spin fields on the sphere and also the conditions on the specific mode, such as the transverse, traceless and symmetric conditions.

is dominated by the QNMs, it is exciting that perturbation theory in curved spacetimes can now be possibly tested in a real experimental system, and that a large number of scientists from all over the world are involved in the data analysis. Related issues of the QNMs within the range of research into cosmology include studying the stability of black holes and probing the dimensionality of spacetime. It is, however, the observable gravitational wave data from the collisions of binary black hole systems, which indicates how the background spacetime will finally become a Kerr black hole spacetime through gravitational wave emission. Perturbation theory on a Kerr black hole spacetime still includes some difficulties in the higher dimensional cases, which will

Methods that are used to obtain QNMs can be both semi-analytic and numerical methods and were introduced by Cho et al. [1], the most famous of these is the WKB approximation methods [2]. Note that, the WKB approximation has been extended to sixth order [3] and is powerful in many cases, but like all methods have several limitations. A new method has been developed in recent years called the asymptotic iteration method (AIM), which is more efficient in some cases. This method was used to solve eigenvalue problems for the second-order homogeneous linear differential equations [4, 5] and also successfully used in calculating QNMs [6]. Reviewing this AIM and providing the tools "in detail" for studying QNMs in the

As such, this chapter is organised as follows: In the next section, we shall review the recent progress on perturbation theory in curved spacetimes. More precisely, we shall present a comparison of the spin-3/2 field in general dimensional Schwarzschild spacetimes with other spin fields, including the spherical harmonics and the radial equations. In Section 3, we shall review the AIM and present an exercise detailing how the QNMs of Dirac fields are obtained in general dimensional Schwarzschild spacetimes. Furthermore, we can also compare to spin-

2. Perturbation theory in a general dimensional Schwarzschild spacetime

For perturbation theory in curved spacetimes, separability can always simplify the equations of motion and plays an important role. For the maximally symmetric spacetime cases, the eigenmodes on spheres allow us to separate the angular part for various spin fields and simplify the equations of motion from the general form into a "radial-time" presentation. For the case of bosonic fields, an earlier study by Rubin and Ordóñez presented a systematic study [7, 8] as well as in a later work by Higuchi [9]. For the case of fermionic fields, Camporesi and Higuchi presented the eigenmodes for spinor fields on arbitrary dimensional spheres [10], and in a recent work by the authors [11], the spinor-vector eigenmodes on arbitrary dimensional spheres were derived using a similar approach to Camporesi and Higuchi's methods. In this section, we review the structure of these eigenmodes, especially for the case of spinor-vector fields, which shall be presented with the characteristics of both spinor and vector fields.

be a challenge for the theoretical community for some time to come.

higher dimensional spacetimes are the key focus of this chapter.

3/2 QNMs results. We shall conclude with a brief summary.

2.1. Eigenvalue problem on spheres

198 Trends in Modern Cosmology

The metric of the N-sphere is given by

Looking first at the longitudinal and non-transverse modes for bosonic fields, the longitudinal and non-transverse eigenfunctions for higher spins are the linear combination of the eigenfunctions for the lower spin one. For example, for the vector fields, the longitudinal eigenvector is the covariant derivative of a scalar eigenfunction. Furthermore, for the symmetric tensor fields, there are three types of non-transverse eigenfunctions. The first one is the metric element multiplied by a scalar eigenfunction, the second one is the longitudinal-longitudinal eigenfunction, which is the linear combination for longitudinal eigenvectors, and the last one is


Table 1. Structure of the eigenvalue problem for various fields on N-sphere.

the longitudinal-transverse eigenfunction, which is the linear combination of transverse eigenvectors. Analogous to the non-transverse-traceless modes for fermionic fields, the nontransverse-traceless eigenspinor-vector is the linear combination of the eigenspinor. We note that there are two non-transverse-traceless eigenspinor-vectors, due to the a(+) and a(�) being different factors where

$$a\_{(\pm)} = -\frac{\mathrm{i}}{2} \left( \mathrm{j} + \frac{N-1}{2} \right) \pm \mathrm{i}\sqrt{\mathrm{j}^2 + (N-1)\mathrm{j} + \frac{1}{4}(N-5)(N-1)}.\tag{2}$$

This indicates a special signature for spinor-vector harmonics that do not have the transversetraceless eigenmodes for S<sup>2</sup> .

Next, we look at the transverse-traceless modes. For the bosonic fields, an unique way to construct this type of eigenfunction was suggested by Higuchi [9]. Analogous to the fermionic case, for the transverse-traceless eigenspinor-vector, ψμ ¼ ðψθ<sup>N</sup> ,ψθ<sup>i</sup> Þ, ψθ<sup>N</sup> behaves like a spinor on N – 1 spheres, and ψθ<sup>i</sup> behaves like a spinor-vector on N – 1 spheres. If we let ψθ<sup>i</sup> be the linear combination of the non-transverse-traceless eigenspinor-vector on N – 1 spheres, ψθ<sup>N</sup> has to be non-zero to satisfy the transverse and traceless conditions. If we let ψθ<sup>i</sup> be the linear combination of the transverse-traceless eigenspinor-vector on N – 1-spheres, ψθ<sup>N</sup> has to be zero, because the ψθ<sup>i</sup> already satisfies the transverse and traceless condition.

On the other hand, we can take a look at the eigenvalue. For the bosonic fields, all of the eigenvalues contain a similar first term, which is the eigenvalue of the scalar fields; however, the starting value of the angular momentum quantum number l has to be considered case by case, as well as the second term for higher spin cases. For the fermionic cases, the eigenvalue of the non-transverse-traceless eigenspinor-vector has a very different value from the spinor eigenvalue, though the eigenvalue of the transverse and traceless eigenspinor-vector is exactly the same as the eigenspinor for the spin-1/2 field.

As a remark on this section, the eigenfunctions and eigenvalues on N-spheres are independent for bosonic and fermionic fields, even though they have a very similar style of structure, which indicates that the spherical harmonics for a bosonic field cannot be constructed by the spherical harmonics of a fermionic field, and vice versa. In this section, we presented a review of the eigenvalue problem for scalar, spinor, vector, spinor-vector, and symmetric tensor fields on spheres, where further details can be found in the papers referred to in this section.

#### 2.2. Effective potentials

In perturbation theory with various fields in the Schwarzschild black hole spacetime, a radial equation (Schrödinger-like equation) will be derived from the equations of motion, which shall be the master equation of this study. In a general way, the studies of perturbation theory in maximally symmetric spacetimes are well established and include the (A)dS and Reissner-Nordström spacetimes, but not for the spin-3/2 fields yet. With our recent progress in the study of spin-3/2 fields, we may now do a comparison of various massless fields in a general dimensional Schwarzschild black hole spacetime in this section.

The metric of a general dimensional Schwarzschild spacetime is given by

$$dS^2 = -f(r)dt^2 + f^{-1}(r)dr^2 + r^2d\Omega\_{D-2'}^2\tag{3}$$

where <sup>f</sup>ðrÞ ¼ <sup>1</sup> � <sup>2</sup>M=rD�3, <sup>D</sup> is the dimensional factor and <sup>M</sup> is the mass of the black hole. The master equation can be obtained from the equations of motion by a change of coordinates to give the Schrödinger-like equation

$$
\frac{d^2}{dr\_\*^2} \Psi\_s + (\omega^2 - V\_s) \Psi\_s = 0,\tag{4}
$$

where the subscript s represents the "spin" and r\* represents the "tortoise" coordinate, which can be defined as follows <sup>d</sup> dr� <sup>¼</sup> <sup>f</sup>ðr<sup>Þ</sup> <sup>d</sup> dr. The mathematical meaning of this coordinate is that a mapping of the location of the event horizon of the Schwarzschild black hole from r<sup>0</sup> (where f (r0) = 0) is taken to minus infinity.

We shall first look at the four-dimensional cases, where, for the bosonic fields, the radial equation can be represented with the potential [12]

$$V\_s = f \left[ \frac{l(l+1)}{r^2} + (1-s^2) \frac{2\mathcal{M}}{r^3} \right],\tag{5}$$

where l is an integer, s = 0 represents the effective potential for scalar fields, s = 1 for the electromagnetic fields, and s = 2 represents the "vector-type" perturbation for gravitational fields (which is the Regge-Wheeler equation). The "scalar-type" perturbation for the gravitational field in the four-dimensional case is the Zerilli equation, with an effective potential

$$V\_{Z,s=2} = \frac{2f}{r^3} \left[ \frac{9M^3 + 3\lambda^2Mr^2 + \lambda^2(1+\lambda)r^3 + 9M^2\lambda r}{\left(3M + \lambda r\right)^2} \right],\tag{6}$$

where λ ¼ ðl � 1Þðl þ 2Þ=2.

For the fermionic fields in the four-dimensional Schwarzschild case, the effective potential can be shown as follows [13, 14]:

$$dV\_s = \pm f \frac{dW\_s}{dr} + W\_{s\prime}^2 \tag{7}$$

where

the longitudinal-transverse eigenfunction, which is the linear combination of transverse eigenvectors. Analogous to the non-transverse-traceless modes for fermionic fields, the nontransverse-traceless eigenspinor-vector is the linear combination of the eigenspinor. We note that there are two non-transverse-traceless eigenspinor-vectors, due to the a(+) and a(�) being

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 4

ðN � 5ÞðN � 1Þ

: ð2Þ

Þ, ψθ<sup>N</sup> behaves like a spinor

different factors where

200 Trends in Modern Cosmology

traceless eigenmodes for S<sup>2</sup>

2.2. Effective potentials

<sup>a</sup>ð�Þ ¼ � <sup>i</sup>

<sup>2</sup> <sup>j</sup> <sup>þ</sup>

.

the same as the eigenspinor for the spin-1/2 field.

N � 1 2 � �

case, for the transverse-traceless eigenspinor-vector, ψμ ¼ ðψθ<sup>N</sup> ,ψθ<sup>i</sup>

zero, because the ψθ<sup>i</sup> already satisfies the transverse and traceless condition.

� i

j

This indicates a special signature for spinor-vector harmonics that do not have the transverse-

Next, we look at the transverse-traceless modes. For the bosonic fields, an unique way to construct this type of eigenfunction was suggested by Higuchi [9]. Analogous to the fermionic

on N – 1 spheres, and ψθ<sup>i</sup> behaves like a spinor-vector on N – 1 spheres. If we let ψθ<sup>i</sup> be the linear combination of the non-transverse-traceless eigenspinor-vector on N – 1 spheres, ψθ<sup>N</sup> has to be non-zero to satisfy the transverse and traceless conditions. If we let ψθ<sup>i</sup> be the linear combination of the transverse-traceless eigenspinor-vector on N – 1-spheres, ψθ<sup>N</sup> has to be

On the other hand, we can take a look at the eigenvalue. For the bosonic fields, all of the eigenvalues contain a similar first term, which is the eigenvalue of the scalar fields; however, the starting value of the angular momentum quantum number l has to be considered case by case, as well as the second term for higher spin cases. For the fermionic cases, the eigenvalue of the non-transverse-traceless eigenspinor-vector has a very different value from the spinor eigenvalue, though the eigenvalue of the transverse and traceless eigenspinor-vector is exactly

As a remark on this section, the eigenfunctions and eigenvalues on N-spheres are independent for bosonic and fermionic fields, even though they have a very similar style of structure, which indicates that the spherical harmonics for a bosonic field cannot be constructed by the spherical harmonics of a fermionic field, and vice versa. In this section, we presented a review of the eigenvalue problem for scalar, spinor, vector, spinor-vector, and symmetric tensor fields on

In perturbation theory with various fields in the Schwarzschild black hole spacetime, a radial equation (Schrödinger-like equation) will be derived from the equations of motion, which shall be the master equation of this study. In a general way, the studies of perturbation theory in maximally symmetric spacetimes are well established and include the (A)dS and Reissner-Nordström spacetimes, but not for the spin-3/2 fields yet. With our recent progress in the study of spin-3/2 fields, we may now do a comparison of various massless fields in a general

spheres, where further details can be found in the papers referred to in this section.

dimensional Schwarzschild black hole spacetime in this section.

r

<sup>2</sup> þ ð<sup>N</sup> � <sup>1</sup>Þ<sup>j</sup> <sup>þ</sup>

$$\mathcal{W}\_{s=\frac{1}{2}} = \frac{\sqrt{f}(j+\frac{1}{2})}{r}, j = \frac{1}{2}, \frac{3}{2}, \dots \tag{8}$$

$$\mathcal{W}\_{s=\frac{3}{2}} = \frac{\sqrt{f}(j - \frac{1}{2})(j + \frac{1}{2})(j + \frac{3}{2})}{r\left[(j + \frac{1}{2})^2 - f\right]}, j = \frac{3}{2}, \frac{5}{2}, \dots \tag{9}$$

Note that the "�" represents two isospectral potentials, which were known as the supersymmetric partner potentials.

For the higher dimensional cases, there are more types of effective potentials, which can be presented in the four-dimensional case, which indicates that some special cases will not exist in the four-dimensional cases but will exist in the higher dimensional one. In the following sections, we shall discuss the higher dimensional effective potentials as presented in Table 2.

Starting with bosonic fields, we have one effective potential for scalar fields in the higher dimensional Schwarzschild spacetime, and it is necessary to satisfy Eq. (5) when D = 4 and s = 0. For the case of electromagnetic fields, there are two types of effective potentials, which are the

Fields Vs Scalar [12] Vs¼<sup>0</sup> <sup>¼</sup> <sup>f</sup> <sup>l</sup>ðlþD�3<sup>Þ</sup> <sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>D</sup>�<sup>2</sup> <sup>4</sup> <sup>D</sup>�<sup>4</sup> <sup>r</sup><sup>2</sup> <sup>f</sup> <sup>þ</sup> <sup>2</sup><sup>f</sup> 0 r h i � � l ¼ 0; 1; 2; …. Dirac [13] Vs¼1=<sup>2</sup> ¼ �<sup>f</sup> dWs¼1=<sup>2</sup> dr <sup>þ</sup> <sup>W</sup><sup>2</sup> s¼1=2 where Ws¼1=<sup>2</sup> ¼ fð <sup>j</sup>þD�<sup>3</sup> 2 <sup>r</sup> Þ, j ¼ 1=2; 3=2; 5=2; …. Electromagnetic [15] Scalar-type perturbation VS,s¼<sup>1</sup> <sup>¼</sup> <sup>f</sup> <sup>l</sup>ðlþD�3ÞþðD�2ÞðD�4<sup>Þ</sup> 4 <sup>r</sup><sup>2</sup> � <sup>ð</sup>3D�8ÞðD�4Þ<sup>M</sup> 2rD�<sup>1</sup> � �. l ¼ 1; 2; 3; …. Vector-type perturbation VV,s¼<sup>1</sup> <sup>¼</sup> <sup>f</sup> <sup>l</sup>ðlþD�3Þþð<sup>D</sup>�2ÞðD�4<sup>Þ</sup> 4 <sup>r</sup><sup>2</sup> � <sup>D</sup>ðD�4Þ<sup>M</sup> 2rD�<sup>1</sup> � �. l ¼ 1; 2; 3; …. Rarita-Schwinger [11] Related to the non-TT eigenmodes VNTT,s¼3=<sup>2</sup> ¼ �<sup>f</sup> dWNTT,s¼3=<sup>2</sup> dr <sup>þ</sup> <sup>W</sup><sup>2</sup> NTT,s¼3=2 where WNTT,s¼3=<sup>2</sup> ¼ ffiffi <sup>f</sup> <sup>p</sup> <sup>ð</sup>jþD�<sup>3</sup> 2 Þ r ð 2 <sup>D</sup>�2Þ 2 <sup>ð</sup>jþD�<sup>3</sup> 2 Þ 2 �1�D�<sup>4</sup> <sup>D</sup>�2<sup>ð</sup> <sup>2</sup><sup>M</sup> rD�<sup>3</sup> Þ ð 2 <sup>D</sup>�2Þ 2 <sup>ð</sup>jþD�<sup>3</sup> 2 Þ 2 �f � �, j <sup>¼</sup> <sup>1</sup>=2; <sup>3</sup>=2; <sup>5</sup>=2…. Related to the TT eigenmodes VTT,s¼3=<sup>2</sup> ¼ �<sup>f</sup> dWTT,s¼3=<sup>2</sup> dr <sup>þ</sup> <sup>W</sup><sup>2</sup> TT,s¼3=2

where

$$W\_{TT,s=3/2} = f(\frac{j+\frac{D-1}{2}}{r}), j = 1/2, 3/2, 5/2, \dots, 1$$

Fields Vs

Note that the "�" represents two isospectral potentials, which were known as the supersym-

For the higher dimensional cases, there are more types of effective potentials, which can be presented in the four-dimensional case, which indicates that some special cases will not exist in the four-dimensional cases but will exist in the higher dimensional one. In the following sections, we shall discuss the higher dimensional effective potentials as presented in Table 2. Starting with bosonic fields, we have one effective potential for scalar fields in the higher dimensional Schwarzschild spacetime, and it is necessary to satisfy Eq. (5) when D = 4 and s = 0. For the case of electromagnetic fields, there are two types of effective potentials, which are the

metric partner potentials.

202 Trends in Modern Cosmology

Fields Vs

Vs¼<sup>0</sup> <sup>¼</sup> <sup>f</sup> <sup>l</sup>ðlþD�3<sup>Þ</sup>

Vs¼1=<sup>2</sup> ¼ �<sup>f</sup> dWs¼1=<sup>2</sup>

<sup>j</sup>þD�<sup>3</sup> 2

Scalar-type perturbation VS,s¼<sup>1</sup> <sup>¼</sup> <sup>f</sup> <sup>l</sup>ðlþD�3ÞþðD�2ÞðD�4<sup>Þ</sup>

Vector-type perturbation VV,s¼<sup>1</sup> <sup>¼</sup> <sup>f</sup> <sup>l</sup>ðlþD�3Þþð<sup>D</sup>�2ÞðD�4<sup>Þ</sup>

VNTT,s¼3=<sup>2</sup> ¼ �<sup>f</sup> dWNTT,s¼3=<sup>2</sup>

l ¼ 0; 1; 2; ….

where

Ws¼1=<sup>2</sup> ¼ fð

l ¼ 1; 2; 3; ….

l ¼ 1; 2; 3; ….

where

where

WNTT,s¼3=<sup>2</sup> ¼

WTT,s¼3=<sup>2</sup> ¼ fð

<sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>D</sup>�<sup>2</sup>

dr <sup>þ</sup> <sup>W</sup><sup>2</sup> s¼1=2

<sup>4</sup> <sup>D</sup>�<sup>4</sup> <sup>r</sup><sup>2</sup> <sup>f</sup> <sup>þ</sup> <sup>2</sup><sup>f</sup> 0 r

<sup>r</sup> Þ, j ¼ 1=2; 3=2; 5=2; ….

� �

<sup>r</sup><sup>2</sup> � <sup>ð</sup>3D�8ÞðD�4Þ<sup>M</sup> 2rD�<sup>1</sup>

.

.

NTT,s¼3=2

, j ¼ 1=2; 3=2; 5=2….

4

4 <sup>r</sup><sup>2</sup> � <sup>D</sup>ðD�4Þ<sup>M</sup> 2rD�<sup>1</sup>

Related to the non-TT eigenmodes

ffiffi <sup>f</sup> <sup>p</sup> <sup>ð</sup>jþD�<sup>3</sup> 2 Þ r

Related to the TT eigenmodes

<sup>j</sup>þD�<sup>3</sup> 2

VTT,s¼3=<sup>2</sup> ¼ �<sup>f</sup> dWTT,s¼3=<sup>2</sup>

� �

dr <sup>þ</sup> <sup>W</sup><sup>2</sup>

ð 2 <sup>D</sup>�2Þ 2 <sup>ð</sup>jþD�<sup>3</sup> 2 Þ 2 �1�D�<sup>4</sup> <sup>D</sup>�2<sup>ð</sup> <sup>2</sup><sup>M</sup> rD�<sup>3</sup> Þ

dr <sup>þ</sup> <sup>W</sup><sup>2</sup>

ð 2 <sup>D</sup>�2Þ 2 <sup>ð</sup>jþD�<sup>3</sup> 2 Þ 2 �f � �

<sup>r</sup> Þ, j ¼ 1=2; 3=2; 5=2;….

TT,s¼3=2

h i � �

Scalar [12]

Dirac [13]

Electromagnetic [15]

Rarita-Schwinger [11]

Gravitational [16, 17]

Scalar-type perturbation VS,s¼<sup>2</sup> <sup>¼</sup> f H <sup>16</sup>r<sup>2</sup> <sup>m</sup>þ<sup>1</sup> ½ � <sup>2</sup>NðNþ1Þð1�f<sup>Þ</sup> <sup>2</sup> , N <sup>¼</sup> <sup>D</sup> � <sup>2</sup> <sup>H</sup> <sup>¼</sup> <sup>N</sup><sup>4</sup>ð<sup>N</sup> <sup>þ</sup> <sup>1</sup><sup>Þ</sup> 2 ð1 � fÞ 3 <sup>þ</sup> <sup>N</sup>ð<sup>N</sup> <sup>þ</sup> <sup>1</sup>Þ½4ð2N<sup>2</sup> � <sup>3</sup><sup>N</sup> <sup>þ</sup> <sup>4</sup>Þ<sup>m</sup> <sup>þ</sup> <sup>N</sup>ð<sup>N</sup> � <sup>2</sup>Þð<sup>N</sup> � <sup>4</sup>Þð<sup>N</sup> <sup>þ</sup> <sup>1</sup>Þ�ð<sup>1</sup> � <sup>f</sup><sup>Þ</sup> 2 � <sup>12</sup>N½ð<sup>N</sup> � <sup>4</sup>Þ<sup>m</sup> <sup>þ</sup> <sup>N</sup>ð<sup>N</sup> <sup>þ</sup> <sup>1</sup>Þð<sup>N</sup> � <sup>2</sup>Þ�mð<sup>1</sup> � <sup>f</sup>Þ þ <sup>16</sup>m<sup>3</sup> <sup>þ</sup> <sup>4</sup>Nð<sup>N</sup> <sup>þ</sup> <sup>2</sup>Þm<sup>2</sup>: m ¼ lðl þ N � 1Þ � N, l ¼ 2; 3; 4…: Vector and tensor type perturbation VV=T,s¼<sup>2</sup> <sup>¼</sup> <sup>f</sup> <sup>r</sup><sup>2</sup> <sup>l</sup>ð<sup>l</sup> <sup>þ</sup> <sup>D</sup> � <sup>3</sup>Þ þ <sup>ð</sup>D�2ÞðD�4<sup>Þ</sup> <sup>4</sup> � <sup>μ</sup>V=<sup>T</sup> 2 ðD�2Þ 2 M rD�<sup>3</sup> h i. μ<sup>V</sup> ¼ 3;lV ¼ 2; 3; 4… for vector-type perturbation. μ<sup>T</sup> ¼ �1;lT ¼ 1; 2; 3;… for tensor-type perturbation.

Table 2. The effective potential for various fields in the higher dimensional cases (D ≥ 5).

"scalar-type perturbation potential" and the "vector-type perturbation potential". It is accidental that both of these effective potentials satisfy Eq. (5) when D = 4 and s = 1, but it has been shown they have different behaviours in the higher dimensional cases. It is believed that these two types of effective potentials are strongly linked to two types of eigenvectors on spheres, which are longitudinal and transverse ones. For the case of gravitational fields, the "scalartype perturbation potential" becomes the potential for the Zerilli equation, and the "vector type perturbation potential" becomes the potential for the Regge-Wheeler equation when D = 4 and s = 2. The "tensor-type perturbation potential" will be present when D ≥ 5 but absent in the four-dimensional case.

For the fermionic fields, the "�" still represents two isospectral supersymmetric partner potentials in the higher dimensional cases. We have one set of effective potentials for the Dirac field, which is strongly related to the eigenspinor on the sphere and reduces to Eq. (8) when D = 4. For the case of the Rarita-Schwinger field, the potentials related to the non-transverse-traceless eigenmodes are the leading equation both for the four-dimensional case and the higher dimensional one, which are strongly linked to the "non-transverse-traceless" eigenspinor-vector on spheres. Another effective potential for the Rarita-Schwinger fields is the one related to the transverse and traceless eigenmodes; however, this type of eigenspinor-vector was absent on the 2-sphere, which indicates that the potentials related to the transverse and traceless eigenmodes exist for the cases when D ≥ 5. We must note that, in this case, the effective potentials are exactly the same as the Dirac case.

Lastly, note that, most of the effective potentials in Table 2 are simple barrier like potentials. Nevertheless, some cases in the higher dimensions, or for the lowest energy state with j = 1/2, for the potentials related to the non-transverse-traceless eigenmodes exhibit special behaviours but not a simple barrier potential, which strongly suggests a link with the instabilities of the black hole [18] and warrants further study.

To summarise, we have in this section provided a brief review of the effective potentials, which play an important role in the perturbation theory of various spin fields in the general dimensional Schwarzschild black hole spacetimes.

#### 3. QNM frequencies by AIM

#### 3.1. AIM methods

The AIM is a well-established approach in solving the eigenvalue problem for the secondorder differential equations, for example, Schrödinger-like equations. As mentioned in the previous section, the radial equations of the perturbation theory with various spin fields in general dimensional Schwarzschild spacetimes were presented as Schrödinger-like equations. The QNMs, which are the signature modes in the black hole perturbation theory, can be obtained naturally by using the AIM. In this subsection, we shall present a brief review of the AIMs, and in the next subsection, we shall present an example calculation, showing the methods used to obtain the quasi normal frequencies. We shall start with the second-order differential equation for the function χðxÞ

$$
\chi^"=\lambda\_0(\mathbf{x})\chi'+\mathbf{s}\_0(\mathbf{x})\chi.\tag{10}
$$

where χ<sup>0</sup> ¼ dχ=dx. The symmetric structure of the right-hand side of Eq. (10) leads to the method, where we differentiate on both sides of the equation we find that

$$\begin{array}{l} \chi'' = \lambda\_0 \chi'' + (\lambda'\_0 + \mathbf{s}\_0)\chi' + \mathbf{s}'\_0 \chi\\ \quad = (\lambda'\_0 + \mathbf{s}\_0 + \lambda\_0^2)\chi' + (\mathbf{s}'\_0 + \mathbf{s}\_0 \lambda\_0)\chi\\ \equiv \lambda\_1 \chi' + \mathbf{s}\_1 \chi. \end{array} \tag{11}$$

Taking the second derivative of Eq. (10) we have

$$
\chi^{(4)} = \lambda\_2 \chi' + s\_2 \chi. \tag{12}
$$

where

$$
\lambda\_2 = \lambda\_1' + \mathbf{s}\_1 + \lambda\_0 \lambda\_1 \; \; \; \; s\_2 = \mathbf{s}\_1' + \mathbf{s}\_0 \lambda\_1 \; \; \; \tag{13}
$$

Differentiating iteratively to the ðn þ 1Þ th and the <sup>ð</sup><sup>n</sup> <sup>þ</sup> <sup>2</sup><sup>Þ</sup> th order, we have

$$
\chi^{(n+1)} = \lambda\_{n-1}\chi' + \mathbf{s}\_{n-1}\chi \; ; \; \chi^{(n+2)} = \lambda\_n \chi' + \mathbf{s}\_n \chi \tag{14}
$$

where

$$
\lambda\_n = \lambda\_{n-1}' + \mathbf{s}\_{n-1} + \lambda\_0 \lambda\_{n-1} \; ; \; \mathbf{s}\_n = \mathbf{s}\_{n-1}' + \mathbf{s}\_0 \lambda\_{n-1} \; . \tag{15}
$$

In the AIM, we suppose that for sufficiently large n, which represents the iterating number, the coefficients λ<sup>n</sup> and sn will have the relation

Semi-Analytic Techniques for Solving Quasi-Normal Modes http://dx.doi.org/10.5772/68114 205

$$\frac{\mathbf{s}\_{\text{n}}}{\lambda\_{\text{n}}} = \frac{\mathbf{s}\_{\text{n}+1}}{\lambda\_{\text{n}+1}} = \beta(\mathbf{x}),\tag{16}$$

where the general solution of Eq. (10) is

To summarise, we have in this section provided a brief review of the effective potentials, which play an important role in the perturbation theory of various spin fields in the general dimen-

The AIM is a well-established approach in solving the eigenvalue problem for the secondorder differential equations, for example, Schrödinger-like equations. As mentioned in the previous section, the radial equations of the perturbation theory with various spin fields in general dimensional Schwarzschild spacetimes were presented as Schrödinger-like equations. The QNMs, which are the signature modes in the black hole perturbation theory, can be obtained naturally by using the AIM. In this subsection, we shall present a brief review of the AIMs, and in the next subsection, we shall present an example calculation, showing the methods used to obtain the quasi normal frequencies. We shall start with the second-order

where χ<sup>0</sup> ¼ dχ=dx. The symmetric structure of the right-hand side of Eq. (10) leads to the

<sup>0</sup> <sup>þ</sup> <sup>s</sup><sup>0</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

<sup>1</sup> þ s<sup>1</sup> þ λ0λ<sup>1</sup> ; s<sup>2</sup> ¼ s

<sup>n</sup>�<sup>1</sup> þ sn�<sup>1</sup> þ λ0λ<sup>n</sup>�<sup>1</sup> ; sn ¼ s

In the AIM, we suppose that for sufficiently large n, which represents the iterating number, the

th and the <sup>ð</sup><sup>n</sup> <sup>þ</sup> <sup>2</sup><sup>Þ</sup>

<sup>0</sup> þ s0Þχ<sup>0</sup> þ s<sup>0</sup>

0Þχ<sup>0</sup> þ ðs<sup>0</sup>

0

<sup>χ</sup>ðnþ1<sup>Þ</sup> <sup>¼</sup> <sup>λ</sup><sup>n</sup>�<sup>1</sup>χ<sup>0</sup> <sup>þ</sup> sn�<sup>1</sup><sup>χ</sup> ; <sup>χ</sup>ðnþ2<sup>Þ</sup> <sup>¼</sup> <sup>λ</sup>nχ<sup>0</sup> <sup>þ</sup> snχ, <sup>ð</sup>14<sup>Þ</sup>

0

method, where we differentiate on both sides of the equation we find that

¼ ðλ<sup>0</sup>

λ<sup>2</sup> ¼ λ<sup>0</sup>

<sup>χ</sup>″ <sup>¼</sup> <sup>λ</sup>0χ″ þ ðλ<sup>0</sup>

� λ1χ<sup>0</sup> þ s1χ:

<sup>χ</sup>″ <sup>¼</sup> <sup>λ</sup>0ðxÞχ<sup>0</sup> <sup>þ</sup> <sup>s</sup>0ðxÞχ, <sup>ð</sup>10<sup>Þ</sup>

<sup>0</sup>χ,

<sup>0</sup> þ s0λ0Þχ,

<sup>χ</sup>ð4<sup>Þ</sup> <sup>¼</sup> <sup>λ</sup>2χ<sup>0</sup> <sup>þ</sup> <sup>s</sup>2χ, <sup>ð</sup>12<sup>Þ</sup>

th order, we have

<sup>1</sup> þ s0λ1: ð13Þ

<sup>n</sup>�<sup>1</sup> þ s0λ<sup>n</sup>�<sup>1</sup>: ð15Þ

ð11Þ

sional Schwarzschild black hole spacetimes.

3. QNM frequencies by AIM

differential equation for the function χðxÞ

Taking the second derivative of Eq. (10) we have

Differentiating iteratively to the ðn þ 1Þ

λ<sup>n</sup> ¼ λ<sup>0</sup>

coefficients λ<sup>n</sup> and sn will have the relation

3.1. AIM methods

204 Trends in Modern Cosmology

where

where

$$\chi(\mathbf{x}) = \exp\left(-\int^{\mathbf{x}} \beta(\mathbf{x}') d\mathbf{x}'\right) [\mathbf{C}\_2 + \mathbf{C}\_1] \int^{\mathbf{x}} \exp(\int^{\mathbf{x}} (\lambda\_0(\mathbf{x}') + 2\beta(\mathbf{x}')) d\mathbf{x}') d\mathbf{x}'].\tag{17}$$

C<sup>1</sup> and C<sup>2</sup> are constants determined by the normalisation, and the QNMs (or the energy eigenstates) can be obtained by the termination condition

$$s\_n \lambda\_{n+1} - s\_{n+1} \lambda\_n = 0.\tag{18}$$

This is the basic idea for the AIM, where to appreciate the effectiveness of this methods, we refer the reader to Ciftci et al. [4, 5], which presented some studies for the constant coefficient, harmonic oscillator, and the energy eigenvalue problem for several well-known potentials. For the study of perturbation theory in curved spacetimes by the AIM, Cho et al. [1] present a review of the QNMs for the bosonic fields in the four-dimensional maximally symmetric and the Kerr black hole spacetimes. In the next subsection, as an example, we shall present how to obtain the QNMs for Dirac fields in the higher dimensional Schwarzschild black hole spacetimes by the AIM and compare these with other numerical semi-analytic results.

#### 3.2. Example: How to obtain the QNMs for Dirac fields in general dimensional Schwarzschild black hole spacetimes by the AIM

The QNMs for Dirac particles in the higher dimensional Schwarzschild black hole spacetimes had been done in the earlier work by some of the authors [13] using the third-order WKB approximation but not the AIM. With recent progress in spin-3/2 fields [11], we find that these results greatly overlap with some of the spin-3/2 particles, which are represented by the relations in the radial equations of the transverse and traceless eigenmodes. In this section, as an example, we are going to show how to reproduce the results by the AIM.

In Table 2, the effective potential of the radial equation, Eq. (4), for the Dirac particle is as follows:

$$W\_{s=1/2} = f \frac{d\mathcal{W}\_{s=1/2}}{dr} + \mathcal{W}\_{s=1/2}^2 \quad , \quad W\_{s=1/2} = f \left(\frac{j + \frac{D-3}{2}}{r}\right) \; , \; j = \frac{1}{2}, \; \frac{3}{2}, \dots \tag{19}$$

As we are going to reproduce the results in Ref. [13], a similar choose of f(r) will be

$$f(r) = 1 - \left(\frac{r\_H}{r}\right)^{D-3}, \quad r\_H^{D-3} = \frac{8\pi M \Gamma((D-1)/2)}{\pi^{(D-1)/2} (D-2)} \quad , \ M \equiv 1,\tag{20}$$

where rH represents the location of the event horizon and M represents the mass of the black hole. By making a coordinate transformation

$$
\xi^2(r) = 1 - \frac{r\_H}{r},
\tag{21}
$$

the radial equation becomes

$$\left[f\frac{d\xi}{dr}\frac{d}{d\xi}\left(f\frac{d\xi}{dr}\frac{d}{d\xi}\right)+\omega^2-V\_{s=1/2}\right]\Psi\_{s=1/2}=0.\tag{22}$$

Simplifying Eq. (22), we have

$$
\left[\frac{d^2}{d\xi^2} + \left(\frac{f'}{f} + \frac{\xi'}{\xi'}\right)\frac{d}{d\xi} + \frac{\omega^2 - V\_{s=1/2}}{f^2 \xi'^2}\right] \Psi\_{s=1/2} = 0,\tag{23}
$$

where

$$f' = \frac{d}{d\xi} f(\xi) \; , \; \xi' = \frac{d}{dr} \xi(r)|\_{r = \frac{r\_H}{1 - \xi^2}} \; , \; \xi^\* = \frac{d}{d\xi} \xi' . \tag{24}$$

Next, by setting the boundary behaviour of Ψ<sup>s</sup>¼1=<sup>2</sup> ¼ αðξÞχðξÞ and together with Eq. (23), we have

$$\frac{d^2}{d\xi}\chi = -\left(\frac{f'}{f} + \frac{\xi'}{\xi} + \frac{2a'}{\alpha}\right)\frac{d}{d\xi}\chi - \left[\frac{a^2 - V\_{s=1/2}}{f'\xi'^2} + \frac{a^\*}{\alpha} + \left(\frac{f'}{f} + \frac{\xi'}{\xi}\right)\frac{a'}{\alpha}\right]\chi. \tag{25}$$

This is the second-order differential equation, which is the same as Eq. (10) with

$$\lambda\_0 = -\left(\frac{f'}{f} + \frac{\xi}{\xi} + \frac{2a'}{a}\right), \quad s\_0 = -\left[\frac{a^2 - V\_{s=1/2}}{f'\xi^{r^2}} + \frac{a^\*}{a} + \left(\frac{f'}{f} + \frac{\xi}{\xi}\right)\frac{a'}{a}\right]. \tag{26}$$

Note that the last parameter we have to define is the asymptotic behaviour function α(ξ), and we approach this by starting with the asymptotic behaviour of Ψ<sup>s</sup>¼1=2, which can be represented as an outgoing plane wave

$$\begin{array}{ll}\Psi\_{s=1/2} \sim e^{i\alpha r\_\*} & \text{for} \quad r \to \text{eq}\_{\prime} \\ \Psi\_{s=1/2} \sim e^{-i\omega r\_\*} & \text{for} \quad r \to -\text{eq}\_{\prime} \end{array} \tag{27}$$

with r\* being the tortoise coordinate, which has the relation with r

$$\frac{d}{dr\_\*} = f(r)\frac{d}{dr}.\tag{28}$$

Solving Eq. (28) in the four-dimensional case, we have

$$r\_\* = r + r\_H \ln\left(\frac{r\_H}{r} - 1\right),\tag{29}$$

together with Eqs. (21) and (27), we have

$$\begin{array}{ll}\Psi\_{s=1/2} \sim \overline{\mathscr{C}^{1-\zeta^{2}}} (1-\xi^{2})^{-i\omega r\_{H}} \xi^{2i\omega r\_{H}} & \text{for} \quad \xi \to 1, \\\Psi\_{s=1/2} \sim \overline{\mathscr{C}^{1-\zeta^{2}}} (1-\xi^{2})^{i\omega r\_{H}} \xi^{-2i\omega r\_{H}} & \text{for} \quad \xi \to 0. \end{array} \tag{30}$$

Collecting the leading terms, we can define α(ξ) in the four-dimensional case as follows:

$$a(\xi) = \overline{e^{i\nu\_H}} (1 - \xi^2)^{-i\nu r\_H} \xi^{-2i\nu r\_H}.\tag{31}$$

If we now consider the higher dimensional cases, Eq. (29) will become more complicated with

$$r\_\* = r + \sum\_{a=1}^{D-3} \frac{r\_{H\mathcal{E}}}{D-3} \ln\left(\frac{r}{r\_H \varepsilon} - 1\right),\tag{32}$$

where ε ¼ e ia2π <sup>D</sup>�3. Eq. (27) in the higher dimensions can be presented in the general form

$$\begin{split} \Psi\_{s=1/2} &\sim e^{\frac{i\nu\mu}{1-\zeta^{2}}} \prod\_{a=1}^{D-3} \left( \varepsilon (1-\xi^{2}) \right)^{-\frac{i\nu\mu\mu}{D-3}} \left( 1 - \varepsilon (1-\xi^{2}) \right)^{\frac{i\nu\mu\mu}{D-3}} \quad \text{for} \quad \xi \to 1, \\\Psi\_{s=1/2} &\sim e^{-\frac{i\nu\mu}{1-\zeta^{2}}} \prod\_{a=1}^{D-3} \left( \varepsilon (1-\xi^{2}) \right)^{\frac{i\nu\mu}{D-3}} \left( 1 - \varepsilon (1-\xi^{2}) \right)^{-\frac{i\nu\mu\mu}{D-3}} \quad \text{for} \quad \xi \to 0. \end{split} \tag{33}$$

Next, we shall consider how to define the asymptotic behaviour function α(ξ) in the higher dimensional cases, where in our experience, finding the dominant terms in Eq. (33) is helpful to the AIM calculation. When D = 5, Eq. (33) becomes

$$\begin{array}{ll} \Psi\_{s=1/2} \sim e^{\frac{i\nu r\_H}{1-\zeta^2}} (\xi^2 - 2)^{-\frac{i\nu r\_H}{2}} (\xi^2)^{\frac{i\nu r\_H}{2}} \quad \text{for} \quad \xi \to 1, \\\Psi\_{s=1/2} \sim e^{-\frac{i\nu r\_H}{1-\zeta^2}} (\xi^2 - 2)^{\frac{i\nu r\_H}{2}} (\xi^2)^{-\frac{i\nu r\_H}{2}} \quad \text{for} \quad \xi \to 0. \end{array} \tag{34}$$

By considering the dominant term for the boundary behaviour, a suitable choice of the asymptotic behaviour function is as follows:

$$\alpha\_{(D=5)}(\xi) = \mathcal{e}^{\frac{i\nu r\_H}{1-\xi^2}} (\xi^2)^{-\frac{i\nu r\_H}{2}}.\tag{35}$$

For D = 6, Eq. (33) becomes

ξ2

� �

d dξ þ

" #

dξ dr d dξ � �

0 <sup>f</sup> <sup>þ</sup> <sup>ξ</sup><sup>00</sup> ξ0

!

<sup>d</sup><sup>ξ</sup> <sup>f</sup>ðξ<sup>Þ</sup> , <sup>ξ</sup><sup>0</sup> <sup>¼</sup> <sup>d</sup>

This is the second-order differential equation, which is the same as Eq. (10) with

the radial equation becomes

206 Trends in Modern Cosmology

Simplifying Eq. (22), we have

d2

<sup>d</sup><sup>ξ</sup> <sup>χ</sup> ¼ � <sup>f</sup>

<sup>λ</sup><sup>0</sup> ¼ � <sup>f</sup>

represented as an outgoing plane wave

together with Eqs. (21) and (27), we have

where

have

f dξ dr d <sup>d</sup><sup>ξ</sup> <sup>f</sup>

d2 <sup>d</sup>ξ<sup>2</sup> <sup>þ</sup> <sup>f</sup>

f <sup>0</sup> <sup>¼</sup> <sup>d</sup>

0 f þ ξ″ ξ þ 2α<sup>0</sup> α

0 f þ ξ″ ξ þ 2α<sup>0</sup> α

� �

with r\* being the tortoise coordinate, which has the relation with r

Solving Eq. (28) in the four-dimensional case, we have

� � d

<sup>ð</sup>rÞ ¼ <sup>1</sup> � rH

<sup>þ</sup> <sup>ω</sup><sup>2</sup> � Vs¼1=<sup>2</sup>

<sup>ω</sup><sup>2</sup> � Vs¼1=<sup>2</sup> f 2 ξ02

dr <sup>ξ</sup>ðrÞj<sup>r</sup><sup>¼</sup> rH

Next, by setting the boundary behaviour of Ψ<sup>s</sup>¼1=<sup>2</sup> ¼ αðξÞχðξÞ and together with Eq. (23), we

<sup>d</sup><sup>ξ</sup> <sup>χ</sup> � <sup>ω</sup><sup>2</sup> � Vs¼1=<sup>2</sup> f 0 <sup>ξ</sup>0<sup>2</sup> <sup>þ</sup>

, s<sup>0</sup> ¼ � <sup>ω</sup><sup>2</sup> � Vs¼1=<sup>2</sup> f 0 <sup>ξ</sup>0<sup>2</sup> <sup>þ</sup>

Note that the last parameter we have to define is the asymptotic behaviour function α(ξ), and we approach this by starting with the asymptotic behaviour of Ψ<sup>s</sup>¼1=2, which can be

<sup>Ψ</sup><sup>s</sup>¼1=<sup>2</sup> ~ e<sup>i</sup>ωr� for <sup>r</sup> ! <sup>∞</sup>,

<sup>¼</sup> <sup>f</sup>ðr<sup>Þ</sup> <sup>d</sup>

<sup>r</sup> � <sup>1</sup> � �

d dr�

<sup>r</sup>� <sup>¼</sup> <sup>r</sup> <sup>þ</sup> rHln rH

<sup>1</sup>�ξ<sup>2</sup>

, <sup>ξ</sup>″ <sup>¼</sup> <sup>d</sup>

α″ α þ f 0 <sup>f</sup> <sup>þ</sup> <sup>ξ</sup>″ ξ � � α<sup>0</sup>

" #

α″ α þ f 0 f þ ξ″ ξ � � α<sup>0</sup>

" #

<sup>Ψ</sup><sup>s</sup>¼1=<sup>2</sup> ~ e�iωr� for <sup>r</sup> ! �∞, <sup>ð</sup>27<sup>Þ</sup>

dr: <sup>ð</sup>28<sup>Þ</sup>

, ð29Þ

<sup>d</sup><sup>ξ</sup> <sup>ξ</sup><sup>0</sup>

<sup>r</sup> , <sup>ð</sup>21<sup>Þ</sup>

Ψ<sup>s</sup>¼1=<sup>2</sup> ¼ 0: ð22Þ

Ψ<sup>s</sup>¼1=<sup>2</sup> ¼ 0; ð23Þ

α

α

: ð24Þ

χ: ð25Þ

: ð26Þ

$$\begin{split} & \Psi\_{s=1/2} \sim e^{\frac{i\nu\eta}{1-\xi^{2}}} \Big( e^{\frac{2\pi}{3}} (1-\xi^{2}) \Big)^{-\frac{i\nu\eta}{3}} \Big( 1 - e^{\frac{2\pi}{3}} (1-\xi^{2}) \Big)^{i\nu\frac{i\nu\eta}{3}} \Big( e^{\frac{i\nu\eta}{3}} (1-\xi^{2}) \Big)^{-\frac{i\nu\eta\beta}{3}} \\ & \Big( 1 - e^{\frac{i\nu\eta}{3}} (1-\xi^{2}) \Big)^{\frac{i\nu\eta}{3}} (1-\xi^{2})^{-\frac{i\nu\eta}{3}} (1-\xi^{2})^{-\frac{i\nu\eta}{3}} \text{ for } \xi \to 1, \\ & \Psi\_{s=1/2} \sim e^{-\frac{i\nu\eta}{1-\xi^{2}}} \Big( e^{\frac{2\pi}{3}} (1-\xi^{2}) \Big)^{-\frac{i\nu\eta\beta}{3}} \Big( 1 - e^{\frac{2\pi}{3}} (1-\xi^{2}) \Big)^{-\frac{i\nu\eta}{3}} \Big( e^{\frac{i\nu\eta}{3}} (1-\xi^{2}) \Big)^{\frac{i\nu\eta\beta}{3}} \\ & \Big( 1 - e^{\frac{i\nu\eta}{3}} (1-\xi^{2}) \Big)^{\frac{i\nu\eta}{3}} (1-\xi^{2})^{-\frac{i\nu\eta}{3}} (1-\xi^{2})^{-\frac{i\nu\eta}{3}} \text{ for } \xi \to 0. \end{split} \tag{36}$$

The dominant term for the boundary behaviour can be chosen as follows:

$$\alpha\_{(D=6)}(\xi) = e^{\frac{i\nu\_H}{1-\xi^2}} (\xi^2)^{-\frac{i\nu\_H}{3}} (1-\xi^2)^{-\frac{i\nu\_H}{3}}.\tag{37}$$

For a similar discussion on the higher dimensional cases, we find that α(ξ) can be defined separately for the odd dimensional and even dimensional cases, where

$$\begin{array}{ll} a(\xi) = \mathcal{e}^{\frac{nr\_H}{1-\zeta^2}} (\xi^2)^{-\frac{inr\_H}{D-3}}, & D \in \text{odd};\\ a(\xi) = \mathcal{e}^{\frac{inr\_H}{1-\zeta^2}} (\xi^2)^{-\frac{inr\_H}{D-3}} (1-\xi^2)^{-\frac{inr\_H}{D-3}}, & D \in \text{even}.\end{array} \tag{38}$$

Substituting Eqs. (19), (20), (21) and (38) into Eq. (26), we find λ<sup>1</sup> and s<sup>1</sup> by the relation in Eq. (15). Next, by defining a suitable initial value of ω0, which represents the parameter ω in s0, with the termination condition Eq. (18), we can solve the parameter ω1, which represents the parameter ω in s1. This loop iterates, as with ω<sup>1</sup> we can solve for ω<sup>2</sup> in the next iteration and so on. For a sufficiently large number of iterations, the ωn-1 and ω<sup>n</sup> will become stable, and it will be the quasi normal frequency we are seeking.

The QNM results are obtained with the iteration number = 200 in Table 3, for the D = 5,6 dimensional Schwarzschild spacetimes, and in Table 4, for D = 7,8. The results using the thirdorder WKB methods are the same as presented in our previous work [13], where for completeness we also list the sixth-order WKB results.



The dominant term for the boundary behaviour can be chosen as follows:

iωrH <sup>1</sup>�ξ<sup>2</sup> <sup>ð</sup>ξ<sup>2</sup> Þ �iωrH

For a similar discussion on the higher dimensional cases, we find that α(ξ) can be defined

�iωrH <sup>D</sup>�<sup>3</sup> <sup>ð</sup><sup>1</sup> � <sup>ξ</sup><sup>2</sup>

Substituting Eqs. (19), (20), (21) and (38) into Eq. (26), we find λ<sup>1</sup> and s<sup>1</sup> by the relation in Eq. (15). Next, by defining a suitable initial value of ω0, which represents the parameter ω in s0, with the termination condition Eq. (18), we can solve the parameter ω1, which represents the parameter ω in s1. This loop iterates, as with ω<sup>1</sup> we can solve for ω<sup>2</sup> in the next iteration and so on. For a sufficiently large number of iterations, the ωn-1 and ω<sup>n</sup> will become stable, and it will

The QNM results are obtained with the iteration number = 200 in Table 3, for the D = 5,6 dimensional Schwarzschild spacetimes, and in Table 4, for D = 7,8. The results using the thirdorder WKB methods are the same as presented in our previous work [13], where for complete-

<sup>3</sup> <sup>ð</sup><sup>1</sup> � <sup>ξ</sup><sup>2</sup>

�iωrH <sup>D</sup>�<sup>3</sup> , D <sup>∈</sup> odd;

Þ

Þ �iωrH

�iωrH <sup>D</sup>�<sup>3</sup> , D <sup>∈</sup>even:

<sup>3</sup> : ð37Þ

ð38Þ

αðD¼<sup>6</sup>ÞðξÞ ¼ e

separately for the odd dimensional and even dimensional cases, where

iωrH <sup>1</sup>�ξ<sup>2</sup> <sup>ð</sup>ξ<sup>2</sup> Þ

αðξÞ ¼ e

iωrH <sup>1</sup>�ξ<sup>2</sup> <sup>ð</sup>ξ<sup>2</sup> Þ

αðξÞ ¼ e

be the quasi normal frequency we are seeking.

208 Trends in Modern Cosmology

ness we also list the sixth-order WKB results.

5 Dimensions

l n Third-order WKB Sixth-order WKB AIM

 0 0.7247–0.3960 i 0.7823–0.3635 i 0.7252–0.3960 i 0 1.3158–0.3839 i 1.3301–0.3852 i 1.3163–0.3839 i 1 1.1490–1.2192 i 1.1800–1.2021 i 1.1495–1.2192 i 0 1.8754–0.3838 i 1.8801–0.3844 i 1.8802–0.3840 i 1 1.7541–1.1818 i 1.7672–1.1791 i 1.7674–1.1779 i 2 1.5588–2.0318 i 1.5620–2.0517 i 1.5593–2.0318 i 0 2.4251–0.3839 i 2.4271–0.3840 i 2.4256–0.3839 i 1 2.3322–1.1686 i 2.3382–1.1674 i 2.3327–1.1686 i 2 2.1704–1.9891 i 2.1691–1.9980 i 2.1709–1.9891 i 3 1.9611–2.8411 i 1.9385–2.9063 i 1.9616–2.8411 i 0 2.9716–0.3839 i 2.9726–0.3839 i 2.9721–0.3839 i 1 2.8963–1.1624 i 2.8994–1.1618 i 2.8968–1.1624 i 2 2.7596–1.9668 i 2.7574–1.9711 i 2.7601–1.9668 i 3 2.5773–2.7984 i 2.5569–2.8330 i 2.5778–2.7984 i 4 2.3583–3.6512 i 2.3126–3.7665 i 2.3588–3.6512 i 0 3.5166–0.3838 i 3.5171–0.3838 i 3.5171–0.3838 i 1 3.4533–1.1591 i 3.4550–1.1588 i 3.4538–1.1591 i 2 3.3353–1.9536 i 3.3333–1.956 i 3.3358–1.9536 i

Table 3. Low-lying (n ≤ l, with l ¼ j � 1=2) spin-1/2 field QNM frequencies using the WKB methods and the AIM with D = 5,6.



Table 4. Low-lying (n ≤ l, with l ¼ j � 1=2) spin-1/2 field QNM frequencies using the WKB methods and the AIM with D = 7,8.

#### 4. A remark on the spin-3/2 field QNMs

In this section, we are going to present a discussion of our recent work [11] on the spin-3/2 fields in general dimensional Schwarzschild spacetimes. Note that, this section will not go into as great a detail as the previous one but shall just list some improvements made in light of recent considerations. Starting with the radial equation related to the "non-TT eigenmodes", the radial equation can still be represented as the Schrödinger like one, Eq. (4), where

$$\begin{array}{rcl} \boldsymbol{V}\_{\text{NTT},s=3/2} & = & \pm f \frac{d \boldsymbol{W}\_{\text{NTT},s=3/2}}{dr} + \boldsymbol{W}\_{\text{NTT},s=3/2}^{2} \\\\ \boldsymbol{W}\_{\text{NTT},s=3/2} & = & \frac{\sqrt{f}(\boldsymbol{j}+\frac{\boldsymbol{D}-3}{2})}{r} \left[ \frac{(\frac{2}{D-2})^{2}(\boldsymbol{j}+\frac{D-3}{2})^{2}-1-\frac{D-4}{D-2}(\frac{2M}{r})^{D-3}}{(\frac{2}{D-2})^{2}(\boldsymbol{j}+\frac{D-3}{2})^{2}-f} \right], \end{array} \tag{39}$$

and

7 Dimensions

210 Trends in Modern Cosmology

8 Dimensions

D = 7,8.

 2 3.3809–4.3622 i 3.4158–4.4006 i 3.3814–4.3622 i 3 2.4624–6.363 i 2.3163–6.6549 i 2.4629–6.363 i 0 5.2968–0.8218 i 5.3035–0.8220 i 5.3033–0.8220 i 1 4.9878–2.5009 i 5.0176–2.4966 i 4.9883–2.5009 i 2 4.4049–4.2771 i 4.4224–4.2810 i 4.4054–4.2772 i 3 3.6066–6.1782 i 3.4954–6.3058 i 3.6071–6.1782 i 4 2.6393–8.1950 i 2.2624–8.7432 i 2.6398–8.1950 i 0 6.1273–0.8221 i 6.1310–0.8219 i 6.1309–0.8221 i 1 5.8671–2.4910 i 5.8847–2.4870 i 5.8676–2.4910 i 2 5.3673–4.2295 i 5.3756–4.2263 i 5.3678–4.2295 i 3 4.6654–6.0660 i 4.5819–6.1252 i 4.6659–6.0660 i 4 3.8002–8.0044 i 3.5047–8.3081 i 3.8007–8.0044 i 5 2.7964–10.035 i 2.1874–10.910 i 2.7969–10.0348i

 0 2.2437–0.9238 i 2.6486–0.8930 i 2.2442–0.9238 i 0 3.2663–0.9359 i 3.3105–1.0469 i 3.2668–0.9359 i 1 2.2397–3.1796 i 2.6722–3.0659 i 2.2402–3.1796 i 0 4.2077–0.9556 i 4.2279–0.9928 i 4.2082–0.9556 i 1 3.4491–3.0091 i 3.6051–3.1428 i 3.4496–3.0091 i 2 2.1453–5.4463 i 2.2891–5.5740 i 2.1458–5.4463 i 0 5.1097–0.9641 i 5.1285–0.9704 i 5.1102–0.9642 i 1 4.5378–2.9607 i 4.6284–2.9895 i 4.5383–2.9607 i 2 3.4576–5.1805 i 3.5481–5.2651 i 3.4581–5.1805 i 3 2.0383–7.6877 i 1.7944–8.0198 i 2.0388–7.6877 i 0 5.9947–0.9669 i 6.0062–0.9669 i 5.9952–0.9669 i 1 5.5365–2.9415 i 5.5943–2.9365 i 5.5370–2.9415 i 2 4.6421–5.0549 i 4.6967–5.0480 i 4.6426–5.0549 i 3 3.4017–7.3877 i 3.2284–7.5066 i 3.4022–7.3877 i 4 1.9199–9.9347 i 1.2118–10.564 i 1.9204–9.9347 i 0 6.8720–0.9677 i 6.8785–0.9669 i 6.8772–0.9659 i 1 6.4875–2.9309 i 6.5231–2.9224 i 6.4880–2.9309 i 2 5.7272–4.9883 i 5.7608–4.9630 i 5.7277–4.9883 i 3 4.6390–7.2092 i 4.5109–7.2241 i 4.6395–7.2092 i 4 3.3023–9.6186 i 2.7516–9.9294 i 3.3028–9.6186 i 5 1.7759–12.195 i 0.5680–13.282 i 1.7764–12.1949 i

Table 4. Low-lying (n ≤ l, with l ¼ j � 1=2) spin-1/2 field QNM frequencies using the WKB methods and the AIM with

$$f(r) = 1 - \left(\frac{2M}{r}\right)^{D-3}.\tag{40}$$

Because the general form of the radial equation is the same as the Dirac case, Eqs. (22)–(38) are still sufficient in this case, but the effective potential will be Eq. (39), not Eq. (19). What we need to consider here is that the asymptotic behaviour function is simpler than what we had used previously and successfully generates better results for the QNMs.

In Table 5, for some lower modes in the seven-dimensional spacetime, the new AIM results (with the current choice of boundary behaviour function, Eq. (38)) are better than the previous


Table 5. Selected QNM frequencies of low-lying (n ≤ l, with l ¼ j � 3=2) for spin-3/2 field related to the non-TT eigenmodes by using the WKB methods and the AIM.

results for the first few modes. This is why we believe Eq. (38) is a suitable choice of the boundary behaviour function for the AIM methods in the higher dimensional spacetimes.

#### 5. Summary

In this chapter, we presented a brief review of recent progress on perturbation theory with various fields in curved spacetimes, especially for a systematic comparison to the fermionic and bosonic fields. Generally, the first step in this topic is the obtaining of the radial equations as discussed in Section 2. There are then various methods, but no unique approach for considering a specific field on a specific spacetime, where we strongly suggest the reader follow the references provided to develop a step-by-step approach. On the other hand, the process for obtaining the radial equations always relates to the separability of a spacetime, this being the main reason that we still have difficulties for perturbation theory in the higher dimensional Kerr spacetimes. As such we mentioned in the introduction section that for the gravitational wave experiments, the QNMs in the higher dimensional Kerr spacetimes are definitely an interesting next stage of study for this area, and it shall be interesting seeing further progress in this direction.

In Section 3, we presented the AIM, giving an in detail example to study the perturbation theory in the higher dimensional spacetimes and also presented a remark for our recent work on the spin-3/2 fields. Since there are several numerical methods for obtaining the QNMs, we believe that the AIM is a straightforward way to study the QNMs due to its simple mathematical structure.

#### Author details

Chun-Hung Chen<sup>2</sup> , Hing-Tong Cho3 and Alan S. Cornell<sup>1</sup> \*

\*Address all correspondence to: alan.cornell@wits.ac.za

1 National Institute for Theoretical Physics, School of Physics, University of the Witwatersrand, Wits, South Africa

2 Institute of Physics, Academia Sinica, Taipei, Taiwan

3 Department of Physics, Tamkang University, Tamsui, Taipei, Taiwan

#### References

[1] H. T. Cho, A. S. Cornell, J. Doukas, T. R. Huang and W. Naylor, Adv. Math. Phys. 2012, 281705 (2012). doi:10.1155/2012/281705 [arXiv:1111.5024 [gr-qc]].


results for the first few modes. This is why we believe Eq. (38) is a suitable choice of the boundary behaviour function for the AIM methods in the higher dimensional spacetimes.

In this chapter, we presented a brief review of recent progress on perturbation theory with various fields in curved spacetimes, especially for a systematic comparison to the fermionic and bosonic fields. Generally, the first step in this topic is the obtaining of the radial equations as discussed in Section 2. There are then various methods, but no unique approach for considering a specific field on a specific spacetime, where we strongly suggest the reader follow the references provided to develop a step-by-step approach. On the other hand, the process for obtaining the radial equations always relates to the separability of a spacetime, this being the main reason that we still have difficulties for perturbation theory in the higher dimensional Kerr spacetimes. As such we mentioned in the introduction section that for the gravitational wave experiments, the QNMs in the higher dimensional Kerr spacetimes are definitely an interesting next stage of study for this area, and it shall be interesting seeing further progress

In Section 3, we presented the AIM, giving an in detail example to study the perturbation theory in the higher dimensional spacetimes and also presented a remark for our recent work on the spin-3/2 fields. Since there are several numerical methods for obtaining the QNMs, we believe that the AIM is a straightforward way to study the QNMs due to its simple mathemat-

\*

, Hing-Tong Cho3 and Alan S. Cornell<sup>1</sup>

1 National Institute for Theoretical Physics, School of Physics, University of the

[1] H. T. Cho, A. S. Cornell, J. Doukas, T. R. Huang and W. Naylor, Adv. Math. Phys. 2012,

3 Department of Physics, Tamkang University, Tamsui, Taipei, Taiwan

281705 (2012). doi:10.1155/2012/281705 [arXiv:1111.5024 [gr-qc]].

\*Address all correspondence to: alan.cornell@wits.ac.za

2 Institute of Physics, Academia Sinica, Taipei, Taiwan

5. Summary

212 Trends in Modern Cosmology

in this direction.

ical structure.

Author details

Chun-Hung Chen<sup>2</sup>

References

Witwatersrand, Wits, South Africa


## *Edited by Abraao Jesse Capistrano de Souza*

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Trends in Modern Cosmology

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*Edited by Abraao Jesse Capistrano de Souza*

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