**Cosmological Constant and Dark Energy: Historical Insights**

## Emilio Elizalde

areas of research not always fully settled, the idea of having different voices presenting these topics and defending their own views is, in my opinion, an excellent scientific and communicative exercise. For this reason, I would like to transmit to the authors of this book my most sincere gratitude for their contributions and my deepest admiration for their work.

VIII Preface

**Gonzalo J. Olmo** Research Associate

Valencia (Spain)

Dept. Theoretical Physics & IFIC University of Valencia - CSIC

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51697

## **1. Introduction**

In this Chapter we are going to discuss about the large scale structure of the Universe. In particular, about the laws of Physics which allow us to describe and try to understand the present Universe behavior as a whole, as a global structure. These physical laws, when they are brought to their most extreme consequences---to their limits in their respective domains of applicability---are able to give us a plausible idea of how the origin of our Universe could happen to occur and also of how, expectedly, its future evolution and its end will finally take place.

The vision we have now of the so-called global or large-scale Universe (what astrophysicists term the extragalactic Universe) began to get shape during the second and third decades of the past Century. We should start by saying that, at that time, everybody thought that the Universe was reduced to just our own galaxy, the Milky Way. It is indeed true that a very large number of nebulae had been observed by then, but there was no clear proof that these objects were not within the domains of our own galaxy. Actually, the first nebulae had been already identified many centuries ago by Ptolemy who, in his celebrated work Almagest [1], reported five in AD 150. Later, Persian, Arabic and Chinese astronomers, among others, dis‐ covered some more nebulae, along several centuries of the History of Mankind. Concerning scientific publications, Edmond Halley [2] was the first to report six nebulae in the year 1715, Charles Messier [3] catalogued 103 of them in 1781 (now called Messier objects), while confessing his interest was "detecting comets, and nebulae could just be mistaken for them, thus wasting time." William Herschel and his sister Caroline published three full catalogues of nebulae, one after the other [4], between 1786 and 1802, where a total of 2510 nebulae where identified. However, in all these cases the dominant belief was that these objects were merely unresolved clusters of stars, in need of more powerful telescopes. On 26 April 1920,

© 2012 Elizalde; licensee InTech. This is an open access article 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. © 2012 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.

in the Baird auditorium of the Smithsonian Museum of Natural History a debate took place (called now by astronomers, in retrospective, the Great Debate), on the basis of two works by Harlow Shapley and Heber Curtis, later published in the Bulletin of the National Re‐ search Council. During the day, the two scientists presented independent technical results on "The Scale of the Universe" and then took part in a joint discussion in the evening. Shap‐ ley defended the Milky Way to be the entirety of the Universe and believed that objects as Andromeda and the Spiral Nebulae were just part of it. Curtis, on the contrary, affirmed that Andromeda and other nebulae were separate galaxies, or "island universes" (a term in‐ vented by the philosopher Immanuel Kant, who also believed that the spiral nebulae were extragalactic). Curtis showed that there were more novae in Andromeda than in the Milky Way and argued that it would be very unlikely within the same galaxy to have so many more novae in one small section of the galaxy than in the other sections. This led him to sup‐ port Andromeda as a separate galaxy with its own signature age and rate of novae occur‐ rences. He also mentioned Doppler redshifts found in many nebulae. Following this debate, by 1922 it had become clear that many nebulae were most probably other galaxies, far away from our own.

**Figure 2.** Edmond Halley, 1656 – 1742. Charles Messier, 1730 – 1817.

**Figure 3.** Sir Frederick William Herschel, 1738 – 1822. Caroline Lucretia Herschel, 1750 – 1848.

But it was Edwin Hubble [5] who, between 1922 and 1924, presented a definite proof that one of this nebulae, Andromeda, was at a distance of some 800.000 light years from us and, therefore, far beyond the limits of our own galaxy, the Milky Way. In this way, he definitely changed the until then predominant vision of our Universe, and opened to human knowl‐

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3

**2. An expanding Universe**

**Figure 1.** Claudius Ptolemaeus, c. AD 90 – c. AD 168.

**Figure 2.** Edmond Halley, 1656 – 1742. Charles Messier, 1730 – 1817.

in the Baird auditorium of the Smithsonian Museum of Natural History a debate took place (called now by astronomers, in retrospective, the Great Debate), on the basis of two works by Harlow Shapley and Heber Curtis, later published in the Bulletin of the National Re‐ search Council. During the day, the two scientists presented independent technical results on "The Scale of the Universe" and then took part in a joint discussion in the evening. Shap‐ ley defended the Milky Way to be the entirety of the Universe and believed that objects as Andromeda and the Spiral Nebulae were just part of it. Curtis, on the contrary, affirmed that Andromeda and other nebulae were separate galaxies, or "island universes" (a term in‐ vented by the philosopher Immanuel Kant, who also believed that the spiral nebulae were extragalactic). Curtis showed that there were more novae in Andromeda than in the Milky Way and argued that it would be very unlikely within the same galaxy to have so many more novae in one small section of the galaxy than in the other sections. This led him to sup‐ port Andromeda as a separate galaxy with its own signature age and rate of novae occur‐ rences. He also mentioned Doppler redshifts found in many nebulae. Following this debate, by 1922 it had become clear that many nebulae were most probably other galaxies, far away

from our own.

2 Open Questions in Cosmology

**Figure 1.** Claudius Ptolemaeus, c. AD 90 – c. AD 168.

**Figure 3.** Sir Frederick William Herschel, 1738 – 1822. Caroline Lucretia Herschel, 1750 – 1848.

## **2. An expanding Universe**

But it was Edwin Hubble [5] who, between 1922 and 1924, presented a definite proof that one of this nebulae, Andromeda, was at a distance of some 800.000 light years from us and, therefore, far beyond the limits of our own galaxy, the Milky Way. In this way, he definitely changed the until then predominant vision of our Universe, and opened to human knowl‐ edge the much more complex extragalactic Universe, whose theoretical study is one of the main goals of this Chapter [6].

The best known, by far, of the equations Einstein discovered (and probably the most famous

has a very deep physical meaning, since it establishes the equivalence between mass and en‐ ergy, as two forms of one and the same physical quantity, thus susceptible to be trans‐ formed one into the other, and vice versa. The conversion factor is enormous (the velocity of light squared), meaning that a very small quantity of mass will give rise to an enormous amount of energy –as nuclear power plants prove every day (and very destructive bombs did in the past, to the shame of the Humankind). In any case, here we are not referring to this Einstein's equation (which will not be discussed any further), but to the so-called Ein‐

stein's field equations [7], actually only one in tensorial language, namely

<sup>2</sup> *gμν<sup>R</sup>* <sup>+</sup> *gμν<sup>Λ</sup>* <sup>=</sup> <sup>8</sup>*π<sup>G</sup>*

which he published in 1915. This is an extraordinary formula: it connects, in a very precise way, Mathematics with Physics, by establishing that the curvature, R, of space-time (a pure mathematical concept, the reference, coordinate system, so to say) is proportional to (name‐ ly, it will be affected or even determined by) the stress-energy tensor, T, which contains the whole of the mass-energy momentum (already unified by SR, as we just said) of the Uni‐ verse. The proportionality factors are the universal Newton constant, G, the speed of light, c, to the fourth inverse power, and the numbers 8 and π, while Λ is the already mentioned cos‐ mological constant, which multiplies g, the metric of space-time itself. This last term is the

*Rμ<sup>ν</sup>* <sup>−</sup> <sup>1</sup>

one that was absent in Einstein's initial formulation of GR.

**Figure 5.** Karl Schwarzschild, 1873 – 1916. Alexander Alexandrovich Friedmann, 1988 – 1925.

Soon Karl Schwarzschild (letter to Einstein from December 1915) found a solution to Ein‐ stein's equations (the original ones, without the cosmological constant), which corre‐ sponds to what is now know as a black hole (see below). Einstein was very surprised to see

and corresponds to his Special Relativity theory (SR). It

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*<sup>c</sup>* <sup>4</sup> *Tμν*, (1)

equation ever written) is: E = m c2

Another very important fact, this one from the theoretical perspective, is that when Albert Einstein constructed, at the beginning of the second decade of last century and starting from very basic physical postulates–as the principles of covariance and equivalence of the laws of Physics–his theory of General Relativity (GR), scientists (himself included) where firmly convinced that our Universe was stationary. Static, in the more appropriate terminology, albeit rather counterintuitive, since this does not mean that celestial bodies do not move, but that stars and their clusters, in their wandering and distribution, would always have remained from the utmost far past, and would continue to do so into the utmost far fu‐ ture, as we see them at present, with no essential changes. No beginning or end of the Universe was foreseeable, nor needed or called for. But, to his extreme disappointment, Einstein realized that a Universe of this sort was not compatible with his equations, that is, the static universe is not a solution of Einstein's field equations for GR. The reason (not difficult to see) is that a universe of this kind cannot be stable: it will ultimately collapse with time owing to the attraction of the gravity force, against which there is no available protection. This led Einstein astray, until he came up with a solution. While keeping all physical principles that led him to construct his equations (there are ten of them, in scalar language, six of which are independent, but only one in tensorial representation), there was still the remaining freedom to introduce an extra term, a constant (with either sign) multi‐ plied by the metric tensor. This is the now famous cosmological constant, but the problem was that it had no physical interpretation, of any sort. However, endowed with the right sign, it did produce a repulsive pressure to exactly counter the gravitational attraction and keep the universe solution static. Einstein was happy with this arrangement for some years (later it was proven that this solution was not stable, but this is considered nowadays to be just a technical detail that played no major role in the scientific discussion of the time).

**Figure 4.** Albert Einstein, 1879 – 1955. Edwin Hubble, 1889 – 1953.

The best known, by far, of the equations Einstein discovered (and probably the most famous equation ever written) is: E = m c2 and corresponds to his Special Relativity theory (SR). It has a very deep physical meaning, since it establishes the equivalence between mass and en‐ ergy, as two forms of one and the same physical quantity, thus susceptible to be trans‐ formed one into the other, and vice versa. The conversion factor is enormous (the velocity of light squared), meaning that a very small quantity of mass will give rise to an enormous amount of energy –as nuclear power plants prove every day (and very destructive bombs did in the past, to the shame of the Humankind). In any case, here we are not referring to this Einstein's equation (which will not be discussed any further), but to the so-called Ein‐ stein's field equations [7], actually only one in tensorial language, namely

edge the much more complex extragalactic Universe, whose theoretical study is one of the

Another very important fact, this one from the theoretical perspective, is that when Albert Einstein constructed, at the beginning of the second decade of last century and starting from very basic physical postulates–as the principles of covariance and equivalence of the laws of Physics–his theory of General Relativity (GR), scientists (himself included) where firmly convinced that our Universe was stationary. Static, in the more appropriate terminology, albeit rather counterintuitive, since this does not mean that celestial bodies do not move, but that stars and their clusters, in their wandering and distribution, would always have remained from the utmost far past, and would continue to do so into the utmost far fu‐ ture, as we see them at present, with no essential changes. No beginning or end of the Universe was foreseeable, nor needed or called for. But, to his extreme disappointment, Einstein realized that a Universe of this sort was not compatible with his equations, that is, the static universe is not a solution of Einstein's field equations for GR. The reason (not difficult to see) is that a universe of this kind cannot be stable: it will ultimately collapse with time owing to the attraction of the gravity force, against which there is no available protection. This led Einstein astray, until he came up with a solution. While keeping all physical principles that led him to construct his equations (there are ten of them, in scalar language, six of which are independent, but only one in tensorial representation), there was still the remaining freedom to introduce an extra term, a constant (with either sign) multi‐ plied by the metric tensor. This is the now famous cosmological constant, but the problem was that it had no physical interpretation, of any sort. However, endowed with the right sign, it did produce a repulsive pressure to exactly counter the gravitational attraction and keep the universe solution static. Einstein was happy with this arrangement for some years (later it was proven that this solution was not stable, but this is considered nowadays to be just a technical detail that played no major role in the scientific discussion of the time).

main goals of this Chapter [6].

4 Open Questions in Cosmology

**Figure 4.** Albert Einstein, 1879 – 1955. Edwin Hubble, 1889 – 1953.

$$R\_{\mu\nu} - \frac{1}{2}g\_{\mu\nu}R + g\_{\mu\nu}\Lambda = \frac{8\pi G}{c^4}T\_{\mu\nu} \tag{1}$$

which he published in 1915. This is an extraordinary formula: it connects, in a very precise way, Mathematics with Physics, by establishing that the curvature, R, of space-time (a pure mathematical concept, the reference, coordinate system, so to say) is proportional to (name‐ ly, it will be affected or even determined by) the stress-energy tensor, T, which contains the whole of the mass-energy momentum (already unified by SR, as we just said) of the Uni‐ verse. The proportionality factors are the universal Newton constant, G, the speed of light, c, to the fourth inverse power, and the numbers 8 and π, while Λ is the already mentioned cos‐ mological constant, which multiplies g, the metric of space-time itself. This last term is the one that was absent in Einstein's initial formulation of GR.

**Figure 5.** Karl Schwarzschild, 1873 – 1916. Alexander Alexandrovich Friedmann, 1988 – 1925.

Soon Karl Schwarzschild (letter to Einstein from December 1915) found a solution to Ein‐ stein's equations (the original ones, without the cosmological constant), which corre‐ sponds to what is now know as a black hole (see below). Einstein was very surprised to see this so beautiful solution and wrote back to Schwarzschild congratulating him and admit‐ ting he had never thought that such a simple and elegant solution to his so complicated equations could exist.

$$ds^2 = \left(1 - \frac{2Gm}{c^2r}\right)^{-1}dr^2 + r^2\left(d\,\theta^2 + \sin^2\theta d\,\phi^2\right) - c^2\left(1 - \frac{2Gm}{c^2r}\right)dt^2\tag{2}$$

Λ, which corresponds to positive vacuum energy density and negative pressure. As a sub‐

being α some positive constant which has dimensions of length. Topologically, de Sitter space is R × Sn−1. A de Sitter universe has no ordinary matter content, but just a positive cos‐

**Figure 6.** Willem de Sitter, 1872 – 1934. Monsignor Georges Henri Joseph Édouard Lemaître, 1894 – 1966.

*a*(*t*)=*e Ht*

where H is Hubble's constant and t is time. This was a very simple solution of Einstein's equations that undoubtedly corresponded to an expanding universe. In fact, in 1917 de Sit‐ ter had theorized, for the first time, that the Universe might be expanding. The big problem with his solution was, however, that it only could describe a universe devoid of matter, just a vacuum, and this seemed to be at that time not very useful or physically meaningful. Nowadays, on the contrary, this solution has gained extreme importance, as an asymptotic case to describe with good approximation the most probable final stages of the evolution of our Universe (if it will go on expanding forever) and also, as we shall see latter in more de‐

It is then immediate to obtain the scale factor as

, (5)

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*Hα Λ*. (6)

, (7)

manifold, de Sitter space is in essence the one sheeted hyperboloid

mological constant which yields the Hubble expansion rate, H, as

− *x*<sup>0</sup> <sup>2</sup> + ∑ *i*=1 *n xi* <sup>2</sup> =*α* <sup>2</sup>

There is now evidence that Einstein himself had been working hard to find such solution but failed, probably because he was looking for a more general one. Schwarzschild's insight was namely to look for the simplest, with spherical symmetry. And Alexander Friedmann, in 1922, obtained another solution, which is derived by solving the now called Friedmann equations:

$$\begin{aligned} \left(\frac{\dot{a}}{a}\right)^2 + \frac{kc^2}{a^2} - \frac{\Lambda c^2}{3} &= \frac{8\pi G}{3}\rho\_\prime\\ 2\frac{\dot{a}}{a} + \left(\frac{\dot{a}}{a}\right)^2 + \frac{kc^2}{a^2} - \Lambda c^2 &= -\frac{8\pi G}{c^2}\rho\_\prime \end{aligned} \tag{3}$$

These are nowadays more commonly written in terms of the Hubble parameter, H,

$$\begin{aligned} \left(H\,^2 = \left(\frac{a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2},\\ \left(H\,^2 + H\,^2 = \frac{a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right). \end{aligned} \tag{4}$$

and they are even much more interesting for cosmology than Schwarzschild's solution, be‐ cause they correspond to the whole Universe. Friedmann's early death in 1925, at the age of 37, from typhoid fever, prevented him from realizing that, indeed, his solution would de‐ scribe an expanding universe. This honor was reserved to the Belgian priest, astronomer and physicist Monsignor Georges Lemaître who, being not aware of Friedmann's important finding, went to re-discover essentially the same solution while he was working at the Mas‐ sachusetts Institute of Technology on his second PhD Thesis, which he submitted in 1925. Before, Lemaître had already obtained a doctorate from Leuven university in 1920 and had been ordained a priest three years later, just before going to Cambridge University, to start working in cosmology under Arthur Eddington. In Cambridge, Massachusetts, he worked with Harlow Shapley, already quite famous (as mentioned above) for his work on nebulae.

The case is that, around the same time, Willem de Sitter had also been working on a uni‐ verse solution (now called de Sitter space), which is the maximally symmetric vacuum solu‐ tion of Einstein's field equations with a positive (therefore repulsive) cosmological constant Λ, which corresponds to positive vacuum energy density and negative pressure. As a sub‐ manifold, de Sitter space is in essence the one sheeted hyperboloid

$$-\left.\chi\_0^2 + \sum\_{i=1}^n \chi\_i^2 \right| = \alpha^2 \tag{5}$$

being α some positive constant which has dimensions of length. Topologically, de Sitter space is R × Sn−1. A de Sitter universe has no ordinary matter content, but just a positive cos‐ mological constant which yields the Hubble expansion rate, H, as

$$
\overline{\text{H}a\sqrt{\Lambda}}.\tag{6}
$$

**Figure 6.** Willem de Sitter, 1872 – 1934. Monsignor Georges Henri Joseph Édouard Lemaître, 1894 – 1966.

It is then immediate to obtain the scale factor as

this so beautiful solution and wrote back to Schwarzschild congratulating him and admit‐ ting he had never thought that such a simple and elegant solution to his so complicated

There is now evidence that Einstein himself had been working hard to find such solution but failed, probably because he was looking for a more general one. Schwarzschild's insight was namely to look for the simplest, with spherical symmetry. And Alexander Friedmann, in 1922, obtained another solution, which is derived by solving the now called Friedmann

> <sup>3</sup> <sup>=</sup> <sup>8</sup>*π<sup>G</sup>* <sup>3</sup> *<sup>ρ</sup>*,

These are nowadays more commonly written in terms of the Hubble parameter, H,

*<sup>a</sup>* <sup>=</sup> <sup>−</sup> <sup>4</sup>*π<sup>G</sup>*

*<sup>a</sup>* <sup>2</sup> <sup>−</sup>*Λ<sup>c</sup>* <sup>2</sup> <sup>=</sup> <sup>−</sup> <sup>8</sup>*π<sup>G</sup>*

<sup>3</sup> *<sup>ρ</sup>* <sup>−</sup> *kc* <sup>2</sup> *a* 2 ,

<sup>3</sup> (*<sup>ρ</sup>* <sup>+</sup>

and they are even much more interesting for cosmology than Schwarzschild's solution, be‐ cause they correspond to the whole Universe. Friedmann's early death in 1925, at the age of 37, from typhoid fever, prevented him from realizing that, indeed, his solution would de‐ scribe an expanding universe. This honor was reserved to the Belgian priest, astronomer and physicist Monsignor Georges Lemaître who, being not aware of Friedmann's important finding, went to re-discover essentially the same solution while he was working at the Mas‐ sachusetts Institute of Technology on his second PhD Thesis, which he submitted in 1925. Before, Lemaître had already obtained a doctorate from Leuven university in 1920 and had been ordained a priest three years later, just before going to Cambridge University, to start working in cosmology under Arthur Eddington. In Cambridge, Massachusetts, he worked with Harlow Shapley, already quite famous (as mentioned above) for his work on nebulae.

The case is that, around the same time, Willem de Sitter had also been working on a uni‐ verse solution (now called de Sitter space), which is the maximally symmetric vacuum solu‐ tion of Einstein's field equations with a positive (therefore repulsive) cosmological constant

3*p <sup>c</sup>* <sup>2</sup> ),

*<sup>c</sup>* <sup>2</sup> *<sup>p</sup>*.

*θdϕ*2) −*c* <sup>2</sup>

(1<sup>−</sup> <sup>2</sup>*Gm c* 2

*<sup>r</sup>* )*dt* <sup>2</sup> (2)

(3)

(4)

*dr* <sup>2</sup> + *r* 2(*dθ* <sup>2</sup> + sin<sup>2</sup>

equations could exist.

6 Open Questions in Cosmology

equations:

*ds* <sup>2</sup> =(1<sup>−</sup> <sup>2</sup>*Gm c* 2 *r* ) −1

> ( *a* · *a* ) 2 + *kc* <sup>2</sup> *<sup>a</sup>* <sup>2</sup> <sup>−</sup> *<sup>Λ</sup><sup>c</sup>* <sup>2</sup>

2 *a* ·· *a* + ( *a* · *a* ) 2 + *kc* <sup>2</sup>

*<sup>H</sup>* <sup>2</sup> =( *<sup>a</sup>* · *a* ) 2 <sup>=</sup> <sup>8</sup>*π<sup>G</sup>*

> <sup>+</sup> *<sup>H</sup>* <sup>2</sup> <sup>=</sup> *<sup>a</sup>* ··

*H* ·

$$a(t) = e^{Ht},\tag{7}$$

where H is Hubble's constant and t is time. This was a very simple solution of Einstein's equations that undoubtedly corresponded to an expanding universe. In fact, in 1917 de Sit‐ ter had theorized, for the first time, that the Universe might be expanding. The big problem with his solution was, however, that it only could describe a universe devoid of matter, just a vacuum, and this seemed to be at that time not very useful or physically meaningful. Nowadays, on the contrary, this solution has gained extreme importance, as an asymptotic case to describe with good approximation the most probable final stages of the evolution of our Universe (if it will go on expanding forever) and also, as we shall see latter in more de‐ tail (even more in other Chapters), the initial stages, as the inflationary epoch: the fact that the de Sitter expansion is exactly exponential is very helpful in the construction of inflation‐ ary models.

lae, including the Andromeda and the Triangulum nebulae. His observations, made in 1922– 1923, proved conclusively that these nebulae were much too distant to be part of the Mil‐ ky Way and should be considered as separate galaxies. This was the first clear evidence of the "island universe" theory. Hubble, who was then 35, found opposition to his results in the astronomy establishment and his finding was first published in the New York Times, on November 23, 1924, before being formally presented in the 1925 meeting of the Ameri‐ can Astronomical Society. As said, most important in this discovery was the identification of the Cepheid variable stars in those nebulae, and this brings us to Henrietta S. Leavitt, who, in 1912, discovered the very important period-luminosity relation: a straight line rela‐ tionship between the luminosity and the logarithm of the period of variability of these brilliant stars. Leavitt was a distinguished member of the so-called "women human comput‐ ers" brought in at Harvard College by Edward C. Pickering to measure and catalog the brightness of stars in the observatory's photographic plate collection. In particular, her re‐ sults came from the study, during several years, of 1,777 variable stars. Hubble did public‐ ly recognize the importance of Leavitt's discovery for his own (saying even that she deserved the Nobel Prize). It is interesting to describe the now common explanation for the pulsa‐ tion of Cepheid variables, which have been for many decades the "standard candles" for measuring distances at galaxy scales and were crucial, e.g., for the precise determination of Hubble's law. It is the so-called Eddington valve mechanism, based on the fact that dou‐ bly ionized helium is more opaque than singly ionized one. At the dimmest part of a Ce‐ pheid's cycle, the ionized gas in the outer layers of the star is more opaque. The gas is then heated by the star's radiation, temperature increases and it begins to expand. As it ex‐ pands, it cools, and becomes single ionized and thus more transparent, allowing the radia‐ tion to escape. Thus the expansion stops, and gas falls back to the star due to gravitational

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In 1929 Hubble derived his important velocity-distance relationship for nebulae using, as he later wrote to Slipher, "your velocities and my distances." Hubble acknowledged Slipher's seminal contribution to his own work by declaring that "the first steps in a new field are the most difficult and the most significant. Once the barrier is forced, further development is relatively simple." Before that, however, we should go back again to Lemaître, who had vis‐ ited in 1924-25 both Slipher and Hubble to learn about their results first hand. He also at‐ tended the meeting of the American Astronomical Society in Washington DC, in 1925, where Hubble announced his discovery that certain spiral nebulae, previously thought to be gaseous clouds within the Milky Way, were actually separate galaxies. Lemaître realized that the new galaxies could be used to test certain predictions of the general relativity equa‐ tions and, soon after the meeting, he started to work on his own cosmological model. He re‐ alized the uniformity of the recession speed of the galaxies (yet nebulae), in different directions, and the fact that the redshift seemed to be proportional to the known distances to them, and concluded that the recession speed of these celestial objects could be better under‐ stood not as proper displacements of the galaxies, but much more naturally as a stretching of space itself, a true expansion of the fabric of our Universe! And this was not as crazy as it could seem at first sight, since his solution to Einstein's equations (recall, the same as Fried‐

attraction, and the process starts again.

But let us continue with Lemaître. During his two-year stay in Cambridge, MA, he visited Vesto Slipher, at the Lowell Observatory in Arizona, and also Edwin Hubble, at Mount Wilson, in California, who had already accumulated at that time important evidence on the spectral displacements towards longer light wavelengths (redshift) of a large number of far distant nebulae. Actually, the most consistent earlier evidence of the redshift of distant nebulae had been gathered by Slipher who, already in 1912, had published his first results on the surpris‐ ingly large recessional velocity of the Andromeda nebula and, in 1914, at the American Astronomical Society's meeting at Evanston, Illinois, had announced radial velocities for fifteen spirals, reporting that "in the great majority of cases the nebula is receding; the largest veloc‐ ities are all positive and the striking preponderance of the positive sign indicates a general fleeing from us or the Milky Way." Slipher was seeing the nebulae recede at up to 1,100 kilometers per second, the greatest celestial velocities that had ever been observed. He was so clear and convincing that chronicles say that when Slipher described his equipment and techniques along with his results, he received an unprecedented standing ovation. But the interpretation of the redshifts as true movements of the galaxies was not generally accepted then. De Sitter, for one, posited that the nebulae might only appear to be moving, the light waves themselves getting longer and longer as the light traveled towards Earth because of some interstellar processes.

**Figure 7.** Vesto Melvin Slipher, 1875 – 1969. Henrietta Swan Leavitt, 1868 – 1921.

When Hubble arrived at Mount Wilson, California, in 1919, the prevailing view of the cosmos was that the universe consisted entirely of the Milky Way Galaxy. Using the new Hooker telescope at Mt. Wilson, Hubble identified Cepheid variable stars in several spiral nebu‐ lae, including the Andromeda and the Triangulum nebulae. His observations, made in 1922– 1923, proved conclusively that these nebulae were much too distant to be part of the Mil‐ ky Way and should be considered as separate galaxies. This was the first clear evidence of the "island universe" theory. Hubble, who was then 35, found opposition to his results in the astronomy establishment and his finding was first published in the New York Times, on November 23, 1924, before being formally presented in the 1925 meeting of the Ameri‐ can Astronomical Society. As said, most important in this discovery was the identification of the Cepheid variable stars in those nebulae, and this brings us to Henrietta S. Leavitt, who, in 1912, discovered the very important period-luminosity relation: a straight line rela‐ tionship between the luminosity and the logarithm of the period of variability of these brilliant stars. Leavitt was a distinguished member of the so-called "women human comput‐ ers" brought in at Harvard College by Edward C. Pickering to measure and catalog the brightness of stars in the observatory's photographic plate collection. In particular, her re‐ sults came from the study, during several years, of 1,777 variable stars. Hubble did public‐ ly recognize the importance of Leavitt's discovery for his own (saying even that she deserved the Nobel Prize). It is interesting to describe the now common explanation for the pulsa‐ tion of Cepheid variables, which have been for many decades the "standard candles" for measuring distances at galaxy scales and were crucial, e.g., for the precise determination of Hubble's law. It is the so-called Eddington valve mechanism, based on the fact that dou‐ bly ionized helium is more opaque than singly ionized one. At the dimmest part of a Ce‐ pheid's cycle, the ionized gas in the outer layers of the star is more opaque. The gas is then heated by the star's radiation, temperature increases and it begins to expand. As it ex‐ pands, it cools, and becomes single ionized and thus more transparent, allowing the radia‐ tion to escape. Thus the expansion stops, and gas falls back to the star due to gravitational attraction, and the process starts again.

tail (even more in other Chapters), the initial stages, as the inflationary epoch: the fact that the de Sitter expansion is exactly exponential is very helpful in the construction of inflation‐

But let us continue with Lemaître. During his two-year stay in Cambridge, MA, he visited Vesto Slipher, at the Lowell Observatory in Arizona, and also Edwin Hubble, at Mount Wilson, in California, who had already accumulated at that time important evidence on the spectral displacements towards longer light wavelengths (redshift) of a large number of far distant nebulae. Actually, the most consistent earlier evidence of the redshift of distant nebulae had been gathered by Slipher who, already in 1912, had published his first results on the surpris‐ ingly large recessional velocity of the Andromeda nebula and, in 1914, at the American Astronomical Society's meeting at Evanston, Illinois, had announced radial velocities for fifteen spirals, reporting that "in the great majority of cases the nebula is receding; the largest veloc‐ ities are all positive and the striking preponderance of the positive sign indicates a general fleeing from us or the Milky Way." Slipher was seeing the nebulae recede at up to 1,100 kilometers per second, the greatest celestial velocities that had ever been observed. He was so clear and convincing that chronicles say that when Slipher described his equipment and techniques along with his results, he received an unprecedented standing ovation. But the interpretation of the redshifts as true movements of the galaxies was not generally accepted then. De Sitter, for one, posited that the nebulae might only appear to be moving, the light waves themselves getting longer and longer as the light traveled towards Earth because of

ary models.

8 Open Questions in Cosmology

some interstellar processes.

**Figure 7.** Vesto Melvin Slipher, 1875 – 1969. Henrietta Swan Leavitt, 1868 – 1921.

When Hubble arrived at Mount Wilson, California, in 1919, the prevailing view of the cosmos was that the universe consisted entirely of the Milky Way Galaxy. Using the new Hooker telescope at Mt. Wilson, Hubble identified Cepheid variable stars in several spiral nebu‐ In 1929 Hubble derived his important velocity-distance relationship for nebulae using, as he later wrote to Slipher, "your velocities and my distances." Hubble acknowledged Slipher's seminal contribution to his own work by declaring that "the first steps in a new field are the most difficult and the most significant. Once the barrier is forced, further development is relatively simple." Before that, however, we should go back again to Lemaître, who had vis‐ ited in 1924-25 both Slipher and Hubble to learn about their results first hand. He also at‐ tended the meeting of the American Astronomical Society in Washington DC, in 1925, where Hubble announced his discovery that certain spiral nebulae, previously thought to be gaseous clouds within the Milky Way, were actually separate galaxies. Lemaître realized that the new galaxies could be used to test certain predictions of the general relativity equa‐ tions and, soon after the meeting, he started to work on his own cosmological model. He re‐ alized the uniformity of the recession speed of the galaxies (yet nebulae), in different directions, and the fact that the redshift seemed to be proportional to the known distances to them, and concluded that the recession speed of these celestial objects could be better under‐ stood not as proper displacements of the galaxies, but much more naturally as a stretching of space itself, a true expansion of the fabric of our Universe! And this was not as crazy as it could seem at first sight, since his solution to Einstein's equations (recall, the same as Fried‐ mann's) could be actually interpreted as corresponding to an expanding Universe. Theory and observations incredibly matched!

nouncing his very famous sentence: "Away with the cosmological constant. This was the biggest blunder in my life" (in German: "Weg mit der kosmologischen Konstante. Dass war die grösste Eselei meines Lebens"). He clearly realized that, had he truly believed in his field equations, he could have predicted that the Universe was actually expanding (and not stat‐ ic), much before anybody else. (Quite in the same way, say, as Dirac predicted the existence of the positive electron, the positron, because it was a second solution of his quantum equa‐

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

11

It is easy to understand that the Church, which had been so disappointed with the findings of Galileo several centuries ago and had condemned him for his defense of a sun-centered universe, was extremely happy with Lemaître's scenario. He was lauded and raised to the

tion for the electron, impossible to get rid of by natural arguments.)

**Figure 8.** Albert Einstein and Georges Lemaître.

### **3. The Big Bang**

Lemaître was still half-way to these conclusions when he submitted his PhD thesis at MIT in 1925, but he completed his work two years later and published it in an obscure Belgian jour‐ nal in 1927. The retreat of distant nebulae, he wrote in his paper, is "a cosmical effect of the expansion of the universe." He even estimated a rate of expansion close to the figure that Hubble eventually calculated and published two years later. And at a scientific meeting in Brussels, in 1927, the young priest cornered Einstein and tried to persuade him. We do not know his exact words, but presumably they must have been something like: "Sehen Sie, Herr Einstein, your static model for the Universe, with a cosmological constant, does not stand, since it is not stable in the far past. But, on the other hand, you do not need a cosmo‐ logical constant, your original equations are all right! In fact, I have found a solution to these equations which can be interpreted as describing an expanding Universe. And, on the other hand, the redshifts of distant galaxies, as found by astronomers, as Slipher and Hubble, most naturally account for an expansion of the space containing the galaxies, and not for ar‐ bitrary displacements of the galaxies themselves, since exactly the same pattern is seen in any direction!" But, as quoted later by Lemaître, Einstein's reply was utterly disappointing to him. He answered: "Monsieur Lemaître, I can find no mistake in your calculations, but your physical insight is abominable." Einstein, the great genius, the master of space and time, was not ready to imagine a universe in which this space-time was stretching! It took him more than two years to accept this. And now, when we are teaching the expanding uni‐ verse to high-school student, or even to popular audiences, we pretend they should get this concept on the spot! Lemaître's paper was finally noticed by Eddington and with his help it was reprinted in 1931 in the Monthly Notices of the Royal Astronomical Society; it explained clearly, using Lemaître's (Friedmann's) solution, why Hubble saw the velocities of the galax‐ ies steadily increase with distance. The same year, in the much prestigious journal Nature, Lemaître suggested that all the mass-energy of the universe was once packed within a "unique quantum," which he later called the "primeval atom." This was the logical conclu‐ sion of his looking back in time in the Universe evolution: an immediate consequence of his model was that long time ago the Universe was much smaller and that, going even more backwards, that it had had an origin. In 1933 he resumed his theory of the expanding Uni‐ verse and published a more detailed version in the Annals of the Scientific Society of Brus‐ sels, finally achieving his greatest popularity (his name is now, however, rather forgotten by the younger generations of astronomers and physicists). From Lemaître's scenario arose the current vision of the Big Bang (albeit not this name, as we will soon see), a model that has shaped there since the thought of cosmologists as strongly as the idea of crystalline spheres, popularized by Ptolemy (who was already mentioned at the beginning), influenced natural philosophers through the Middle Ages. It took Einstein over two years to understand that Lemaître's model was right and, then, he abhorred of the cosmological constant by pro‐ nouncing his very famous sentence: "Away with the cosmological constant. This was the biggest blunder in my life" (in German: "Weg mit der kosmologischen Konstante. Dass war die grösste Eselei meines Lebens"). He clearly realized that, had he truly believed in his field equations, he could have predicted that the Universe was actually expanding (and not stat‐ ic), much before anybody else. (Quite in the same way, say, as Dirac predicted the existence of the positive electron, the positron, because it was a second solution of his quantum equa‐ tion for the electron, impossible to get rid of by natural arguments.)

**Figure 8.** Albert Einstein and Georges Lemaître.

mann's) could be actually interpreted as corresponding to an expanding Universe. Theory

Lemaître was still half-way to these conclusions when he submitted his PhD thesis at MIT in 1925, but he completed his work two years later and published it in an obscure Belgian jour‐ nal in 1927. The retreat of distant nebulae, he wrote in his paper, is "a cosmical effect of the expansion of the universe." He even estimated a rate of expansion close to the figure that Hubble eventually calculated and published two years later. And at a scientific meeting in Brussels, in 1927, the young priest cornered Einstein and tried to persuade him. We do not know his exact words, but presumably they must have been something like: "Sehen Sie, Herr Einstein, your static model for the Universe, with a cosmological constant, does not stand, since it is not stable in the far past. But, on the other hand, you do not need a cosmo‐ logical constant, your original equations are all right! In fact, I have found a solution to these equations which can be interpreted as describing an expanding Universe. And, on the other hand, the redshifts of distant galaxies, as found by astronomers, as Slipher and Hubble, most naturally account for an expansion of the space containing the galaxies, and not for ar‐ bitrary displacements of the galaxies themselves, since exactly the same pattern is seen in any direction!" But, as quoted later by Lemaître, Einstein's reply was utterly disappointing to him. He answered: "Monsieur Lemaître, I can find no mistake in your calculations, but your physical insight is abominable." Einstein, the great genius, the master of space and time, was not ready to imagine a universe in which this space-time was stretching! It took him more than two years to accept this. And now, when we are teaching the expanding uni‐ verse to high-school student, or even to popular audiences, we pretend they should get this concept on the spot! Lemaître's paper was finally noticed by Eddington and with his help it was reprinted in 1931 in the Monthly Notices of the Royal Astronomical Society; it explained clearly, using Lemaître's (Friedmann's) solution, why Hubble saw the velocities of the galax‐ ies steadily increase with distance. The same year, in the much prestigious journal Nature, Lemaître suggested that all the mass-energy of the universe was once packed within a "unique quantum," which he later called the "primeval atom." This was the logical conclu‐ sion of his looking back in time in the Universe evolution: an immediate consequence of his model was that long time ago the Universe was much smaller and that, going even more backwards, that it had had an origin. In 1933 he resumed his theory of the expanding Uni‐ verse and published a more detailed version in the Annals of the Scientific Society of Brus‐ sels, finally achieving his greatest popularity (his name is now, however, rather forgotten by the younger generations of astronomers and physicists). From Lemaître's scenario arose the current vision of the Big Bang (albeit not this name, as we will soon see), a model that has shaped there since the thought of cosmologists as strongly as the idea of crystalline spheres, popularized by Ptolemy (who was already mentioned at the beginning), influenced natural philosophers through the Middle Ages. It took Einstein over two years to understand that Lemaître's model was right and, then, he abhorred of the cosmological constant by pro‐

and observations incredibly matched!

**3. The Big Bang**

10 Open Questions in Cosmology

It is easy to understand that the Church, which had been so disappointed with the findings of Galileo several centuries ago and had condemned him for his defense of a sun-centered universe, was extremely happy with Lemaître's scenario. He was lauded and raised to the rank of monsignor and was made a fellow, and later president, of the Pontifical Academy of Sciences. But Lemaître always recoiled from any suggestion that his primeval atom had been inspired by the biblical story of Genesis. He insisted, throughout his life, that his theo‐ ry about the origin of space and time sprang solely from the equations before him. Howev‐ er, the name Bib Bang, after which this theory is known today, was actually an occurrence of a rival scientist, Fred Hoyle, who in a BBC radio program, broadcasted on March 28th, 1949, pronounced these magic two words for the first time. Just the year before, Hoyle, Thomas Gold and Hermann Bondi had issued a theory, that was to became quite famous under the name of the Steady State theory, which involved a creation field (called the C-field), which created matter and energy constantly in wider regions of the Universe in a rather smooth manner. These researchers had realized the impossibility of the whole matter-energy of our Universe having been all packed once within a unique quantum or primeval atom. This could have no sense and a creation process needed to be involved. They were very clever to solve the question how to create matter-energy from 'nothing', constantly and at 'zero-cost,' since they realized that any positive amount of ordinary matter and energy would be com‐ pensated by the same amount of negative energy which corresponds to the associated gravi‐ tational potential (which in GR does also have negative energy content!). This observation was extremely important, since it anticipated the physical principles involved in inflationary theories; indeed, it has been widely recognized that the steady state theory anticipated infla‐ tion. Actually, the possibility that the negative energy of gravity could supply the positive energy for the matter of the universe was suggested by Richard Tolman already in 1932, al‐ though a viable mechanism for the energy transfer was not indicated. In any case, just trans‐ lating this physics to Lemaître's scenario would mean that an unbelievably enormous amount of matter and energy should be created instantly, at the very moment of the origin of our Universe. After these considerations the reader should be prepared to understand the words that Fred Hoyle uttered on that occasion. In the BBC program Hoyle tried to push up his theory, as being much more reasonable, in contraposition to Lemaître's one. At a point, he refuted, in a very disrespectful manner, that: "The whole of the matter in the universe was created in one Big Bang in a particular time in the remote past." Hoyle could never imagine that these two words, pronounced with the purpose to absolutely discredit the rival theory, would serve from that moment on to identify what is nowadays the most accepted theory of the Universe, a name that any school child knows. This was clearly not Hoyle's intention. Before going on, a last word on Lemaître's primeval atom scenario. It may seem incredible that this wrong, physically unsustainable idea (again, the whole energy of the universe could never in the past have been concentrated in a nutshell) can be still found nowadays in popular books on cosmology that are being issued by scientific writers having no idea about the physical principles underlying inflation, quantum gravity, or even the more primitive steady state theory. The creation of matter and energy from a void, de Sitter state is key to inflationary models and, as already said, the steady state theory gave a first clear hint to how this could be done while respecting all basic physical principles including energy conservation.

employed for a number of years for satellite work, and which they wanted to prepare for use in radioastronomy. Even if, at that time, there were at several other places much more pow‐ erful radio-telescopes available, this seven meter, very modest horn reflector had some spe‐ cial features they wanted to exploit for high-precision measurements in the 21 cm band, a wavelength at which the galactic halo would be bright enough in order to be detected, and at which the line corresponding to neutral hydrogen atoms could be observed. They wanted, in particular, to detect the presence of hydrogen in clusters of galaxies (this development is very nicely described, and in much more detail, in the Nobel Lecture by Wilson [8]). After having carried out a number of measurements during several months, Penzias and Wilson did not manage to get rid of a very light but persistent noise, which translated into temperature was an excess of some 2 to 4 K, and which it was exactly the same in all directions, day and night. Indeed, the antenna temperature should have been only the sum of the atmospheric contribu‐ tion, so-called temperature of the sky (due to microwave absorption by the terrestrial atmos‐ phere), of 2.3 K, and the radiation from the walls of the antenna and ground, of 1 K. Unless Penzias and Wilson could understand what they first called "antenna problem" their 21 cm galactic halo experiment would not be feasible. So they went through a number of possible reasons for the temperature excess and tested for them. They considered the possibility of some terrestrial source and pointed their antenna towards different directions, in particular to New York City, but the variation was always insignificant. They also took into account the possible influence of the radiation from our Galaxy, but they checked this could not contrib‐ ute decisively, either. They also ruled out discrete extraterrestrial radio sources as the source of the excess radiation as they had a spectrum similar to that of the Galaxy. For some time they lived with the antenna temperature problem and concentrated on measurements in which it was not critical. One day they discovered that a pair of pigeons was roosting up in the horn and had covered part of it with (in their own words) 'what all city dwellers know well.' They cleaned the mess, and later, in the spring of 1965, they thoroughly cleaned out the hornreflector and put aluminum tape over the riveted joints, but only a small reduction in anten‐

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

13

na temperature was obtained. In this way, a whole year passed.

**Figure 9.** Arno Allan Penzias (born April 26, 1933). Robert Woodrow Wilson (born January 10, 1936).

In the year 1963, Arno Allan Penzias and Robert Woodrow Wilson started a project, at the Bell Labs in New Jersey, on the recalibration of a 20-foot horn-reflector, that had been previously employed for a number of years for satellite work, and which they wanted to prepare for use in radioastronomy. Even if, at that time, there were at several other places much more pow‐ erful radio-telescopes available, this seven meter, very modest horn reflector had some spe‐ cial features they wanted to exploit for high-precision measurements in the 21 cm band, a wavelength at which the galactic halo would be bright enough in order to be detected, and at which the line corresponding to neutral hydrogen atoms could be observed. They wanted, in particular, to detect the presence of hydrogen in clusters of galaxies (this development is very nicely described, and in much more detail, in the Nobel Lecture by Wilson [8]). After having carried out a number of measurements during several months, Penzias and Wilson did not manage to get rid of a very light but persistent noise, which translated into temperature was an excess of some 2 to 4 K, and which it was exactly the same in all directions, day and night. Indeed, the antenna temperature should have been only the sum of the atmospheric contribu‐ tion, so-called temperature of the sky (due to microwave absorption by the terrestrial atmos‐ phere), of 2.3 K, and the radiation from the walls of the antenna and ground, of 1 K. Unless Penzias and Wilson could understand what they first called "antenna problem" their 21 cm galactic halo experiment would not be feasible. So they went through a number of possible reasons for the temperature excess and tested for them. They considered the possibility of some terrestrial source and pointed their antenna towards different directions, in particular to New York City, but the variation was always insignificant. They also took into account the possible influence of the radiation from our Galaxy, but they checked this could not contrib‐ ute decisively, either. They also ruled out discrete extraterrestrial radio sources as the source of the excess radiation as they had a spectrum similar to that of the Galaxy. For some time they lived with the antenna temperature problem and concentrated on measurements in which it was not critical. One day they discovered that a pair of pigeons was roosting up in the horn and had covered part of it with (in their own words) 'what all city dwellers know well.' They cleaned the mess, and later, in the spring of 1965, they thoroughly cleaned out the hornreflector and put aluminum tape over the riveted joints, but only a small reduction in anten‐ na temperature was obtained. In this way, a whole year passed.

rank of monsignor and was made a fellow, and later president, of the Pontifical Academy of Sciences. But Lemaître always recoiled from any suggestion that his primeval atom had been inspired by the biblical story of Genesis. He insisted, throughout his life, that his theo‐ ry about the origin of space and time sprang solely from the equations before him. Howev‐ er, the name Bib Bang, after which this theory is known today, was actually an occurrence of a rival scientist, Fred Hoyle, who in a BBC radio program, broadcasted on March 28th, 1949, pronounced these magic two words for the first time. Just the year before, Hoyle, Thomas Gold and Hermann Bondi had issued a theory, that was to became quite famous under the name of the Steady State theory, which involved a creation field (called the C-field), which created matter and energy constantly in wider regions of the Universe in a rather smooth manner. These researchers had realized the impossibility of the whole matter-energy of our Universe having been all packed once within a unique quantum or primeval atom. This could have no sense and a creation process needed to be involved. They were very clever to solve the question how to create matter-energy from 'nothing', constantly and at 'zero-cost,' since they realized that any positive amount of ordinary matter and energy would be com‐ pensated by the same amount of negative energy which corresponds to the associated gravi‐ tational potential (which in GR does also have negative energy content!). This observation was extremely important, since it anticipated the physical principles involved in inflationary theories; indeed, it has been widely recognized that the steady state theory anticipated infla‐ tion. Actually, the possibility that the negative energy of gravity could supply the positive energy for the matter of the universe was suggested by Richard Tolman already in 1932, al‐ though a viable mechanism for the energy transfer was not indicated. In any case, just trans‐ lating this physics to Lemaître's scenario would mean that an unbelievably enormous amount of matter and energy should be created instantly, at the very moment of the origin of our Universe. After these considerations the reader should be prepared to understand the words that Fred Hoyle uttered on that occasion. In the BBC program Hoyle tried to push up his theory, as being much more reasonable, in contraposition to Lemaître's one. At a point, he refuted, in a very disrespectful manner, that: "The whole of the matter in the universe was created in one Big Bang in a particular time in the remote past." Hoyle could never imagine that these two words, pronounced with the purpose to absolutely discredit the rival theory, would serve from that moment on to identify what is nowadays the most accepted theory of the Universe, a name that any school child knows. This was clearly not Hoyle's intention. Before going on, a last word on Lemaître's primeval atom scenario. It may seem incredible that this wrong, physically unsustainable idea (again, the whole energy of the universe could never in the past have been concentrated in a nutshell) can be still found nowadays in popular books on cosmology that are being issued by scientific writers having no idea about the physical principles underlying inflation, quantum gravity, or even the more primitive steady state theory. The creation of matter and energy from a void, de Sitter state is key to inflationary models and, as already said, the steady state theory gave a first clear hint to how this could be done while respecting all basic physical principles including

In the year 1963, Arno Allan Penzias and Robert Woodrow Wilson started a project, at the Bell Labs in New Jersey, on the recalibration of a 20-foot horn-reflector, that had been previously

energy conservation.

12 Open Questions in Cosmology

**Figure 9.** Arno Allan Penzias (born April 26, 1933). Robert Woodrow Wilson (born January 10, 1936).

visited Penzias and Wilson and were quickly convinced of the accuracy of their measure‐ ments. They agreed to a side-by-side publication of two letters in the Astrophysical Journal a letter on the theory from Princeton [9] and one on the measured excess temperature from Bell Laboratories [10]. Penzias and Wilson were careful to exclude any discussion of the cos‐ mological theory of the origin of background radiation from their letter, because they had not been involved in any of that work and thought, that their measurement was independ‐ ent of the theory and might outlive it. After the meeting, an experimental group was set up in Princeton to complete their own measurement with the expectation that the background temperature would be about 3 K. There was the great expectation that what Penzias and Wilson had detected could be in fact the Big Bang itself! However, the final confirmation of

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

15

And the first additional evidence did not actually come from the experimental group at Princeton, but from a totally different, indirect measurement. Indeed, it came out from res‐ cuing from oblivion a measurement that had been made thirty years earlier by W.S. Adams and T. Dunhan Jr., who had discovered several faint optical interstellar absorption lines which were later identified with the molecules CH, CH+, and CN. In the case of CN, in addi‐ tion to the ground state, absorption was seen from the first rotationally excited state. This was reanalyzed in 1965-66, and it was realized that the CN is in equilibrium with the back‐ ground radiation, since there is no other significant source of excitation where these mole‐ cules are located. In December 1965, P.G. Roll and D.T. Wilkinson [11] completed their measurement of 3.0 ± 0.5 K at 3.2 cm, the first confirming microwave measurement, which was followed shortly by T.F. Howell and J.R. Shakeshaft's value of 2.8 ± 0.6 K at 20.7 cm [12] and then by Penzias and Wilson's one of 3.2 K ± 1 K at 21.1 cm [13]. By mid 1966 the intensi‐ ty of the microwave background radiation had been shown to be close to 3 K between 21 cm and 2.6 mm, almost two orders of magnitude in wavelength. This was already very close to

In the same way that the first experimental evidence for the cosmic microwave background radiation was obtained (but unrecognized) long before 1965, it soon was realized that the theoretical prediction had been made, at least sixteen years before Penzias and Wilson's de‐ tection, by George Gamow (a former student of Friedmann) in 1948, and improved by R.A. Alpher and R.C. Herman, in 1949 [14]. Those authors are now recognized as the first who theoretically predicted the cosmic radiation associated to the Big Bang, for which they calcu‐ lated a value of 5 K, approximately (a very nice figure that they later spoiled, bringing it to 28 K). We will finish this section with the well known fact that Arno Penzias and Robert Wilson were laureated with the 1978 Nobel Prize in Physics by their very important discov‐ ery, which can be considered as one the milestone findings in Human History. The Universe had indeed an origin, the fabric of space was stretching and, as clearly understood by Le‐ maître, Friedmann's solution to Einstein's equations was a unique, real description of our Universe. The stationary universe, also under its more modern form of the steady state theo‐

this extraordinarily important cosmological discovery took several years yet.

the present, highly accurate value of 2,725 K.

ry, was dead.

**Figure 10.** Big Bang detection: Penzias and Wilson's 20-foot horn-reflector.

At the same time, in Princeton, only 60 km away, R.H. Dicke, P.J.E. Peebles and D.T.

**Figure 11.** Robert H. Dicke, 1916 – 1997. Philip J. E. Peebles, born April 25, 1935. David T. Wilkinson, 1935 – 2002.

Wilkinson where working on a paper where they tried to guess the characteristics that a mi‐ crowave radiation that would come from a very dense universe, in its origin (possibly pul‐ sating), should have; that is to say, under the conditions that they thought could correspond to those of the Big Bang. The sequence of events which led to the unraveling of the mystery began one day when Penzias was talking to Bernard Burke of MIT about other matters and mentioned the unexplained noise. Burke recalled hearing about the work of the theoretical group in Princeton on radiation in the universe. In the preprint, Peebles, following Dicke's suggestion calculated that the universe should be filled with a relic blackbody radiation at a minimum temperature of 10 K. Shortly after sending the preprint, Dicke and his coworkers visited Penzias and Wilson and were quickly convinced of the accuracy of their measure‐ ments. They agreed to a side-by-side publication of two letters in the Astrophysical Journal a letter on the theory from Princeton [9] and one on the measured excess temperature from Bell Laboratories [10]. Penzias and Wilson were careful to exclude any discussion of the cos‐ mological theory of the origin of background radiation from their letter, because they had not been involved in any of that work and thought, that their measurement was independ‐ ent of the theory and might outlive it. After the meeting, an experimental group was set up in Princeton to complete their own measurement with the expectation that the background temperature would be about 3 K. There was the great expectation that what Penzias and Wilson had detected could be in fact the Big Bang itself! However, the final confirmation of this extraordinarily important cosmological discovery took several years yet.

And the first additional evidence did not actually come from the experimental group at Princeton, but from a totally different, indirect measurement. Indeed, it came out from res‐ cuing from oblivion a measurement that had been made thirty years earlier by W.S. Adams and T. Dunhan Jr., who had discovered several faint optical interstellar absorption lines which were later identified with the molecules CH, CH+, and CN. In the case of CN, in addi‐ tion to the ground state, absorption was seen from the first rotationally excited state. This was reanalyzed in 1965-66, and it was realized that the CN is in equilibrium with the back‐ ground radiation, since there is no other significant source of excitation where these mole‐ cules are located. In December 1965, P.G. Roll and D.T. Wilkinson [11] completed their measurement of 3.0 ± 0.5 K at 3.2 cm, the first confirming microwave measurement, which was followed shortly by T.F. Howell and J.R. Shakeshaft's value of 2.8 ± 0.6 K at 20.7 cm [12] and then by Penzias and Wilson's one of 3.2 K ± 1 K at 21.1 cm [13]. By mid 1966 the intensi‐ ty of the microwave background radiation had been shown to be close to 3 K between 21 cm and 2.6 mm, almost two orders of magnitude in wavelength. This was already very close to the present, highly accurate value of 2,725 K.

**Figure 10.** Big Bang detection: Penzias and Wilson's 20-foot horn-reflector.

14 Open Questions in Cosmology

At the same time, in Princeton, only 60 km away, R.H. Dicke, P.J.E. Peebles and D.T.

**Figure 11.** Robert H. Dicke, 1916 – 1997. Philip J. E. Peebles, born April 25, 1935. David T. Wilkinson, 1935 – 2002.

Wilkinson where working on a paper where they tried to guess the characteristics that a mi‐ crowave radiation that would come from a very dense universe, in its origin (possibly pul‐ sating), should have; that is to say, under the conditions that they thought could correspond to those of the Big Bang. The sequence of events which led to the unraveling of the mystery began one day when Penzias was talking to Bernard Burke of MIT about other matters and mentioned the unexplained noise. Burke recalled hearing about the work of the theoretical group in Princeton on radiation in the universe. In the preprint, Peebles, following Dicke's suggestion calculated that the universe should be filled with a relic blackbody radiation at a minimum temperature of 10 K. Shortly after sending the preprint, Dicke and his coworkers

In the same way that the first experimental evidence for the cosmic microwave background radiation was obtained (but unrecognized) long before 1965, it soon was realized that the theoretical prediction had been made, at least sixteen years before Penzias and Wilson's de‐ tection, by George Gamow (a former student of Friedmann) in 1948, and improved by R.A. Alpher and R.C. Herman, in 1949 [14]. Those authors are now recognized as the first who theoretically predicted the cosmic radiation associated to the Big Bang, for which they calcu‐ lated a value of 5 K, approximately (a very nice figure that they later spoiled, bringing it to 28 K). We will finish this section with the well known fact that Arno Penzias and Robert Wilson were laureated with the 1978 Nobel Prize in Physics by their very important discov‐ ery, which can be considered as one the milestone findings in Human History. The Universe had indeed an origin, the fabric of space was stretching and, as clearly understood by Le‐ maître, Friedmann's solution to Einstein's equations was a unique, real description of our Universe. The stationary universe, also under its more modern form of the steady state theo‐ ry, was dead.

shaping of the concept of inflation constitutes, for different reasons, another brilliant page in

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

17

The first to come up to this very revolutionary idea was Allan Guth, born in 1947 and who studied (both graduation and PhD) at MIT, from 1964 to 1971. During the following nine years he was, successively, a PostDoc at Princeton, Columbia, Cornell and Stanford (SLAC), all of them top class Universities. But Guth did not manage to jump over this level and get a real contract. In 1978 he was at Cornell while his career was up in the air and he badly need‐ ed to find a permanent job to support his wife and son. Someday, a fellow PostDoc called Henry Tye (now a professor at Cornell) proposed him to study jointly the problem of mo‐ nopole production in the very early Universe. Guth got interested in this subject so that when Robert Dicke (whom we have already mentioned before) came to give a seminar, he attended it with much interest. Guth was very intrigued by Dicke's conclusion that the tra‐ ditional Big Bang theory had severe problems and that it was leaving out something impor‐ tant. There was the problem of flatness (also called Dicke's coincidence): the fact that the matter density of the Universe was so close to the critical mass corresponding to a flat (Eu‐ clidean) Universe. Also the horizon problem, namely the fact that the Universe is so perfect‐ ly homogeneous and isotropic at large enough scales, which is in absolutely good agreement with the cosmological principle. And to these problems Guth and Tye added, as a results of their specific study, the problem of absence of magnetic monopoles, which should actually be very abundant in the present Universe, but it is the case that (with the only exception of Blas Cabrera, who reported finding one in 1982) nobody has ever seen any of them. One should note, however, that John Preskill, at Harvard at that time, had published a result in the same direction before Guth and Tye. Anyway, all these problems and the sudden inter‐ est of Guth on cosmology kept him busy for two years. His personal description of how, in a sleepless night, he suddenly thought of a mechanism in order to solve these severe prob‐ lems, all of them at once, is better than any science fiction story. Looking back at the situa‐ tion, now we can say that it was very risky on his side, being just a PostDoc fellow without a tenured job, to propose such a revolutionary mechanism as the inflationary model, what he did in 1980. Guth first made public his ideas in a seminar at SLAC on January 23, 1980. In August, he submitted his paper, entitled "The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems," to the Physical Review, and was published in January 1981 [15]. Soon after, he captured the interest of several universities and got several offers which he rejected, until he had the possibility to come back to MIT, as an associate visiting professor in 1980. His scientific career has been growing since then. Not without some prob‐ lems at the beginning, however: it was discovered that his initial model had an important flaw, which was corrected by Andrei Linde (now at Stanford) and, independently, by Paul Steinhardt (Princeton) and Andreas Albrecht (Davies). The modified theory was given the name of "new inflation." The works of Katsuhiko Sato, who about the same time as Guth proposed a much related theory and of Alexei Starobinsky, who argued at about the same time that quantum corrections to gravity would replace the initial singularity of the universe with an exponentially expanding de Sitter phase, must be mentioned, as well. In particular the last one is getting more and more popular recently, in a unifying context to be explained

the history leading to our present knowledge of the cosmos.

**Figure 12.** George Gamow, 1904 – 1968. Ralph Asher Alpher, 1921 – 2007 (with Victor S. Alpher, in Tampa, Florida).

## **4. The Big Bang modified: Inflation**

However, the original Big Bang theory had to be modified, what occurred at the beginning of the eighties, in order to solve several very serious discrepancies it had accumulated when comparing it with the most accurate astronomical observations of the cosmos, specifically, concerning what happened during the very first second in the history of the Universe. It was realized that the expansion during this first second could by no means be an ordinary one, understanding by this the one that has taken place later in its evolution, say, kind of a linear one. A very special stage had to be devised to account for what occurred in this initial in‐ stant of time (well, in fact one second is a very, very long time at this scale). This stage is generically called inflation, and its formulation is mainly due to Allan Guth, Katsuhiko Sato, Andrei Linde, Andreas Albrecht, Paul Steinhardt, Alexei Starobinsky, Slava Mukhanov, G.V. Chibisov, and a large list of other scientist (the number and classes of models are ac‐ tually still growing, nowadays). The name inflation comes from the fact that the Universe expansion had to be enormous, incredibly big during an extremely small instant of time (of the order of 10-33 seconds). In this infinitesimal fraction of a second the Universe expanded from the size of a peanut to that of the present Milky Way (in volume, an increase of at least 75 orders of magnitude). Actually, in the inflationary theory the Universe begins incredibly small, some 10-24 cm, a hundred billion times smaller than a proton. And, at the same time, during inflation it cools down abruptly (supercooling) by 5 orders of magnitude, from some 1027 K to 1022 K. This relatively low temperature is maintained during the inflationary phase. When inflation ends the temperature returns to the pre-inflationary temperature; this is called reheating or thermalization because the large potential energy of the inflaton field de‐ cays into particles and electromagnetic radiation, which fills the universe, starting in this way the radiation dominated phase of the Universe. Because the very nature of inflation is not known, this process is still poorly understood. As explained before, energy conservation is consistent with physics during the whole process: this lies in the subtle behavior of gravi‐ ty, already present in Newtonian physics, where we know that the energy of the gravitation‐ al potential is always negative, a fact which is maintained in GR. The development and shaping of the concept of inflation constitutes, for different reasons, another brilliant page in the history leading to our present knowledge of the cosmos.

**Figure 12.** George Gamow, 1904 – 1968. Ralph Asher Alpher, 1921 – 2007 (with Victor S. Alpher, in Tampa, Florida).

However, the original Big Bang theory had to be modified, what occurred at the beginning of the eighties, in order to solve several very serious discrepancies it had accumulated when comparing it with the most accurate astronomical observations of the cosmos, specifically, concerning what happened during the very first second in the history of the Universe. It was realized that the expansion during this first second could by no means be an ordinary one, understanding by this the one that has taken place later in its evolution, say, kind of a linear one. A very special stage had to be devised to account for what occurred in this initial in‐ stant of time (well, in fact one second is a very, very long time at this scale). This stage is generically called inflation, and its formulation is mainly due to Allan Guth, Katsuhiko Sato, Andrei Linde, Andreas Albrecht, Paul Steinhardt, Alexei Starobinsky, Slava Mukhanov, G.V. Chibisov, and a large list of other scientist (the number and classes of models are ac‐ tually still growing, nowadays). The name inflation comes from the fact that the Universe expansion had to be enormous, incredibly big during an extremely small instant of time (of the order of 10-33 seconds). In this infinitesimal fraction of a second the Universe expanded from the size of a peanut to that of the present Milky Way (in volume, an increase of at least 75 orders of magnitude). Actually, in the inflationary theory the Universe begins incredibly small, some 10-24 cm, a hundred billion times smaller than a proton. And, at the same time, during inflation it cools down abruptly (supercooling) by 5 orders of magnitude, from some 1027 K to 1022 K. This relatively low temperature is maintained during the inflationary phase. When inflation ends the temperature returns to the pre-inflationary temperature; this is called reheating or thermalization because the large potential energy of the inflaton field de‐ cays into particles and electromagnetic radiation, which fills the universe, starting in this way the radiation dominated phase of the Universe. Because the very nature of inflation is not known, this process is still poorly understood. As explained before, energy conservation is consistent with physics during the whole process: this lies in the subtle behavior of gravi‐ ty, already present in Newtonian physics, where we know that the energy of the gravitation‐ al potential is always negative, a fact which is maintained in GR. The development and

**4. The Big Bang modified: Inflation**

16 Open Questions in Cosmology

The first to come up to this very revolutionary idea was Allan Guth, born in 1947 and who studied (both graduation and PhD) at MIT, from 1964 to 1971. During the following nine years he was, successively, a PostDoc at Princeton, Columbia, Cornell and Stanford (SLAC), all of them top class Universities. But Guth did not manage to jump over this level and get a real contract. In 1978 he was at Cornell while his career was up in the air and he badly need‐ ed to find a permanent job to support his wife and son. Someday, a fellow PostDoc called Henry Tye (now a professor at Cornell) proposed him to study jointly the problem of mo‐ nopole production in the very early Universe. Guth got interested in this subject so that when Robert Dicke (whom we have already mentioned before) came to give a seminar, he attended it with much interest. Guth was very intrigued by Dicke's conclusion that the tra‐ ditional Big Bang theory had severe problems and that it was leaving out something impor‐ tant. There was the problem of flatness (also called Dicke's coincidence): the fact that the matter density of the Universe was so close to the critical mass corresponding to a flat (Eu‐ clidean) Universe. Also the horizon problem, namely the fact that the Universe is so perfect‐ ly homogeneous and isotropic at large enough scales, which is in absolutely good agreement with the cosmological principle. And to these problems Guth and Tye added, as a results of their specific study, the problem of absence of magnetic monopoles, which should actually be very abundant in the present Universe, but it is the case that (with the only exception of Blas Cabrera, who reported finding one in 1982) nobody has ever seen any of them. One should note, however, that John Preskill, at Harvard at that time, had published a result in the same direction before Guth and Tye. Anyway, all these problems and the sudden inter‐ est of Guth on cosmology kept him busy for two years. His personal description of how, in a sleepless night, he suddenly thought of a mechanism in order to solve these severe prob‐ lems, all of them at once, is better than any science fiction story. Looking back at the situa‐ tion, now we can say that it was very risky on his side, being just a PostDoc fellow without a tenured job, to propose such a revolutionary mechanism as the inflationary model, what he did in 1980. Guth first made public his ideas in a seminar at SLAC on January 23, 1980. In August, he submitted his paper, entitled "The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems," to the Physical Review, and was published in January 1981 [15]. Soon after, he captured the interest of several universities and got several offers which he rejected, until he had the possibility to come back to MIT, as an associate visiting professor in 1980. His scientific career has been growing since then. Not without some prob‐ lems at the beginning, however: it was discovered that his initial model had an important flaw, which was corrected by Andrei Linde (now at Stanford) and, independently, by Paul Steinhardt (Princeton) and Andreas Albrecht (Davies). The modified theory was given the name of "new inflation." The works of Katsuhiko Sato, who about the same time as Guth proposed a much related theory and of Alexei Starobinsky, who argued at about the same time that quantum corrections to gravity would replace the initial singularity of the universe with an exponentially expanding de Sitter phase, must be mentioned, as well. In particular the last one is getting more and more popular recently, in a unifying context to be explained later. The Spanish researcher Jaume Garriga, at Barcelona University, has published influen‐ tial papers in this area, too.

er, that two years before Zwicky, Einstein and de Sitter had already published a paper where they considered a most probable theoretical existence of enormous amounts of matter in the Universe which did not emit light. It had also been postulated by Jan Oort, one year before Zwicky, to account for the orbital velocities of some stars in the Milky Way. But Zwicky's calculations, based on the use of the virial theorem, where much more convincing. According to them, the gravity of the visible galaxies in the Coma cluster was too small in order to possibly account for the large speeds of the more exterior galaxies. A big amount of mass was missing! This was called the missing mass problem and Zwicky referred to this unseen matter as dunkle Materie (dark matter). Since those years, more and more different observations indicated the presence of dark matter in the universe, such as the anomalous rotational speeds of galaxies, gravitational lensing by galaxy clusters, such as the Bullet

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

19

Cluster, the temperature distribution of hot gas in galaxies and clusters, and other.

is extremely dark in relation with the central part.

lively nowadays.

Very famous astronomers in this context now are Vera Rubin and Kent Ford for their semi‐ nal papers published around 1975. It so happened that during some forty years after Zwicky's discovery no other corroborating observations appeared and the problem was al‐ most forgotten. But in the early 70s, Vera Rubin, a young astronomer at the Carnegie Institu‐ tion of Washington, presented findings based on a new, very sensitive spectrograph that could measure the velocity curve of edge-on spiral galaxies to a great degree of accuracy. In 1975, in a meeting of the American Astronomical Society, Rubin and Ford announced their important discovery that most stars in spiral galaxies orbit at roughly the same speed, im‐ plying that their mass densities were uniform well beyond the locations of most of the visi‐ ble stars in the galaxy. In 1980 they published a paper [16] which has had enormous influence in modern cosmology, where they summarized the results of over a decade of work on this subject. Their results have shaken the very grounds of Newton's universal law of gravity since they undoubtedly indicate either that Newton's results are not applicable to the Universe at large distances (the error obtained is certainly enormous) or that a very im‐ portant part of the mass of spiral galaxies must be located in the galactic halo region, which

At the beginning and for some time these results met very strong skepticism by the com‐ munity of astronomers. But Rubin, a brave and stubborn scientist, never changed her con‐ viction that her results were correct. They have been subsequently checked to enormous precision and there is now no more doubt that an important problem to be explained is fac‐ ing us. The most accepted conclusion is the existence of dark matter, that is, ordinary matter made up of particles that we cannot see for some reason. There are many candidates for dark matter but, while this is the most generally accepted conclusion, there still remains open the other mentioned possibility, namely that Newton's laws need to be modified at large distances (modified gravities, MOG, MOND, and other theories). Actually, Rubin her‐ self is a convinced supporter of this second possibility. The debate continues and it is very

To finish with this point let us summarize that, talking in terms of dark matter, for what we now know it must constitute an enormous amount of ordinary (that is, gravitating) matter, ten times as abundant as visible galaxies. And we infer its existence not just by the clear

**Figure 13.** Allan Guth Andrei Linde Fritz Zwicky, 1898 – 1974

Nowadays, under the name of inflation there are over fifty different theories which have evolved from Guth's original idea. Borrowing of energy from the gravitational field is the basic principle of the inflationary paradigm, completely different from the classical Big Bang theory, where all matter-energy in the universe was assumed to be there from the beginning (as explained above). In Guth's words: "Inflation provides a mechanism by which the entire universe can develop from just a few ounces of primordial matter." As a final consequence of all these developments, the so called standard cosmological model, or FLRW (Friedmann-Lemaître-Robertson-Walker) model emerged. The two last names appear here because, be‐ tween 1935 and 1937, the mathematicians Howard P. Robertson and Arthur G. Walker rigorously proved that the FLRW metric is the only one possible, on a spacetime that is spa‐ tially homogeneous and isotropic. In other words, they showed that the solution to Ein‐ stein's equations found by Friedmann and later by Lemaître was unique in describing the Universe we live in. Let us pause to ponder, for a second, the extraordinary beauty of this cosmological model as a description of the Universe: to the uniqueness of Einstein's field equation (the only freedom being the cosmological constant) we add up the fact that the sol‐ ution is also single. We have arrived to just one possible mathematical description of our Universe, and the inflation paradigm opens a possible way to understand how it could be created, without violating the basic conservation principles of Physics. This last point will however require further elaboration.

## **5. Dark matter**

Before that, however, we need go back in time and explain about another very important problem in cosmology which appeared for the first time, in a compelling, clear way, in 1933 when the Swiss astrophysicist Fritz Zwicky, at CALTECH, unveiled it from his detailed ob‐ servations of the most exterior galaxies of the Coma cluster. It should be mentioned, howev‐ er, that two years before Zwicky, Einstein and de Sitter had already published a paper where they considered a most probable theoretical existence of enormous amounts of matter in the Universe which did not emit light. It had also been postulated by Jan Oort, one year before Zwicky, to account for the orbital velocities of some stars in the Milky Way. But Zwicky's calculations, based on the use of the virial theorem, where much more convincing. According to them, the gravity of the visible galaxies in the Coma cluster was too small in order to possibly account for the large speeds of the more exterior galaxies. A big amount of mass was missing! This was called the missing mass problem and Zwicky referred to this unseen matter as dunkle Materie (dark matter). Since those years, more and more different observations indicated the presence of dark matter in the universe, such as the anomalous rotational speeds of galaxies, gravitational lensing by galaxy clusters, such as the Bullet Cluster, the temperature distribution of hot gas in galaxies and clusters, and other.

later. The Spanish researcher Jaume Garriga, at Barcelona University, has published influen‐

Nowadays, under the name of inflation there are over fifty different theories which have evolved from Guth's original idea. Borrowing of energy from the gravitational field is the basic principle of the inflationary paradigm, completely different from the classical Big Bang theory, where all matter-energy in the universe was assumed to be there from the beginning (as explained above). In Guth's words: "Inflation provides a mechanism by which the entire universe can develop from just a few ounces of primordial matter." As a final consequence of all these developments, the so called standard cosmological model, or FLRW (Friedmann-Lemaître-Robertson-Walker) model emerged. The two last names appear here because, be‐ tween 1935 and 1937, the mathematicians Howard P. Robertson and Arthur G. Walker rigorously proved that the FLRW metric is the only one possible, on a spacetime that is spa‐ tially homogeneous and isotropic. In other words, they showed that the solution to Ein‐ stein's equations found by Friedmann and later by Lemaître was unique in describing the Universe we live in. Let us pause to ponder, for a second, the extraordinary beauty of this cosmological model as a description of the Universe: to the uniqueness of Einstein's field equation (the only freedom being the cosmological constant) we add up the fact that the sol‐ ution is also single. We have arrived to just one possible mathematical description of our Universe, and the inflation paradigm opens a possible way to understand how it could be created, without violating the basic conservation principles of Physics. This last point will

Before that, however, we need go back in time and explain about another very important problem in cosmology which appeared for the first time, in a compelling, clear way, in 1933 when the Swiss astrophysicist Fritz Zwicky, at CALTECH, unveiled it from his detailed ob‐ servations of the most exterior galaxies of the Coma cluster. It should be mentioned, howev‐

tial papers in this area, too.

18 Open Questions in Cosmology

**Figure 13.** Allan Guth Andrei Linde Fritz Zwicky, 1898 – 1974

however require further elaboration.

**5. Dark matter**

Very famous astronomers in this context now are Vera Rubin and Kent Ford for their semi‐ nal papers published around 1975. It so happened that during some forty years after Zwicky's discovery no other corroborating observations appeared and the problem was al‐ most forgotten. But in the early 70s, Vera Rubin, a young astronomer at the Carnegie Institu‐ tion of Washington, presented findings based on a new, very sensitive spectrograph that could measure the velocity curve of edge-on spiral galaxies to a great degree of accuracy. In 1975, in a meeting of the American Astronomical Society, Rubin and Ford announced their important discovery that most stars in spiral galaxies orbit at roughly the same speed, im‐ plying that their mass densities were uniform well beyond the locations of most of the visi‐ ble stars in the galaxy. In 1980 they published a paper [16] which has had enormous influence in modern cosmology, where they summarized the results of over a decade of work on this subject. Their results have shaken the very grounds of Newton's universal law of gravity since they undoubtedly indicate either that Newton's results are not applicable to the Universe at large distances (the error obtained is certainly enormous) or that a very im‐ portant part of the mass of spiral galaxies must be located in the galactic halo region, which is extremely dark in relation with the central part.

At the beginning and for some time these results met very strong skepticism by the com‐ munity of astronomers. But Rubin, a brave and stubborn scientist, never changed her con‐ viction that her results were correct. They have been subsequently checked to enormous precision and there is now no more doubt that an important problem to be explained is fac‐ ing us. The most accepted conclusion is the existence of dark matter, that is, ordinary matter made up of particles that we cannot see for some reason. There are many candidates for dark matter but, while this is the most generally accepted conclusion, there still remains open the other mentioned possibility, namely that Newton's laws need to be modified at large distances (modified gravities, MOG, MOND, and other theories). Actually, Rubin her‐ self is a convinced supporter of this second possibility. The debate continues and it is very lively nowadays.

To finish with this point let us summarize that, talking in terms of dark matter, for what we now know it must constitute an enormous amount of ordinary (that is, gravitating) matter, ten times as abundant as visible galaxies. And we infer its existence not just by the clear gravitational effects we have mentioned, as the observed anomalies in the rotation curves of spiral galaxies (just described), and which account for the rotational speeds of the exterior stars of the galaxy as a function of the distance of the start to the galactic center, but also from the rotation of the so-called satellite galaxies of our Milky Way (and of Andromeda, too), some of which can already be measured with enough accuracy as they turn around our own galaxy (resp. Andromeda), in a way very similar to how planets describe orbits around the Sun. The extraordinary regularities found in the trajectories of such satellite galaxies constitute a really thrilling, very active research field at present. A different way to trace the presence of dark matter is through gravitational lensing (both macro and micro lensing). Its effects are very apparent there, as a notorious amplification of the power of gravitational lenses, compared with the case that the effect would be just due to the visible stellar objects. In clusters as, for instance, Abell 1689, the observed, very strong effects cannot by any means be explained as being produced by its visible mass only. And in the case of the Bullet cluster one clearly detects an enormous mass acting as a gravitational lens and which is completely separated from the barionic, visible mass which emits X rays.

that the generic name WIMPs (weakly interacting massive particles) has been proposed to

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

21

Another very important landmark in the knowledge of the cosmos at large scale was the publication, in 1986 of the first map of the Universe in three dimensions. In fact, it was only a very thin slice of an angular sector of the same but it was extremely important and com‐ pletely changed the vision astronomers and other scientists had of it. Up to then, the only representations of the cosmos were in form of two dimensional projections on the celestial sphere, as still is (and serves as a very good example) the APM Galaxy Survey, which con‐ tains two million galaxies. Even if, in comparison, the Harvard CfA strip of Valérie de Lap‐ parent, Margaret Geller and John Huchra [17], contained only a total of 1,100 galaxies, what was most important was that for 584 of them their distance from us could be determined (through the observation of their cosmological redshift). And this allowed, for the first time in History, to see a part of our Universe in the elusive third dimension: the distance from us. Actually the plot looks again two-dimensional, since the slice is represented as flat but, again, the spatial structures created by the disposition of the galaxies and clusters, away

**Figure 15.** The first slice of the CfA Survey, by Valerie de Lapparent, Margaret Geller and John Huchra, published in 1986.

The impact of this work was spectacular, also due in part to the shapes of these point struc‐ tures, showing that the distribution of galaxies in space was anything but random, with gal‐ axies actually appearing to be distributed on surfaces, almost bubble like, surrounding large empty regions, or "voids." Anyone could easily identify what looked like a human being (the man), another shape looked like a thumb imprint (God's thumb) pointing towards us, and so on. But the most intriguing fact, for scientists, was the presence in the whole picture of such very large regions devoid of any galaxy (voids), while they concentrated on the verge of these voids, and forming filaments and large walls (as the so-called Great Wall).

generically name any particle of this sort as a dark matter candidate.

**6. The Universe in depth**

from us, had never been seen before.

**Figure 14.** Two Million Galaxies: S. Maddox (Nottingham U.) et al., APM Survey, Astrophys. Dept. Oxford U.

We certainly do not know yet what dark matter is made of, neither why we cannot see it. But we do know that the discovered neutrino mass (neutrinos being indeed invisible!) is not enough to account for it; and also that adding up the masses of big, Jupiter like planets (so called MACHOs, which are also very difficult to see) is again not enough in order to explain the missing amount of mass. But astroparticle physicists got indeed a good number of other possible candidates, as axions, neutralinos and other (they come from the breakdown of cer‐ tain fundamental symmetries in particle and quantum field theories). What we know is that they must be elusive particles, very weekly coupled with any of the known physical fields since, on the contrary, its presence would have been detected already. It is for this reason that the generic name WIMPs (weakly interacting massive particles) has been proposed to generically name any particle of this sort as a dark matter candidate.

## **6. The Universe in depth**

gravitational effects we have mentioned, as the observed anomalies in the rotation curves of spiral galaxies (just described), and which account for the rotational speeds of the exterior stars of the galaxy as a function of the distance of the start to the galactic center, but also from the rotation of the so-called satellite galaxies of our Milky Way (and of Andromeda, too), some of which can already be measured with enough accuracy as they turn around our own galaxy (resp. Andromeda), in a way very similar to how planets describe orbits around the Sun. The extraordinary regularities found in the trajectories of such satellite galaxies constitute a really thrilling, very active research field at present. A different way to trace the presence of dark matter is through gravitational lensing (both macro and micro lensing). Its effects are very apparent there, as a notorious amplification of the power of gravitational lenses, compared with the case that the effect would be just due to the visible stellar objects. In clusters as, for instance, Abell 1689, the observed, very strong effects cannot by any means be explained as being produced by its visible mass only. And in the case of the Bullet cluster one clearly detects an enormous mass acting as a gravitational lens and which is

completely separated from the barionic, visible mass which emits X rays.

20 Open Questions in Cosmology

**Figure 14.** Two Million Galaxies: S. Maddox (Nottingham U.) et al., APM Survey, Astrophys. Dept. Oxford U.

We certainly do not know yet what dark matter is made of, neither why we cannot see it. But we do know that the discovered neutrino mass (neutrinos being indeed invisible!) is not enough to account for it; and also that adding up the masses of big, Jupiter like planets (so called MACHOs, which are also very difficult to see) is again not enough in order to explain the missing amount of mass. But astroparticle physicists got indeed a good number of other possible candidates, as axions, neutralinos and other (they come from the breakdown of cer‐ tain fundamental symmetries in particle and quantum field theories). What we know is that they must be elusive particles, very weekly coupled with any of the known physical fields since, on the contrary, its presence would have been detected already. It is for this reason

Another very important landmark in the knowledge of the cosmos at large scale was the publication, in 1986 of the first map of the Universe in three dimensions. In fact, it was only a very thin slice of an angular sector of the same but it was extremely important and com‐ pletely changed the vision astronomers and other scientists had of it. Up to then, the only representations of the cosmos were in form of two dimensional projections on the celestial sphere, as still is (and serves as a very good example) the APM Galaxy Survey, which con‐ tains two million galaxies. Even if, in comparison, the Harvard CfA strip of Valérie de Lap‐ parent, Margaret Geller and John Huchra [17], contained only a total of 1,100 galaxies, what was most important was that for 584 of them their distance from us could be determined (through the observation of their cosmological redshift). And this allowed, for the first time in History, to see a part of our Universe in the elusive third dimension: the distance from us. Actually the plot looks again two-dimensional, since the slice is represented as flat but, again, the spatial structures created by the disposition of the galaxies and clusters, away from us, had never been seen before.

**Figure 15.** The first slice of the CfA Survey, by Valerie de Lapparent, Margaret Geller and John Huchra, published in 1986.

The impact of this work was spectacular, also due in part to the shapes of these point struc‐ tures, showing that the distribution of galaxies in space was anything but random, with gal‐ axies actually appearing to be distributed on surfaces, almost bubble like, surrounding large empty regions, or "voids." Anyone could easily identify what looked like a human being (the man), another shape looked like a thumb imprint (God's thumb) pointing towards us, and so on. But the most intriguing fact, for scientists, was the presence in the whole picture of such very large regions devoid of any galaxy (voids), while they concentrated on the verge of these voids, and forming filaments and large walls (as the so-called Great Wall). Many astronomers, but also a good number of prestigious theoretical physicists and even mathematicians who had never before dealt with cosmological issues started to work on this point distribution, trying to find some fundamental model that could possibly generate such peculiar pattern in the Universe evolution. Astronomers, on their side, tried to find new ob‐ servational confirmation of this large-scale behavior of galaxies and clusters. Collaborations of pure theoreticians and astronomers flourished, as was the case of Edward Witten with Jeremiah Ostriker. That same year, in Spain, in the historical Peñíscola Castle, we had a fiveday workshop of GIFT (Interuniversity Group of Theoretical Physics) where Ricky Kolb and Mike Turner were invited to present such recent and astonishing developments. This author was there and felt immediately captivated by such map. Coming back from the workshop he handed a problem to Enrique Gaztañaga (who was, by the way, in search of a subject for his PhD Thesis): to provide an effective mathematical characterization of the point distribu‐ tion, more simple than the usual higher-order point correlation statistics, and to try to gener‐ ate such point distribution from a phenomenological model by taking into account, essentially, the gravitational attraction. This was the origin of our large-scale cosmology group in our Institute ICE-CSIC and IEEC, in Barcelona. When more and more precise sur‐ veys, of millions of galaxies with redshifts, as the 2d Field, where carried out, all these spec‐ tacular forms have smoothly disappeared:

very rich and marvelous web structure. However, the problem to obtain this large scale pat‐

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

23

Summarizing a lot, cosmologists know now that our Universe is not static nor in a steady state. Quite on the contrary, it had a very spectacular origin some 13,730 million years ago, what we know with an error of less than 1%, according to the most recent (7th year) data from the WMAP (Wilkinson Microwave Anisotropy Probe) satellite, and to the first data coming from the PLANCK mission, and from different terrestrial observatories. All astro‐ nomical tests that have been carried out until now have confirmed, without the slightest doubt and with increasing accuracy, the new Big Bang theory, that is, the one which in‐ cludes inflation (although, concerning this last, a too-large number of different, competing

Until the end of last Century, cosmologists were convinced that the expansion of the fabric of the Universe, originated in the Big Bang, was uniform. Up to then the main challenge of cosmology at large scale was to determine if the mass-energy density, ρ, of our cosmos was large enough (above critical) so that it would be able to completely stop this expansion at some point in the future—an instant after which the Universe would begin to contract, to finally finish in a so-called Big Crunch—or if, quite on the contrary, this energy density ρ was smaller, subcritical, and thus unable to stop the Universe expansion completely, ever in the future. In this case, expansion would continue forever, even if, of course, there was no doubt that the action of gravity would certainly decelerate the expansion rate, this was crys‐ tal clear. The most precise observations carried out until then indicated that the actual value of ρ was indeed very close to the critical value, ρc, being in fact quite difficult to determine if

This situation radically changed just before the end of the Century, because of two different analyses of very precise observations carried out —with the big Hubble Space Telescope on type Ia supernovae by two teams, each comprising some thirty scientists. The two groups wanted to measure with high precision the deceleration, caused by gravity attraction, on the expansion rate of the Universe, by calibrating the variation in this expansion rate with dis‐ tance. To their enormous surprise, the values obtained by both teams were completely unex‐ pected, and matched with each other. The first to issue results, in 1998, was the High-z Supernova Search Team an Australian-American project, led by Brian Schmidt and Adam Riess, while the other group, with the name Supernova Cosmology Project and led by Saul Perlmutter at Lawrence Berkeley National Laboratory, published independent results the year after, 1999. The author cannot help mentioning that one of the members in this last col‐ laboration is the Spanish astronomer Pilar Ruiz Lapuente, from Barcelona University. The common and very clear conclusion of the two observations was that the expansion of the Universe is nowadays accelerating and not decelerating, and that it has been accelerating for a long period of time in the past. This was one of these moments in History where you have

tern starting from a fundamental theory remains, to large extent, open.

models still remains). But this is by no means the last word.

**7. The expansion accelerates**

it was, in fact, above or below such value.

**Figure 16.** The 2dF Galaxy Redshift Survey (2dFGRS), by M.M. Colless, et al, 2001.

almost all were generated by errors in the computation of distances, due to the fact that the redshift produced by the Universe expansion gets mixed with the redshift coming from the proper movements of the galaxy with respect to other celestial bodies in its neighborhood and from the movement of the observer, which are sometimes not easy to disentangle. Some of the big structures remain, however, as is the case of the Great Wall, and of the voids sur‐ rounded by galaxies on their surfaces. Moreover, on top of slices we have now true 3-dimen‐ sional representations of the observed data, together with computer simulations depicting a very rich and marvelous web structure. However, the problem to obtain this large scale pat‐ tern starting from a fundamental theory remains, to large extent, open.

Summarizing a lot, cosmologists know now that our Universe is not static nor in a steady state. Quite on the contrary, it had a very spectacular origin some 13,730 million years ago, what we know with an error of less than 1%, according to the most recent (7th year) data from the WMAP (Wilkinson Microwave Anisotropy Probe) satellite, and to the first data coming from the PLANCK mission, and from different terrestrial observatories. All astro‐ nomical tests that have been carried out until now have confirmed, without the slightest doubt and with increasing accuracy, the new Big Bang theory, that is, the one which in‐ cludes inflation (although, concerning this last, a too-large number of different, competing models still remains). But this is by no means the last word.

## **7. The expansion accelerates**

Many astronomers, but also a good number of prestigious theoretical physicists and even mathematicians who had never before dealt with cosmological issues started to work on this point distribution, trying to find some fundamental model that could possibly generate such peculiar pattern in the Universe evolution. Astronomers, on their side, tried to find new ob‐ servational confirmation of this large-scale behavior of galaxies and clusters. Collaborations of pure theoreticians and astronomers flourished, as was the case of Edward Witten with Jeremiah Ostriker. That same year, in Spain, in the historical Peñíscola Castle, we had a fiveday workshop of GIFT (Interuniversity Group of Theoretical Physics) where Ricky Kolb and Mike Turner were invited to present such recent and astonishing developments. This author was there and felt immediately captivated by such map. Coming back from the workshop he handed a problem to Enrique Gaztañaga (who was, by the way, in search of a subject for his PhD Thesis): to provide an effective mathematical characterization of the point distribu‐ tion, more simple than the usual higher-order point correlation statistics, and to try to gener‐ ate such point distribution from a phenomenological model by taking into account, essentially, the gravitational attraction. This was the origin of our large-scale cosmology group in our Institute ICE-CSIC and IEEC, in Barcelona. When more and more precise sur‐ veys, of millions of galaxies with redshifts, as the 2d Field, where carried out, all these spec‐

tacular forms have smoothly disappeared:

22 Open Questions in Cosmology

**Figure 16.** The 2dF Galaxy Redshift Survey (2dFGRS), by M.M. Colless, et al, 2001.

almost all were generated by errors in the computation of distances, due to the fact that the redshift produced by the Universe expansion gets mixed with the redshift coming from the proper movements of the galaxy with respect to other celestial bodies in its neighborhood and from the movement of the observer, which are sometimes not easy to disentangle. Some of the big structures remain, however, as is the case of the Great Wall, and of the voids sur‐ rounded by galaxies on their surfaces. Moreover, on top of slices we have now true 3-dimen‐ sional representations of the observed data, together with computer simulations depicting a

Until the end of last Century, cosmologists were convinced that the expansion of the fabric of the Universe, originated in the Big Bang, was uniform. Up to then the main challenge of cosmology at large scale was to determine if the mass-energy density, ρ, of our cosmos was large enough (above critical) so that it would be able to completely stop this expansion at some point in the future—an instant after which the Universe would begin to contract, to finally finish in a so-called Big Crunch—or if, quite on the contrary, this energy density ρ was smaller, subcritical, and thus unable to stop the Universe expansion completely, ever in the future. In this case, expansion would continue forever, even if, of course, there was no doubt that the action of gravity would certainly decelerate the expansion rate, this was crys‐ tal clear. The most precise observations carried out until then indicated that the actual value of ρ was indeed very close to the critical value, ρc, being in fact quite difficult to determine if it was, in fact, above or below such value.

This situation radically changed just before the end of the Century, because of two different analyses of very precise observations carried out —with the big Hubble Space Telescope on type Ia supernovae by two teams, each comprising some thirty scientists. The two groups wanted to measure with high precision the deceleration, caused by gravity attraction, on the expansion rate of the Universe, by calibrating the variation in this expansion rate with dis‐ tance. To their enormous surprise, the values obtained by both teams were completely unex‐ pected, and matched with each other. The first to issue results, in 1998, was the High-z Supernova Search Team an Australian-American project, led by Brian Schmidt and Adam Riess, while the other group, with the name Supernova Cosmology Project and led by Saul Perlmutter at Lawrence Berkeley National Laboratory, published independent results the year after, 1999. The author cannot help mentioning that one of the members in this last col‐ laboration is the Spanish astronomer Pilar Ruiz Lapuente, from Barcelona University. The common and very clear conclusion of the two observations was that the expansion of the Universe is nowadays accelerating and not decelerating, and that it has been accelerating for a long period of time in the past. This was one of these moments in History where you have something in front of your eyes that you really do not believe. You cannot explain it with the scientific tools at your hand. The impact of this discovery on our knowledge of the Universe was extraordinary and the three researchers who led the teams have been awarded the 2011 Nobel Prize in Physics. The first conclusion seems quite clear: in order that this acceleration can occur a force must be present, as we already know since Galileo, XVI C, and Newton, XVII C, but in this case the force must be acting constantly at the level of the whole cosmos! The question is now, what kind of force can have this property in order to produce the de‐ sired acceleration?

chanics. This is probably the most radical change in our conception of the world that has ev‐ er happened. In spite of Richard Feynman saying that "nobody can understand QM," the fact that it works to enormous precision for the description of nature is witnessed by the un‐ challenged 14 to 15 digit matching in the results of some particle physics experiments. Al‐ ready Wolfgang Pauli in the 20's, and then Yakov Zel'dovich in the 60's, among others, clearly realized that if the fluctuations of the quantum vacuum—which are always there ow‐ ing to W. Heisenberg's uncertainty principle and have a magnitude of the order of Planck's constant— are taken into account in Einstein's equation (as a valid form of energy satisfying the equivalence principle), then their contribution at cosmological scale (which happens to go together with Λ) would be enormously big. In principle, infinitely so, albeit we know that through a regularization and renormalization process the number is rendered finite. But even then it is still enormous: some 60 to 120 orders of magnitude larger than needed in or‐ der to explain the observed Universe acceleration. This is the famous cosmological constant problem, which was around since the first attempts to reconcile General Relativity and Quantum Physics appeared (although, at first, the problem was just to explain why vacuum fluctuations yielded a zero contribution, not a very small one, as now). Some very important physicists, as the Nobel Prize laureate Steven Weinberg [18], have been working for years on this problem, without real success. The reader must be adverted that, in these discussions, the concepts of cosmological constant and of quantum vacuum fluctuations are taken as one and the same thing, the reason being that there is no other possible contribution to Λ which

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

25

Another possible explanation is that there might exist some peculiar energy fluid filling the Universe (of course not of ordinary nature, as in the case of dark matter). There are many different models, with fancy names, for this fluid: quintessence, k-essence, Chaplygin gas, Galileons, and many more. The third possible explanation is the most radical of all, from the theoretical viewpoint and as seen from the whole description of the History of the Universe as summarized in the present article: maybe something is in error when trying to apply Ein‐ stein's General Relativity to cosmological scales, so that this marvelous theory may need be modified at these scales (as also Newton's equations might have to be modified too, in order to account for the missing dark matter). The reader will find full details of these thrilling is‐ sues in the other chapters of the Book. Let me here just note that modifications of Einstein's equation usually proceed by way of introducing additional terms with higher order powers on the Ricci curvature, R, a general function f(R), and/or higher order derivatives. In fact some of these terms are difficult to avoid when one considers quantum corrections to Ein‐ stein's equation, as Alexey Starobinsky and collaborators did already at the beginning of the eighties, finding in this way a model that would, to start, produce inflation and which can be modified to possibly account for its present acceleration, too. Scientists do not know yet the right answer, nor if the Universe acceleration is actually constant. To this end, the deriv‐ ative of the acceleration should be obtained, what is still impossible with the quality of

We should add that, for some time, the interpretation of the Ia Supernovae results as imply‐ ing a Universal acceleration were controverted, some possible explanations involving a non-

is known up to now.

present data.

**Figure 17.** Saul Perlmutter Brian Schmidt Adam Riess

Thinking for a while, it is not difficult to explain the problem even to a non-specialist. An expanding Universe, as in the case of the Bing Bang theory, does not need any force to ex‐ pand forever, just an initial impulse, for a short interval of time, as when we throw a stone in the air. In this case, owing to the enormous mass of the Earth we are sure the stone will stop flying and come back; but if the Earth was the size of a mountain this same stone would never return. As already explained, at cosmological level everything just depends on the mass density of the whole Universe being larger or smaller than the critical value, ρc, which marks the difference between the situation when the Universe would continue expanding forever and the one in which it would stop expanding, to start contracting back. But now, in order that the stone can accelerate, a force must act on it all the time, as with an accelerating car. As in the case of dark matter, nobody knows yet what produces this acceleration of the Universe expansion, and this missing energy is generically called dark energy. In fact a (too large) number of possible explanations have arisen, which can be roughly classified into three types.

The first one is the most natural and immediate, but in no way the simplest to match. Com‐ ing back where we started, with Einstein's equation, the only possibility to provide a repul‐ sive force there is by introducing again the cosmological constant, Λ, with the appropriate sign. There is no other freedom but, fortunately, we still have this one! Regretfully, however, as with Einstein's, there is a big question mark behind it, namely, what is the physical nature of Λ? Where does it come from? This brings us to explain about another crucial revolution which took place in Physics during the first thirty years of the past Century: Quantum Me‐ chanics. This is probably the most radical change in our conception of the world that has ev‐ er happened. In spite of Richard Feynman saying that "nobody can understand QM," the fact that it works to enormous precision for the description of nature is witnessed by the un‐ challenged 14 to 15 digit matching in the results of some particle physics experiments. Al‐ ready Wolfgang Pauli in the 20's, and then Yakov Zel'dovich in the 60's, among others, clearly realized that if the fluctuations of the quantum vacuum—which are always there ow‐ ing to W. Heisenberg's uncertainty principle and have a magnitude of the order of Planck's constant— are taken into account in Einstein's equation (as a valid form of energy satisfying the equivalence principle), then their contribution at cosmological scale (which happens to go together with Λ) would be enormously big. In principle, infinitely so, albeit we know that through a regularization and renormalization process the number is rendered finite. But even then it is still enormous: some 60 to 120 orders of magnitude larger than needed in or‐ der to explain the observed Universe acceleration. This is the famous cosmological constant problem, which was around since the first attempts to reconcile General Relativity and Quantum Physics appeared (although, at first, the problem was just to explain why vacuum fluctuations yielded a zero contribution, not a very small one, as now). Some very important physicists, as the Nobel Prize laureate Steven Weinberg [18], have been working for years on this problem, without real success. The reader must be adverted that, in these discussions, the concepts of cosmological constant and of quantum vacuum fluctuations are taken as one and the same thing, the reason being that there is no other possible contribution to Λ which is known up to now.

something in front of your eyes that you really do not believe. You cannot explain it with the scientific tools at your hand. The impact of this discovery on our knowledge of the Universe was extraordinary and the three researchers who led the teams have been awarded the 2011 Nobel Prize in Physics. The first conclusion seems quite clear: in order that this acceleration can occur a force must be present, as we already know since Galileo, XVI C, and Newton, XVII C, but in this case the force must be acting constantly at the level of the whole cosmos! The question is now, what kind of force can have this property in order to produce the de‐

Thinking for a while, it is not difficult to explain the problem even to a non-specialist. An expanding Universe, as in the case of the Bing Bang theory, does not need any force to ex‐ pand forever, just an initial impulse, for a short interval of time, as when we throw a stone in the air. In this case, owing to the enormous mass of the Earth we are sure the stone will stop flying and come back; but if the Earth was the size of a mountain this same stone would never return. As already explained, at cosmological level everything just depends on the mass density of the whole Universe being larger or smaller than the critical value, ρc, which marks the difference between the situation when the Universe would continue expanding forever and the one in which it would stop expanding, to start contracting back. But now, in order that the stone can accelerate, a force must act on it all the time, as with an accelerating car. As in the case of dark matter, nobody knows yet what produces this acceleration of the Universe expansion, and this missing energy is generically called dark energy. In fact a (too large) number of possible explanations have arisen, which can be roughly classified into

The first one is the most natural and immediate, but in no way the simplest to match. Com‐ ing back where we started, with Einstein's equation, the only possibility to provide a repul‐ sive force there is by introducing again the cosmological constant, Λ, with the appropriate sign. There is no other freedom but, fortunately, we still have this one! Regretfully, however, as with Einstein's, there is a big question mark behind it, namely, what is the physical nature of Λ? Where does it come from? This brings us to explain about another crucial revolution which took place in Physics during the first thirty years of the past Century: Quantum Me‐

sired acceleration?

24 Open Questions in Cosmology

three types.

**Figure 17.** Saul Perlmutter Brian Schmidt Adam Riess

Another possible explanation is that there might exist some peculiar energy fluid filling the Universe (of course not of ordinary nature, as in the case of dark matter). There are many different models, with fancy names, for this fluid: quintessence, k-essence, Chaplygin gas, Galileons, and many more. The third possible explanation is the most radical of all, from the theoretical viewpoint and as seen from the whole description of the History of the Universe as summarized in the present article: maybe something is in error when trying to apply Ein‐ stein's General Relativity to cosmological scales, so that this marvelous theory may need be modified at these scales (as also Newton's equations might have to be modified too, in order to account for the missing dark matter). The reader will find full details of these thrilling is‐ sues in the other chapters of the Book. Let me here just note that modifications of Einstein's equation usually proceed by way of introducing additional terms with higher order powers on the Ricci curvature, R, a general function f(R), and/or higher order derivatives. In fact some of these terms are difficult to avoid when one considers quantum corrections to Ein‐ stein's equation, as Alexey Starobinsky and collaborators did already at the beginning of the eighties, finding in this way a model that would, to start, produce inflation and which can be modified to possibly account for its present acceleration, too. Scientists do not know yet the right answer, nor if the Universe acceleration is actually constant. To this end, the deriv‐ ative of the acceleration should be obtained, what is still impossible with the quality of present data.

We should add that, for some time, the interpretation of the Ia Supernovae results as imply‐ ing a Universal acceleration were controverted, some possible explanations involving a nonCopernican view of the position of our local galaxy group in the cosmos were published (we could be in one of these enormous voids surrounded by very massive structures), and even very recently some alternative interpretation has appeared. However, Type Ia Supernovae are very good standard candles for the redshift range where the observations were carried out, since they have a very strong and consistent brightness along considerable cosmological distances; moreover, since 1990 several other independent proofs have been added to check the results. Among them, the impact of acceleration on the fluctuations of the cosmic micro‐ wave background, where measures have been carried out on the imprint of the acceleration on the gravitational potential wells which contribute to the integrated Sachs-Wolfe effect (and translate into colder and hotter spots in the CMB map). Also, the effect of acceleration on the gravitational lenses, and the one that it has on the large scale structures of the Uni‐ verse, on the basis of the phenomenon known as acoustic baryon oscillations (BAO). All these observations are absolutely independent from each other and this contributes to the fact that there remains little doubt today that the Universe expansion accelerates. The Dark Energy Survey project (DES) is being set to provide new measurements, integrating all these different techniques, with participation of a group of our Institute ICE-CSIC and IEEC, led by Enrique Gaztañaga.

to scalar-tensor ones is still open today. At the classical level they are most probably equiva‐ lent, but at the quantum level the answer seems to be clearly negative. The recent and ex‐ tremely important discovery of the Higgs field will surely give a spectacular thrust to this kind of models. In fact, in a paper of 2004 by Elizalde, Odintsov and Shin'ichi Nojiri (now at Nagoya University, Japan)[19] there was an independent proposal of the so-called quintom dark energy: one phantom plus one quintessence scalar which could have a relation with the

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

27

And with this we have reached the very final stage of our general description of our knowl‐ edge of the cosmos at large scales. There are still no observations to confirm or disprove these last theories. A lot more about them and all the most recent developments is to be found in the other chapters of this Book. Some very promising results seem to indicate that, within f(R) theories, there is the possibility to build, with blocks of a really fundamental theory, as string or M theory, a fully-fledged model which could describe all the stages of the evolution of the cosmos, from the Big Bang through inflation, reheating and recombina‐ tion, to the present accelerated expansion and on towards the end of the Universe in a de Sitter asymptotic phase, which is the most plausible one (although some compelling models with future singularities, as the Big Rip, or either pulsating universes, cannot be excluded with present data). Adding up our knowledge of the Universe, we must shamefully confess that over 95% of it is, as of today, 'terra ignota.' But this is actually good for Science, since it

Even more uncertain is the explanation of the creation of the Universe, of the very instant when it came to being. We are more or less acquainted with the corresponding passage of the Bible. Looking now at the descriptions of scientists, Stephen Hawking and Roger Pen‐ rose did important work on the subject, which has been influential for several decades, with the conclusion (obtained again under very general and natural conditions) that such instant is (or it was until recently) a mathematical singularity and, therefore, beyond reach of any kind of physical interpretation. This result was quite disappointing but, fortunately, it just affects classical theories and does not take into account quantum corrections which generi‐ cally soften the singularities, or even make them completely disappear. Making the story short, there are new models (Alex Vilenkin and also Andrei Linde have been working on them since over twenty years ago) in which one can sidestep the singularity problem: by combining inflation with quantum fluctuations of the vacuum state of a primordial system in which a spark or miniscule particle---a "twist in matter and space-time" so-called "Hawk‐ ing-Turok instanton"---would be able, at zero-energy cost (as explained already), to ignite inflation which, on its turn, would amplify the negligibly small quantum fluctuations (of Planck's constant magnitude) of the vacuum, giving rise, in this way, to the cosmic fluctua‐ tions (of order 10-5) which we clearly observe on the CMB plot below. This is the most an‐ cient map of the Universe that we have been able to capture until now. It corresponds to when it was some 370,000 years old. Just before that, the Universe was like a very dense and

means that, in front of us, there is a lot to be discovered, hopefully soon!

discovered Higgs.

**8. The origin of the Universe**

**Figure 18.** Jaume Garriga, UB Enrique Gaztañaga, CSIC Sergei D. Odintsov, ICREA Emilio Elizalde, CSIC

As already mentioned, a promising possibility to explain the acceleration consists in modify‐ ing Einstein's equation, that is GR itself, at least at large scales, entering the so-called f(R) or scalar-tensor theories, in their different variants. In our group of the Institute for Space Sci‐ ence (ICE-CSIC) and of the Catalan Institute of Space Research (IEEC), led by Sergei D. Odi‐ ntsov and the author of this Chapter, we are presently working on this kind of models with a long list of international collaborators. As clearly stated at the beginning, all present day cosmology is based on Einstein's equation, thus, in making this step we are entering a new age in our knowledge of the cosmos. Yet to be seen is if it will finally be a successful one. As advanced, there are different ways to depart from GR, one of the most popular is by extend‐ ing the Hilbert-Einstein action by the addition of a function, f(R), in principle arbitrary, of the Ricci curvature, R. A theory of this kind was first proposed by Hagen Kleinert and Hans-Jürgen Schmidt, and independently by Salvatore Capozziello, in 2002. Already from the be‐ ginning this theory was related with quintessence, in which a scalar field with time evolution is incorporated to GR. The discussion about f(R) theories being in fact equivalent to scalar-tensor ones is still open today. At the classical level they are most probably equiva‐ lent, but at the quantum level the answer seems to be clearly negative. The recent and ex‐ tremely important discovery of the Higgs field will surely give a spectacular thrust to this kind of models. In fact, in a paper of 2004 by Elizalde, Odintsov and Shin'ichi Nojiri (now at Nagoya University, Japan)[19] there was an independent proposal of the so-called quintom dark energy: one phantom plus one quintessence scalar which could have a relation with the discovered Higgs.

And with this we have reached the very final stage of our general description of our knowl‐ edge of the cosmos at large scales. There are still no observations to confirm or disprove these last theories. A lot more about them and all the most recent developments is to be found in the other chapters of this Book. Some very promising results seem to indicate that, within f(R) theories, there is the possibility to build, with blocks of a really fundamental theory, as string or M theory, a fully-fledged model which could describe all the stages of the evolution of the cosmos, from the Big Bang through inflation, reheating and recombina‐ tion, to the present accelerated expansion and on towards the end of the Universe in a de Sitter asymptotic phase, which is the most plausible one (although some compelling models with future singularities, as the Big Rip, or either pulsating universes, cannot be excluded with present data). Adding up our knowledge of the Universe, we must shamefully confess that over 95% of it is, as of today, 'terra ignota.' But this is actually good for Science, since it means that, in front of us, there is a lot to be discovered, hopefully soon!

## **8. The origin of the Universe**

Copernican view of the position of our local galaxy group in the cosmos were published (we could be in one of these enormous voids surrounded by very massive structures), and even very recently some alternative interpretation has appeared. However, Type Ia Supernovae are very good standard candles for the redshift range where the observations were carried out, since they have a very strong and consistent brightness along considerable cosmological distances; moreover, since 1990 several other independent proofs have been added to check the results. Among them, the impact of acceleration on the fluctuations of the cosmic micro‐ wave background, where measures have been carried out on the imprint of the acceleration on the gravitational potential wells which contribute to the integrated Sachs-Wolfe effect (and translate into colder and hotter spots in the CMB map). Also, the effect of acceleration on the gravitational lenses, and the one that it has on the large scale structures of the Uni‐ verse, on the basis of the phenomenon known as acoustic baryon oscillations (BAO). All these observations are absolutely independent from each other and this contributes to the fact that there remains little doubt today that the Universe expansion accelerates. The Dark Energy Survey project (DES) is being set to provide new measurements, integrating all these different techniques, with participation of a group of our Institute ICE-CSIC and IEEC, led

**Figure 18.** Jaume Garriga, UB Enrique Gaztañaga, CSIC Sergei D. Odintsov, ICREA Emilio Elizalde, CSIC

As already mentioned, a promising possibility to explain the acceleration consists in modify‐ ing Einstein's equation, that is GR itself, at least at large scales, entering the so-called f(R) or scalar-tensor theories, in their different variants. In our group of the Institute for Space Sci‐ ence (ICE-CSIC) and of the Catalan Institute of Space Research (IEEC), led by Sergei D. Odi‐ ntsov and the author of this Chapter, we are presently working on this kind of models with a long list of international collaborators. As clearly stated at the beginning, all present day cosmology is based on Einstein's equation, thus, in making this step we are entering a new age in our knowledge of the cosmos. Yet to be seen is if it will finally be a successful one. As advanced, there are different ways to depart from GR, one of the most popular is by extend‐ ing the Hilbert-Einstein action by the addition of a function, f(R), in principle arbitrary, of the Ricci curvature, R. A theory of this kind was first proposed by Hagen Kleinert and Hans-Jürgen Schmidt, and independently by Salvatore Capozziello, in 2002. Already from the be‐ ginning this theory was related with quintessence, in which a scalar field with time evolution is incorporated to GR. The discussion about f(R) theories being in fact equivalent

by Enrique Gaztañaga.

26 Open Questions in Cosmology

Even more uncertain is the explanation of the creation of the Universe, of the very instant when it came to being. We are more or less acquainted with the corresponding passage of the Bible. Looking now at the descriptions of scientists, Stephen Hawking and Roger Pen‐ rose did important work on the subject, which has been influential for several decades, with the conclusion (obtained again under very general and natural conditions) that such instant is (or it was until recently) a mathematical singularity and, therefore, beyond reach of any kind of physical interpretation. This result was quite disappointing but, fortunately, it just affects classical theories and does not take into account quantum corrections which generi‐ cally soften the singularities, or even make them completely disappear. Making the story short, there are new models (Alex Vilenkin and also Andrei Linde have been working on them since over twenty years ago) in which one can sidestep the singularity problem: by combining inflation with quantum fluctuations of the vacuum state of a primordial system in which a spark or miniscule particle---a "twist in matter and space-time" so-called "Hawk‐ ing-Turok instanton"---would be able, at zero-energy cost (as explained already), to ignite inflation which, on its turn, would amplify the negligibly small quantum fluctuations (of Planck's constant magnitude) of the vacuum, giving rise, in this way, to the cosmic fluctua‐ tions (of order 10-5) which we clearly observe on the CMB plot below. This is the most an‐ cient map of the Universe that we have been able to capture until now. It corresponds to when it was some 370,000 years old. Just before that, the Universe was like a very dense and hot soup of quarks, gluons and elementary particles. It was absolutely dark, light being un‐ able to travel in it, since photons, even if continuously created, where destroyed immediate‐ ly, through recombination with the neighboring particles at such high densities. But the Universe was expanding and the temperature went down until it reached a value below the ionization threshold of the lightest of all atoms: that of hydrogen. All of a sudden, hydrogen precipitated at cosmic scale and, in this way, for the very first time in History, the very first light of the first cosmic dawn started to fill out the entire Universe. And this light is still reaching us from the most remote corners of the cosmos, and we can see it in all its bright‐ ness with the very curious eyes of our satellites as COBE, WMAP and PLANCK, which have transformed it into images, each time more and more clear, of the most ancient map of the Universe we now have. Putting all pieces together the so-called standard cosmological mod‐ el, or ΛCDM (Cold Dark Matter with a cosmological constant, Λ) could be constructed and remains unchallenged till now.

to two decades from now (projects LISA, BBO, DECIGO, etc.). In that way we will obtain pictures of a much younger Universe and inflation could be eventually confirmed. But what is a real challenge for present day Physics, at least without involving any form of the an‐ thropic principle (which in its strong version states that the properties of the Universe, the universal constants must be such that they need allow intelligent life to exist, that is, our presence as observers), is to develop a model for the origin and evolution of a single Uni‐ verse like ours. The most advanced, and only feasible, theories will always produce a multi‐ verse, that is, an uncountable collection of universes, of all possible kinds of sizes and properties, one of which, by mere chance, would be the one we happen to live in. But, until no observational proof of the existence of a multiverse is obtained, these theories will yet stay beyond the frontiers of Physics, and rather in the domain of science fiction. It must be acknowledged that these theories have been built up by very competent scientists and that they do not contravene any of the basic laws of nature. But in order to enter its realm, as in the case of the other theories discussed in this Chapter, compelling observational evidence

Cosmological Constant and Dark Energy: Historical Insights

http://dx.doi.org/10.5772/51697

29

As last word, among theorists there is still the much extended idea that the ultimate answer will be found, sooner or later, within string (or M) theory, the so-called "theory of every‐ thing." But a too common mistake at different moments in the History of Science has been the strong belief that one already had on its hands the final theory, that all what was left to do was just polish it a bit, fill up some small holes, and carry out more precise calculations. Errors in the past have been flagrant and were committed by some of the most brilliant sci‐ entists of each generation. The author of this Chapter defends the idea that a new theory will emerge, sooner or later, which will be very different from the ones we now have at dis‐ posal, and which will radically change our vision of the world, as much as General Relativi‐

Consejo Superior de Investigaciones Científicas, ICE/CSIC and IEEC Campus UAB, Facultat

[1] Ptolemy, Claudius. (1515). *University of Vienna: Almagest*, http://www.univie.ac.at/

ty and Quantum Mechanics did one hundred years ago.

Address all correspondence to: elizalde@ieec.uab.es

hwastro/rare/1515\_ptolemae.htm.

must first be found.

**Author details**

Emilio Elizalde\*

de Ciències, Spain

**References**

**Figure 19.** Stephen Hawking Roger Penrose Alex Vilenkin

**Figure 20.** CMB Seven Year Microwave Sky, NASA/WMAP Science Team.

In order to proceed further into the observation of the origin --- eventually until the very ori‐ gin of time --- we will need much better eyes. To start, those capable of processing the infor‐ mation hidden in the primordial gravitational waves, what we expect to be able to do in one to two decades from now (projects LISA, BBO, DECIGO, etc.). In that way we will obtain pictures of a much younger Universe and inflation could be eventually confirmed. But what is a real challenge for present day Physics, at least without involving any form of the an‐ thropic principle (which in its strong version states that the properties of the Universe, the universal constants must be such that they need allow intelligent life to exist, that is, our presence as observers), is to develop a model for the origin and evolution of a single Uni‐ verse like ours. The most advanced, and only feasible, theories will always produce a multi‐ verse, that is, an uncountable collection of universes, of all possible kinds of sizes and properties, one of which, by mere chance, would be the one we happen to live in. But, until no observational proof of the existence of a multiverse is obtained, these theories will yet stay beyond the frontiers of Physics, and rather in the domain of science fiction. It must be acknowledged that these theories have been built up by very competent scientists and that they do not contravene any of the basic laws of nature. But in order to enter its realm, as in the case of the other theories discussed in this Chapter, compelling observational evidence must first be found.

As last word, among theorists there is still the much extended idea that the ultimate answer will be found, sooner or later, within string (or M) theory, the so-called "theory of every‐ thing." But a too common mistake at different moments in the History of Science has been the strong belief that one already had on its hands the final theory, that all what was left to do was just polish it a bit, fill up some small holes, and carry out more precise calculations. Errors in the past have been flagrant and were committed by some of the most brilliant sci‐ entists of each generation. The author of this Chapter defends the idea that a new theory will emerge, sooner or later, which will be very different from the ones we now have at dis‐ posal, and which will radically change our vision of the world, as much as General Relativi‐ ty and Quantum Mechanics did one hundred years ago.

## **Author details**

hot soup of quarks, gluons and elementary particles. It was absolutely dark, light being un‐ able to travel in it, since photons, even if continuously created, where destroyed immediate‐ ly, through recombination with the neighboring particles at such high densities. But the Universe was expanding and the temperature went down until it reached a value below the ionization threshold of the lightest of all atoms: that of hydrogen. All of a sudden, hydrogen precipitated at cosmic scale and, in this way, for the very first time in History, the very first light of the first cosmic dawn started to fill out the entire Universe. And this light is still reaching us from the most remote corners of the cosmos, and we can see it in all its bright‐ ness with the very curious eyes of our satellites as COBE, WMAP and PLANCK, which have transformed it into images, each time more and more clear, of the most ancient map of the Universe we now have. Putting all pieces together the so-called standard cosmological mod‐ el, or ΛCDM (Cold Dark Matter with a cosmological constant, Λ) could be constructed and

remains unchallenged till now.

28 Open Questions in Cosmology

**Figure 19.** Stephen Hawking Roger Penrose Alex Vilenkin

**Figure 20.** CMB Seven Year Microwave Sky, NASA/WMAP Science Team.

In order to proceed further into the observation of the origin --- eventually until the very ori‐ gin of time --- we will need much better eyes. To start, those capable of processing the infor‐ mation hidden in the primordial gravitational waves, what we expect to be able to do in one

Emilio Elizalde\*

Address all correspondence to: elizalde@ieec.uab.es

Consejo Superior de Investigaciones Científicas, ICE/CSIC and IEEC Campus UAB, Facultat de Ciències, Spain

## **References**

[1] Ptolemy, Claudius. (1515). *University of Vienna: Almagest*, http://www.univie.ac.at/ hwastro/rare/1515\_ptolemae.htm.

[2] Halley, Edmund. (1714-16). An account of several nebulae or lucid spots like clouds, lately discovered among the fixt stars by help of the telescope. *Philosophical Transac‐ tions*, 39, 390-392.

**Chapter 2**

**Provisional chapter**

**Exploring the Components of the Universe Through**

**Exploring the Components of the Universe Through**

Our current cosmological model, backed by a large body of evidence from a variety of different cosmological probes (for example, see [1, 2]), describes a Universe comprised of around 5% normal baryonic matter, 22% cold dark matter and 73% dark energy. While many cosmologists accept this so-called concordance cosmology – the Λ*CDM* cosmological model – as accurate, very little is known about the nature and properties of these dark components

Studies of the cosmic microwave background (CMB), combined with other observational evidence of big bang nucleosynthesis indicate that dark matter is non-baryonic. This supports measurements on galaxy and cluster scales, which found evidence of a large proportion of dark matter. This dark matter appears to be cold and collisionless, apparent

While dark matter is largely responsible for the growth of the largest structures in the Universe, dark energy – dominant at late times – appears to have a negative pressure, and to be responsible for an accelerated expansion of the Universe [3]. It is usually parameterised by its equation of state parameter *w* = *p*/*ρ*, where *p* is the pressure associated with the dark energy and *ρ* is its energy density. An equation of state parameter of *w* = −1 would indicate a cosmological constant, consistent with general relativistic theory, but deviations from this value would suggest a rather more exotic dark energy, and might perhaps imply a

Gravitational lensing – the deflection of light rays by massive objects due to gravitational effects, which gives rise to distortions in images of galaxies – is an ideal probe of the dark universe, as it can probe the evolution of the dark matter power spectrum in an unbiased way, and offers complementary constraints to those obtained from the CMB and other probes

and reproduction in any medium, provided the original work is properly cited.

©2012 Leonard et al., licensee InTech. This is an open access chapter 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. © 2012 Leonard et al.; licensee InTech. This is an open access article 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.

© 2012 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,

**Higher-Order Weak Lensing Statistics**

**Higher-Order Weak Lensing Statistics**

Adrienne Leonard, Jean-Luc Starck,

Adrienne Leonard, Jean-Luc Starck, Sandrine Pires and François-Xavier Dupé

http://dx.doi.org/10.5772/51871

only through its gravitational effects.

modification to our current theory of gravity.

of large-scale structure.

**1. Introduction**

of the Universe.

Sandrine Pires and François-Xavier Dupé

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter


**Provisional chapter**

## **Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics**

Adrienne Leonard, Jean-Luc Starck, Sandrine Pires and François-Xavier Dupé Sandrine Pires and François-Xavier Dupé Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51871

Adrienne Leonard, Jean-Luc Starck,

## **1. Introduction**

[2] Halley, Edmund. (1714-16). An account of several nebulae or lucid spots like clouds, lately discovered among the fixt stars by help of the telescope. *Philosophical Transac‐*

[3] Messier, Charles. (1781). Catalogue des Nébuleuses & des amas d'Étoiles. *Connois‐*

[4] Herschel, William, & Herschel, Caroline. (1786). Catalogue of One Thousand New Nebulae and Clusters of Stars. *Philosophical transactions*, Vol.Royal Society GB.

[6] Bartusiak, Marcia. ,(2010). The Day We Found the Universe. (Random House Digital)

[7] Einstein, Albert. (1915). Die Feldgleichungen der Gravitation. *Sitzungsberichte der Pre‐ ussischen Akademie der Wissenschaften zu Berlin*, 844-847, http://nausikaa2.mpiwg-ber‐

[8] Wilson, Robert W. (1978). The Cosmic Microwave Background Radiation. *Nobel Lec‐ ture*, http://nobelprize.org/nobel\_prizes/physics/laureates/1978/wilson-lecture.pdf.

[9] Dicke, R. H., Peebles, P. J. E., Roll, P. G., & Wilkinson, D. T. (1965). *Ap. J.*, 142, 4-14.

[14] Gamow, G. (1948). The Origin of Elements and the Separation of Galaxies. The evolu‐ tion of the universe. On the Relative Abundance of the Elements. One, Two, Three...Infinity. *Physical Review*, 74, 505, *Nature*, 162, 680, AlpherR.A.HermanR., 1948,

*Physical Review*, 74, 1577, GamowG, 1974, *Viking Press*, revised 1961; Dover P.

[15] Guth, Alan H. (1981). Inflationary universe: A possible solution to the horizon and

[16] Rubin, V., Thonnard, N., & Ford, W. K. , Jr. (1980). Rotational Properties of 21 Sc Gal‐ axies with a Large Range of Luminosities and Radii from NGC 4605 (R=4kpc) to

[17] de Lapparent, V., Geller, M. J., & Huchra, J. P. (1986). A slice of the universe. *Astro‐*

[18] Weinberg, Steven. (1989). The cosmological constant problem. Dreams of a Final Theory: the search for the fundamental laws of nature. The First Three Minutes. *Rev.*

[19] Elizalde, E., Nojiri, S., & Odintsov, S. D. (2004). Late-time cosmology in (phantom) scalar-tensor theory: dark energy and the cosmic speed-up. *Phys. Rev.*, D70, 04359.

*Mod. Phys.*, 61, 1-23, *Vintage Press*, 1993, *Basic Books, 1977; updated 1993*.

*sance des Temps*, 1784, 227-267, [Bibcode: 1781CdT.1784.227M].

lin.mpg.de/cgi-bin/toc/toc.x.cgi?dir=6E3MAXK4&step=thumb.

[11] Roll, P. G., & Wilkinson, D. T. (1966). *Physical Review Letters*, 16, 405.

[10] Penzias, A. A., & Wilson, R. W. (1965). *Ap. J.*, 142, 420.

[12] Howell, T. F., & Shakeshaft, J. R. (1966). *Nature*, 210, 138.

[13] Penzias, A. A., & Wilson, R. W. (1967). *Astron. J.*, 72, 315.

UGC 2885 (R=122kpc). *Astrophysical Journal*, 238, 471.

*physical Journal, Letters to the Editor*, 302, L1-L5, March 1.

flatness problems. *Phys. Rev. D*, 23, 347.

[5] Hubble, Edwin P. (1937). *The Observational Approach to Cosmology (Oxford)*.

*tions*, 39, 390-392.

30 Open Questions in Cosmology

Our current cosmological model, backed by a large body of evidence from a variety of different cosmological probes (for example, see [1, 2]), describes a Universe comprised of around 5% normal baryonic matter, 22% cold dark matter and 73% dark energy. While many cosmologists accept this so-called concordance cosmology – the Λ*CDM* cosmological model – as accurate, very little is known about the nature and properties of these dark components of the Universe.

Studies of the cosmic microwave background (CMB), combined with other observational evidence of big bang nucleosynthesis indicate that dark matter is non-baryonic. This supports measurements on galaxy and cluster scales, which found evidence of a large proportion of dark matter. This dark matter appears to be cold and collisionless, apparent only through its gravitational effects.

While dark matter is largely responsible for the growth of the largest structures in the Universe, dark energy – dominant at late times – appears to have a negative pressure, and to be responsible for an accelerated expansion of the Universe [3]. It is usually parameterised by its equation of state parameter *w* = *p*/*ρ*, where *p* is the pressure associated with the dark energy and *ρ* is its energy density. An equation of state parameter of *w* = −1 would indicate a cosmological constant, consistent with general relativistic theory, but deviations from this value would suggest a rather more exotic dark energy, and might perhaps imply a modification to our current theory of gravity.

Gravitational lensing – the deflection of light rays by massive objects due to gravitational effects, which gives rise to distortions in images of galaxies – is an ideal probe of the dark universe, as it can probe the evolution of the dark matter power spectrum in an unbiased way, and offers complementary constraints to those obtained from the CMB and other probes of large-scale structure.

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. © 2012 Leonard et al.; licensee InTech. This is an open access article 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. © 2012 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.

©2012 Leonard et al., licensee InTech. This is an open access chapter distributed under the terms of the

Moreover, gravitational lensing probes cosmological perturbations on smaller angular scales than CMB studies, and is thus sensitive to the non-Gaussianity induced by the late-time non-linear evolution of structures such as clusters of galaxies, as well as any primordial non-Gaussianity arising, for example, due to inflation very early in the evolution of the Universe.

originating at angle β is deflected such that it appears to the observer to originate at an

where *ψ* is the two-dimensional deflection potential associated with the lens and *α* is the

For extended sources, photons from different angular positions in the source plane are deflected differently, giving rise to a distortion in the observed galaxy image, which is

=


described – to first order – by the Jacobian of the lens equation (1):

*<sup>∂</sup>*<sup>θ</sup> <sup>=</sup> *<sup>δ</sup>i*,*<sup>j</sup>* <sup>−</sup> *<sup>∂</sup>*2*<sup>ψ</sup>*

*<sup>γ</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> 2 � *∂*2 <sup>1</sup>*<sup>ψ</sup>* <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> 2*ψ* �

<sup>D</sup>(θ) <sup>≡</sup> *<sup>θ</sup>*<sup>2</sup>

<sup>0</sup>Ω*<sup>M</sup>* 2*c*<sup>2</sup>

*fK*(*w*) =

 

� *<sup>w</sup>* 0

*<sup>γ</sup>*(θ) = <sup>1</sup>

*π* � *<sup>d</sup>*2θ′

<sup>2</sup> <sup>−</sup> *<sup>θ</sup>*<sup>2</sup>

<sup>1</sup> <sup>−</sup> <sup>2</sup>*iθ*1*θ*<sup>2</sup>

The convergence, in turn, can be related to the 3D density contrast *δ*(r) ≡ *ρ*(r)/*ρ* − 1 by

where *ρ* is the mean density of the Universe, *H*<sup>0</sup> is the Hubble parameter, Ω*<sup>M</sup>* is the matter density parameter, *c* is the speed of light, *a*(*w*) is the scale parameter evaluated at comoving

(−*K*)<sup>−</sup>1/2sinh([−*K*]

gives the comoving angular diameter distance as a function of the comoving distance and

*dw*′ *fK*(*w*′

*∂θi∂θj*

lensed image. The components of the shear are related to the potential via:

where *<sup>∂</sup><sup>i</sup>* = *<sup>∂</sup>*/*∂θi*, and the shear is related to the convergence through the relation:

A = *<sup>∂</sup>*<sup>β</sup>

where the convolution kernel D is given by

*<sup>κ</sup>*(θ, *<sup>w</sup>*) = <sup>3</sup>*H*<sup>2</sup>

β = θ − α(θ) = θ − ∇*ψ*(θ), (1)

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

� <sup>1</sup> − *<sup>κ</sup>* − *<sup>γ</sup>*<sup>1</sup> −*γ*<sup>2</sup>

<sup>2</sup> <sup>∇</sup>2*<sup>ψ</sup>* is the convergence, or dimensionless surface density, and *<sup>γ</sup>* <sup>=</sup> *<sup>γ</sup>*<sup>1</sup> <sup>+</sup> *<sup>i</sup>γ*<sup>2</sup> <sup>=</sup>

D(<sup>θ</sup> − <sup>θ</sup>′

<sup>|</sup>θ|<sup>4</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup>

)*fK*(*<sup>w</sup>* <sup>−</sup> *<sup>w</sup>*′

*fK*(*w*)

*<sup>K</sup>*−1/2 sin(*K*1/2*w*), *<sup>K</sup>* > <sup>0</sup> *w*, *K* = 0

)

1/2*w*) *K* < 0

*<sup>δ</sup>*[ *fK*(*w*′

)*κ*(θ′

−*γ*<sup>2</sup> <sup>1</sup> − *<sup>κ</sup>* + *<sup>γ</sup>*<sup>1</sup>

�

, *<sup>γ</sup>*<sup>2</sup> = *<sup>∂</sup>*1*∂*2*<sup>ψ</sup>* , (3)

) , (4)

(*θ*<sup>1</sup> <sup>−</sup> *<sup>i</sup>θ*2)<sup>2</sup> . (5)

(6)

, (7)

)θ, *<sup>w</sup>*′ ]

*<sup>a</sup>*(*w*′) ,

, (2)

http://dx.doi.org/10.5772/51871

33

angular position θ, where

deflection angle.

where *κ* = <sup>1</sup>

distance *w*, and

the curvature, *K*, of the Universe.

Traditionally, constraints from gravitational lensing are obtained by considering two-point statistics of the lensing shear field, which encodes the small elliptical distortions applied to the images of galaxies as a result of the gravitational potential field of structures along the light's path. Such two-point statistics are only weakly sensitive to the dark energy density parameter ΩΛ 1, and depend on a degenerate combination of the amplitude of the matter power spectrum *σ*<sup>8</sup> and the matter density parameter Ω*M*. In addition, two-point statistics probe only the Gaussian part of the shear field, therefore in considering such statistics alone, information about nonlinear structure evolution and primordial non-Gaussianity is lost.

Similarly, attempts to reconstruct a map of the two-dimensional projected surface mass density (the convergence *κ*) and three dimensional density field have often involved the use of Gaussian priors to constrain the reconstruction (for example, see [4]), thus again having limited application in studies of non-Gaussianity.

In this chapter, we present a review of recently developed gravitational lensing techniques that go beyond the standard two-point statistics, both in the arena of map-making in two and three dimensions and that of higher-order statistics of the shear field or, equivalently, the convergence field *κ*.

The methods we present are all based on the concept of sparse recovery, which has been found to be a very powerful tool in signal processing [5, 6]. Such methods are based on the assumption that a given image or observation can be represented sparsely in an appropriate basis (such as a Fourier or wavelet basis). Sparse priors using a wavelet basis have been used in many areas of signal processing in astronomy; of particular interest in this chapter will be the areas of denoising and in the reconstruction of the 3D density field from lensing measurements, but other applications include deconvolution and inpainting.

There is much information to be gained by considering non-Gaussian statistics and nonlinear signal processing methods, and this is an exciting and active area of research. With new surveys such as the Euclid satellite [7] coming online within the next decade, a wealth of high-quality data will soon be available. These new techniques will therefore prove invaluable in constraining the cosmological model, and allow us to better understand the nature of the primary constituents of the Universe.

## **2. Weak gravitational lensing**

We begin with a brief overview of weak lensing theory. For a more complete description and discussion of the subject, see [8, 9]

## **2.1. Weak lensing theory**

The basic idea underlying the theory of gravitational lensing is that massive objects distort the spacetime around them, and thus bend the path of light in their vicinity. A light ray

<sup>1</sup> The energy density parameters are defined by Ω*<sup>i</sup>* = *ρi*/*ρ<sup>c</sup>* , where *ρ<sup>c</sup>* is the critical density of the Universe

originating at angle β is deflected such that it appears to the observer to originate at an angular position θ, where

$$
\beta = \theta - \alpha(\theta) = \theta - \nabla \psi(\theta),
\tag{1}
$$

where *ψ* is the two-dimensional deflection potential associated with the lens and *α* is the deflection angle.

For extended sources, photons from different angular positions in the source plane are deflected differently, giving rise to a distortion in the observed galaxy image, which is described – to first order – by the Jacobian of the lens equation (1):

$$\mathcal{A} = \frac{\partial \mathcal{B}}{\partial \boldsymbol{\theta}} = \delta\_{i,j} - \frac{\partial^2 \boldsymbol{\psi}}{\partial \theta\_i \partial \theta\_j} = \begin{pmatrix} 1 - \kappa - \gamma\_1 & -\gamma\_2 \\ -\gamma\_2 & 1 - \kappa + \gamma\_1 \end{pmatrix} \tag{2}$$

where *κ* = <sup>1</sup> <sup>2</sup> <sup>∇</sup>2*<sup>ψ</sup>* is the convergence, or dimensionless surface density, and *<sup>γ</sup>* <sup>=</sup> *<sup>γ</sup>*<sup>1</sup> <sup>+</sup> *<sup>i</sup>γ*<sup>2</sup> <sup>=</sup> |*γ*|*e*2*i<sup>φ</sup>* is the complex shear, which gives rise to an anisotropic elliptical distortion in the lensed image. The components of the shear are related to the potential via:

$$
\gamma\_1 = \frac{1}{2} \left( \partial\_1^2 \psi - \partial\_2^2 \psi \right) \,, \quad \gamma\_2 = \partial\_1 \partial\_2 \psi \,, \tag{3}
$$

where *<sup>∂</sup><sup>i</sup>* = *<sup>∂</sup>*/*∂θi*, and the shear is related to the convergence through the relation:

$$\gamma(\boldsymbol{\theta}) = \frac{1}{\pi} \int d^2 \boldsymbol{\theta}' \mathcal{D}(\boldsymbol{\theta} - \boldsymbol{\theta}') \kappa(\boldsymbol{\theta}') \,. \tag{4}$$

where the convolution kernel D is given by

2 Open Questions in Cosmology

Universe.

parameter ΩΛ

the convergence field *κ*.

limited application in studies of non-Gaussianity.

nature of the primary constituents of the Universe.

**2. Weak gravitational lensing**

discussion of the subject, see [8, 9]

**2.1. Weak lensing theory**

Moreover, gravitational lensing probes cosmological perturbations on smaller angular scales than CMB studies, and is thus sensitive to the non-Gaussianity induced by the late-time non-linear evolution of structures such as clusters of galaxies, as well as any primordial non-Gaussianity arising, for example, due to inflation very early in the evolution of the

Traditionally, constraints from gravitational lensing are obtained by considering two-point statistics of the lensing shear field, which encodes the small elliptical distortions applied to the images of galaxies as a result of the gravitational potential field of structures along the light's path. Such two-point statistics are only weakly sensitive to the dark energy density

power spectrum *σ*<sup>8</sup> and the matter density parameter Ω*M*. In addition, two-point statistics probe only the Gaussian part of the shear field, therefore in considering such statistics alone, information about nonlinear structure evolution and primordial non-Gaussianity is lost. Similarly, attempts to reconstruct a map of the two-dimensional projected surface mass density (the convergence *κ*) and three dimensional density field have often involved the use of Gaussian priors to constrain the reconstruction (for example, see [4]), thus again having

In this chapter, we present a review of recently developed gravitational lensing techniques that go beyond the standard two-point statistics, both in the arena of map-making in two and three dimensions and that of higher-order statistics of the shear field or, equivalently,

The methods we present are all based on the concept of sparse recovery, which has been found to be a very powerful tool in signal processing [5, 6]. Such methods are based on the assumption that a given image or observation can be represented sparsely in an appropriate basis (such as a Fourier or wavelet basis). Sparse priors using a wavelet basis have been used in many areas of signal processing in astronomy; of particular interest in this chapter will be the areas of denoising and in the reconstruction of the 3D density field from lensing

There is much information to be gained by considering non-Gaussian statistics and nonlinear signal processing methods, and this is an exciting and active area of research. With new surveys such as the Euclid satellite [7] coming online within the next decade, a wealth of high-quality data will soon be available. These new techniques will therefore prove invaluable in constraining the cosmological model, and allow us to better understand the

We begin with a brief overview of weak lensing theory. For a more complete description and

The basic idea underlying the theory of gravitational lensing is that massive objects distort the spacetime around them, and thus bend the path of light in their vicinity. A light ray

<sup>1</sup> The energy density parameters are defined by Ω*<sup>i</sup>* = *ρi*/*ρ<sup>c</sup>* , where *ρ<sup>c</sup>* is the critical density of the Universe

measurements, but other applications include deconvolution and inpainting.

1, and depend on a degenerate combination of the amplitude of the matter

$$\mathcal{D}(\boldsymbol{\theta}) \equiv \frac{\theta\_2^2 - \theta\_1^2 - 2\dot{\imath}\theta\_1\theta\_2}{|\boldsymbol{\theta}|^4} = -\frac{1}{(\theta\_1 - i\theta\_2)^2} \,. \tag{5}$$

The convergence, in turn, can be related to the 3D density contrast *δ*(r) ≡ *ρ*(r)/*ρ* − 1 by

$$\kappa(\theta, w) = \frac{3H\_0^2 \Omega\_M}{2c^2} \int\_0^w dw' \frac{f\_\mathcal{K}(w') f\_\mathcal{K}(w - w')}{f\_\mathcal{K}(w)} \frac{\delta[f\_\mathcal{K}(w')\theta, w']}{a(w')},\tag{6}$$

where *ρ* is the mean density of the Universe, *H*<sup>0</sup> is the Hubble parameter, Ω*<sup>M</sup>* is the matter density parameter, *c* is the speed of light, *a*(*w*) is the scale parameter evaluated at comoving distance *w*, and

$$f\_K(w) = \begin{cases} K^{-1/2} \sin(K^{1/2} w), & K > 0 \\ w, & K = 0 \\ (-K)^{-1/2} \sinh([-K]^{1/2} w) & K < 0 \end{cases},\tag{7}$$

gives the comoving angular diameter distance as a function of the comoving distance and the curvature, *K*, of the Universe.

## **2.2. Two-point statistics**

The typical gravitational shear applied to a galaxy as a result of lensing by large-scale structure in the Universe – so-called cosmic shear – is of order |*γ*| ∼ 0.01. However, galaxies are intrinsically elliptical in shape, with a typical ellipticity of order |*ε*| ∼ 0.2 − 0.3, therefore the gravitational lensing effect can only be measured statistically; under the assumption that galaxy shapes are intrinsically uncorrelated, the mean ellipticity computed over a large number of sources will yield the gravitational shear: �*ε*� ≃ *γ*.

The most common method for constraining cosmological parameters in weak lensing studies is to use two-point statistics of the shear field. The power spectrum of the shear or convergence, *<sup>P</sup><sup>κ</sup>* (ℓ) = *<sup>P</sup>γ*(ℓ), can be related directly to the 3D matter power spectrum of density fluctuations *δ* by:

$$P\_{\mathbb{K}}(\ell) = \frac{9H\_0^4 \Omega\_M^2}{4c^4} \int dw \, \frac{W^2(w)}{a^2(w)} P\_\delta \left(\frac{\ell}{f\_K(w)}, w\right) \,\,\,\,\tag{8}$$

the shear data [13]. In both cases, a large degeneracy is seen between the matter density parameter Ω*<sup>M</sup>* and *σ*8, the amplitude of the matter power spectrum. The CFHT results also show constraints on the dark energy equation of state parameter *w*. Here, again, a

> 3d 2d

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35

0.0 0.2 0.4 0.6 0.8 1.0 0.2

(c) Joint constraints on Ω*<sup>M</sup>* and *σ*<sup>8</sup> from a 2D and 3D analysis of the Cosmos

Ω<sup>m</sup>

0.4 0.6 0.8 1.0 1.2 1.4

survey data [13].

*d*2ϑ *κ*(ϑ)*U*(|ϑ|) . (11)

σ8

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

(b) Joint constraints on Ω*<sup>M</sup>* and *w* from a 2D analysis of the CFHT deep

**Figure 1.** Cosmological parameter constraints from recent weak lensing surveys. Contours are plotted at the 68% (cyan), 95% (blue) and 99.9% (green) confidence levels for the CFHT data, and at the 68% (inner) and 95% (outer) confidence levels for the

While it is clear from the figures that two-point shear statistics hold a wealth of information, it is also evident that these statistics depend on degenerate combinations of the cosmological model parameters. In order to break these degeneracies, and thus more tightly constrain our cosmological model, we must therefore consider higher-order statistics of the shear or convergence. Such higher-order statistics will enable us to capture information about the non-Gaussian part of the lensing spectrum due to a combination of primordial non-Gaussianity and late-time nonlinear evolution of structures, which two-point statistics

The shear *γ* is a spin-2 field, therefore while the two-point correlations of the shear field can be reduced to a scalar quantity for parity reasons, this is not the case for higher-order moments of the shear field [14–16]. For this reason, it is useful to compute a scalar quantity

For this purpose, the aperture mass statistic *Map* [17, 18] is widely-used. The *Map* statistic is defined as the convolution of the convergence *κ* with a radially symmetrical filter function

survey [12].

degeneracy between parameters is seen.

(a) Joint constraints on Ω*<sup>M</sup>* and *σ*<sup>8</sup> from a 2D analysis of the combined CFHT wide and deep surveys [12].

Cosmos data.

are unable to probe.

*U*(|ϑ|) of width *θ*:

**3. Scalar fields associated with the shear**

**3.1. The aperture mass statistic**

from the shear field before computing higher-order statistics.

*Map*(θ) =

where

$$\mathcal{W}(w) = \int\_{w}^{\infty} dw' \frac{f\_K(w'-w)}{f\_k(w')} p(z) \left[\frac{dz}{dw'}\right] \,\,\,\,\,\tag{9}$$

is a weighting function integrated over the probability distribution of sources *p*(*z*) in the sample as a function of redshift *z*.

When working with real data, it is often more convenient to consider statistics computed in real space, namely the shear correlation functions, which are defined in terms of the convergence power spectrum as [10, 11]:

$$
\zeta\_{\pm}(\theta) = \langle \gamma\_{\ell}\gamma\_{\ell} \rangle \pm \langle \gamma\_{\times}\gamma\_{\times} \rangle = \int\_{0}^{\infty} \frac{d\ell}{2\pi} I\_{0\mathcal{A}}(\ell\theta) P\_{\mathbf{x}}(\ell) \, , \tag{10}
$$

where *Jn* is an *n*-th order Bessel function of the first kind, and *γt* and *γ*× are the tangential and cross shear components, respectively, which are defined relative to the vector connecting the two galaxies separated by angular distance *θ*.

#### **2.3. Cosmological constraints**

The measured two-point statistics described above can, by virtue of their relationship to the power spectrum of the underlying matter density fluctuations, be used to place constraints on cosmological parameters. These constraints may be improved if information about the distances to source galaxies is used to bin the galaxies into redshift slices, and compute the correlation functions tomographically.

Figure 1 shows cosmological parameter constraints obtained using two state-of-the-art telescopes: the Canada-France-Hawaii Telescope (CFHT), and the Hubble Space Telescope (specifically, the Cosmos survey). While the CFHT results were obtained using a 2D analysis [12], the Cosmos results show constraints obtained by both a 2D and a 3D analysis of the shear data [13]. In both cases, a large degeneracy is seen between the matter density parameter Ω*<sup>M</sup>* and *σ*8, the amplitude of the matter power spectrum. The CFHT results also show constraints on the dark energy equation of state parameter *w*. Here, again, a degeneracy between parameters is seen.

(a) Joint constraints on Ω*<sup>M</sup>* and *σ*<sup>8</sup> from a 2D analysis of the combined CFHT wide and deep surveys [12].

4 Open Questions in Cosmology

**2.2. Two-point statistics**

density fluctuations *δ* by:

sample as a function of redshift *z*.

**2.3. Cosmological constraints**

correlation functions tomographically.

convergence power spectrum as [10, 11]:

the two galaxies separated by angular distance *θ*.

where

The typical gravitational shear applied to a galaxy as a result of lensing by large-scale structure in the Universe – so-called cosmic shear – is of order |*γ*| ∼ 0.01. However, galaxies are intrinsically elliptical in shape, with a typical ellipticity of order |*ε*| ∼ 0.2 − 0.3, therefore the gravitational lensing effect can only be measured statistically; under the assumption that galaxy shapes are intrinsically uncorrelated, the mean ellipticity computed over a large

The most common method for constraining cosmological parameters in weak lensing studies is to use two-point statistics of the shear field. The power spectrum of the shear or convergence, *<sup>P</sup><sup>κ</sup>* (ℓ) = *<sup>P</sup>γ*(ℓ), can be related directly to the 3D matter power spectrum of

> *dw <sup>W</sup>*2(*w*) *<sup>a</sup>*2(*w*) *<sup>P</sup><sup>δ</sup>*

*dw*′ *fK*(*w*′ <sup>−</sup> *<sup>w</sup>*)

is a weighting function integrated over the probability distribution of sources *p*(*z*) in the

When working with real data, it is often more convenient to consider statistics computed in real space, namely the shear correlation functions, which are defined in terms of the

where *Jn* is an *n*-th order Bessel function of the first kind, and *γt* and *γ*× are the tangential and cross shear components, respectively, which are defined relative to the vector connecting

The measured two-point statistics described above can, by virtue of their relationship to the power spectrum of the underlying matter density fluctuations, be used to place constraints on cosmological parameters. These constraints may be improved if information about the distances to source galaxies is used to bin the galaxies into redshift slices, and compute the

Figure 1 shows cosmological parameter constraints obtained using two state-of-the-art telescopes: the Canada-France-Hawaii Telescope (CFHT), and the Hubble Space Telescope (specifically, the Cosmos survey). While the CFHT results were obtained using a 2D analysis [12], the Cosmos results show constraints obtained by both a 2D and a 3D analysis of

*fk*(*w*′) *<sup>p</sup>*(*z*)

 <sup>∞</sup> 0

*d*ℓ ℓ

 ℓ *fK*(*w*) , *w* 

> *dz dw*′

, (8)

, (9)

<sup>2</sup>*<sup>π</sup> <sup>J</sup>*0,4(ℓ*θ*)*P<sup>κ</sup>* (ℓ) , (10)

number of sources will yield the gravitational shear: �*ε*� ≃ *γ*.

*<sup>P</sup><sup>κ</sup>* (ℓ) = <sup>9</sup>*H*<sup>4</sup>

*W*(*w*) =

0Ω<sup>2</sup> *M* 4*c*<sup>4</sup>

> <sup>∞</sup> *w*

*ξ*±(*θ*) = �*γtγt*�±�*γ*×*γ*� =

(b) Joint constraints on Ω*<sup>M</sup>* and *w* from a 2D analysis of the CFHT deep survey [12].

(c) Joint constraints on Ω*<sup>M</sup>* and *σ*<sup>8</sup> from a 2D and 3D analysis of the Cosmos survey data [13].

**Figure 1.** Cosmological parameter constraints from recent weak lensing surveys. Contours are plotted at the 68% (cyan), 95% (blue) and 99.9% (green) confidence levels for the CFHT data, and at the 68% (inner) and 95% (outer) confidence levels for the Cosmos data.

While it is clear from the figures that two-point shear statistics hold a wealth of information, it is also evident that these statistics depend on degenerate combinations of the cosmological model parameters. In order to break these degeneracies, and thus more tightly constrain our cosmological model, we must therefore consider higher-order statistics of the shear or convergence. Such higher-order statistics will enable us to capture information about the non-Gaussian part of the lensing spectrum due to a combination of primordial non-Gaussianity and late-time nonlinear evolution of structures, which two-point statistics are unable to probe.

## **3. Scalar fields associated with the shear**

The shear *γ* is a spin-2 field, therefore while the two-point correlations of the shear field can be reduced to a scalar quantity for parity reasons, this is not the case for higher-order moments of the shear field [14–16]. For this reason, it is useful to compute a scalar quantity from the shear field before computing higher-order statistics.

#### **3.1. The aperture mass statistic**

For this purpose, the aperture mass statistic *Map* [17, 18] is widely-used. The *Map* statistic is defined as the convolution of the convergence *κ* with a radially symmetrical filter function *U*(|ϑ|) of width *θ*:

$$M\_{ap}(\theta) = \int d^2 \theta \,\kappa(\vartheta) \mathcal{U}(|\vartheta|) \,. \tag{11}$$

By considering the relationship between the shear, *γ*, and the convergence, one can reformulate equation (11) in terms of the measured shear as

$$M\_{ap}(\boldsymbol{\theta}) = \int d^2 \boldsymbol{\vartheta} \ \gamma\_t(\boldsymbol{\vartheta}) Q(|\boldsymbol{\vartheta}|) \ . \tag{12}$$

than the typical scale of the noise (which is typically dominant on pixel scales, depending on the binning chosen for the shear data). Indeed, optimal signal-to-noise is obtained when the filter is chosen such that its angular scale matches that of the structures we aim to detect,

The wavelet transform is a multiscale transform, where the wavelet coefficients of an image are computed at each position in the image at various different scales simultaneously. In one dimension, the wavelet coefficient of a function *f*(*x*), evaluated at position *b* and scale *a* is

*<sup>f</sup>*(*x*)*ψ*<sup>∗</sup>

where *ψ*(*x*) is the analysing wavelet. The analysis is analogous in 2 dimensions, with

By definition, wavelets are compensated functions; i.e. the wavelet function *ψ*(*x*) is

According to the definition in equation (16), the continuous wavelet transform of an image is therefore nothing more than the convolution of that image with compensated filter functions of various characteristic scales. If the image *f*(*x*, *y*) is taken to be the convergence *κ*(*x*, *y*), then for an appropriate choice of (radially-symmetric, local) wavelet, the wavelet transform is formally identical to the aperture mass statistic at the corresponding scales, the only

In practice in application, we use the starlet transform algorithm [6, 23–25], which simultaneously computes the wavelet transform on dyadic scales corresponding to 2*<sup>j</sup>* pixels. This algorithm decomposes the convergence map of size *N* × *N* into *J* = *j*max + 1 sub-arrays

where *j*max represents the number of wavelet bands (or, equivalently, aperture mass maps) considered, *cJ* represents a smooth (or continuum) version of the original image *κ*, and *wj* represents the input map filtered at scale 2*<sup>j</sup>* (i.e. the aperture mass map at *θ* = 2*<sup>j</sup>* pixels).

Using the wavelet formalism to derive the aperture mass statistic presents a number of advantages. Many families of wavelet functions have been studied in the statistical literature, and all these wavelet functions could be applied to the aperture mass statistic. This allows us to tune our filter function to optimise the signal-to-noise in the resulting maps. In addition, for some specific wavelet functions, discrete and very fast algorithms exist, allowing us to

*j*max ∑ *j*=1

*<sup>κ</sup>*(*x*, *<sup>y</sup>*) = *cJ*(*x*, *<sup>y</sup>*) +

**<sup>R</sup>**<sup>1</sup> *ψ*(*x*)*dx* = 0 and hence, by extension

 **R**2  *x* − *b a*

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

*ψ*(*x*, *y*)*dx dy* = 0. (17)

*wj*(*x*, *<sup>y</sup>*) , (18)

*dx* , (16)

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37

and its shape matches closely the expected profile of these structures.

*<sup>W</sup>*(*a*, *<sup>b</sup>*) = <sup>1</sup>

√*a* 

**3.2. The wavelet transform**

defined as [23, 24]:

*ψ*(*x*, *y*) = *ψ*(*x*)*ψ*(*y*).

constrained such that

of size *N* × *N* as follows:

difference being the choice of filter functions.

where *γt*(ϑ) is the tangential component of the shear at position ϑ relative to the centre of the aperture, *Q*(|ϑ|) is a second radially-symmetric function, related to *U*(|ϑ|) by:

$$Q(\theta) \equiv \frac{2}{\theta^2} \int\_0^{\theta} \theta' \mathcal{U}(\theta') d\theta' - \mathcal{U}(\theta) \; , \tag{13}$$

and *U*(*ϑ*) is required to be compensated, i.e.

$$\int\_{0}^{\theta\_{cut}} \theta \,\,\mathrm{U}(\theta) \,\, d\theta = 0 \,\,\mathrm{} \,\mathrm{} \,\tag{14}$$

with *ϑcut* often taken to be the radius of the aperture, *θ*. Furthermore, *Q*(*ϑ*) and *U*(*ϑ*) are required to go to zero smoothly at *ϑcut*. It is also preferable that the power spectrum of *U*(*ϑ*) is local in the frequency domain, and shows no oscillatory behaviour. This ensures that the filter function acts as a band-pass filter, allowing detection of structures at the scale of interest only.

Several authors [19, 20] have advocated a filter function of the form:

$$U(\theta) = \frac{A}{\theta^2} \left( 1 - \frac{b\theta^2}{\theta^2} \right) \exp\left( -\frac{b\theta^2}{\theta^2} \right) \text{ .}$$

$$Q(\theta) = \frac{A}{\theta^2} \frac{b\theta^2}{\theta^2} \exp\left( -\frac{b\theta^2}{\theta^2} \right) \text{ .} \tag{15}$$

where various choices for the constant *b* and the overall normalisation *A* have been used in the literature [20–22]. Of specific interest is the form used in [20] and [22], where *b* = 1/2 is chosen.2 This form is considered to be optimal for higher-order weak lensing statistics, such as the skewness of the aperture mass statistic [19, 22].

Note that the *<sup>Q</sup>*(*ϑ*) filter function described in equation (15) shows a peak at *<sup>ϑ</sup>* <sup>=</sup> <sup>√</sup>2*θ*, and tends to zero as *ϑ* → ∞. In practice, any algorithm used to generate an aperture mass map will need to truncate this filter function at some finite radius. This will involve a trade-off between accuracy and algorithm speed, and truncation may affect the effective width of the filter.

One advantage of this method is that the aperture mass statistic can be computed directly from the shear catalogue. Moreover, as the filter acts as a band-pass filter, it is possible to boost the signal relative to the noise by considering a filter with a scale substantially larger

<sup>2</sup> This form is analogous to the Mexican Hat wavelet function

than the typical scale of the noise (which is typically dominant on pixel scales, depending on the binning chosen for the shear data). Indeed, optimal signal-to-noise is obtained when the filter is chosen such that its angular scale matches that of the structures we aim to detect, and its shape matches closely the expected profile of these structures.

#### **3.2. The wavelet transform**

6 Open Questions in Cosmology

only.

filter.

By considering the relationship between the shear, *γ*, and the convergence, one can

where *γt*(ϑ) is the tangential component of the shear at position ϑ relative to the centre of

with *ϑcut* often taken to be the radius of the aperture, *θ*. Furthermore, *Q*(*ϑ*) and *U*(*ϑ*) are required to go to zero smoothly at *ϑcut*. It is also preferable that the power spectrum of *U*(*ϑ*) is local in the frequency domain, and shows no oscillatory behaviour. This ensures that the filter function acts as a band-pass filter, allowing detection of structures at the scale of interest

> −*bϑ*<sup>2</sup> *θ*2

where various choices for the constant *b* and the overall normalisation *A* have been used in the literature [20–22]. Of specific interest is the form used in [20] and [22], where *b* = 1/2 is chosen.2 This form is considered to be optimal for higher-order weak lensing statistics, such

Note that the *<sup>Q</sup>*(*ϑ*) filter function described in equation (15) shows a peak at *<sup>ϑ</sup>* <sup>=</sup> <sup>√</sup>2*θ*, and tends to zero as *ϑ* → ∞. In practice, any algorithm used to generate an aperture mass map will need to truncate this filter function at some finite radius. This will involve a trade-off between accuracy and algorithm speed, and truncation may affect the effective width of the

One advantage of this method is that the aperture mass statistic can be computed directly from the shear catalogue. Moreover, as the filter acts as a band-pass filter, it is possible to boost the signal relative to the noise by considering a filter with a scale substantially larger

*d*2ϑ *γt*(ϑ)*Q*(|ϑ|) , (12)

)*dϑ*′ <sup>−</sup> *<sup>U</sup>*(*ϑ*) , (13)

, (15)

*ϑ U*(*ϑ*) *dϑ* = 0 , (14)

the aperture, *Q*(|ϑ|) is a second radially-symmetric function, related to *U*(|ϑ|) by:

*ϑ*2 *ϑ* 0 *ϑ*′ *<sup>U</sup>*(*ϑ*′

 *ϑcut* 0

reformulate equation (11) in terms of the measured shear as

*Map*(θ) =

*<sup>Q</sup>*(*ϑ*) <sup>≡</sup> <sup>2</sup>

Several authors [19, 20] have advocated a filter function of the form:

*θ*2 <sup>1</sup> <sup>−</sup> *<sup>b</sup>ϑ*<sup>2</sup> *θ*2 exp −*bϑ*<sup>2</sup> *θ*2 ,

*θ*2 *bϑ*<sup>2</sup> *<sup>θ</sup>*<sup>2</sup> exp

*<sup>U</sup>*(*ϑ*) = *<sup>A</sup>*

*<sup>Q</sup>*(*ϑ*) = *<sup>A</sup>*

as the skewness of the aperture mass statistic [19, 22].

<sup>2</sup> This form is analogous to the Mexican Hat wavelet function

and *U*(*ϑ*) is required to be compensated, i.e.

The wavelet transform is a multiscale transform, where the wavelet coefficients of an image are computed at each position in the image at various different scales simultaneously. In one dimension, the wavelet coefficient of a function *f*(*x*), evaluated at position *b* and scale *a* is defined as [23, 24]:

$$\mathcal{W}(a,b) = \frac{1}{\sqrt{a}} \int f(\mathbf{x}) \psi^\* \left(\frac{\mathbf{x} - b}{a}\right) \, d\mathbf{x} \, \tag{16}$$

where *ψ*(*x*) is the analysing wavelet. The analysis is analogous in 2 dimensions, with *ψ*(*x*, *y*) = *ψ*(*x*)*ψ*(*y*).

By definition, wavelets are compensated functions; i.e. the wavelet function *ψ*(*x*) is constrained such that **<sup>R</sup>**<sup>1</sup> *ψ*(*x*)*dx* = 0 and hence, by extension

$$\int\_{\mathbb{R}^2} \psi(x, y) dx \, dy = 0. \tag{17}$$

According to the definition in equation (16), the continuous wavelet transform of an image is therefore nothing more than the convolution of that image with compensated filter functions of various characteristic scales. If the image *f*(*x*, *y*) is taken to be the convergence *κ*(*x*, *y*), then for an appropriate choice of (radially-symmetric, local) wavelet, the wavelet transform is formally identical to the aperture mass statistic at the corresponding scales, the only difference being the choice of filter functions.

In practice in application, we use the starlet transform algorithm [6, 23–25], which simultaneously computes the wavelet transform on dyadic scales corresponding to 2*<sup>j</sup>* pixels. This algorithm decomposes the convergence map of size *N* × *N* into *J* = *j*max + 1 sub-arrays of size *N* × *N* as follows:

$$\kappa(\mathbf{x}, y) = c\_I(\mathbf{x}, y) + \sum\_{j=1}^{j\_{\text{max}}} w\_j(\mathbf{x}, y) \; \prime \tag{18}$$

where *j*max represents the number of wavelet bands (or, equivalently, aperture mass maps) considered, *cJ* represents a smooth (or continuum) version of the original image *κ*, and *wj* represents the input map filtered at scale 2*<sup>j</sup>* (i.e. the aperture mass map at *θ* = 2*<sup>j</sup>* pixels).

Using the wavelet formalism to derive the aperture mass statistic presents a number of advantages. Many families of wavelet functions have been studied in the statistical literature, and all these wavelet functions could be applied to the aperture mass statistic. This allows us to tune our filter function to optimise the signal-to-noise in the resulting maps. In addition, for some specific wavelet functions, discrete and very fast algorithms exist, allowing us to compute a set of wavelet scales through the use of a filter bank with a very limited number of operations. See [6] for a full review of the different wavelet transform algorithms.

In the starlet transform algorithm, the wavelet *ψ*(*x*, *y*) is separable and can be defined by:

$$
\psi\left(\frac{\chi}{2}, \frac{y}{2}\right) = 4 \left[\varphi(\mathbf{x}, y) - \frac{1}{4}\varphi\left(\frac{\chi}{2}, \frac{y}{2}\right)\right] \tag{19}
$$

(a) *Map* and wavelet filter functions in real space (b) Filter response curves in Fourier space

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

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39

the shear catalogue directly. This avoids the need to generate an aperture mass map, which can be computationally intensive, and furthermore does not require the computation of the

Computing the convergence from the shear measurements can be tricky for a number of reasons. Equation (4) describes a convolution over all 2D space. Given that we aim to invert this equation for an image of a finite size, a direct inversion in Fourier space will give rise to significant edge effects, and a leakage of power into so-called B-modes, which effectively imply a spurious (non-lensing) cross-component to the shear field *γ*, and usually only arise

Several methods have been devised to invert equation (4) whilst minimising these undesirable effects [26–29]. Most recently, the authors in [30] have presented a method based on a wavelet-Helmholtz decomposition, with which they demonstrate a reconstruction error at the few percent level, as compared to an error of ∼ 30% seen with Fourier-based methods. This implies that there is little advantage to the shear catalogue as opposed to the convergence map. Indeed, working on the convergence map and under the wavelet

The first advantage comes directly from the nature of the wavelet transform. The aperture mass filter is a bandpass filter. Because the *Map* reconstruction is only computed for a discrete set of aperture scales, information on intermediate scales may be lost, particularly

**Figure 2.** A comparison of the *Map* and starlet wavelet filters in real and Fourier space.

due to systematic effects in the lensing measurements.

framework may offer some distinct advantages.

convergence, *κ*.

where *ϕ*(*x*, *y*) = *ϕ*(*x*)*ϕ*(*y*) and *ϕ*(*x*) is a scaling function from which the wavelet is generated. In the case of the starlet wavelet, *ϕ*(*x*) is a B3-spline:

$$\varphi(\mathbf{x}) = \frac{1}{12}(|\mathbf{x} - \mathbf{2}|^3 - 4|\mathbf{x} - \mathbf{1}|^3 + 6|\mathbf{x}|^3 - 4|\mathbf{x} + \mathbf{1}|^3 + |\mathbf{x} + \mathbf{2}|^3), \tag{20}$$

which is a compact function that is identically zero for |*x*| > 2.

This wavelet function has a compact support in real space, is well localized in Fourier domain, and the wavelet decomposition of an image can be obtained with a very fast algorithm (see [24] for a full description).

Figure 2 shows the starlet wavelet and aperture mass filters defined above in both real and Fourier space. Notice that the two filters presented have very similar shapes in real space, but different widths. While the starlet filter function goes to zero identically at *ϑ* = 2*θ*, and remains zero beyond this value, the *Map* filter function goes to zero as *ϑ* → ∞, and must therefore be truncated when applied in practice. Clearly at *ϑ* = 5*θ*, the *Map* filters are sufficiently close to zero, so that this is an appropriate truncation radius; however this will impact the computation time of the *Map* statistic.

In Fourier space, we show the response of the *Map* filter for truncation radii of *ϑcut* = *θ* and 5*θ*. Notice that the shape of the response curves in Fourier space for both the *Map* and wavelet filter functions are similar, but that the peak of the response curve for the *Map* filter function shifts when *ϑcut* is varied. This is because the effective scale of the filter function is being changed. Also notice that in the case of *ϑcut* = *θ*, high-frequency oscillations are present in the response curve. This is a direct result of the truncation of the filter function, and will occur whenever such truncation is applied. Indeed, imperceptibly small oscillations are still present in the response curve for the filter truncated at *ϑcut* = 5*θ*.

Such oscillations are not present at all with the starlet filter function, as no truncation is applied to this function whatsoever. This gives the starlet transform the distinct advantage of being directly applicable, without any consideration needed regarding truncation radii and the associated impact on the Fourier-space response of the filter functions.

#### **3.3. Advantages of the wavelet formalism**

The aperture mass formalism tends to be the preferred method for weak lensing studies for a number of reasons. Firstly, the filter functions can be simply expressed analytically, and any associated statistics of the aperture mass can therefore be straightforwardly computed from

(a) *Map* and wavelet filter functions in real space (b) Filter response curves in Fourier space

8 Open Questions in Cosmology

compute a set of wavelet scales through the use of a filter bank with a very limited number

*<sup>ϕ</sup>*(*x*, *<sup>y</sup>*) <sup>−</sup> <sup>1</sup>

where *ϕ*(*x*, *y*) = *ϕ*(*x*)*ϕ*(*y*) and *ϕ*(*x*) is a scaling function from which the wavelet is generated.

This wavelet function has a compact support in real space, is well localized in Fourier domain, and the wavelet decomposition of an image can be obtained with a very fast

Figure 2 shows the starlet wavelet and aperture mass filters defined above in both real and Fourier space. Notice that the two filters presented have very similar shapes in real space, but different widths. While the starlet filter function goes to zero identically at *ϑ* = 2*θ*, and remains zero beyond this value, the *Map* filter function goes to zero as *ϑ* → ∞, and must therefore be truncated when applied in practice. Clearly at *ϑ* = 5*θ*, the *Map* filters are sufficiently close to zero, so that this is an appropriate truncation radius; however this will

In Fourier space, we show the response of the *Map* filter for truncation radii of *ϑcut* = *θ* and 5*θ*. Notice that the shape of the response curves in Fourier space for both the *Map* and wavelet filter functions are similar, but that the peak of the response curve for the *Map* filter function shifts when *ϑcut* is varied. This is because the effective scale of the filter function is being changed. Also notice that in the case of *ϑcut* = *θ*, high-frequency oscillations are present in the response curve. This is a direct result of the truncation of the filter function, and will occur whenever such truncation is applied. Indeed, imperceptibly small oscillations

Such oscillations are not present at all with the starlet filter function, as no truncation is applied to this function whatsoever. This gives the starlet transform the distinct advantage of being directly applicable, without any consideration needed regarding truncation radii

The aperture mass formalism tends to be the preferred method for weak lensing studies for a number of reasons. Firstly, the filter functions can be simply expressed analytically, and any associated statistics of the aperture mass can therefore be straightforwardly computed from

are still present in the response curve for the filter truncated at *ϑcut* = 5*θ*.

and the associated impact on the Fourier-space response of the filter functions.

<sup>3</sup> + 6|*x*|

<sup>3</sup> − 4|*x* − 1|

4 *ϕ x* 2 , *y* 2 

<sup>3</sup> − 4|*x* + 1|

<sup>3</sup> + |*x* + 2|

, (19)

<sup>3</sup>), (20)

In the starlet transform algorithm, the wavelet *ψ*(*x*, *y*) is separable and can be defined by:

of operations. See [6] for a full review of the different wavelet transform algorithms.

*ψ x* 2 , *y* 2 = 4 

In the case of the starlet wavelet, *ϕ*(*x*) is a B3-spline:

<sup>12</sup> (|*<sup>x</sup>* <sup>−</sup> <sup>2</sup><sup>|</sup>

which is a compact function that is identically zero for |*x*| > 2.

*<sup>ϕ</sup>*(*x*) = <sup>1</sup>

algorithm (see [24] for a full description).

impact the computation time of the *Map* statistic.

**3.3. Advantages of the wavelet formalism**

**Figure 2.** A comparison of the *Map* and starlet wavelet filters in real and Fourier space.

the shear catalogue directly. This avoids the need to generate an aperture mass map, which can be computationally intensive, and furthermore does not require the computation of the convergence, *κ*.

Computing the convergence from the shear measurements can be tricky for a number of reasons. Equation (4) describes a convolution over all 2D space. Given that we aim to invert this equation for an image of a finite size, a direct inversion in Fourier space will give rise to significant edge effects, and a leakage of power into so-called B-modes, which effectively imply a spurious (non-lensing) cross-component to the shear field *γ*, and usually only arise due to systematic effects in the lensing measurements.

Several methods have been devised to invert equation (4) whilst minimising these undesirable effects [26–29]. Most recently, the authors in [30] have presented a method based on a wavelet-Helmholtz decomposition, with which they demonstrate a reconstruction error at the few percent level, as compared to an error of ∼ 30% seen with Fourier-based methods. This implies that there is little advantage to the shear catalogue as opposed to the convergence map. Indeed, working on the convergence map and under the wavelet framework may offer some distinct advantages.

The first advantage comes directly from the nature of the wavelet transform. The aperture mass filter is a bandpass filter. Because the *Map* reconstruction is only computed for a discrete set of aperture scales, information on intermediate scales may be lost, particularly at both the small and large frequency extremes. In contrast, the wavelet transform retains information on all scales. The first wavelet scale is effectively a high-pass filter, retaining all the high-frequency information in the image, while the remaining wavelet scales are bandpass filtered as with the *Map*-filtered images. Finally, the wavelet transform retains *cJ*, which encodes the large-scale information in the image, and therefore consists of a smoothed version of the image (or an image with a low-pass filter applied). Figure 3(a) demonstrates this point for a wavelet transform with *J* = 5, showing the Fourier-space response of the wavelet filter as a function of scale *j*.

(a) The filter response of each of 5 scales of the wavelet transform

(b) Comparison of a the processing time of a brute-force aperture mass algorithm and the starlet transform algorithm

As discussed in [31], this time advantage further extends to higher-order statistics of the *Map*, which are typically related directly to *n*-point correlation functions of the shear. In recent years, tree codes have been employed to speed up computation of *n*-point correlation functions. Typical tree codes to compute *<sup>n</sup>*-point correlation functions are O(*Ngal* log(*Ngal*)) [32] on a shear catalogue. For a single Euclid exposure of 0.5 deg2, we can expect *Ngal* ∼ 54, 000 (30 galaxies/arcmin2) - 180, 000 (100 galaxies/arcmin2). Tree codes exist that act on

*pixnbin*) where *<sup>N</sup>*<sup>2</sup>

*pix* <sup>=</sup> 1800, and *nbin* will be dependent on the required resolution

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

*nbin* is the number of bins in the correlation function. For a Euclid exposure, assuming pixels

The wavelet method acts on pixellated data, and is O(*N*<sup>2</sup> *J*) in computation time, so our algorithm will be comparable for computation of 2-point statistics, if the 2-point correlation function is computed on pixellated data, but if a shear catalogue is used, we will have a faster algorithm by at least an order of magnitude. For higher-order statistics, this advantage is even more pronounced. Furthermore, while optimised software is freely and publicly available to compute the wavelet transform, such optimised software is not available for

Another advantage of the wavelet formalism is that it is possible to carry out an explicit denoising of the convergence field using thresholding based on a False Discovery Rate method. For details on this method, see [34], where the *MRLens* software package encoding this method is presented3. This allows one to derive robust detection levels in wavelet space,

In addition, wavelet-based methods offer more flexibility than aperture mass filters. Whilst we have thus far discussed only the starlet wavelet function, many other wavelet dictionaries may be used. The starlet filter seems ideal for lensing studies, due to its similarity to the aperture mass filter presented here, which was deemed to be optimal in [22]. However, different dictionaries may be optimal in different applications; for example, if one were attempting to study filamentary structure, ridgelets or curvelets might be a more appropriate basis. The vastness of the wavelet libraries and the public availability of fast algorithms to

As noted previously, the optimality of second-order statistics to constrain cosmological parameters depends heavily on the assumption of Gaussianity of the field. However the weak lensing field is composed, at small scales, of non-Gaussian features such as clusters of galaxies. These non-Gaussian signatures carry additional information that cannot be extracted with second-order statistics. Many studies [35–39] have shown that combining second-order statistics with higher-order statistics tightens the constraints on cosmological parameters. We now consider the higher-order statistics most commonly used in weak lensing studies aimed at detecting and constraining non-Gaussianity in the lensing field.

<sup>3</sup> The MRLens software, along with many other software packages for astronomical applications, is freely available

compute these transforms are major strengths of wavelet-based approaches.

**4. Weak lensing beyond two-point shear statistics**

here: http://cosmostat.org/software.html

*pix* is the total number of pixels and

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41

pixelated data [33] which run at O(*N*<sup>2</sup>

of 1 arcminute, we have *N*<sup>2</sup>

of the correlation function.

*n*-point correlation functions.

and to produce high-fidelity denoised mass maps.

**Figure 3.** Illustration of the various advantages offered by the wavelet transform in terms of (a) retention of information on all scales and (b) algorithm processing time.

In addition, from a mapping perspective, the starlet wavelet transform algorithm is substantially faster than a brute-force computation of the aperture mass statistic in real space. Such a naive implementation has complexity ∝ O(*N*2*ϑ*<sup>2</sup> *cut*), where *<sup>N</sup>* <sup>×</sup> *<sup>N</sup>* is the dimension of the image, and *ϑcut* is the chosen truncation radius. This scaling means that for large apertures or, equivalently, for high-resolution images, the algorithm may prove to be very time-consuming. The starlet wavelet transform algorithm is of complexity ∝ O(*N*<sup>2</sup> *J*), where *J* is the number of scales considered, and is limited by *N* ≥ 2*<sup>J</sup>* . This means that the processing time for the wavelet transform algorithm is sensitive only to the number of scales considered, rather than the size of the filter functions involved, and depends linearly on this number.

In Figure 3(b), we compare the processing time for the aperture mass algorithm and the starlet transform algorithm, both programmed in C++, to analyse an image of 1024 × 1024 pixels on a 2 × 2.66GHz Intel Xeon Dual-Core processor. We consider aperture scales *θ* = [2, 4, 8, 16, 32, 64], which correspond to *J* = *j*max + 1 = [2, 3, 4, 5, 6, 7] wavelet scales in the wavelet transform. In the aperture mass algorithm, the filters are truncated at a radius of *ϑcut* = *θ*. For the application of filters which necessitate truncation at a much larger radius, we expect the computation time to be roughly an order of magnitude longer. Even at the smallest aperture radius, the wavelet transform is ∼ 5× faster than the aperture mass algorithm. At *θ* = 64 pixels, the wavelet transform is ∼ 1200× faster than the aperture mass algorithm. Note that the wavelet transform for a given value of *J* simultaneously computes the wavelet transform at all scales 2*<sup>j</sup>* , 0 < *j* ≤ *J* − 1, in addition to the smoothed continuum map *cJ*, whilst the aperture mass algorithm computes the transform at a single scale *θ*. We note further that the computational time for the wavelet transform for *J* = 7 wavelet scales is still a factor of ∼ 2× less than the computational time for the aperture mass algorithm at *θ* = 2 pixels.

As discussed in [31], this time advantage further extends to higher-order statistics of the *Map*, which are typically related directly to *n*-point correlation functions of the shear. In recent years, tree codes have been employed to speed up computation of *n*-point correlation functions. Typical tree codes to compute *<sup>n</sup>*-point correlation functions are O(*Ngal* log(*Ngal*)) [32] on a shear catalogue. For a single Euclid exposure of 0.5 deg2, we can expect *Ngal* ∼ 54, 000 (30 galaxies/arcmin2) - 180, 000 (100 galaxies/arcmin2). Tree codes exist that act on pixelated data [33] which run at O(*N*<sup>2</sup> *pixnbin*) where *<sup>N</sup>*<sup>2</sup> *pix* is the total number of pixels and *nbin* is the number of bins in the correlation function. For a Euclid exposure, assuming pixels of 1 arcminute, we have *N*<sup>2</sup> *pix* <sup>=</sup> 1800, and *nbin* will be dependent on the required resolution of the correlation function.

10 Open Questions in Cosmology

transform

wavelet filter as a function of scale *j*.

(a) The filter response of each of 5 scales of the wavelet

Such a naive implementation has complexity ∝ O(*N*2*ϑ*<sup>2</sup>

*J* is the number of scales considered, and is limited by *N* ≥ 2*<sup>J</sup>*

scales and (b) algorithm processing time.

the wavelet transform at all scales 2*<sup>j</sup>*

*θ* = 2 pixels.

at both the small and large frequency extremes. In contrast, the wavelet transform retains information on all scales. The first wavelet scale is effectively a high-pass filter, retaining all the high-frequency information in the image, while the remaining wavelet scales are bandpass filtered as with the *Map*-filtered images. Finally, the wavelet transform retains *cJ*, which encodes the large-scale information in the image, and therefore consists of a smoothed version of the image (or an image with a low-pass filter applied). Figure 3(a) demonstrates this point for a wavelet transform with *J* = 5, showing the Fourier-space response of the

> (b) Comparison of a the processing time of a brute-force aperture mass algorithm and the starlet

, 0 < *j* ≤ *J* − 1, in addition to the smoothed continuum

*cut*), where *<sup>N</sup>* <sup>×</sup> *<sup>N</sup>* is the dimension

. This means that the processing

transform algorithm

**Figure 3.** Illustration of the various advantages offered by the wavelet transform in terms of (a) retention of information on all

In addition, from a mapping perspective, the starlet wavelet transform algorithm is substantially faster than a brute-force computation of the aperture mass statistic in real space.

of the image, and *ϑcut* is the chosen truncation radius. This scaling means that for large apertures or, equivalently, for high-resolution images, the algorithm may prove to be very time-consuming. The starlet wavelet transform algorithm is of complexity ∝ O(*N*<sup>2</sup> *J*), where

time for the wavelet transform algorithm is sensitive only to the number of scales considered, rather than the size of the filter functions involved, and depends linearly on this number. In Figure 3(b), we compare the processing time for the aperture mass algorithm and the starlet transform algorithm, both programmed in C++, to analyse an image of 1024 × 1024 pixels on a 2 × 2.66GHz Intel Xeon Dual-Core processor. We consider aperture scales *θ* = [2, 4, 8, 16, 32, 64], which correspond to *J* = *j*max + 1 = [2, 3, 4, 5, 6, 7] wavelet scales in the wavelet transform. In the aperture mass algorithm, the filters are truncated at a radius of *ϑcut* = *θ*. For the application of filters which necessitate truncation at a much larger radius, we expect the computation time to be roughly an order of magnitude longer. Even at the smallest aperture radius, the wavelet transform is ∼ 5× faster than the aperture mass algorithm. At *θ* = 64 pixels, the wavelet transform is ∼ 1200× faster than the aperture mass algorithm. Note that the wavelet transform for a given value of *J* simultaneously computes

map *cJ*, whilst the aperture mass algorithm computes the transform at a single scale *θ*. We note further that the computational time for the wavelet transform for *J* = 7 wavelet scales is still a factor of ∼ 2× less than the computational time for the aperture mass algorithm at The wavelet method acts on pixellated data, and is O(*N*<sup>2</sup> *J*) in computation time, so our algorithm will be comparable for computation of 2-point statistics, if the 2-point correlation function is computed on pixellated data, but if a shear catalogue is used, we will have a faster algorithm by at least an order of magnitude. For higher-order statistics, this advantage is even more pronounced. Furthermore, while optimised software is freely and publicly available to compute the wavelet transform, such optimised software is not available for *n*-point correlation functions.

Another advantage of the wavelet formalism is that it is possible to carry out an explicit denoising of the convergence field using thresholding based on a False Discovery Rate method. For details on this method, see [34], where the *MRLens* software package encoding this method is presented3. This allows one to derive robust detection levels in wavelet space, and to produce high-fidelity denoised mass maps.

In addition, wavelet-based methods offer more flexibility than aperture mass filters. Whilst we have thus far discussed only the starlet wavelet function, many other wavelet dictionaries may be used. The starlet filter seems ideal for lensing studies, due to its similarity to the aperture mass filter presented here, which was deemed to be optimal in [22]. However, different dictionaries may be optimal in different applications; for example, if one were attempting to study filamentary structure, ridgelets or curvelets might be a more appropriate basis. The vastness of the wavelet libraries and the public availability of fast algorithms to compute these transforms are major strengths of wavelet-based approaches.

## **4. Weak lensing beyond two-point shear statistics**

As noted previously, the optimality of second-order statistics to constrain cosmological parameters depends heavily on the assumption of Gaussianity of the field. However the weak lensing field is composed, at small scales, of non-Gaussian features such as clusters of galaxies. These non-Gaussian signatures carry additional information that cannot be extracted with second-order statistics. Many studies [35–39] have shown that combining second-order statistics with higher-order statistics tightens the constraints on cosmological parameters. We now consider the higher-order statistics most commonly used in weak lensing studies aimed at detecting and constraining non-Gaussianity in the lensing field.

<sup>3</sup> The MRLens software, along with many other software packages for astronomical applications, is freely available here: http://cosmostat.org/software.html

## **4.1. Higher-order lensing statistics**

As we have already noted, it is more convenient to consider higher-order statistics of the *Map* or wavelet transform of the convergence field, as opposed to statistics of the shear field directly. The obvious first extension to two-point statistics is to consider the three-point correlation function or, equivalently, the bispectrum, usually considered as a function of aperture or wavelet scale.

of the aperture mass map for a given cosmological model, which is related to the four-point

Again, this statistic can be computed from the wavelet transform of the convergence map

be computed either on a noisy or a denoised convergence map, the latter expected to yield

We define a peak as set of connected pixels above a detection threshold T , and denote the

is carried out on a denoised map, the detection threshold T is automatically set by the denoising algorithm, and a small threshold *ǫ* is used when identifying peaks in the denoised

We consider all pixels that are connected via the sides or the corners of a pixel as one structure. In a two-dimensional projected map of the convergence, we are therefore unable to discriminate between peaks due to massive halos and peaks due to projections of large-scale

While theory exists to predict cluster counts from the halo model and a cosmological model encoding the growth and evolution of structures in the Universe, there is no analytic formalism to predict the fraction of spurious detections in lensing maps arising from projections of large-scale structure. In [44], an attempt is made to derive an analytic formalism for predicting peak counts in projected maps. However, this method is based on an assumption of Gaussianity in the lensing field and, predictably, underestimates counts at the high end of the mass function, where clusters of galaxies are dominant. This means that predicting peak counts for a given cosmological model relies on considering N-body simulations generated under a range of cosmological parameters (for example, see [45]).

The question now arises: which of these statistics provides the most information about the underlying cosmology and – specifically – non-Gaussianity in the density field? In order to asses the performance of these statistics, we consider the ability of each statistic to discriminate between cosmological models using N-body simulations under a range of

To this end, we consider N-body simulations carried out for 5 different cosmological models along the <sup>Ω</sup>*<sup>M</sup>* − *<sup>σ</sup>*<sup>8</sup> degeneracy, the parameters of which are summarised in Table 1 below [46]. These simulations were carried out using the RAMSES N-body code [47], and full details of the models considered are given in [38]. For each model, 100 realizations of weak lensing maps were generated for a field of 3.95◦ × 3.95◦, downsampled to 1024 × 1024 pixels (0.23′ per pixel). Noise was added to the simulations at a level consistent with predictions for deep space-based observations, with a number density of galaxies of *ng* = 100 gal/arcmin2. This is somewhat optimistic; however, as seen in [46], increasing the noise level in the simulations does not change the conclusions of the comparison test between the higher-order statistics.

*<sup>j</sup>* � of the wavelet band *<sup>j</sup>* corresponding to the aperture scale *<sup>θ</sup>*. This can

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

*Map* and *<sup>P</sup>*<sup>T</sup>

*wj*

, respectively. If peak counting

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43

correlation function or trispectrum of the 3D density contrast [20].

maps, in order to reject spurious detections in these denoised maps.

peak count in the *Map* and wavelet maps by *<sup>P</sup>*<sup>T</sup>

**4.2. Optimal capture of non-Gaussianity**

as the kurtosis �*w*<sup>4</sup>

improved constraints.

structures such as filaments.

different cosmologies [38, 46].

*4.1.3. Peak Counts*

Interesting analytical results relative to the shear three-point correlation function or the convergence bispectrum have been reported (for example, see [40–43]). However, it has been shown that in using only the equilateral configuration of the bispectrum, the ability to discriminate between cosmological models is relatively poor [38]. An analytical comparison has been performed in [39] between the full bispectrum and an optimal match-filter peak count for a Euclid-like survey, and both approaches were found to provide similar results. However, as the full bispectrum calculation has a much higher complexity than other statistics, and no public software exists to compute it, we will not consider the full bispectrum here. Rather, we will restrict ourselves to statistics that are more straightforward to compute from the *Map* or wavelet transform maps.

## *4.1.1. The Skewness*

The skewness of the aperture mass map, *M*<sup>3</sup> *ap*, is the third-order moment of the aperture mass *Map*(*θ*) and can be computed directly from shear maps filtered with different aperture mass. The skewness is a measure of the asymmetry of the probability distribution function. The probability distribution function will be more or less skewed positively depending on the abundance of dark matter haloes at the *θ* scale. The formalism exists to predict the skewness of the aperture mass map for a given cosmological model, which is related to the three-point correlation function or the bispectrum of density fluctuations *δ* [20, 37].

In [37], it is argued that the skewness as a function of scale is a preferable statistic to three-point correlation functions of the shear, as the integral relations between *M*<sup>3</sup> *ap* and the bispectrum are much easier and faster to compute than the three-point correlation function. This is because the skewness is a local measure of the bispectrum, whereas the integral kernel for the three-point correlation function is a highly oscillating function with infinite support.

This statistic can equivalently be computed from the wavelet transform of the convergence map as the skewness *w*<sup>3</sup> *<sup>j</sup>* of the wavelet band *<sup>j</sup>* corresponding to the aperture scale *<sup>θ</sup>*. This can be computed either on the noisy convergence map estimated, for example, by the Fourier space relationship between the shear and convergence – in which case the results should be comparable to the *Map* results – or on a denoised convergence map using the method of [34], which should provide improved results.

#### *4.1.2. The Kurtosis*

The kurtosis of the aperture mass map, *M*<sup>4</sup> *ap*, is the fourth-order moment of the aperture mass *Map*(*θ*) and can be computed directly from the different aperture mass maps. The kurtosis is a measure of the peakedness of the probability distribution function. The presence of dark matter haloes at a given *θ* scale will flatten the probability distribution function and widen its shoulders leading to a larger kurtosis. The formalism exists to predict the kurtosis of the aperture mass map for a given cosmological model, which is related to the four-point correlation function or trispectrum of the 3D density contrast [20].

Again, this statistic can be computed from the wavelet transform of the convergence map as the kurtosis �*w*<sup>4</sup> *<sup>j</sup>* � of the wavelet band *<sup>j</sup>* corresponding to the aperture scale *<sup>θ</sup>*. This can be computed either on a noisy or a denoised convergence map, the latter expected to yield improved constraints.

## *4.1.3. Peak Counts*

12 Open Questions in Cosmology

aperture or wavelet scale.

*4.1.1. The Skewness*

map as the skewness *w*<sup>3</sup>

*4.1.2. The Kurtosis*

which should provide improved results.

The kurtosis of the aperture mass map, *M*<sup>4</sup>

**4.1. Higher-order lensing statistics**

from the *Map* or wavelet transform maps.

The skewness of the aperture mass map, *M*<sup>3</sup>

As we have already noted, it is more convenient to consider higher-order statistics of the *Map* or wavelet transform of the convergence field, as opposed to statistics of the shear field directly. The obvious first extension to two-point statistics is to consider the three-point correlation function or, equivalently, the bispectrum, usually considered as a function of

Interesting analytical results relative to the shear three-point correlation function or the convergence bispectrum have been reported (for example, see [40–43]). However, it has been shown that in using only the equilateral configuration of the bispectrum, the ability to discriminate between cosmological models is relatively poor [38]. An analytical comparison has been performed in [39] between the full bispectrum and an optimal match-filter peak count for a Euclid-like survey, and both approaches were found to provide similar results. However, as the full bispectrum calculation has a much higher complexity than other statistics, and no public software exists to compute it, we will not consider the full bispectrum here. Rather, we will restrict ourselves to statistics that are more straightforward to compute

mass *Map*(*θ*) and can be computed directly from shear maps filtered with different aperture mass. The skewness is a measure of the asymmetry of the probability distribution function. The probability distribution function will be more or less skewed positively depending on the abundance of dark matter haloes at the *θ* scale. The formalism exists to predict the skewness of the aperture mass map for a given cosmological model, which is related to the three-point

In [37], it is argued that the skewness as a function of scale is a preferable statistic to

bispectrum are much easier and faster to compute than the three-point correlation function. This is because the skewness is a local measure of the bispectrum, whereas the integral kernel for the three-point correlation function is a highly oscillating function with infinite support. This statistic can equivalently be computed from the wavelet transform of the convergence

can be computed either on the noisy convergence map estimated, for example, by the Fourier space relationship between the shear and convergence – in which case the results should be comparable to the *Map* results – or on a denoised convergence map using the method of [34],

mass *Map*(*θ*) and can be computed directly from the different aperture mass maps. The kurtosis is a measure of the peakedness of the probability distribution function. The presence of dark matter haloes at a given *θ* scale will flatten the probability distribution function and widen its shoulders leading to a larger kurtosis. The formalism exists to predict the kurtosis

*<sup>j</sup>* of the wavelet band *<sup>j</sup>* corresponding to the aperture scale *<sup>θ</sup>*. This

three-point correlation functions of the shear, as the integral relations between *M*<sup>3</sup>

correlation function or the bispectrum of density fluctuations *δ* [20, 37].

*ap*, is the third-order moment of the aperture

*ap*, is the fourth-order moment of the aperture

*ap* and the

We define a peak as set of connected pixels above a detection threshold T , and denote the peak count in the *Map* and wavelet maps by *<sup>P</sup>*<sup>T</sup> *Map* and *<sup>P</sup>*<sup>T</sup> *wj* , respectively. If peak counting is carried out on a denoised map, the detection threshold T is automatically set by the denoising algorithm, and a small threshold *ǫ* is used when identifying peaks in the denoised maps, in order to reject spurious detections in these denoised maps.

We consider all pixels that are connected via the sides or the corners of a pixel as one structure. In a two-dimensional projected map of the convergence, we are therefore unable to discriminate between peaks due to massive halos and peaks due to projections of large-scale structures such as filaments.

While theory exists to predict cluster counts from the halo model and a cosmological model encoding the growth and evolution of structures in the Universe, there is no analytic formalism to predict the fraction of spurious detections in lensing maps arising from projections of large-scale structure. In [44], an attempt is made to derive an analytic formalism for predicting peak counts in projected maps. However, this method is based on an assumption of Gaussianity in the lensing field and, predictably, underestimates counts at the high end of the mass function, where clusters of galaxies are dominant. This means that predicting peak counts for a given cosmological model relies on considering N-body simulations generated under a range of cosmological parameters (for example, see [45]).

## **4.2. Optimal capture of non-Gaussianity**

The question now arises: which of these statistics provides the most information about the underlying cosmology and – specifically – non-Gaussianity in the density field? In order to asses the performance of these statistics, we consider the ability of each statistic to discriminate between cosmological models using N-body simulations under a range of different cosmologies [38, 46].

To this end, we consider N-body simulations carried out for 5 different cosmological models along the <sup>Ω</sup>*<sup>M</sup>* − *<sup>σ</sup>*<sup>8</sup> degeneracy, the parameters of which are summarised in Table 1 below [46]. These simulations were carried out using the RAMSES N-body code [47], and full details of the models considered are given in [38]. For each model, 100 realizations of weak lensing maps were generated for a field of 3.95◦ × 3.95◦, downsampled to 1024 × 1024 pixels (0.23′ per pixel). Noise was added to the simulations at a level consistent with predictions for deep space-based observations, with a number density of galaxies of *ng* = 100 gal/arcmin2. This is somewhat optimistic; however, as seen in [46], increasing the noise level in the simulations does not change the conclusions of the comparison test between the higher-order statistics.


*<sup>θ</sup><sup>i</sup> <sup>M</sup>*<sup>3</sup>

Scale *w*<sup>3</sup>

obtained on the wavelet transform of the noisy convergence maps.

applied to the convergence maps before the statistics are computed.

as compared to the *nσ* thresholding applied previously.

*ap <sup>M</sup>*<sup>4</sup>

*<sup>j</sup> <sup>w</sup>*<sup>4</sup>

*ap <sup>P</sup>*2*<sup>σ</sup>*

0.46' 04.60 % 02.30 % 39.70 % 54.95 % 0.92' **33.40 %** 03.70 % 79.30 % 76.40 % 1.85' 03.45 % 01.45 % **91.25 % 89.20** % 3.70' 15.15 % 23.00 % 69.40 % 86.70 % 7.40' 26.95 % **24.30** % 4.90 % 60.50 %

(a) Mean discrimination efficiency of *Map* statistics

0.46' 02.00 % 01.15 % 12.05% 00.70 % 0.92' **37.95 %** 04.75 % 86.30 % 73.05 % 1.85' 03.55 % 02.10 % **94.40 % 93.85 %** 3.70' 18.25 % 25.65 % 84.05 % 87.05 % 7.40' 36.40 % **30.90** % 24.60 % 66.35 %

(b) Mean discrimination efficiency of wavelet statistics **Table 2.** Mean discrimination efficiency (in percent) from noisy *Map* reconstructions of the shear field, compared with that

In Table 3 below, we present the mean discrimination efficiency of statistics computed on the denoised convergence maps, as well as the peak discrimination efficiency between each pair of models in our sample. There is a clear improvement on the discrimination efficiency of all the higher-order statistics when denoising is applied to the convergence maps. This is unsurprising, as the presence of Gaussian (or near-Gaussian) noise within the shear and convergence maps will make the whole field appear more Gaussian, masking the non-Gaussian features in the map and pushing the skewness and kurtosis values closer to zero. The more noise that is present in the maps, the stronger this effect will be. It is for this reason that such a dramatic improvement is seen in these statistics when denoising is

The improvement seen in the discrimination efficiency of the peak statistics is also significant, and this arises due to the fact that while the peak counts in the noisy maps involve application of a simple *nσ* detection threshold to the maps, the denoising algorithm involves a more sophisticated discriminant between signal and noise, again making use of the FDR method. This technique is more effective and distinguishing between signal and noise, and therefore more information about the true peaks in the map is retained using this denoising method

It is clear that peak statistics are much more efficient at capturing non-Gaussianity in the shear field than other higher-order statistics, and that it is advantageous to measure this statistic on denoised convergence maps. However, in order to combine constraints from peak counting with other probes, such as two-point statistics, in order to better constrain the cosmological model, it is important to be able to predict peak statistics as a function of cosmology in order to be able to compute the likelihood function for all models within the parameter space. As we have discussed, this is not possible to do analytically, due to contamination by projections of large scale structures, and it is too computationally-intensive

*<sup>j</sup> <sup>P</sup>*2*<sup>σ</sup>*

*Map <sup>P</sup>*3*<sup>σ</sup>*

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

*wj <sup>P</sup>*3*<sup>σ</sup>*

*Map*

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45

*wj*

**Table 1.** Parameters of the five cosmological models that have been chosen along the Ω*<sup>M</sup>* − *σ*<sup>8</sup> degeneracy. The simulations have <sup>256</sup><sup>3</sup> particles, *<sup>h</sup>* is equal to *<sup>H</sup>*0/100 km s−<sup>1</sup> Mpc−1, where *<sup>H</sup>*<sup>0</sup> is the Hubble constant.

To find the best statistic, we need to characterise quantitatively for each statistic the discrimination between two different models *m*<sup>1</sup> and *m*2. To do this, we consider the distribution functions for the different statistics estimated on the 100 realisations of each model. These distributions are expected to overlap, so False Discovery Rate (FDR) method is used to determine the threshold *τ*<sup>1</sup> in the distribution function of model *m*1, such that the distribution function of model *<sup>m</sup>*<sup>2</sup> accounts for fewer than a fraction *<sup>α</sup>* = 0.05 of the total counts. A similar threshold *τ*<sup>2</sup> is defined for the distribution function of model *m*2. This is illustrated in Figure 4. The discrimination efficiency is then defined as the percentage of counts remaining under the hatched area for each model.

**Figure 4.** Illustration of the discrimination efficiency criterion. *τ*<sup>1</sup> and *τ*<sup>2</sup> are defined such that the yellow area under the curve represents 0.05 of the total area under the curves for models *m*<sup>2</sup> and *m*1, respectively. The discrimination efficiency is defined as the area under the *mi* curve delimited by the threshold *τi*, as a percentage of the total area under the model *mi* curve.

In Table 2 below, we show the mean discrimination efficiency obtained considering all pairs of models for the higher-order statistics of interest. These are presented as a function of *Map* or wavelet scale, and shown for statistics computed on the aperture mass map and the wavelet transform of the noisy convergence map. The results overall are comparable, with the wavelet transform statistics appearing to offer a slight improvement over the *Map* statistics in all cases. This is perhaps related to the fact that, when computing the *Map* statistics, we truncate the filter at *ϑcut* = *θ*, which may give rise to some small systematics in the resulting maps. Furthermore, it is clear that, in all cases, the peak statistics are much more efficient at discriminating between cosmological models than either the skewness or the kurtosis.

<sup>44</sup> Open Questions in Cosmology Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics 15 Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics http://dx.doi.org/10.5772/51871 45


14 Open Questions in Cosmology

**Model Box Length**

(*h*<sup>−</sup>1Mpc)

Ω*<sup>M</sup>* ΩΛ *h σ*<sup>8</sup>

*m*<sup>1</sup> 165.8 0.23 0.77 0.594 1 *m*<sup>2</sup> 159.5 0.3 0.7 0.7 0.9 *m*<sup>3</sup> 152.8 0.36 0.64 0.798 0.8 *m*<sup>4</sup> 145.7 0.47 0.53 0.894 0.7 *m*<sup>5</sup> 137.5 0.64 0.36 0.982 0.6

**Table 1.** Parameters of the five cosmological models that have been chosen along the Ω*<sup>M</sup>* − *σ*<sup>8</sup> degeneracy. The simulations

To find the best statistic, we need to characterise quantitatively for each statistic the discrimination between two different models *m*<sup>1</sup> and *m*2. To do this, we consider the distribution functions for the different statistics estimated on the 100 realisations of each model. These distributions are expected to overlap, so False Discovery Rate (FDR) method is used to determine the threshold *τ*<sup>1</sup> in the distribution function of model *m*1, such that the distribution function of model *<sup>m</sup>*<sup>2</sup> accounts for fewer than a fraction *<sup>α</sup>* = 0.05 of the total counts. A similar threshold *τ*<sup>2</sup> is defined for the distribution function of model *m*2. This is illustrated in Figure 4. The discrimination efficiency is then defined as the percentage of

**Figure 4.** Illustration of the discrimination efficiency criterion. *τ*<sup>1</sup> and *τ*<sup>2</sup> are defined such that the yellow area under the curve represents 0.05 of the total area under the curves for models *m*<sup>2</sup> and *m*1, respectively. The discrimination efficiency is defined as the area under the *mi* curve delimited by the threshold *τi*, as a percentage of the total area under the model *mi* curve. In Table 2 below, we show the mean discrimination efficiency obtained considering all pairs of models for the higher-order statistics of interest. These are presented as a function of *Map* or wavelet scale, and shown for statistics computed on the aperture mass map and the wavelet transform of the noisy convergence map. The results overall are comparable, with the wavelet transform statistics appearing to offer a slight improvement over the *Map* statistics in all cases. This is perhaps related to the fact that, when computing the *Map* statistics, we truncate the filter at *ϑcut* = *θ*, which may give rise to some small systematics in the resulting maps. Furthermore, it is clear that, in all cases, the peak statistics are much more efficient at discriminating between cosmological models than either the skewness or the kurtosis.

have <sup>256</sup><sup>3</sup> particles, *<sup>h</sup>* is equal to *<sup>H</sup>*0/100 km s−<sup>1</sup> Mpc−1, where *<sup>H</sup>*<sup>0</sup> is the Hubble constant.

counts remaining under the hatched area for each model.

(a) Mean discrimination efficiency of *Map* statistics


(b) Mean discrimination efficiency of wavelet statistics

**Table 2.** Mean discrimination efficiency (in percent) from noisy *Map* reconstructions of the shear field, compared with that obtained on the wavelet transform of the noisy convergence maps.

In Table 3 below, we present the mean discrimination efficiency of statistics computed on the denoised convergence maps, as well as the peak discrimination efficiency between each pair of models in our sample. There is a clear improvement on the discrimination efficiency of all the higher-order statistics when denoising is applied to the convergence maps. This is unsurprising, as the presence of Gaussian (or near-Gaussian) noise within the shear and convergence maps will make the whole field appear more Gaussian, masking the non-Gaussian features in the map and pushing the skewness and kurtosis values closer to zero. The more noise that is present in the maps, the stronger this effect will be. It is for this reason that such a dramatic improvement is seen in these statistics when denoising is applied to the convergence maps before the statistics are computed.

The improvement seen in the discrimination efficiency of the peak statistics is also significant, and this arises due to the fact that while the peak counts in the noisy maps involve application of a simple *nσ* detection threshold to the maps, the denoising algorithm involves a more sophisticated discriminant between signal and noise, again making use of the FDR method. This technique is more effective and distinguishing between signal and noise, and therefore more information about the true peaks in the map is retained using this denoising method as compared to the *nσ* thresholding applied previously.

It is clear that peak statistics are much more efficient at capturing non-Gaussianity in the shear field than other higher-order statistics, and that it is advantageous to measure this statistic on denoised convergence maps. However, in order to combine constraints from peak counting with other probes, such as two-point statistics, in order to better constrain the cosmological model, it is important to be able to predict peak statistics as a function of cosmology in order to be able to compute the likelihood function for all models within the parameter space. As we have discussed, this is not possible to do analytically, due to contamination by projections of large scale structures, and it is too computationally-intensive


Therefore the 3D lensing problem is effectively one of observing the density contrast convolved with the linear operator **R**, and contaminated by noise, which is assumed to be

The general idea behind linear inversion methods is to find a linear operator **H** that acts on the data vector to yield a solution that minimises some functional, such as the variance of the residual between the estimated signal and the true signal, subject to some regularisation or prior-based constraints. Two different linear approaches have been described in recent

The most competitive method is that proposed in [4], in which the authors propose a Saskatoon filter [50, 51], which combines a Wiener filter and an inverse variance filter, with a tuning parameter *α* introduced that allows switching between the two. This gives rise to a

ss† encodes prior information about the signal covariance, Σ ≡

This switching is designed to allow a balance between the increased constraining power offered by the Wiener filter over the inverse variance filter – which yields an improved signal-to-noise in the reconstruction – and the biasing that the Wiener filter imposes on the

As discussed extensively in [4] and [49], linear methods give rise to a significant bias in the location of detected peaks, damping of the peak signal, and a substantial smearing of the density along the line of sight. Furthermore, the resolution attainable in the reconstructions obtained using linear methods is limited to the resolution of the data. In other words, we cannot reconstruct the density contrast field at higher resolution than the resolution of our

Figure 5 shows the 3D reconstruction obtained in this way for a simulated cluster of galaxies at a redshift of *<sup>z</sup>*cl = 0.25. The tuning parameter used for this reconstruction was *<sup>α</sup>* <sup>=</sup> 0.05. The cluster was simulated according to an NFW halo with *<sup>M</sup>* <sup>=</sup> 1015*h*−1*M*<sup>⊙</sup> and *c* = 3, and the shear data were assumed to come from a galaxy distribution given by *<sup>p</sup>*(*z*) <sup>∝</sup> *<sup>z</sup>*2e<sup>−</sup>(1.4*z*)1.5 [52, 53], with a maximum redshift of *zmax* <sup>=</sup> 2.0 and a galaxy density of *ng* <sup>=</sup> 100 galaxies/arcmin2. The simulation covers a 1◦ <sup>×</sup> <sup>1</sup>◦ field binned into 60 <sup>×</sup> <sup>60</sup> angular pixels, and 20 redshift bins. Shown in the figure are both the 3D rendering of the reconstruction, and a 1D plot showing the four central lines of sight through the cluster as a function of redshift. The smearing, damping and redshift bias effects are all clearly visible in

the 1D plot, where the amplitude of the cluster density contrast should be *δ* ∼ 36.

<sup>s</sup>ˆ*MV* = [*α***<sup>1</sup>** <sup>+</sup> **SR**†**Σ**−1**R**]

the covariance matrix of the noise, and **1** is the identity matrix.

input data which, in turn, is limited by the noise properties of the data.

where d is the observation, s the real density and *ε* the Gaussian noise.

d = **R**s + *ε*, *ε* ∼ N (0, *σ*2) , (23)

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

<sup>−</sup>1**SR**†**Σ**−1<sup>d</sup> , (24)

http://dx.doi.org/10.5772/51871

47

nn† gives

Gaussian. Formally, we can write

literature [4, 49].

where **S** ≡

solution.

**5.1. Linear approaches to 3D map-making**

minimum variance filter, expressed as


(a) Mean discrimination efficiency

(b) Discrimination efficiency for all models obtained using peak statistics

**Table 3.** Discrimination efficiency for statistics computed on denoised convergence maps.

to consider obtaining the predictions, and associated covariances, from N-body simulations if we wish to sample the parameter space completely and at high resolution.

As the quality of data available to astronomers continues to improve, it has recently become possible to consider the shear field in three dimensions, using the colours of galaxies to determine their redshifts and, therefore, distances from us. If this information may be used to deproject the lensing signal, and thus recover information about the full three-dimensional density field, it should then be possible offer predictions for peak statistics as a function of cosmology, uncontaminated by projection effects which will, in turn, allow us to improve our constraints on our cosmological model.

## **5. Reconstructing the density contrast in 3D**

The measured shear can be related to the 2D convergence via a simple linear relationship (equation (4)), and the convergence is related to the density contrast by another linear mapping (equation (6)). We can express these relationships conveniently in matrix notation as

$$\kappa(\theta, z) = \mathbf{Q}\delta(\theta, z) \tag{21}$$

$$\gamma(\theta, z) = \mathbf{P}\_{\gamma \kappa} \mathbf{Q} \delta(\theta, z) \,. \tag{22}$$

The 3D lensing problem is therefore one of finding an estimator to invert equation (21) or (22) in the presence of noise. For simplicity, in what follows we assume the shear measurement noise is Gaussian and uncorrelated between redshift bins. In practice, errors in the photometric estimates of the redshifts of galaxies will introduce correlations between redshift bins. Methods have been developed to account for such errors, however, and the problem is therefore readily tractable [48].

Therefore the 3D lensing problem is effectively one of observing the density contrast convolved with the linear operator **R**, and contaminated by noise, which is assumed to be Gaussian. Formally, we can write

$$\mathbf{d} = \mathbf{R}\mathbf{s} + \varepsilon, \quad \varepsilon \sim \mathcal{N}(\mathbf{0}, \sigma^2) \; , \tag{23}$$

where d is the observation, s the real density and *ε* the Gaussian noise.

## **5.1. Linear approaches to 3D map-making**

16 Open Questions in Cosmology

Scale *w*˜ <sup>3</sup>

**Table 3.** Discrimination efficiency for statistics computed on denoised convergence maps.

constraints on our cosmological model.

problem is therefore readily tractable [48].

as

**5. Reconstructing the density contrast in 3D**

if we wish to sample the parameter space completely and at high resolution.

*<sup>j</sup> <sup>w</sup>*˜ <sup>4</sup>

0.46' 53.40 % 43.20 % 68.35 % 0.92' 47.90 % 41.15 % 92.45 % 1.85' 58.80 % 44.70 % **96.75 %** 3.70' **63.30 % 48.05** % 90.40 % 7.40' 54.90 % 40.45 % 63.45 %

(a) Mean discrimination efficiency

(b) Discrimination efficiency for all models obtained using peak statistics

to consider obtaining the predictions, and associated covariances, from N-body simulations

As the quality of data available to astronomers continues to improve, it has recently become possible to consider the shear field in three dimensions, using the colours of galaxies to determine their redshifts and, therefore, distances from us. If this information may be used to deproject the lensing signal, and thus recover information about the full three-dimensional density field, it should then be possible offer predictions for peak statistics as a function of cosmology, uncontaminated by projection effects which will, in turn, allow us to improve our

The measured shear can be related to the 2D convergence via a simple linear relationship (equation (4)), and the convergence is related to the density contrast by another linear mapping (equation (6)). We can express these relationships conveniently in matrix notation

The 3D lensing problem is therefore one of finding an estimator to invert equation (21) or (22) in the presence of noise. For simplicity, in what follows we assume the shear measurement noise is Gaussian and uncorrelated between redshift bins. In practice, errors in the photometric estimates of the redshifts of galaxies will introduce correlations between redshift bins. Methods have been developed to account for such errors, however, and the

*κ*(θ, *z*) = **Q***δ*(θ, *z*) , (21) *<sup>γ</sup>*(θ, *<sup>z</sup>*) = **<sup>P</sup>***γκ***Q***δ*(θ, *<sup>z</sup>*) . (22)

*m*<sup>1</sup> *m*<sup>2</sup> *m*<sup>3</sup> *m*<sup>4</sup> *m*<sup>5</sup> *m*<sup>1</sup> x 85 % 100 % 100 % 100 % *m*<sup>2</sup> 89 % x 92 % 100 % 100 % *m*<sup>3</sup> 100 % 92 % x 89 % 100 % *m*<sup>4</sup> 100 % 100 % 92 % x 98 % *m*<sup>5</sup> 100 % 100 % 100 % 98 % x

*<sup>j</sup> Pw*˜*<sup>j</sup>*

The general idea behind linear inversion methods is to find a linear operator **H** that acts on the data vector to yield a solution that minimises some functional, such as the variance of the residual between the estimated signal and the true signal, subject to some regularisation or prior-based constraints. Two different linear approaches have been described in recent literature [4, 49].

The most competitive method is that proposed in [4], in which the authors propose a Saskatoon filter [50, 51], which combines a Wiener filter and an inverse variance filter, with a tuning parameter *α* introduced that allows switching between the two. This gives rise to a minimum variance filter, expressed as

$$\hat{\mathbf{s}}\_{MV} = [\mathbf{a}\mathbf{1} + \mathbf{S}\mathbf{R}^\dagger \Sigma^{-1} \mathbf{R}]^{-1} \mathbf{S} \mathbf{R}^\dagger \Sigma^{-1} \mathbf{d} \,\tag{24}$$

where **S** ≡ ss† encodes prior information about the signal covariance, Σ ≡ nn† gives the covariance matrix of the noise, and **1** is the identity matrix.

This switching is designed to allow a balance between the increased constraining power offered by the Wiener filter over the inverse variance filter – which yields an improved signal-to-noise in the reconstruction – and the biasing that the Wiener filter imposes on the solution.

As discussed extensively in [4] and [49], linear methods give rise to a significant bias in the location of detected peaks, damping of the peak signal, and a substantial smearing of the density along the line of sight. Furthermore, the resolution attainable in the reconstructions obtained using linear methods is limited to the resolution of the data. In other words, we cannot reconstruct the density contrast field at higher resolution than the resolution of our input data which, in turn, is limited by the noise properties of the data.

Figure 5 shows the 3D reconstruction obtained in this way for a simulated cluster of galaxies at a redshift of *<sup>z</sup>*cl = 0.25. The tuning parameter used for this reconstruction was *<sup>α</sup>* <sup>=</sup> 0.05. The cluster was simulated according to an NFW halo with *<sup>M</sup>* <sup>=</sup> 1015*h*−1*M*<sup>⊙</sup> and *c* = 3, and the shear data were assumed to come from a galaxy distribution given by *<sup>p</sup>*(*z*) <sup>∝</sup> *<sup>z</sup>*2e<sup>−</sup>(1.4*z*)1.5 [52, 53], with a maximum redshift of *zmax* <sup>=</sup> 2.0 and a galaxy density of *ng* <sup>=</sup> 100 galaxies/arcmin2. The simulation covers a 1◦ <sup>×</sup> <sup>1</sup>◦ field binned into 60 <sup>×</sup> <sup>60</sup> angular pixels, and 20 redshift bins. Shown in the figure are both the 3D rendering of the reconstruction, and a 1D plot showing the four central lines of sight through the cluster as a function of redshift. The smearing, damping and redshift bias effects are all clearly visible in the 1D plot, where the amplitude of the cluster density contrast should be *δ* ∼ 36.

**Figure 5.** The reconstruction of a simulated cluster of galaxies using the method of [4]. Coloured broken lines in the 1D plot show the reconstruction of the four central lines of sight, while the solid line shows the true cluster density contrast.

## **5.2. Compressive sensing**

We consider some data *Yi* ( *<sup>i</sup>* ∈ [1, .., *<sup>m</sup>*]) acquired through the linear system

*Y* = **Θ***X* , (25)

In real life, signals are generally not "strictly" sparse, but are *compressible*; i.e. we can represent the signal in a basis or frame (Fourier, Wavelets, Curvelets, etc.) in which the curve obtained by plotting the obtained coefficients, sorted by their decreasing absolute values, exhibits a polynomial decay. Note that most natural signals and images are compressible in

We can therefore reformulate the CS equation above (Equation (26)) to include the data

where *<sup>X</sup>* <sup>=</sup> **<sup>Φ</sup>**∗*α*, and *<sup>α</sup>* are the coefficients of the transformed solution *<sup>X</sup>* in **<sup>Φ</sup>**, which is generally referred to as the *dictionary*. Each column represents a vector (also called an *atom*), which ideally should be chosen to match the features contained in *X*. If **Φ** admits a fast implicit transform (e.g. Fourier transform, Wavelet transform), fast algorithms exist to

One problem we face when considering CS in a given application is that very few matrices meet the RIP criterion. However, it has been shown that accurate recovery can be obtained

The mutual coherence measures the degree of similarity between the sparsifying basis and the sensing operator. Hence, in its relaxed definition, we consider a linear inverse problem

3. the mutual coherence *<sup>µ</sup>***Θ**,**<sup>Φ</sup>** is low. This will happen every time the matrix **<sup>A</sup>** = **ΘΦ** has the effect of spreading out the coefficients *α<sup>j</sup>* of the sparse signal on all measurements *Yi*.

Most CS applications described in the literature are based on such a soft CS definition. Compressed sensing was introduced for the first time in astronomy for data compression [57, 58], and a direct link between CS and radio-interferometric image reconstruction was recently established in [59], leading to dramatic improvement thanks to the sparse ℓ<sup>1</sup> recovery

The 3D weak lensing reconstruction problem can be seen to completely meet the soft-CS criteria above. The problem is underdetermined, as we seek a higher resolution than can be attained in the noise-limited observations, the matter density in the Universe is sparsely distributed, and the lensing operator spreads out the underlying density in a compressed

In particular, for the reconstruction of clusters of galaxies, we are in a perfect situation for sparse recovery because clusters are localised in the angular domain, and are not resolved along the line of sight owing to the bin size. They can therefore be modelled as Dirac *δ*−functions along the line of sight, while an isotropic wavelet basis can be used in the

as long the mutual coherence between **<sup>Θ</sup>** and **<sup>Φ</sup>**, *<sup>µ</sup>***Θ**,**<sup>Φ</sup>** = max*i*,*<sup>k</sup>*

*Y* = **ΘΦ***X* as being an instance of CS when

2. the signal is compressible in a given dictionary **Φ**,

1. the problem is underdetermined,

min*<sup>α</sup>* �*α*�<sup>1</sup> *<sup>s</sup>*.*t*. *<sup>Y</sup>* <sup>=</sup> **ΘΦ***<sup>α</sup>* , (27)

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

 **<sup>Θ</sup>***i*, **<sup>Φ</sup>***k*, http://dx.doi.org/10.5772/51871

49

, is low [56].

an appropriate basis.

transformation matrix **Φ**:

minimise Equation (27).

[60].

sensing way.

angular domain.

where **Θ** is an *m* × *n* matrix. Compressed sensing [54, 55] is a sampling/compression theory based on the sparsity of the observed signal, which shows that, under certain conditions, one can exactly recover a *k*-sparse signal (a signal for which only *k* pixels have values different from zero, out of *n* total pixels, where *k* < *n*) from *m* < *n* measurements.

This recovery is possible from undersampled data only if the sensing matrix **Θ** verifies the *restricted isometry property* (RIP) [see 54, for more details]. This property has the effect that each measurement *Yi* contains some information about the entire pixels of *X*; in other words, the sensing operator **Θ** acts to spread the information contained in *X* across many measurements *Yi*.

Under these two constraints – sparsity and a transformation meeting the RIP criterion – a signal can be recovered exactly even if the number of measurements *m* is much smaller than the number of unknown *n*. This means that, using CS methods, we will be able to outperform the well-known Shannon sampling criterion by far.

The solution *X* of (25) is obtained by minimizing

$$\min\_{X} \|X\|\_{1} \quad \text{s.t.} \ Y = \Theta X \,\,\,\,\tag{26}$$

where the <sup>ℓ</sup><sup>1</sup> norm is defined by �*X*�<sup>1</sup> <sup>=</sup> <sup>∑</sup>*<sup>i</sup>* <sup>|</sup> *Xi* <sup>|</sup>. The <sup>ℓ</sup><sup>1</sup> norm is well-known to be a sparsity-promoting function; i.e. minimisation of the ℓ<sup>1</sup> norm yields the most sparse solution to the inverse problem. Many optimisation methods have been proposed in recent years to minimise this equation. More details about CS and ℓ<sup>1</sup> minimisation algorithms can be found in [6].

In real life, signals are generally not "strictly" sparse, but are *compressible*; i.e. we can represent the signal in a basis or frame (Fourier, Wavelets, Curvelets, etc.) in which the curve obtained by plotting the obtained coefficients, sorted by their decreasing absolute values, exhibits a polynomial decay. Note that most natural signals and images are compressible in an appropriate basis.

We can therefore reformulate the CS equation above (Equation (26)) to include the data transformation matrix **Φ**:

$$\min\_{\mathfrak{a}} \||\mathfrak{a}||\_1 \quad \text{s.t. } Y = \Theta \Phi \mathfrak{a} \text{ .} \tag{27}$$

where *<sup>X</sup>* <sup>=</sup> **<sup>Φ</sup>**∗*α*, and *<sup>α</sup>* are the coefficients of the transformed solution *<sup>X</sup>* in **<sup>Φ</sup>**, which is generally referred to as the *dictionary*. Each column represents a vector (also called an *atom*), which ideally should be chosen to match the features contained in *X*. If **Φ** admits a fast implicit transform (e.g. Fourier transform, Wavelet transform), fast algorithms exist to minimise Equation (27).

One problem we face when considering CS in a given application is that very few matrices meet the RIP criterion. However, it has been shown that accurate recovery can be obtained as long the mutual coherence between **<sup>Θ</sup>** and **<sup>Φ</sup>**, *<sup>µ</sup>***Θ**,**<sup>Φ</sup>** = max*i*,*<sup>k</sup>* **<sup>Θ</sup>***i*, **<sup>Φ</sup>***k*, , is low [56]. The mutual coherence measures the degree of similarity between the sparsifying basis and the sensing operator. Hence, in its relaxed definition, we consider a linear inverse problem *Y* = **ΘΦ***X* as being an instance of CS when

1. the problem is underdetermined,

18 Open Questions in Cosmology

**5.2. Compressive sensing**

measurements *Yi*.

in [6].

(a) 3D plot of the cluster reconstruction. (b) The four central lines of sight

**Figure 5.** The reconstruction of a simulated cluster of galaxies using the method of [4]. Coloured broken lines in the 1D plot show the reconstruction of the four central lines of sight, while the solid line shows the true cluster density contrast.

where **Θ** is an *m* × *n* matrix. Compressed sensing [54, 55] is a sampling/compression theory based on the sparsity of the observed signal, which shows that, under certain conditions, one can exactly recover a *k*-sparse signal (a signal for which only *k* pixels have values different

This recovery is possible from undersampled data only if the sensing matrix **Θ** verifies the *restricted isometry property* (RIP) [see 54, for more details]. This property has the effect that each measurement *Yi* contains some information about the entire pixels of *X*; in other words, the sensing operator **Θ** acts to spread the information contained in *X* across many

Under these two constraints – sparsity and a transformation meeting the RIP criterion – a signal can be recovered exactly even if the number of measurements *m* is much smaller than the number of unknown *n*. This means that, using CS methods, we will be able to outperform

where the <sup>ℓ</sup><sup>1</sup> norm is defined by �*X*�<sup>1</sup> <sup>=</sup> <sup>∑</sup>*<sup>i</sup>* <sup>|</sup> *Xi* <sup>|</sup>. The <sup>ℓ</sup><sup>1</sup> norm is well-known to be a sparsity-promoting function; i.e. minimisation of the ℓ<sup>1</sup> norm yields the most sparse solution to the inverse problem. Many optimisation methods have been proposed in recent years to minimise this equation. More details about CS and ℓ<sup>1</sup> minimisation algorithms can be found

We consider some data *Yi* ( *<sup>i</sup>* ∈ [1, .., *<sup>m</sup>*]) acquired through the linear system

from zero, out of *n* total pixels, where *k* < *n*) from *m* < *n* measurements.

min

the well-known Shannon sampling criterion by far. The solution *X* of (25) is obtained by minimizing

through the cluster.

*Y* = **Θ***X* , (25)

*<sup>X</sup>* �*X*�<sup>1</sup> *<sup>s</sup>*.*t*. *<sup>Y</sup>* <sup>=</sup> **<sup>Θ</sup>***<sup>X</sup>* , (26)


Most CS applications described in the literature are based on such a soft CS definition. Compressed sensing was introduced for the first time in astronomy for data compression [57, 58], and a direct link between CS and radio-interferometric image reconstruction was recently established in [59], leading to dramatic improvement thanks to the sparse ℓ<sup>1</sup> recovery [60].

The 3D weak lensing reconstruction problem can be seen to completely meet the soft-CS criteria above. The problem is underdetermined, as we seek a higher resolution than can be attained in the noise-limited observations, the matter density in the Universe is sparsely distributed, and the lensing operator spreads out the underlying density in a compressed sensing way.

In particular, for the reconstruction of clusters of galaxies, we are in a perfect situation for sparse recovery because clusters are localised in the angular domain, and are not resolved along the line of sight owing to the bin size. They can therefore be modelled as Dirac *δ*−functions along the line of sight, while an isotropic wavelet basis can be used in the angular domain.

## **5.3. Results and future prospects**

In [61], we present an algorithm to solve the 3D lensing problem. In this method, the 3D lensing problem is reduced to a one-dimensional problem, by taking as the data vector the (noisy) lensing convergence along each line of sight, which is related to the density contrast through Equation (21). Each line of sight can therefore be considered independently.

Clearly, a one-dimensional implementation throws away information, because we do not account at all for the correlation between neighbouring lines of sight that will arise in the presence of a large structure in the image; however, reducing the problem to a single dimension is fast and easy to implement, and allows us to test the efficacy of the algorithm using a particularly simple basis function through which we impose sparsity.

In Figure 6, we show the reconstruction obtained in this way for the simulated cluster described in section 5.1. The line of sight plot shows a clear improvement in all the target areas, reducing the bias, smearing and damping effects seen using linear methods. Small-scale noise is present, particularly at high redshifts, but this is likely due to overfitting in the algorithm used, and may be reduced by considering the whole 3D field, rather than each line of sight independently (for a full discussion of this issue, see [61]).

**Figure 6.** The reconstruction of a simulated cluster of galaxies using the method of [61]. Coloured broken lines in the 1D plot show the reconstruction of the four central lines of sight, while the solid line shows the true cluster density contrast.

through the cluster.

(a) 3D plot of the cluster reconstruction. (b) The four central lines of sight

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

(a) 3D plot of the cluster reconstruction. (b) The four central lines of sight

**Figure 8.** The reconstruction of a simulated cluster of galaxies at a redshift of *z*cl = 0.2 using a 3D CS approach, and with

This marked improvement in the cluster reconstruction seen in the 3D CS approach represents a definite step in the right direction for weak lensing studies. There is much work to be done, however. The application of this CS approach to the 3D lensing problem is a very recent development, and many questions remain: the choice of algorithm, for example; how best to control the noise in the reconstruction; how to deal with photometric redshift errors; by what factor the reconstruction resolution might be improved as compared to the

improved resolution on the reconstruction as compared to the input data.

data.

**Figure 7.** The reconstruction of a simulated cluster of galaxies at a redshift of *z*cl = 0.2, and with improved resolution on the

This is demonstrated in Figure 8, which shows reconstructions of the cluster at redshift *<sup>z</sup>*cl = 0.2, with the same improved resolution in the reconstruction as before, but this time using the three-dimensional CS approach. Dramatic improvement is seen on all fronts, with the reconstructions showing no bias or redshift smearing, and very little amplitude damping,

and with none of the small-scale false detections seen in the 1D CS approach.

reconstruction as compared to the input data.

through the cluster.

http://dx.doi.org/10.5772/51871

51

through the cluster.

This reconstruction was undertaken using the same resolution on the reconstruction as on the input data. However, we know that the CS approach is particularly well-adapted to dealing with ill-posed inversion problems. In order to test this, we consider a cluster at a redshifts of *<sup>z</sup>*cl = 0.2, simulated as before. We use *<sup>N</sup>*sp = 20 redshift bins in our data, but now aim to reconstruct our density contrast with a redshift resolution of *<sup>N</sup>*lp = 25. The results are shown in Figure 7. Here, again, we see small scale noise at high-redshift, but the overall smearing and redshift bias issues seen in the linear reconstructions is absent.

As noted previously, the one-dimensional solver employed here throws away a wealth of information about the angular correlation of the lensing signal, and is thus not optimal. Indeed, a simple algorithm based on this CS approach, but implemented as a full three-dimensional treatment, offers marked improvements in the quality of the reconstructions [62].

<sup>50</sup> Open Questions in Cosmology Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics 21 Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics http://dx.doi.org/10.5772/51871 51

20 Open Questions in Cosmology

**5.3. Results and future prospects**

In [61], we present an algorithm to solve the 3D lensing problem. In this method, the 3D lensing problem is reduced to a one-dimensional problem, by taking as the data vector the (noisy) lensing convergence along each line of sight, which is related to the density contrast

Clearly, a one-dimensional implementation throws away information, because we do not account at all for the correlation between neighbouring lines of sight that will arise in the presence of a large structure in the image; however, reducing the problem to a single dimension is fast and easy to implement, and allows us to test the efficacy of the algorithm

In Figure 6, we show the reconstruction obtained in this way for the simulated cluster described in section 5.1. The line of sight plot shows a clear improvement in all the target areas, reducing the bias, smearing and damping effects seen using linear methods. Small-scale noise is present, particularly at high redshifts, but this is likely due to overfitting in the algorithm used, and may be reduced by considering the whole 3D field, rather than

(a) 3D plot of the cluster reconstruction. (b) The four central lines of sight

**Figure 6.** The reconstruction of a simulated cluster of galaxies using the method of [61]. Coloured broken lines in the 1D plot show the reconstruction of the four central lines of sight, while the solid line shows the true cluster density contrast.

This reconstruction was undertaken using the same resolution on the reconstruction as on the input data. However, we know that the CS approach is particularly well-adapted to dealing with ill-posed inversion problems. In order to test this, we consider a cluster at a redshifts of *<sup>z</sup>*cl = 0.2, simulated as before. We use *<sup>N</sup>*sp = 20 redshift bins in our data, but now aim to reconstruct our density contrast with a redshift resolution of *<sup>N</sup>*lp = 25. The results are shown in Figure 7. Here, again, we see small scale noise at high-redshift, but the overall smearing

As noted previously, the one-dimensional solver employed here throws away a wealth of information about the angular correlation of the lensing signal, and is thus not optimal. Indeed, a simple algorithm based on this CS approach, but implemented as a full three-dimensional treatment, offers marked improvements in the quality of the

and redshift bias issues seen in the linear reconstructions is absent.

reconstructions [62].

through the cluster.

through Equation (21). Each line of sight can therefore be considered independently.

using a particularly simple basis function through which we impose sparsity.

each line of sight independently (for a full discussion of this issue, see [61]).

**Figure 7.** The reconstruction of a simulated cluster of galaxies at a redshift of *z*cl = 0.2, and with improved resolution on the reconstruction as compared to the input data.

This is demonstrated in Figure 8, which shows reconstructions of the cluster at redshift *<sup>z</sup>*cl = 0.2, with the same improved resolution in the reconstruction as before, but this time using the three-dimensional CS approach. Dramatic improvement is seen on all fronts, with the reconstructions showing no bias or redshift smearing, and very little amplitude damping, and with none of the small-scale false detections seen in the 1D CS approach.

**Figure 8.** The reconstruction of a simulated cluster of galaxies at a redshift of *z*cl = 0.2 using a 3D CS approach, and with improved resolution on the reconstruction as compared to the input data.

This marked improvement in the cluster reconstruction seen in the 3D CS approach represents a definite step in the right direction for weak lensing studies. There is much work to be done, however. The application of this CS approach to the 3D lensing problem is a very recent development, and many questions remain: the choice of algorithm, for example; how best to control the noise in the reconstruction; how to deal with photometric redshift errors; by what factor the reconstruction resolution might be improved as compared to the data.

Yet it is clear that this approach opens up the possibility of being able to generate accurate reconstructions of the density contrast using weak lensing measurements, and perhaps using information such as the 3D peak count – in combination with constraints from other probes – to place ever-tighter constraints on our cosmological model. This, in turn, will offer a unique insight into the nature and properties of the dark components of the Universe.

[6] J.-L. Starck, F. Murtagh, and M.J. Fadili. *Sparse Image and Signal Processing*. Cambridge

Exploring the Components of the Universe Through Higher-Order Weak Lensing Statistics

http://dx.doi.org/10.5772/51871

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## **Acknowledgements**

This work is supported by the European Research Council grant SparseAstro (ERC-228261). The authors would like to thank Jalal Fadili for useful discussions, and Jake VanderPlas for helpful comments and use of his code to generate the linear reconstructions.

## **Author details**

Adrienne Leonard1,<sup>⋆</sup>, Jean-Luc Starck1, Sandrine Pires1 and François-Xavier Dupé<sup>2</sup>

<sup>⋆</sup> Address all correspondence to: adrienne.leonard@cea.fr

1 Laboratoire AIM (UMR 7158), CEA/DSM-CNRS-Université Paris Diderot, IRFU, SEDI-SAP, Service d'Astrophysique, Centre de Saclay, France

2 Laboratoire d'Informatique Fondamentale, Centre de Mathématique et d'Informatique, UMR CNRS, Aix-Marseille Université, Marseille, France

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22 Open Questions in Cosmology

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Adrienne Leonard1,<sup>⋆</sup>, Jean-Luc Starck1, Sandrine Pires1 and François-Xavier Dupé<sup>2</sup>

<sup>⋆</sup> Address all correspondence to: adrienne.leonard@cea.fr

UMR CNRS, Aix-Marseille Université, Marseille, France

Service d'Astrophysique, Centre de Saclay, France

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This work is supported by the European Research Council grant SparseAstro (ERC-228261). The authors would like to thank Jalal Fadili for useful discussions, and Jake VanderPlas for

1 Laboratoire AIM (UMR 7158), CEA/DSM-CNRS-Université Paris Diderot, IRFU, SEDI-SAP,

2 Laboratoire d'Informatique Fondamentale, Centre de Mathématique et d'Informatique,

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**Chapter 3**

**Provisional chapter**

**Extragalactic Compact Sources in the** *Planck* **Sky and**

As of mid August 2012, the *Planck* cosmic microwave background anisotropy probe1 [1,2] – launched into space on 14 May 2009 at 13:12:02 UTC, by an Ariane 5 ECA launcher, from the Guiana Space Centre, Kourou, French Guiana – is still successfully operating. The spacecraft accumulated data with its two instruments, the High Frequency Instrument (HFI) [3], based on bolometers working between 100 and 857 GHz, and the Low Frequency Instrument (LFI) [4], based on radiometers working between 30 and 70 GHz, up to the consumption of the cryogenic liquids on January 2012, achieving ≃ 29.5 months of integration, corresponding to about five complete sky surveys. A further 12 months extension is on-going for observations with LFI only, cooled down with the cryogenic system provided by HFI. Moreover, *Planck* is

Thanks to its great sensitivity and resolution on the whole sky and to its wide frequency coverage that allows a substantial progress in foreground modeling and removal, *Planck* will open a new era in our understanding of the Universe and of its astrophysical structures (see [5] for a full description of the *Planck* Scientific programme). *Planck* will improve the accuracy of current measures of a wide set of cosmological parameters by a factor from ∼ 3 to ∼ 10 and will characterize the geometry of the Universe with unprecedented accuracy. *Planck* will shed light on many of the open issues in the connection between the early stages of the Universe and the evolution of the cosmic structures, from the characterization of primordial

<sup>⋆</sup> This paper is based largely on the *Planck* Early Release Compact Source Catalogue, a product of ESA and the *Planck* Collaboration. Any material presented in this review that is not already described in *Planck* Collaboration papers

<sup>1</sup> *Planck* (http://www.esa.int/Planck) is a project of the European Space Agency - ESA - with instruments provided by two scientific Consortia funded by ESA member states (in particular the lead countries: France and Italy) with contributions from NASA (USA), and telescope reflectors provided in a collaboration between ESA and a scientific

and reproduction in any medium, provided the original work is properly cited.

©2012 Toffolatti et al., licensee InTech. This is an open access chapter 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. © 2012 Toffolatti et al.; licensee InTech. This is an open access article 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.

© 2012 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,

conditions and perturbations, to the late phases of cosmological reionization.

represents the views of the authors and not necessarily those of the *Planck* Collaboration.

**Extragalactic Compact Sources in the** *Planck* **Sky**

**Their Cosmological Implications**

**and Their Cosmological Implications**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Luigi Toffolatti, Carlo Burigana,

Luigi Toffolatti, Carlo Burigana, Francisco Argüeso and José M. Diego

http://dx.doi.org/10.5772/52908

**1. Introduction**

Francisco Argüeso and José M. Diego

sensitive to linear polarization up to 353 GHz.

Consortium led and funded by Denmark.


**Provisional chapter**

## **Extragalactic Compact Sources in the** *Planck* **Sky and Their Cosmological Implications Extragalactic Compact Sources in the** *Planck* **Sky and Their Cosmological Implications**

Luigi Toffolatti, Carlo Burigana, Francisco Argüeso and José M. Diego Luigi Toffolatti, Carlo Burigana, Francisco Argüeso and José M. Diego

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52908

## **1. Introduction**

26 Open Questions in Cosmology

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*Astronomical Society*, 395:1733–1742, 2009.

As of mid August 2012, the *Planck* cosmic microwave background anisotropy probe1 [1,2] – launched into space on 14 May 2009 at 13:12:02 UTC, by an Ariane 5 ECA launcher, from the Guiana Space Centre, Kourou, French Guiana – is still successfully operating. The spacecraft accumulated data with its two instruments, the High Frequency Instrument (HFI) [3], based on bolometers working between 100 and 857 GHz, and the Low Frequency Instrument (LFI) [4], based on radiometers working between 30 and 70 GHz, up to the consumption of the cryogenic liquids on January 2012, achieving ≃ 29.5 months of integration, corresponding to about five complete sky surveys. A further 12 months extension is on-going for observations with LFI only, cooled down with the cryogenic system provided by HFI. Moreover, *Planck* is sensitive to linear polarization up to 353 GHz.

Thanks to its great sensitivity and resolution on the whole sky and to its wide frequency coverage that allows a substantial progress in foreground modeling and removal, *Planck* will open a new era in our understanding of the Universe and of its astrophysical structures (see [5] for a full description of the *Planck* Scientific programme). *Planck* will improve the accuracy of current measures of a wide set of cosmological parameters by a factor from ∼ 3 to ∼ 10 and will characterize the geometry of the Universe with unprecedented accuracy. *Planck* will shed light on many of the open issues in the connection between the early stages of the Universe and the evolution of the cosmic structures, from the characterization of primordial conditions and perturbations, to the late phases of cosmological reionization.

<sup>1</sup> *Planck* (http://www.esa.int/Planck) is a project of the European Space Agency - ESA - with instruments provided by two scientific Consortia funded by ESA member states (in particular the lead countries: France and Italy) with contributions from NASA (USA), and telescope reflectors provided in a collaboration between ESA and a scientific Consortium led and funded by Denmark.

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. © 2012 Toffolatti et al.; licensee InTech. This is an open access article 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. © 2012 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.

©2012 Toffolatti et al., licensee InTech. This is an open access chapter distributed under the terms of the

<sup>⋆</sup> This paper is based largely on the *Planck* Early Release Compact Source Catalogue, a product of ESA and the *Planck* Collaboration. Any material presented in this review that is not already described in *Planck* Collaboration papers represents the views of the authors and not necessarily those of the *Planck* Collaboration.

The *Planck* perspectives on some crucial selected topics linking cosmology to fundamental physics (the neutrino masses and effective number of species, the primordial helium abundance, various physics fundamental constants, the parity property of CMB maps and its connection with CPT symmetry with emphasis to the Cosmic Birefringence, the detection of the stochastic field of gravitational waves) will also show how *Planck* represents an extremely powerful *fundamental and particle physics laboratory*. Some of these analyses will be carried out mainly through a precise measure of CMB anisotropy angular power spectrum (APS) in temperature, polarization and in their correlations, whereas others, in particular those related to the geometry of the Universe and to the research of non-Gaussianity signatures, are based on the exploitation of the anisotropy pattern. The most ambitious goal is the possible detection of the so-called B-mode APS.

LFI

HFI

by 1/*σ*<sup>2</sup>

*<sup>j</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup>* 1/*σ*<sup>2</sup>

*j*,*i*

Frequency (GHz) 30 44 70 FWHM 33.34 26.81 13.03 N of R (or feeds) 4 (2) 6 (3) 12 (6) EB 6 8.8 14 NET 159 197 158 *δ*T/T [*µ*K/K] (in *T*) 2.04 3.14 5.17 *δ*T/T [*µ*K/K] (in *P*) 2.88 4.44 7.32

Frequency (GHz) 217 353 FWHM in *T* (*P*) 4.6 (4.6) 4.7 (4.6) N of B in *T* (*P*) 4 (8) 4 (8) EB in *T* (*P*) 72 (63) 99 (102) NET in *T* (*P*) 91 (132) 277 (404) *δ*T/T [*µ*K/K] in *T* (*P*) 3.31 (6.24) 13.7 (26.2)

and per detector for HFI) in 1 sec of integration (NET, in *µ*K ·

a Gaussian symmetric beam, is *W*<sup>ℓ</sup> = exp(−ℓ(ℓ + 1)*σ*<sup>2</sup>

instrumental noise only determines the capability of detecting the B mode.

cosmological channels (i.e., 70 GHz channel), respectively (see Table 1).

by combining polarization measurements.

HFI

**Table 1.** *Planck* performance. The average sensitivity, *δ*T/T, per (FWHM)<sup>2</sup> resolution element (FWHM: Full Width at Half Maximum of the beam response function, is indicated in arcmin) is given in CMB temperature units (i.e., equivalent thermodynamic temperature) for 29.5 (plus 12 for LFI) months of integration. The white noise (per frequency channel for LFI

√

here used are: N of R (or B) = number of radiometers (or bolometers), EB = effective bandwidth (in GHz). Adapted from [6, 7] and consistent with [3, 4]. Note that at 100 GHz all bolometers are polarized and the equivalent temperature value is obtained

beamwidth which defines the angular resolution of the experiment. For *fsky* = 1 the first term in parenthesis defines the "cosmic variance", an intrinsic limit on the accuracy at which the APS of a certain cosmological model defined by a suitable set of parameters can be derived with CMB anisotropy measurements5. It typically dominates the uncertainty on the APS at low ℓ because of the small, 2ℓ + 1, number of modes *m* for each ℓ. The second term in parenthesis characterizes the instrumental noise, that never vanishes in the case of real experiments. Note also the coupling between experiment sensitivity and resolution, the former defining the low ℓ experimental uncertainty, namely for *W*<sup>ℓ</sup> close to unit, the latter determining the exponential loss in sensitivity at angular scales comparable with the beamwidth. We computed an overall sensitivity value, weighted over the channels, defined

listed in Table 1. FWHM values of 13 and 33 arcmin are used in Fig.1 to define the overall combination of *Planck* sensitivity and resolution, i.e. the computation of the effective beam window function6, relevant for the sensitivity at high ℓ. Finally, to improve the signal to noise ratio in the APS sensitivity, especially at high multipoles, a multipole binning is usually applied. Of course, the real sensitivity of the whole mission will have to also include the potential residuals of systematic effects. The *Planck* mission has been designed to suppress

<sup>5</sup> Note that the cosmic and sampling variance (74% sky coverage excluding the sky regions mostly affected by Galactic emission) implies a dependence of the overall sensitivity on *r* at low multipoles, relevant to the parameter estimation;

<sup>6</sup> In fact, it is possible to smooth maps acquired at higher frequencies with smaller beamwidths to the lowest resolution corresponding to a given experiment. We adopt here FWHM values of 33 and 13 arcmin, which correspond to the lowest resolution of all the *Planck* instruments (i.e., 30 GHz channel) and to the lowest resolution of the so called

, where *j* = *T* and *i* indicates the sensitivity of each frequency channel,

HFI

Extragalactic Compact Sources in the *Planck* Sky and Their Cosmological Implications

Frequency (GHz) 100 143 FWHM in *T* (*P*) 9.6 (9.6) 7.1 (6.9) N of B in *T* (*P*) – (8) 4 (8) EB in *T* (*P*) 33 (33) 43 (46) NET in *T* (*P*) 100 (100) 62 (82) *δ*T/T [*µ*K/K] (in *T*) 2.04 1.56 *δ*T/T [*µ*K/K] (in *P*) 3.31 2.83

http://dx.doi.org/10.5772/52908

59

Frequency (GHz) 545 857 FWHM in *T* 4.7 4.3 N of B in *T* 4 4 EB in *T* 169 257 NET in *T* 2000 91000 *δ*T/T [*µ*K/K] in *T* 103 4134

s) is also given in CMB temperature units. The other acronyms

*<sup>B</sup>*), with *<sup>σ</sup><sup>B</sup>* <sup>=</sup> *FWHM*/

√

8 ln 2 the

The first scientific results2, the so-called *Planck* Early Papers <sup>3</sup> have been released in January 2011 and published by Astronomy and Astrophysics (EDP sciences), in the dedicated Volume 536 (December 2011). A further set of astrophysical results has been presented on the occasion of the Conference *Astrophysics from radio to sub-millimeter wavelengths: the Planck view and other experiments*<sup>4</sup> held in Bologna on 13-17 February 2012. Several articles have been already submitted in 2012 and others are in preparation, constituting the set of so-called *Planck* Intermediate Papers.

The outline of this Chapter is as follows: in Section 2 we briefly sketch the main characteristics and the capabilities of the ESA *Planck* mission; in Section 3 we discuss the most recent detection methods for compact source detection; in Section 4 the SZ effect, detected by *Planck* in many cluster of galaxies and its importance for cosmological studies are analyzed; Section 5 is dedicated to summarize current results obtained by *Planck* data on the properties of EPS; finally, Section 6, discusses the very important results up to now achieved by the analysis of CIB anisotropies detected by *Planck*.

## **2. The ESA** *Planck* **mission: Overview**

CMB experimental data are affected by uncertainties due to instrumental noise (crucial at high multipoles, ℓ, i.e. small angular scales), cosmic and sampling variance (crucial at low ℓ, i.e. large angular scales) and from systematic effects. The uncertainty on the angular power spectrum is given by the combination of three components, cosmic and sampling variance, and instrumental noise, and it is approximately given by [9]:

$$\frac{\delta \mathcal{C}\_{\ell}}{\mathcal{C}\_{\ell}} = \sqrt{\frac{2}{f\_{sky}(2\ell + 1)}} \left( 1 + \frac{A\sigma^2}{N\mathcal{C}\_{\ell}W\_{\ell}} \right) \,. \tag{1}$$

Here *fsky* is the sky coverage, *A* is the surveyed area, *σ* is the instrumental rms noise per pixel, *N* is the total pixel number, *W*ℓ is the beam window function that, in the case of

<sup>2</sup> http://www.sciops.esa.int/index.php?project=PLANCK&page=Planck\_Published\_Papers

<sup>3</sup> The *Planck* Early papers describe the instrument performance in flight including thermal behaviour (papers I–IV), the LFI and HFI data analysis pipelines (papers V–VI), and the main astrophysical results (papers VII-XXVI). These papers have complemented by a subsequent work, published in 2012, based on a combination of high energy and *Planck* observations (see [8]).

<sup>4</sup> http://www.iasfbo.inaf.it/events/planck-2012/


2 Open Questions in Cosmology

possible detection of the so-called B-mode APS.

analysis of CIB anisotropies detected by *Planck*.

**2. The ESA** *Planck* **mission: Overview**

and instrumental noise, and it is approximately given by [9]:

*δC*ℓ *C*ℓ = 2

<sup>2</sup> http://www.sciops.esa.int/index.php?project=PLANCK&page=Planck\_Published\_Papers

*fsky*(2ℓ + <sup>1</sup>)

Here *fsky* is the sky coverage, *A* is the surveyed area, *σ* is the instrumental rms noise per pixel, *N* is the total pixel number, *W*ℓ is the beam window function that, in the case of

<sup>3</sup> The *Planck* Early papers describe the instrument performance in flight including thermal behaviour (papers I–IV), the LFI and HFI data analysis pipelines (papers V–VI), and the main astrophysical results (papers VII-XXVI). These papers have complemented by a subsequent work, published in 2012, based on a combination of high energy and

*Planck* Intermediate Papers.

*Planck* observations (see [8]).

<sup>4</sup> http://www.iasfbo.inaf.it/events/planck-2012/

The *Planck* perspectives on some crucial selected topics linking cosmology to fundamental physics (the neutrino masses and effective number of species, the primordial helium abundance, various physics fundamental constants, the parity property of CMB maps and its connection with CPT symmetry with emphasis to the Cosmic Birefringence, the detection of the stochastic field of gravitational waves) will also show how *Planck* represents an extremely powerful *fundamental and particle physics laboratory*. Some of these analyses will be carried out mainly through a precise measure of CMB anisotropy angular power spectrum (APS) in temperature, polarization and in their correlations, whereas others, in particular those related to the geometry of the Universe and to the research of non-Gaussianity signatures, are based on the exploitation of the anisotropy pattern. The most ambitious goal is the

The first scientific results2, the so-called *Planck* Early Papers <sup>3</sup> have been released in January 2011 and published by Astronomy and Astrophysics (EDP sciences), in the dedicated Volume 536 (December 2011). A further set of astrophysical results has been presented on the occasion of the Conference *Astrophysics from radio to sub-millimeter wavelengths: the Planck view and other experiments*<sup>4</sup> held in Bologna on 13-17 February 2012. Several articles have been already submitted in 2012 and others are in preparation, constituting the set of so-called

The outline of this Chapter is as follows: in Section 2 we briefly sketch the main characteristics and the capabilities of the ESA *Planck* mission; in Section 3 we discuss the most recent detection methods for compact source detection; in Section 4 the SZ effect, detected by *Planck* in many cluster of galaxies and its importance for cosmological studies are analyzed; Section 5 is dedicated to summarize current results obtained by *Planck* data on the properties of EPS; finally, Section 6, discusses the very important results up to now achieved by the

CMB experimental data are affected by uncertainties due to instrumental noise (crucial at high multipoles, ℓ, i.e. small angular scales), cosmic and sampling variance (crucial at low ℓ, i.e. large angular scales) and from systematic effects. The uncertainty on the angular power spectrum is given by the combination of three components, cosmic and sampling variance,

> 1 +

*Aσ*<sup>2</sup> *NC*ℓ*W*ℓ

. (1)

**Table 1.** *Planck* performance. The average sensitivity, *δ*T/T, per (FWHM)<sup>2</sup> resolution element (FWHM: Full Width at Half Maximum of the beam response function, is indicated in arcmin) is given in CMB temperature units (i.e., equivalent thermodynamic temperature) for 29.5 (plus 12 for LFI) months of integration. The white noise (per frequency channel for LFI and per detector for HFI) in 1 sec of integration (NET, in *µ*K · √s) is also given in CMB temperature units. The other acronyms here used are: N of R (or B) = number of radiometers (or bolometers), EB = effective bandwidth (in GHz). Adapted from [6, 7] and consistent with [3, 4]. Note that at 100 GHz all bolometers are polarized and the equivalent temperature value is obtained by combining polarization measurements.

a Gaussian symmetric beam, is *W*<sup>ℓ</sup> = exp(−ℓ(ℓ + 1)*σ*<sup>2</sup> *<sup>B</sup>*), with *<sup>σ</sup><sup>B</sup>* <sup>=</sup> *FWHM*/ √ 8 ln 2 the beamwidth which defines the angular resolution of the experiment. For *fsky* = 1 the first term in parenthesis defines the "cosmic variance", an intrinsic limit on the accuracy at which the APS of a certain cosmological model defined by a suitable set of parameters can be derived with CMB anisotropy measurements5. It typically dominates the uncertainty on the APS at low ℓ because of the small, 2ℓ + 1, number of modes *m* for each ℓ. The second term in parenthesis characterizes the instrumental noise, that never vanishes in the case of real experiments. Note also the coupling between experiment sensitivity and resolution, the former defining the low ℓ experimental uncertainty, namely for *W*<sup>ℓ</sup> close to unit, the latter determining the exponential loss in sensitivity at angular scales comparable with the beamwidth. We computed an overall sensitivity value, weighted over the channels, defined by 1/*σ*<sup>2</sup> *<sup>j</sup>* <sup>=</sup> <sup>∑</sup>*<sup>i</sup>* 1/*σ*<sup>2</sup> *j*,*i* , where *j* = *T* and *i* indicates the sensitivity of each frequency channel, listed in Table 1. FWHM values of 13 and 33 arcmin are used in Fig.1 to define the overall combination of *Planck* sensitivity and resolution, i.e. the computation of the effective beam window function6, relevant for the sensitivity at high ℓ. Finally, to improve the signal to noise ratio in the APS sensitivity, especially at high multipoles, a multipole binning is usually applied. Of course, the real sensitivity of the whole mission will have to also include the potential residuals of systematic effects. The *Planck* mission has been designed to suppress

<sup>5</sup> Note that the cosmic and sampling variance (74% sky coverage excluding the sky regions mostly affected by Galactic emission) implies a dependence of the overall sensitivity on *r* at low multipoles, relevant to the parameter estimation; instrumental noise only determines the capability of detecting the B mode.

<sup>6</sup> In fact, it is possible to smooth maps acquired at higher frequencies with smaller beamwidths to the lowest resolution corresponding to a given experiment. We adopt here FWHM values of 33 and 13 arcmin, which correspond to the lowest resolution of all the *Planck* instruments (i.e., 30 GHz channel) and to the lowest resolution of the so called cosmological channels (i.e., 70 GHz channel), respectively (see Table 1).

**Figure 1.** CMB E polarization modes (black long dashes) compatible with *WMAP* data and CMB B polarization modes (black solid lines) for different tensor-to-scalar ratios *T*/*S* = *r* of primordial perturbations are compared to the *Planck* overall sensitivity to the APS assuming two different FWHM angular resolutions (33 and 13 arcmin) and the overall sensitivity corresponding to the whole mission duration (and also to two surveys only: upper curve in black thick dots, labeled 13 arcmin). The expected noise is assumed to be properly subtracted. The plots include cosmic and sampling variance plus instrumental noise (green dots for B modes, green long dashes for E modes, labeled with cv+sv+n; black thick dots, noise only) assuming a multipole binning of 30%.The B mode induced by lensing (blue dots) is also shown. Galactic synchrotron (purple dashes) and dust (purple dot-dashes) polarized emissions produce the overall Galactic foreground (purple three dots-dashes). *WMAP* 3-yr power-law fits for uncorrelated dust and synchrotron have been used. For comparison, *WMAP* 3-yr results (http://lambda.gsfc.nasa.gov/) derived from the foreground maps using HEALPix tools (http://healpix.jpl.nasa.gov/) [12] are shown (red three dots-dashes broken line): power-law fits provide (generous) upper limits to the power at low multipoles. Residual contamination levels by Galactic foregrounds (purple three dot-dashes) are shown for 10%, 5%, and 3% of the map level, at increasing thickness. We plot also as thick and thin green dashes realistic estimates of the residual contribution of un-subtracted extragalactic sources, *C*res,PS <sup>ℓ</sup> and the corresponding uncertainty, *δC*res,PS <sup>ℓ</sup> .

**Figure 2.** CMB removed *Planck* full-sky maps. From left to right and from top to bottom: 30, 44, 70; 100, 143, 217; 353, 545, and 857 GHz, respectively. Credits: Zacchei, et al., A&A, Vol. 536, A5, 2011; *Planck* HFI Core Team, A&A, Vol. 536, A6, 2011b,

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of the foreground signal *at map level* (or, equivalently, its square at power spectrum level) contaminates CMB maps. Of course, one can easily rescale the following results to any fraction of residual foreground contamination. The frequency of 70 GHz, i.e. the *Planck* channel where Galactic foregrounds are expected to be at their minimum level, at least at

For what concerns CMB temperature fluctuations produced by undetected EPS [13], we adopt the recent (conservative) estimate of their Poisson contribution to the (polarized) APS [14] at 100 GHz<sup>7</sup> by assuming a detection threshold of ≃ 0.1 Jy. We also assume a potential residual coming from an uncertainty in the subtraction of this contribution computed by assuming a relative uncertainty of ≃ 10% in the knowledge of their degree of polarization and in the determination of the source detection threshold, implying a reduction to ≃ 30% of the original level. Except at very high multipoles, their residual is likely significantly below

The first publications of the main cosmological (i.e. properly based on *Planck* CMB maps) implications are expected in early 2013, together with the delivery of a first set of *Planck* maps and cosmological products coming from the first 15 months of data. They will be mainly based on temperature data. Waiting for these products, a first multifrequency view of the *Planck* astrophysical sky has been presented in the Early Papers: Fig. 2 reports the first LFI and HFI frequency (CMB subtracted) maps. These maps are the basis of the construction of the *Planck* Early Release Compact Source Catalog (ERCSC) (see [15] and *The Explanatory Supplement to the Planck Early Release Compact Source Catalogue*), the first *Planck*

<sup>7</sup> We adopt here a frequency slightly larger than that considered for Galactic foregrounds (70 GHz) because at small angular scales, where point sources are more critical, the minimum of foreground contamination is likely shifted to

angular scales above ∼ one degree, is adopted as reference.

reproduced with permission © ESO.

that coming from Galactic foregrounds.

product delivered to the scientific community.

higher frequencies.

potential systematic effects down to ∼ *µK* level or below. Fig.1 compares CMB polarization modes with the ideal sensitivity of *Planck* (including also a 15% level of HFI data loss because of cosmic rays; see [10]) and the signals coming from astrophysical foregrounds as discussed below.

CMB anisotropy maps are contaminated by a significant level of foreground emission of both Galactic and extragalactic origin. For polarization, the most critical Galactic foregrounds are certainly synchrotron and thermal dust emission, whereas free-free emission gives a negligible contribution. Other components, like spinning dust and "haze", are still poorly known, particularly in polarization. Synchrotron emission is the dominant Galactic foreground signal at low frequencies, up to ∼60 GHz, where dust emission starts to dominate. External galaxies are critical only at high ℓ, and extragalactic radio sources are likely the most crucial in polarization up to frequencies ∼200 GHz, the most suitable for CMB anisotropy experiments. We parameterize a potential residual from non perfect cleaning of CMB maps from Galactic foregrounds simply assuming that a certain fraction <sup>60</sup> Open Questions in Cosmology Extragalactic Compact Sources in the *Planck* Sky and Their Cosmological Implications 5 Extragalactic Compact Sources in the *Planck* Sky and Their Cosmological Implications http://dx.doi.org/10.5772/52908 61

4 Open Questions in Cosmology

*C*res,PS

below.

<sup>ℓ</sup> and the corresponding uncertainty, *δC*res,PS

<sup>ℓ</sup> .

**Figure 1.** CMB E polarization modes (black long dashes) compatible with *WMAP* data and CMB B polarization modes (black solid lines) for different tensor-to-scalar ratios *T*/*S* = *r* of primordial perturbations are compared to the *Planck* overall sensitivity to the APS assuming two different FWHM angular resolutions (33 and 13 arcmin) and the overall sensitivity corresponding to the whole mission duration (and also to two surveys only: upper curve in black thick dots, labeled 13 arcmin). The expected noise is assumed to be properly subtracted. The plots include cosmic and sampling variance plus instrumental noise (green dots for B modes, green long dashes for E modes, labeled with cv+sv+n; black thick dots, noise only) assuming a multipole binning of 30%.The B mode induced by lensing (blue dots) is also shown. Galactic synchrotron (purple dashes) and dust (purple dot-dashes) polarized emissions produce the overall Galactic foreground (purple three dots-dashes). *WMAP* 3-yr power-law fits for uncorrelated dust and synchrotron have been used. For comparison, *WMAP* 3-yr results (http://lambda.gsfc.nasa.gov/) derived from the foreground maps using HEALPix tools (http://healpix.jpl.nasa.gov/) [12] are shown (red three dots-dashes broken line): power-law fits provide (generous) upper limits to the power at low multipoles. Residual contamination levels by Galactic foregrounds (purple three dot-dashes) are shown for 10%, 5%, and 3% of the map level, at increasing thickness. We plot also as thick and thin green dashes realistic estimates of the residual contribution of un-subtracted extragalactic sources,

potential systematic effects down to ∼ *µK* level or below. Fig.1 compares CMB polarization modes with the ideal sensitivity of *Planck* (including also a 15% level of HFI data loss because of cosmic rays; see [10]) and the signals coming from astrophysical foregrounds as discussed

CMB anisotropy maps are contaminated by a significant level of foreground emission of both Galactic and extragalactic origin. For polarization, the most critical Galactic foregrounds are certainly synchrotron and thermal dust emission, whereas free-free emission gives a negligible contribution. Other components, like spinning dust and "haze", are still poorly known, particularly in polarization. Synchrotron emission is the dominant Galactic foreground signal at low frequencies, up to ∼60 GHz, where dust emission starts to dominate. External galaxies are critical only at high ℓ, and extragalactic radio sources are likely the most crucial in polarization up to frequencies ∼200 GHz, the most suitable for CMB anisotropy experiments. We parameterize a potential residual from non perfect cleaning of CMB maps from Galactic foregrounds simply assuming that a certain fraction

**Figure 2.** CMB removed *Planck* full-sky maps. From left to right and from top to bottom: 30, 44, 70; 100, 143, 217; 353, 545, and 857 GHz, respectively. Credits: Zacchei, et al., A&A, Vol. 536, A5, 2011; *Planck* HFI Core Team, A&A, Vol. 536, A6, 2011b, reproduced with permission © ESO.

of the foreground signal *at map level* (or, equivalently, its square at power spectrum level) contaminates CMB maps. Of course, one can easily rescale the following results to any fraction of residual foreground contamination. The frequency of 70 GHz, i.e. the *Planck* channel where Galactic foregrounds are expected to be at their minimum level, at least at angular scales above ∼ one degree, is adopted as reference.

For what concerns CMB temperature fluctuations produced by undetected EPS [13], we adopt the recent (conservative) estimate of their Poisson contribution to the (polarized) APS [14] at 100 GHz<sup>7</sup> by assuming a detection threshold of ≃ 0.1 Jy. We also assume a potential residual coming from an uncertainty in the subtraction of this contribution computed by assuming a relative uncertainty of ≃ 10% in the knowledge of their degree of polarization and in the determination of the source detection threshold, implying a reduction to ≃ 30% of the original level. Except at very high multipoles, their residual is likely significantly below that coming from Galactic foregrounds.

The first publications of the main cosmological (i.e. properly based on *Planck* CMB maps) implications are expected in early 2013, together with the delivery of a first set of *Planck* maps and cosmological products coming from the first 15 months of data. They will be mainly based on temperature data. Waiting for these products, a first multifrequency view of the *Planck* astrophysical sky has been presented in the Early Papers: Fig. 2 reports the first LFI and HFI frequency (CMB subtracted) maps. These maps are the basis of the construction of the *Planck* Early Release Compact Source Catalog (ERCSC) (see [15] and *The Explanatory Supplement to the Planck Early Release Compact Source Catalogue*), the first *Planck* product delivered to the scientific community.

<sup>7</sup> We adopt here a frequency slightly larger than that considered for Galactic foregrounds (70 GHz) because at small angular scales, where point sources are more critical, the minimum of foreground contamination is likely shifted to higher frequencies.

NRAO VLA Sky Survey (NVSS), the quasar catalogue of Sloan Digital Sky Survey Release Six (SDSS DR6 QSOs) and the MegaZ-LRG (DR7), the final SDSS II Luminous Red Galaxy (LRG) photometric redshift survey, have been recently analysed by [21] (see this work and references therein for a thorough analysis on the subject). Through a global analysis of the constraints on the amplitude of primordial non-Gaussianity by the angular power spectra obtained from extragalactic radio sources (mapped by these surveys) and, moreover, from their cross-correlation power spectra with the WMAP CMB temperature map, [21] set limits on *fNL* = <sup>48</sup> ± 20, *fNL* = <sup>50</sup> ± 265 and *fNL* = <sup>183</sup> ± 95 at 68% confidence level for local, equilateral and enfolded templates, respectively. These results have been found to be stable with respect to potential systematic errors: the source number density and the contamination by Galactic emissions, for NVSS sources; the use of different CMB temperature fluctuation templates and the contamination of stars in the SDSS and LRG samples. Such tests of non–Gaussianity would have profound implications for inflationary mechanisms – such as single-field slow roll, multifields, curvaton (local type) – and for models which have effects on the halo clustering can be described by the equilateral template (related to higher-order derivative type non-Gaussianity) and by the enfolded template (related to modified initial state or higher-derivative interactions). Fundamental progress on this topic will be achieved

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by combining forthcoming LSS surveys with the CMB maps provided by *Planck*.

Compact sources, in CMB literature, are defined as spatially bounded sources which subtend very small angular scales in the images, such as galaxies and galaxy clusters. On the other hand, diffuse components, such as the CMB itself and Galactic foregrounds, do not show clear spatial boundaries and extend over large areas of the sky. Due to the fact that compact sources are spatially localized, the techniques for detecting them differ from those applied for the separation of the diffuse components. Most of the detection methods use scale diversity, i.e. different power at different angular scales, to enhance compact sources against diffuse components. Sources must be detected against a combination of CMB, instrumental noise and Galactic foregrounds. From the point of view of signal processing, the source is the

Point sources are "compact" sources in the sense that their typical observed angular size is much smaller than the beam resolution of the experiment. Therefore, they appear as point-like objects convolved with the instrumental beam. Radio sources and far–IR sources are usually seen as point-like sources. Galaxy clusters, which are detected through the thermal SZ effect [22], have a shape that is obtained as the convolution of the instrumental beam with the cluster profile. In contrast to point sources, the cluster profile has to be taken into account for cluster detection. However, since the projected angular scale of clusters is generally small, techniques that are useful for point sources can be adapted for clusters, too. The thermal SZ effect has a general dependence with frequency, that makes the use of multichannel images very convenient for cluster detection. On the contrary, the flux of each individual point source has its own frequency dependence. Despite this fact, the combination of several channels can also improve point source detection. We will review techniques applied to single-frequency channels in a first subsection and then we will discuss more

**3. Methods for compact source detection in CMB maps**

signal and the other components are the noise.

recent methods, that use multichannel information.

**Figure 3.** The *Planck* ERCSC flux density limit quantified as the faintest ERCSC sources at |*b*| < 10◦ (dashed black line) and at |*b*| > 30◦ (solid black line) is shown relative to other wide area surveys. See Fig. 5 of [15] for more details. Credit: *Planck* Collaboration, A&A, vol. 536, A7, 2011, reproduced with permission © ESO.

Fig. 3 compares the sensitivity of *Planck* ERCSC with those of other surveys from radio to far-infrared wavelengths. Of course, by accumulating sky surveys and refining data analysis, the *Planck* sensitivity to point sources will significantly improve in the coming years. The forthcoming *Planck* Legacy Catalog of Compact Sources (PCCS), to be released in early 2013 and to be updated in subsequent years, will represent one of the major *Planck* products relevant for multi-frequency studies of compact or point–like sources.

## **2.1. Extragalactic point sources vs. non-Gaussianity**

The cosmological evolution of extragalactic sources and its implications for the CMB and the CIB will be discussed in the following Sections, 5 and 6. On the other hand, statistical analyses of the extragalactic source distribution in the sky can be applied to test cosmological models. In this context, the possibility of probing the Gaussianity of primordial perturbations appears particularly promising. Primordial perturbations at the origin of the large scale structure (LSS) may leave their imprint in the form of small deviations from a Gaussian distribution, [16, 17] that can appear in different kinds of configurations, such as the so-called local type, equilateral, enfolded, orthogonal. For example, the local type of deviation from Gaussianity is parameterized by a constant dimensionless parameter *fNL* (see, e.g., [18–20]) <sup>Φ</sup> = *<sup>φ</sup>* + *fNL*(*φ*<sup>2</sup> − *φ*<sup>2</sup> ), where Φ denotes Bardeen's gauge-invariant potential (evaluated deep in the matter era in the CMB convention) and *φ* is a Gaussian random field. Extragalactic radio sources are particularly interesting as tracers of the LSS since they span large volumes extending out to substantial redshifts. The radio sources from the NRAO VLA Sky Survey (NVSS), the quasar catalogue of Sloan Digital Sky Survey Release Six (SDSS DR6 QSOs) and the MegaZ-LRG (DR7), the final SDSS II Luminous Red Galaxy (LRG) photometric redshift survey, have been recently analysed by [21] (see this work and references therein for a thorough analysis on the subject). Through a global analysis of the constraints on the amplitude of primordial non-Gaussianity by the angular power spectra obtained from extragalactic radio sources (mapped by these surveys) and, moreover, from their cross-correlation power spectra with the WMAP CMB temperature map, [21] set limits on *fNL* = <sup>48</sup> ± 20, *fNL* = <sup>50</sup> ± 265 and *fNL* = <sup>183</sup> ± 95 at 68% confidence level for local, equilateral and enfolded templates, respectively. These results have been found to be stable with respect to potential systematic errors: the source number density and the contamination by Galactic emissions, for NVSS sources; the use of different CMB temperature fluctuation templates and the contamination of stars in the SDSS and LRG samples. Such tests of non–Gaussianity would have profound implications for inflationary mechanisms – such as single-field slow roll, multifields, curvaton (local type) – and for models which have effects on the halo clustering can be described by the equilateral template (related to higher-order derivative type non-Gaussianity) and by the enfolded template (related to modified initial state or higher-derivative interactions). Fundamental progress on this topic will be achieved by combining forthcoming LSS surveys with the CMB maps provided by *Planck*.

## **3. Methods for compact source detection in CMB maps**

6 Open Questions in Cosmology

**Figure 3.** The *Planck* ERCSC flux density limit quantified as the faintest ERCSC sources at |*b*| < 10◦ (dashed black line) and at |*b*| > 30◦ (solid black line) is shown relative to other wide area surveys. See Fig. 5 of [15] for more details. Credit: *Planck*

Fig. 3 compares the sensitivity of *Planck* ERCSC with those of other surveys from radio to far-infrared wavelengths. Of course, by accumulating sky surveys and refining data analysis, the *Planck* sensitivity to point sources will significantly improve in the coming years. The forthcoming *Planck* Legacy Catalog of Compact Sources (PCCS), to be released in early 2013 and to be updated in subsequent years, will represent one of the major *Planck* products

The cosmological evolution of extragalactic sources and its implications for the CMB and the CIB will be discussed in the following Sections, 5 and 6. On the other hand, statistical analyses of the extragalactic source distribution in the sky can be applied to test cosmological models. In this context, the possibility of probing the Gaussianity of primordial perturbations appears particularly promising. Primordial perturbations at the origin of the large scale structure (LSS) may leave their imprint in the form of small deviations from a Gaussian distribution, [16, 17] that can appear in different kinds of configurations, such as the so-called local type, equilateral, enfolded, orthogonal. For example, the local type of deviation from Gaussianity is parameterized by a constant dimensionless parameter *fNL* (see, e.g.,

(evaluated deep in the matter era in the CMB convention) and *φ* is a Gaussian random field. Extragalactic radio sources are particularly interesting as tracers of the LSS since they span large volumes extending out to substantial redshifts. The radio sources from the

), where Φ denotes Bardeen's gauge-invariant potential

Collaboration, A&A, vol. 536, A7, 2011, reproduced with permission © ESO.

**2.1. Extragalactic point sources vs. non-Gaussianity**

*φ*<sup>2</sup>

[18–20]) <sup>Φ</sup> = *<sup>φ</sup>* + *fNL*(*φ*<sup>2</sup> −

relevant for multi-frequency studies of compact or point–like sources.

Compact sources, in CMB literature, are defined as spatially bounded sources which subtend very small angular scales in the images, such as galaxies and galaxy clusters. On the other hand, diffuse components, such as the CMB itself and Galactic foregrounds, do not show clear spatial boundaries and extend over large areas of the sky. Due to the fact that compact sources are spatially localized, the techniques for detecting them differ from those applied for the separation of the diffuse components. Most of the detection methods use scale diversity, i.e. different power at different angular scales, to enhance compact sources against diffuse components. Sources must be detected against a combination of CMB, instrumental noise and Galactic foregrounds. From the point of view of signal processing, the source is the signal and the other components are the noise.

Point sources are "compact" sources in the sense that their typical observed angular size is much smaller than the beam resolution of the experiment. Therefore, they appear as point-like objects convolved with the instrumental beam. Radio sources and far–IR sources are usually seen as point-like sources. Galaxy clusters, which are detected through the thermal SZ effect [22], have a shape that is obtained as the convolution of the instrumental beam with the cluster profile. In contrast to point sources, the cluster profile has to be taken into account for cluster detection. However, since the projected angular scale of clusters is generally small, techniques that are useful for point sources can be adapted for clusters, too.

The thermal SZ effect has a general dependence with frequency, that makes the use of multichannel images very convenient for cluster detection. On the contrary, the flux of each individual point source has its own frequency dependence. Despite this fact, the combination of several channels can also improve point source detection. We will review techniques applied to single-frequency channels in a first subsection and then we will discuss more recent methods, that use multichannel information.

## **3.1. Single channel detection**

We now focus on techniques for detecting point–like sources. Galaxy clusters can be detected by similar methods, but taking into account the cluster profile (see, e.g., [23-25]). Since multichannel methods for cluster detection improve significantly the performance of single channel techniques, we leave a more detailed study of clusters for the next subsection.

As mentioned before, compact source detection techniques make use of scale diversity. For example, SEXTRACTOR [26] – where maps are pre–filtered by a Gaussian kernel the same size as the beam – approximates the image background by a low-order polynomial and then subtracts the background from the image. The object is detected after connecting the pixels above a given flux density threshold. SEXTRACTOR has been used for elaborating the *Planck* ERCSC [15] in the highest frequency channels, from 217 to 857 GHz. However, CMB emission and diffuse foregrounds are complex and cannot be modeled in a straightforward way. Thus, apart from this important application, SEXTRACTOR has had a limited use in CMB astronomy.

A standard method which has been used often for compact source detection is the common matched filter (MF) [27]. The MF is just a linear filter with suitable characteristics for amplifying the source against the background. The image *y*(*x*) is convolved with a filter *ψ*(*x*):

$$
\omega(\vec{\mathfrak{x}}) = \int y(\vec{u})\psi(\vec{x} - \vec{u}) \, d\vec{u} \tag{2}
$$

The MHW has been succesfully applied to simulated CMB data [30]. The scale *R* is fixed in order to obtain the maximum amplification and the determination of the power spectrum is not necessary. A family of wavelets that generalize the MHW, the Mexican Hat Wavelet Family (MHWF) was presented in [31]. The performance of this family was compared with the MF in [28] and it produced similar results when implemented on Planck simulations. The MHWF was also applied to point source detection in WMAP images [32] by using a non-blind method. This method yielded a larger number of detections than the MF used by

*ψ*(*q*) ∝ (*qR*)2*<sup>n</sup>* exp

*n* being a natural number. Further improvements can be obtained if we admit any real

The MF and the diverse types of wavelets do not use any prior information about the average number of sources in the surveyed patch, the flux distribution of the sources or other properties. Therefore, useful information is being neglected by these methods. In contrast, Bayesian methods provide a natural way to incorporate information about the statistical properties of both the source and the noise. Several Bayesian methods have been proposed in the literature for the detection problem [24, 34, 35]. These methods construct a posterior

*Pr*(*θ*|*D*, *<sup>H</sup>*) = *Pr*(*D*|*θ*, *<sup>H</sup>*)*Pr*(*θ*|*H*)

where *θ* are the relevant parameters (positions, fluxes, sizes, etc.), *D* the data, and *H* the underlying hypothesis. In Bayesian terminology *Pr*(*D*|*θ*, *H*) is the likelihood, *Pr*(*θ*|*H*) is the prior and *Pr*(*D*|*H*) is the Bayesian evidence. Different Bayesian techniques can differ in the priors or in the way of exploring the complicated posterior probability. PowellSnakes [34] is an interesting method, which has been applied with success to the compilation of the ERCSC for *Planck* frequencies between 30 and 143 GHz [15]. It uses Powell's minimization and physically motivated priors. This method can be also applied to cluster detection just by

A simple Bayesian way of determining the position of the sources and estimating their number and flux densities has been presented in [35]. Whereas by the MF, or by wavelets, sources are detected above an arbitrary threshold, Bayesian methods select them in a more natural way, for instance by comparing the posterior probability of two hypothesis: presence or absence of the source. In the next subsection we will explore multichannel methods that

The flux density distribution, *fν*, of extragalactic radio sources as a function of frequency, *ν*, is usually approximated by a power law, although this approximation is only valid in

<sup>−</sup> (*qR*)<sup>2</sup> 2

Extragalactic Compact Sources in the *Planck* Sky and Their Cosmological Implications

(5)

http://dx.doi.org/10.5772/52908

65

*Pr*(*D*|*H*) (6)

the WMAP group. The general expression of the MHWF is:

exponent such as in the Bi-parametric Adaptive Filter (BAF) [33].

probability *Pr*(*θ*|*D*, *H*) by using Bayes' theorem

introducing a suitable prior on the cluster size.

help improve the detection performance

**3.2. Multi-channel detection**

The MF is defined as the linear filter that is an unbiased estimator of the source flux and minimizes the variance of the filtered map. In order to satisfy these mathematical constraints, if we assume that the beam is circularly symmetric and the background a homogeneous and isotropic random field, the MF must be defined in Fourier space as

$$
\psi(q) = k \frac{\tau(q)}{P(q)} \tag{3}
$$

where *τ*(*q*) is the Fourier transform of the beam, *P*(*q*) the background power spectrum and k the normalization constant. With this definition, the MF gives the maximum amplification of the compact source with respect to the background. Once the source has been amplified, point sources are detected in the filtered map as peaks above a given threshold, typically5*σ*, with *σ* the r.m.s deviation of the filtered map. The MF has been used both with simulations [28] and real data [29]. In this last paper, the WMAP team estimated the fluxes by fitting to a Gaussian profile plus a planar baseline. In [24] the MF was also applied to the detection of clusters. The use of wavelets for source detection is an interesting alternative to the MF. Wavelets are compensated linear filters, i.e their integral is zero, that help to remove the background contribution and yield a high source amplification. Since the beam is approximately Gaussian, the Mexican Hat Wavelet (MHW), constructed as the Laplacian of a Gaussian function, adapts itself very well to the detection problem. The MHW depends on the scale *R*, a parameter that determines the width of the wavelet:

$$\psi(q) \propto \left(q\mathbb{R}\right)^2 \exp\left(-\frac{\left(q\mathbb{R}\right)^2}{2}\right) \tag{4}$$

The MHW has been succesfully applied to simulated CMB data [30]. The scale *R* is fixed in order to obtain the maximum amplification and the determination of the power spectrum is not necessary. A family of wavelets that generalize the MHW, the Mexican Hat Wavelet Family (MHWF) was presented in [31]. The performance of this family was compared with the MF in [28] and it produced similar results when implemented on Planck simulations. The MHWF was also applied to point source detection in WMAP images [32] by using a non-blind method. This method yielded a larger number of detections than the MF used by the WMAP group. The general expression of the MHWF is:

$$
\psi(q) \propto (q\mathbb{R})^{2n} \exp\left(-\frac{(q\mathbb{R})^2}{2}\right) \tag{5}
$$

*n* being a natural number. Further improvements can be obtained if we admit any real exponent such as in the Bi-parametric Adaptive Filter (BAF) [33].

The MF and the diverse types of wavelets do not use any prior information about the average number of sources in the surveyed patch, the flux distribution of the sources or other properties. Therefore, useful information is being neglected by these methods. In contrast, Bayesian methods provide a natural way to incorporate information about the statistical properties of both the source and the noise. Several Bayesian methods have been proposed in the literature for the detection problem [24, 34, 35]. These methods construct a posterior probability *Pr*(*θ*|*D*, *H*) by using Bayes' theorem

$$Pr(\theta|D, H) = \frac{Pr(D|\theta, H)Pr(\theta|H)}{Pr(D|H)}\tag{6}$$

where *θ* are the relevant parameters (positions, fluxes, sizes, etc.), *D* the data, and *H* the underlying hypothesis. In Bayesian terminology *Pr*(*D*|*θ*, *H*) is the likelihood, *Pr*(*θ*|*H*) is the prior and *Pr*(*D*|*H*) is the Bayesian evidence. Different Bayesian techniques can differ in the priors or in the way of exploring the complicated posterior probability. PowellSnakes [34] is an interesting method, which has been applied with success to the compilation of the ERCSC for *Planck* frequencies between 30 and 143 GHz [15]. It uses Powell's minimization and physically motivated priors. This method can be also applied to cluster detection just by introducing a suitable prior on the cluster size.

A simple Bayesian way of determining the position of the sources and estimating their number and flux densities has been presented in [35]. Whereas by the MF, or by wavelets, sources are detected above an arbitrary threshold, Bayesian methods select them in a more natural way, for instance by comparing the posterior probability of two hypothesis: presence or absence of the source. In the next subsection we will explore multichannel methods that help improve the detection performance

## **3.2. Multi-channel detection**

8 Open Questions in Cosmology

CMB astronomy.

*ψ*(*x*):

**3.1. Single channel detection**

We now focus on techniques for detecting point–like sources. Galaxy clusters can be detected by similar methods, but taking into account the cluster profile (see, e.g., [23-25]). Since multichannel methods for cluster detection improve significantly the performance of single channel techniques, we leave a more detailed study of clusters for the next subsection.

As mentioned before, compact source detection techniques make use of scale diversity. For example, SEXTRACTOR [26] – where maps are pre–filtered by a Gaussian kernel the same size as the beam – approximates the image background by a low-order polynomial and then subtracts the background from the image. The object is detected after connecting the pixels above a given flux density threshold. SEXTRACTOR has been used for elaborating the *Planck* ERCSC [15] in the highest frequency channels, from 217 to 857 GHz. However, CMB emission and diffuse foregrounds are complex and cannot be modeled in a straightforward way. Thus, apart from this important application, SEXTRACTOR has had a limited use in

A standard method which has been used often for compact source detection is the common matched filter (MF) [27]. The MF is just a linear filter with suitable characteristics for amplifying the source against the background. The image *y*(*x*) is convolved with a filter

The MF is defined as the linear filter that is an unbiased estimator of the source flux and minimizes the variance of the filtered map. In order to satisfy these mathematical constraints, if we assume that the beam is circularly symmetric and the background a homogeneous and

*ψ*(*q*) = *k*

where *τ*(*q*) is the Fourier transform of the beam, *P*(*q*) the background power spectrum and k the normalization constant. With this definition, the MF gives the maximum amplification of the compact source with respect to the background. Once the source has been amplified, point sources are detected in the filtered map as peaks above a given threshold, typically5*σ*, with *σ* the r.m.s deviation of the filtered map. The MF has been used both with simulations [28] and real data [29]. In this last paper, the WMAP team estimated the fluxes by fitting to a Gaussian profile plus a planar baseline. In [24] the MF was also applied to the detection of clusters. The use of wavelets for source detection is an interesting alternative to the MF. Wavelets are compensated linear filters, i.e their integral is zero, that help to remove the background contribution and yield a high source amplification. Since the beam is approximately Gaussian, the Mexican Hat Wavelet (MHW), constructed as the Laplacian of a Gaussian function, adapts itself very well to the detection problem. The MHW depends

*τ*(*q*)

<sup>−</sup> (*qR*)<sup>2</sup> 2

*y*(*u*)*ψ*(*x* −*u*) *du* (2)

*<sup>P</sup>*(*q*) (3)

(4)

*ω*(*x*) =

isotropic random field, the MF must be defined in Fourier space as

on the scale *R*, a parameter that determines the width of the wavelet:

*ψ*(*q*) ∝ (*qR*)<sup>2</sup> exp

The flux density distribution, *fν*, of extragalactic radio sources as a function of frequency, *ν*, is usually approximated by a power law, although this approximation is only valid in limited frequency intervals, i.e. *<sup>f</sup><sup>ν</sup>* <sup>∝</sup> (*ν*/*ν*0)*α*, with *<sup>ν</sup>*<sup>0</sup> being some frequency of reference. Nevertheless, the so called "spectral index", *α*, changes from source to source and this formula is not reliable when the range of frequencies is wide enough. In [36] a scheme for channel combination was proposed that makes the spectral behavior irrelevant. This method is called matrix multifilters (MTXFs) and relies on the application of a set (*N* × *N* matrix) of linear filters which combine the information of the *N* channels in such a way that: 1) an unbiased estimator of the source flux at each channel is obtained; and 2) the variance of the estimator is minimum. Note that the method does not mix the images in a single map, but it produces *N* maps in which the sources are conveniently amplified. The method defaults to the MF when there is no cross-correlation among the channels. When there is a non negligible correlation among the channels, as is the case for microwave images taken at different frequencies where CMB and Galactic foregrounds are present in all the images, this method gives a clear increase of the amplification when compared with the MF.

Now, we discuss a method tailored for cluster detection through the thermal SZ effect. Matched Multifilters (MMF) [37] combine *N* channels in a single image, incorporating the information about the spectral behavior (thermal SZ effect) and with the *N* filters depending on a scale parameter *S*. The filters are constructed in the usual way, by imposing unbiasedness and minimum variance. The MMF is given (in matrix notation) by

$$\mathbf{Y}(q) = \alpha \mathbf{P}^{-1} \mathbf{F}, \; \alpha^{-1} = \int d\mathbf{q} \, F^{\mathbf{f}} \mathbf{P}^{-1} \mathbf{F}, \tag{7}$$

blending, amounts to using the MMF presented above. Here we have briefly summarized the most important topics on the subject: a more detailed review can be found in [41].

Extragalactic Compact Sources in the *Planck* Sky and Their Cosmological Implications

The Sunyaev-Zeldovich effect (SZ, [22]) accounts for the interaction between a hot plasma (in a cluster environment) and the photons of the CMB. When CMB photons cross a galaxy cluster, some of these photons interact with the free electrons in the hot plasma through inverse Compton scattering. The temperature change observed in a given direction, *θ*, and

where *<sup>C</sup>*<sup>0</sup> contains all the relevant constants including the frequency dependence (*gx* = *x*(*e<sup>x</sup>* + 1)/(*e<sup>x</sup>* − 1) − 4, with *x* = *hν*/*kT*), *ne* is the electron density and *T* is the electron

The same electrons that interact with the CMB photons emit X-rays through the

where *<sup>D</sup>*ℓ(*z*) is the luminosity distance. The quantity *<sup>S</sup>*<sup>0</sup> contains all the relevant constants and frequency dependence. Combining X–ray and SZ observations it is thus possible to reduce the degeneracy between different models due to their different dependency on *T* and

Due to the nature of the microwave radiation, water vapour (and hence our atmosphere) presents a challenge for studying this radiation from the ground. Observations have to be carried out through several windows where the transmission of the microwave light is maximized. In recent years, ground–based experiments have benefited from important progress in the development of very sensitive bolometers. These bolometer arrays when combined with superb atmospheric conditions – found in places like the South Pole and the Atacama desert (with extremely low levels of water vapour) – have allowed, for the first time at all, galaxy clusters to be mapped in great detail through the SZ effect. The South Pole Telescope (or SPT; see, e.g., [42]) and the Atacama Cosmology Telescope (or ACT; see, e.g., [43]) are today the most important ground–based experiments carrying out these

From space, the *Planck* satellite – even though it lacks the spatial resolution of ground–based experiments – complements them by applying a full–sky coverage, a wider frequency range and a better understanding of Galactic and extragalactic foregrounds. In particular, *Planck* is better suited than ground–based experiments to detect large angular scale SZ signals like nearby galaxy clusters or the diffuse SZ effect. In fact, ground–based experiments can have their large angular scales affected by atmospheric fluctuations that need to be removed, carefully. This removal process can distort the modes that include the large angular scales signal. On the contrary, *Planck* data does not suffer from these limitations

*n*<sup>2</sup>

*<sup>e</sup> T*1/2*dl*

*ne*(*l*)*T*(*l*)*dl* (8)

http://dx.doi.org/10.5772/52908

67

*<sup>D</sup>*ℓ(*z*)<sup>2</sup> (9)

<sup>∆</sup>*T*(*θ*, *<sup>ν</sup>*) = *<sup>C</sup>*<sup>0</sup>

*Sx*(*θ*, *<sup>ν</sup>*) = *<sup>S</sup>*<sup>0</sup>

**4. Sunyaev-Zeldovich effect in clusters of galaxies**

temperature. The integral is performed along the line of sight.

at the frequency *ν*, can be described as

bremstrahlung process:

especially with *ne*.

observations.

where *<sup>F</sup>* is the column vector [ *<sup>f</sup>ντν*], which incorporates the spectral behavior *<sup>f</sup><sup>ν</sup>* and the shape of the cluster at each frequency *τν* and **P** is the cross-power spectrum. Since the cluster size is not known a priori, the images are convolved with a set of filters with different scales *Si*, and it has been proven that the amplification is maximum when the chosen scale coincides with the cluster size. A common pressure profile is assumed for the clusters. The detection is performed by searching for the maxima of the filtered map above a given threshold. The estimated amplitude of the thermal SZ effect is given by the amplitude at the maxima.

MMF can also be adapted to detect the fainter kinematic SZ effect. In this case an orthogonality condition with respect to the spectral behavior of the thermal SZ effect is imposed. Together with the usual unbiasedness and minimum variance conditions, this last constraint helps cancel out the thermal SZ effect contamination [38]. A MMF can also be designed for point source detection, by incorporating the (unknown) spectral behavior of the sources as a set of parameters in the filter, it has also been proven that the amplification is maximum when these parameters coincide with the real source spectrum. By changing the parameters and selecting those which give the maximum amplification, in [39] the authors were able to increase the number of point source detections in the WMAP 7-year maps.

Finally, a multi-channel Bayesian method has been developed recently, Powell-Snakes II [40]. This method constructs a posterior distribution by combining the likelihood and the prior information of the different channels. It is an extension of Powell-Snakes I and uses prior information on the positions, number of sources, intensities, sizes and spectral parameters. The method is suitable both for point sources and for clusters. It is worth noting that maximizing the likelihood when the sources are well separated, i.e. in the absence of source blending, amounts to using the MMF presented above. Here we have briefly summarized the most important topics on the subject: a more detailed review can be found in [41].

## **4. Sunyaev-Zeldovich effect in clusters of galaxies**

10 Open Questions in Cosmology

limited frequency intervals, i.e. *<sup>f</sup><sup>ν</sup>* <sup>∝</sup> (*ν*/*ν*0)*α*, with *<sup>ν</sup>*<sup>0</sup> being some frequency of reference. Nevertheless, the so called "spectral index", *α*, changes from source to source and this formula is not reliable when the range of frequencies is wide enough. In [36] a scheme for channel combination was proposed that makes the spectral behavior irrelevant. This method is called matrix multifilters (MTXFs) and relies on the application of a set (*N* × *N* matrix) of linear filters which combine the information of the *N* channels in such a way that: 1) an unbiased estimator of the source flux at each channel is obtained; and 2) the variance of the estimator is minimum. Note that the method does not mix the images in a single map, but it produces *N* maps in which the sources are conveniently amplified. The method defaults to the MF when there is no cross-correlation among the channels. When there is a non negligible correlation among the channels, as is the case for microwave images taken at different frequencies where CMB and Galactic foregrounds are present in all the images, this

method gives a clear increase of the amplification when compared with the MF.

unbiasedness and minimum variance. The MMF is given (in matrix notation) by

<sup>Υ</sup>(*q*) = *<sup>α</sup>***P**−1*F*, *<sup>α</sup>*−<sup>1</sup> <sup>=</sup>

Now, we discuss a method tailored for cluster detection through the thermal SZ effect. Matched Multifilters (MMF) [37] combine *N* channels in a single image, incorporating the information about the spectral behavior (thermal SZ effect) and with the *N* filters depending on a scale parameter *S*. The filters are constructed in the usual way, by imposing

where *<sup>F</sup>* is the column vector [ *<sup>f</sup>ντν*], which incorporates the spectral behavior *<sup>f</sup><sup>ν</sup>* and the shape of the cluster at each frequency *τν* and **P** is the cross-power spectrum. Since the cluster size is not known a priori, the images are convolved with a set of filters with different scales *Si*, and it has been proven that the amplification is maximum when the chosen scale coincides with the cluster size. A common pressure profile is assumed for the clusters. The detection is performed by searching for the maxima of the filtered map above a given threshold. The estimated amplitude of the thermal SZ effect is given by the amplitude at the maxima.

MMF can also be adapted to detect the fainter kinematic SZ effect. In this case an orthogonality condition with respect to the spectral behavior of the thermal SZ effect is imposed. Together with the usual unbiasedness and minimum variance conditions, this last constraint helps cancel out the thermal SZ effect contamination [38]. A MMF can also be designed for point source detection, by incorporating the (unknown) spectral behavior of the sources as a set of parameters in the filter, it has also been proven that the amplification is maximum when these parameters coincide with the real source spectrum. By changing the parameters and selecting those which give the maximum amplification, in [39] the authors were able to increase the number of point source detections in the WMAP 7-year maps.

Finally, a multi-channel Bayesian method has been developed recently, Powell-Snakes II [40]. This method constructs a posterior distribution by combining the likelihood and the prior information of the different channels. It is an extension of Powell-Snakes I and uses prior information on the positions, number of sources, intensities, sizes and spectral parameters. The method is suitable both for point sources and for clusters. It is worth noting that maximizing the likelihood when the sources are well separated, i.e. in the absence of source

 *d***q** *F<sup>t</sup>*

**<sup>P</sup>**−1*F*, (7)

The Sunyaev-Zeldovich effect (SZ, [22]) accounts for the interaction between a hot plasma (in a cluster environment) and the photons of the CMB. When CMB photons cross a galaxy cluster, some of these photons interact with the free electrons in the hot plasma through inverse Compton scattering. The temperature change observed in a given direction, *θ*, and at the frequency *ν*, can be described as

$$
\Delta T(\theta, \nu) = \mathbb{C}\_0 \int n\_\varepsilon(l) T(l) dl \tag{8}
$$

where *<sup>C</sup>*<sup>0</sup> contains all the relevant constants including the frequency dependence (*gx* = *x*(*e<sup>x</sup>* + 1)/(*e<sup>x</sup>* − 1) − 4, with *x* = *hν*/*kT*), *ne* is the electron density and *T* is the electron temperature. The integral is performed along the line of sight.

The same electrons that interact with the CMB photons emit X-rays through the bremstrahlung process:

$$S\_{\mathcal{X}}(\theta, \nu) = S\_0 \frac{\int n\_{\mathcal{e}}^2 T^{1/2} dl}{D\_{\ell}(z)^2} \tag{9}$$

where *<sup>D</sup>*ℓ(*z*) is the luminosity distance. The quantity *<sup>S</sup>*<sup>0</sup> contains all the relevant constants and frequency dependence. Combining X–ray and SZ observations it is thus possible to reduce the degeneracy between different models due to their different dependency on *T* and especially with *ne*.

Due to the nature of the microwave radiation, water vapour (and hence our atmosphere) presents a challenge for studying this radiation from the ground. Observations have to be carried out through several windows where the transmission of the microwave light is maximized. In recent years, ground–based experiments have benefited from important progress in the development of very sensitive bolometers. These bolometer arrays when combined with superb atmospheric conditions – found in places like the South Pole and the Atacama desert (with extremely low levels of water vapour) – have allowed, for the first time at all, galaxy clusters to be mapped in great detail through the SZ effect. The South Pole Telescope (or SPT; see, e.g., [42]) and the Atacama Cosmology Telescope (or ACT; see, e.g., [43]) are today the most important ground–based experiments carrying out these observations.

From space, the *Planck* satellite – even though it lacks the spatial resolution of ground–based experiments – complements them by applying a full–sky coverage, a wider frequency range and a better understanding of Galactic and extragalactic foregrounds. In particular, *Planck* is better suited than ground–based experiments to detect large angular scale SZ signals like nearby galaxy clusters or the diffuse SZ effect. In fact, ground–based experiments can have their large angular scales affected by atmospheric fluctuations that need to be removed, carefully. This removal process can distort the modes that include the large angular scales signal. On the contrary, *Planck* data does not suffer from these limitations and its relatively poor angular resolution (if compared to some ground experiments) can be used to its advantage. The wide frequency coverage and extremely high sensitivity of *Planck* allows for detailed foreground (and CMB) removal that could overwhelm the weak signal of the SZ effect. *Planck* data will help improve the understanding of the distribution and the characteristics of the plasma in clusters. The conclusions derived on the internal structure of clusters will ultimately have an impact on other works that focus on deriving cosmological parameters. In fact, cosmological studies cannot by themselves disentangle among the uncertainties in the physics inside galaxy clusters.

**Figure 5.** Fig. 3 from [44]. Distribution of Planck clusters that were already known (blue) and the new cluster candidates (green

Extragalactic Compact Sources in the *Planck* Sky and Their Cosmological Implications

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69

new clusters, that have been later confirmed by X-ray and/or optical follows up. Fig. 5 shows the distribution in the sky of clusters of galaxies as seen by *Planck*. This includes the most massive clusters in the nearby and intermediate distance Universe. The redshift independence of the SZ effect can be appreciated in Fig. 6 (next page), which shows the relative flatness of the selection function of *Planck* as compared to cluster selections based on X-ray luminosity. New analysis based on better data will improve the selection function by reducing the limiting mass as a function of redshift. It is expected [5] that *Planck* will increase the number of known clusters in a significant way and, in particular, it will explore

the high–redshift regime, detecting the most massive clusters at these high redshifts.

finding them.

One of the most interesting conclusions derived by these earlier results comes from the combination of X-ray and SZ data. Earlier studies based on X-ray data were able to conclude that there exists a universal profile that accurately describes the gas pressure in galaxy clusters [51]. The newly discovered (by *Planck*) SZ clusters seem to follow well this profile but small deviations were observed when comparing the mean SZ profile with the average profile derived from X-ray observations. Fig. 7 summarizes one of the main results of the paper [45] where it can be appreciated how the average profile of the SZ observations (red thick line) flattens towards the cluster center (*R*<sup>500</sup> ≪ 1) when compared to the average of a sample of cool, core relaxed X-ray clusters (thick blue line). This fact suggests that the new clusters detected by *Planck* tend to be non–cool core, morphologically disturbed clusters. This would explain why these clusters where not found by previous X-ray surveys but *Planck*, that is sensitive to the total pressure rather than to the distribution of the gas, has no problem in

Many other relevant results can be found in the first series of papers from the *Planck* ERCSC, including studies of scaling relations between SZ quantities and optical or X-ray ones. More recently, a new analysis based on 2.5 full sky surveys has studied the relationship between the Compton parameter and weak lensing mass estimates [50]. As shown in Fig. 8, this work is very promising and could allow – in the near future – the use of the Compton parameter

and red). Credit: *Planck* Collaboration, A&A, Vol. 536, A8, 2011, reproduced with permission © ESO.

**Figure 4.** Fig. 1 from [44]. Coma cluster as seen by *Planck*. Contours show the the X-ray emission from Coma. Credit: *Planck* Collaboration, A&A, Vol. 536, A8, 2011, reproduced with permission © ESO.

The *Planck* satellite is currently detecting hundreds of clusters of galaxies through the thermal SZ effect. One of the peculiarities of the SZ effect is that the change in the CMB temperature in the direction of a cluster is independent of the distance to that cluster. This makes the SZ an ideal tool to explore the high redshift Universe. *Planck* is perfect for studying the most massive clusters in the Universe and is expected to see clusters beyond *z* = 1. Earlier results on galaxy clusters obtained by *Planck* have been presented in a subset of the *Planck* Early results papers and, more specifically, can be found in [44–49] and also in [50], where new results based on additional data are starting to be presented.

Among the first results published by the *Planck* collaboration on the SZ effect, the Coma cluster (see Fig. 4) constitutes one of the most spectacular ones. Coma is a nearby massive cluster that is well resolved by *Planck*. Fig. 4 shows the power of *Planck* to study the SZ effect with unprecedented quality. In the near future, studies based on *Planck* data alone or combined with X-ray data will reveal new details about the internal structure of this and other clusters.

*Planck*'s earlier results include the detection of almost 200 clusters through their SZ signature ([44]). *Planck* is particularly sensitive to phenomena that increase the pressure, like mergers or superclusters. One such supercluster was detected by *Planck* [45]. Most of the clusters seen by *Planck* in the early analysis were known nearby objects but some of them were

12 Open Questions in Cosmology

and its relatively poor angular resolution (if compared to some ground experiments) can be used to its advantage. The wide frequency coverage and extremely high sensitivity of *Planck* allows for detailed foreground (and CMB) removal that could overwhelm the weak signal of the SZ effect. *Planck* data will help improve the understanding of the distribution and the characteristics of the plasma in clusters. The conclusions derived on the internal structure of clusters will ultimately have an impact on other works that focus on deriving cosmological parameters. In fact, cosmological studies cannot by themselves disentangle

> **13:00:00.0**

**Figure 4.** Fig. 1 from [44]. Coma cluster as seen by *Planck*. Contours show the the X-ray emission from Coma. Credit: *Planck*

The *Planck* satellite is currently detecting hundreds of clusters of galaxies through the thermal SZ effect. One of the peculiarities of the SZ effect is that the change in the CMB temperature in the direction of a cluster is independent of the distance to that cluster. This makes the SZ an ideal tool to explore the high redshift Universe. *Planck* is perfect for studying the most massive clusters in the Universe and is expected to see clusters beyond *z* = 1. Earlier results on galaxy clusters obtained by *Planck* have been presented in a subset of the *Planck* Early results papers and, more specifically, can be found in [44–49] and also in [50], where new

Among the first results published by the *Planck* collaboration on the SZ effect, the Coma cluster (see Fig. 4) constitutes one of the most spectacular ones. Coma is a nearby massive cluster that is well resolved by *Planck*. Fig. 4 shows the power of *Planck* to study the SZ effect with unprecedented quality. In the near future, studies based on *Planck* data alone or combined with X-ray data will reveal new details about the internal structure of this and

*Planck*'s earlier results include the detection of almost 200 clusters through their SZ signature ([44]). *Planck* is particularly sensitive to phenomena that increase the pressure, like mergers or superclusters. One such supercluster was detected by *Planck* [45]. Most of the clusters seen by *Planck* in the early analysis were known nearby objects but some of them were

**58:00.0** **12:56:00.0**

**26:59:59.9**

**29:59.9**

**27:59:59.9**

among the uncertainties in the physics inside galaxy clusters.

**02:00.0**

Collaboration, A&A, Vol. 536, A8, 2011, reproduced with permission © ESO.

results based on additional data are starting to be presented.

other clusters.

**Figure 5.** Fig. 3 from [44]. Distribution of Planck clusters that were already known (blue) and the new cluster candidates (green and red). Credit: *Planck* Collaboration, A&A, Vol. 536, A8, 2011, reproduced with permission © ESO.

new clusters, that have been later confirmed by X-ray and/or optical follows up. Fig. 5 shows the distribution in the sky of clusters of galaxies as seen by *Planck*. This includes the most massive clusters in the nearby and intermediate distance Universe. The redshift independence of the SZ effect can be appreciated in Fig. 6 (next page), which shows the relative flatness of the selection function of *Planck* as compared to cluster selections based on X-ray luminosity. New analysis based on better data will improve the selection function by reducing the limiting mass as a function of redshift. It is expected [5] that *Planck* will increase the number of known clusters in a significant way and, in particular, it will explore the high–redshift regime, detecting the most massive clusters at these high redshifts.

One of the most interesting conclusions derived by these earlier results comes from the combination of X-ray and SZ data. Earlier studies based on X-ray data were able to conclude that there exists a universal profile that accurately describes the gas pressure in galaxy clusters [51]. The newly discovered (by *Planck*) SZ clusters seem to follow well this profile but small deviations were observed when comparing the mean SZ profile with the average profile derived from X-ray observations. Fig. 7 summarizes one of the main results of the paper [45] where it can be appreciated how the average profile of the SZ observations (red thick line) flattens towards the cluster center (*R*<sup>500</sup> ≪ 1) when compared to the average of a sample of cool, core relaxed X-ray clusters (thick blue line). This fact suggests that the new clusters detected by *Planck* tend to be non–cool core, morphologically disturbed clusters. This would explain why these clusters where not found by previous X-ray surveys but *Planck*, that is sensitive to the total pressure rather than to the distribution of the gas, has no problem in finding them.

Many other relevant results can be found in the first series of papers from the *Planck* ERCSC, including studies of scaling relations between SZ quantities and optical or X-ray ones. More recently, a new analysis based on 2.5 full sky surveys has studied the relationship between the Compton parameter and weak lensing mass estimates [50]. As shown in Fig. 8, this work is very promising and could allow – in the near future – the use of the Compton parameter

to reduce the uncertainties of the models. Also, the combination of X-ray and SZ data can be particularly powerful to study the clumpiness of the gas and deviations from spherical symmetry. An area where future *Planck* data will be used extensively will be the detection of new cluster candidates. The legacy *Planck* cluster catalogue will contain the most significant cluster signals. Hundreds of weaker SZ (and unknown) clusters will still be present in the public data but not in the legacy catalogue. Many groups will dig into the *Planck* data searching for these weaker signals. Among them there will be the most distant clusters at

Extragalactic Compact Sources in the *Planck* Sky and Their Cosmological Implications

http://dx.doi.org/10.5772/52908

71

**Figure 8.** Fig. 1 from [50]. Correlation between the Compton parameter and the weak lensing mass. Credit: *Planck*

Another area where *Planck* data might contribute significantly is in the study of energetic phenomena in galaxy clusters. The SZ effect is sensitive to the temperature of the Plasma (or more generally, to the speed distribution of the electrons). Hot clusters have an SZ spectrum that deviates from the standard shape. The shift (or relativistic correction) is more dramatic at higher frequencies (*ν* > 100 GHz). Current X-ray missions like *Chandra* have some trouble determining the temperature of the plasma for clusters with high temperatures. On the contrary, the relativistic corrections to the SZ effect can dramatically boost the SZ signal in *Planck* at frequencies *ν* > 500 GHz making, it easier to detect these clusters at these frequencies and also to derive constraints on the physical state of the plasma. A strong deviation in the spectrum could be an indication that very energetic phenomena (very high

The *Planck* ERCSC [15] provides positions and flux densities of compact sources found in each of the nine *Planck* frequency maps. The flux densities are calculated using aperture photometry, with careful modeling of *Planck'*s elliptical beams8. These data on sources detected during the first 1.6 full-sky surveys offers, among other things, the opportunity of studying the statistical and emission properties of extragalactic radio and far–IR sources

<sup>8</sup> Flux densities taken from the ERCSC should be divided by the appropriate colour correction to give the correct flux

Collaboration, A&A, submitted (ms AA/2012/19398), 2012, reproduced with permission © ESO.

temperatures, shock waves, etc.) are operating on the cluster at large scales.

over a broad frequency range, never before fully explored by blind surveys.

values for an assumed narrow band measurement at the central frequency.

**5. Extragalactic radio and far–IR sources**

*z* > 1 that will be crucial for future cosmological studies.

**Figure 6.** Fig. 21 from [44]. Redshift distribution as a function of luminosity for the 158 clusters from the Planck Early SZ sample (diamonds and triangles) identified with known X-ray clusters, compared with serendipitous and RASS clusters (crosses). See [44] for more details. Credit: *Planck* Collaboration, A&A, Vol. 536, A8, 2011, reproduced with permission © ESO.

**Figure 7.** Fig. 10 from [45]. Scaled density profiles, derived from X-ray data, of the new *Planck* SZ clusters compared to those of similar mass systems from representative X–ray samples. Thick lines show the mean profile of each sample. Credit: *Planck* Collaboration, A&A, Vol. 536, A9, 2011, reproduced with permission © ESO.

as a mass proxy in cosmological studies, where a good mass estimator is crucial to derive accurate cosmological parameters from the analysis of a cluster sample.

After *Planck*'s data release, science based on the SZ effect will change dramatically. *Planck* is expected to release more than 2 full sky surveys of data early in 2013, thus opening the door for multiple studies to be carried out by the scientific community. Cluster science based on the SZ effect will motivate many of these studies. Of particular interest will be those works that combine SZ effect and X-ray data. The different dependence of the SZ effect and X-ray emission with electron density and temperature allows for deprojection techniques to reduce the uncertainties of the models. Also, the combination of X-ray and SZ data can be particularly powerful to study the clumpiness of the gas and deviations from spherical symmetry. An area where future *Planck* data will be used extensively will be the detection of new cluster candidates. The legacy *Planck* cluster catalogue will contain the most significant cluster signals. Hundreds of weaker SZ (and unknown) clusters will still be present in the public data but not in the legacy catalogue. Many groups will dig into the *Planck* data searching for these weaker signals. Among them there will be the most distant clusters at *z* > 1 that will be crucial for future cosmological studies.

**Figure 8.** Fig. 1 from [50]. Correlation between the Compton parameter and the weak lensing mass. Credit: *Planck* Collaboration, A&A, submitted (ms AA/2012/19398), 2012, reproduced with permission © ESO.

Another area where *Planck* data might contribute significantly is in the study of energetic phenomena in galaxy clusters. The SZ effect is sensitive to the temperature of the Plasma (or more generally, to the speed distribution of the electrons). Hot clusters have an SZ spectrum that deviates from the standard shape. The shift (or relativistic correction) is more dramatic at higher frequencies (*ν* > 100 GHz). Current X-ray missions like *Chandra* have some trouble determining the temperature of the plasma for clusters with high temperatures. On the contrary, the relativistic corrections to the SZ effect can dramatically boost the SZ signal in *Planck* at frequencies *ν* > 500 GHz making, it easier to detect these clusters at these frequencies and also to derive constraints on the physical state of the plasma. A strong deviation in the spectrum could be an indication that very energetic phenomena (very high temperatures, shock waves, etc.) are operating on the cluster at large scales.

## **5. Extragalactic radio and far–IR sources**

14 Open Questions in Cosmology

0.0 0.2 0.4 0.6 redshift

**Figure 6.** Fig. 21 from [44]. Redshift distribution as a function of luminosity for the 158 clusters from the Planck Early SZ sample (diamonds and triangles) identified with known X-ray clusters, compared with serendipitous and RASS clusters (crosses). See [44] for more details. Credit: *Planck* Collaboration, A&A, Vol. 536, A8, 2011, reproduced with permission © ESO.

**Figure 7.** Fig. 10 from [45]. Scaled density profiles, derived from X-ray data, of the new *Planck* SZ clusters compared to those of similar mass systems from representative X–ray samples. Thick lines show the mean profile of each sample. Credit: *Planck*

as a mass proxy in cosmological studies, where a good mass estimator is crucial to derive

After *Planck*'s data release, science based on the SZ effect will change dramatically. *Planck* is expected to release more than 2 full sky surveys of data early in 2013, thus opening the door for multiple studies to be carried out by the scientific community. Cluster science based on the SZ effect will motivate many of these studies. Of particular interest will be those works that combine SZ effect and X-ray data. The different dependence of the SZ effect and X-ray emission with electron density and temperature allows for deprojection techniques

RASS clusters in mask Serendipitous clusters in mask Planck/ESZ XMM confirmed Planck/ESZ known clusters

10-3

Collaboration, A&A, Vol. 536, A9, 2011, reproduced with permission © ESO.

accurate cosmological parameters from the analysis of a cluster sample.

10-2

10-1

L500 [0.1-2.4]keV [1044 ergs/s]

1

10<sup>1</sup>

10<sup>2</sup>

The *Planck* ERCSC [15] provides positions and flux densities of compact sources found in each of the nine *Planck* frequency maps. The flux densities are calculated using aperture photometry, with careful modeling of *Planck'*s elliptical beams8. These data on sources detected during the first 1.6 full-sky surveys offers, among other things, the opportunity of studying the statistical and emission properties of extragalactic radio and far–IR sources over a broad frequency range, never before fully explored by blind surveys.

<sup>8</sup> Flux densities taken from the ERCSC should be divided by the appropriate colour correction to give the correct flux values for an assumed narrow band measurement at the central frequency.

As shown by [15], their Table 1, the full-sky surveys of the *Planck* satellite are – and will be, for years to come – unique in the millimetre, at *λ* ≤ 3 mm, and submillimetre domains. The lack of data in this frequency range represented the largest remaining gap in our knowledge of bright extragalactic sources (i.e., normal and star–forming galaxies and Active Galactic Nuclei, AGNs) across the electromagnetic spectrum. In the course of its planned surveys, *Planck* has been able to measure the integrated flux of many hundreds of "radio" sources – i.e., sources at intermediate to high–redshift, dominated by synchrotron emission due to hot electrons in the inner jets of the Active Galactic Nucleus (AGN) of the source – and of many thousands "'far–IR" sources – i.e., low–redshift dusty galaxies or sources with emission dominated by interstellar dust in thermal equilibrium with the radiation field – thus providing the *first complete full-sky catalogue* (ERCSC) of bright submillimetre sources. Thanks to this huge amount of new data it is thus possible to investigate the spectral energy distributions (SEDs) of EPS in a spectral domain very poorly explored before and, at the same time, their cosmological evolution, at least for some relevant source populations.

error estimated at each *Planck* frequency. Finally, for estimating spectral index distributions, these upper limits have been redistributed among the flux density bins by using a Survival Analysis technique and, more specifically, by adopting the Kaplan-Meyer estimator10[59].

In the sample analyzed by [59], the 30–100 GHz median spectral index is very close to the *α* ≃ −0.39 found by [60] between 20 and 95 GHz, for a sample with 20 GHz flux density *S* > 150 mJy. Moreover, the 30–143 GHz median spectral index is in very good agreement with the one found by [56] for their bright (*S<sup>ν</sup>* > 50 mJy) 148 GHz-selected sample with complete

Analysis) in the form of a 2D probability field: the colour scale can be interpreted as the probability of a given pair of spectral indices and a bending down, i.e. *α* < −0.5, at high frequencies is displayed. In the whole, the results of [59] show that in their sample selected at 30 GHz a moderate steepening of spectral indices of EPS at high radio frequencies, i.e.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2

account the upper limits to flux densities at each frequency. The colour scale can be interpreted as the probability of having any

As already noted, at high radio frequencies (*ν* > 30 GHz) most of the bright extragalactic radio-sources are blazars. From the contour plot of Fig. 9 it is possible to see that the

i.e. [61], a detailed discussion on the modelling of the spectra of this source class is also presented. In this paper, spectral energy distributions (SEDs) and radio continuum spectra are presented for a northern sample of 104 extragalactic radio sources, based on the *Planck* ERCSC and simultaneous multifrequency data12. The nine *Planck* frequencies, from 30 to 857 GHz, are complemented by a set of quasi–simultaneous observations ranging from radio to gamma-rays. SED modelling methods are discussed, with an emphasis on proper, physical

<sup>10</sup> Since the fraction of upper limits is found to be always small (it reaches approximately 30% only in the less sensitive

<sup>11</sup> Some hints in this direction were previously found by other works on the subject. Additional evidence of spectral

<sup>12</sup> The great amount of data present in the *Planck* ERCSC complemented with quasi-simultaneous ground–based observations at mm wavelengths have also enabled the study of the very interesting spectral properties of the rare

<sup>70</sup> vs. *α*<sup>70</sup>

α<sup>70</sup> 30 [S∝να ] <sup>70</sup> vs. *<sup>α</sup>*<sup>70</sup>

Extragalactic Compact Sources in the *Planck* Sky and Their Cosmological Implications

0.05

<sup>30</sup> obtained by Survival Analysis, i.e., taking into

<sup>70</sup> ≃ −1.2. In a companion paper,

<sup>30</sup> ≃ −0.18 and *<sup>α</sup>*<sup>143</sup>

<sup>70</sup> ≃ −0.5.

<sup>30</sup> ≃ −0.18 and

0.1

0.15

0.2

0.25

<sup>20</sup> <sup>=</sup> <sup>−</sup>0.39 <sup>±</sup> 0.04.

<sup>30</sup> (obtained using Survival

http://dx.doi.org/10.5772/52908

73

cross-identifications from the Australia Telescope 20 GHz survey, i.e *α*<sup>148</sup>

Fig.9 presents the contour levels of the distribution of *α*<sup>143</sup>

−1.5 −1 −0.5 0 0.5 1 1.5 2

particular pair of values of the two spectral indices. The maximum probability corresponds to *α*<sup>70</sup>

maximum probability of the spectral indices of blazars corresponds to *α*<sup>70</sup>

channel at 44GHz), the spectral index distributions are reliably reconstructed at each frequency.

steepening is also presented in [61] by the analysis of a complete sample of blazars selected at 37 GHz.

Credit: Planck Collaboration, A&A, Vol. 536, A13, 2011, reproduced with permission © ESO.

<sup>70</sup> ≃ −0.5. A secondary maximum can also be seen at *<sup>α</sup>*<sup>143</sup>

peculiar and/or extreme radio sources detected by the *Planck* surveys [62].

α143 70 [S∝να

**Figure 9.** Fig. 7 from [59]. Contour levels of the distribution of *α*<sup>143</sup>

*α*<sup>143</sup>

]

70 − 100 GHz, is clearly apparent11.

## **5.1. Synchrotron sources: "blazars"**

The most recent estimates on source number counts of radio (synchrotron) sources up to ∼ 50 − 70 GHz, and the optical identifications of the corresponding point sources (see, e.g., [52]), show that these counts are dominated by radio sources whose average spectral index is "flat", i.e., *<sup>α</sup>* ≃ 0.0 (with the usual convention *<sup>S</sup><sup>ν</sup>* <sup>∝</sup> *<sup>ν</sup>α*). This result confirms that the underlying source population is essentially made of Flat Spectrum Radio Quasars (FSRQ) and BL Lac objects, collectively called "blazars"9, with minor contributions coming from other source populations [13, 54]. At frequencies > 100 GHz, however, there is now new information for sources with flux densities below about 1 Jy, coming from the South Pole Telescope (SPT) collaboration [55], with surveys over 87 deg2 at 150 and 220 GHz, and from the Atacama Cosmology Telescope (ACT) survey over 455 deg2 at 148 GHz [56].

The "flat" spectra of blazars are generally believed to result from the superposition of different components in the inner part of AGN relativistic jets, each with a different synchrotron self-absorption frequency [57]. At a given frequency, the observed flux density is thus dominated by the synchrotron-emitting component which becomes self-absorbed, and, in the equipartition regime, the resulting spectrum is approximately flat. Given their sensitivity and full sky coverage, *Planck* surveys are uniquely able to shed light on this transition from an almost "flat" to a "steep" regime in the spectra of blazar sources, which can be very informative on the ages of sources and on the inner jet processes which determine the acceleration of the electrons [58].

To study the spectral properties of the extragalactic radio sources in the *Planck* ERCSC [59] used a reference 30 GHz sample above an estimated completeness limit *Slim* ≃ 1.0 Jy. Not all of these sources were detected at the ≥ 5*σ* level in each of the *Planck* frequency channels considered. Whenever a source was not detected in a given channel they replaced its (unknown) flux density by a 5*σ* upper limit, where for *σ* they used the average r.m.s.

<sup>9</sup> Blazars are jet-dominated extragalactic objects characterized by a strongly variable and polarized emission of the non-thermal radiation, from low radio energies up to high energy gamma rays [53].

error estimated at each *Planck* frequency. Finally, for estimating spectral index distributions, these upper limits have been redistributed among the flux density bins by using a Survival Analysis technique and, more specifically, by adopting the Kaplan-Meyer estimator10[59].

16 Open Questions in Cosmology

**5.1. Synchrotron sources: "blazars"**

the acceleration of the electrons [58].

As shown by [15], their Table 1, the full-sky surveys of the *Planck* satellite are – and will be, for years to come – unique in the millimetre, at *λ* ≤ 3 mm, and submillimetre domains. The lack of data in this frequency range represented the largest remaining gap in our knowledge of bright extragalactic sources (i.e., normal and star–forming galaxies and Active Galactic Nuclei, AGNs) across the electromagnetic spectrum. In the course of its planned surveys, *Planck* has been able to measure the integrated flux of many hundreds of "radio" sources – i.e., sources at intermediate to high–redshift, dominated by synchrotron emission due to hot electrons in the inner jets of the Active Galactic Nucleus (AGN) of the source – and of many thousands "'far–IR" sources – i.e., low–redshift dusty galaxies or sources with emission dominated by interstellar dust in thermal equilibrium with the radiation field – thus providing the *first complete full-sky catalogue* (ERCSC) of bright submillimetre sources. Thanks to this huge amount of new data it is thus possible to investigate the spectral energy distributions (SEDs) of EPS in a spectral domain very poorly explored before and, at the same time, their cosmological evolution, at least for some relevant source populations.

The most recent estimates on source number counts of radio (synchrotron) sources up to ∼ 50 − 70 GHz, and the optical identifications of the corresponding point sources (see, e.g., [52]), show that these counts are dominated by radio sources whose average spectral index is "flat", i.e., *<sup>α</sup>* ≃ 0.0 (with the usual convention *<sup>S</sup><sup>ν</sup>* <sup>∝</sup> *<sup>ν</sup>α*). This result confirms that the underlying source population is essentially made of Flat Spectrum Radio Quasars (FSRQ) and BL Lac objects, collectively called "blazars"9, with minor contributions coming from other source populations [13, 54]. At frequencies > 100 GHz, however, there is now new information for sources with flux densities below about 1 Jy, coming from the South Pole Telescope (SPT) collaboration [55], with surveys over 87 deg2 at 150 and 220 GHz, and from

The "flat" spectra of blazars are generally believed to result from the superposition of different components in the inner part of AGN relativistic jets, each with a different synchrotron self-absorption frequency [57]. At a given frequency, the observed flux density is thus dominated by the synchrotron-emitting component which becomes self-absorbed, and, in the equipartition regime, the resulting spectrum is approximately flat. Given their sensitivity and full sky coverage, *Planck* surveys are uniquely able to shed light on this transition from an almost "flat" to a "steep" regime in the spectra of blazar sources, which can be very informative on the ages of sources and on the inner jet processes which determine

To study the spectral properties of the extragalactic radio sources in the *Planck* ERCSC [59] used a reference 30 GHz sample above an estimated completeness limit *Slim* ≃ 1.0 Jy. Not all of these sources were detected at the ≥ 5*σ* level in each of the *Planck* frequency channels considered. Whenever a source was not detected in a given channel they replaced its (unknown) flux density by a 5*σ* upper limit, where for *σ* they used the average r.m.s.

<sup>9</sup> Blazars are jet-dominated extragalactic objects characterized by a strongly variable and polarized emission of the

non-thermal radiation, from low radio energies up to high energy gamma rays [53].

the Atacama Cosmology Telescope (ACT) survey over 455 deg2 at 148 GHz [56].

In the sample analyzed by [59], the 30–100 GHz median spectral index is very close to the *α* ≃ −0.39 found by [60] between 20 and 95 GHz, for a sample with 20 GHz flux density *S* > 150 mJy. Moreover, the 30–143 GHz median spectral index is in very good agreement with the one found by [56] for their bright (*S<sup>ν</sup>* > 50 mJy) 148 GHz-selected sample with complete cross-identifications from the Australia Telescope 20 GHz survey, i.e *α*<sup>148</sup> <sup>20</sup> <sup>=</sup> <sup>−</sup>0.39 <sup>±</sup> 0.04. Fig.9 presents the contour levels of the distribution of *α*<sup>143</sup> <sup>70</sup> vs. *<sup>α</sup>*<sup>70</sup> <sup>30</sup> (obtained using Survival Analysis) in the form of a 2D probability field: the colour scale can be interpreted as the probability of a given pair of spectral indices and a bending down, i.e. *α* < −0.5, at high frequencies is displayed. In the whole, the results of [59] show that in their sample selected at 30 GHz a moderate steepening of spectral indices of EPS at high radio frequencies, i.e. 70 − 100 GHz, is clearly apparent11.

**Figure 9.** Fig. 7 from [59]. Contour levels of the distribution of *α*<sup>143</sup> <sup>70</sup> vs. *α*<sup>70</sup> <sup>30</sup> obtained by Survival Analysis, i.e., taking into account the upper limits to flux densities at each frequency. The colour scale can be interpreted as the probability of having any particular pair of values of the two spectral indices. The maximum probability corresponds to *α*<sup>70</sup> <sup>30</sup> ≃ −0.18 and *<sup>α</sup>*<sup>143</sup> <sup>70</sup> ≃ −0.5. Credit: Planck Collaboration, A&A, Vol. 536, A13, 2011, reproduced with permission © ESO.

As already noted, at high radio frequencies (*ν* > 30 GHz) most of the bright extragalactic radio-sources are blazars. From the contour plot of Fig. 9 it is possible to see that the maximum probability of the spectral indices of blazars corresponds to *α*<sup>70</sup> <sup>30</sup> ≃ −0.18 and *α*<sup>143</sup> <sup>70</sup> ≃ −0.5. A secondary maximum can also be seen at *<sup>α</sup>*<sup>143</sup> <sup>70</sup> ≃ −1.2. In a companion paper, i.e. [61], a detailed discussion on the modelling of the spectra of this source class is also presented. In this paper, spectral energy distributions (SEDs) and radio continuum spectra are presented for a northern sample of 104 extragalactic radio sources, based on the *Planck* ERCSC and simultaneous multifrequency data12. The nine *Planck* frequencies, from 30 to 857 GHz, are complemented by a set of quasi–simultaneous observations ranging from radio to gamma-rays. SED modelling methods are discussed, with an emphasis on proper, physical

<sup>10</sup> Since the fraction of upper limits is found to be always small (it reaches approximately 30% only in the less sensitive channel at 44GHz), the spectral index distributions are reliably reconstructed at each frequency.

<sup>11</sup> Some hints in this direction were previously found by other works on the subject. Additional evidence of spectral steepening is also presented in [61] by the analysis of a complete sample of blazars selected at 37 GHz.

<sup>12</sup> The great amount of data present in the *Planck* ERCSC complemented with quasi-simultaneous ground–based observations at mm wavelengths have also enabled the study of the very interesting spectral properties of the rare peculiar and/or extreme radio sources detected by the *Planck* surveys [62].

modelling of the synchrotron bump using multiple components, and a thorough discussion on the original accelerated electron energy spectrum in blazar jets is presented. The main conclusion is that, al least for a fraction of the observed mm/sub-mm blazar spectra, the energy spectrum could be much harder than commonly thought, with a power-law index ∼ 1.5 and the implications of this hard value are discussed for the acceleration mechanisms effective in blazar shocks.

It has also been shown by [59] that differential number counts at 30, 44, and 70 GHz are in good agreement with those derived from *WMAP* [29] data at nearby frequencies. The model proposed by [54] is consistent with the present counts at frequencies up to 70 GHz, but over-predicts the counts at higher frequencies by a factor of about 2.0 at 143 GHz and about 2.6 at 217 GHz13. As shown above, the analysis of the spectral index distribution over different frequency intervals, within the uniquely broad range covered by *Planck* in the mm and sub-mm domain, has highlighted an average *steepening* of source spectra above about 70 GHz. This steepening accounts for the discrepancy between the model predictions of [54] and the observed differential number counts at HFI frequencies.

> **Figure 10.** Comparison between predicted and observed differential number counts at 148 GHz (*left panel*) and at 220 GHz (*right panel*). Filled circles: ACT data; open black circles: SPT data; open blue circles: *Planck* ERCSC counts [59] at 143 GHz (left panel) and 217 GHz (right panel). The plotted lines indicate predictions of different models, as follows: **C0**(dotted lines), **C1** (thick continuous lines), **C2Ex** (lower red long–dashed lines) and **C2Co** (upper red long–dashed lines) and the [54] model (blue

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model **C2Ex**. According to this, most of the FSRQs (which are the dominant population at low frequencies and at Jy flux densities), differently from BL Lacs, should bend their flat spectrum before or around 100 GHz. The **C2Ex** model also predicts a substantial increase of the BL Lac fraction at high frequencies and bright flux densities16. On the whole, the results of [63] imply that the parameter *rM* should be of parsec–scales, at least for FSRQs, in agreement with the theoretical predictions of [67], whereas values of *rM* ≪ 1 pc should be

The full-sky coverage of the *Planck* ERCSC provides an unsurpassed survey of galaxies at submillimetre (submm) wavelengths, representing a major improvement in the numbers of galaxies detected, as well as the range of far-IR/submm wavelengths over which they have been observed. The analysis done by [68] presented the first results on the properties of nearby galaxies using these data. They matched the ERCSC catalogue to IRAS-detected galaxies in the Imperial IRAS Faint Source Redshift Catalogue (IIFSCz) [69], so that they could measure the SEDs of these objects from 60 to 850 *µ*m. This produced a list of 1717 galaxies with reliable associations between *Planck* and IRAS, from which they selected a subset of 468 for SED studies, namely those with strong detections in the three highest frequency Planck bands and no evidence of cirrus contamination. This selection has thus provided a first *Planck* sample of local, i.e. at redshift < 0.1, dusty galaxies, very important for determining their emission properties and, in particular, the presence of different dust components contributing to their submm SEDs. Moreover, the richness of data on

<sup>16</sup> This is indeed observed: a clear dichotomy between FSRQs and BL Lac objects has been found in the *Planck* ERCSC. Almost all radio sources show very flat spectral indices at LFI frequencies, i.e. *αLFI* ≥ −0.2, whereas at HFI frequencies, BL Lacs keep flat spectra, i.e. *αHFI* ≥ −0.5, with a high fraction of FSRQs showing steeper spectra,

dash–dotted line). Credit: Tucci M., et al., A&A, Vol. 533, A57, 2011, reproduced with permission © ESO.

only typical of BL Lac objects or of rare, and compact, quasar sources.

**5.2. Far–IR sources: Local dusty galaxies**

i.e. *αHFI* ≤ −0.5.

In the fall of 2011, a successful explanation of the change detected in the spectral behavior of extragalactic radio sources (ERS) at frequencies above 70-80 GHz has been proposed by [63]. This paper makes a first attempt at constraining the most relevant physical parameters that characterize the emission of blazar sources by using the number counts and the spectral properties of extragalactic radio sources estimated from high–frequency radio surveys14. As noted before, a relevant steepening in blazar spectra with emerging spectral indices in the interval between −0.5 and −1.2, is commonly observed at mm/sub-mm wavelengths. [63] interpreted this spectral behavior as caused, at least partially, by the transition from the optically–thick to the optically–thin regime in the observed synchrotron emission of AGN jets [64]. Indeed, a "break" in the synchrotron spectrum of blazars, above which the spectrum bends down, thus becoming "steep", is predicted by models of synchrotron emission from inhomogeneous unresolved relativistic jets [65, 66]. Based on these models, [63] estimated the value of the frequency *ν<sup>M</sup>* (and of the corresponding radius *rM*) at which the break occurs on the basis of the flux densities of ERS measured at 5 GHz and of the most typical values for the relevant physical parameters of AGNs.

As displayed in Fig. 10, high frequency (*ν* ≥ 100 GHz) data on source number counts are the most powerful for distinguishing among different cosmological evolution models (see [63] for more details on the models plotted in Fig. 10)15. As clearly shown, these most recent data on number counts require spectral "breaks" in blazars' spectra and clearly favor the

<sup>13</sup> This implies that the contamination of the CMB APS by radio sources below the 1 Jy detection limit is lower than previously estimated. No significant changes are found, however, if we consider fainter source detection limits, i.e., 100 mJy, given the convergence between predicted and observed number counts.

<sup>14</sup> The main goal of [63] was to present physically grounded models to extrapolate the number counts of ERS, observationally determined over very large flux density intervals at cm wavelengths down to mm wavelengths, where experiments aimed at accurately measuring CMB anisotropies are carried out.

<sup>15</sup> The two most relevant models of [63], i.e. **C2Co** and **C2Ex**, assume different distributions of *rM* – i.e., the smallest radius in the AGN jet from which optically-thin synchrotron emission can be observed – for BL Lacs and FSRQs, with the former objects that generate, in general, the synchrotron emission from more compact regions, implying higher values of *ν<sup>M</sup>* (above 100 GHz for bright objects). These two models differ only in the *rM* distributions for FSRQs: in the **C2Co** model the emitting regions are more compact, implying values of *ν<sup>M</sup>* partially overlapping with those for BL Lacs, whereas in the **C2Ex** model they are more extended, thus predicting very different values of *ν<sup>M</sup>* for FSRQs and BL Lacs.

**Figure 10.** Comparison between predicted and observed differential number counts at 148 GHz (*left panel*) and at 220 GHz (*right panel*). Filled circles: ACT data; open black circles: SPT data; open blue circles: *Planck* ERCSC counts [59] at 143 GHz (left panel) and 217 GHz (right panel). The plotted lines indicate predictions of different models, as follows: **C0**(dotted lines), **C1** (thick continuous lines), **C2Ex** (lower red long–dashed lines) and **C2Co** (upper red long–dashed lines) and the [54] model (blue dash–dotted line). Credit: Tucci M., et al., A&A, Vol. 533, A57, 2011, reproduced with permission © ESO.

model **C2Ex**. According to this, most of the FSRQs (which are the dominant population at low frequencies and at Jy flux densities), differently from BL Lacs, should bend their flat spectrum before or around 100 GHz. The **C2Ex** model also predicts a substantial increase of the BL Lac fraction at high frequencies and bright flux densities16. On the whole, the results of [63] imply that the parameter *rM* should be of parsec–scales, at least for FSRQs, in agreement with the theoretical predictions of [67], whereas values of *rM* ≪ 1 pc should be only typical of BL Lac objects or of rare, and compact, quasar sources.

## **5.2. Far–IR sources: Local dusty galaxies**

18 Open Questions in Cosmology

effective in blazar shocks.

modelling of the synchrotron bump using multiple components, and a thorough discussion on the original accelerated electron energy spectrum in blazar jets is presented. The main conclusion is that, al least for a fraction of the observed mm/sub-mm blazar spectra, the energy spectrum could be much harder than commonly thought, with a power-law index ∼ 1.5 and the implications of this hard value are discussed for the acceleration mechanisms

It has also been shown by [59] that differential number counts at 30, 44, and 70 GHz are in good agreement with those derived from *WMAP* [29] data at nearby frequencies. The model proposed by [54] is consistent with the present counts at frequencies up to 70 GHz, but over-predicts the counts at higher frequencies by a factor of about 2.0 at 143 GHz and about 2.6 at 217 GHz13. As shown above, the analysis of the spectral index distribution over different frequency intervals, within the uniquely broad range covered by *Planck* in the mm and sub-mm domain, has highlighted an average *steepening* of source spectra above about 70 GHz. This steepening accounts for the discrepancy between the model predictions of [54]

In the fall of 2011, a successful explanation of the change detected in the spectral behavior of extragalactic radio sources (ERS) at frequencies above 70-80 GHz has been proposed by [63]. This paper makes a first attempt at constraining the most relevant physical parameters that characterize the emission of blazar sources by using the number counts and the spectral properties of extragalactic radio sources estimated from high–frequency radio surveys14. As noted before, a relevant steepening in blazar spectra with emerging spectral indices in the interval between −0.5 and −1.2, is commonly observed at mm/sub-mm wavelengths. [63] interpreted this spectral behavior as caused, at least partially, by the transition from the optically–thick to the optically–thin regime in the observed synchrotron emission of AGN jets [64]. Indeed, a "break" in the synchrotron spectrum of blazars, above which the spectrum bends down, thus becoming "steep", is predicted by models of synchrotron emission from inhomogeneous unresolved relativistic jets [65, 66]. Based on these models, [63] estimated the value of the frequency *ν<sup>M</sup>* (and of the corresponding radius *rM*) at which the break occurs on the basis of the flux densities of ERS measured at 5 GHz and of the most typical values

As displayed in Fig. 10, high frequency (*ν* ≥ 100 GHz) data on source number counts are the most powerful for distinguishing among different cosmological evolution models (see [63] for more details on the models plotted in Fig. 10)15. As clearly shown, these most recent data on number counts require spectral "breaks" in blazars' spectra and clearly favor the

<sup>13</sup> This implies that the contamination of the CMB APS by radio sources below the 1 Jy detection limit is lower than previously estimated. No significant changes are found, however, if we consider fainter source detection limits, i.e.,

<sup>14</sup> The main goal of [63] was to present physically grounded models to extrapolate the number counts of ERS, observationally determined over very large flux density intervals at cm wavelengths down to mm wavelengths,

<sup>15</sup> The two most relevant models of [63], i.e. **C2Co** and **C2Ex**, assume different distributions of *rM* – i.e., the smallest radius in the AGN jet from which optically-thin synchrotron emission can be observed – for BL Lacs and FSRQs, with the former objects that generate, in general, the synchrotron emission from more compact regions, implying higher values of *ν<sup>M</sup>* (above 100 GHz for bright objects). These two models differ only in the *rM* distributions for FSRQs: in the **C2Co** model the emitting regions are more compact, implying values of *ν<sup>M</sup>* partially overlapping with those for BL Lacs, whereas in the **C2Ex** model they are more extended, thus predicting very different values of *ν<sup>M</sup>* for FSRQs

100 mJy, given the convergence between predicted and observed number counts.

where experiments aimed at accurately measuring CMB anisotropies are carried out.

and the observed differential number counts at HFI frequencies.

for the relevant physical parameters of AGNs.

and BL Lacs.

The full-sky coverage of the *Planck* ERCSC provides an unsurpassed survey of galaxies at submillimetre (submm) wavelengths, representing a major improvement in the numbers of galaxies detected, as well as the range of far-IR/submm wavelengths over which they have been observed. The analysis done by [68] presented the first results on the properties of nearby galaxies using these data. They matched the ERCSC catalogue to IRAS-detected galaxies in the Imperial IRAS Faint Source Redshift Catalogue (IIFSCz) [69], so that they could measure the SEDs of these objects from 60 to 850 *µ*m. This produced a list of 1717 galaxies with reliable associations between *Planck* and IRAS, from which they selected a subset of 468 for SED studies, namely those with strong detections in the three highest frequency Planck bands and no evidence of cirrus contamination. This selection has thus provided a first *Planck* sample of local, i.e. at redshift < 0.1, dusty galaxies, very important for determining their emission properties and, in particular, the presence of different dust components contributing to their submm SEDs. Moreover, the richness of data on

<sup>16</sup> This is indeed observed: a clear dichotomy between FSRQs and BL Lac objects has been found in the *Planck* ERCSC. Almost all radio sources show very flat spectral indices at LFI frequencies, i.e. *αLFI* ≥ −0.2, whereas at HFI frequencies, BL Lacs keep flat spectra, i.e. *αHFI* ≥ −0.5, with a high fraction of FSRQs showing steeper spectra, i.e. *αHFI* ≤ −0.5.

extragalactic point sources gathered by *Planck* has allowed the measurement of the submm number density of bright (*S* > 0.5 − 2 Jy) dusty galaxies (and of synchrotron-dominated sources) for the first time.

[68] it is also found that some local galaxies are both luminous and cool, with properties similar to those of the distant submm galaxies uncovered in deep surveys. This suggests that previous studies of dust in local galaxies have been biased away from such luminous cool objects. In most galaxies the dust SEDs are found to be better described by parametric models containing two dust components, one warm and one cold, with the cold component reaching temperatures as low as 10 K17. The main conclusion of [68] is that cold (*T* < 20 K) dust is thus a significant and largely unexplored component of many nearby galaxies. Furthermore, a new population of cool submm galaxies is detected, with presence of very cold dust (*T* = 10 − 13 K) showing a more extended spatial distribution than generally

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**Figure 12.** Fig. 9 from [70]. *Planck* differential number counts, normalized to the Euclidean value (i.e. *S*2.5*dN*/*dS*), compared with models and other data sets. *Planck* counts: total (black filled circles); dusty (red circles); synchrotron (blue circles). Four models are also plotted: [54], dealing only with synchrotron sources – solid line; [63] dealing only with synchrotron sources – dots; [71] dealing only with dusty sources – long dashes; [72] dealing only with local dusty sources – short dashes. Other data sets: *Planck* early counts for 30 GHz-selected radio galaxies [59] at 100, 143 and 217 GHz (open diamonds); *Herschel* ATLAS and HerMES counts at 350 and 500*µm* from [73] and [74]; BLAST at the same two wavelengths, from [75], all shown as triangles. Left vertical axes are in units of Jy1.5 sr−1, and the right vertical axis in Jy1.5.deg−2. Credit: Planck Collaboration, A&A, submitted

Very recently, using EPS samples selected from the first *Planck* 1.6 full-sky surveys, i.e. from the *Planck* ERCSC, [70] have derived number counts of extragalactic sources from 100 to 857 GHz (3 mm to 350 *µ*m). Three zones (deep, medium and shallow) of approximately

<sup>17</sup> Fits to SEDs of selected objects using more sophisticated templates derived from radiative transfer models confirm

assumed for the gas and dust in galaxies.

(ms AA/2012/20053), 2012, reproduced with permission © ESO.

the presence of the colder dust found through parametric fitting.

**Figure 11.** Fig. 2 from [68]. Sky plot of ERCSC sources in Galactic coordinates. ERCSC point-sources (black filled hexagons) and ERCSC sources flagged as extended (blue filled hexagons) are shown. Red hexagons are sources associated with IIFSCz IRAS FSC galaxies. Green hexagons are ERCSC sources not associated with IIFSCz, but associated with bright galaxies in NED (only for |*b*| > 60◦ for extended sources). Credit: Planck Collaboration, A&A, Vol. 536, A16, 2011, reproduced with permission © ESO.

Fig. 11 shows the sky distribution of ERCSC sources at |*b*| > <sup>20</sup>◦, with sources flagged as extended in the ERCSC shown as blue filled hexagons, and point-sources shown in black. Associations with the IIFSCz are shown as red circles. The extended sources not associated with IIFSCz sources have a strikingly clustered distribution, which matches the areas of our Galaxy with strong cirrus emission, as evidenced by IRAS 100 *µ*m maps. Therefore, the majority of these are cirrus sources and not extragalactic (see [68] for more details).

The studies of nearby galaxies detected by *Planck* [68] confirm the presence of cold dust in local galaxies and also largely in dwarf galaxies. The SEDs are fitted using parametric dust models to determine the range of dust temperatures and emissivities. They found evidence for colder dust than has previously been found in external galaxies, with temperatures *T* < 20 K. Such cold temperatures are found by using both the standard single temperature dust model with variable emissivity *β*, or a two dust temperature model with *β* fixed at 2. In [68] it is also found that some local galaxies are both luminous and cool, with properties similar to those of the distant submm galaxies uncovered in deep surveys. This suggests that previous studies of dust in local galaxies have been biased away from such luminous cool objects. In most galaxies the dust SEDs are found to be better described by parametric models containing two dust components, one warm and one cold, with the cold component reaching temperatures as low as 10 K17. The main conclusion of [68] is that cold (*T* < 20 K) dust is thus a significant and largely unexplored component of many nearby galaxies. Furthermore, a new population of cool submm galaxies is detected, with presence of very cold dust (*T* = 10 − 13 K) showing a more extended spatial distribution than generally assumed for the gas and dust in galaxies.

20 Open Questions in Cosmology

sources) for the first time.

ESO.

extragalactic point sources gathered by *Planck* has allowed the measurement of the submm number density of bright (*S* > 0.5 − 2 Jy) dusty galaxies (and of synchrotron-dominated

**Figure 11.** Fig. 2 from [68]. Sky plot of ERCSC sources in Galactic coordinates. ERCSC point-sources (black filled hexagons) and ERCSC sources flagged as extended (blue filled hexagons) are shown. Red hexagons are sources associated with IIFSCz IRAS FSC galaxies. Green hexagons are ERCSC sources not associated with IIFSCz, but associated with bright galaxies in NED (only for |*b*| > 60◦ for extended sources). Credit: Planck Collaboration, A&A, Vol. 536, A16, 2011, reproduced with permission ©

Fig. 11 shows the sky distribution of ERCSC sources at |*b*| > <sup>20</sup>◦, with sources flagged as extended in the ERCSC shown as blue filled hexagons, and point-sources shown in black. Associations with the IIFSCz are shown as red circles. The extended sources not associated with IIFSCz sources have a strikingly clustered distribution, which matches the areas of our Galaxy with strong cirrus emission, as evidenced by IRAS 100 *µ*m maps. Therefore, the

The studies of nearby galaxies detected by *Planck* [68] confirm the presence of cold dust in local galaxies and also largely in dwarf galaxies. The SEDs are fitted using parametric dust models to determine the range of dust temperatures and emissivities. They found evidence for colder dust than has previously been found in external galaxies, with temperatures *T* < 20 K. Such cold temperatures are found by using both the standard single temperature dust model with variable emissivity *β*, or a two dust temperature model with *β* fixed at 2. In

majority of these are cirrus sources and not extragalactic (see [68] for more details).

**Figure 12.** Fig. 9 from [70]. *Planck* differential number counts, normalized to the Euclidean value (i.e. *S*2.5*dN*/*dS*), compared with models and other data sets. *Planck* counts: total (black filled circles); dusty (red circles); synchrotron (blue circles). Four models are also plotted: [54], dealing only with synchrotron sources – solid line; [63] dealing only with synchrotron sources – dots; [71] dealing only with dusty sources – long dashes; [72] dealing only with local dusty sources – short dashes. Other data sets: *Planck* early counts for 30 GHz-selected radio galaxies [59] at 100, 143 and 217 GHz (open diamonds); *Herschel* ATLAS and HerMES counts at 350 and 500*µm* from [73] and [74]; BLAST at the same two wavelengths, from [75], all shown as triangles. Left vertical axes are in units of Jy1.5 sr−1, and the right vertical axis in Jy1.5.deg−2. Credit: Planck Collaboration, A&A, submitted (ms AA/2012/20053), 2012, reproduced with permission © ESO.

Very recently, using EPS samples selected from the first *Planck* 1.6 full-sky surveys, i.e. from the *Planck* ERCSC, [70] have derived number counts of extragalactic sources from 100 to 857 GHz (3 mm to 350 *µ*m). Three zones (deep, medium and shallow) of approximately

<sup>17</sup> Fits to SEDs of selected objects using more sophisticated templates derived from radiative transfer models confirm the presence of the colder dust found through parametric fitting.

homogeneous coverage are used to ensure a clean completeness correction18. For the first time, bright number counts of EPS at 353, 545 and 857 GHz (i.e., 850, 550 and 350 *µ*m) have been calculated19. *Planck* number counts are found to be in the Euclidean regime in this frequency range, since the ERCSC comprises only bright sources (*S* > 0.3 Jy). The estimated number counts appear generally in agreement with other data sets, when available (see [70] for more details).

[75], about 10% by *Herschel* at 350 *µ*m [73]. Thus, in the absence of foreground (Galactic dust) and CMB emissions, and when the instrument noise is subdominant, maps of the diffuse emission at the angular resolution probed by the current surveys reveal a web of structures, characteristic of CIB anisotropies. With the advent of large area far-IR to millimeter surveys (*Herschel*, *Planck*, SPT, and ACT), CIB anisotropies thus constitute a new tool for structure

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CIB anisotropies are expected to trace large-scale structures and probe the clustering properties of galaxies, which in turn are linked to those of their hosting dark matter halos. Because the clustering of dark matter is well understood, observations of anisotropies in the CIB constrain the relationship between dusty, star-forming galaxies at high redshift, i.e. *z* ≥ 2, and the underlying dark matter distribution. The angular power spectrum of CIB anisotropies has two contributions, a white-noise component caused by shot noise and an additional component caused by spatial correlations between the sources of the CIB. Correlated CIB anisotropies have already been measured by many space–borne as well as ground–based experiments (see [80] for more details). Depending on the frequency, the angular resolution and size of the survey, these measurements can probe two different clustering regimes. On small angular scales (ℓ ≥ 2000), they measure the clustering within a single dark matter halo and, accordingly, the physics governing how dusty, star–forming galaxies form within a halo. On large angular scales, i.e. 200 ≤ ℓ ≤ 2000, CIB anisotropies measure clustering between galaxies in different dark matter halos. These measurements primarily constrain the large-scale, linear bias, *b*, of dusty galaxies, which is usually assumed

Thanks to the exceptional quality of the *Planck* data, [80] were able to measure the clustering of dusty, star-forming galaxies at 217, 353, 545, and 857 GHz with unprecedented precision. The CIB maps were cleaned using templates: HI for Galactic cirrus; and the *Planck* 143 GHz maps for CMB. Having HI data is necessary to cleanly separate CIB and cirrus fluctuations. After careful cleaning, they obtained CIB anisotropy maps that reveal structures produced by the cumulative emission of high-redshift, dusty, star–forming galaxies. The maps are highly correlated at high *Planck* frequencies, whereas they decorrelate at lower *Planck* HFI frequencies. [80] then computed the power spectra of the maps and their associated errors using a dedicated pipeline and ended up with measurements of the APS of the CIB anisotropy, *C*ℓ, at 217, 353, 545, and 857 GHz, with high signal-to-noise ratio over the range 200 < *l* < 2000. These measurements compare very well with previous measurements at

Moreover, from *Planck* data alone [80] could exclude a model where galaxies trace the (linear theory) matter power spectrum with a scale-independent bias: that model requires an *unrealistic* high level of shot noise to match the small-scale power they observed. Consequently, an alternative model that couples the dusty galaxy, parametric evolution model of [71] with a halo model approach has been developed (see [80], again, for more details). Characterized by only two parameters, this model provides an excellent fit to our measured anisotropy angular power spectrum for each frequency treated independently. In the near future, modelling and interpretation of the CIB anisotropy will be aided by the use

<sup>22</sup> The SED of CIB anisotropies is not different from the CIB mean SED, even at 217 GHz. This is expected from the model of [71] and reflects the fact that the CIB intensity and anisotropies are produced by the same population of

formation and evolution studies.

to be scale-independent over the relevant range.

higher ℓ22.

sources.

Using multi-frequency information to classify the sources as dusty- or synchrotron-dominated (and measure their spectral indices), the most striking result of [70] is the estimated contribution to the number counts by each population. These new estimates of number counts of synchrotron and of dust–dominated EPS (displayed in Fig. 12) have allowed new constraints to be placed on models which extend their predictions to bright flux densities, i.e. *S* > 1 Jy. A very relevant result is that the model **C2Ex** of [63] (see Section 5.1) is performing particularly well at reproducing the number counts of synchrotron-dominated sources up to 545 GHz. On the contrary, [70] highlights the failure of many models for number count predictions of dusty sources to reproduce all the high-frequency counts. The model of [71] agrees marginally at 857 GHz but is too low at 545 GHz and also at lower frequencies, whereas the model of [72] is marginally lower at 857 GHz, fits the data well at 545 GHz, but is too low at 353 GHz. The likely origin of the discrepancies is an inaccurate description of the galaxy SEDs used at low redshift in these models. Indeed a cold dust component, detected by [68], is rarely included in the models of galaxy SEDs at low redshift. On the whole, these results already obtained by the exploitation of the *Planck* ERCSC data are providing valuable information about the ubiquity of cold dust in the local Universe, at least in statistical terms, and are guiding to a better understanding of the cosmological evolution of EPS at mm/sub-mm wavelengths.

## **6. Cosmic Infrared Background anisotropies**

The Cosmic Infrared Background (CIB) is the relic emission, at wavelengths larger than a few microns, of the formation and evolution of the galaxies of all types, including AGNs and star-forming systems [76–78]20. The CIB accounts for roughly half of the total energy in the optical/infrared Extragalactic Background Light (EBL) [77], although with some uncertainty, and its SED peaks near 150 *µ*m. Since local galaxies give rise to an integrated infrared output that amounts to only about a third of the optical one [79], there must have been a strong evolution of galaxy properties towards enhanced far–IR output in the past. Therefore, the CIB, made up by high density, faint and distant galaxies21 is barely resolved into its constituents. Indeed, less than 10% of the CIB is resolved by the *Spitzer* satellite at 160 *µ*m

<sup>18</sup> The sample, prior to the 80 % completeness cut, contains between 217 sources at 100 GHz and 1058 sources at 857 GHz over about 12,800 to 16,550 deg<sup>2</sup> (31 to 40 % of the sky). After the 80 % completeness cut, between 122 and 452 and sources remain, with flux densities above 0.3 and 1.9 Jy, at 100 and 857 GHz, respectively.

<sup>19</sup> More specifically, number counts have been provided of synchrotron-dominated sources at high frequency (353 to 857 GHz) and of dusty-dominated galaxies at lower frequencies (217 and 353 GHz).

<sup>20</sup> An important goal of studies of galaxy formation has thus been the characterization of the statistical behavior of galaxies responsible for the CIB - such as the number counts, redshift distribution, mean SED, luminosity function, clustering – and their physical properties, such as the roles of star-forming vs. accreting systems, the density of star formation, and the number density of very hot stars

<sup>21</sup> The CIB records much of the radiant energy released by processes of structure formation occurred since the last scattering epoch, four hundred thousand years after the Big Bang, when the CMB was produced.

[75], about 10% by *Herschel* at 350 *µ*m [73]. Thus, in the absence of foreground (Galactic dust) and CMB emissions, and when the instrument noise is subdominant, maps of the diffuse emission at the angular resolution probed by the current surveys reveal a web of structures, characteristic of CIB anisotropies. With the advent of large area far-IR to millimeter surveys (*Herschel*, *Planck*, SPT, and ACT), CIB anisotropies thus constitute a new tool for structure formation and evolution studies.

22 Open Questions in Cosmology

for more details).

homogeneous coverage are used to ensure a clean completeness correction18. For the first time, bright number counts of EPS at 353, 545 and 857 GHz (i.e., 850, 550 and 350 *µ*m) have been calculated19. *Planck* number counts are found to be in the Euclidean regime in this frequency range, since the ERCSC comprises only bright sources (*S* > 0.3 Jy). The estimated number counts appear generally in agreement with other data sets, when available (see [70]

Using multi-frequency information to classify the sources as dusty- or synchrotron-dominated (and measure their spectral indices), the most striking result of [70] is the estimated contribution to the number counts by each population. These new estimates of number counts of synchrotron and of dust–dominated EPS (displayed in Fig. 12) have allowed new constraints to be placed on models which extend their predictions to bright flux densities, i.e. *S* > 1 Jy. A very relevant result is that the model **C2Ex** of [63] (see Section 5.1) is performing particularly well at reproducing the number counts of synchrotron-dominated sources up to 545 GHz. On the contrary, [70] highlights the failure of many models for number count predictions of dusty sources to reproduce all the high-frequency counts. The model of [71] agrees marginally at 857 GHz but is too low at 545 GHz and also at lower frequencies, whereas the model of [72] is marginally lower at 857 GHz, fits the data well at 545 GHz, but is too low at 353 GHz. The likely origin of the discrepancies is an inaccurate description of the galaxy SEDs used at low redshift in these models. Indeed a cold dust component, detected by [68], is rarely included in the models of galaxy SEDs at low redshift. On the whole, these results already obtained by the exploitation of the *Planck* ERCSC data are providing valuable information about the ubiquity of cold dust in the local Universe, at least in statistical terms, and are guiding to a better understanding

The Cosmic Infrared Background (CIB) is the relic emission, at wavelengths larger than a few microns, of the formation and evolution of the galaxies of all types, including AGNs and star-forming systems [76–78]20. The CIB accounts for roughly half of the total energy in the optical/infrared Extragalactic Background Light (EBL) [77], although with some uncertainty, and its SED peaks near 150 *µ*m. Since local galaxies give rise to an integrated infrared output that amounts to only about a third of the optical one [79], there must have been a strong evolution of galaxy properties towards enhanced far–IR output in the past. Therefore, the CIB, made up by high density, faint and distant galaxies21 is barely resolved into its constituents. Indeed, less than 10% of the CIB is resolved by the *Spitzer* satellite at 160 *µ*m

<sup>18</sup> The sample, prior to the 80 % completeness cut, contains between 217 sources at 100 GHz and 1058 sources at 857 GHz over about 12,800 to 16,550 deg<sup>2</sup> (31 to 40 % of the sky). After the 80 % completeness cut, between 122 and 452 and

<sup>19</sup> More specifically, number counts have been provided of synchrotron-dominated sources at high frequency (353 to

<sup>20</sup> An important goal of studies of galaxy formation has thus been the characterization of the statistical behavior of galaxies responsible for the CIB - such as the number counts, redshift distribution, mean SED, luminosity function, clustering – and their physical properties, such as the roles of star-forming vs. accreting systems, the density of star

<sup>21</sup> The CIB records much of the radiant energy released by processes of structure formation occurred since the last

sources remain, with flux densities above 0.3 and 1.9 Jy, at 100 and 857 GHz, respectively.

scattering epoch, four hundred thousand years after the Big Bang, when the CMB was produced.

857 GHz) and of dusty-dominated galaxies at lower frequencies (217 and 353 GHz).

formation, and the number density of very hot stars

of the cosmological evolution of EPS at mm/sub-mm wavelengths.

**6. Cosmic Infrared Background anisotropies**

CIB anisotropies are expected to trace large-scale structures and probe the clustering properties of galaxies, which in turn are linked to those of their hosting dark matter halos. Because the clustering of dark matter is well understood, observations of anisotropies in the CIB constrain the relationship between dusty, star-forming galaxies at high redshift, i.e. *z* ≥ 2, and the underlying dark matter distribution. The angular power spectrum of CIB anisotropies has two contributions, a white-noise component caused by shot noise and an additional component caused by spatial correlations between the sources of the CIB. Correlated CIB anisotropies have already been measured by many space–borne as well as ground–based experiments (see [80] for more details). Depending on the frequency, the angular resolution and size of the survey, these measurements can probe two different clustering regimes. On small angular scales (ℓ ≥ 2000), they measure the clustering within a single dark matter halo and, accordingly, the physics governing how dusty, star–forming galaxies form within a halo. On large angular scales, i.e. 200 ≤ ℓ ≤ 2000, CIB anisotropies measure clustering between galaxies in different dark matter halos. These measurements primarily constrain the large-scale, linear bias, *b*, of dusty galaxies, which is usually assumed to be scale-independent over the relevant range.

Thanks to the exceptional quality of the *Planck* data, [80] were able to measure the clustering of dusty, star-forming galaxies at 217, 353, 545, and 857 GHz with unprecedented precision. The CIB maps were cleaned using templates: HI for Galactic cirrus; and the *Planck* 143 GHz maps for CMB. Having HI data is necessary to cleanly separate CIB and cirrus fluctuations. After careful cleaning, they obtained CIB anisotropy maps that reveal structures produced by the cumulative emission of high-redshift, dusty, star–forming galaxies. The maps are highly correlated at high *Planck* frequencies, whereas they decorrelate at lower *Planck* HFI frequencies. [80] then computed the power spectra of the maps and their associated errors using a dedicated pipeline and ended up with measurements of the APS of the CIB anisotropy, *C*ℓ, at 217, 353, 545, and 857 GHz, with high signal-to-noise ratio over the range 200 < *l* < 2000. These measurements compare very well with previous measurements at higher ℓ22.

Moreover, from *Planck* data alone [80] could exclude a model where galaxies trace the (linear theory) matter power spectrum with a scale-independent bias: that model requires an *unrealistic* high level of shot noise to match the small-scale power they observed. Consequently, an alternative model that couples the dusty galaxy, parametric evolution model of [71] with a halo model approach has been developed (see [80], again, for more details). Characterized by only two parameters, this model provides an excellent fit to our measured anisotropy angular power spectrum for each frequency treated independently. In the near future, modelling and interpretation of the CIB anisotropy will be aided by the use

<sup>22</sup> The SED of CIB anisotropies is not different from the CIB mean SED, even at 217 GHz. This is expected from the model of [71] and reflects the fact that the CIB intensity and anisotropies are produced by the same population of sources.

of cross-power spectra between bands, and by the combination of the *Planck* and *Herschel* data at 857 and 545/600 GHz and *Planck* and SPT/ACT data at 220 GHz.

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## **7. Acknowledgements**

LT, FA and JMD acknowledge partial financial support from the Spanish Ministry of Science and Innovation (MICINN) under project AYA2010–21766–C03-01. CB acknowledges partial financial support by ASI through ASI/INAF Agreement I/072/09/0 for the Planck LFI Activity of Phase E2 and by MIUR through PRIN 2009 grant n. 2009XZ54H2. The analysis of the *Planck* full-sky maps has been performed by means of the HEALPix package [12]. We thank the Editorial Board of Astronomy and Astrophysics (European Southern Observatory; ESO) for having granted us the permission to reproduce many Figures originally published in (or submitted to) the same Journal. Credits are indicated in each one of the Figures we used. Many thanks are due to Douglas Scott, for carefully reading the original manuscript and for his very useful suggestions. We also warmly thank the *Planck* Collaboration and, in particular, all the members of the *Planck* Working Groups 2, 5 and 6 and of the LFI Core Team A09, with whom we shared the analysis and the interpretation of *Planck* data as for the subject here discussed, i.e. "extragalactic compact sources". Finally, we also thank the members of the *Planck* Science Team (ST) and of the *Planck* Editorial Board (EB) for granting us the permission of publishing this Chapter.

## **Author details**

Luigi Toffolatti1,2,<sup>⋆</sup>, Carlo Burigana3,4, Francisco Argüeso2,5 and José M. Diego<sup>2</sup>

<sup>⋆</sup> Address all correspondence to: ltoffolatti@uniovi.es

1 Department of Physics, University of Oviedo, Oviedo, Spain

2 IFCA-CSIC, University of Cantabria, Santander, Spain

3 National Institute of Astrophysics - Institute of Space Astrophysics and Cosmic Physics (INAF-IASF), Bologna, Italy

4 Department of Physics, University of Ferrara, Ferrara, Italy

5 Department of Mathematics, University of Oviedo, Oviedo, Spain

## **References**


24 Open Questions in Cosmology

**7. Acknowledgements**

**Author details**

**References**

A4:1-20.

us the permission of publishing this Chapter.

<sup>⋆</sup> Address all correspondence to: ltoffolatti@uniovi.es

1 Department of Physics, University of Oviedo, Oviedo, Spain 2 IFCA-CSIC, University of Cantabria, Santander, Spain

4 Department of Physics, University of Ferrara, Ferrara, Italy 5 Department of Mathematics, University of Oviedo, Oviedo, Spain

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Luigi Toffolatti1,2,<sup>⋆</sup>, Carlo Burigana3,4, Francisco Argüeso2,5 and José M. Diego<sup>2</sup>

(INAF-IASF), Bologna, Italy

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LT, FA and JMD acknowledge partial financial support from the Spanish Ministry of Science and Innovation (MICINN) under project AYA2010–21766–C03-01. CB acknowledges partial financial support by ASI through ASI/INAF Agreement I/072/09/0 for the Planck LFI Activity of Phase E2 and by MIUR through PRIN 2009 grant n. 2009XZ54H2. The analysis of the *Planck* full-sky maps has been performed by means of the HEALPix package [12]. We thank the Editorial Board of Astronomy and Astrophysics (European Southern Observatory; ESO) for having granted us the permission to reproduce many Figures originally published in (or submitted to) the same Journal. Credits are indicated in each one of the Figures we used. Many thanks are due to Douglas Scott, for carefully reading the original manuscript and for his very useful suggestions. We also warmly thank the *Planck* Collaboration and, in particular, all the members of the *Planck* Working Groups 2, 5 and 6 and of the LFI Core Team A09, with whom we shared the analysis and the interpretation of *Planck* data as for the subject here discussed, i.e. "extragalactic compact sources". Finally, we also thank the members of the *Planck* Science Team (ST) and of the *Planck* Editorial Board (EB) for granting

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**Chapter 4**

**Provisional chapter**

**Uncertainties in Dark Matter Indirect Detection**

**Uncertainties in Dark Matter Indirect Detection**

Astrophysical observations interpreted in the standard (ΛCDM) cosmological framework indicate that about eighty percent of matter in the Universe is non-luminous. This dark matter poses a major problem for particle physics: no known particle explains its inferred properties. Observations are most consistent with the assumption that dark matter is composed of weakly interacting massive particles. The discovery of these particles is vital to validate the prevailing dark matter paradigm. In this work, we examine the uncertainties affecting the astrophysical discovery of dark matter particles via secondary cosmic ray

Before trying to discover dark matter particles, we should know some of their properties. These properties of dark matter are reconstructed from astrophysical observations, most of which (including the galactic rotational velocities, galactic structure formation, and weak gravitational lensing) indicate that dark matter particles have mass and are present in large numbers around us. Measurements of the cosmic microwave background radiation and the abundance of light elements further suggests that dark matter is not composed of baryonic particles (that is quarks). Since electromagnetic interactions imply light emission, we are led to conclude that dark matter particles may only interact with ordinary matter weakly, either via the standard W and Z bosons, or via an unknown force. Since they are electrically neutral, the simplest assumption is that dark matter particles are their own anti-particles. Their diffuse distribution, inferred from their gravitational effects, indicates that they probably interact weakly with each other. To be present over the observed distance scales, dark matter particles have to be stable on the timescale of the age of the Universe. Lastly, the observed large scale structure indicates that dark matter is cold - its particles are non-relativistic at its

Based on the above properties, dark matter particles are being searched for in three major types of experiments. First, since the CERN Large Hadron Collider (LHC) in Geneva was built specifically to explore the electroweak sector of the standard particle model, it is an

and reproduction in any medium, provided the original work is properly cited.

©2012 Auchettl and Balázs, licensee InTech. This is an open access chapter 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. © 2012 Auchettl and Balázs; licensee InTech. This is an open access article 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.

© 2012 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,

Katie Auchettl and Csaba Balázs

Katie Auchettl and Csaba Balázs

http://dx.doi.org/10.5772/52052

**1. Introduction**

emission.

present temperature.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter


**Chapter 4**

**Provisional chapter**

## **Uncertainties in Dark Matter Indirect Detection**

**Uncertainties in Dark Matter Indirect Detection**

Katie Auchettl and Csaba Balázs Additional information is available at the end of the chapter

Katie Auchettl and Csaba Balázs

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52052

## **1. Introduction**

30 Open Questions in Cosmology

86 Open Questions in Cosmology

L21-L25.

Vol. 516, A43:1-15.

A18:1-30.

Society, Vol. 356:192–204.

and Astrophysics, Vol. 518, L8-L12.

and Astrophysics, Vol. 451:417-429.

universe. The Astronomical Journal, Vol. 101:354-361.

[72] Serjeant S., and Harrison D. (2005) The local submillimetre luminosity functions and predictions from Spitzer to Herschel. Monthly Notices of the Royal Astronomical

[73] Oliver S.J., Wang L., Smith A.J., Altieri B., Amblard A., et al. (2010) HerMES: SPIRE galaxy number counts at 250, 350, and 500 *µ*m. Astronomy and Astrophysics, Vol. 518,

[74] Clements D.L., Rigby E., Maddox S., Dunne L., Mortier A., et al. (2010) Herschel-ATLAS: Extragalactic number counts from 250 to 500 microns. Astronomy

[75] Bethermin M., Dole H., Cousin M., and Bavouzet N. (2010) Submillimeter number counts at 250 *µ*m, 350 *µ*m and 500 *µ*m in BLAST data. Astronomy and Astrophysics,

[76] Puget J.L., Abergel A., Bernard J.P., et al. (1996) Tentative detection of a cosmic far-infrared background with COBE. Astronomy and Astrophysics, Vol. 308, L5-L9.

[77] Hauser M.G., and Dwek E. (2001) The Cosmic Infrared Background: Measurements and Implications. Annual Review of Astronomy and Astrophysics, Vol. 39:249-307.

[78] Dole H., Lagache G., Puget J.L., et al. (2006) The cosmic infrared background resolved by Spitzer. Contributions of mid-infrared galaxies to the far-IR background. Astronomy

[79] Soifer B.T., and Neugebauer G. (1991) The properties of infrared galaxies in the local

[80] Planck Collaboration XVIII. (2011) Planck early results. XVIII. The power spectrum of cosmic infrared background anisotropies. Astronomy and Astrophysics, Vol. 536, Astrophysical observations interpreted in the standard (ΛCDM) cosmological framework indicate that about eighty percent of matter in the Universe is non-luminous. This dark matter poses a major problem for particle physics: no known particle explains its inferred properties. Observations are most consistent with the assumption that dark matter is composed of weakly interacting massive particles. The discovery of these particles is vital to validate the prevailing dark matter paradigm. In this work, we examine the uncertainties affecting the astrophysical discovery of dark matter particles via secondary cosmic ray emission.

Before trying to discover dark matter particles, we should know some of their properties. These properties of dark matter are reconstructed from astrophysical observations, most of which (including the galactic rotational velocities, galactic structure formation, and weak gravitational lensing) indicate that dark matter particles have mass and are present in large numbers around us. Measurements of the cosmic microwave background radiation and the abundance of light elements further suggests that dark matter is not composed of baryonic particles (that is quarks). Since electromagnetic interactions imply light emission, we are led to conclude that dark matter particles may only interact with ordinary matter weakly, either via the standard W and Z bosons, or via an unknown force. Since they are electrically neutral, the simplest assumption is that dark matter particles are their own anti-particles. Their diffuse distribution, inferred from their gravitational effects, indicates that they probably interact weakly with each other. To be present over the observed distance scales, dark matter particles have to be stable on the timescale of the age of the Universe. Lastly, the observed large scale structure indicates that dark matter is cold - its particles are non-relativistic at its present temperature.

Based on the above properties, dark matter particles are being searched for in three major types of experiments. First, since the CERN Large Hadron Collider (LHC) in Geneva was built specifically to explore the electroweak sector of the standard particle model, it is an

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. © 2012 Auchettl and Balázs; licensee InTech. This is an open access article 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. © 2012 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.

©2012 Auchettl and Balázs, licensee InTech. This is an open access chapter distributed under the terms of the

obvious place for trying to create dark matter particles. While we are in (almost) full control of this experiment, without knowing the exact mass and interaction strength between ordinary and dark matter we can only hope that the energy and luminosity of the LHC are high enough to produce the latter. Second, because it appears that the Solar System is immersed in a high flux stream of dark matter particles, it is natural to try to detect collisions between them and well shielded nuclei in underground laboratories. These experiments have the potential to probe interactions between dark and ordinary matter even beyond the reach of the LHC. However, these experiments are also limited by the unknown mass, interaction strength, flux and velocity distribution of the dark matter particles. Finally, perhaps the most general and unconstrained way to discover dark matter particles is to find traces of their annihilation or decay products in cosmic rays bombarding Earth.

• CAPRICE [3], • MASS [4], and • HEAT [5, 6]

• AMS [12],

• HESS [14, 15]

• PPB-BETS [13], and

and momentum loss rates.

collaborations. Measurements of the PAMELA satellite stirred great interest by showing an unmistakable rise of the local *<sup>e</sup>*+/*e*<sup>−</sup> fraction, that deviates significantly from theoretical predictions for *Ee*<sup>+</sup> > 10 GeV [7]. The combined experimental and theoretical uncertainties

Uncertainties in Dark Matter Indirect Detection

http://dx.doi.org/10.5772/52052

89

The summed flux of electrons and positrons also indicates an anomalous excess of

collaborations. The Fermi Large Area Telescope (LAT) satellite confirmed the excess of the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> flux for energies over 100 GeV [16, 17]. The Fermi-LAT results are consistent with those of the PAMELA collaboration, which measured the cosmic ray electron flux up to 625 GeV [18]. To date the Fermi-LAT data are the most precise indication of such an anomaly in the electron-positron spectrum. The Fermi-LAT data differ by several standard deviations

Between 2008 and 2011, of the order of a thousand papers were devoted to explaining the difference between the experimental measurements of Fermi-LAT and PAMELA and theoretical calculations. Speculation ranged from the modification of the cosmic ray propagation through supernova remnants, to dark matter annihilation. A concise summary

Before drawing conclusions from the electron-positron anomaly, however, one has to carefully examine the status of the theoretical understanding of Galactic cosmic rays. Unfortunately, even the origin of the cosmic rays is uncertain. The theory describing the propagation of cosmic ray particles from their birthplace through the Milky Way is based on the diffusion-convection model. The quantitative description of propagation is facilitated by the transport equation. This is a partial differential equation for each cosmic ray species which requires fixing the distribution of initial sources and the boundary conditions. Specifying the initial source distribution is a source of significant uncertainty in these calculations. The local cosmic ray fluxes are obtained as the self-consistent solutions of the set of transport equations. Obtaining these solutions is challenging due to the large number of free parameters such as the convection velocities, spatial diffusion coefficients,

In the rest of this chapter we show how to determine those uncertainties of the electron-positron cosmic ray flux that originate from the propagation parameters of the diffusion equation. First we find the set of propagation parameters that the electron-positron flux is most sensitive to. Then we extract the values of these propagation parameters from

do not seem to account for such a large excess [8–11].

from the theoretical calculations encoded in GALPROP by [19].

**3. The problem of cosmic ray background calculation**

of this literature with detailed references can be found in [20] and [21].

observation over theory, as measured by the

This last type of experiment is called indirect dark matter detection and it is a sensible way to find dark matter particles if they are either their own anti-particles or the matter-antimatter asymmetry in the dark sector is not pronounced. In this case, via weak interactions, dark matter particles self annihilate into standard ones and create secondary cosmic rays. Alternatively, if dark matter decays into standard particles (with the lifetime of about 10<sup>26</sup> sec or more) its decay products can contribute to the secondaries. The most promising detection modes are the photon final states or the ones that contribute to anti-matter cosmic rays, such as positrons or anti-protons. In the last few years several anomalies were found in the cosmic positron fluxes by the PAMELA and Fermi-LAT satellites, which could be the first glimpses of dark matter.

However, various factors make indirect detection of dark matter challenging and less straightforward than we would like it to be. First, the immense cosmic ray background originating from ordinary astrophysical sources makes it hard to find the signal contributed by dark matter. Next, in many cases sources of the cosmic ray background are not known or not understood well enough. Finally, an important source of uncertainty and the main subject of our study is the cosmic ray propagation through the galaxy. This propagation is described by the diffusion equation; an equation with many unknown parameters. We review the state of this field of research. We show that using state of the art numerical codes, CPU intensive statistical inference, and the latest cosmic ray observations, the most important of these propagation parameters can be determined with a certain precision. Then we show how to propagate these uncertainties into recent cosmic ray measurements of Fermi-LAT and PAMELA. In the light of these findings we quantify the statistical significance of the present hints of signals in dark matter indirect detection. Finally, we contrast the experimental standing with some of the theoretical dark matter models proposed in the recent literature to explain cosmic ray 'anomalies'.

## **2. Experimental status of cosmic electrons and positrons**

Experiments detecting cosmic rays near Earth have been finding various unexpected deviations from theoretical predictions over the last twenty years. The local flux of high energy positrons is notoriously anomalous as reported by the


• CAPRICE [3],

2 Open Questions in Cosmology

of dark matter.

• TS [1], • AMS [2],

to explain cosmic ray 'anomalies'.

obvious place for trying to create dark matter particles. While we are in (almost) full control of this experiment, without knowing the exact mass and interaction strength between ordinary and dark matter we can only hope that the energy and luminosity of the LHC are high enough to produce the latter. Second, because it appears that the Solar System is immersed in a high flux stream of dark matter particles, it is natural to try to detect collisions between them and well shielded nuclei in underground laboratories. These experiments have the potential to probe interactions between dark and ordinary matter even beyond the reach of the LHC. However, these experiments are also limited by the unknown mass, interaction strength, flux and velocity distribution of the dark matter particles. Finally, perhaps the most general and unconstrained way to discover dark matter particles is to find traces of their

This last type of experiment is called indirect dark matter detection and it is a sensible way to find dark matter particles if they are either their own anti-particles or the matter-antimatter asymmetry in the dark sector is not pronounced. In this case, via weak interactions, dark matter particles self annihilate into standard ones and create secondary cosmic rays. Alternatively, if dark matter decays into standard particles (with the lifetime of about 10<sup>26</sup> sec or more) its decay products can contribute to the secondaries. The most promising detection modes are the photon final states or the ones that contribute to anti-matter cosmic rays, such as positrons or anti-protons. In the last few years several anomalies were found in the cosmic positron fluxes by the PAMELA and Fermi-LAT satellites, which could be the first glimpses

However, various factors make indirect detection of dark matter challenging and less straightforward than we would like it to be. First, the immense cosmic ray background originating from ordinary astrophysical sources makes it hard to find the signal contributed by dark matter. Next, in many cases sources of the cosmic ray background are not known or not understood well enough. Finally, an important source of uncertainty and the main subject of our study is the cosmic ray propagation through the galaxy. This propagation is described by the diffusion equation; an equation with many unknown parameters. We review the state of this field of research. We show that using state of the art numerical codes, CPU intensive statistical inference, and the latest cosmic ray observations, the most important of these propagation parameters can be determined with a certain precision. Then we show how to propagate these uncertainties into recent cosmic ray measurements of Fermi-LAT and PAMELA. In the light of these findings we quantify the statistical significance of the present hints of signals in dark matter indirect detection. Finally, we contrast the experimental standing with some of the theoretical dark matter models proposed in the recent literature

Experiments detecting cosmic rays near Earth have been finding various unexpected deviations from theoretical predictions over the last twenty years. The local flux of high

annihilation or decay products in cosmic rays bombarding Earth.

**2. Experimental status of cosmic electrons and positrons**

energy positrons is notoriously anomalous as reported by the


collaborations. Measurements of the PAMELA satellite stirred great interest by showing an unmistakable rise of the local *<sup>e</sup>*+/*e*<sup>−</sup> fraction, that deviates significantly from theoretical predictions for *Ee*<sup>+</sup> > 10 GeV [7]. The combined experimental and theoretical uncertainties do not seem to account for such a large excess [8–11].

The summed flux of electrons and positrons also indicates an anomalous excess of observation over theory, as measured by the


collaborations. The Fermi Large Area Telescope (LAT) satellite confirmed the excess of the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> flux for energies over 100 GeV [16, 17]. The Fermi-LAT results are consistent with those of the PAMELA collaboration, which measured the cosmic ray electron flux up to 625 GeV [18]. To date the Fermi-LAT data are the most precise indication of such an anomaly in the electron-positron spectrum. The Fermi-LAT data differ by several standard deviations from the theoretical calculations encoded in GALPROP by [19].

## **3. The problem of cosmic ray background calculation**

Between 2008 and 2011, of the order of a thousand papers were devoted to explaining the difference between the experimental measurements of Fermi-LAT and PAMELA and theoretical calculations. Speculation ranged from the modification of the cosmic ray propagation through supernova remnants, to dark matter annihilation. A concise summary of this literature with detailed references can be found in [20] and [21].

Before drawing conclusions from the electron-positron anomaly, however, one has to carefully examine the status of the theoretical understanding of Galactic cosmic rays. Unfortunately, even the origin of the cosmic rays is uncertain. The theory describing the propagation of cosmic ray particles from their birthplace through the Milky Way is based on the diffusion-convection model. The quantitative description of propagation is facilitated by the transport equation. This is a partial differential equation for each cosmic ray species which requires fixing the distribution of initial sources and the boundary conditions. Specifying the initial source distribution is a source of significant uncertainty in these calculations. The local cosmic ray fluxes are obtained as the self-consistent solutions of the set of transport equations. Obtaining these solutions is challenging due to the large number of free parameters such as the convection velocities, spatial diffusion coefficients, and momentum loss rates.

In the rest of this chapter we show how to determine those uncertainties of the electron-positron cosmic ray flux that originate from the propagation parameters of the diffusion equation. First we find the set of propagation parameters that the electron-positron flux is most sensitive to. Then we extract the values of these propagation parameters from cosmic ray data (different from the Fermi-LAT and PAMELA measurements). Based on the values of the propagation parameters most favored by the data we calculate theoretical predictions for the electron-positron fluxes and compare these to Fermi-LAT and PAMELA. By calculating the difference between our predictions and the observed fluxes we are able to isolate the anomalous part of the cosmic *<sup>e</sup>*<sup>−</sup> and *<sup>e</sup>*<sup>+</sup> fluxes.

If the time-scale of cosmic ray propagation (which is of the order of 1 Myr at 100 GeV energies) is much longer than the typical time scales of the galactic collapse of dark matter and the variation in the propagation conditions, then one can assume that the steady state condition holds. In this case, the left hand side of equation 1 can be set to zero and the time dependence of all quantities can be dropped. For our analysis we focus on a simplified version of the transport equation which to a first order approximation is sufficient to describe the propagation of electrons, positrons or anti-protons through the Galaxy and

*<sup>b</sup>*(*E*) *<sup>ψ</sup><sup>a</sup>* − *<sup>K</sup>*EE(*E*) *<sup>ψ</sup><sup>a</sup>*

where *E* is the energy of the secondary particle species *a*. To ensure that on the outer surface of the cylinder the cosmic ray density vanishes, boundary conditions are imposed. Similarly, outside of the diffusive region, these boundary condtions allow the particles to freely propagate and escape. This ensures that the modelling is consistent with the physical picture described above. One also imposes the symmetric condition *∂ψa*/*∂r*(*r* = 0) = 0 at

The transport of cosmic ray species through turbulent magnetic fields, the energy losses experienced by these particles due to Inverse Compton scattering (ICS), synchrotron radiation, Coulomb scattering or bremsstrahlung and their re-acceleration due to their interaction with moving magnetised scattering targets in the Galaxy is defined by the spatial diffusion coefficient *K*(*E*), the energy loss rate *b*(*E*) and the diffusive re-acceleration coefficient *<sup>K</sup>*EE(*E*) respectively. The effect of Galactic winds propagating vertically from stars in the Galactic disk can be incorporated by defining the convective velocity *VC*. The source of the cosmic rays is defined by *Qa*(*r*, *E*) in equation 2, with a standard source term resulting

> *ρg*(*r*) *m<sup>χ</sup>*

*fimi* (mean mass), (4)

*fi fj*�*σa*,*ijvij*� (mean cross section times relative velocity), (5)

, (annihilation spectrum per annihilation). (6)

2

 − *∂*

*<sup>∂</sup><sup>z</sup>* (sign(*z*) *VC <sup>ψ</sup>a*), (2)

Uncertainties in Dark Matter Indirect Detection

http://dx.doi.org/10.5772/52052

91

. (3)

*∂ ∂E* 

*r* = 0. In momentum space, null boundary conditions are imposed.

from the annihilation of dark matter which can be written as:

*Qa*(*<sup>r</sup>*, *<sup>E</sup>*) = <sup>1</sup>

2 *dNa dE* �*σav*�<sup>0</sup>

Here �*σav*�<sup>0</sup> corresponds to the thermally averaged annihilation cross section of the relevant species, and *ρg*(*r*) is the energy density of dark matter in the Galaxy. The energy distribution of the secondary particle *a* is defined as *dNa*/*dE* and is normalised per annihilation. This formula applies to self-conjugated annihilating dark matter. In the case of non-self-conjugated dark matter, or of multicomponent dark matter, the quantities in Eq. (3) should be replaced as follows, where an index *i* denotes a charge state and/or particle species (indeed any particle property, collectively called "component") and *fi* = *ni*/*<sup>n</sup>* is the number

their corresponding spectrum at Earth:

fraction of the *i*-th component:

*ij*

<sup>∑</sup>*ij fi fjσa*,*ijvij* (*dNa*,*ij*/*dE*) ∑*ij fi fjσa*,*ijvij*

*<sup>m</sup><sup>χ</sup>* <sup>→</sup> ∑ *i*

�*σav*� → ∑

*dNa*/*dE* →

0 = *Qa*(*r*, *E*) + *K*(*E*) ∇2*ψ<sup>a</sup>* +

Similar results have been published in the literature before. However, our results supersede these in two important aspects. First, we show that when analyzed in the framework of the standard propagation model there exists a statistically significant tension between the *<sup>e</sup>*−,*e*<sup>+</sup> and the rest of the charged cosmic ray fluxes. Second, unlike anyone else before us, we isolate the anomalous contribution within the *<sup>e</sup>*<sup>−</sup> and *<sup>e</sup>*<sup>+</sup> spectrum together with its theoretical uncertainty.

Our analysis uses more charged cosmic ray spectral data points than similar studies before us such as [22]. Unlike us [23] use gamma ray data when extracting the background, however this may bias the analysis since gamma rays originating from anomalous electrons or positrons are not part of the background. Our numerical treatment, similar to that of [24], is more complete than the one of [25–27].

Our statistical analysis can be considered as an extension of [24] since we calculate the *<sup>e</sup>*−,*e*<sup>+</sup> background with a theoretical uncertainty. Ref. [24] use 76 spectral data points, while we use 219 which gives us a significant edge over their analyses. The parameters that we freely vary are somewhat different from those of [24]. Before we choose the parameter space, we analyzed the sensitivity of the electron and positron spectrum to the parameters to maximize the efficiency of our parameter extraction. Our choice and treatment of the nuisance parameters also differs from [24]. Finally, we use a different scanning technique from the one they use.

## **4. Cosmic ray propagation through the Galaxy**

The propagation of charged particles through the Galaxy can be well described using the diffusion-convection model [28]. This model assumes that the charged particles propagate homogeneously within a defined region of diffusion (similar to the leaky box propagation model), while taking the effects of energy loss into account. The diffusive region is assumed to be a solid flat cylinder defined with a radius *R* and a half-height of *L*. Its shape is such that it encloses the Galactic plane which confines charged cosmic rays to the Galactic magnetic fields inside it, while cosmic rays outside are free to stream away. The solar system in this diffusive region is defined in cylindrical coordinates as*r*<sup>⊙</sup> = (8.33 kpc, 0 kpc, 0 kpc) [29]. The phase-space density *ψa*(*r*, *p*, *t*) of a particular cosmic ray species *a* at time *t*, Galactic position *r* and with momentum *p* can be determined by solving the cosmic ray transport equation, which has the general form [30]

$$\begin{split} \frac{\partial \psi\_{a}(\vec{r}, p, t)}{\partial t} &= Q\_{a}(\vec{r}, p, t) + \nabla \cdot (D\_{\text{xx}} \nabla \psi\_{a} - \vec{V} \psi\_{a}) \\ &+ \frac{\partial}{\partial p} \left( p^{2} D\_{pp} \frac{\partial}{\partial p} \frac{1}{p^{2}} \psi\_{a} \right) - \frac{\partial}{\partial p} \left( \dot{p} \psi\_{a} - \frac{p}{3} \left( \nabla \cdot \vec{V} \right) \psi\_{a} \right) - \frac{1}{\tau\_{f}} \psi\_{a} - \frac{1}{\tau\_{r}} \psi\_{a}. \end{split} \tag{1}$$

If the time-scale of cosmic ray propagation (which is of the order of 1 Myr at 100 GeV energies) is much longer than the typical time scales of the galactic collapse of dark matter and the variation in the propagation conditions, then one can assume that the steady state condition holds. In this case, the left hand side of equation 1 can be set to zero and the time dependence of all quantities can be dropped. For our analysis we focus on a simplified version of the transport equation which to a first order approximation is sufficient to describe the propagation of electrons, positrons or anti-protons through the Galaxy and their corresponding spectrum at Earth:

4 Open Questions in Cosmology

theoretical uncertainty.

from the one they use.

which has the general form [30]

+ *∂ ∂p p*2*Dpp*

*∂ψa*(*r*, *p*, *t*)

is more complete than the one of [25–27].

cosmic ray data (different from the Fermi-LAT and PAMELA measurements). Based on the values of the propagation parameters most favored by the data we calculate theoretical predictions for the electron-positron fluxes and compare these to Fermi-LAT and PAMELA. By calculating the difference between our predictions and the observed fluxes we are able to

Similar results have been published in the literature before. However, our results supersede these in two important aspects. First, we show that when analyzed in the framework of the standard propagation model there exists a statistically significant tension between the *<sup>e</sup>*−,*e*<sup>+</sup> and the rest of the charged cosmic ray fluxes. Second, unlike anyone else before us, we isolate the anomalous contribution within the *<sup>e</sup>*<sup>−</sup> and *<sup>e</sup>*<sup>+</sup> spectrum together with its

Our analysis uses more charged cosmic ray spectral data points than similar studies before us such as [22]. Unlike us [23] use gamma ray data when extracting the background, however this may bias the analysis since gamma rays originating from anomalous electrons or positrons are not part of the background. Our numerical treatment, similar to that of [24],

Our statistical analysis can be considered as an extension of [24] since we calculate the *<sup>e</sup>*−,*e*<sup>+</sup> background with a theoretical uncertainty. Ref. [24] use 76 spectral data points, while we use 219 which gives us a significant edge over their analyses. The parameters that we freely vary are somewhat different from those of [24]. Before we choose the parameter space, we analyzed the sensitivity of the electron and positron spectrum to the parameters to maximize the efficiency of our parameter extraction. Our choice and treatment of the nuisance parameters also differs from [24]. Finally, we use a different scanning technique

The propagation of charged particles through the Galaxy can be well described using the diffusion-convection model [28]. This model assumes that the charged particles propagate homogeneously within a defined region of diffusion (similar to the leaky box propagation model), while taking the effects of energy loss into account. The diffusive region is assumed to be a solid flat cylinder defined with a radius *R* and a half-height of *L*. Its shape is such that it encloses the Galactic plane which confines charged cosmic rays to the Galactic magnetic fields inside it, while cosmic rays outside are free to stream away. The solar system in this diffusive region is defined in cylindrical coordinates as*r*<sup>⊙</sup> = (8.33 kpc, 0 kpc, 0 kpc) [29]. The phase-space density *ψa*(*r*, *p*, *t*) of a particular cosmic ray species *a* at time *t*, Galactic position *r* and with momentum *p* can be determined by solving the cosmic ray transport equation,

*<sup>p</sup>*˙*ψ<sup>a</sup>* <sup>−</sup> *<sup>p</sup>*

<sup>3</sup> (∇ · *<sup>V</sup>* )*ψ<sup>a</sup>*

 − 1 *τf* *<sup>ψ</sup><sup>a</sup>* <sup>−</sup> <sup>1</sup> *τr*

*ψa*. (1)

isolate the anomalous part of the cosmic *<sup>e</sup>*<sup>−</sup> and *<sup>e</sup>*<sup>+</sup> fluxes.

**4. Cosmic ray propagation through the Galaxy**

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *Qa*(*<sup>r</sup>*, *<sup>p</sup>*, *<sup>t</sup>*) + ∇ · (*Dxx*∇*ψ<sup>a</sup>* <sup>−</sup> *<sup>V</sup> <sup>ψ</sup>a*)

*∂ ∂p* 1 *<sup>p</sup>*<sup>2</sup> *<sup>ψ</sup><sup>a</sup>* − *∂ ∂p* 

$$0 = Q\_a(\vec{r}, E) + K(E) \, \nabla^2 \psi\_a + \frac{\partial}{\partial E} \left( b(E) \, \psi\_a - K\_{EE}(E) \, \psi\_a \right) - \frac{\partial}{\partial z} \left( \text{sign}(z) \, V\_{\mathbb{C}} \, \psi\_a \right), \tag{2}$$

where *E* is the energy of the secondary particle species *a*. To ensure that on the outer surface of the cylinder the cosmic ray density vanishes, boundary conditions are imposed. Similarly, outside of the diffusive region, these boundary condtions allow the particles to freely propagate and escape. This ensures that the modelling is consistent with the physical picture described above. One also imposes the symmetric condition *∂ψa*/*∂r*(*r* = 0) = 0 at *r* = 0. In momentum space, null boundary conditions are imposed.

The transport of cosmic ray species through turbulent magnetic fields, the energy losses experienced by these particles due to Inverse Compton scattering (ICS), synchrotron radiation, Coulomb scattering or bremsstrahlung and their re-acceleration due to their interaction with moving magnetised scattering targets in the Galaxy is defined by the spatial diffusion coefficient *K*(*E*), the energy loss rate *b*(*E*) and the diffusive re-acceleration coefficient *<sup>K</sup>*EE(*E*) respectively. The effect of Galactic winds propagating vertically from stars in the Galactic disk can be incorporated by defining the convective velocity *VC*. The source of the cosmic rays is defined by *Qa*(*r*, *E*) in equation 2, with a standard source term resulting from the annihilation of dark matter which can be written as:

$$Q\_d(\vec{r}, E) = \frac{1}{2} \frac{dN\_d}{dE} \langle \sigma\_d v \rangle\_0 \left(\frac{\rho\_\mathcal{S}(\vec{r})}{m\_\mathcal{X}}\right)^2. \tag{3}$$

Here �*σav*�<sup>0</sup> corresponds to the thermally averaged annihilation cross section of the relevant species, and *ρg*(*r*) is the energy density of dark matter in the Galaxy. The energy distribution of the secondary particle *a* is defined as *dNa*/*dE* and is normalised per annihilation. This formula applies to self-conjugated annihilating dark matter. In the case of non-self-conjugated dark matter, or of multicomponent dark matter, the quantities in Eq. (3) should be replaced as follows, where an index *i* denotes a charge state and/or particle species (indeed any particle property, collectively called "component") and *fi* = *ni*/*<sup>n</sup>* is the number fraction of the *i*-th component:

$$m\_{\chi} \to \sum f\_{i} m\_{i} \tag{12a-mean mass} \} \tag{4}$$

$$
\langle \sigma\_a v \rangle \to \sum\_{i\bar{j}}^l f\_{\bar{l}} f\_{\bar{j}} \langle \sigma\_{a\bar{j}\bar{j}} v\_{\bar{l}\bar{j}} \rangle \qquad \text{(mean cross section times relative velocity)}, \tag{5}
$$

$$
dN\_d / d\mathbb{E} \to \frac{\sum\_{\bar{l}\bar{j}} f\_{\bar{l}} f\_{\bar{l}\bar{j}} \sigma\_{a\bar{j}\bar{j}} v\_{\bar{l}\bar{j}} (dN\_{a\bar{i}\bar{j}}/dE)}{\sum\_{\bar{l}\bar{j}} f\_{\bar{l}\bar{j}} f\_{\bar{l}\bar{j}} \sigma\_{a\bar{i}\bar{j}} v\_{\bar{l}\bar{j}}}, \quad \text{(anihilation spectrum per annihilation)}. \tag{6}
$$

The spatial diffusion coefficient *K*(*E*) is assumed to have the form

$$K(E) = K\_0 \ v^{\eta} \left(\frac{R}{\text{GeV}}\right)^{\delta} \tag{7}$$

*<sup>ψ</sup>a*(*<sup>r</sup>*, *<sup>E</sup>*) = *<sup>m</sup><sup>χ</sup>*

**5. Statistical framework**

Here

Here we only highlight the main concepts used.

parameter space its value can be recovered as

L(*D*|*P*) =

*χ*2

flat priors P(*P*), we then construct the posterior probability distribution

<sup>E</sup>(*D*) =

The differential flux is related to the solution in Eq. (10) via

*E*

*dES* 

*d*Φ*<sup>a</sup>*

(2) should be introduced to account for spallations on the gas in the disk.

*M* ∏ *i*=1

*dE* <sup>=</sup> *<sup>v</sup>*(*E*)

For the propagation of protons or anti-protons in the Galactic halo, additional terms in Eq.

We use standard Bayesian parameter inference to determine the statistically favored regions of the propagation parameters *<sup>P</sup>* = {*p*1, ..., *pN*} that the electron and positron cosmic ray fluxes are the most sensitive to. For full mathematical details we refer the reader to [21].

Using the experimental data *<sup>D</sup>* = {*d*1, ..., *dM*} and their corresponding theoretical predictions *<sup>T</sup>* = {*t*1(*P*), ..., *tM*(*P*)}, as the fuction of the parameters, we construct the likelihood function

exp(−*χ*<sup>2</sup>

*σi*

2

*N* ∏ *j*=1

P(*P*|*D*) = L(*D*|*P*)P(*P*)/E(*D*). (14)

1 √2*πσ<sup>i</sup>*

*<sup>i</sup>* (*D*, *<sup>P</sup>*) = *di* <sup>−</sup> *ti*(*P*)

and *σ<sup>i</sup>* are the corresponding combined theoretical and experimental uncertainties. Assuming

At this stage the value of the evidence E(*D*) is unknown. After integrating over the whole

L(*D*|*P*)P(*P*)

More relevant to our purpose is the adaptive scan of the likelihood function during this integration which will give us the shape of the posterior distribution over the relevant part

*<sup>d</sup>*<sup>3</sup>*rS <sup>G</sup>*(*<sup>r</sup>*, *<sup>E</sup>*;*rS*, *ES*) *Qa*(*rS*, *ES*). (10)

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<sup>4</sup>*<sup>π</sup> <sup>ψ</sup>a*(*<sup>r</sup>*, *<sup>E</sup>*). (11)

*<sup>i</sup>* (*D*, *<sup>P</sup>*)/2). (12)

, (13)

*dpj*. (15)

where *v* is the speed (in units of *c*) and *R* = *p*/*eZ* is the magnetic rigidity of the cosmic ray particles. Here *Z* is the effective nuclear charge of the particle and *e* is the absolute value of its electric charge (if we considered particles other than electrons, positrons, protons or anti-protons then this quantity would be different from 1). At low energies the behaviour of the cosmic rays as they diffuse is controlled using the parameter *η*. Traditionally one will set *η* = 1 but departures from this traditional value to other values (either positive or negative) have been suggested. More detailed treatments allow one to incorporate spatial dependence into the diffusion coefficient (*K*(*r*, *E*)) and the influence of particle motion on the diffusion of these particles (which leads to anisotropic diffusion).

Synchrotron radiation and Inverse Compton scattering are position dependent phenomena. Synchrotron radiation arises from the interaction of a charged particle with Galactic magnetic fields and thus it depends on the strength of the magnetic field which changes in the Galaxy. Similarly, inverse Compton scattering is dependent on the distribution of background light which varies in the Galaxy. If one neglects this position dependence of energy losses in the Galactic halo and assumes that all energy losses can be described using a relationship that is proportional to *E*<sup>2</sup> (which is only valid if one neglects energy losses, such as Coulomb losses and bremsstrahlung, and considers only inverse Compton scattering for electrons with relatively low energy - Thomson scattering regime), then the energy loss rate can be parametrized as

$$
\partial \mathcal{B}(E) = b\_0 \to^2. \tag{8}
$$

In more detailed treatments the spatial dependence associated with the energy loss rate *b*(*r*, *E*) would be considered and a more general energy dependence relationship for the energy loss rate would be obtained. Additionally Coulomb losses (*dE*/*dt* ∼ const) and bremsstrahlung losses (*dE*/*dt* ∼ *bE*) could also be taken into account. These losses can be calculated using functions dependent on position and energy as well as gas, interstellar radiation and magnetic field distributions [31].

Finally, the diffusive re-acceleration coefficient *<sup>K</sup>*EE(*E*) is usually parametrized as

$$K\_{\rm EE}(E) = \frac{2}{9} v\_{\rm A}^2 \frac{v^4 E^2}{K(E)},\tag{9}$$

where *v*<sup>A</sup> is the Alfvén speed.

A propagator, or Green's function *G*, is used to describe the evolution of the cosmic ray that originates from a source *<sup>Q</sup>* at *rS* with energy *ES* through the diffusive halo and reaches the Earth at point*r* with energy *E*. This allows the general solution for Eq. (2) to be written as

$$\Psi\_{a}(\vec{r},E) = \int\_{E}^{m\_{\vec{\lambda}}} dE\_{\mathcal{S}} \int d^{3}r\_{\mathcal{S}} \, G(\vec{r},E; \vec{r}\_{\mathcal{S}}, E\_{\mathcal{S}}) \, \mathcal{Q}\_{a}(\vec{r}\_{\mathcal{S}}, E\_{\mathcal{S}}).\tag{10}$$

The differential flux is related to the solution in Eq. (10) via

$$\frac{d\Phi\_a}{dE} = \frac{v(E)}{4\pi} \psi\_a(\vec{r}, E). \tag{11}$$

For the propagation of protons or anti-protons in the Galactic halo, additional terms in Eq. (2) should be introduced to account for spallations on the gas in the disk.

#### **5. Statistical framework**

We use standard Bayesian parameter inference to determine the statistically favored regions of the propagation parameters *<sup>P</sup>* = {*p*1, ..., *pN*} that the electron and positron cosmic ray fluxes are the most sensitive to. For full mathematical details we refer the reader to [21]. Here we only highlight the main concepts used.

Using the experimental data *<sup>D</sup>* = {*d*1, ..., *dM*} and their corresponding theoretical predictions *<sup>T</sup>* = {*t*1(*P*), ..., *tM*(*P*)}, as the fuction of the parameters, we construct the likelihood function

$$\mathcal{L}(D|P) = \prod\_{i=1}^{M} \frac{1}{\sqrt{2\pi}\sigma\_i} \exp(-\chi\_i^2(D\_\prime P)/2). \tag{12}$$

Here

6 Open Questions in Cosmology

parametrized as

The spatial diffusion coefficient *K*(*E*) is assumed to have the form

these particles (which leads to anisotropic diffusion).

radiation and magnetic field distributions [31].

where *v*<sup>A</sup> is the Alfvén speed.

*<sup>K</sup>*(*E*) = *<sup>K</sup>*<sup>0</sup> *<sup>v</sup><sup>η</sup>*

 *R* GeV

where *v* is the speed (in units of *c*) and *R* = *p*/*eZ* is the magnetic rigidity of the cosmic ray particles. Here *Z* is the effective nuclear charge of the particle and *e* is the absolute value of its electric charge (if we considered particles other than electrons, positrons, protons or anti-protons then this quantity would be different from 1). At low energies the behaviour of the cosmic rays as they diffuse is controlled using the parameter *η*. Traditionally one will set *η* = 1 but departures from this traditional value to other values (either positive or negative) have been suggested. More detailed treatments allow one to incorporate spatial dependence into the diffusion coefficient (*K*(*r*, *E*)) and the influence of particle motion on the diffusion of

Synchrotron radiation and Inverse Compton scattering are position dependent phenomena. Synchrotron radiation arises from the interaction of a charged particle with Galactic magnetic fields and thus it depends on the strength of the magnetic field which changes in the Galaxy. Similarly, inverse Compton scattering is dependent on the distribution of background light which varies in the Galaxy. If one neglects this position dependence of energy losses in the Galactic halo and assumes that all energy losses can be described using a relationship that is proportional to *E*<sup>2</sup> (which is only valid if one neglects energy losses, such as Coulomb losses and bremsstrahlung, and considers only inverse Compton scattering for electrons with relatively low energy - Thomson scattering regime), then the energy loss rate can be

In more detailed treatments the spatial dependence associated with the energy loss rate *b*(*r*, *E*) would be considered and a more general energy dependence relationship for the energy loss rate would be obtained. Additionally Coulomb losses (*dE*/*dt* ∼ const) and bremsstrahlung losses (*dE*/*dt* ∼ *bE*) could also be taken into account. These losses can be calculated using functions dependent on position and energy as well as gas, interstellar

> 9 *v*2 A *v*<sup>4</sup> *E*<sup>2</sup> *K*(*E*)

A propagator, or Green's function *G*, is used to describe the evolution of the cosmic ray that originates from a source *<sup>Q</sup>* at *rS* with energy *ES* through the diffusive halo and reaches the Earth at point*r* with energy *E*. This allows the general solution for Eq. (2) to be written as

Finally, the diffusive re-acceleration coefficient *<sup>K</sup>*EE(*E*) is usually parametrized as

*<sup>K</sup>*EE(*E*) = <sup>2</sup>

*δ*

, (7)

*<sup>b</sup>*(*E*) = *<sup>b</sup>*<sup>0</sup> *<sup>E</sup>*2. (8)

, (9)

$$
\chi^2\_i(D, P) = \left(\frac{d\_i - t\_i(P)}{\sigma\_i}\right)^2,\tag{13}
$$

and *σ<sup>i</sup>* are the corresponding combined theoretical and experimental uncertainties. Assuming flat priors P(*P*), we then construct the posterior probability distribution

$$\mathcal{P}(P|D) = \mathcal{L}(D|P)\mathcal{P}(P)/\mathcal{E}(D). \tag{14}$$

At this stage the value of the evidence E(*D*) is unknown. After integrating over the whole parameter space its value can be recovered as

$$\mathcal{E}(D) = \int \mathcal{L}(D|P)\mathcal{P}(P)\prod\_{j=1}^{N} dp\_j. \tag{15}$$

More relevant to our purpose is the adaptive scan of the likelihood function during this integration which will give us the shape of the posterior distribution over the relevant part of the parameter space (where the likelihood is the highest). Having this shape we can calculate the probability density of a certain theoretical parameter *pi* acquiring a given value by marginalization

$$\mathcal{P}(p\_i|D) = \int \mathcal{P}(P|D) \prod\_{i \neq j=1}^{N} dp\_j. \tag{16}$$

[14, 15]. The AMS collaboration reported an excess in positrons with energies greater than 10 GeV [12], while the HESS collaboration measured a significant steepening of the electron plus photon spectrum above one TeV as measured by HESS's atmospheric Cherenkov telescope (ACT). Using the Large Area Telescope (LAT) on the Fermi Satellite, the Fermi-LAT collaboration released a high precision measurement of the *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> spectrum for energies from 7 GeV to 1 TeV [17], extending the energy range of their previously published results. We defined our *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> spectrum by using data from these three experiments. The PAMELA collaboration recently released their measurement of the *<sup>e</sup>*<sup>−</sup> only spectrum [18] confirming the behaviour of the *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> spectrum as measured by Fermi-LAT. The energy ranges that

**Measured flux Experiment Energy Number of**

*<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> Fermi-LAT [17] 7.05 - 886 47

*<sup>e</sup>*+/(*e*<sup>+</sup> + *<sup>e</sup>*−) PAMELA [32] 1.65 - 82.40 16 *<sup>e</sup>*<sup>−</sup> PAMELA [18] 1.11 - 491.4 39 anti-proton/proton PAMELA [33] 0.28 - 129 23

Boron/Carbon [36] 0.30 - 0.50 2

(Sc+Ti+V)/Fe ACE [40] 0.14 - 35 20

Be-10/Be-9 ISOMAX98 [44] 0.08 - 0.08 1

**Table 1.** In this table we have listed the cosmic ray experiments and the energy ranges of the corresponding data points that we selected for our analysis. There are two sets of cosmic ray data listed in this table. Electron positron flux related experiments make up the first five lines of the table, while all other experiments make up the rest of the table. On these two

Apart from measuring the electron positron sum, collaborations such as PAMELA have measured the differential positron fraction *<sup>e</sup>*+/(*e*<sup>+</sup> + *<sup>e</sup>*−) between energies of 1.5 and 100 GeV [7]. If one assumes that all secondary positrons are produced during the propagation of cosmic rays in the Galaxy, one would expect the positron fraction to decrease, however, the

How primary and secondary cosmic rays are produced and transported throughout the Galaxy can be studied by using cosmic-ray particles such as anti-protons. One requires a

sets of data, we perform a Bayesian analysis to highlight the tension between the two sets of data.

observed fraction increases for energies greater than 10 GeV.

AMS [12] 0.60 - 0.91 3

HESS [14, 15] 918 - 3480 9

IMP8 [34] 0.03 - 0.11 7 ISEE3 [35] 0.12 - 0.18 6

HEAO3 [37] 0.62 - 0.99 3 PAMELA [38] 1.24 - 72.36 8 CREAM [39] 91 - 1433 3

SANRIKU [41] 46 - 460 6 [42] 0.003 - 0.029 3 [43] 0.034 - 0.034 1 [42] 0.06 - 0.06 1

ACE-CRIS [45] 0.11 - 0.11 1 ACE [46] 0.13 - 0.13 1 AMS-02 [47] 0.15 - 9.03 15

(GeV) **data points**

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these data points were selected over are listed in Table 1.

Here the integral is carried out over the full range of the parameters. We can also determine Bayesian credibility regions R*<sup>x</sup>* for each of the parameters:

$$\infty = \int\_{\mathcal{R}\_x} \mathcal{P}(p\_i|D) \, dp\_i. \tag{17}$$

The above relation expresses the fact that *x* % of the total posterior probability lies within the region R*x*.

After examining the electron and positron fluxes for parameter sensitivity we found that the relevant parameters are:

$$P = \{ \gamma^{\varepsilon^-}, \gamma^{nucleus}, \delta\_1, \delta\_2, D\_{0\text{xx}} \}. \tag{18}$$

These parameters enter into the diffusion calculation as follows; *<sup>γ</sup>e*<sup>−</sup> and *γnucleus* are the primary electron and nucleus injection indices parameterizing an injection spectrum without a break; *δ*<sup>1</sup> and *δ*<sup>2</sup> are spatial diffusion coefficients below and above a reference rigidity *ρ*0; and *D*0*xx* determines the normalization of the spatial diffusion coefficient.

We treat the normalizations of all charged cosmic ray fluxes as theoretical nuisance parameters:

$$P\_{\text{musance}} = \{ \Phi^0\_{\varepsilon-\prime} \Phi^0\_{\varepsilon^+\prime} \Phi^0\_{\mathfrak{p}/p^\prime} \Phi^0\_{\text{B}/\text{C}^\prime} \Phi^0\_{(\text{SC}+\text{Ti}+V)/\text{Fe}^\prime} \Phi^0\_{\text{Be}-10/\text{Be}-9} \}. \tag{19}$$

We discuss other statistical and numerical issues, such as the choice of priors, the systematic uncertainties, sampling and convergence, in detail in [21].

## **6. Experimental data used in this analysis**

In our statistical analysis we use 219 of the most recent data points corresponding to five different types of cosmic ray experiment. A majority of these data points (114 in total) come from electron-positron related experiments while the other 105 are made up of Boron/Carbon, anti-proton/proton and (Sc+Ti+V)/Fe and Be-10/Be-9 cosmic ray flux measurements. If any of the energy ranges of the experiments overlap, the most recent experimental data point was chosen in that energy range.

There are three main experiments that have measured electrons and positrons over different decades of energy. These experiments include AMS by [12], Fermi-LAT by [17] and HESS by [14, 15]. The AMS collaboration reported an excess in positrons with energies greater than 10 GeV [12], while the HESS collaboration measured a significant steepening of the electron plus photon spectrum above one TeV as measured by HESS's atmospheric Cherenkov telescope (ACT). Using the Large Area Telescope (LAT) on the Fermi Satellite, the Fermi-LAT collaboration released a high precision measurement of the *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> spectrum for energies from 7 GeV to 1 TeV [17], extending the energy range of their previously published results. We defined our *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> spectrum by using data from these three experiments. The PAMELA collaboration recently released their measurement of the *<sup>e</sup>*<sup>−</sup> only spectrum [18] confirming the behaviour of the *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> spectrum as measured by Fermi-LAT. The energy ranges that these data points were selected over are listed in Table 1.

8 Open Questions in Cosmology

by marginalization

region R*x*.

parameters:

relevant parameters are:

*Pnuisance* = {Φ<sup>0</sup>

of the parameter space (where the likelihood is the highest). Having this shape we can calculate the probability density of a certain theoretical parameter *pi* acquiring a given value

P(*P*|*D*)

Here the integral is carried out over the full range of the parameters. We can also determine

The above relation expresses the fact that *x* % of the total posterior probability lies within the

After examining the electron and positron fluxes for parameter sensitivity we found that the

primary electron and nucleus injection indices parameterizing an injection spectrum without a break; *δ*<sup>1</sup> and *δ*<sup>2</sup> are spatial diffusion coefficients below and above a reference rigidity *ρ*0;

We treat the normalizations of all charged cosmic ray fluxes as theoretical nuisance

*<sup>B</sup>*/*C*, <sup>Φ</sup><sup>0</sup>

We discuss other statistical and numerical issues, such as the choice of priors, the systematic

In our statistical analysis we use 219 of the most recent data points corresponding to five different types of cosmic ray experiment. A majority of these data points (114 in total) come from electron-positron related experiments while the other 105 are made up of Boron/Carbon, anti-proton/proton and (Sc+Ti+V)/Fe and Be-10/Be-9 cosmic ray flux measurements. If any of the energy ranges of the experiments overlap, the most recent

There are three main experiments that have measured electrons and positrons over different decades of energy. These experiments include AMS by [12], Fermi-LAT by [17] and HESS by

(*SC*+*Ti*+*V*)/*Fe*, <sup>Φ</sup><sup>0</sup>

*N* ∏ *i*�=*j*=1

*dpj*. (16)

and *γnucleus* are the

*Be*−10/*Be*−9}. (19)

P(*pi*|*D*) *dpi*. (17)

, *<sup>γ</sup>nucleus*, *<sup>δ</sup>*1, *<sup>δ</sup>*2, *<sup>D</sup>*0*xx*}. (18)

P(*pi*|*D*) =

*x* = R*x*

*<sup>P</sup>* <sup>=</sup> {*γe*<sup>−</sup>

These parameters enter into the diffusion calculation as follows; *<sup>γ</sup>e*<sup>−</sup>

and *D*0*xx* determines the normalization of the spatial diffusion coefficient.

*<sup>e</sup>*<sup>+</sup> , <sup>Φ</sup><sup>0</sup>

*<sup>p</sup>*¯/*p*, <sup>Φ</sup><sup>0</sup>

*<sup>e</sup>*<sup>−</sup> , <sup>Φ</sup><sup>0</sup>

uncertainties, sampling and convergence, in detail in [21].

experimental data point was chosen in that energy range.

**6. Experimental data used in this analysis**

Bayesian credibility regions R*<sup>x</sup>* for each of the parameters:


**Table 1.** In this table we have listed the cosmic ray experiments and the energy ranges of the corresponding data points that we selected for our analysis. There are two sets of cosmic ray data listed in this table. Electron positron flux related experiments make up the first five lines of the table, while all other experiments make up the rest of the table. On these two sets of data, we perform a Bayesian analysis to highlight the tension between the two sets of data.

Apart from measuring the electron positron sum, collaborations such as PAMELA have measured the differential positron fraction *<sup>e</sup>*+/(*e*<sup>+</sup> + *<sup>e</sup>*−) between energies of 1.5 and 100 GeV [7]. If one assumes that all secondary positrons are produced during the propagation of cosmic rays in the Galaxy, one would expect the positron fraction to decrease, however, the observed fraction increases for energies greater than 10 GeV.

How primary and secondary cosmic rays are produced and transported throughout the Galaxy can be studied by using cosmic-ray particles such as anti-protons. One requires a large number of measurements with good statistics over a large energy range to produce detailed anti-proton spectra for study. Anti-proton spectra obtained from previous balloon borne experiments such as CAPRISE98 [3] and HEAT [48] had very low statistics, but recently the PAMELA satellite experiment [33] released a high-quality measurement of the anti-proton/proton flux ratio for an energy range of 1-100 GeV. This spectrum confirmed the behaviour of the anti-proton/proton ratio as observed by previous experiments.

Additionally one can use stable secondary to primary cosmic ray ratios such as Boron/Carbon and (Sc+Ti+V)/Fe ratio to study the variation experienced by cosmic rays as they propagate through the Galaxy. These ratios are particularly sensitive to the properties of cosmic ray propagation as the element in the numerator is produced by a different mechanism to the element that defines the denominator. Primary cosmic rays are produced by the original source of the cosmic rays such as a supernova remnant, while the secondary cosmic rays are generated by the interaction of their primaries with the interstellar medium [49]. Ratios that are defined by a numerator and denominator which are produced by the same mechanism, such as a primary/primary or a secondary/secondary cosmic ray ratio, have a low sensitivity to any variation in the propagation parameters. Analysing Galactic Boron/Carbon and (Sc+Ti+V)/Fe ratios allows one to determine the amount of interstellar material transversed by the primary cosmic ray and its energy dependence [49].

Unstable isotopes such as Beryllium-10/Beryllium-9 are also beneficial to analyse as they produce a constraint on the time it takes for a cosmic ray to propagate through the Galaxy [50]. Various experiments such as ISOMAX98 [44], ACE-CRIS [45], ACE [46] and AMS-02 [47] have all measured Be-10/Be-9 data with varying statistics.

In Table 1 we state over what energy range and from which experiment we selected the data points that define our spectrum of anti-proton/proton, Boron/Carbon, (Sc+Ti+V)/Fe and Be-10/Be-9 ratio.

**Figure 1.** Best fit curves plotted against non-electron-positron related data. These curves were calculated using the most probable parameter values obtained from the peak values of the posterior probabilities (inferred from p¯/p, B/C, (Sc+Ti+V)/Fe and Be data) shown in red in Fig. 2. The best fit curves pass through the estimated systematic error bands, shown in gray.

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solar modulation potential was not substantial over the period of PAMELA's data taking, and approximately the same average value for the potential can be used for Fermi-LAT.

There has been a plethora of experiments which have hinted at the existence of an anomaly in the electron-positron spectrum. The most notable measurements are the Fermi-LAT electron positron sum and the PAMELA positron fraction. Ref. [53] and [24] have all questioned the reality of the anomaly in the PAMELA and Fermi-LAT data as well as the absence of an anomaly in the anti-proton flux. Ref. [24] suggested that one requires only to readjust the diffusion parameters that define the propagation model as encoded in GALPROP to reproduce the Fermi-LAT data. This conclusion is highlighted clearly in figure 8 of [24], where their propagation model obtained from the best fit to 76 cosmic ray spectral data points agrees well with the Fermi-LAT data. Interestingly, the corresponding positron fraction obtained from their best fit does not agree with the PAMELA data, indicating that one cannot fit the PAMELA data by simply adjusting the parameters of the propagation model. This indicates to us that the anomaly observed in cosmic electron-positron data is real and rather than adjusting the propagation parameters, one has to perform a detailed investigation of its

**7.1. The presence or absence of a cosmic ray anomaly**

**7. Results**

existence and characteristics.

For energies below E < 10 GeV, solar magnetic and coronal activities perturb the low energy part of the cosmic ray spectrum. This is called solar modulation and has an important role in determining the observed spectral shape(s) of cosmic rays measured at earth [51, 52]. Solar modulation is accounted for in GALPROP by using a force field approximation. It should be noted that this is an approximation and does not include important influences such as the structure of the heliospheric magnetic field. To incorporate these effects into our analysis we vary the value of the modulation potential in GALPROP. Following Gast & Schael (2009), we also assume that the positively and negatively charged cosmic rays are modulated differently by solar activities (charge-sign dependent modulation). This charge dependent modulation has a significant effect on positrons and its effect on the anti-proton/proton ratio can be comparable to the experiment's statistical uncertainties. The modulation effect on heavy nuclei such as B, C, Sc, Ti, V, Fe and Be is mild even though these nuclei have a higher positive charge compared to the proton. The reason for this is that the modulation potential is proportional to the charge-to-mass ratio and these heavy nuclei have a much lower charge-to-mass ratio than the proton therefore the effect is minute. Regardless, as we use the ratio of their fluxes, most of the effect of solar modulation on these nuclei is cancelled, thus we can safely absorb this modulation effect into the systematic uncertainties of the experiment. To be able to compare experimental data we set the modulation potential in GALPROP for positrons (electrons) to the value determined by Gast & Schael (2009), *φ*+ = 442(2) MeV. Previous work by Usoskin et al. (2011) showed that the time dependence of the

**Figure 1.** Best fit curves plotted against non-electron-positron related data. These curves were calculated using the most probable parameter values obtained from the peak values of the posterior probabilities (inferred from p¯/p, B/C, (Sc+Ti+V)/Fe and Be data) shown in red in Fig. 2. The best fit curves pass through the estimated systematic error bands, shown in gray.

solar modulation potential was not substantial over the period of PAMELA's data taking, and approximately the same average value for the potential can be used for Fermi-LAT.

## **7. Results**

10 Open Questions in Cosmology

Be-10/Be-9 ratio.

large number of measurements with good statistics over a large energy range to produce detailed anti-proton spectra for study. Anti-proton spectra obtained from previous balloon borne experiments such as CAPRISE98 [3] and HEAT [48] had very low statistics, but recently the PAMELA satellite experiment [33] released a high-quality measurement of the anti-proton/proton flux ratio for an energy range of 1-100 GeV. This spectrum confirmed the

Additionally one can use stable secondary to primary cosmic ray ratios such as Boron/Carbon and (Sc+Ti+V)/Fe ratio to study the variation experienced by cosmic rays as they propagate through the Galaxy. These ratios are particularly sensitive to the properties of cosmic ray propagation as the element in the numerator is produced by a different mechanism to the element that defines the denominator. Primary cosmic rays are produced by the original source of the cosmic rays such as a supernova remnant, while the secondary cosmic rays are generated by the interaction of their primaries with the interstellar medium [49]. Ratios that are defined by a numerator and denominator which are produced by the same mechanism, such as a primary/primary or a secondary/secondary cosmic ray ratio, have a low sensitivity to any variation in the propagation parameters. Analysing Galactic Boron/Carbon and (Sc+Ti+V)/Fe ratios allows one to determine the amount of interstellar

Unstable isotopes such as Beryllium-10/Beryllium-9 are also beneficial to analyse as they produce a constraint on the time it takes for a cosmic ray to propagate through the Galaxy [50]. Various experiments such as ISOMAX98 [44], ACE-CRIS [45], ACE [46] and AMS-02

In Table 1 we state over what energy range and from which experiment we selected the data points that define our spectrum of anti-proton/proton, Boron/Carbon, (Sc+Ti+V)/Fe and

For energies below E < 10 GeV, solar magnetic and coronal activities perturb the low energy part of the cosmic ray spectrum. This is called solar modulation and has an important role in determining the observed spectral shape(s) of cosmic rays measured at earth [51, 52]. Solar modulation is accounted for in GALPROP by using a force field approximation. It should be noted that this is an approximation and does not include important influences such as the structure of the heliospheric magnetic field. To incorporate these effects into our analysis we vary the value of the modulation potential in GALPROP. Following Gast & Schael (2009), we also assume that the positively and negatively charged cosmic rays are modulated differently by solar activities (charge-sign dependent modulation). This charge dependent modulation has a significant effect on positrons and its effect on the anti-proton/proton ratio can be comparable to the experiment's statistical uncertainties. The modulation effect on heavy nuclei such as B, C, Sc, Ti, V, Fe and Be is mild even though these nuclei have a higher positive charge compared to the proton. The reason for this is that the modulation potential is proportional to the charge-to-mass ratio and these heavy nuclei have a much lower charge-to-mass ratio than the proton therefore the effect is minute. Regardless, as we use the ratio of their fluxes, most of the effect of solar modulation on these nuclei is cancelled, thus we can safely absorb this modulation effect into the systematic uncertainties of the experiment. To be able to compare experimental data we set the modulation potential in GALPROP for positrons (electrons) to the value determined by Gast & Schael (2009), *φ*+ = 442(2) MeV. Previous work by Usoskin et al. (2011) showed that the time dependence of the

behaviour of the anti-proton/proton ratio as observed by previous experiments.

material transversed by the primary cosmic ray and its energy dependence [49].

[47] have all measured Be-10/Be-9 data with varying statistics.

#### **7.1. The presence or absence of a cosmic ray anomaly**

There has been a plethora of experiments which have hinted at the existence of an anomaly in the electron-positron spectrum. The most notable measurements are the Fermi-LAT electron positron sum and the PAMELA positron fraction. Ref. [53] and [24] have all questioned the reality of the anomaly in the PAMELA and Fermi-LAT data as well as the absence of an anomaly in the anti-proton flux. Ref. [24] suggested that one requires only to readjust the diffusion parameters that define the propagation model as encoded in GALPROP to reproduce the Fermi-LAT data. This conclusion is highlighted clearly in figure 8 of [24], where their propagation model obtained from the best fit to 76 cosmic ray spectral data points agrees well with the Fermi-LAT data. Interestingly, the corresponding positron fraction obtained from their best fit does not agree with the PAMELA data, indicating that one cannot fit the PAMELA data by simply adjusting the parameters of the propagation model. This indicates to us that the anomaly observed in cosmic electron-positron data is real and rather than adjusting the propagation parameters, one has to perform a detailed investigation of its existence and characteristics.

One of our important results is that just by adjusting the parameters in Eq.(18) and (19) it is possible to generate a theoretical prediction which is fully consistent with the Fermi-LAT *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> flux measurement. Similarly, we found that both the PAMELA positron fraction and the electron flux can be reproduced by theory. This means that the theory has enough flexibility to accommodate the experimentally measured fluxes.

If one assumes that all types of cosmic ray data (electron-positron related and non-electron-positron related) can be described well using a single set of propagation parameters it quickly becomes obvious that one cannot fit the data simultaneously. To analyse this behaviour, we first divided the cosmic ray data listed in table 1 into two groups: electron and/or positron fluxes as measured by AMS, Fermi, HESS and PAMELA and non-electron and/or positron fluxes such as anti-proton/proton, Boron/Carbon, (Sc+Ti+V)/Fe, Be-10/Be-9. We then attempted to fit only the second group of cosmic ray data, that is, excluding the electron-positron related data. We obtain a *χ*<sup>2</sup> per degree of freedom of 0.34 from this fit and as a consequence the corresponding best fit curves each pass through all the estimated systematic error bands shown in grey in Fig. 1. When we apply the best-fit parameters to the electron and/or positron flux data, however, we obtain a *χ*<sup>2</sup> per degree of freedom of 24. Similarly, when we do the converse, i.e. find the best-fit propagation parameters using only the electron-positron related flux, we obtain an excellent *χ*<sup>2</sup> per degree of freedom of 1.0, but the best-fit parameters gave a larger *χ*<sup>2</sup> per degree of freedom (3.1) for the non-electron-positron data. As we have a large number (105) of data points the deviations observed between these two sets of data is significant which signals statistically significant tension between electron-positron and non-electron-positron measurements. These results support the conclusion highlighted in Fig. 7 and 8 of [24], that one requires something more than simply adjusting the propagation parameters to accommodate for the cosmic ray anomaly.

This tension between the electron-positron and non-electron-positron measurements was further investigated by performing an independent Bayesian analysis on the two groups of data. This allows us to extract the values of the propagation parameters as preferred by the different sets of data. Interestingly, one can derive information about the propagation parameters of the electron-positron related data from the non-electron-positron related data. This arises due to a number of reasons. Firstly, the value of some propagation parameters such as *D*0*xx* is highly dependent on the species that one is modelling the propagation of. Secondly, to model the propagation of cosmic rays one uses the transport equation (equation 1). In this equation a large number of processes, including nuclear fragmentation and decay, are incorporated, which directly affects the predicted secondary electron-positron flux. Thirdly, as the energy density of cosmic rays is comparable to the energy density of the interstellar radiation field and the local magnetic field, different cosmic ray species will influence the dynamics of each species non-negligibly.

**Figure 2.** Marginalized posterior probability distributions of propagation parameters listed in Eq.(18). The likelihood functions containing electron and/or positron flux data are plotted as blue dashed lines while the likelihood functions for the rest of the comic ray data are plotted as solid red lines. The 68 % credibility regions are highlighted by the shaded areas of the posteriors. In the lower three frames it is evident that there exists a statistically significant tension between the electron-positron data and

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solar modulation on this data is minor; and (4) the theoretical prediction for the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> is insensitive to the value of the propagation parameter for mid-range energies (this is

In addition to using non-*e*<sup>±</sup> related data points (i.e. p/p, B/C, (Sc+Ti+V)/Fe and Be data), ¯ we also selected four *<sup>e</sup>*<sup>±</sup> related data points to use in our analysis. This included the lowest energy data point from AMS, the highest energy data point from HESS and the 19.40 GeV and 29.20 GeV data points of Fermi-LAT. We checked that this selection of *<sup>e</sup>*<sup>±</sup> related data points does not bias the final conclusion and the results that we obtained with this selection

In figure 2 we have plotted the marginalized posterior probability densities of our selected propagation parameters as obtained by completing a Bayesian analysis on the two sets

highlighted by the distinct bow-tie shape of the theoretical uncertainty band).

the rest.

are robust.

As a consequence, even if no electron-positron related data is used in our fit one can still constrain some of the propagation parameters of the electron-positron data. Unfortunately this method does not constrain the value of injection indices sufficiently, so in order to fix these parameters we have to include a minimal amount of information about the electron-positron related fluxes in our analysis. We selected data points from the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> spectrum for four reasons: (1) these points cover the largest energy range; (2) before setting out to find the optimal parameter value, within uncertainties the end points of the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> spectrum agree with theoretical predictions; (3) for low energies the effect of

12 Open Questions in Cosmology

anomaly.

One of our important results is that just by adjusting the parameters in Eq.(18) and (19) it is possible to generate a theoretical prediction which is fully consistent with the Fermi-LAT *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> flux measurement. Similarly, we found that both the PAMELA positron fraction and the electron flux can be reproduced by theory. This means that the theory has enough

If one assumes that all types of cosmic ray data (electron-positron related and non-electron-positron related) can be described well using a single set of propagation parameters it quickly becomes obvious that one cannot fit the data simultaneously. To analyse this behaviour, we first divided the cosmic ray data listed in table 1 into two groups: electron and/or positron fluxes as measured by AMS, Fermi, HESS and PAMELA and non-electron and/or positron fluxes such as anti-proton/proton, Boron/Carbon, (Sc+Ti+V)/Fe, Be-10/Be-9. We then attempted to fit only the second group of cosmic ray data, that is, excluding the electron-positron related data. We obtain a *χ*<sup>2</sup> per degree of freedom of 0.34 from this fit and as a consequence the corresponding best fit curves each pass through all the estimated systematic error bands shown in grey in Fig. 1. When we apply the best-fit parameters to the electron and/or positron flux data, however, we obtain a *χ*<sup>2</sup> per degree of freedom of 24. Similarly, when we do the converse, i.e. find the best-fit propagation parameters using only the electron-positron related flux, we obtain an excellent *χ*<sup>2</sup> per degree of freedom of 1.0, but the best-fit parameters gave a larger *χ*<sup>2</sup> per degree of freedom (3.1) for the non-electron-positron data. As we have a large number (105) of data points the deviations observed between these two sets of data is significant which signals statistically significant tension between electron-positron and non-electron-positron measurements. These results support the conclusion highlighted in Fig. 7 and 8 of [24], that one requires something more than simply adjusting the propagation parameters to accommodate for the cosmic ray

This tension between the electron-positron and non-electron-positron measurements was further investigated by performing an independent Bayesian analysis on the two groups of data. This allows us to extract the values of the propagation parameters as preferred by the different sets of data. Interestingly, one can derive information about the propagation parameters of the electron-positron related data from the non-electron-positron related data. This arises due to a number of reasons. Firstly, the value of some propagation parameters such as *D*0*xx* is highly dependent on the species that one is modelling the propagation of. Secondly, to model the propagation of cosmic rays one uses the transport equation (equation 1). In this equation a large number of processes, including nuclear fragmentation and decay, are incorporated, which directly affects the predicted secondary electron-positron flux. Thirdly, as the energy density of cosmic rays is comparable to the energy density of the interstellar radiation field and the local magnetic field, different cosmic ray species will

As a consequence, even if no electron-positron related data is used in our fit one can still constrain some of the propagation parameters of the electron-positron data. Unfortunately this method does not constrain the value of injection indices sufficiently, so in order to fix these parameters we have to include a minimal amount of information about the electron-positron related fluxes in our analysis. We selected data points from the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> spectrum for four reasons: (1) these points cover the largest energy range; (2) before setting out to find the optimal parameter value, within uncertainties the end points of the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> spectrum agree with theoretical predictions; (3) for low energies the effect of

flexibility to accommodate the experimentally measured fluxes.

influence the dynamics of each species non-negligibly.

**Figure 2.** Marginalized posterior probability distributions of propagation parameters listed in Eq.(18). The likelihood functions containing electron and/or positron flux data are plotted as blue dashed lines while the likelihood functions for the rest of the comic ray data are plotted as solid red lines. The 68 % credibility regions are highlighted by the shaded areas of the posteriors. In the lower three frames it is evident that there exists a statistically significant tension between the electron-positron data and the rest.

solar modulation on this data is minor; and (4) the theoretical prediction for the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> is insensitive to the value of the propagation parameter for mid-range energies (this is highlighted by the distinct bow-tie shape of the theoretical uncertainty band).

In addition to using non-*e*<sup>±</sup> related data points (i.e. p/p, B/C, (Sc+Ti+V)/Fe and Be data), ¯ we also selected four *<sup>e</sup>*<sup>±</sup> related data points to use in our analysis. This included the lowest energy data point from AMS, the highest energy data point from HESS and the 19.40 GeV and 29.20 GeV data points of Fermi-LAT. We checked that this selection of *<sup>e</sup>*<sup>±</sup> related data points does not bias the final conclusion and the results that we obtained with this selection are robust.

In figure 2 we have plotted the marginalized posterior probability densities of our selected propagation parameters as obtained by completing a Bayesian analysis on the two sets of data. The blue dashed curve represents the likelihood functions generated for the electron-positron related data (AMS, Fermi-LAT, HESS, and PAMELA), while the red solid curves represent the likelihood functions obtained for the rest of the cosmic ray data (anti-proton/proton, Boron/Carbon, (Sc+Ti+V)/Fe, Be-10/Be-9) listed in table 1. The 68% credibility regions of the likelihood functions are highlighted by the shaded areas of figure 2 and table 2 lists the numerical values of these credibility regions as well as the best-fit values of each propagation parameter. By looking at figure 2 it is obvious that the electron-positron related data and the non-electron-positron related data are inconsistent with the hypothesis that the model of cosmic ray propagation and/or the sources encoded in GALPROP provide a sufficient theoretical description.

analysis. We noticed that the tension we observe is significantly milder if it is not included. This, and the effect of using a larger amount of data compared to previous studies, suggests

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As we conclude that new physics is buried within the electron-positron fluxes, we now attempt to extract from the data the size of this new physics signal. Assuming that the new physics affects only the electron-positron fluxes but its influence on the rest of the cosmic ray data is negligible, we can determine the central value and credibility regions of the cosmic ray propagation parameters from the unbiased data: anti-proton/proton, Boron/Carbon, (Sc+Ti+V)/Fe, Be-10/Be-9 to generate a background prediction for all cosmic ray data including the electron-positron fluxes. Once we calculate the theoretical background prediction we can subtract this background from the electron-positron data and determine if

To do this we calculate the prediction for the PAMELA and Fermi-LAT electron-positron fluxes by using the central values of the propagation parameters determined using p/p, ¯ B/C, (Sc+Ti+V)/Fe, Be-10/Be-9. Then using all the scanned values of all five propagation parameters lying within the 68 % credibility region we generate a 1-*σ* uncertainty band for the background around this central value. In figure 3 we overlay the uncertainty background in gray over the Fermi-LAT electron+positron and the PAMELA electron and positron fraction fluxes. For the Fermi-LAT and PAMELA *<sup>e</sup>*<sup>−</sup> the statistical and systematic uncertainties were combined in quadrature, while as the PAMELA *<sup>e</sup>*+/(*e*<sup>+</sup> + *<sup>e</sup>*−) only had statistical uncertainties, we scaled these uncertainties using *τ* = 0.2 to produce the experimental error bands. The magenta bands correspond to our background predictions, while the green dashed lines and band correspond respectively to the central value and the 1-*σ* uncertainty

In figure 3 one can see that our background prediction deviates from the data at energies below ≈ 10 GeV and above 100 GeV. For this analysis we focus on the deviation between the background and the data for energies greater than 100 GeV, while for the deviation observed at low energies we leave this to future research, but we note that this deviation could arise from inadequacies of the propagation model. Based on our background prediction we obtain a weak but statistically significant anomaly signal which we interpret as the presence of a new physics in the Fermi-LAT electron+positron flux. A similar conclusion can be drawn about the PAMELA positron fraction when taking the difference between the central values of the data and the background, but due to sizeable uncertainties of the PAMELA measurement we

To determine the size of the new physics signal in the electron positron data we subtract the central value of the corresponding background prediction from the central value of the data. The 1-*σ* uncertainty band of the signal is obtained by combining the experimental <sup>1</sup> The Fermi-LAT collaboration recently presented a preliminary measurement of the positron fraction [54] which confirmed the results of PAMELA. At first glance this measurement appears to have smaller systematic uncertainties than that of PAMELA. If the officially published Fermi-LAT measurement has systematic errors of approximately the same size as the statistical errors of PAMELA then the data will also deviate from our background unveiling an

why the tension we observe was not detected by authors such as [24].

**7.2. The size of the anomaly**

of the calculated anomaly.

a statistically significant signal can be extracted.

cannot claim a statistically significant deviation. <sup>1</sup>

anomalous signal in the positron fraction.


**Table 2.** Best fit values of the propagation parameters and their 68 % credibility ranges. Numerical values are shown for both fits: including the electron-positron related cosmic ray data only, and including the rest of the data.

For the posterior densities of the electron and nucleus injection indices *<sup>γ</sup>e*<sup>−</sup> and *γnucleus* shown in the first two frames of figure 2, there is a mild but tolerable tension between the two data sets. In the final three frames of figure 2 the posterior densities for *δ*1, *δ*<sup>2</sup> and *D*0*xx* are shown. These frames indicate a statistically significant tension between the two sets of data as the 68 % credibility regions of each set for the two spatial diffusion coefficients *δ*<sup>1</sup> and *δ*2, as well as *D*0*xx* do not overlap each other. Although not shown it is easily extrapolated that not even the 99% credibility regions of these posteriors will overlap. As a consequence we can conclude that if one adjusts the values of the cosmic ray parameters, one can indeed obtain a good fit for either the electron-positron related data or to the rest of the data individually, however, you cannot obtain a good fit for both sets of data simultaneously.

This tension can be interpreted to mean that the data measured by the PAMELA and Fermi-LAT collaborations is affected by new physics that is unaccounted for by the propagation model and/or cosmic ray sources encoded in GALPROP. Based on simple theoretical arguments, the observed behaviour of the PAMELA positron fraction is unexpected. If one attempts to fit this data by simply adjusting the value of the propagation parameters this will lead to a bad fit of the non-electron-positron related data. One also expects that the anomaly in the PAMELA *<sup>e</sup>*+/(*e*<sup>+</sup> + *<sup>e</sup>*−) would also produce an observable anomaly in other electron-positron related data such as the Fermi-LAT *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> and the PAMELA *<sup>e</sup>*<sup>−</sup> spectra. This conclusion agrees with the argument of [24] that "secondary positron production in the general ISM is not capable of producing an abundance that rises with energy".

The observed tension in our data is dramatically increased when one incorporates the recently released PAMELA *<sup>e</sup>*<sup>−</sup> flux [18] in our electron-positron related data. For consistency we checked the result that we would obtain if we excluded this new electron flux data into our analysis. We noticed that the tension we observe is significantly milder if it is not included. This, and the effect of using a larger amount of data compared to previous studies, suggests why the tension we observe was not detected by authors such as [24].

## **7.2. The size of the anomaly**

14 Open Questions in Cosmology

a sufficient theoretical description.

*γe*−

with energy".

of data. The blue dashed curve represents the likelihood functions generated for the electron-positron related data (AMS, Fermi-LAT, HESS, and PAMELA), while the red solid curves represent the likelihood functions obtained for the rest of the cosmic ray data (anti-proton/proton, Boron/Carbon, (Sc+Ti+V)/Fe, Be-10/Be-9) listed in table 1. The 68% credibility regions of the likelihood functions are highlighted by the shaded areas of figure 2 and table 2 lists the numerical values of these credibility regions as well as the best-fit values of each propagation parameter. By looking at figure 2 it is obvious that the electron-positron related data and the non-electron-positron related data are inconsistent with the hypothesis that the model of cosmic ray propagation and/or the sources encoded in GALPROP provide

**parameter Fit for the** *<sup>e</sup>*<sup>±</sup> **related data Fit for the rest of the data**

*γnucleus* 1.60 {1.51, 1.69} 2.10 {1.88, 2.92} *δ*<sup>1</sup> 0.24 {0.23, 0.26} 0.06 {0.04, 0.08} *δ*<sup>2</sup> 0.10 {0.08, 0.12} 0.35 {0.32, 0.39} *<sup>D</sup>*0*xx* [×1028] 2.17 {1.85, 2.19} 11.49 {8.86,13.48}

**Table 2.** Best fit values of the propagation parameters and their 68 % credibility ranges. Numerical values are shown for both

shown in the first two frames of figure 2, there is a mild but tolerable tension between the two data sets. In the final three frames of figure 2 the posterior densities for *δ*1, *δ*<sup>2</sup> and *D*0*xx* are shown. These frames indicate a statistically significant tension between the two sets of data as the 68 % credibility regions of each set for the two spatial diffusion coefficients *δ*<sup>1</sup> and *δ*2, as well as *D*0*xx* do not overlap each other. Although not shown it is easily extrapolated that not even the 99% credibility regions of these posteriors will overlap. As a consequence we can conclude that if one adjusts the values of the cosmic ray parameters, one can indeed obtain a good fit for either the electron-positron related data or to the rest of the data individually,

This tension can be interpreted to mean that the data measured by the PAMELA and Fermi-LAT collaborations is affected by new physics that is unaccounted for by the propagation model and/or cosmic ray sources encoded in GALPROP. Based on simple theoretical arguments, the observed behaviour of the PAMELA positron fraction is unexpected. If one attempts to fit this data by simply adjusting the value of the propagation parameters this will lead to a bad fit of the non-electron-positron related data. One also expects that the anomaly in the PAMELA *<sup>e</sup>*+/(*e*<sup>+</sup> + *<sup>e</sup>*−) would also produce an observable anomaly in other electron-positron related data such as the Fermi-LAT *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> and the PAMELA *<sup>e</sup>*<sup>−</sup> spectra. This conclusion agrees with the argument of [24] that "secondary positron production in the general ISM is not capable of producing an abundance that rises

The observed tension in our data is dramatically increased when one incorporates the recently released PAMELA *<sup>e</sup>*<sup>−</sup> flux [18] in our electron-positron related data. For consistency we checked the result that we would obtain if we excluded this new electron flux data into our

fits: including the electron-positron related cosmic ray data only, and including the rest of the data.

however, you cannot obtain a good fit for both sets of data simultaneously.

For the posterior densities of the electron and nucleus injection indices *<sup>γ</sup>e*<sup>−</sup>

**best fit value 68**% **Cr range best fit value 68**% **Cr range**

2.55 {2.45, 2.60} 2.71 {2.54, 2.92}

and *γnucleus*

As we conclude that new physics is buried within the electron-positron fluxes, we now attempt to extract from the data the size of this new physics signal. Assuming that the new physics affects only the electron-positron fluxes but its influence on the rest of the cosmic ray data is negligible, we can determine the central value and credibility regions of the cosmic ray propagation parameters from the unbiased data: anti-proton/proton, Boron/Carbon, (Sc+Ti+V)/Fe, Be-10/Be-9 to generate a background prediction for all cosmic ray data including the electron-positron fluxes. Once we calculate the theoretical background prediction we can subtract this background from the electron-positron data and determine if a statistically significant signal can be extracted.

To do this we calculate the prediction for the PAMELA and Fermi-LAT electron-positron fluxes by using the central values of the propagation parameters determined using p/p, ¯ B/C, (Sc+Ti+V)/Fe, Be-10/Be-9. Then using all the scanned values of all five propagation parameters lying within the 68 % credibility region we generate a 1-*σ* uncertainty band for the background around this central value. In figure 3 we overlay the uncertainty background in gray over the Fermi-LAT electron+positron and the PAMELA electron and positron fraction fluxes. For the Fermi-LAT and PAMELA *<sup>e</sup>*<sup>−</sup> the statistical and systematic uncertainties were combined in quadrature, while as the PAMELA *<sup>e</sup>*+/(*e*<sup>+</sup> + *<sup>e</sup>*−) only had statistical uncertainties, we scaled these uncertainties using *τ* = 0.2 to produce the experimental error bands. The magenta bands correspond to our background predictions, while the green dashed lines and band correspond respectively to the central value and the 1-*σ* uncertainty of the calculated anomaly.

In figure 3 one can see that our background prediction deviates from the data at energies below ≈ 10 GeV and above 100 GeV. For this analysis we focus on the deviation between the background and the data for energies greater than 100 GeV, while for the deviation observed at low energies we leave this to future research, but we note that this deviation could arise from inadequacies of the propagation model. Based on our background prediction we obtain a weak but statistically significant anomaly signal which we interpret as the presence of a new physics in the Fermi-LAT electron+positron flux. A similar conclusion can be drawn about the PAMELA positron fraction when taking the difference between the central values of the data and the background, but due to sizeable uncertainties of the PAMELA measurement we cannot claim a statistically significant deviation. <sup>1</sup>

To determine the size of the new physics signal in the electron positron data we subtract the central value of the corresponding background prediction from the central value of the data. The 1-*σ* uncertainty band of the signal is obtained by combining the experimental

<sup>1</sup> The Fermi-LAT collaboration recently presented a preliminary measurement of the positron fraction [54] which confirmed the results of PAMELA. At first glance this measurement appears to have smaller systematic uncertainties than that of PAMELA. If the officially published Fermi-LAT measurement has systematic errors of approximately the same size as the statistical errors of PAMELA then the data will also deviate from our background unveiling an anomalous signal in the positron fraction.

If one assumes that the propagation model satisfactorily describes the propagation of cosmic rays through our Galaxy it is only natural to suspect that local effects are modifying the distribution of electrons and positrons. The lack of sources included in the GALPROP calculation seem to confirm this suspicion. There have been a plethora of papers that account for this anomaly by proposing various new sources of cosmic rays. There are two major categories of new cosmic ray sources that have been proposed. The first involves known astrophysical objects with uncertain parameters such as supernova remnants, pulsars, or various other objects in the Galactic centre, while the second involves more exotic astronomical and/or particle physics phenomena such as dark matter. Literature discussing

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For energies greater than 100 GeV, energy losses such as inverse Compton scattering of interstellar dust and cosmic microwave background light or synchrotron radiation become important. These effects result in a relatively short lifetime of the electron and positron while simultaneously this causes a decrease in the intensity of these particles as energy increases. As a result it is hypothesised that a large number of the electrons and positrons detected at Earth with an energy above 100 GeV come from individual sources within a few kilo-parsecs of Earth [51, 52]. Random fluctuations in the injection spectrum and spatial distribution of these nearby sources can produce detectable differences between the predicted background and the most energetic part of the observed electron and positron spectrum. This deviation could indicate the presence of new physics arising from either an astrophysical object(s) or

If the size of the anomalous signal can be isolated from the experimental data then, regardless of the origin of the anomaly, the source will have to produce a signal with those characteristics. In Fig. 4 we compare our extracted signal to a few randomly selected attempts from the literature to match this anomaly. The first frame features the spectrum of electrons and positrons unaccounted for from local supernovae as calculated by [59] . The top right frame shows the contribution from additional primary cosmic ray sources such as pulsars or annihilation of particle dark matter as calculated by [52]. The bottom left frame contains the predictions of [60] for anomalous electron-positron sources from dark matter annihilations, while the last frame shows the dark matter annihilation contributions calculated by [61].

If the theoretical uncertainty of a new cosmic ray source and its contribution to cosmic ray measurements at Earth is unknown it can be difficult to draw any conclusion about its contribution to our isolated signal. In the case where the theoretical uncertainty of a new cosmic ray source is known it usually tends to be of significant size that it can prevent us from judging whether it is a valid explanation of our signal. Regardless, we can select a few scenarios that are more likely to be favoured than some others based on the present amount of information we have obtained from our analysis. With more data it will be possible to reduce the size of the uncertainty of our signal, while with more detailed calculations we can produce a more precise prediction of the cosmic ray spectrum as measured at Earth. This may enable the various suggestions of the source of the electron-positron anomaly to be

these cases is extensively cited by [21].

dark matter.

confirmed or ruled out.

**Figure 3.** The anomalous signal we extracted for various electron-positron fluxes. The green dotted curves (marking the central values) and bands (showing 68 % credible intervals) correspond to the extracted size of the anomaly. The data points correspond to the spectra measured by Fermi-LAT and PAMELA. Combined statistical and systematic uncertainties are shown (by the gray bands) for Fermi-LAT and PAMELA *<sup>e</sup>*−, while (*<sup>τ</sup>* = 0.2) scaled statistical uncertainties are shown for PAMELA *e*+/(*e*<sup>+</sup> + *e*−). Overlaid in magenta is our background prediction (central value curve and 68 % credible intervals).

and background uncertainties quadratically. In Fig. 3 the results for the electron-positron anomaly are shown. Based on our background predictions, we obtain a non-vanishing anomalous signal for the Fermi-LAT *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> flux, while for the PAMELA data we cannot claim the presence of a statistically significant anomaly due to the large uncertainties of the data.

#### **7.3. The source of the anomaly**

Since the publication of the PAMELA positron fraction [7], there have been numerous publications that have speculated on the origin of the discrepancy between the theoretical prediction of electron-positron spectra and the experimental data. Based on the available evidence we can only postulate on the origin of this deviation. An obvious guess would be that the model used to describe the propagation of electrons and positrons in our Galaxy is insufficient in some respect, which if correct would mean that there exists no anomalous signal in the data. One such reasonable effect which is not incorporated in the two dimensional GALPROP calculation is the spectral hardening of cosmic ray spectra due to the presence of non-steady sources. To confirm these possibilities it would be an interesting exercise to repeat our analysis using different calculation tools such as DRAGON by [55], USINE by [56], PPPC4DMID by [57] or the code of [58].

If one assumes that the propagation model satisfactorily describes the propagation of cosmic rays through our Galaxy it is only natural to suspect that local effects are modifying the distribution of electrons and positrons. The lack of sources included in the GALPROP calculation seem to confirm this suspicion. There have been a plethora of papers that account for this anomaly by proposing various new sources of cosmic rays. There are two major categories of new cosmic ray sources that have been proposed. The first involves known astrophysical objects with uncertain parameters such as supernova remnants, pulsars, or various other objects in the Galactic centre, while the second involves more exotic astronomical and/or particle physics phenomena such as dark matter. Literature discussing these cases is extensively cited by [21].

16 Open Questions in Cosmology

data.

**7.3. The source of the anomaly**

USINE by [56], PPPC4DMID by [57] or the code of [58].

**Figure 3.** The anomalous signal we extracted for various electron-positron fluxes. The green dotted curves (marking the central values) and bands (showing 68 % credible intervals) correspond to the extracted size of the anomaly. The data points correspond to the spectra measured by Fermi-LAT and PAMELA. Combined statistical and systematic uncertainties are shown (by the gray bands) for Fermi-LAT and PAMELA *<sup>e</sup>*−, while (*<sup>τ</sup>* = 0.2) scaled statistical uncertainties are shown for PAMELA

and background uncertainties quadratically. In Fig. 3 the results for the electron-positron anomaly are shown. Based on our background predictions, we obtain a non-vanishing anomalous signal for the Fermi-LAT *<sup>e</sup>*<sup>+</sup> + *<sup>e</sup>*<sup>−</sup> flux, while for the PAMELA data we cannot claim the presence of a statistically significant anomaly due to the large uncertainties of the

Since the publication of the PAMELA positron fraction [7], there have been numerous publications that have speculated on the origin of the discrepancy between the theoretical prediction of electron-positron spectra and the experimental data. Based on the available evidence we can only postulate on the origin of this deviation. An obvious guess would be that the model used to describe the propagation of electrons and positrons in our Galaxy is insufficient in some respect, which if correct would mean that there exists no anomalous signal in the data. One such reasonable effect which is not incorporated in the two dimensional GALPROP calculation is the spectral hardening of cosmic ray spectra due to the presence of non-steady sources. To confirm these possibilities it would be an interesting exercise to repeat our analysis using different calculation tools such as DRAGON by [55],

*e*+/(*e*<sup>+</sup> + *e*−). Overlaid in magenta is our background prediction (central value curve and 68 % credible intervals).

For energies greater than 100 GeV, energy losses such as inverse Compton scattering of interstellar dust and cosmic microwave background light or synchrotron radiation become important. These effects result in a relatively short lifetime of the electron and positron while simultaneously this causes a decrease in the intensity of these particles as energy increases. As a result it is hypothesised that a large number of the electrons and positrons detected at Earth with an energy above 100 GeV come from individual sources within a few kilo-parsecs of Earth [51, 52]. Random fluctuations in the injection spectrum and spatial distribution of these nearby sources can produce detectable differences between the predicted background and the most energetic part of the observed electron and positron spectrum. This deviation could indicate the presence of new physics arising from either an astrophysical object(s) or dark matter.

If the size of the anomalous signal can be isolated from the experimental data then, regardless of the origin of the anomaly, the source will have to produce a signal with those characteristics. In Fig. 4 we compare our extracted signal to a few randomly selected attempts from the literature to match this anomaly. The first frame features the spectrum of electrons and positrons unaccounted for from local supernovae as calculated by [59] . The top right frame shows the contribution from additional primary cosmic ray sources such as pulsars or annihilation of particle dark matter as calculated by [52]. The bottom left frame contains the predictions of [60] for anomalous electron-positron sources from dark matter annihilations, while the last frame shows the dark matter annihilation contributions calculated by [61].

If the theoretical uncertainty of a new cosmic ray source and its contribution to cosmic ray measurements at Earth is unknown it can be difficult to draw any conclusion about its contribution to our isolated signal. In the case where the theoretical uncertainty of a new cosmic ray source is known it usually tends to be of significant size that it can prevent us from judging whether it is a valid explanation of our signal. Regardless, we can select a few scenarios that are more likely to be favoured than some others based on the present amount of information we have obtained from our analysis. With more data it will be possible to reduce the size of the uncertainty of our signal, while with more detailed calculations we can produce a more precise prediction of the cosmic ray spectrum as measured at Earth. This may enable the various suggestions of the source of the electron-positron anomaly to be confirmed or ruled out.

the electron+positron flux in a model independent fashion. Using the statistical techniques we were able to extract the size, shape and uncertainty of the anomalous contribution in the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> cosmic ray spectrum. We briefly compared the extracted signal to some theoretical

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105

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and electrons above 7-GeV. *Astron. Astrophys.*, 392:287–294, 2002.

results predicting such an anomaly.

Katie Auchettl and Csaba Balázs Monash University, Australia

*J.*, 561:787–799, 2001.

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*Astrophys.*, 524:A51, 2010.

component. *arXiv:1102.0744*, 2011.

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**Author details**

**References**

**Figure 4.** Comparing the detected cosmic ray flux (data points and gray band) and the signal extracted in this work (green dotted curve and band) to potential explanations of the electron-positron cosmic ray anomaly (solid curves). The various theoretical predictions come from [59], [52], [60] and [61]. Currently the comparison is fairly inconclusive but with more data it will be possible to shrink the uncertainty in the determination of the signal. Then various suggestions can be confirmed or ruled out.

## **8. Conclusions**

Motivated by the possibility of new physics contributing to the measurements of PAMELA and Fermi-LAT, we subjected a wide range of cosmic ray observations to a Bayesian likelihood analysis. In the context of the propagation model coded in GALPROP, we found a significant tension between the *<sup>e</sup>*−\*e*<sup>+</sup> related data and the rest of the cosmic ray fluxes. This tension can be interpreted as the failure of the model to describe all the data simultaneously or as the effect of a missing source component.

Since the PAMELA and Fermi-LAT data are suspected to contain a component unaccounted for in GALPROP, we extracted the preferred values of the cosmic ray propagation parameters from the non-electron-positron related measurements. Based on these parameter values we calculated background predictions, with uncertainties, for PAMELA and Fermi-LAT. We found a deviation between the PAMELA and Fermi-LAT data and the predicted background even when uncertainties, including systematics, were taken into account. Interpreting this as an indication of new physics we subtracted the background from the data isolating the size of the anomalous component.

The signal of new physics in the electron+positron spectrum was found to be non-vanishing within the calculated uncertainties. Thus the use of 219 cosmic ray spectral data points within the Bayesian framework allowed us to confirm the existence of new physics effects in the electron+positron flux in a model independent fashion. Using the statistical techniques we were able to extract the size, shape and uncertainty of the anomalous contribution in the *<sup>e</sup>*<sup>−</sup> + *<sup>e</sup>*<sup>+</sup> cosmic ray spectrum. We briefly compared the extracted signal to some theoretical results predicting such an anomaly.

## **Author details**

18 Open Questions in Cosmology

ruled out.

**8. Conclusions**

or as the effect of a missing source component.

of the anomalous component.

**Figure 4.** Comparing the detected cosmic ray flux (data points and gray band) and the signal extracted in this work (green dotted curve and band) to potential explanations of the electron-positron cosmic ray anomaly (solid curves). The various theoretical predictions come from [59], [52], [60] and [61]. Currently the comparison is fairly inconclusive but with more data it will be possible to shrink the uncertainty in the determination of the signal. Then various suggestions can be confirmed or

Motivated by the possibility of new physics contributing to the measurements of PAMELA and Fermi-LAT, we subjected a wide range of cosmic ray observations to a Bayesian likelihood analysis. In the context of the propagation model coded in GALPROP, we found a significant tension between the *<sup>e</sup>*−\*e*<sup>+</sup> related data and the rest of the cosmic ray fluxes. This tension can be interpreted as the failure of the model to describe all the data simultaneously

Since the PAMELA and Fermi-LAT data are suspected to contain a component unaccounted for in GALPROP, we extracted the preferred values of the cosmic ray propagation parameters from the non-electron-positron related measurements. Based on these parameter values we calculated background predictions, with uncertainties, for PAMELA and Fermi-LAT. We found a deviation between the PAMELA and Fermi-LAT data and the predicted background even when uncertainties, including systematics, were taken into account. Interpreting this as an indication of new physics we subtracted the background from the data isolating the size

The signal of new physics in the electron+positron spectrum was found to be non-vanishing within the calculated uncertainties. Thus the use of 219 cosmic ray spectral data points within the Bayesian framework allowed us to confirm the existence of new physics effects in Katie Auchettl and Csaba Balázs

Monash University, Australia

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**Chapter 5**

**Provisional chapter**

**Extended Theories of Gravitation and the Curvature of**

Cosmology is a young science. Less than a century ago cosmology stopped to be a branch of philosophy and it crossbred with General Relativity to become a science. Until very recently cosmological observations were quite rough and qualitative. Until the 80s one was quite

Since then a number of extremely precise surveys have been carried over producing a massive amount of very precise data. The current picture that emerged from those data is quite awkward. In order to fit observations and maintain standard GR as general framework for gravity one is forced to introduce dark sources, at least in a large amount different from the matter that can be seen in the universe and which has somehow odd behavior; see [1], [2],

Actually, following this direction one is led to assume that about 70% of gravitational sources in the universe is constituted by some strange kind of dark energy, closely resembling a (small and positive) cosmological constant, about 25% of gravitational sources is constituted by some kind of dark matter (for which different models have been proposed and discussed), while visible matter amounts to few percents (about 4-5% depending on the model) of the total amount of matter. It is important to notice that we do not have any direct evidence or data about dark energy and dark matter other than their supposed gravitational effects on visible matter. Moreover, the best models for dark energy and dark matter are often definitely

On the other hand, it has been suggested that the description of the gravitational field given by standard GR may fail at cosmological scale and we missed something, so that a good agreement with data can be obtained by modifying the description of gravity more than adding exotic sources. In any event it is now clear that something has to be changed in our

and reproduction in any medium, provided the original work is properly cited.

©2012 Fatibene and Francaviglia, licensee InTech. This is an open access chapter 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. © 2012 Fatibene and Francaviglia; licensee InTech. This is an open access article 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.

© 2012 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,

**of the Universe – Do We Really Need Dark Matter?**

**Extended Theories of Gravitation and the Curvature**

**the Universe – Do We Really Need Dark Matter?**

L. Fatibene and M. Francaviglia

L. Fatibene and M. Francaviglia

http://dx.doi.org/10.5772/52041

**1. Introduction**

[3], [4].

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

satisfied with data with error bars of few percent.

unsatisfactory from a fundamental viewpoint; see [5].

standard framework in order to understand the universe out there.

**Provisional chapter**

## **Extended Theories of Gravitation and the Curvature of the Universe – Do We Really Need Dark Matter? of the Universe – Do We Really Need Dark Matter?**

**Extended Theories of Gravitation and the Curvature**

L. Fatibene and M. Francaviglia Additional information is available at the end of the chapter

L. Fatibene and M. Francaviglia

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52041

## **1. Introduction**

Cosmology is a young science. Less than a century ago cosmology stopped to be a branch of philosophy and it crossbred with General Relativity to become a science. Until very recently cosmological observations were quite rough and qualitative. Until the 80s one was quite satisfied with data with error bars of few percent.

Since then a number of extremely precise surveys have been carried over producing a massive amount of very precise data. The current picture that emerged from those data is quite awkward. In order to fit observations and maintain standard GR as general framework for gravity one is forced to introduce dark sources, at least in a large amount different from the matter that can be seen in the universe and which has somehow odd behavior; see [1], [2], [3], [4].

Actually, following this direction one is led to assume that about 70% of gravitational sources in the universe is constituted by some strange kind of dark energy, closely resembling a (small and positive) cosmological constant, about 25% of gravitational sources is constituted by some kind of dark matter (for which different models have been proposed and discussed), while visible matter amounts to few percents (about 4-5% depending on the model) of the total amount of matter. It is important to notice that we do not have any direct evidence or data about dark energy and dark matter other than their supposed gravitational effects on visible matter. Moreover, the best models for dark energy and dark matter are often definitely unsatisfactory from a fundamental viewpoint; see [5].

On the other hand, it has been suggested that the description of the gravitational field given by standard GR may fail at cosmological scale and we missed something, so that a good agreement with data can be obtained by modifying the description of gravity more than adding exotic sources. In any event it is now clear that something has to be changed in our standard framework in order to understand the universe out there.

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. © 2012 Fatibene and Francaviglia; licensee InTech. This is an open access article 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. © 2012 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.

©2012 Fatibene and Francaviglia, licensee InTech. This is an open access chapter distributed under the terms of

Besides these obvious considerations let us add quite a trivial remark. Our understanding of the meaning of observations is generally weak and often depending on the model. Standard GR has a good set of protocols which allow one to make predictions and tests. The theory is extremely well tested at Solar system scales, while it is known to require corrections (by adding dark sources or by modifying dynamics) at galactic, astrophysical and cosmological scale (oddly enough whenever non-vacuum solutions are considered).

However, what we observe when we measure the distance of a supernova is not clear at all. GR is a relativistic theory with a huge symmetry group, namely all spacetime diffeomorphisms. The observable quantities should be then invariant with respect to spacetime diffeomorphisms, i.e. gauge invariant. Unfortunately, due to the particular nature of diffeomorphisms and their action on the geometry of spacetime, we do not know any non-trivial quantity which is diff-invariant. Also scalars are not (unless they are constant) since the Lie derivative of a scalar with respect to a generic spacetime vector field (i.e. a generic generator of spacetime diffeomorphisms) is

$$
\delta \mathcal{L}\_{\xi} f = \mathfrak{f}^{\mu} \partial\_{\mu} f \tag{1}
$$

**2. EPS structures on spacetime**

is nothing like future directed spacelike vectors!).

*cones* without resorting to a metric structure we did not define yet.

light rays are given as straight lines in Minkowski spacetime).

and cosmology.

cone).

hitting *P* within *Vp*.

metric structure on *M*.

As already mentioned, in the early 70s Ehlers, Pirani and Schild (EPS) proposed an axiomatic framework for relativistic theories in which they showed how one can derive the geometric structure of spacetime from potentially observable quantities, i.e. worldlines of particles and light rays; see [7]. Accordingly, in the EPS framework the geometry of spacetime is not assumed but derived by more fundamental objects. We shall first briefly review EPS formalism; in the next Sections we shall discuss its consequences in gravitational theories

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113

Let *M* be an (orientable, time orientable, connected, paracompact, smooth) *m*-dimensional manifold. Points in *M* are called *events* and *M* is called accordingly a *spacetime*. Let us stress that although *M* is chosen so that it allows global Lorentzian metrics, we do not fix any

On *M* we consider two congruences of trajectories. Let P be the congruence of all possible motions of massive particles and L be the congruence of all possible light rays. Of course there are reasonable physical requirement to be asked about P and L since we expect they cannot be chosen to be completely generic or unrelated since we expect photons to feel the gravitational field as well as we expect matter to interact with the electromagnetic field. If we restrict ourselves to particles and light rays passing through an event *x* ∈ *M*, we *know* that the directions of light rays form a *cone* (the *light cone*). We can express this experimental fact by asking that the directions of light rays divide spacetime directions (i.e. the projective space of *TxM*) into two connected components (i.e. the directions *inside* and *outside* the light

We also know that the set of vectors *inside* the light cone is topologically different from the set of vectors *outside* the light cone. If one removes the zero vector then the set of vectors *inside* the light cone disconnects into two connected components (namely, *future* and *past* directed timelike vectors), while the set of vectors *outside* the light cone keeps being connected (there

Moreover, we know that one has two kinds of vectors tangent to light rays: the ones pointing to the future and the ones pointing to the past. Thus we assume that (once the zero vector is removed) the set of vectors tangent to light rays also splits into two connected components (namely, *future* and *past* directed vectors). Let us stress that past and future are defined *at a point x* and it does not really matter which one of them is called future or past. These three requirements are physically well founded and in the end they constrain the light cones to be

Then we have a number of regularity conditions. We need axioms to certify that one has enough light rays to account for physical standard messaging. Let us thus assume that for any particle *P* ∈ P and for any event *p* ∈ *P* ⊂ *M* there exists a neighbourhood *Vp* and a neighbourhood *Up* ⊂ *Vp* such that for any event *x* ∈ *Up* there are two light rays through *x*

Let us remark that in Minkowski spacetime one can set *Up* = *Vp* = *M* and there are always two such light rays (as one can check by direct calculation remembering that particles and

which is in general not zero, showing that in fact the quantity is not gauge invariant. This is known since the very beginning and it is the starting point of the celebrated hole argument; see [6]. Since we do measure quantities that are not gauge invariant, the only possible explanation is that we set observational protocols which as a matter of fact break gauge invariance on a conventional basis (possibly using matter references, as suggested in [6]).

That would not be too bad, if we clearly understood the details of such conventions and gauge fixing, that we do not. Instead, standard GR mixes from its very beginning physical quantities (i.e. the gravitational field) and the observational protocols (e.g. for measuring distances and times) in the same object (namely, the metric tensor). Originally, Einstein had not many options, since at that time the only way to describe curvature was through a metric structure and general (linear) connections were still to be fully described. As a consequence it becomes very difficult to keep the two things separated as they should.

In the 70s Ehlers, Pirani and Schild (EPS) gave a fundamental contribution to the understanding of the foundations of any reasonable theory of spacetime and gravity. They proposed an axiomatic approach to gravitational theories which, instead of assuming a metric or a connection on spacetime, assumed as fundamental potentially observable quantities (namely the worldlines of particles and light rays) and derived from them the geometry of spacetime; see [7], [8]. The original project was to obtain standard GR. However, the proposal finally turned out to give us a fundamental insight about what is to be considered observable and which geometrical structures are really essential for gravity.

In particular EPS framework allows a more general geometric structure on spacetime in which standard GR comes out to be just one of many possible theories of gravitation. Moreover, a more general framework potentially allows to test which geometric structure on spacetime is actually physically realized. As a side effect, EPS has an impact on observational protocols (not all standard protocols can be trivially extended to a general extended theory).

We shall hereafter review the EPS framework, define extended theories of gravitation and attempt a rough classification of possible extended theories. Finally we shall discuss some simple application to cosmology and observational protocols.

## **2. EPS structures on spacetime**

2 Open Questions in Cosmology

Besides these obvious considerations let us add quite a trivial remark. Our understanding of the meaning of observations is generally weak and often depending on the model. Standard GR has a good set of protocols which allow one to make predictions and tests. The theory is extremely well tested at Solar system scales, while it is known to require corrections (by adding dark sources or by modifying dynamics) at galactic, astrophysical and cosmological

However, what we observe when we measure the distance of a supernova is not clear at all. GR is a relativistic theory with a huge symmetry group, namely all spacetime diffeomorphisms. The observable quantities should be then invariant with respect to spacetime diffeomorphisms, i.e. gauge invariant. Unfortunately, due to the particular nature of diffeomorphisms and their action on the geometry of spacetime, we do not know any non-trivial quantity which is diff-invariant. Also scalars are not (unless they are constant) since the Lie derivative of a scalar with respect to a generic spacetime vector field (i.e. a

which is in general not zero, showing that in fact the quantity is not gauge invariant. This is known since the very beginning and it is the starting point of the celebrated hole argument; see [6]. Since we do measure quantities that are not gauge invariant, the only possible explanation is that we set observational protocols which as a matter of fact break gauge invariance on a conventional basis (possibly using matter references, as suggested in [6]). That would not be too bad, if we clearly understood the details of such conventions and gauge fixing, that we do not. Instead, standard GR mixes from its very beginning physical quantities (i.e. the gravitational field) and the observational protocols (e.g. for measuring distances and times) in the same object (namely, the metric tensor). Originally, Einstein had not many options, since at that time the only way to describe curvature was through a metric structure and general (linear) connections were still to be fully described. As a consequence

In the 70s Ehlers, Pirani and Schild (EPS) gave a fundamental contribution to the understanding of the foundations of any reasonable theory of spacetime and gravity. They proposed an axiomatic approach to gravitational theories which, instead of assuming a metric or a connection on spacetime, assumed as fundamental potentially observable quantities (namely the worldlines of particles and light rays) and derived from them the geometry of spacetime; see [7], [8]. The original project was to obtain standard GR. However, the proposal finally turned out to give us a fundamental insight about what is to be considered observable

In particular EPS framework allows a more general geometric structure on spacetime in which standard GR comes out to be just one of many possible theories of gravitation. Moreover, a more general framework potentially allows to test which geometric structure on spacetime is actually physically realized. As a side effect, EPS has an impact on observational protocols (not all standard protocols can be trivially extended to a general extended theory). We shall hereafter review the EPS framework, define extended theories of gravitation and attempt a rough classification of possible extended theories. Finally we shall discuss some

*£<sup>ξ</sup> <sup>f</sup>* <sup>=</sup> *<sup>ξ</sup>µ∂µ <sup>f</sup>* (1)

scale (oddly enough whenever non-vacuum solutions are considered).

it becomes very difficult to keep the two things separated as they should.

and which geometrical structures are really essential for gravity.

simple application to cosmology and observational protocols.

generic generator of spacetime diffeomorphisms) is

As already mentioned, in the early 70s Ehlers, Pirani and Schild (EPS) proposed an axiomatic framework for relativistic theories in which they showed how one can derive the geometric structure of spacetime from potentially observable quantities, i.e. worldlines of particles and light rays; see [7]. Accordingly, in the EPS framework the geometry of spacetime is not assumed but derived by more fundamental objects. We shall first briefly review EPS formalism; in the next Sections we shall discuss its consequences in gravitational theories and cosmology.

Let *M* be an (orientable, time orientable, connected, paracompact, smooth) *m*-dimensional manifold. Points in *M* are called *events* and *M* is called accordingly a *spacetime*. Let us stress that although *M* is chosen so that it allows global Lorentzian metrics, we do not fix any metric structure on *M*.

On *M* we consider two congruences of trajectories. Let P be the congruence of all possible motions of massive particles and L be the congruence of all possible light rays. Of course there are reasonable physical requirement to be asked about P and L since we expect they cannot be chosen to be completely generic or unrelated since we expect photons to feel the gravitational field as well as we expect matter to interact with the electromagnetic field.

If we restrict ourselves to particles and light rays passing through an event *x* ∈ *M*, we *know* that the directions of light rays form a *cone* (the *light cone*). We can express this experimental fact by asking that the directions of light rays divide spacetime directions (i.e. the projective space of *TxM*) into two connected components (i.e. the directions *inside* and *outside* the light cone).

We also know that the set of vectors *inside* the light cone is topologically different from the set of vectors *outside* the light cone. If one removes the zero vector then the set of vectors *inside* the light cone disconnects into two connected components (namely, *future* and *past* directed timelike vectors), while the set of vectors *outside* the light cone keeps being connected (there is nothing like future directed spacelike vectors!).

Moreover, we know that one has two kinds of vectors tangent to light rays: the ones pointing to the future and the ones pointing to the past. Thus we assume that (once the zero vector is removed) the set of vectors tangent to light rays also splits into two connected components (namely, *future* and *past* directed vectors). Let us stress that past and future are defined *at a point x* and it does not really matter which one of them is called future or past. These three requirements are physically well founded and in the end they constrain the light cones to be *cones* without resorting to a metric structure we did not define yet.

Then we have a number of regularity conditions. We need axioms to certify that one has enough light rays to account for physical standard messaging. Let us thus assume that for any particle *P* ∈ P and for any event *p* ∈ *P* ⊂ *M* there exists a neighbourhood *Vp* and a neighbourhood *Up* ⊂ *Vp* such that for any event *x* ∈ *Up* there are two light rays through *x* hitting *P* within *Vp*.

Let us remark that in Minkowski spacetime one can set *Up* = *Vp* = *M* and there are always two such light rays (as one can check by direct calculation remembering that particles and light rays are given as straight lines in Minkowski spacetime).

*Up* such that for any *x* ∈ *Up* there is a (future directed) light ray through *x* then hitting the clock *Pi* at its parameter value *si*. The values of the parameters *si* do form a good coordinate system in *Up*. We assume that the spacetime differential structure on *M* is the one compatible with these charts. Let us remark that parallax coordinates mimick how astronomers define

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115

*P1 P2 P3*

*s2*

*s2*

*gp* = *<sup>g</sup>µν*(*p*) *dx<sup>µ</sup>* ⊗ *dx<sup>ν</sup>* = *∂µν*Φ(*p*) *dx<sup>µ</sup>* ⊗ *dx<sup>ν</sup>* (2)

*p*

One can show that as a consequence of these assumptions a class of Lorentzian metrics *g* is defined on *M*. Let us then fix a clock *P* though an event *p*. For any event *x* ∈ *Up* one has two light rays through *x* intersecting *P*, say at events *p*± which correspond to the parameter values *s*±. Then we can define a local function Φ : *Up* → **R** : *x* �→ −*s*<sup>+</sup> · *s*−. As one can easily show, if there exists a light ray through *x* and *p* then Φ(*x*) = 0. According to the topological assumptions made on the light cones then one can show that Φ(*p*) = 0 and *d*Φ(*p*) = 0. Then one can consider the Hessian *∂µν*Φ(*p*) as the first non-zero term in the Taylor expansion of

For any light ray direction *v* at *p* one has *g*(*v*, *v*) = 0. One can also easily show that for *u*

Accordingly, *g* cannot be definite positive. In order not to contradict again assumptions about light cones, one can show that *g* is necessarily non-degenerate and Lorentzian (see [10]). Of course the the tensor *g* depends on the conventional choice of the clock. If one changes clock one defines a different tensor *g*˜ which is related to the previous one by a conformal

Let us now consider the set Lor(*M*) of all (global) Lorentzian metrics on *M*. Let us say that two metrics *<sup>g</sup>*(1), *<sup>g</sup>*(2) <sup>∈</sup> Lor(*M*) are *conformally equivalent* iff there exists a positive scalar field

Φ around the event *p*. In this case it defines a tensor field (a bilinear form)

transformation, namely *g*˜ = *ϕ*(*x*) · *g* for some positive scalar field *ϕ*.

*s1*

positions of objects.

**Figure 2.** Parallax coordinates in dimension *m* = 3

tangent to the clock *P* one has *g*(*u*, *u*) < 0.

Given two particles *P*, *Q* ∈ P we can consider the family of light rays *λ* ∈ L intersecting *P* and *Q*. By the above assumption, when *P* and *Q* are close enough such family is not empty. This family of light rays does define a local one-to-one map between *P* and *Q* which is called a *message* which is denoted by *µ* : *P* → *Q*. If one takes the composition of a message from *P* to *Q* and a message from *Q* to *P* the resulting map *ǫ* : *P* → *P* is called and *echo* of *P* on *Q*. Both messages and echoes are assumed to be smooth maps.

**Figure 1.** Messages and Echoes

Finally, we have to guarantee that there exist enough particles in P and light rays in L (which until now could be empty, as far as we know). We assume that there is a particle for each vector inside the light cone and a light ray for each vector on the light cone.

Let us now define a *clock* to be a parametrized particle, the parametrization accounting for the time maintained by the clock; [9]. For any clock *P* ∈ P, for any event *p* ∈ *P* one can set the parameter to be *s* = 0 at *p*. Using echoes one can use a number of clocks to define a special class of local coordinates, *called radar coordinates* or *parallax coodinates*. If dim(*M*) = *m* one can always choose *<sup>m</sup>* clocks *Pi* near an event *<sup>p</sup>* ∈ *<sup>M</sup>* so that there exists a neibourhood *Up* such that for any *x* ∈ *Up* there is a (future directed) light ray through *x* then hitting the clock *Pi* at its parameter value *si*. The values of the parameters *si* do form a good coordinate system in *Up*. We assume that the spacetime differential structure on *M* is the one compatible with these charts. Let us remark that parallax coordinates mimick how astronomers define positions of objects.

**Figure 2.** Parallax coordinates in dimension *m* = 3

4 Open Questions in Cosmology

*P*

*Up*

Given two particles *P*, *Q* ∈ P we can consider the family of light rays *λ* ∈ L intersecting *P* and *Q*. By the above assumption, when *P* and *Q* are close enough such family is not empty. This family of light rays does define a local one-to-one map between *P* and *Q* which is called a *message* which is denoted by *µ* : *P* → *Q*. If one takes the composition of a message from *P* to *Q* and a message from *Q* to *P* the resulting map *ǫ* : *P* → *P* is called and *echo* of *P* on *Q*.

Finally, we have to guarantee that there exist enough particles in P and light rays in L (which until now could be empty, as far as we know). We assume that there is a particle for each

Let us now define a *clock* to be a parametrized particle, the parametrization accounting for the time maintained by the clock; [9]. For any clock *P* ∈ P, for any event *p* ∈ *P* one can set the parameter to be *s* = 0 at *p*. Using echoes one can use a number of clocks to define a special class of local coordinates, *called radar coordinates* or *parallax coodinates*. If dim(*M*) = *m* one can always choose *<sup>m</sup>* clocks *Pi* near an event *<sup>p</sup>* ∈ *<sup>M</sup>* so that there exists a neibourhood

*x*

*P Q*

ε: *P* → *P*

*p*

*Vp*

Both messages and echoes are assumed to be smooth maps.

*P Q*

μ: *P* → *Q*

vector inside the light cone and a light ray for each vector on the light cone.

**Figure 1.** Messages and Echoes

One can show that as a consequence of these assumptions a class of Lorentzian metrics *g* is defined on *M*. Let us then fix a clock *P* though an event *p*. For any event *x* ∈ *Up* one has two light rays through *x* intersecting *P*, say at events *p*± which correspond to the parameter values *s*±. Then we can define a local function Φ : *Up* → **R** : *x* �→ −*s*<sup>+</sup> · *s*−. As one can easily show, if there exists a light ray through *x* and *p* then Φ(*x*) = 0. According to the topological assumptions made on the light cones then one can show that Φ(*p*) = 0 and *d*Φ(*p*) = 0. Then one can consider the Hessian *∂µν*Φ(*p*) as the first non-zero term in the Taylor expansion of Φ around the event *p*. In this case it defines a tensor field (a bilinear form)

$$\mathcal{g}\_p = \mathcal{g}\_{\mu\nu}(p) \, d\mathfrak{x}^{\mu} \otimes d\mathfrak{x}^{\nu} = \partial\_{\mu\nu} \Phi(p) \, d\mathfrak{x}^{\mu} \otimes d\mathfrak{x}^{\nu} \tag{2}$$

For any light ray direction *v* at *p* one has *g*(*v*, *v*) = 0. One can also easily show that for *u* tangent to the clock *P* one has *g*(*u*, *u*) < 0.

Accordingly, *g* cannot be definite positive. In order not to contradict again assumptions about light cones, one can show that *g* is necessarily non-degenerate and Lorentzian (see [10]). Of course the the tensor *g* depends on the conventional choice of the clock. If one changes clock one defines a different tensor *g*˜ which is related to the previous one by a conformal transformation, namely *g*˜ = *ϕ*(*x*) · *g* for some positive scalar field *ϕ*.

Let us now consider the set Lor(*M*) of all (global) Lorentzian metrics on *M*. Let us say that two metrics *<sup>g</sup>*(1), *<sup>g</sup>*(2) <sup>∈</sup> Lor(*M*) are *conformally equivalent* iff there exists a positive scalar field *ϕ* such that *g*˜*<sup>x</sup>* = *ϕ*(*x*) · *gx*. The construction above shows that one can define out of light rays (i.e. out of the electromagnetic field) a *conformal class* of metrics C = [*g*]. Let us remark that the choice of a representative *g*˜ ∈ C is conventional and in fact part of the specification of the observer; conformal transformations are gauge transformations.

Notice that light cones are invariant with respect to conformal transformations; given a conformal structure C on *M* one can define *lightlike* (*timelike* and *spacelike*, respectively) vectors being *g*(*v*, *v*) = 0 (*g*(*u*, *u*) < 0 and *g*(*w*, *w*) > 0, respectively) recovering standard notations used in GR.

Finally, we have to focus on particles. Let us first assume that we have one particle through *p* ∈ *M* for any timelike direction and a light ray for any lightlike direction. Some constraint on particles must be set. Originally, EPS resorted to the equivalence principle and special relativity (SR) assuming that particle worldlines are geodesics of some connection Γ˜.

We cannot, for various reasons, be totally satisfied with this assumption, even if we accept of course the result. First of all relativistic theories are more fundamental than SR, which should hence be obtained in some limit from GR rather than being used to define it. Then the equivalence principle is an experimental fact and we would like to keep the possibility to test it rather than assuming it as a must. Luckily enough, one can obtain geodesic equations (together with a better insight on the nature of gravitational field) also without resorting to SR and equivalence principle. In fact, if one assumes that free fall must be described by differential equations of the second order, deterministic, covariant with respect to spacetime diffeomorphisms and with respect to arbitrary reparametrizations of worldlines those candidate equations are strongly constrained; see [11], [12]. If one then defines gravitational interaction to be the one which cannot be cancelled in a way independent of the coordinates and parametrizations then the equation uniquely determined are geodesic equations

$$
\ddot{q}^{\lambda} + \tilde{\Gamma}^{\lambda}\_{\alpha\beta} \dot{q}^{\alpha} \dot{q}^{\beta} = \lambda \dot{q}^{\lambda} \tag{3}
$$

Γ˜-autoparallel trajectories. According to EPS-compatibility condition one can show that a representative Γ˜ ∈ P of the projective structure can be always (and uniquely) chosen so that

Extended Theories of Gravitation and the Curvature of the Universe – Do We Really Need Dark Matter?

where *g* is a representative of the conformal structure *g* ∈ C and the covariant derivative ∇˜

*βµ* + (*gαǫgβµ* <sup>−</sup> <sup>2</sup>*δ<sup>α</sup>*

To summarize, by assuming particles and light rays one can define on spacetime a *EPS structure*, i.e. a triple (*M*, C, P). The conformal structure P describes light cones and it is associated to light rays. Notice that having just a conformal structure one cannot yet define distances (that are not conformally invariant) and this not being a gauge covariant must resort to a *convention* which corresponds to the choice of a representative *g* ∈ C. On the other hand, the projective structure P is associated to free fall so that one can make a canonical gauge fixing by choosing the only representative in the form (6) or, equivalently, the 1-form

The triple (*M*, C, Γ˜) (or, equivalently, the triple (*M*, C, *A*)) is called a *Weyl geometry* on spacetime. This setting is more general than the setting for standard GR where one has just a Lorentzian metric *g* determining both the conformal structure *g* ∈ C and the free fall Γ˜ = {*g*} (i.e. the Levi-Civita connection uniquely associated to *g*). Hence standard GR is a very peculiar case of EPS framework, where there is a gauge fixing of the conformal gauge. Such a fixing is possible iff the covector *<sup>A</sup>* = *<sup>A</sup>µdx<sup>µ</sup>* is exact, i.e. *<sup>A</sup>* = *<sup>d</sup>ϕ*. In this case, there exists a Lorentzian metric *g*˜ ∈ C also determining free fall by Γ˜ = {*g*˜}. When this happens the Weyl geometry (*M*, [*g*˜], {*g*˜}) is called a *metric Weyl geometry*. Notice that this is still more general than standard GR in the sense that the metric determining free fall and light cones is not the original *g* chosen to describe dynamics, but a conformal one *g*˜ ∈ [*g*]. Reverting to standard GR in a sense amounts to choose *ϕ* to be a constant (so that *A* vanishes identically). At this point the reader could argue that in a metric Weyl geometry one could fix the conformal metric *g*˜ at the beginning and use it to describe dynamics, thus obtaining a framework which *exactly* reproduces standard GR. We shall discuss this issue below in greater details. Now we simply notice that this is not the case. The choice of a representative

In fact, since astronomers do measure distances, we do have a protocol (or better a number of protocols) to measure distances. As a matter of fact, such a protocol selects (in a rather obscure way) a precise representative *<sup>g</sup>*′ of the conformal structure which is the one that corresponds to the distances that we measure. If one metric *g* geometrically accounts for a given physical distance measured between two events then obviously no other conformal metric *g*˜ (i.e. no other representative of the same conformal factor) can geometrically account for the same measure (modulo constant conformal factors which can be treated as a definition

(*βδ<sup>ǫ</sup>*

∇˜ *g* = 2*A* ⊗ *g* (5)

*<sup>µ</sup>*))*A<sup>ǫ</sup>* (6)

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117

there exists a covector *<sup>A</sup>* = *<sup>A</sup><sup>µ</sup> dx<sup>µ</sup>* such that

*A*.

of units).

is the one associated to Γ˜; see [14]. Equivalently one has

Γ˜ *α*

*βµ* <sup>=</sup> {*g*}*<sup>α</sup>*

of the conformal structure is, in fact, what allows us to define distances.

for some (global torsionless) connection Γ˜(*x*) and some function *λ*(*s*). In this way free fall is naturally associated to a connection Γ˜ and one is considering the Einstein's lift experiment as showing that *there are* observers who see a gravitational field rather than a *gedanken* experiment showing that there is a class of observers who do not (approximately) observe it.

As is well known, different connections can define the same autoparallel trajectories. In fact the connection

$$
\tilde{\Gamma}^{\prime \mu}\_{\beta \mu} = \tilde{\Gamma}^{a}\_{\beta \mu} + \delta^{a}\_{(\beta} V\_{\mu)} \tag{4}
$$

defines the same geodesic trajectories as Γ˜ *<sup>α</sup> βµ* for any covector *Vµ*. In this case we say that <sup>Γ</sup>˜ and <sup>Γ</sup>˜′ are *projectively equivalent*. Accordingly, free fall corresponds to a projective class P = [Γ˜]; see [13].

Finally, we need (as we said above) a compatibility condition between the conformal class C associated to light cones and the projective class P associated to free fall. This is due by the simple fact that we know that light rays (and hence light cones) feel the gravitational fields as mass particles. Noticing that *g*-lightlike *g*-geodesics are conformally invariant (unlike general *g*-geodesics), we have then to assume that *g*-lightlike *g*-geodesics are a subset of Γ˜-autoparallel trajectories. According to EPS-compatibility condition one can show that a representative Γ˜ ∈ P of the projective structure can be always (and uniquely) chosen so that there exists a covector *<sup>A</sup>* = *<sup>A</sup><sup>µ</sup> dx<sup>µ</sup>* such that

6 Open Questions in Cosmology

used in GR.

equations

the connection

P = [Γ˜]; see [13].

defines the same geodesic trajectories as Γ˜ *<sup>α</sup>*

*ϕ* such that *g*˜*<sup>x</sup>* = *ϕ*(*x*) · *gx*. The construction above shows that one can define out of light rays (i.e. out of the electromagnetic field) a *conformal class* of metrics C = [*g*]. Let us remark that the choice of a representative *g*˜ ∈ C is conventional and in fact part of the specification

Notice that light cones are invariant with respect to conformal transformations; given a conformal structure C on *M* one can define *lightlike* (*timelike* and *spacelike*, respectively) vectors being *g*(*v*, *v*) = 0 (*g*(*u*, *u*) < 0 and *g*(*w*, *w*) > 0, respectively) recovering standard notations

Finally, we have to focus on particles. Let us first assume that we have one particle through *p* ∈ *M* for any timelike direction and a light ray for any lightlike direction. Some constraint on particles must be set. Originally, EPS resorted to the equivalence principle and special relativity (SR) assuming that particle worldlines are geodesics of some connection Γ˜.

We cannot, for various reasons, be totally satisfied with this assumption, even if we accept of course the result. First of all relativistic theories are more fundamental than SR, which should hence be obtained in some limit from GR rather than being used to define it. Then the equivalence principle is an experimental fact and we would like to keep the possibility to test it rather than assuming it as a must. Luckily enough, one can obtain geodesic equations (together with a better insight on the nature of gravitational field) also without resorting to SR and equivalence principle. In fact, if one assumes that free fall must be described by differential equations of the second order, deterministic, covariant with respect to spacetime diffeomorphisms and with respect to arbitrary reparametrizations of worldlines those candidate equations are strongly constrained; see [11], [12]. If one then defines gravitational interaction to be the one which cannot be cancelled in a way independent of the coordinates and parametrizations then the equation uniquely determined are geodesic

for some (global torsionless) connection Γ˜(*x*) and some function *λ*(*s*). In this way free fall is naturally associated to a connection Γ˜ and one is considering the Einstein's lift experiment as showing that *there are* observers who see a gravitational field rather than a *gedanken* experiment showing that there is a class of observers who do not (approximately) observe it. As is well known, different connections can define the same autoparallel trajectories. In fact

*βµ* <sup>+</sup> *<sup>δ</sup><sup>α</sup>*

<sup>Γ</sup>˜ and <sup>Γ</sup>˜′ are *projectively equivalent*. Accordingly, free fall corresponds to a projective class

Finally, we need (as we said above) a compatibility condition between the conformal class C associated to light cones and the projective class P associated to free fall. This is due by the simple fact that we know that light rays (and hence light cones) feel the gravitational fields as mass particles. Noticing that *g*-lightlike *g*-geodesics are conformally invariant (unlike general *g*-geodesics), we have then to assume that *g*-lightlike *g*-geodesics are a subset of

*<sup>λ</sup>* (3)

(*βVµ*) (4)

*βµ* for any covector *Vµ*. In this case we say that

of the observer; conformal transformations are gauge transformations.

*q*¨ *λ* + Γ˜ *λ αβq*˙ *αq*˙ *<sup>β</sup>* = *λq*˙

Γ˜′*α βµ* <sup>=</sup> <sup>Γ</sup>˜ *<sup>α</sup>*

$$
\tilde{\nabla}\mathfrak{g} = \mathfrak{L}A \otimes \mathfrak{g} \tag{5}
$$

where *g* is a representative of the conformal structure *g* ∈ C and the covariant derivative ∇˜ is the one associated to Γ˜; see [14]. Equivalently one has

$$\tilde{\Gamma}^{a}\_{\beta\mu} = \{g\}^{a}\_{\beta\mu} + (g^{a\varepsilon} g\_{\beta\mu} - 2\delta^{\mu}\_{(\beta} \delta^{\varepsilon}\_{\mu)}) A\_{\varepsilon} \tag{6}$$

To summarize, by assuming particles and light rays one can define on spacetime a *EPS structure*, i.e. a triple (*M*, C, P). The conformal structure P describes light cones and it is associated to light rays. Notice that having just a conformal structure one cannot yet define distances (that are not conformally invariant) and this not being a gauge covariant must resort to a *convention* which corresponds to the choice of a representative *g* ∈ C. On the other hand, the projective structure P is associated to free fall so that one can make a canonical gauge fixing by choosing the only representative in the form (6) or, equivalently, the 1-form *A*.

The triple (*M*, C, Γ˜) (or, equivalently, the triple (*M*, C, *A*)) is called a *Weyl geometry* on spacetime. This setting is more general than the setting for standard GR where one has just a Lorentzian metric *g* determining both the conformal structure *g* ∈ C and the free fall Γ˜ = {*g*} (i.e. the Levi-Civita connection uniquely associated to *g*). Hence standard GR is a very peculiar case of EPS framework, where there is a gauge fixing of the conformal gauge. Such a fixing is possible iff the covector *<sup>A</sup>* = *<sup>A</sup>µdx<sup>µ</sup>* is exact, i.e. *<sup>A</sup>* = *<sup>d</sup>ϕ*. In this case, there exists a Lorentzian metric *g*˜ ∈ C also determining free fall by Γ˜ = {*g*˜}. When this happens the Weyl geometry (*M*, [*g*˜], {*g*˜}) is called a *metric Weyl geometry*. Notice that this is still more general than standard GR in the sense that the metric determining free fall and light cones is not the original *g* chosen to describe dynamics, but a conformal one *g*˜ ∈ [*g*]. Reverting to standard GR in a sense amounts to choose *ϕ* to be a constant (so that *A* vanishes identically).

At this point the reader could argue that in a metric Weyl geometry one could fix the conformal metric *g*˜ at the beginning and use it to describe dynamics, thus obtaining a framework which *exactly* reproduces standard GR. We shall discuss this issue below in greater details. Now we simply notice that this is not the case. The choice of a representative of the conformal structure is, in fact, what allows us to define distances.

In fact, since astronomers do measure distances, we do have a protocol (or better a number of protocols) to measure distances. As a matter of fact, such a protocol selects (in a rather obscure way) a precise representative *<sup>g</sup>*′ of the conformal structure which is the one that corresponds to the distances that we measure. If one metric *g* geometrically accounts for a given physical distance measured between two events then obviously no other conformal metric *g*˜ (i.e. no other representative of the same conformal factor) can geometrically account for the same measure (modulo constant conformal factors which can be treated as a definition of units).

In standard GR one assumes that such a representative *<sup>g</sup>*′ *also* determines light cones and free fall. In metric Weyl geometries there is nothing ensuring that the canonical representative *g*˜ also gives us the measured distances, that as far as we can see could as well be related to any other conformally equivalent metric *g*. Fixing the metric that we use to calculate distances is, in the end, a choice that we can do only *a posteriori*, on the basis of observations.

distances were defined using *g* (which by the way is the only way to select a conformal gauge to write a non conformally invariant dynamics) the new model is only *similar* to standard GR but inequivalent as far as distances are concerned. If the original theory is recognized to be inequivalent to standard GR based on the Hilbert-Einstein-Palatini Lagrangian *<sup>L</sup>* <sup>=</sup> <sup>√</sup>*ggµνRµν*(Γ) *ds*, let us remark that as a matter of fact dark sources are precisely related to

Extended Theories of Gravitation and the Curvature of the Universe – Do We Really Need Dark Matter?

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119

Now let us suppose for the sake of argument that standard GR is still a perfect theory to describe the actual universe (something that we know to be strongly questioned by actual observations). Still we believe that analyzing it within a wider framework as the one of EPS structures and Weyl geometries is in any case useful if not even necessary. If we can understand observations in this wider framework, in fact, we can better test gravity and maybe eventually show that standard GR is compatible with observations. If we assume standard GR setting and we build observational protocols for it then it may become difficult to understand which data come from assumptions and which data come instead from real physical facts, especially in a theory of gravitation in which we clearly made exceptions about

Having said that, we see now that EPS formalism points to a Weyl geometry on spacetime in which one has a conformal structure C defining light cones and a compatible (torsionless) connection Γ˜ defining free fall. Of course, whenever interaction with matter of half-integer spin is considered nothing prevents from relaxing the symmetry requirements on connections. However, until only test particles are considered matter is unaffected by torsion and one can drop it from the beginning. Then our protocols for measuring distances select a representative *g* ∈ C for the conformal structure. In particular there is no reason why one should assume *a priori* that the connection Γ˜ is metric or, if such, that it is metric for the

Accordingly, one can use the kinematic and interpretation suggested by EPS to constrain dynamics. In a Palatini or *metric-affine* formalism the metric and connection are completely unrelated *a priori*, so that only dynamics may give their reciprocal relations. Then field equations may force a relation between the metric and the connection. That is exactly what happens in vacuum standard Palatini GR: field equations force the connection to be the Levi-Civita connection of the given metric. The same happens with some specific kind of matter, but for general matter such a feature is generally lost, and in general the connection

However, EPS analysis shows that the connection and the metric cannot be completely arbitrary if one wants a theory that fits fundamental principles; in fact there must be a covector *A<sup>µ</sup>* for which (6) holds true. If the compatibility condition is not already imposed at the kinematical level —for example writing the theory for the fundamental fields (*gµν*, *<sup>A</sup>µ*)

We shall thence call *extended theory of gravitation* any field theory for independent variables

In these models the geometry of spacetime is described by a Weyl geometry.

*βµ*) in which field equations imply the compatibility condition (6) as a consequence.

*βµ*)— then the only option is that field equations impose the compatibility

mismatches in observed distances. . . !

gauge invariant observables.

instead of (*gµν*, <sup>Γ</sup>˜ *<sup>α</sup>*

(*gµν*, <sup>Γ</sup>˜ *<sup>α</sup>*

same metric one happens to have selected for distances.

cannot be the one associated to the original metric.

conditions *a posteriori* as a consequence of field equations.

At a fundamental level one can either decide to be strict on the interpretation of conformal gauge symmetry (and accordingly quantities that are not gauge invariant, such as distances, cannot be really observable) or one accepts conventional gauge fixing to define such quantities as observable, thus restricting symmetries of the system to the conformal transformations which preserves these gauge fixing. In the first case standard GR is equivalent to metric Weyl geometry (though we cannot measure distances) or, in the second case, we define distances but standard GR is not necessarily equivalent to metric Weyl geometries. Again, deciding which is the metric that really enters observational protocols is something that should not be imposed *a priori* but rather something to be tested locally.

## **3. Extended theories of gravitation**

EPS analysis sets a number of constraints to any theoretical framework that can be called a *reasonable theory of gravitation*. Such constraints are much weaker than the strong metricity assumptions done in standard GR.

Before explicitly analyzing these constraints, let us first discuss about the interpretation of a relativistic theory. One usually chooses fundamental fields in kinematics and then considers dynamics. In gauge theories these two levels are usually quite disconnected since one is free to change fundamental field variables; this induces a change of dynamics (which is in fact assumed to be gauge covariant) and it does not affect observable quantities (which are also assumed to be gauge invariant).

However, the situation in gravitational theories is quite different. In relativistic theories, as we already discussed in the introduction, there are no known non-trivial gauge-invariant quantities. If we want to retain some connection with astrophysics and cosmology we are forced to assume as a fact that matter allows some conventional (partial) gauge fixing. Strictly speaking observables are not gauge invariant and accordingly they are not preserved under changes of fundamental field coordinates. When discussing the equivalence between different formalisms one must additionally declare how observational and measuring protocols are modified by the transformations allowed and/or chosen. For example, let us consider a metric Weyl theory in which the dynamics is described in terms of a metric *g* and a connection Γ˜. When a solution of field equations is found then one can determine both light cones and free fall by a single conformal representative *g*˜. Of course one can rewrite the dynamics in terms of *g*˜ only. Is this metric theory fully equivalent to GR, especially in presence of matter?

To better understand this apparently trivial question, let us recall that changing a metric *g* to a conformally equivalent one *g*˜ = *ϕ* · *g* will not change electromagnetism but will certainly change the coupling with non-electromegnetic matter (e.g., a cosmological fluid).

We cannot answer this simple question, before considering which metric is used to define distances. In standard GR, one makes the *a priori* (unjustified) ansatz that distances would be defined by the same metric which defines free fall, i.e. in this case *g*˜. If in the original model distances were defined using *g* (which by the way is the only way to select a conformal gauge to write a non conformally invariant dynamics) the new model is only *similar* to standard GR but inequivalent as far as distances are concerned. If the original theory is recognized to be inequivalent to standard GR based on the Hilbert-Einstein-Palatini Lagrangian *<sup>L</sup>* <sup>=</sup> <sup>√</sup>*ggµνRµν*(Γ) *ds*, let us remark that as a matter of fact dark sources are precisely related to mismatches in observed distances. . . !

8 Open Questions in Cosmology

**3. Extended theories of gravitation**

assumptions done in standard GR.

assumed to be gauge invariant).

presence of matter?

In standard GR one assumes that such a representative *<sup>g</sup>*′ *also* determines light cones and free fall. In metric Weyl geometries there is nothing ensuring that the canonical representative *g*˜ also gives us the measured distances, that as far as we can see could as well be related to any other conformally equivalent metric *g*. Fixing the metric that we use to calculate distances is,

At a fundamental level one can either decide to be strict on the interpretation of conformal gauge symmetry (and accordingly quantities that are not gauge invariant, such as distances, cannot be really observable) or one accepts conventional gauge fixing to define such quantities as observable, thus restricting symmetries of the system to the conformal transformations which preserves these gauge fixing. In the first case standard GR is equivalent to metric Weyl geometry (though we cannot measure distances) or, in the second case, we define distances but standard GR is not necessarily equivalent to metric Weyl geometries. Again, deciding which is the metric that really enters observational protocols is something that should not be imposed *a priori* but rather something to be tested locally.

EPS analysis sets a number of constraints to any theoretical framework that can be called a *reasonable theory of gravitation*. Such constraints are much weaker than the strong metricity

Before explicitly analyzing these constraints, let us first discuss about the interpretation of a relativistic theory. One usually chooses fundamental fields in kinematics and then considers dynamics. In gauge theories these two levels are usually quite disconnected since one is free to change fundamental field variables; this induces a change of dynamics (which is in fact assumed to be gauge covariant) and it does not affect observable quantities (which are also

However, the situation in gravitational theories is quite different. In relativistic theories, as we already discussed in the introduction, there are no known non-trivial gauge-invariant quantities. If we want to retain some connection with astrophysics and cosmology we are forced to assume as a fact that matter allows some conventional (partial) gauge fixing. Strictly speaking observables are not gauge invariant and accordingly they are not preserved under changes of fundamental field coordinates. When discussing the equivalence between different formalisms one must additionally declare how observational and measuring protocols are modified by the transformations allowed and/or chosen. For example, let us consider a metric Weyl theory in which the dynamics is described in terms of a metric *g* and a connection Γ˜. When a solution of field equations is found then one can determine both light cones and free fall by a single conformal representative *g*˜. Of course one can rewrite the dynamics in terms of *g*˜ only. Is this metric theory fully equivalent to GR, especially in

To better understand this apparently trivial question, let us recall that changing a metric *g* to a conformally equivalent one *g*˜ = *ϕ* · *g* will not change electromagnetism but will certainly

We cannot answer this simple question, before considering which metric is used to define distances. In standard GR, one makes the *a priori* (unjustified) ansatz that distances would be defined by the same metric which defines free fall, i.e. in this case *g*˜. If in the original model

change the coupling with non-electromegnetic matter (e.g., a cosmological fluid).

in the end, a choice that we can do only *a posteriori*, on the basis of observations.

Now let us suppose for the sake of argument that standard GR is still a perfect theory to describe the actual universe (something that we know to be strongly questioned by actual observations). Still we believe that analyzing it within a wider framework as the one of EPS structures and Weyl geometries is in any case useful if not even necessary. If we can understand observations in this wider framework, in fact, we can better test gravity and maybe eventually show that standard GR is compatible with observations. If we assume standard GR setting and we build observational protocols for it then it may become difficult to understand which data come from assumptions and which data come instead from real physical facts, especially in a theory of gravitation in which we clearly made exceptions about gauge invariant observables.

Having said that, we see now that EPS formalism points to a Weyl geometry on spacetime in which one has a conformal structure C defining light cones and a compatible (torsionless) connection Γ˜ defining free fall. Of course, whenever interaction with matter of half-integer spin is considered nothing prevents from relaxing the symmetry requirements on connections. However, until only test particles are considered matter is unaffected by torsion and one can drop it from the beginning. Then our protocols for measuring distances select a representative *g* ∈ C for the conformal structure. In particular there is no reason why one should assume *a priori* that the connection Γ˜ is metric or, if such, that it is metric for the same metric one happens to have selected for distances.

Accordingly, one can use the kinematic and interpretation suggested by EPS to constrain dynamics. In a Palatini or *metric-affine* formalism the metric and connection are completely unrelated *a priori*, so that only dynamics may give their reciprocal relations. Then field equations may force a relation between the metric and the connection. That is exactly what happens in vacuum standard Palatini GR: field equations force the connection to be the Levi-Civita connection of the given metric. The same happens with some specific kind of matter, but for general matter such a feature is generally lost, and in general the connection cannot be the one associated to the original metric.

However, EPS analysis shows that the connection and the metric cannot be completely arbitrary if one wants a theory that fits fundamental principles; in fact there must be a covector *A<sup>µ</sup>* for which (6) holds true. If the compatibility condition is not already imposed at the kinematical level —for example writing the theory for the fundamental fields (*gµν*, *<sup>A</sup>µ*) instead of (*gµν*, <sup>Γ</sup>˜ *<sup>α</sup> βµ*)— then the only option is that field equations impose the compatibility conditions *a posteriori* as a consequence of field equations.

We shall thence call *extended theory of gravitation* any field theory for independent variables (*gµν*, <sup>Γ</sup>˜ *<sup>α</sup> βµ*) in which field equations imply the compatibility condition (6) as a consequence. In these models the geometry of spacetime is described by a Weyl geometry.

Let us call *extended metric theory of gravitation* any extended theory of gravitation in which field equations imply dynamically that the connection is a metric connection, so that in that case the geometry of spacetime is described by a *metric* Weyl geometry.

No relevant progress in this direction is still at hands, although it corresponds to an even more physically reasonable situation; only few concrete examples have been worked out

Extended Theories of Gravitation and the Curvature of the Universe – Do We Really Need Dark Matter?

Let us remark that *a priori* the Ricci tensor *<sup>R</sup>*˜ *µν* of the connection <sup>Γ</sup>˜ is not necessarily symmetric since the connection is not necessarily metric. As we said the matter Lagrangian *Lm* is here assumed to depend only on matter and metric (together with their derivatives up to order 1). Thus if one needs covariant derivatives of matter fields they are explicitly defined with respect to the metric field. Requiring that the matter Lagrangian does not depend on the connection Γ˜ is a standard requirement to simplify the analysis of field equation below although (as we said above) it would correspond to more reasonable physical situations. Let us here notice that what follows can be in fact extended to a more general framework; there are in fact matter Lagrangians depending on the connection Γ˜ in which field equations still

imply the EPS-compatibility condition (6); see [15], [16], [17].

*<sup>f</sup>* ′

∇˜ *α*

(R)*R*˜

Notice that *Tµν* depends both on the matter fields and on the metric *g*.

(R)) <sup>2</sup>

<sup>√</sup>*g f* ′

*g*˜*g*˜

us consider in fact a conformal transformation *<sup>g</sup>*˜*µν* = (*<sup>f</sup>* ′

Thus the second field equation in (9) can be recast as

∇˜ *α*

*g*˜ *µν* = (*<sup>f</sup>* ′

which by the Levi-Civita theorem implies

<sup>√</sup>*g f* ′

(*µν*) <sup>−</sup> <sup>1</sup>

(R)*gβµ*

We do not write the matter field equations (which will be considered as matter equations of state). The constant *κ* = 8*πG*/*c*<sup>4</sup> is the coupling constant between matter and gravity. The

of the connection <sup>Γ</sup>˜. The first stress tensor *<sup>T</sup>µν* arises since the matter Lagrangian is a function

<sup>√</sup>*gTµν* <sup>=</sup> *<sup>δ</sup>Lm*

Under these simplifying assumptions the second field equations can be solved explicitly. Let

<sup>2</sup>−*<sup>m</sup>* · *<sup>g</sup>µν <sup>g</sup>*˜ = (*<sup>f</sup>* ′

= ∇˜ *<sup>α</sup>*

*βµ* <sup>=</sup> {*g*˜}*<sup>α</sup>*

*g*˜*g*˜

*βµ*

*βµ* <sup>=</sup> <sup>√</sup>*g f* ′

(R)*gβµ*

Γ˜ *α*

<sup>2</sup> *<sup>f</sup>*(R)*gµν* <sup>=</sup> *<sup>κ</sup>Tµν*

= *Tβµ*

*<sup>α</sup>* vanishes since the matter Lagrangian is assumed to be independent

(R)) <sup>2</sup>

(R)) *<sup>m</sup> m*−2

*<sup>α</sup>* <sup>=</sup> <sup>0</sup> (9)

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121

*<sup>δ</sup>gµν* (10)

(R) · *gβµ* (12)

*βµ* (14)

*<sup>m</sup>*−<sup>2</sup> · *<sup>g</sup>µν* (with *<sup>m</sup>* > 2). One has

= 0 (13)

<sup>√</sup>*<sup>g</sup>* (11)

insofar; see [15], [22], [23].

Field equations of (7) are

second stress tensor *Tβµ*

of the metric

and then

We know a class of dynamics which are in fact extended metric theories of gravitation. As is well known, any Palatini *f*(R)-theory is in fact an extended metric theory of gravitation; see [8], [15], [19], [20], [21]. Standard GR is a specific extended metric theory of gravitation in which field equations imply that *<sup>A</sup><sup>µ</sup>* = 0 (and then <sup>Γ</sup> = {*g*}).

Of course it is well known that general Weyl geometries may have unpleasant holonomy problems in the definition of length (namely, the length of a ruler depends on the path). However, metric Weyl geometries are not affected by these problems and they are still more general than standard GR as we shall discuss hereafter for *f*(R)-models.

With such theories rulers cannot change length when parallel transported, although one has to be careful to notice that the metric scales can change point by point because of conformal rescaling.

## **3.1. Palatini** *f*(R)**-theories**

In order to fix notation let us briefly review a generic *f*(R)-theory with matter.

Let us consider a Lorentzian metric *<sup>g</sup>µν* and a torsionless connection <sup>Γ</sup>˜ *<sup>α</sup> βµ* on spacetime *M* of dimension *m* > 2. The conformal class [*g*] = C of metrics defines light cones. The representative *g* ∈ C is chosen to define distances. The connection Γ˜ is associated to free fall and it is chosen to be torsionless since geodesic equations is insensitive to torsion.

Let us remark that in this context conformal transformations are defined to be *g*˜(*x*) = *ϕ*(*x*) · *g*(*x*) and they leave the connection unchanged. One is *forced* to leave the connection unchanged (as it is possible in Palatini formalism) since our connection Γ˜ is uniquely selected to describe free fall (and by the projective gauge fixing ∇*g* = 2*A* ⊗ *g*). One could say that this definition of conformal transformations preserves the interpretation of fields.

Let us restrict our analysis to dynamics induced by a Lagrangian in the form

$$L = \sqrt{\mathfrak{g}}f(\mathcal{R}) + L\_m(\mathfrak{g}, \mathfrak{g}) \tag{7}$$

where *f* is a generic (analytic or *sufficiently regular*) function, *φ* is a collection of matter fields and we set R := *<sup>g</sup>µνR*˜ *µν*.

With this choice we are implicitly assuming that matter fields *ψ* minimally couple with the metric *g* which in turn encodes electromagnetic properties (photons and light cones). Since gravity, according to EPS formalism, is mostly inherent with the equivalence principle and free fall, that is encoded in the projective structure P, one should better assume that matter couples *also* with Γ˜ and investigate the more general case in which the matter Lagrangian has the form *Lm*(*ψ*, *g*, Γ˜). However, this case in much harder to be investigated since it entails that a second stress tensor is generated by the variational derivative

$$
\sqrt{\mathcal{g}}T^{\mu\nu}\_{\mu} = \frac{\delta L\_m}{\delta \tilde{\Gamma}^a\_{\mu\nu}}\tag{8}
$$

No relevant progress in this direction is still at hands, although it corresponds to an even more physically reasonable situation; only few concrete examples have been worked out insofar; see [15], [22], [23].

Let us remark that *a priori* the Ricci tensor *<sup>R</sup>*˜ *µν* of the connection <sup>Γ</sup>˜ is not necessarily symmetric since the connection is not necessarily metric. As we said the matter Lagrangian *Lm* is here assumed to depend only on matter and metric (together with their derivatives up to order 1). Thus if one needs covariant derivatives of matter fields they are explicitly defined with respect to the metric field. Requiring that the matter Lagrangian does not depend on the connection Γ˜ is a standard requirement to simplify the analysis of field equation below although (as we said above) it would correspond to more reasonable physical situations. Let us here notice that what follows can be in fact extended to a more general framework; there are in fact matter Lagrangians depending on the connection Γ˜ in which field equations still imply the EPS-compatibility condition (6); see [15], [16], [17].

Field equations of (7) are

10 Open Questions in Cosmology

rescaling.

**3.1. Palatini** *f*(R)**-theories**

and we set R := *<sup>g</sup>µνR*˜ *µν*.

Let us call *extended metric theory of gravitation* any extended theory of gravitation in which field equations imply dynamically that the connection is a metric connection, so that in that

We know a class of dynamics which are in fact extended metric theories of gravitation. As is well known, any Palatini *f*(R)-theory is in fact an extended metric theory of gravitation; see [8], [15], [19], [20], [21]. Standard GR is a specific extended metric theory of gravitation

Of course it is well known that general Weyl geometries may have unpleasant holonomy problems in the definition of length (namely, the length of a ruler depends on the path). However, metric Weyl geometries are not affected by these problems and they are still more

With such theories rulers cannot change length when parallel transported, although one has to be careful to notice that the metric scales can change point by point because of conformal

of dimension *m* > 2. The conformal class [*g*] = C of metrics defines light cones. The representative *g* ∈ C is chosen to define distances. The connection Γ˜ is associated to free fall

Let us remark that in this context conformal transformations are defined to be *g*˜(*x*) = *ϕ*(*x*) · *g*(*x*) and they leave the connection unchanged. One is *forced* to leave the connection unchanged (as it is possible in Palatini formalism) since our connection Γ˜ is uniquely selected to describe free fall (and by the projective gauge fixing ∇*g* = 2*A* ⊗ *g*). One could say that

where *f* is a generic (analytic or *sufficiently regular*) function, *φ* is a collection of matter fields

With this choice we are implicitly assuming that matter fields *ψ* minimally couple with the metric *g* which in turn encodes electromagnetic properties (photons and light cones). Since gravity, according to EPS formalism, is mostly inherent with the equivalence principle and free fall, that is encoded in the projective structure P, one should better assume that matter couples *also* with Γ˜ and investigate the more general case in which the matter Lagrangian has the form *Lm*(*ψ*, *g*, Γ˜). However, this case in much harder to be investigated since it entails

<sup>√</sup>*gTµν*

*<sup>α</sup>* <sup>=</sup> *<sup>δ</sup>Lm δ*Γ˜ *<sup>α</sup> µν*

*<sup>L</sup>* <sup>=</sup> <sup>√</sup>*g f*(R) + *Lm*(*φ*, *<sup>g</sup>*) (7)

*βµ* on spacetime *M*

(8)

case the geometry of spacetime is described by a *metric* Weyl geometry.

general than standard GR as we shall discuss hereafter for *f*(R)-models.

In order to fix notation let us briefly review a generic *f*(R)-theory with matter.

and it is chosen to be torsionless since geodesic equations is insensitive to torsion.

this definition of conformal transformations preserves the interpretation of fields.

Let us restrict our analysis to dynamics induced by a Lagrangian in the form

that a second stress tensor is generated by the variational derivative

Let us consider a Lorentzian metric *<sup>g</sup>µν* and a torsionless connection <sup>Γ</sup>˜ *<sup>α</sup>*

in which field equations imply that *<sup>A</sup><sup>µ</sup>* = 0 (and then <sup>Γ</sup> = {*g*}).

$$\begin{cases} f'(\mathcal{R})\tilde{R}\_{(\mu\nu)} - \frac{1}{2}f(\mathcal{R})g\_{\mu\nu} = \kappa T\_{\mu\nu} \\ \tilde{\nabla}\_{\alpha} \left( \sqrt{g} f'(\mathcal{R})g^{\beta\mu} \right) = T\_{\alpha}^{\beta\mu} = 0 \end{cases} \tag{9}$$

We do not write the matter field equations (which will be considered as matter equations of state). The constant *κ* = 8*πG*/*c*<sup>4</sup> is the coupling constant between matter and gravity. The second stress tensor *Tβµ <sup>α</sup>* vanishes since the matter Lagrangian is assumed to be independent of the connection <sup>Γ</sup>˜. The first stress tensor *<sup>T</sup>µν* arises since the matter Lagrangian is a function of the metric

$$
\sqrt{\mathfrak{g}}T\_{\mu\nu} = \frac{\delta L\_m}{\delta g^{\mu\nu}}\tag{10}
$$

Notice that *Tµν* depends both on the matter fields and on the metric *g*.

Under these simplifying assumptions the second field equations can be solved explicitly. Let us consider in fact a conformal transformation *<sup>g</sup>*˜*µν* = (*<sup>f</sup>* ′ (R)) <sup>2</sup> *<sup>m</sup>*−<sup>2</sup> · *<sup>g</sup>µν* (with *<sup>m</sup>* > 2). One has

$$\overline{\mathfrak{g}}^{\mu\nu} = (f'(\mathcal{R}))^{\frac{2}{2-n}} \cdot \mathfrak{g}^{\mu\nu} \qquad \qquad \sqrt{\overline{\mathfrak{g}}} = (f'(\mathcal{R}))^{\frac{n}{n-2}} \sqrt{\mathcal{g}} \tag{11}$$

and then

$$
\sqrt{\xi}\tilde{\mathfrak{g}}^{\beta\mu} = \sqrt{\mathfrak{g}}f'(\mathcal{R}) \cdot \mathcal{g}^{\beta\mu} \tag{12}
$$

Thus the second field equation in (9) can be recast as

$$\left(\nabla\_{\mathfrak{a}}\left(\sqrt{\mathfrak{g}}f'(\mathcal{R})g^{\mathfrak{f}\mu}\right) = \nabla\_{\mathfrak{a}}\left(\sqrt{\mathfrak{g}}\mathfrak{g}^{\mathfrak{f}\mu}\right) = 0\tag{13}$$

which by the Levi-Civita theorem implies

$$
\tilde{\Gamma}^{a}\_{\beta\mu} = \{ \tilde{\mathfrak{g}} \}^{a}\_{\beta\mu} \tag{14}
$$

i.e. the connection Γ˜ is the Levi-Civita connection of the conformal metric *g*˜. Thus in these theories the connection is *a posteriori* metric and the geometry of spacetime is described by a metric Weyl geometry. As a consequence the Ricci tensor *R*˜ *µν* is symmetric being the Ricci tensor of the metric *g*˜.

The first field equation now reads as

$$f'(\mathcal{R})\tilde{\mathcal{R}}\_{\mu\nu} - \frac{1}{2}f(\mathcal{R})g\_{\mu\nu} = \kappa T\_{\mu\nu} \tag{15}$$

are defined by *<sup>g</sup>* in both cases); however, the conformal factor *<sup>ϕ</sup>* = (*<sup>f</sup>* ′

� 1 *<sup>f</sup>* ′(*r*(*T*)) �

*<sup>G</sup>*˜*µν* <sup>=</sup> *<sup>R</sup>*˜ *µν* <sup>−</sup> <sup>1</sup>

conformal metric *<sup>g</sup>*˜ = *<sup>f</sup>* ′

effect induced by *<sup>T</sup>* = *<sup>g</sup>µνTµν* �= 0.

**3.2. Equivalence with Brans-Dicke theories**

hereafter restrict to the case in dimension *m* = 4.

described by a Lagrangian in the following form

Field equations for such a theory are

*<sup>ϕ</sup>*<sup>2</sup> <sup>∇</sup>*αϕ*∇*αϕ* <sup>−</sup> <sup>2</sup> *<sup>ω</sup>*

*<sup>ϕ</sup>Rµν* = ∇*µνϕ* +

*LBD* <sup>=</sup> <sup>√</sup>*<sup>g</sup>*

� *<sup>ϕ</sup><sup>R</sup>* <sup>−</sup> *<sup>ω</sup> ϕ*

where *ω* is a real parameter and *U*(*ϕ*) is a potential function.

<sup>2</sup>*Rgµν*) = *<sup>κ</sup>Tµν* <sup>+</sup> <sup>∇</sup>*µνϕ* <sup>+</sup> *<sup>ϕ</sup>gµν* <sup>+</sup> *<sup>ω</sup>*

(*ϕ*)

the original metric *<sup>g</sup>µν* <sup>=</sup> *<sup>ϕ</sup>*−<sup>1</sup> · *<sup>g</sup>*˜*µν* and the conformal factor *<sup>ϕ</sup>* <sup>=</sup> *<sup>f</sup>* ′

*<sup>ϕ</sup>gµν* <sup>−</sup> <sup>3</sup>

2*ϕ*

*<sup>ϕ</sup> <sup>ϕ</sup>* <sup>−</sup> *<sup>U</sup>*′

1 2

data.

 

as

*<sup>ϕ</sup>*(*Rµν* − <sup>1</sup>

*R* = *<sup>ω</sup>*

2

*<sup>R</sup>*˜ *<sup>g</sup>*˜*µν* = *<sup>κ</sup>*

it does not affect geodesics and it can be compensated by a change of units.

However, when *real matter* is present the situation is completely different. In this more general case, we have that R = *r*(*T*) depends on *x* ∈ *M*. The first field equation becomes then

*<sup>T</sup>µν* <sup>−</sup> <sup>1</sup>

so that a Palatini *f*(R)-theory with *real* matter behaves like standard GR with a strongly modified source stress tensor. Naively speaking, one can reasonably hope that the modifications dictated by the choice of the function *f* can be chosen to fit observational

In a sense, whenever *T* �= 0 the presence of standard visible matter *ψ* (assumed to generate, through the matter Lagrangian *Lm*(*g*, *<sup>ψ</sup>*), an energy momentum stress tensor *<sup>T</sup>µν*) would produce by gravitational interaction with Γ˜ (i.e. with the Levi-Civita connection of the

standard matter *ψ* is seen to exist together with *dark (virtual)* matter generated by the gauging of the rulers imposed by the *T*-dependent conformal transformations on *g*. In a sense, the *dark side* of Einstein equations can be mimicked by suitably choosing *f* and *Lm*, as a curvature

Let us hereafter briefly review the mathematical equivalence between Palatini *f*(R)-theories and Brans-Dicke theories and discuss about how physical is such an equivalence. Let us

A *Brans-Dicke* theory is a theory for a metric *gµν* and a scalar field *ϕ*. The dynamics is

∇*µϕ*∇*µϕ* + *<sup>U</sup>*(*ϕ*)

*ϕ* �

1

(21) If one considers now the field equation (19) for a Palatini *<sup>f</sup>*(R)-theory and writes them for

∇*µϕ*∇*νϕ* +

�

∇*µϕ*∇*νϕ* − <sup>1</sup>

<sup>4</sup> *<sup>ϕ</sup>r*ˆ(*T*)*gµν* <sup>+</sup> *<sup>κ</sup>*

*<sup>m</sup> Tgµν*�

Extended Theories of Gravitation and the Curvature of the Universe – Do We Really Need Dark Matter?

(*T*) · *g*) a kind *effective* energy momentum stress tensor *T*˜

+ 2 − *m* 2*m*

(*ρ*)) <sup>2</sup>

*<sup>r</sup>*ˆ(*T*)*gµν*�

<sup>2</sup>−*<sup>m</sup>* is constant and

*µν* (19)

123

*µν* in which

= *κT*˜

http://dx.doi.org/10.5772/52041

+ *Lm*(*g*, *ψ*) (20)

+ 1 <sup>2</sup>*Ugµν*

(R) field equation reads

(22)

<sup>2</sup> <sup>∇</sup>*αϕ*∇*αϕgµν*�

�

*<sup>T</sup>µν* <sup>−</sup> <sup>1</sup> 4 *Tgµν*�

The trace of this equation (with respect to *gµν*) is so important in the analysis of these models that it is called the *master equation*. It reads

$$f'(\mathcal{R})\mathcal{R} - \frac{m}{2}f(\mathcal{R}) = \mathfrak{x}T := \mathfrak{x}g^{\mu\nu}T\_{\mu\nu} \tag{16}$$

For any given (analytic) function *f* , the master equation is an *algebraic* (i.e. not differential) equation between R and *T*. Assuming that *m* �= 2 and excluding the degenerate case in which the following holds

$$f''(\mathcal{R})\mathcal{R} + \frac{2-m}{2}f'(\mathcal{R}) = 0 \quad \Rightarrow f'(\mathcal{R}) = \frac{m}{2}\mathbb{C}\_1\mathcal{R}^{\frac{m-2}{2}} \quad \Rightarrow f(\mathcal{R}) = \mathbb{C}\_1\mathcal{R}^{\frac{m}{2}} + \mathbb{C}\_2\tag{17}$$

we see that the function *<sup>F</sup>*(R, *<sup>T</sup>*) := *<sup>f</sup>* ′ (R)R − *m* <sup>2</sup> *<sup>f</sup>*(R) <sup>−</sup> *<sup>κ</sup><sup>T</sup>* is also analytic and can be generically (i.e. except a discrete set of values for R) solved for R = *r*(*T*) = *κr*ˆ(*T*).

In vacuum or for purely electromagnetic matter obeying Maxwell equations, one has *<sup>T</sup>* = 0, i.e. the trace *<sup>T</sup>* of *<sup>T</sup>µν* is zero and R takes a constant value from a discrete set R ≡ *<sup>ρ</sup>* ∈ {*ρ*0, *<sup>ρ</sup>*1,... } that of course depends on *<sup>f</sup>* . In this vacuum (as well as in purely electromagnetic) case the field equations simplify to

$$\tilde{\mathbf{G}}\_{\mu\nu} = \tilde{\mathbf{R}}\_{\mu\nu} - \frac{1}{2} \tilde{\mathbf{R}} \tilde{\mathbf{g}}\_{\mu\nu} = \left( \frac{2 - m}{2m} (f'(\rho))^{\frac{2}{2 - n}} \rho \right) \tilde{\mathbf{g}}\_{\mu\nu} = \Lambda(\rho) \tilde{\mathbf{g}}\_{\mu\nu} \tag{18}$$

Accordingly, vacuum (or purely electromagnetic) Palatini *f*(R)-theories are generically equivalent to Einstein models with cosmological constant and the possible value of the cosmological constant is chosen in a discrete set which depends on the function *f* . This is known as *universality theorem* for Einstein equations (see [18]). The meaning of this result is not to be overestimated; the equivalence is important but one has a huge freedom in choosing the function *f* (which depends on countable infinite parameters) so that any value for the cosmological constant can be in principle attained. Let us stress once more that this includes all cases in which matter is present but the trace *T* = 0 as it happens for the electromagnetic field.

Accordingly, the physics described by Palatini *f*(R)-theories in vacuum is not richer than standard GR physics with cosmological constant. Still one should notice that in these vacuum *f*(R)-theories free fall is given by *g*˜ while in standard GR it is given by *g* (while distances are defined by *<sup>g</sup>* in both cases); however, the conformal factor *<sup>ϕ</sup>* = (*<sup>f</sup>* ′ (*ρ*)) <sup>2</sup> <sup>2</sup>−*<sup>m</sup>* is constant and it does not affect geodesics and it can be compensated by a change of units.

However, when *real matter* is present the situation is completely different. In this more general case, we have that R = *r*(*T*) depends on *x* ∈ *M*. The first field equation becomes then

$$\tilde{\mathbf{G}}\_{\mu\nu} = \tilde{\mathbf{R}}\_{\mu\nu} - \frac{1}{2} \tilde{\mathbf{R}} \tilde{\mathbf{g}}\_{\mu\nu} = \mathbf{x} \left( \frac{1}{f'(r(T))} \left( T\_{\mu\nu} - \frac{1}{m} T g\_{\mu\nu} \right) + \frac{2-m}{2m} \mathbf{\hat{r}}(T) g\_{\mu\nu} \right) = \mathbf{x} \tilde{T}\_{\mu\nu} \tag{19}$$

so that a Palatini *f*(R)-theory with *real* matter behaves like standard GR with a strongly modified source stress tensor. Naively speaking, one can reasonably hope that the modifications dictated by the choice of the function *f* can be chosen to fit observational data.

In a sense, whenever *T* �= 0 the presence of standard visible matter *ψ* (assumed to generate, through the matter Lagrangian *Lm*(*g*, *<sup>ψ</sup>*), an energy momentum stress tensor *<sup>T</sup>µν*) would produce by gravitational interaction with Γ˜ (i.e. with the Levi-Civita connection of the conformal metric *<sup>g</sup>*˜ = *<sup>f</sup>* ′ (*T*) · *g*) a kind *effective* energy momentum stress tensor *T*˜ *µν* in which standard matter *ψ* is seen to exist together with *dark (virtual)* matter generated by the gauging of the rulers imposed by the *T*-dependent conformal transformations on *g*. In a sense, the *dark side* of Einstein equations can be mimicked by suitably choosing *f* and *Lm*, as a curvature effect induced by *<sup>T</sup>* = *<sup>g</sup>µνTµν* �= 0.

## **3.2. Equivalence with Brans-Dicke theories**

12 Open Questions in Cosmology

tensor of the metric *g*˜.

which the following holds

*<sup>f</sup>* ′′(R)<sup>R</sup> <sup>+</sup>

field.

The first field equation now reads as

that it is called the *master equation*. It reads

2 − *m* <sup>2</sup> *<sup>f</sup>* ′

we see that the function *<sup>F</sup>*(R, *<sup>T</sup>*) := *<sup>f</sup>* ′

electromagnetic) case the field equations simplify to

2

*<sup>R</sup>*˜ *<sup>g</sup>*˜*µν* =

*<sup>G</sup>*˜*µν* <sup>=</sup> *<sup>R</sup>*˜ *µν* <sup>−</sup> <sup>1</sup>

*f* ′

*f* ′

(R) = <sup>0</sup> <sup>⇒</sup> *<sup>f</sup>* ′

(R)R − *m*

(R)*R*˜ *µν* <sup>−</sup> <sup>1</sup>

i.e. the connection Γ˜ is the Levi-Civita connection of the conformal metric *g*˜. Thus in these theories the connection is *a posteriori* metric and the geometry of spacetime is described by a metric Weyl geometry. As a consequence the Ricci tensor *R*˜ *µν* is symmetric being the Ricci

The trace of this equation (with respect to *gµν*) is so important in the analysis of these models

For any given (analytic) function *f* , the master equation is an *algebraic* (i.e. not differential) equation between R and *T*. Assuming that *m* �= 2 and excluding the degenerate case in

(R) = *m*

(R)R − *m*

In vacuum or for purely electromagnetic matter obeying Maxwell equations, one has *<sup>T</sup>* = 0, i.e. the trace *<sup>T</sup>* of *<sup>T</sup>µν* is zero and R takes a constant value from a discrete set R ≡ *<sup>ρ</sup>* ∈ {*ρ*0, *<sup>ρ</sup>*1,... } that of course depends on *<sup>f</sup>* . In this vacuum (as well as in purely

Accordingly, vacuum (or purely electromagnetic) Palatini *f*(R)-theories are generically equivalent to Einstein models with cosmological constant and the possible value of the cosmological constant is chosen in a discrete set which depends on the function *f* . This is known as *universality theorem* for Einstein equations (see [18]). The meaning of this result is not to be overestimated; the equivalence is important but one has a huge freedom in choosing the function *f* (which depends on countable infinite parameters) so that any value for the cosmological constant can be in principle attained. Let us stress once more that this includes all cases in which matter is present but the trace *T* = 0 as it happens for the electromagnetic

Accordingly, the physics described by Palatini *f*(R)-theories in vacuum is not richer than standard GR physics with cosmological constant. Still one should notice that in these vacuum *f*(R)-theories free fall is given by *g*˜ while in standard GR it is given by *g* (while distances

generically (i.e. except a discrete set of values for R) solved for R = *r*(*T*) = *κr*ˆ(*T*).

2 − *m* <sup>2</sup>*<sup>m</sup>* (*<sup>f</sup>* ′

<sup>2</sup> *<sup>C</sup>*1<sup>R</sup> *<sup>m</sup>*−<sup>2</sup>

(*ρ*)) <sup>2</sup> <sup>2</sup>−*<sup>m</sup> ρ* 

<sup>2</sup> *<sup>f</sup>*(R)*gµν* <sup>=</sup> *<sup>κ</sup>Tµν* (15)

<sup>2</sup> *<sup>f</sup>*(R) = *<sup>κ</sup><sup>T</sup>* :<sup>=</sup> *<sup>κ</sup>gµνTµν* (16)

<sup>2</sup> <sup>⇒</sup> *<sup>f</sup>*(R) = *<sup>C</sup>*1<sup>R</sup> *<sup>m</sup>*

<sup>2</sup> *<sup>f</sup>*(R) <sup>−</sup> *<sup>κ</sup><sup>T</sup>* is also analytic and can be

*<sup>g</sup>*˜*µν* = <sup>Λ</sup>(*ρ*)*g*˜*µν* (18)

<sup>2</sup> + *<sup>C</sup>*<sup>2</sup> (17)

Let us hereafter briefly review the mathematical equivalence between Palatini *f*(R)-theories and Brans-Dicke theories and discuss about how physical is such an equivalence. Let us hereafter restrict to the case in dimension *m* = 4.

A *Brans-Dicke* theory is a theory for a metric *gµν* and a scalar field *ϕ*. The dynamics is described by a Lagrangian in the following form

$$L\_{BD} = \sqrt{\mathcal{g}} \left[ \mathcal{q} \mathcal{R} - \frac{\omega}{\mathcal{q}} \nabla\_{\mu} \mathcal{q} \nabla^{\mu} \mathcal{q} + \mathcal{U}(\mathcal{q}) \right] + L\_{\mathfrak{m}}(\mathcal{g}, \psi) \tag{20}$$

where *ω* is a real parameter and *U*(*ϕ*) is a potential function.

Field equations for such a theory are

$$\begin{cases} \boldsymbol{\varrho}(\boldsymbol{R}\_{\mu\nu} - \frac{1}{2}\mathbf{R}\boldsymbol{\varrho}\_{\mu\nu}) = \boldsymbol{\kappa}\boldsymbol{T}\_{\mu\nu} + \boldsymbol{\nabla}\_{\mu\nu}\boldsymbol{\varrho} + \boldsymbol{\Box}\boldsymbol{\varrho}\boldsymbol{\varrho}\_{\mu\nu} + \frac{\omega}{\boldsymbol{\varrho}}\left(\boldsymbol{\nabla}\_{\mu}\boldsymbol{\varrho}\boldsymbol{\nabla}\_{\nu}\boldsymbol{\varrho} - \frac{1}{2}\boldsymbol{\nabla}\_{\mu}\boldsymbol{\varrho}\boldsymbol{\nabla}^{\mu}\boldsymbol{\varrho}\boldsymbol{\varrho}\_{\mu\nu}\right) + \frac{1}{2}\boldsymbol{\mathcal{U}}\boldsymbol{\varrho}\_{\mu\nu} \\ \boldsymbol{R} = \frac{\omega}{\boldsymbol{\varrho}^{2}}\boldsymbol{\nabla}\_{\mu}\boldsymbol{\varrho}\boldsymbol{\nabla}^{\mu}\boldsymbol{\varrho} - 2\frac{\omega}{\boldsymbol{\varrho}}\boldsymbol{\Box}\boldsymbol{\varrho} - \boldsymbol{\mathcal{U}}^{\prime}(\boldsymbol{\varrho}) \end{cases}$$

(21) If one considers now the field equation (19) for a Palatini *<sup>f</sup>*(R)-theory and writes them for the original metric *<sup>g</sup>µν* <sup>=</sup> *<sup>ϕ</sup>*−<sup>1</sup> · *<sup>g</sup>*˜*µν* and the conformal factor *<sup>ϕ</sup>* <sup>=</sup> *<sup>f</sup>* ′ (R) field equation reads as

$$\mathcal{q}\mathcal{R}\_{\mu\nu} = \nabla\_{\mu\nu}\mathcal{q} + \frac{1}{2}\Box\mathcal{q}\mathcal{g}\_{\mu\nu} - \frac{3}{2\varrho}\nabla\_{\mu}\mathcal{q}\nabla\_{\nu}\mathcal{q} + \frac{1}{4}\varrho\mathfrak{f}(T)\mathcal{g}\_{\mu\nu} + \varkappa\left(T\_{\mu\nu} - \frac{1}{4}T\mathcal{g}\_{\mu\nu}\right) \tag{22}$$

while the master equation reads as

$$
\delta\varphi\text{R} = 3\Box\varphi - \frac{3}{2\varrho}\nabla\_{\mathfrak{a}}\varrho\nabla^{\mathfrak{a}}\varrho + \kappa T + 2f\tag{23}
$$

**4. Extended cosmologies**

*T*˜ *µν* <sup>=</sup> <sup>1</sup> *f* ′ �

where we set *<sup>u</sup>*˜*<sup>µ</sup>* = (*<sup>f</sup>* ′

three fluids with

� *<sup>ρ</sup>*˜1 = *<sup>ρ</sup> <sup>p</sup>*˜1 = *<sup>p</sup>*

field.

can set

 

*<sup>p</sup>*˜2 = � 1 *<sup>m</sup>* (*<sup>f</sup>* ′ ) *<sup>m</sup>* <sup>2</sup>−*<sup>m</sup>* − 1 � *p* + <sup>1</sup> *<sup>m</sup>* (*<sup>f</sup>* ′ ) *<sup>m</sup>* <sup>2</sup>−*<sup>m</sup> ρ* =

*<sup>ρ</sup>*˜2 <sup>=</sup> *<sup>m</sup>*−<sup>1</sup> *<sup>m</sup>* (*<sup>f</sup>* ′

momentum tensor of a (perfect) fluid

*p* = *wρ* for some (constant) *w*. Then one has

=(*<sup>ρ</sup>* <sup>+</sup> *<sup>p</sup>*)(*<sup>f</sup>* ′

) <sup>1</sup>

�

�

*<sup>T</sup>µν* <sup>−</sup> <sup>1</sup>

=(*ρ*˜ + *<sup>p</sup>*˜)*u*˜*µu*˜*<sup>ν</sup>* + *<sup>p</sup>*˜*g*˜*µν*

*<sup>m</sup>*−<sup>2</sup> *u<sup>µ</sup>* and

*<sup>p</sup>*˜ <sup>=</sup> *<sup>p</sup>*+*<sup>ρ</sup> <sup>m</sup>* (*<sup>f</sup>* ′ ) *<sup>m</sup>* <sup>2</sup>−*<sup>m</sup>* − *<sup>m</sup>*−<sup>2</sup>

*<sup>ρ</sup>*˜2 <sup>=</sup> *<sup>m</sup>*−<sup>1</sup> *<sup>m</sup>* (*<sup>p</sup>* <sup>+</sup> *<sup>ρ</sup>*)(*<sup>f</sup>* ′

*<sup>m</sup>* (*<sup>f</sup>* ′ ) *<sup>m</sup>* <sup>2</sup>−*<sup>m</sup>* − *p*

*<sup>p</sup>*˜2 <sup>=</sup> *<sup>p</sup>*+*<sup>ρ</sup>*

) *<sup>m</sup>* <sup>2</sup>−*<sup>m</sup> p* + � *<sup>m</sup>*−<sup>1</sup> *<sup>m</sup>* (*<sup>f</sup>* ′

) *<sup>m</sup>* <sup>2</sup>−*<sup>m</sup>* − 1 � *ρ* =

) *<sup>m</sup>*

*<sup>m</sup> Tgµν*�

<sup>2</sup>−*<sup>m</sup> <sup>u</sup>*˜*µu*˜*<sup>ν</sup>* +

*<sup>ρ</sup>*˜ <sup>=</sup> *<sup>m</sup>*−<sup>1</sup> *<sup>m</sup>* (*<sup>p</sup>* <sup>+</sup> *<sup>ρ</sup>*)(*<sup>f</sup>* ′

+ 2 − *m* 2*m*

> � *p* + *ρ <sup>m</sup>* (*<sup>f</sup>* ′

> > ) *<sup>m</sup>* <sup>2</sup>−*<sup>m</sup>* + *<sup>m</sup>*−<sup>2</sup>

Thus the effect of a Palatini *f*(R)-dynamics is to modify the fluid tensor representing sources into another stress tensor which is again in the form of a (perfect) fluid, with modified pressure and density. This can be split quite naturally (though of course non-uniquely) into

The first fluid accounts for what we see as visible matter and it has standard equation of states *<sup>p</sup>*<sup>1</sup> = *<sup>w</sup>*1*ρ*<sup>1</sup> with *<sup>w</sup>*<sup>1</sup> = *<sup>w</sup>*, i.e. the same state equation chosen for the visible matter. The third fluid has equation of states in the form *<sup>p</sup>*<sup>3</sup> = *<sup>w</sup>*3*ρ*<sup>3</sup> with *<sup>w</sup>*<sup>3</sup> = −1, i.e. it is a quintessence

For the second fluid, taking into account the equation of state *p* = *wρ* of visible matter, one

� 1 *<sup>m</sup>* (*<sup>f</sup>* ′ ) *<sup>m</sup>*

) *<sup>m</sup>* <sup>2</sup>−*<sup>m</sup>* − *ρ*

<sup>2</sup>*<sup>m</sup> <sup>r</sup>*ˆ(*<sup>f</sup>* ′

*f*(R)-model.

Let us now apply to a cosmological situation the discussion above for a general Palatini

Extended Theories of Gravitation and the Curvature of the Universe – Do We Really Need Dark Matter?

In a cosmological setting let us assume that the matter stress tensor *Tµν* is the energy

where *<sup>u</sup>αuβgαβ* <sup>=</sup> <sup>−</sup>1 and we set *<sup>ρ</sup>* for the fluid density and *<sup>p</sup>* for its pressure. Matter field equations are assumed to provide a relation between pressure and density under the form

*<sup>r</sup>*ˆ(*T*)*gµν* =

) *<sup>m</sup>*

<sup>2</sup>−*<sup>m</sup>* <sup>−</sup> *<sup>m</sup>* <sup>−</sup> <sup>2</sup> 2*m*

<sup>2</sup>*<sup>m</sup> <sup>r</sup>*ˆ(*<sup>f</sup>* ′

) <sup>2</sup> 2−*m*

�

� *<sup>m</sup>*−<sup>1</sup> *<sup>m</sup>* (*<sup>f</sup>* ′

) *<sup>m</sup>*

<sup>2</sup>−*<sup>m</sup>* (*w* + 1) − *w*

<sup>2</sup>−*<sup>m</sup>* (*w* + 1) − 1

� *ρ*

) <sup>2</sup> 2−*m*

*<sup>ρ</sup>*˜3 <sup>=</sup> *<sup>m</sup>*−<sup>2</sup>

*<sup>p</sup>*˜3 <sup>=</sup> <sup>−</sup> *<sup>m</sup>*−<sup>2</sup>

<sup>2</sup>*<sup>m</sup> <sup>r</sup>*ˆ(*<sup>f</sup>* ′

<sup>2</sup>*<sup>m</sup> <sup>r</sup>*ˆ(*<sup>f</sup>* ′

) <sup>2</sup> 2−*m*

) <sup>2</sup>

<sup>2</sup>−*<sup>m</sup>* = −*ρ*˜3

http://dx.doi.org/10.5772/52041

� *ρ*

*<sup>r</sup>*ˆ(*<sup>f</sup>* ′ ) <sup>2</sup> 2−*m* � *<sup>g</sup>*˜*µν* =

(27)

125

(28)

(29)

(30)

*<sup>T</sup>µν* := *pgµν* + (*<sup>p</sup>* + *<sup>ρ</sup>*)*uµu<sup>ν</sup>* (26)

Within the framework for *f*(R)-theory, one can generically invert the definition of the conformal factor

$$
\varphi = f'(\mathcal{R}) \quad \Rightarrow \mathcal{R} = \sigma(\varphi) \tag{24}
$$

and define a potential function

$$\mathcal{U}(\varphi) = -\varrho \sigma(\varphi) + f(\sigma(\varphi)) \qquad \text{(} \mathcal{U}'(\varphi) = -\sigma'\varphi - \sigma + f'\sigma' = -\sigma\text{)}\tag{25}$$

Then one has a manifest correspondence between a Palatini *f*(R)-theory and a Brans-Dicke theory with the potential *U*(*ϕ*) = −*ϕσ*(*ϕ*) + *f*(*σ*(*ϕ*)) and *ω* = −<sup>3</sup> <sup>2</sup> . This correspondence holds at the level of field equations (and solutions) but it can be shown at the level of action principles as well; see [20].

This equivalence is sometimes used against *f*(R)-theories since Brans-Dicke theories go to standard GR for *ω* → ∞ and the value *ω* = −<sup>3</sup> <sup>2</sup> is ruled out by standard tests in the solar system, e.g. by precession of perihelia of Mercury. In view of the correspondence shown above the same tests would rule out *f*(R)-theories as well.

Letting aside the fact that tests rule out Brans-Dicke theories *without potential*, there is a further aspect that we believe is worth discussing here. In Brans-Dicke theory the gravitational interaction is mediated by a scalar field as well as the metric field. That means that *g* determines light cones, free fall and distances while the scalar field *ϕ* just participates to the dynamics.

In the corresponding Palatini *f*(R)-theory the metric *g* defines distances, it defines light cones (as well as *g*˜ does), but free fall is described by *g not* ˜ by *g*!

The standard tests (as the precession of perihelia of Mercury) which rule out Brans-Dicke theories (see e.g. [24]) simply do not apply to the corresponding *f*(R)-theory since, in the two different models, Mercury moves along the geodesics of two different metrics. In Brans-Dicke theories it moves along the geodesics of a metric *g* which can be expanded in series of *<sup>ω</sup>*−<sup>1</sup> around the standard Schwarzschild solution of standard GR; in the corresponding *f*(R)-theory it moves along geodesics of a different metric *g*˜ which, being in vacuum and in view of universality theorem, is a Schwarzschild-AdS solution; see also [25]. Since it is reasonable to assume a value for the cosmological constant which has no measurable effect at solar system scales, Mercury can be assumed move with good approximation along geodesics of the standard Schwarzschild metric and, despite the mathematical equivalence, *f*(R)-theories pass the tests while Brans-Dicke does not.

This is a pretty neat example in which a mathematical equivalence between two field theories is broken by the interpretation of the theories since the physical assumptions are not preserved by the transformation mapping one framework into the other; see [26], [27].

## **4. Extended cosmologies**

14 Open Questions in Cosmology

conformal factor

while the master equation reads as

and define a potential function

principles as well; see [20].

to the dynamics.

*<sup>ϕ</sup><sup>R</sup>* <sup>=</sup> <sup>3</sup>*<sup>ϕ</sup>* <sup>−</sup> <sup>3</sup>

*<sup>ϕ</sup>* <sup>=</sup> *<sup>f</sup>* ′

*<sup>U</sup>*(*ϕ*) = <sup>−</sup>*ϕσ*(*ϕ*) + *<sup>f</sup>*(*σ*(*ϕ*)) (*U*′

standard GR for *ω* → ∞ and the value *ω* = −<sup>3</sup>

above the same tests would rule out *f*(R)-theories as well.

cones (as well as *g*˜ does), but free fall is described by *g not* ˜ by *g*!

*f*(R)-theories pass the tests while Brans-Dicke does not.

theory with the potential *U*(*ϕ*) = −*ϕσ*(*ϕ*) + *f*(*σ*(*ϕ*)) and *ω* = −<sup>3</sup>

2*ϕ*

Within the framework for *f*(R)-theory, one can generically invert the definition of the

Then one has a manifest correspondence between a Palatini *f*(R)-theory and a Brans-Dicke

holds at the level of field equations (and solutions) but it can be shown at the level of action

This equivalence is sometimes used against *f*(R)-theories since Brans-Dicke theories go to

system, e.g. by precession of perihelia of Mercury. In view of the correspondence shown

Letting aside the fact that tests rule out Brans-Dicke theories *without potential*, there is a further aspect that we believe is worth discussing here. In Brans-Dicke theory the gravitational interaction is mediated by a scalar field as well as the metric field. That means that *g* determines light cones, free fall and distances while the scalar field *ϕ* just participates

In the corresponding Palatini *f*(R)-theory the metric *g* defines distances, it defines light

The standard tests (as the precession of perihelia of Mercury) which rule out Brans-Dicke theories (see e.g. [24]) simply do not apply to the corresponding *f*(R)-theory since, in the two different models, Mercury moves along the geodesics of two different metrics. In Brans-Dicke theories it moves along the geodesics of a metric *g* which can be expanded in series of *<sup>ω</sup>*−<sup>1</sup> around the standard Schwarzschild solution of standard GR; in the corresponding *f*(R)-theory it moves along geodesics of a different metric *g*˜ which, being in vacuum and in view of universality theorem, is a Schwarzschild-AdS solution; see also [25]. Since it is reasonable to assume a value for the cosmological constant which has no measurable effect at solar system scales, Mercury can be assumed move with good approximation along geodesics of the standard Schwarzschild metric and, despite the mathematical equivalence,

This is a pretty neat example in which a mathematical equivalence between two field theories is broken by the interpretation of the theories since the physical assumptions are not preserved by the transformation mapping one framework into the other; see [26], [27].

(*ϕ*) = <sup>−</sup>*σ*′

∇*αϕ*∇*αϕ* + *<sup>κ</sup><sup>T</sup>* + <sup>2</sup> *<sup>f</sup>* (23)

(R) ⇒ R = *σ*(*ϕ*) (24)

*<sup>ϕ</sup>* <sup>−</sup> *<sup>σ</sup>* <sup>+</sup> *<sup>f</sup>* ′

<sup>2</sup> is ruled out by standard tests in the solar

*<sup>σ</sup>*′ <sup>=</sup> <sup>−</sup>*σ*) (25)

<sup>2</sup> . This correspondence

Let us now apply to a cosmological situation the discussion above for a general Palatini *f*(R)-model.

In a cosmological setting let us assume that the matter stress tensor *Tµν* is the energy momentum tensor of a (perfect) fluid

$$T\_{\mu\nu} := \mathcal{p}\mathbb{g}\_{\mu\nu} + (\mathcal{p} + \rho)u\_{\mu}u\_{\nu} \tag{26}$$

where *<sup>u</sup>αuβgαβ* <sup>=</sup> <sup>−</sup>1 and we set *<sup>ρ</sup>* for the fluid density and *<sup>p</sup>* for its pressure. Matter field equations are assumed to provide a relation between pressure and density under the form *p* = *wρ* for some (constant) *w*. Then one has

$$\begin{split} \mathcal{T}\_{\mu\nu} &= \frac{1}{f'} \left( T\_{\mu\nu} - \frac{1}{m} T g\_{\mu\nu} \right) + \frac{2-m}{2m} \mathfrak{r}(T) g\_{\mu\nu} = \\ &= (\rho+p)(f')^{\frac{m}{2-m}} \tilde{u}\_{\mu} \tilde{u}\_{\nu} + \left( \frac{p+\rho}{m} (f')^{\frac{m}{2-m}} - \frac{m-2}{2m} \mathfrak{r}(f')^{\frac{2}{2-m}} \right) \tilde{g}\_{\mu\nu} = \\ &= (\tilde{\rho}+\tilde{p}) \tilde{u}\_{\mu} \tilde{u}\_{\nu} + \tilde{p} \tilde{g}\_{\mu\nu} \end{split} \tag{27}$$

where we set *<sup>u</sup>*˜*<sup>µ</sup>* = (*<sup>f</sup>* ′ ) <sup>1</sup> *<sup>m</sup>*−<sup>2</sup> *u<sup>µ</sup>* and

$$\begin{cases} \tilde{\rho} = \frac{m-1}{m} (p+\rho) (f')^{\frac{m}{2-m}} + \frac{m-2}{2m} \mathfrak{f} (f')^{\frac{2}{2-m}}\\ \tilde{p} = \frac{p+\rho}{m} (f')^{\frac{m}{2-m}} - \frac{m-2}{2m} \mathfrak{f} (f')^{\frac{2}{2-m}} \end{cases} \tag{28}$$

Thus the effect of a Palatini *f*(R)-dynamics is to modify the fluid tensor representing sources into another stress tensor which is again in the form of a (perfect) fluid, with modified pressure and density. This can be split quite naturally (though of course non-uniquely) into three fluids with

$$ \begin{cases} \not p\_1 = \rho \\ \not p\_1 = p \end{cases} \qquad \begin{cases} \not p\_2 = \frac{m-1}{m}(p+\rho)(f')^{\frac{m}{2-m}} - \rho \\ \not p\_2 = \frac{p+\rho}{m}(f')^{\frac{m}{2-m}} - p \end{cases} \qquad \begin{cases} \not p\_3 = \frac{m-2}{2m}\flat(f')^{\frac{2}{2-m}} \\ \not p\_3 = -\frac{m-2}{2m}\flat(f')^{\frac{2}{2-m}} = -\not p\_3 \end{cases} \tag{29} $$

The first fluid accounts for what we see as visible matter and it has standard equation of states *<sup>p</sup>*<sup>1</sup> = *<sup>w</sup>*1*ρ*<sup>1</sup> with *<sup>w</sup>*<sup>1</sup> = *<sup>w</sup>*, i.e. the same state equation chosen for the visible matter. The third fluid has equation of states in the form *<sup>p</sup>*<sup>3</sup> = *<sup>w</sup>*3*ρ*<sup>3</sup> with *<sup>w</sup>*<sup>3</sup> = −1, i.e. it is a quintessence field.

For the second fluid, taking into account the equation of state *p* = *wρ* of visible matter, one can set

$$\begin{cases} \check{\rho}\_{2} = \frac{m-1}{m} (f')^{\frac{m}{2-m}} p + \left( \frac{m-1}{m} (f')^{\frac{m}{2-m}} - 1 \right) \rho = \left( \frac{m-1}{m} (f')^{\frac{m}{2-m}} (w+1) - 1 \right) \rho \\ \check{p}\_{2} = \left( \frac{1}{m} (f')^{\frac{m}{2-m}} - 1 \right) p + \frac{1}{m} (f')^{\frac{m}{2-m}} \rho = \left( \frac{1}{m} (f')^{\frac{m}{2-m}} (w+1) - w \right) \rho \end{cases} \tag{30}$$

which corresponds to an equation of state of the form *<sup>p</sup>*<sup>2</sup> = *<sup>w</sup>*2*ρ*<sup>2</sup> with

$$w\_2 = \frac{\frac{1}{m} (f')^{\frac{m}{2-m}} (w+1) - w}{\frac{m-1}{m} (f')^{\frac{m}{2-m}} (w+1) - 1} \tag{31}$$

provided one rescales the cosmological time with the conformal factor (which depends only

and rescales the Friedmann-Lemaître-Robertson-Walker (FLRW) scale factor accordingly

*<sup>a</sup>*˜(˜*t*) = �

The equations for the scale factor are the celebrated Friedmann equations

<sup>3</sup> *<sup>ρ</sup>*˜ <sup>=</sup> *<sup>κ</sup>* 12 *f* ′ �

<sup>6</sup> (*ρ*˜ <sup>+</sup> <sup>3</sup>*p*˜) <sup>=</sup> *<sup>κ</sup>*

*<sup>f</sup>* ′ *dt* <sup>⇒</sup> ˜*t*(*t*) = � �

*f* ′ *a*(*t*)

Extended Theories of Gravitation and the Curvature of the Universe – Do We Really Need Dark Matter?

*<sup>r</sup>*ˆ(*ρ*) + <sup>3</sup>*<sup>ρ</sup> f* ′ �

> *<sup>r</sup>*ˆ(*ρ*) − <sup>3</sup>*<sup>ρ</sup> f* ′ �

12 *f* ′ �

For a given *f* (and the associated *r*ˆ(*ρ*)) these are two equations for the two unknowns *a*˜(˜*t*)

There is no much one can say in general without specifying *f* . Nevertheless, one can still notice that the worldlines *<sup>γ</sup>* : *<sup>s</sup>* �→ (*t*<sup>0</sup> + *<sup>s</sup>*,*r*0, *<sup>θ</sup>*0, *<sup>φ</sup>*0) are always geodesics (something that depends on the cosmological principle, not on Friedmann equations). Also the curves *γ*¯ : *s* �→ (*t*0,*r*0*s*, *<sup>θ</sup>*0, *<sup>φ</sup>*0) are geodesic trajectories and their length is thence related to spacial distances

Let us thus consider a point (*t*0,*r*0, *<sup>θ</sup>*0, *<sup>φ</sup>*0) representing for example a galaxy, and let us suppose we want to compute its distance from us. If we defined distances by *g*˜ (as one

> � <sup>1</sup> 0

�

*ds*

1 − *Ks*2*r*<sup>2</sup> 0

*ds*

1 − *Ks*2*r*<sup>2</sup> 0

> = 1 �*<sup>f</sup>* ′ ˜*<sup>d</sup>*

would probably do in scalar tensor theories) such a distance would be given by

� <sup>1</sup> 0

�

Then these *f*(R)-theories have an extra time-dependent mismatch in measuring distances. Being the conformal factor dependent on time, it would affect non-trivially the measured

To the best of our knowledge such a possible effect has not only been totally ignored in

˜*<sup>d</sup>* = *<sup>a</sup>*˜(˜*t*0)*r*<sup>0</sup>

However, we defined distances by using *g*. Accordingly, one has

*<sup>d</sup>* = *<sup>a</sup>*(*t*0)*r*<sup>0</sup>

interpreting raw data, but it has not been discussed or proved to vanish.

acceleration of faraway galaxies.

*f* ′ *dt*

http://dx.doi.org/10.5772/52041

127

*d*˜*t* = �

 

˙ *a*˜<sup>2</sup>+*K <sup>a</sup>*˜2 <sup>=</sup> *<sup>κ</sup>*

¨ *a*˜ *<sup>a</sup>*˜ <sup>=</sup> <sup>−</sup>*<sup>κ</sup>*

and *ρ*(*t*) which in principle should be determined as functions of *t*.

on time)

at time *t*0.

Within the standard viewpoint this kind of matter is quite puzzling. Its equation of state is changing in time (since in cosmology *<sup>f</sup>* ′ (*r*((*m* − 1)*p*(*t*) − *ρ*(*t*))) is a function of time).

It is reasonable to assume that at present time visible matter is dominated by dust (*w* = 0) and *m* = 4, in which case we have

$$w\_2^{\text{dust}} = \frac{1}{3 - 4(f')^2} \tag{32}$$

Of course, the splitting of the fluid is not canonical or unique. In particular the second fluid can be further split in different components (for example in order to isolate components which are dominant in various regimes).

This is probably the main reason to consider Palatini *f*(R)-theories as good as models also for cosmology: although we assumed only dust at fundamental level, from the gravitational viewpoint that behaves effectively as a more general fluid the characteristics of which depend on the extended gravitational theories chosen, i.e. on *f* .

Morever, let us also remark that this simple toy model can be easily tested and falsified by current data and it makes predictions about near future surveys. In the standard Λ*CDM* one assumes a cosmological constant Λ which is here modeled by the third fluid. Thus in order to fit data one has to fix the current value for *<sup>f</sup>* ′ , which in turn fixes the current equation of state for the *CDM* dark matter which is also observed. Of course one can consider other reasonable models considering more realistic and finer descriptions of visible matter. Near future surveys will provide data about the evolution of the cosmological constant in time allowing in principle to observe *<sup>f</sup>* ′ (*t*) directly.

Let us now set *m* = 4, *w* = 0 and impose the cosmological principle ansazt, i.e. homogeneity and isotropy. Again should we impose it for *g* or *g*˜? It is fortunate that this does not matter at all! If one does that for *g*˜ assuming the form

$$\tilde{g} = -d\tilde{t}^2 - \tilde{a}^2(\tilde{t}) \left( \frac{dr^2}{1 - Kr^2} + r^2 d\Omega^2 \right).$$

then also *g* is homogeneous and isotropic, i.e. in the form

$$g = -dt^2 - a^2(t) \left( \frac{dr^2}{1 - Kr^2} + r^2 d\Omega^2 \right)^2$$

provided one rescales the cosmological time with the conformal factor (which depends only on time)

$$d\vec{t} = \sqrt{f'} \, dt \qquad \Rightarrow \vec{t}(t) = \int \sqrt{f'} \, dt'$$

and rescales the Friedmann-Lemaître-Robertson-Walker (FLRW) scale factor accordingly

$$
\tilde{a}(\tilde{t}) = \sqrt{f'} \, a(t).
$$

The equations for the scale factor are the celebrated Friedmann equations

16 Open Questions in Cosmology

changing in time (since in cosmology *<sup>f</sup>* ′

which are dominant in various regimes).

on the extended gravitational theories chosen, i.e. on *f* .

to fit data one has to fix the current value for *<sup>f</sup>* ′

at all! If one does that for *g*˜ assuming the form

*g*˜ = −*d*˜*t*

then also *g* is homogeneous and isotropic, i.e. in the form

allowing in principle to observe *<sup>f</sup>* ′

and *m* = 4, in which case we have

which corresponds to an equation of state of the form *<sup>p</sup>*<sup>2</sup> = *<sup>w</sup>*2*ρ*<sup>2</sup> with

*<sup>w</sup>*<sup>2</sup> =

1 *<sup>m</sup>* (*<sup>f</sup>* ′ ) *<sup>m</sup>*

*w*dust

*<sup>m</sup>*−<sup>1</sup> *<sup>m</sup>* (*<sup>f</sup>* ′) *<sup>m</sup>*

Within the standard viewpoint this kind of matter is quite puzzling. Its equation of state is

It is reasonable to assume that at present time visible matter is dominated by dust (*w* = 0)

<sup>2</sup> <sup>=</sup> <sup>1</sup>

Of course, the splitting of the fluid is not canonical or unique. In particular the second fluid can be further split in different components (for example in order to isolate components

This is probably the main reason to consider Palatini *f*(R)-theories as good as models also for cosmology: although we assumed only dust at fundamental level, from the gravitational viewpoint that behaves effectively as a more general fluid the characteristics of which depend

Morever, let us also remark that this simple toy model can be easily tested and falsified by current data and it makes predictions about near future surveys. In the standard Λ*CDM* one assumes a cosmological constant Λ which is here modeled by the third fluid. Thus in order

of state for the *CDM* dark matter which is also observed. Of course one can consider other reasonable models considering more realistic and finer descriptions of visible matter. Near future surveys will provide data about the evolution of the cosmological constant in time

Let us now set *m* = 4, *w* = 0 and impose the cosmological principle ansazt, i.e. homogeneity and isotropy. Again should we impose it for *g* or *g*˜? It is fortunate that this does not matter

> *dr*<sup>2</sup> <sup>1</sup> <sup>−</sup> *Kr*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*

> *dr*<sup>2</sup> <sup>1</sup> <sup>−</sup> *Kr*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*

<sup>2</sup>*d*Ω<sup>2</sup> 

<sup>2</sup>*d*Ω<sup>2</sup> 

(*t*) directly.

<sup>2</sup> − *a*˜ <sup>2</sup>(˜*t*)

*g* = −*dt*<sup>2</sup> − *a*2(*t*)

<sup>2</sup>−*<sup>m</sup>* (*w* + 1) − *w*

<sup>2</sup>−*<sup>m</sup>* (*<sup>w</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> <sup>1</sup> (31)

<sup>3</sup> <sup>−</sup> <sup>4</sup>(*<sup>f</sup>* ′)<sup>2</sup> (32)

, which in turn fixes the current equation

(*r*((*m* − 1)*p*(*t*) − *ρ*(*t*))) is a function of time).

$$\begin{cases} \frac{\vec{d}^2 + K}{d^2} = \frac{\kappa}{3}\tilde{\rho} = \frac{\kappa}{12f'} \left( \hat{r}(\rho) + \frac{3\rho}{f'} \right) \\\frac{\vec{d}}{d} = -\frac{\kappa}{5} \left( \vec{\rho} + 3\vec{\rho} \right) = \frac{\kappa}{12f'} \left( \hat{r}(\rho) - \frac{3\rho}{f'} \right) \end{cases}$$

For a given *f* (and the associated *r*ˆ(*ρ*)) these are two equations for the two unknowns *a*˜(˜*t*) and *ρ*(*t*) which in principle should be determined as functions of *t*.

There is no much one can say in general without specifying *f* . Nevertheless, one can still notice that the worldlines *<sup>γ</sup>* : *<sup>s</sup>* �→ (*t*<sup>0</sup> + *<sup>s</sup>*,*r*0, *<sup>θ</sup>*0, *<sup>φ</sup>*0) are always geodesics (something that depends on the cosmological principle, not on Friedmann equations). Also the curves *γ*¯ : *s* �→ (*t*0,*r*0*s*, *<sup>θ</sup>*0, *<sup>φ</sup>*0) are geodesic trajectories and their length is thence related to spacial distances at time *t*0.

Let us thus consider a point (*t*0,*r*0, *<sup>θ</sup>*0, *<sup>φ</sup>*0) representing for example a galaxy, and let us suppose we want to compute its distance from us. If we defined distances by *g*˜ (as one would probably do in scalar tensor theories) such a distance would be given by

$$\tilde{d} = \vec{a}(\tilde{t}\_0) r\_0 \int\_0^1 \frac{ds}{\sqrt{1 - Ks^2 r\_0^2}}$$

However, we defined distances by using *g*. Accordingly, one has

$$d = a(t\_0)r\_0 \int\_0^1 \frac{ds}{\sqrt{1 - Ks^2r\_0^2}} = \frac{1}{\sqrt{f'}} \,\tilde{d}$$

Then these *f*(R)-theories have an extra time-dependent mismatch in measuring distances. Being the conformal factor dependent on time, it would affect non-trivially the measured acceleration of faraway galaxies.

To the best of our knowledge such a possible effect has not only been totally ignored in interpreting raw data, but it has not been discussed or proved to vanish.

For example, in this context the *measured* acceleration of the universe (which is defined to be the acceleration of galaxies per unit of distance) would be

constrains possible dynamics and it leads to *extended theories of gravitation*. Standard GR is a

Extended Theories of Gravitation and the Curvature of the Universe – Do We Really Need Dark Matter?

http://dx.doi.org/10.5772/52041

129

In extended gravitational theories one can also recast the fields so that an extended gravitational theory looks like standard GR with additional effective sources. This scenario thoroughly agrees with observations as long as dark matter and dark energy will be detected only through their gravitational effects. This scenario is reasonable also in view of cosmological observations which clearly suggest that the spacetime geometrical structure at large scale might be substantially different from the simple standard GR that we observe

Even if in the end standard GR were the correct theory and dark energy and matter will be understood at a fundamental level, this wider framework would be fundamental. It provides an extended framework in which one could test *directly* the assumptions of standard GR on

In this paper we reviewed EPS formalism and defined *extended theories of gravitation* and *extended metric theories of gravitation*. Then we showed that Palatini *f*(R)-theories provide a

If we restrict and apply *f*(R)-theories to cosmology we showed that matter naturally induces effective sources which can naturally modelled by fluid energy momentum source tensors which at least qualitatively present the main features of dark source models used phenomenologically to fit data. A (running) cosmological constant naturally emerges as well as a fluid with a running equation of states which depends explicitly of the *f*(R) dynamics chosen. We also briefly discussed how one should define distances (as well as velocities and

We acknowledge the contribution of INFN (Iniziativa Specifica NA12) the local research project *Metodi Geometrici in Fisica Matematica e Applicazioni* (2011) of Dipartimento di Matematica of University of Torino (Italy). This paper is also supported by INdAM-GNFM.

model within this extended family of *reasonable* gravitational models.

an experimental basis without resorting to uncertain approximations.

family of such metric extended theories of gravitation.

acceleration parameteres) in this extended framework.

<sup>⋆</sup> Address all correspondence to: lorenzo.fatibene@unito.it

1 Department of Mathematics, University of Torino, Italy 2 INFN- Sezione Torino, Iniziativa Specifica Na12, Italy

[1] D. J. Mortlock, R. L. Webster, Mon. Not. Roy. Astron. Soc. 319 872 (2000)

[2] A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116 1009 (1998); S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517 565

J. L. Tonry et al. [Supernova Search Team Collaboration], Astrophys. J. 594 1 (2003)

at Solar System scale.

**Acknowledgements**

**Author details**

**References**

(1999);

L. Fatibene1,2,<sup>⋆</sup> and M. Francaviglia1,2

$$\frac{\ddot{a}}{a} = \Phi^2 \frac{\ddot{a}}{\ddot{a}} - \frac{\dot{\Phi}}{\Phi} \frac{\dot{a}}{a} - \frac{\ddot{\Phi}}{\Phi} + \left(\frac{\dot{\Phi}}{\Phi}\right)^2$$

where we set <sup>Φ</sup>2(*t*) = *<sup>f</sup>* ′ for the conformal factor.

It is therefore not difficult to find whole classes of functions *f* for which a solution in the FLRW form is allowed ¨ *a*˜/*a*˜ is negative (corresponding to an ever slower expansion) while *a*¨/*a* is positive (corresponding to an accelerating expansion). When this were the case part of the effect of dark energy would be explained as a simple aberration of distance measurement.

Whether for some *f* this can fit experimental data better than the acceleration ¨ *a*˜/*a*˜ is something to be discussed on the observational ground. We just remark on a fundamental ground that extrapolating *our terrestrial current* rulers to 10 billion years ago and 10 billion light years away (in a theory in which geometry is dynamical and measurement protocols depends on all sorts of physical assumptions on the behavior of electromagnetic and gravitational fields) could be slightly hasty.

Of course we are not claiming these effect to be real. However, they are plausible and hence they should be considered in data analysis (and possibly eventually shown to be null). They were not introduced by *ad hoc* argument. On the contrary they are quite natural in metric extended theories of gravitation.

## **5. Conclusions and perspectives**

The astrophysical and cosmological observations of the last decade clearly point to a deep reconsideration of standard scenarios based on standard GR, either on the source side or on the gravitational dynamics; or both. Basically, all observations about gravity in non-vacuum situations need to be somehow corrected.

If one decides to keep stuck to standard gravitational dynamics, then observations force us to modify the matter energy momentum tensor by adding dark sources. If one decides to modify gravity dynamics, the family of different available (covariant, variational, . . . ) theories is huge. Moreover, one variational model for gravity usually may support (for example when the model contains more than one metric) many inequivalent definitions of observational protocols. It is quite natural that in such a huge family one can find (many) models which fit the observations.

Thus usually one has to choose which of these two ways is preferable. In any event, one should reconsider foundations of gravitational theories from a more general perspective. EPS formalism provides us with such a reconsideration. It clearly shows that on spacetime coexist a conformal structure (associated to light rays and defining light cones and causality), a (torsionless) projective structure (associated to particles and free fall), and a metric structure (associated to our definition of clocks and rulers). These three structures can be assumed to be *a priori* independent, provided that dynamics forces *a posteriori* some compatibility conditions. The metric structure should also define the conformal structure and the projective structure can be represented by an affine (torsionless) connection so that lightlike geodesics of the metric structure are *also* autoparallel curves of the connection. This framework strongly constrains possible dynamics and it leads to *extended theories of gravitation*. Standard GR is a model within this extended family of *reasonable* gravitational models.

In extended gravitational theories one can also recast the fields so that an extended gravitational theory looks like standard GR with additional effective sources. This scenario thoroughly agrees with observations as long as dark matter and dark energy will be detected only through their gravitational effects. This scenario is reasonable also in view of cosmological observations which clearly suggest that the spacetime geometrical structure at large scale might be substantially different from the simple standard GR that we observe at Solar System scale.

Even if in the end standard GR were the correct theory and dark energy and matter will be understood at a fundamental level, this wider framework would be fundamental. It provides an extended framework in which one could test *directly* the assumptions of standard GR on an experimental basis without resorting to uncertain approximations.

In this paper we reviewed EPS formalism and defined *extended theories of gravitation* and *extended metric theories of gravitation*. Then we showed that Palatini *f*(R)-theories provide a family of such metric extended theories of gravitation.

If we restrict and apply *f*(R)-theories to cosmology we showed that matter naturally induces effective sources which can naturally modelled by fluid energy momentum source tensors which at least qualitatively present the main features of dark source models used phenomenologically to fit data. A (running) cosmological constant naturally emerges as well as a fluid with a running equation of states which depends explicitly of the *f*(R) dynamics chosen. We also briefly discussed how one should define distances (as well as velocities and acceleration parameteres) in this extended framework.

## **Acknowledgements**

18 Open Questions in Cosmology

FLRW form is allowed ¨

For example, in this context the *measured* acceleration of the universe (which is defined to be

It is therefore not difficult to find whole classes of functions *f* for which a solution in the

is positive (corresponding to an accelerating expansion). When this were the case part of the effect of dark energy would be explained as a simple aberration of distance measurement. Whether for some *f* this can fit experimental data better than the acceleration ¨

something to be discussed on the observational ground. We just remark on a fundamental ground that extrapolating *our terrestrial current* rulers to 10 billion years ago and 10 billion light years away (in a theory in which geometry is dynamical and measurement protocols depends on all sorts of physical assumptions on the behavior of electromagnetic and

Of course we are not claiming these effect to be real. However, they are plausible and hence they should be considered in data analysis (and possibly eventually shown to be null). They were not introduced by *ad hoc* argument. On the contrary they are quite natural in metric

The astrophysical and cosmological observations of the last decade clearly point to a deep reconsideration of standard scenarios based on standard GR, either on the source side or on the gravitational dynamics; or both. Basically, all observations about gravity in non-vacuum

If one decides to keep stuck to standard gravitational dynamics, then observations force us to modify the matter energy momentum tensor by adding dark sources. If one decides to modify gravity dynamics, the family of different available (covariant, variational, . . . ) theories is huge. Moreover, one variational model for gravity usually may support (for example when the model contains more than one metric) many inequivalent definitions of observational protocols. It is quite natural that in such a huge family one can find (many)

Thus usually one has to choose which of these two ways is preferable. In any event, one should reconsider foundations of gravitational theories from a more general perspective. EPS formalism provides us with such a reconsideration. It clearly shows that on spacetime coexist a conformal structure (associated to light rays and defining light cones and causality), a (torsionless) projective structure (associated to particles and free fall), and a metric structure (associated to our definition of clocks and rulers). These three structures can be assumed to be *a priori* independent, provided that dynamics forces *a posteriori* some compatibility conditions. The metric structure should also define the conformal structure and the projective structure can be represented by an affine (torsionless) connection so that lightlike geodesics of the metric structure are *also* autoparallel curves of the connection. This framework strongly

 Φ˙ Φ 2

*a*˜/*a*˜ is negative (corresponding to an ever slower expansion) while *a*¨/*a*

*a*˜/*a*˜ is

the acceleration of galaxies per unit of distance) would be

where we set <sup>Φ</sup>2(*t*) = *<sup>f</sup>* ′ for the conformal factor.

gravitational fields) could be slightly hasty.

extended theories of gravitation.

**5. Conclusions and perspectives**

situations need to be somehow corrected.

models which fit the observations.

*a*¨ *<sup>a</sup>* <sup>=</sup> <sup>Φ</sup><sup>2</sup> ¨ *a*˜ *<sup>a</sup>*˜ <sup>−</sup> <sup>Φ</sup>˙ Φ *a*˙ *<sup>a</sup>* <sup>−</sup> <sup>Φ</sup>¨ <sup>Φ</sup> <sup>+</sup>

> We acknowledge the contribution of INFN (Iniziativa Specifica NA12) the local research project *Metodi Geometrici in Fisica Matematica e Applicazioni* (2011) of Dipartimento di Matematica of University of Torino (Italy). This paper is also supported by INdAM-GNFM.

## **Author details**


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**Chapter 6**

**Provisional chapter**

**Extending Cosmology: The Metric Approach**

**Extending Cosmology: The Metric Approach**

In this chapter it is reviewed a possible physical scenario for which the introduction of a fundamental constant of nature with dimensions of acceleration into the theory of gravity makes it possible to extend gravity in a very consistent manner. In the non-relativistic regime a MOND-like theory with a modification in the force sector is obtained. This description turns out to be the the weak-field limit of a more general metric relativistic theory of gravity. The mass and length scales involved in the dynamics of the whole universe require small accelerations which are of the order of Milgrom's acceleration constant and so, it turns out that this relativistic theory of gravity can be used to explain the expansion of the universe. In this work it is explained how to build that relativistic theory of gravity in such a way that the overall large-scale dynamics of the universe can be treated in a pure metric approach

Cosmological and astrophysical observations are generally explained introducing two unknown mysterious dark components, namely dark matter and dark energy. These ad hoc hypothesis represent a big cosmological paradigm, since they arise due to the fact that Einstein's field equations are forced to remain unchanged under certain observed

A natural alternative scenario would be to see whether viable cosmological solutions can be found if dark unknown entities are assumed non-existent. The price to pay with this assumption is that the field equations of the theory of gravity need to be extended and so, new Friedmann-like equations will arise. The most natural approach to extend gravity arises when a metric extension *f*(*R*) is introduced into the theory [see e.g. 9, and references therein]. In a series of recent articles, Bernal, Capozziello, Cristofano & de Laurentis [4], Bernal, Capozziello, Hidalgo & Mendoza [5], Carranza et al. [10], Hernandez et al. [17, 18], Mendoza et al. [22, 23] have shown how relevant the introduction of a new fundamental physical constant <sup>a</sup><sup>0</sup> <sup>≈</sup> <sup>10</sup><sup>−</sup>10m/s2 with dimensions of acceleration is in excellent agreement with

> ©2012 Mendoza, licensee InTech. This is an open access chapter 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. © 2012 Mendoza; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 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,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

without the need to introduce dark matter and/or dark energy components.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Sergio Mendoza

**1. Introduction**

10.5772/53878

Sergio Mendoza

http://dx.doi.org/10.5772/53878

astrophysical phenomenology.

**Provisional chapter**
