Statistical Analysis Near The Singularity
The sequencing order of era lengths k(s), measured by the number of Kasner epochs contained in them, exhibits the character of a random process. The source of this stochasticity is the rule eqs. 41–42 according to which the transition from one era to the next is determined from an infinite numerical sequence of u values.
In the statistical description of this sequence, instead of a fixed initial value u_{max} = k(0) + x(0), BKL consider values of x(0) that are distributed in the interval from 0 to 1 by some probabilistic distributional law. Then the values of x(s) that finish each (sth) number series will also be distributed according to some laws. It can be shown that with growing s these distributions converge to a definite static (sindependent) distribution of probabilities w(x) in which the initial conditions are completely "forgotten":

(eq._74">eq. 74)
This allows to find the distribution of probabilities for length k:

(eq. 75)
The above formulae are the basis on which the statistical properties of the model evolution are studied.
This study is complicated by the slow decrease of the distribution function eq. 75 at large k:

(eq. 76)
The mean value, calculated from this distribution, diverges logarithmically. For a sequence, cut off at a very large, but still finite number N, one has . The usefulness of the mean in this case is very limited because of its instability: because of the slow decrease of W(k), fluctuations in k diverge faster than its mean. A more adequate characteristic of this sequence is the probability that a randomly chosen number from it belongs to a series of length K where K is large. This probability is lnK/lnN. It is small if . In this respect one can say that a randomly chosen number from the given sequence belongs to the long series with a high probability.
The recurrent formulae defining transitions between eras are rewritten and detailed below. Index s numbers the successive eras (not the Kasner epochs in a given era!), beginning from some era (s = 0) defined as initial. Ω(s) and ε(s) are, respectively, the initial moment and initial matter density in the sth era; δ_{s}Ω_{s} is the initial oscillation amplitude of that pair of functions α, β, γ, which oscillates in the given era: k(s) is the length of sth era, and x(s) determines the length of the next era according to k(s+1) = . According to eqs. 71–73

(eq. 77)

(eq. 78)

(eq. 79)
(ξ_{s} is introduced in eq. 77 to be used further on).
The values of δ(s) (ranging from 0 to 1) have their own static statistical distribution. It satisfies an integral equation expressing the fact that δ(s) and δ(s+1) which are related through eq. 78 have an identical distribution; this equation can be solved numerically (cf.). Since eq. 78 does not contain a singularity, the distribution is perfectly stable; the mean values of δ or its powers calculated through it are definite finite numbers. In particular, the mean value of δ is
The statistical relation between large time intervals Ω and the number of eras s contained in them is found by repeated application of eq. 77:

(eq. 80)
Direct averaging of this equation, however, does not make sense: because of the slow decrease of function W(k) mean values of exp(ξ_{s}) are unstable in the above sense. This instability is removed by taking logarithm: the "doublelogarithmic" time interval

(eq. 81)
is expressed by the sum of values ξ_{p} which have a stable statistical distribution. The mean values of ξ_{s} and their powers (calculated from the distributions of values x, k and δ) are finite; numeric calculation gives
Averaging eq. 81 at a given s obtains

(eq. 82)
which determines the mean doublelogarithmic time interval containing s successive eras.
In order to calculate the mean square of fluctuations of this value one writes
In the last equation, it is taken into account that in the static limit the statistical correlation between ξ(s) and ξ′(s) depends only on the difference  s − s′ . Due to the existing recurrent relationship between x(s), k(s), δ(s) and x(s+1), k(s+1), δ(s+1) this correlation is, strictly speaking, different from zero. It, however, quickly decreases with increasing  s − s′  and numeric calculation shows that even at  s − s′  = 1, = −0.4. Leaving the first two terms in the sum by p, one obtains

(eq. 83)
At s → ∞ the relative fluctuation (i.e., the ratio between the mean squared fluctuations eq. 83 and the mean value eq. 82), therefore, approaches zero as s−1/2. In other words, the statistical relationship eq. 82 at large s becomes close to certainty. This is a corollary that according to eq. 81 τ_{s} can be presented as a sum of a large number of quasiindependent additives (i.e., it has the same origin as the certainty of the values of additive thermodynamic properties of macroscopic bodies). Therefore, the probabilities of various τ_{s} values (at given s) have a Gaussian distribution:

(eq. 84)
Certainty of relationship eq. 82 allows its reversal, i.e., express it as a dependence of the mean number of eras contained in a given interval of doublelogarithmic time τ:

(eq. 85)
The respective statistical distribution is given by the same Gaussian distribution in which the random variable is now s_{τ} at a given τ:

(eq. 86)
Respective to matter density, eq. 79 can be rewritten with account of eq. 80 in the form
and then, for the complete energy change during s eras,

(eq. 87)
The term with the sum by p gives the main contribution to this expression because it contains an exponent with a large power. Leaving only this term and averaging eq. 87, one gets in its right hand side the expression which coincides with eq. 82; all other terms in the sum (also terms with η_{s} in their powers) lead only to corrections of a relative order 1/s. Therefore

(eq. 88)
Thanks to the above established almost certain character of the relation between τ_{s} and s eq. 88 can be written as
which determines the value of the double logarithm of density increase averaged by given doublelogarithmic time intervals τ or by a given number of eras s.
These stable statistical relationships exist specifically for doublelogarithmic time intervals and for the density increase. For other characteristics, e.g., ln (ε(s)/ε(0)) the relative fluctuation increase by a power law with the increase of the averaging range thereby devoiding the term mean value of its sense of stability.
As shown below, in the limiting asymptotic case the abovementioned "dangerous" cases that disturb the regular course of evolution expressed by the recurrent relationships eqs. 77–79, do not occur in reality.
Dangerous are cases when at the end of an era the value of the parameter u = x (and with it also p_{1} ≈ x). A criterion for selection of such cases is the inequality

(eq. 89)
where  α(s)  is the initial minima depth of the functions that oscillate in era s (it would have been better to take the final amplitude, but that would only strengthen the selection criterion).
The value of x(0) in the first era is determined by the initial conditions. Dangerous are values in the interval δx(0) ~ exp ( −  α(0)  ), and also in intervals that could result in dangerous cases in the next eras. In order that x(s) comes into the dangerous interval δx(s) ~ exp ( −  α(s)  ), the initial value x(0) should lie into an interval of a width δx(0) ~ δx(s) / k(1)^2 ... k(s)^2. Therefore, from a unit interval of all possible values of x(0), dangerous cases will appear in parts λ of this interval:

(eq. 90)
(the inner sum is taken by all values k(1), k(2), ..., k(s) from 1 to ∞). It is easy to show that this series converges to the value λ 1 whose order of magnitude is determined by the first term in eq. 90. This can be shown by a strong majoration of the series for which one substitutes  α(s)  = (s + 1)  α(0) , regardless of the lengths of eras k(1), k(2), ... (In fact  α(s)  increase much faster; even in the most unfavorable case k(1) = k(2) = ... = 1 values of  α(s)  increase as qs  α(0)  with q > 1.) Noting that
one obtains
If the initial value of x(0) lies outside the dangerous region λ there will be no dangerous cases. If it lies inside this region dangerous cases occur, but upon their completion the model resumes a "regular" evolution with a new initial value which only occasionally (with a probability λ) may come into the dangerous interval. Repeated dangerous cases occur with probabilities λ2, λ3, ..., asymptopically converging to zero.
Read more about this topic: BKL Singularity, Metric Evolution
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