**6. Modeling the dynamics**

To construct a miniature multiverse for computational purposes, one can proceed as follows: first of all, in the very small multiverses studied here, the extreme phases will only last for one single unit of time. Also, for ease of notation, let us put *T*<sup>1</sup> ¼ *m*, so that the moments of time can in this context be denoted as

$$-m-1, \ -m, \ -m+1, \ \dots, m-1, m, m+1. \tag{5}$$

The dynamics is specified by randomly choosing for each state at time *t* with entropy *S*, *K* edges to states at time *t* þ 1 with entropy *S* þ 1, and similarly *K* edges to states at time *t* � 1 with entropy *S* þ 1 (with obvious modifications at the endpoints). In this section, again to make everything as simple as possible, *K* will be set equal to 2. These random choices are in practice carried out by the random number generator in, e.g., Mathematica or MATLAB. But once these are chosen, they specify a model for the dynamics for the miniature multiverse, and we are faced with the problem of computing the number of paths of different kinds. It should be observed that if *K* ≪ *W*, then only a small fraction of all states will be connected to states with lower entropy at the next or previous moment, in spite of the fact that all states are connected to several states with higher entropy, just as in the Principle 1 in Section 5.

significant variable will be the average number of universes when we consider

ular, in this case all paths joining two given nodes all have the same length. **Definition 3.** The *adjacency matrix* of the (directed) graph *G* with nodes *<sup>v</sup>*1, *<sup>v</sup>*2, … , *vm* is the *<sup>m</sup>* � *<sup>m</sup>*-matrix *<sup>A</sup>* <sup>¼</sup> *aij* , where *aij* <sup>¼</sup> 1 if the pair *vi*, *vj* determines

, equals the number of paths of length *k* starting at *vi* and ending at *vj*.

The basic combinatorial tool for making these computations is the adjacency

Thus, recall that a (directed) path ½ � *v*1, *v*2, … , *vm* from *v*<sup>1</sup> to *vm* is a sequence of nodes such that for each *j* ¼ 1, 2, … , *m* � 1, the pair *vj*, *v <sup>j</sup>*þ<sup>1</sup> belongs to the set of (directed) edges of the graph. A lot of work has been done in combinatorics to calculate the number of paths of a given length between two nodes. In general, this is a hard problem or at least a time-consuming one. But for graphs with the special time-related properties in this chapter, the task may be somewhat easier. In partic-

The reason why this matrix is useful to us lies in the following classical result: **Theorem 1**. The element at position *ij* in the kth power of the adjacency matrix,

*Remark* 2. The fact that I have chosen to work with directed graphs here should not be confused with some kind of preferred direction of time. It would in fact be possible to work with two-sided paths as well. This would however introduce more elements different from zero in the adjacency matrix and hence slow down the computations. In other words, the choice to work with directed graphs is just for technical reasons. In fact, when considering the universes in this chapter, the number of directed paths from *t* ¼ �*m* to *t* ¼ *m* is precisely the same as the number

When considering powers of the adjacency matrix below, everything we need to know about paths starting with *S* ¼ 0 at �*m* can be obtained from the first row of *A*<sup>2</sup>*<sup>m</sup>*. Thus, this can all essentially be done by simple linear algebra. Although simple in principle, the size of *A* grows very fast with the size of the model, i.e., primarily

In view of our simple choice for the dynamics and in particular of the fact the entropy can only change by �1 at each step during the normal phase, it suffices to

Starting from *S* ¼ 0 at time �*m* � 2, we observe that at time *t* ¼ �1, only states which have *S*≤1 have to be considered, which gives 1 + 4 = 5 states. In the same way, we get for *t* ¼ 0, 1 + 4 + 16 = 21 states; for *t* ¼ 1, 1 + 4 + 16 + 64 = 85 states; and

The adjacency matrix can now be written as a block matrix in the following way:

Here empty blocks should be understood as containing just zeros. Each of the five-block rows/columns correspond to a moment of time, i.e., to �2, � 1, 0, 1, 2 as in **Figure 2**. Inside each such row/column, the states are ordered according to entropy: the first element is the unique state with *S* ¼ 0. Then (if *t*≥ � 1) the four elements with *S* ¼ 1 follow, thereafter (if *t*≥ 0) the 16 elements with *S* ¼ 2, and so on. The internal order between all the elements with equal *S* and *t* is not at all

ð6Þ

many graphs at the same time.

*Combinatorial Cosmology*

*Ak*

with *m* and *W*.

important in the following.

**315**

matrix (see [13] or [14]) of the graph.

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

a (directed) edge in *G* and *aij* ¼ 0 otherwise.

of nondirected path between *t* ¼ �*m* and *t* ¼ *m*.

consider nodes in the graph with *S*≤*t* þ *m*.

finally for *t* ¼ 2, 1 + 4 + 16 + 64 + 256 = 341 states.

As an illustration, a schematic picture of the set of the possible states in the case of a very small multiverse with only 5 moments of time between the Big Bang and the Big Crunch and with *W* ¼ 4 is shown in **Figure 2**.

Note that due to the way we have set up the dynamics, the entropy can grow with at most one unit during each unit of time. This means that if we start from an ordered state with *S* ¼ 0 at one end of the normal phase, then only values of *S* less than or equal to four can occur during the life-span of the corresponding universe. This means that the part of the multiverse graph displayed in **Figure 2** is sufficient for computing the number of all possible universes with zero entropy at one end. To actually carry out the computation, we can proceed as follows: it is easy to compute the number of paths with monotonically increasing entropy. According to the above assumptions, each state with entropy *S* is connected to exactly two states with entropy *S* þ 1, both at the next and at the previous moment (with an obvious restriction to just one side at times �*m* and *m*). This clearly implies that for each unit of time, the number of paths doubles: from the state with *S* ¼ 0 at time �*m*, there are precisely two edges to states with *S* ¼ 1 at time �*m* þ 1, and for each of these, there will also be precisely two edges to states with *S* ¼ 2 at time �*m* þ 2 which gives in total four paths. At the next step, there will then be eight paths to states with *S* ¼ 3 at time �*m* þ 3 and so on.

In the case *<sup>m</sup>* <sup>¼</sup> 2, we obtain 2<sup>4</sup> <sup>¼</sup> 16 such universes, since there are in this case four unit intervals of time. For *<sup>m</sup>* <sup>¼</sup> 3, we get in an analogous way 26 <sup>¼</sup> 64 universes since there are in this case six unit intervals of time.

One has to work harder to compute the number of paths with zero entropy at both ends, at least if we want exact results and not just heuristic ones. The number of such universes must be considered as a statistical variable which depends on the random choices of the edges which defines the dynamics. In fact, the most

#### **Figure 2.**

*A schematic picture of a universe in a very small multiverse with only five moments of time between the endpoints (i.e., m* ¼ *2). In this case, the universe has a monotonically increasing entropy [4].*

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

generator in, e.g., Mathematica or MATLAB. But once these are chosen, they specify a model for the dynamics for the miniature multiverse, and we are faced with the problem of computing the number of paths of different kinds. It should be observed that if *K* ≪ *W*, then only a small fraction of all states will be connected to states with lower entropy at the next or previous moment, in spite of the fact that all states are connected to several states with higher entropy, just as in the Principle 1

As an illustration, a schematic picture of the set of the possible states in the case of a very small multiverse with only 5 moments of time between the Big Bang and

Note that due to the way we have set up the dynamics, the entropy can grow with at most one unit during each unit of time. This means that if we start from an ordered state with *S* ¼ 0 at one end of the normal phase, then only values of *S* less than or equal to four can occur during the life-span of the corresponding universe. This means that the part of the multiverse graph displayed in **Figure 2** is sufficient for computing the number of all possible universes with zero entropy at one end. To actually carry out the computation, we can proceed as follows: it is easy to compute the number of paths with monotonically increasing entropy. According to the above assumptions, each state with entropy *S* is connected to exactly two states with entropy *S* þ 1, both at the next and at the previous moment (with an obvious restriction to just one side at times �*m* and *m*). This clearly implies that for each unit of time, the number of paths doubles: from the state with *S* ¼ 0 at time �*m*, there are precisely two edges to states with *S* ¼ 1 at time �*m* þ 1, and for each of these, there will also be precisely two edges to states with *S* ¼ 2 at time �*m* þ 2 which gives in total four paths. At the next step, there will then be eight paths to

In the case *<sup>m</sup>* <sup>¼</sup> 2, we obtain 2<sup>4</sup> <sup>¼</sup> 16 such universes, since there are in this case four unit intervals of time. For *<sup>m</sup>* <sup>¼</sup> 3, we get in an analogous way 26 <sup>¼</sup> 64 universes

One has to work harder to compute the number of paths with zero entropy at both ends, at least if we want exact results and not just heuristic ones. The number of such universes must be considered as a statistical variable which depends on the

random choices of the edges which defines the dynamics. In fact, the most

*A schematic picture of a universe in a very small multiverse with only five moments of time between the endpoints (i.e., m* ¼ *2). In this case, the universe has a monotonically increasing entropy [4].*

the Big Crunch and with *W* ¼ 4 is shown in **Figure 2**.

states with *S* ¼ 3 at time �*m* þ 3 and so on.

since there are in this case six unit intervals of time.

in Section 5.

*Probability, Combinatorics and Control*

**Figure 2.**

**314**

significant variable will be the average number of universes when we consider many graphs at the same time.

The basic combinatorial tool for making these computations is the adjacency matrix (see [13] or [14]) of the graph.

Thus, recall that a (directed) path ½ � *v*1, *v*2, … , *vm* from *v*<sup>1</sup> to *vm* is a sequence of nodes such that for each *j* ¼ 1, 2, … , *m* � 1, the pair *vj*, *v <sup>j</sup>*þ<sup>1</sup> belongs to the set of (directed) edges of the graph. A lot of work has been done in combinatorics to calculate the number of paths of a given length between two nodes. In general, this is a hard problem or at least a time-consuming one. But for graphs with the special time-related properties in this chapter, the task may be somewhat easier. In particular, in this case all paths joining two given nodes all have the same length.

**Definition 3.** The *adjacency matrix* of the (directed) graph *G* with nodes *<sup>v</sup>*1, *<sup>v</sup>*2, … , *vm* is the *<sup>m</sup>* � *<sup>m</sup>*-matrix *<sup>A</sup>* <sup>¼</sup> *aij* , where *aij* <sup>¼</sup> 1 if the pair *vi*, *vj* determines a (directed) edge in *G* and *aij* ¼ 0 otherwise.

The reason why this matrix is useful to us lies in the following classical result: **Theorem 1**. The element at position *ij* in the kth power of the adjacency matrix, *Ak* , equals the number of paths of length *k* starting at *vi* and ending at *vj*.

*Remark* 2. The fact that I have chosen to work with directed graphs here should not be confused with some kind of preferred direction of time. It would in fact be possible to work with two-sided paths as well. This would however introduce more elements different from zero in the adjacency matrix and hence slow down the computations. In other words, the choice to work with directed graphs is just for technical reasons. In fact, when considering the universes in this chapter, the number of directed paths from *t* ¼ �*m* to *t* ¼ *m* is precisely the same as the number of nondirected path between *t* ¼ �*m* and *t* ¼ *m*.

When considering powers of the adjacency matrix below, everything we need to know about paths starting with *S* ¼ 0 at �*m* can be obtained from the first row of *A*<sup>2</sup>*<sup>m</sup>*. Thus, this can all essentially be done by simple linear algebra. Although simple in principle, the size of *A* grows very fast with the size of the model, i.e., primarily with *m* and *W*.

In view of our simple choice for the dynamics and in particular of the fact the entropy can only change by �1 at each step during the normal phase, it suffices to consider nodes in the graph with *S*≤*t* þ *m*.

Starting from *S* ¼ 0 at time �*m* � 2, we observe that at time *t* ¼ �1, only states which have *S*≤1 have to be considered, which gives 1 + 4 = 5 states. In the same way, we get for *t* ¼ 0, 1 + 4 + 16 = 21 states; for *t* ¼ 1, 1 + 4 + 16 + 64 = 85 states; and finally for *t* ¼ 2, 1 + 4 + 16 + 64 + 256 = 341 states.

The adjacency matrix can now be written as a block matrix in the following way:

Here empty blocks should be understood as containing just zeros. Each of the five-block rows/columns correspond to a moment of time, i.e., to �2, � 1, 0, 1, 2 as in **Figure 2**. Inside each such row/column, the states are ordered according to entropy: the first element is the unique state with *S* ¼ 0. Then (if *t*≥ � 1) the four elements with *S* ¼ 1 follow, thereafter (if *t*≥ 0) the 16 elements with *S* ¼ 2, and so on. The internal order between all the elements with equal *S* and *t* is not at all important in the following.

With this setup and the random dynamics introduced earlier, each *B*-matrix contains all the information about the edges from all the states at one moment of time to the states at the next one. For example, *B*<sup>12</sup> contains the information about all edges from the single state with *S* ¼ 0 at time *t* ¼ �2 to the five states with *S*≤ 1 when *t* ¼ �1. In the same way, *B*<sup>23</sup> gives a complete description of the edges from the 5 states with *S*≤1 at time *t* ¼ �1 to the 21 states with *S*≤ 2 when *t* ¼ 0.

As for the normal phase, the choice will, to start with, be the simplest possible one: each path is either possible or not, corresponding to the probability weights 1 and 0. During the extreme phases, this assumption is no longer reasonable. Again the model will be extremely simplified, but still it is based on physical intuition and, most importantly, completely time symmetric. Assume that the only types of edges

½ � �*m* � 1, �*m* are of the following two kinds: The first scenario is that the universe passes through the extreme phase into a state of zero entropy. The other scenario is that it passes into a state with high entropy (equal to 2*m*). Universes of one of these two types will be given the (un-normalized) probability 1 or *p*, respectively. Here *p*> 0 should be thought of as a very small number, at least when the size of the model becomes large. During the other extreme phase ½ � *m*, *m* þ 1 , near the Big

*Remark* 3. These assumptions may perhaps seem somewhat arbitrary. And to a certain extent, this may be so. However, they do represent the following viewpoint of what may happen at the full cosmological scale: we may think of the Big Bang and the Big Crunch as states of complete order with zero volume and entropy. Such states can very well be metastable, very much like an oversaturated gas at a temperature below the point of condensation. If no disturbance takes place, such metastable states can very well continue to exist for a substantial period of time. In particular, a low-entropy state can have a very good chance of surviving the intense but extremely short extreme phase. On the other hand, if a sufficiently large disturbance occurs, then the metastable state may almost immediately decay into a

It is not my intension to further argue in favor of this viewpoint here. The main thing in this chapter is to show that completely symmetric boundary conditions at

having a non-neglectable chance of occurring during the extreme phase

Crunch, we make the completely symmetric assumption.

the endpoints may give rise to a broken time symmetry.

• LH: The entropy is 0 at �*m* and 2*m* at *m*.

• HL: The entropy is 2*m* at �*m* and 0 at *m*.

probability weights for the corresponding types as

**317**

• LL: The entropy is low (=0) at both ends (�*m* and *m*).

• HH: The entropy is high (¼ 2*m*) at both ends (�*m* and *m*).

*PLL* <sup>¼</sup> *NLL*, *PLH* <sup>¼</sup> *pNLH*, *PHL* <sup>¼</sup> *pNHL*, *PHH* <sup>¼</sup> *<sup>p</sup>*<sup>2</sup>

If we now denote by *NLL*, *NLH*, *NHL* and *NHH* the number of paths of the indicated kinds, then with the above assumptions we also get the corresponding

We can now consider the following two types of broken time symmetry: **Definition 4.** A multiverse is said to exhibit a *weak* broken time symmetry if

**Definition 5.** A multiverse is said to exhibit a *strong* broken time symmetry if

Both these definitions should of course be made more precise when applied to specific models for the multiverse, e.g., by showing that the corresponding limits

*NHH:* (10)

*PLL* ≪ *PLH* þ *PHL:* (11)

*PLL* þ *PHH* ≪ *PLH* þ *PHL:* (12)

The multiverse now splits up into four different kinds of paths:

very disordered state of high entropy.

*Combinatorial Cosmology*

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

The number of rows and columns in the *B*-matrices are now given as follows:

$$B\_{12}: \mathbb{1} \times \mathbb{5}, \qquad B\_{23}: \mathbb{5} \times \mathbb{2} \mathbb{1}, \qquad B\_{34}: \mathbb{2} \mathbb{1} \times \mathbb{8} \mathbb{5}, \qquad B\_{45}: \mathbb{8} \mathbb{5} \times \mathbb{3} \mathbb{4} \mathbb{1}.\tag{7}$$

For the quadratic adjacency matrix *A*, this gives the format 453 � 453. The matrices *Bk*,*k*þ<sup>1</sup> can also be described as block matrices in the following way: *B*<sup>12</sup> ¼ ð Þ 0j0101 (the first element is always a 0 and among the other four, two randomly chosen elements will be one instead of zero). For the following matrix, we obtain (with certain random choices of ones as before)

$$B\_{23} = \left(\begin{array}{c|c|c} & \mathbf{C\_1} & \\ \hline \mathbf{C\_2} & & \mathbf{C\_3} \end{array}\right) -$$


Both *C*<sup>1</sup> and *C*<sup>3</sup> have rows containing only zeros, except for two randomly chosen positions where there are ones instead (these are the edges which connect to states with higher entropy one unit of time later), and *C*<sup>2</sup> is a column of zeros with two randomly chosen ones instead (these are the edges which connect to states with lower entropy one unit of time later).

The structures of *B*<sup>34</sup> and *B*<sup>45</sup> are similar:

$$B\_{34} = \left(\begin{array}{c|c|c|c} & D\_1 & & & \\ \hline D\_2 & & D\_3 & & \\ \hline & D\_4 & & & D\_5 \end{array}\right), \qquad B\_{45} = \left(\begin{array}{c|c|c} & E\_1 & & & \\ \hline E\_2 & & E\_3 & & \\ \hline & E\_4 & & E\_5 & \\ \hline & & E\_6 & & E\_7 \end{array}\right), \tag{9}$$

where now all *D*:s and *E*:s with odd indices have rows with two randomly chosen ones and those with even indices have columns with two randomly chosen ones.
