content: text: Combinatorial Cosmology
DOI: http://dx.doi.org/10.5772/intechopen.90696

content: text: 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
having a non-neglectable chance of occurring during the extreme phase
[-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 2m). 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 [m, m + 1], near the Big
Crunch, we make the completely symmetric assumption.

content: text: 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 tem-
perature below the point of condensation. If no disturbance takes place, such meta-
stable 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 dis-
turbance occurs, then the metastable state may almost immediately decay into a
very disordered state of high entropy.

content: text: 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
the endpoints may give rise to a broken time symmetry.

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

content: text: · LL: The entropy is low (=0) at both ends ( -m and m).

content: text: · LH: The entropy is 0 at -m and 2m at m.

content: text: · HL: The entropy is 2m at -m and 0 at m.

content: text: · HH: The entropy is high (= 2m) at both ends (-m and m).

content: text: 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
probability weights for the corresponding types as

content: text: P_{LL}=N_{LL},\quadP_{LH}=pN_{LH},\quadP_{HL}=pN_{HL},\quadP_{HH}=p^2N_{HH}.

content: text: (10)

content: text: 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

content: text: P_{LL}\llP_{LH}+P_{HL}.

content: text: (11)

content: text: Definition 5. A multiverse is said to exhibit a strong broken time symmetry if

content: text: P_{LL}+P_{HH}\llP_{LH}+P_{HL}.

content: text: (12)

content: text: 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