**1. Introduction**

Applications of combinatorics have in recent years invaded many new areas of research. Still, cosmology is probably not the first such area which comes to your mind. Traditional cosmology is usually based on differential geometry and general relativity, often in combination with various ideas from fundamental physics and high-precision astronomical measurements. However, it is very much at the heart of cosmology that any model that we study must be based on rather drastic simplifications. In fact, when the object of study in a sense contains everything, finding the right way to discard nonessential information becomes a fundamental problem. From this point of view, the combinatorial approach is just one of several possible ways to proceed. For a discussion of this question from a more general point of view, see [1].

Different problems may of course call for different kinds of simplifications. As a rather extreme example, I will in this chapter discuss the long open problem of time's arrow, where it can be argued that the best method of attack may be to discard almost everything we know about the universe, just to uncover the underlying combinatorial skeleton. In other words, we should forget almost everything we know about ordinary physics and instead consider all the possible states that a universe could be in as the nodes of a huge graph. Each possible universe then becomes a path in this graph, and our mission becomes to try to decide what kinds of paths are the most common ones.

The ambition here is not to claim any kind of final solution to "the riddle of time." Rather, the ambition is to give a new angle to a well-known problem. And also to show that from this point of view, it may even make sense to study models which are ridiculously small in comparison with the real universe.

be broken in the sense that in one direction of time the entropy is increasing and in the other it is decreasing. Another way to express this would be to say that all the universes in the multiverse would share the same endpoints, the Big Bang and the Big Crunch. But only half of them would have the same Big Bang as we have. In the

This is not the place to try to describe all possible combinatorial models for cosmology. Rather, I have chosen to just discuss the simple case of a closed, finite universe. Many cosmologists these days support open models, and it is of course possible to apply combinatorial methods to them too. However, since such models tend to be infinite, they may be considerably more complicated from a probabilistic

To model the set of all universes in the simplest possible way, let us for each moment of time between the endpoints �*T*<sup>0</sup> and *T*<sup>0</sup> (i.e., the Big Bang and the Big Crunch) consider the finite set of all possible "states" of a universe. To make everything extremely simple, let us suppose that time is discrete in the sense that we

Thus, we can measure time just by counting the number of time intervals, which

means that time can be viewed as integer valued. At the endpoints �*T*<sup>0</sup> and *T*0, there will just be one unique state (with zero volume), but in between, there will be many states for each *t*. All such states will be the nodes of an enormous graph, and a universe will then be just a path in this graph with the property that there is exactly one state for each moment of time. The dynamics of the model can then be specified by choosing at certain collection of edges between adjacent moments of time, say *t* and *t*+1, which correspond to those time developments which are possible. A quite

*Remark* 1. For the readers taking interest in the underlying physics: the word "state" is not referring to quantum states as they are usually interpreted. A better way of thinking of them is to say that they represent "distinguishable configurations." This is in fact a kind of semiclassical approximation (see Tamm [2]).

The important point here is that a given state can lead to different states in the

future. This is very much what actually happens when, say, a particle decays: whether or not this happens may, according to the multiverse interpretation, lead to very different futures within a rather short time. And there is no contradiction

ð1Þ

other half, our Big Bang would instead be the Big Crunch.

only consider it at a finite number of points as follows:

schematic picture is displayed in **Figure 1**.

*One universe in the combinatorial multiverse [3].*

**3. The combinatorial multiverse**

*DOI: http://dx.doi.org/10.5772/intechopen.90696*

*Combinatorial Cosmology*

point of view.

**Figure 1.**

**311**

After a short introduction to the problem of time's arrow in Section 2, the basic structure of the combinatorial multiverse is presented in Section 3. But in order to study the asymmetry of time, we also need the concept of entropy which is introduced in Section 4, and in Section 5 we then turn to the dynamics. Both the entropy and the dynamics are treated in a very simplified way. However, the essential point here is to try to explain how the time-asymmetric development of the entropy that we observe in our universe can arise from time-symmetric dynamical laws and boundary conditions. In Section 6, the object is to show how standard methods from combinatorics can be used to make computations in the combinatorial multiverse. In Sections 7 and 8, we then consider some very simple probabilistic assumptions which turn the combinatorial multiverse into a probability space and discuss the consequences for time asymmetry.

In the simple model discussed so far, it is not difficult to obtain similar results by heuristic reasoning. However, the approach here should mainly be considered as a preparation for more complicated models, where the same combinatorial methods could be used, but heuristics would be difficult to apply.

Thus, this should more be considered as a starting point for further research than as an endpoint. In Section 9, I will therefore take a step in this direction by suggesting one such possible generalization (out of many), which could be used to obtain a stronger kind of time asymmetry. Finally, some conclusions are then discussed in Section 10.

Many of the ideas presented here have appeared before, e.g., in [2, 3], and in particular [4], although from a somewhat different angle.

## **2. The arrow of time**

The term "time's arrow" was coined by Eddington [5] and refers to the fact that macroscopic time is asymmetric. In fact, we all know that the future is very different from the past. For instance, how does it come that we can remember yesterday but we cannot remember tomorrow? This can also be expressed by saying that we all agree that there is a well-defined direction from the past toward the future. Ever since the time of Ludwig Boltzmann, it has been clear that this has something to do with the growth of entropy and the second law of thermodynamics, although it may still not be quite obvious exactly what the connection is.

What is mysterious about time's arrow is that somehow the macroscopic laws that we observe must emerge from the underlying microscopic laws of motion, and these are in general considered to be essentially time invariant. So how can asymmetric macroscopic laws arise from symmetric microscopic ones?

Few questions in physics have generated such a variety of completely different answers (see, e.g., Barbour [6], Halliwell et al. [7], Zeh [8]), and the problem is still wide open. But, as has repeatedly been pointed out by Price [9], most such tentative answers seem to contain some (more or less hidden) asymmetry from the beginning, either in the boundary conditions or in the dynamical laws.

Here I will advocate a different viewpoint. We can consider the set of all possible universes as a probability space, a "multiverse," and this probability space will be completely time symmetric in the sense that reversing the direction of time would generate the same probability space. But for observers, like ourselves, who are by necessity confined to our own universe, it can still be that the symmetry appears to

The ambition here is not to claim any kind of final solution to "the riddle of time." Rather, the ambition is to give a new angle to a well-known problem. And also to show that from this point of view, it may even make sense to study models

After a short introduction to the problem of time's arrow in Section 2, the basic structure of the combinatorial multiverse is presented in Section 3. But in order to study the asymmetry of time, we also need the concept of entropy which is introduced in Section 4, and in Section 5 we then turn to the dynamics. Both the entropy and the dynamics are treated in a very simplified way. However, the essential point here is to try to explain how the time-asymmetric development of the entropy that we observe in our universe can arise from time-symmetric dynamical laws and boundary conditions. In Section 6, the object is to show how standard methods from combinatorics can be used to make computations in the combinatorial multiverse. In Sections 7 and 8, we then consider some very simple probabilistic assumptions which turn the combinatorial multiverse into a probability space and discuss the

In the simple model discussed so far, it is not difficult to obtain similar results by heuristic reasoning. However, the approach here should mainly be considered as a preparation for more complicated models, where the same combinatorial methods

Thus, this should more be considered as a starting point for further research than as an endpoint. In Section 9, I will therefore take a step in this direction by suggesting one such possible generalization (out of many), which could be used to obtain a stronger kind of time asymmetry. Finally, some conclusions are then

Many of the ideas presented here have appeared before, e.g., in [2, 3], and in

The term "time's arrow" was coined by Eddington [5] and refers to the fact that macroscopic time is asymmetric. In fact, we all know that the future is very different from the past. For instance, how does it come that we can remember yesterday but we cannot remember tomorrow? This can also be expressed by saying that we all agree that there is a well-defined direction from the past toward the future. Ever since the time of Ludwig Boltzmann, it has been clear that this has something to do with the growth of entropy and the second law of thermodynamics, although it may

What is mysterious about time's arrow is that somehow the macroscopic laws that we observe must emerge from the underlying microscopic laws of motion, and these are in general considered to be essentially time invariant. So how can asym-

Few questions in physics have generated such a variety of completely different answers (see, e.g., Barbour [6], Halliwell et al. [7], Zeh [8]), and the problem is still wide open. But, as has repeatedly been pointed out by Price [9], most such tentative answers seem to contain some (more or less hidden) asymmetry from the begin-

Here I will advocate a different viewpoint. We can consider the set of all possible universes as a probability space, a "multiverse," and this probability space will be completely time symmetric in the sense that reversing the direction of time would generate the same probability space. But for observers, like ourselves, who are by necessity confined to our own universe, it can still be that the symmetry appears to

which are ridiculously small in comparison with the real universe.

consequences for time asymmetry.

*Probability, Combinatorics and Control*

discussed in Section 10.

**2. The arrow of time**

**310**

could be used, but heuristics would be difficult to apply.

particular [4], although from a somewhat different angle.

still not be quite obvious exactly what the connection is.

metric macroscopic laws arise from symmetric microscopic ones?

ning, either in the boundary conditions or in the dynamical laws.

be broken in the sense that in one direction of time the entropy is increasing and in the other it is decreasing. Another way to express this would be to say that all the universes in the multiverse would share the same endpoints, the Big Bang and the Big Crunch. But only half of them would have the same Big Bang as we have. In the other half, our Big Bang would instead be the Big Crunch.
