Very large
u values correspond to Kasner powers {{NumBlk|:|p_1 \approx -\frac{1}{u},\ p_2 \approx \frac{1}{u},\ p_3 \approx 1-\frac{1}{u^2},|}} which are close to the values (0, 0, 1). Two values that are close to zero, are also close to each other, and therefore the changes in two out of the three types of "perturbations" (the terms with λ, μ and ν in the right hand sides of ) are also very similar. If in the beginning of such long era these terms are very close in absolute values in the moment of transition between two Kasner epochs (or made artificially such by assigning initial conditions) then they will remain close during the greatest part of the length of the whole era. In this case (BKL call this the case of
small oscillations), analysis based on the action of one type of perturbations becomes incorrect; one must take into account the simultaneous effect of two perturbation types.
Two perturbations Consider a long era, during which two of the functions
a,
b,
c (let them be
a and
b) undergo small oscillations while the third function (
c) decreases monotonously. The latter function quickly becomes small; consider the solution just in the region where one can ignore
c in comparison to
a and
b. The calculations are first done for the Type IX space model by substituting accordingly λ = μ = ν = 1. From one obtains :\gamma_\xi=\frac{1}{4}\xi\left(2\chi_\xi^2+\chi^2\right)=A^2,\ \gamma=A^2\left(\xi-\xi_0\right)+\mathrm{const}. After determining α and β from and and expanding
eα and
eβ in series according to the above approximation, one obtains finally: {{NumBlk|:|\begin{cases} a\\ b \end{cases}=a_0\sqrt{\frac{\xi}{\xi_0}}\left[1\pm \frac{A}{\sqrt{\xi}}\sin \left(\xi-\xi_0\right)\right],|}} {{NumBlk|:|c=c_0 e^{-A^2\left(\xi_0-\xi\right)}.|}} The relation between the variable ξ and time
t is obtained by integration of the definition
dt =
abc dτ which gives {{NumBlk|:|\frac{t}{t_0}=e^{-A^2\left(\xi_0-\xi\right)}.|}} The constant
c0 (the value of
с at ξ = ξ0) should be now
c0 \ll α0· Let us now consider the domain ξ \ll 1. Here the major terms in the solution of are: :\chi=\alpha-\beta=k\ln \xi+\mathrm{const},\, where
k is a constant in the range − 1 2
k and ξ−2
k). Then, after determining α, β, and
t, one obtains {{NumBlk|:|a \sim \xi^{\frac{1+k}{2}},\ b \sim \xi^{\frac{1-k}{2}},\ c \sim \xi^{-\frac{1-k^2}{4}},\ t \sim \xi^{\frac{3+k^2}{4}}.|}} This is again a Kasner mode with the negative
t power present in the function
c(
t). These results picture an evolution that is qualitatively similar to that, described above. During a long period of time that corresponds to a large decreasing ξ value, the two functions
a and
b oscillate, remaining close in magnitude \tfrac{a-b}{a} \sim \tfrac{1}{\sqrt{\xi}}; in the same time, both functions
a and
b slowly (\sim \sqrt{\xi}) decrease. The period of oscillations is constant by the variable ξ : Δξ = 2π (or, which is the same, with a constant period by logarithmic time: Δ ln
t = 2π
Α2). The third function,
c, decreases monotonously by a law close to
c =
c0
t/
t0. This evolution continues until ξ ≈1 and formulas and are no longer applicable. Its time duration corresponds to change of
t from
t0 to the value
t1, related to ξ0 according to {{NumBlk|:|A^2\xi_0=\ln \frac{t_0}{t_1}.|}} The relationship between ξ and
t during this time can be presented in the form {{NumBlk|:|\frac{\xi}{\xi_0}=\frac{\ln \tfrac{t}{t_1}}{\ln \tfrac{t_0}{t_1}}.|}} After that, as seen from , the decreasing function
c starts to increase while functions
a and
b start to decrease. This Kasner epoch continues until terms
c2/
a2
b2 in become ~
t2 and a next series of oscillations begins. The law for density change during the long era under discussion is obtained by substitution of in : {{NumBlk|:|\varepsilon \sim \left(\frac{\xi_0}{\xi}\right)^2.|}} When ξ changes from ξ0 to ξ ≈1, the density increases \xi^2_0 times. It must be stressed that although the function
c(
t) changes by a law, close to
c ~
t, the metric does not correspond to a Kasner metric with powers (0, 0, 1). The latter corresponds to an exact solution found by Taub which is allowed by eqs.
– and in which {{NumBlk|:|a^2=b^2=\frac{p}{2}\frac{\mathrm{ch}(2p\tau+\delta_1)}{\mathrm{ch}^2(p\tau+\delta_2)}, \; c^2=\frac{2p}{\mathrm{ch}(2p\tau+\delta_1)},|}} where
p, δ1, δ2 are constant. In the asymptotic region τ → −∞, one can obtain from here
a =
b = const,
c = const.
t after the substitution
ерτ =
t. In this metric, the singularity at
t = 0 is non-physical. Let us now describe the analogous study of the Type VIII model, substituting in eqs.
– λ = −1, μ = ν = 1. During each Kasner epoch
abc = Λ
t,
i. e. α + β + γ = ln Λ + ln
t. On changing over from one epoch (with a given value of the parameter
u) to the next epoch the constant Λ is multiplied by 1 + 2
p1 = (1 –
u +
u2)/(1 +
u +
u2) 1) ≈ −2/
u. For a large
k the maximal value
u(max) =
k +
x ≈ k. Hence the entire variation of ln Λ during an era is given by a sum of the form \sum \ln \left ( 1 + 2p_1 \right ) = \dots + \frac{1}{k-2} + \frac{1}{k-1} + \frac{1}{k} with only the terms that correspond to large values of
u written down. When
k increases this sum increases as ln
k. But the probability for an appearance of an era of a large length
k decreases as 1/
k2 according to ; hence the mean value of the sum above is finite. Consequently, the systematic variation of the quantity ln Λ over a large number of eras will be proportional to this number. But it is seen in that with
t → 0 the number
s increases merely as ln |ln
t|. Thus in the asymptotic limit of arbitrarily small
t the term ln Λ can indeed be neglected as compared to ln
t. In this approximation where Ω denotes the "logarithmic time" and the process of epoch transitions can be regarded as a series of brief time flashes. The magnitudes of maxima of the oscillating scale functions are also subject to a systematic variation. From for u ≫ 1 it follows that a_{\max}^\prime - a_{\max} \approx -1/2 u . In the same way as it was done above for the quantity ln Λ, one can hence deduce that the mean decrease in the height of the maxima during an era is finite and the total decrease over a large number of eras increases with
t → 0 merely as ln Ω. At the same time the lowering of the minima, and by the same token the increase of the
amplitude of the oscillations, proceed () proportional to Ω. In correspondence with the adopted approximation the lowering of the maxima is neglected in comparison with the increase of the amplitudes so for the maximal values of all oscillating functions and the quantities α, β, γ run only through negative values that are connected with one another at each instant of time by the relation . Considering such instant change of epochs, the transition periods are ignored as small in comparison to the epoch length; this condition is actually fulfilled. Replacement of α, β, and γ maxima with zeros requires that quantities ln (|
p1|Λ) be small in comparison with the amplitudes of oscillations of the respective functions. As mentioned
above, during transitions between eras |
p1| values can become very small while their magnitude and probability for occurrence are not related to the oscillation amplitudes in the respective moment. Therefore, in principle, it is possible to reach so small |
p1| values that the above condition (zero maxima) is violated. Such drastic drop of can lead to various special situations in which the transition between Kasner epochs by the rule becomes incorrect (including the situations described
above). These "dangerous" situations could break the laws used for the statistical analysis below. As mentioned, however, the probability for such deviations converges asymptotically to zero; this issue will be discussed below. Consider an era that contains
k Kasner epochs with a parameter
u running through the values and let α and β are the oscillating functions during this era (Fig. 4). Initial moments of Kasner epochs with parameters
un are Ω
n. In each initial moment, one of the values α or β is zero, while the other has a minimum. Values α or β in consecutive minima, that is, in moments Ω
n are (not distinguishing minima α and β). Values δ
n that measure those minima in respective Ω
n units can run between 0 and 1. Function γ monotonously decreases during this era; according to its value in moment Ω
n is During the epoch starting at moment Ω
n and ending at moment Ω
n+1 one of the functions α or β increases from −δ
nΩ
n to zero while the other decreases from 0 to −δ
n+1Ω
n+1 by linear laws, respectively: :\mathrm{const} + |p_1(u_n)|\Omega \, and \mathrm{const} - p_2(u_n)\Omega \, resulting in the
recurrence relation {{NumBlk|:|\delta_{n+1}\Omega_{n+1} = \frac{1+u_n}{u_n} \delta_n\Omega_n = \frac{1+u_0}{u_n} \delta_0\Omega_0|}} and for the logarithmic epoch length {{NumBlk|:|\Delta_{n+1} \equiv \Omega_{n+1} - \Omega_n = \frac{f(u_n)}{u_n} \delta_n\Omega_n = \frac{f(u_n)(1+u_{n-1})}{f(u_{n-1})u_n}\Delta_n,|}} where, for short,
f(
u) = 1 +
u +
u2. The sum of
n epoch lengths is obtained by the formula {{NumBlk|:|\Omega_n - \Omega_0 = \left [n(n-1) + \frac{nf(u_{n-1})}{u_{n-1}}\right ] \delta_0\Omega_0. |}} It can be seen from that |α
n+1| > |α
n|, i.e., the oscillation amplitudes of functions α and β increase during the whole era although the factors δ
n may be small. If the minimum at the beginning of an era is deep, the next minima will not become shallower; in other words, the residue |α — β| at the moment of transition between Kasner epochs remains large. This assertion does not depend upon era length
k because transitions between epochs are determined by the common rule also for long eras. The last oscillation amplitude of functions α or β in a given era is related to the amplitude of the first oscillation by the relationship |α
k−1| = |α0| (
k +
x) / (1 +
x). Even at
ks as small as several units
x can be ignored in comparison to
k so that the increase of α and β oscillation amplitudes becomes proportional to the era length. For functions
a =
eα and
b =
eβ this means that if the amplitude of their oscillations in the beginning of an era was
A0, at the end of this era the amplitude will become A_0^{k/(1+x)}. The length of Kasner epochs (in logarithmic time) also increases inside a given era; it is easy to calculate from that Δ
n+1 > Δ
n. The total era length is {{NumBlk|:|\Omega^\prime_0 - \Omega_0 \equiv \Omega_k - \Omega_0 = k \left ( k + x + \frac{1}{x} \right ) \delta_0\Omega_0|}} (the term with 1/
x arises from the last,
k-th, epoch whose length is great at small
x; cf. Fig. 2). Moment Ω
n when the
k-th epoch of a given era ends is at the same time moment Ω'0 of the beginning of the next era. In the first Kasner epoch of the new era function γ is the first to rise from the minimal value γ
k = − Ω
k (1 − δ
k) that it reached in the previous era; this value plays the role of a starting amplitude δ'0Ω'0 for the new series of oscillations. It is easily obtained that: {{NumBlk|:|\delta^\prime_0 \Omega^\prime_0 = \left ( \delta_0^{-1} + k^2 + kx - 1 \right ) \delta_0 \Omega_0.|}} It is obvious that δ'0Ω'0 > δ0Ω0. Even at not very great
k the amplitude increase is very significant: function
c =
eγ begins to oscillate from amplitude A_0 ' \sim A_0^{k^2}. The issue about the above-mentioned "dangerous" cases of drastic lowering of the upper oscillation limit is left aside for now. According to the increase in matter density during the first (
k − 1) epochs is given by the formula :\ln \left ( \frac{\varepsilon_{n+1}}{\varepsilon_n} \right ) = 2 \left [ 1 - p_3 ( u_n ) \right ] \Delta_{n+1}. For the last
k epoch of a given era, at
u =
x 2(
x) (not
p3(
x) ). Therefore, for the density increase over the whole era one obtains {{NumBlk|:|\ln \left ( \frac{ \varepsilon_k }{ \varepsilon_0 } \right ) \equiv \ln \left ( \frac{ \varepsilon_0 ' }{ \varepsilon_0 } \right ) = 2 (k - 1 + x ) \delta_0 \Omega_0.|}} Therefore, even at not very great
k values, \varepsilon_0' / \varepsilon_0 \sim A_0^{2k}. During the next era (with a length
k ' ) density will increase faster because of the increased starting amplitude
A0': \varepsilon_0'' / \varepsilon_0' \sim A_0'^{2k''} \sim A_0^{2k^2 k'}, etc. These formulae illustrate the steep increase in matter density.
Statistical analysis near the singularity The sequence of era lengths
k(
s), measured by the number of Kasner epochs contained in them, acquires asymptotically the character of a random process. The same pertains also to the sequence of the interchanges of the pairs of oscillating functions on going over from one era to the next (it depends on whether the numbers
k(
s) are even or odd). A source of this stochasticity is the rule – according to which the transition from one era to the next is determined in an infinite numerical sequence of
u values. This rule states, in other words, that if the entire infinite sequence begins with a certain initial value u_\max^{(0)} = k^{(0)} + x^{(0)}, then the lengths of the eras
k(0),
k(1), ..., are the numbers in the
simple continued fraction expansion {{NumBlk|:|k^{(0)} + x^{(0)} = k^{(0)} + \frac{1}{k^{(1)} + \frac{1}{k^{(2)} + \dots}}.|}} This expansion corresponds to the mapping transformation of the interval [0, 1] onto itself by the formula
Tx = {1/
x}, i.e.,
xs+1 = {1/
xs}. This transformation belongs to the so-called expanding transformations of the interval [0, 1], i.e., transformations
x →
f(
x) with |
f′(
x)| > 1. Such transformations possess the property of exponential instability: if we take initially two close points their mutual distance increases exponentially under the iterations of the transformations. It is well known that the exponential instability leads to the appearance of strong stochastic properties. It is possible to change over to a probabilistic description of such a sequence by considering not a definite initial value
x(0) but the values
x(0) = x distributed in the interval from 0 to 1 in accordance with a certain
probabilistic distributional law w0(
x). Then the values of
x(s) terminating each era will also have distributions that follow certain laws
ws(x). Let
ws(x)dx be the probability that the
s-th era terminates with the value u_\max^{(s)} = x lying in a specified interval
dx. The value
x(s) =
x, which terminates the
s-th era, can result from initial (for this era) values u_\max^{(s)} = x + k, where
k = 1, 2, ...; these values of u_\max^{(s)} correspond to the values
x(
s–1) = 1/(
k +
x) for the preceding era. Noting this, one can write the following recurrence relation, which expresses the distribution of the probabilities
ws(x) in terms of the distribution
ws–1(
x): w_{s}(x)dx = \sum_{k=1}^\infty w_{s-1} \left (\frac{1}{k+x} \right ) \left\vert d \frac{1}{k+x} \right\vert or {{NumBlk|:|w_{s}(x) = \sum_{k=1}^\infty \frac{1}{ \left (k+x \right )^{2}} w_{s-1} \left ( \frac{1}{k+x} \right ) .|}} If the distribution
ws(
x) tends with increasing
s to a stationary (independent of
s) limiting distribution
w(
x), then the latter should satisfy an equation obtained from by dropping the indices of the functions
ws−1(
x) and
ws(
x). This equation has a solution {{NumBlk|:|w(x) = \frac{1}{\left (1+x \right )\ln 2}|}} (normalized to unity and taken to the first order of
x). In order for the
s-th era to have a length
k, the preceding era must terminate with a number
x in the interval between 1/(
k + 1) and 1/
k. Therefore, the probability that the era will have a length
k is equal to (in the stationary limit) {{NumBlk|:|W(k) = \int\limits_{1/(k+1)}^{1/k}w(x)\, dx = \frac{1}{\ln 2} \ln \frac{( k+1 )^2}{k (k+2)} .|}} At large values of
k {{NumBlk|:|W(k) \approx \frac{1}{k^2 \ln 2}. |}} In relating the statistical properties of the cosmological model with the
ergodic properties of the transformation
xs+1 = {1/
xs} an important point must be mentioned. In an infinite sequence of numbers
x constructed in accordance with this rule, arbitrarily small (but never vanishing) values of
x will be observed corresponding to arbitrarily large lengths k. Such cases can (by no means necessarily!) give rise to certain specific situations when the notion of eras, as of sequences of Kasner epochs interchanging each other according to the rule , loses its meaning (although the oscillatory mode of evolution of the model still persists). Such an "anomalous" situation can be manifested, for instance, in the necessity to retain in the right-hand side of terms not only with one of the functions
a,
b,
c (say,
a4), as is the case in the "regular" interchange of the Kasner epochs, but simultaneously with two of them (say,
a4,
b4,
a2
b2). On emerging from an "anomalous" series of oscillations a succession of regular eras is restored. Statistical analysis of the behavior of the model which is entirely based on regular iterations of the transformations is corroborated by an important theorem: the probability of the appearance of anomalous cases tends asymptotically to zero as the number of iterations
s → ∞ (i.e., the time
t → 0) which is proved at the end of this section. The validity of this assertion is largely due to a very rapid rate of increase of the oscillation amplitudes during every era and especially in transition from one era to the next one. The process of the relaxation of the cosmological model to the "stationary" statistical regime (with t → 0 starting from a given "initial instant") is less interesting, however, than the properties of this regime itself with due account taken for the concrete laws of the variation of the physical characteristics of the model during the successive eras. An idea of the rate at which the stationary distribution sets in is obtained from the following example. Let the initial values
x(0) be distributed in a narrow interval of width δ
x(0) about some definite number. From the recurrence relation (or directly from the expansion ) it is easy to conclude that the widths of the distributions
ws(
x) (about other definite numbers) will then be equal to {{NumBlk|:|\delta x^{(s)} \approx \delta x^{(0)} \cdot k^{(1)2} k^{(2)2} \dots k^{(s)2}|}} (this expression is valid only so long as it defines quantities δ
x(s) ≪ 1). The mean value \bar k, calculated from this distribution, diverges logarithmically. For a sequence, cut off at a very large, but still finite number
N, one has \bar k \sim \ln N. 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 an era of length
K where
K is large. This probability is ln
K / ln
N. It is small if 1 \ll K \ll N. In this respect one can say that a randomly chosen number from the given sequence belongs to the long era with a high probability. It convenient to average expressions that depend simultaneously on
k(
s) and
x(
s). Since both these quantities are derived from the same quantity
x(
s–1) (which terminates the preceding era), in accordance with the formula
k(
s) +
x(
s) = 1/
x(
s–1), their
statistical distributions cannot be regarded as independent. The joint distribution
Ws(
k,
x)
dx of both quantities can be obtained from the distribution
ws–1(
x)
dx by making in the latter the substitution
x → 1/(
x +
k). In other words, the function
Ws(
k,
x) is given by the very expression under the summation sign in the right side of . In the stationary limit, taking
w from , one obtains {{NumBlk|:|W(k, x) = \frac{k + x + 1}{\left (k + x \right ) \ln 2}. |}} Summation of this distribution over
k brings us back to , and integration with respect to
dx to . The recurrent formulas defining transitions between eras are re-written with index
s numbering 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
s-th 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
s-th era, and
x(
s) determines the length (number of Kasner epochs) of the next era according to
k(
s+1) = [1/
x(
s)]. According to – {{NumBlk|:|\Omega^{(s+1)} / \Omega^{(s)} = 1 + \delta^{(s)} k^{(s)} \left ( k^{(s)} + x^{(s)} + \frac{1}{x^{(s)}} \right ) \equiv \exp \xi^{(s)},|}} {{NumBlk|:|\delta^{(s+1)} = 1 - \frac{\left ( k^{(s)} / x^{(s)} + 1 \right ) \delta^{(s)}}{1 + \delta^{(s)} k^{(s)} \left ( 1 + x^{(s)} + 1 / x^{(s)} \right )},|}} {{NumBlk|:|\ln \left ( \frac{\varepsilon^{(s+1)}}{\varepsilon^{(s)}} \right ) = 2 \left ( k^{(s)} + x^{(s)} - 1 \right ) \delta^{(s)} \Omega^{(s)} |}} (ξ(
s) is introduced in to be used further on). The quantities δ(
s) have a stable stationary statistical distribution
P(δ) and a stable (small relative fluctuations) mean value. For their determination KL showed that the distribution
P(δ) can actually be found exactly by an analytical method (see
Fig. 5). For the statistical properties in the stationary limit, it is reasonable to introduce the so-called natural extension of the transformation
Tx = {1/
x} by continuing it without limit to negative indices. Otherwise stated, this is a transition from a one-sided infinite sequence of the numbers (
x0,
x1,
x2, ...), connected by the equalities
Tx = {1/
x}, to a "doubly infinite" sequence
X = (...,
x−1,
x0,
x1,
x2, ...) of the numbers which are connected by the same equalities for all –∞
s–1 is not determined uniquely by
xs), but all statistical properties of the extended sequence are uniform over its entire length, i.e., are invariant with respect to arbitrary shift (and
x0 loses its meaning of an "initial" condition). The sequence
X is equivalent to a sequence of integers
K = (...,
k−1,
k0,
k1,
k2, ...), constructed by the rule
ks = [1/
xs–1]. Inversely, every number of X is determined by the integers of K as an infinite
continued fraction {{NumBlk|:|x_{s} = \frac{1}{k_{s+1} + \frac{1}{k_{s+2} + \dots}} \equiv x_{s+1}^{+}|}} (the convenience of introducing the notation x_{s+1}^{+} with an index shifted by 1 will become clear in the following). For concise notation the continuous fraction is denoted simply by enumeration (in square brackets) of its denominators; then the definition of x_{s}^{+} can be written as {{NumBlk|:|x_{s}^{+} = \left [ k_{s}, k_{s+1}, \dots \right ]|}} Reverse quantities are defined by a continued fraction with a retrograde (in the direction of diminishing indices) sequence of denominators {{NumBlk|:|x_{s}^{-} = \left [ k_{s-1}, k_{s-2}, \dots \right ]|}} The recurrence relation is transformed by introducing temporarily the notation
ηs = (1 − δ
s)/δ
s. Then can be rewritten as \eta_{s+1} = \frac{1}{\eta_{s} x_{s-1} + k_{s}} By iteration an infinite continuous fraction is obtained \eta_{s+1} x_s = \left [ k_{s}, k_{s-1}, \dots \right ] = x_{s+1}^{-} Hence \eta_{s} = x_{s}^{-} / x_{s}^{+} and finally {{NumBlk|:|\delta_{s} = \frac{x_{s}^{+}}{x_{s}^{+} + x_{s}^{-}}|}} This expression for δ
s contains only two (instead of the three in Since by δs is expressed in terms of the random quantities
x+ and
x−, the knowledge of their joint distribution makes it possible to calculate the statistical distribution
P(δ) by integrating
P(
x+,
x−) over one of the variables at a constant value of δ. Due to symmetry of the function with respect to the variables
x+ and
x−,
P(δ) =
P(1 − δ), i.e., the function
P(δ) is symmetrical with respect to the point δ = 1/2. Then P(\delta)\ d\delta = d\delta \int_0^1 P \left ( x^{+}, \frac{x^{+} \delta}{1 - \delta} \right ) \left ( \frac{\partial x^{-}}{\partial \delta} \right )_{x^{+}} d x^{+} On evaluating this integral (for 0 ≤ δ ≤ 1/2 and then making use of the aforementioned symmetry), finally {{NumBlk|:|P(\delta) = \frac{1}{\left (|1 - 2 \delta| + 1 \right ) \ln 2}|}} The mean value \bar{\delta}= 1/2 already as a result of the symmetry of the function
P(δ). Thus the mean value of the initial (in every era) amplitude of oscillations of the functions α, β, γ increases as Ω/2. The statistical relation between large time intervals Ω and the number of eras
s contained in them is found by repeated application of : {{NumBlk|:|\frac{\Omega^{(s)}}{\Omega^{(0)}} = \exp \left ( \sum_{p=0}^{s-1} \xi^{(p)} \right ).|}} Direct averaging of this equation, however, does not make sense: because of the slow decrease of function
W(
k) , the average values of the quantity exp ξ(
s) are unstable in the above sense – the fluctuations increase even more rapidly than the mean value itself with increasing region of averaging. This instability is eliminated by taking the logarithm: the "doubly-logarithmic" time interval {{NumBlk|:|\tau_s \equiv \ln \left ( \frac{\Omega^{(s)}}{\Omega^{(0)}} \right ) = \ln | \ln t_s | - \ln | \ln t_0 | = \sum_{p=0}^{s-1} \xi^{(p)} |}} is expressed by the sum of quantities ξ(
p) which have a stable statistical distribution. The mean value of τ is \bar{\tau} = s \bar{\xi}. To calculate \bar{\xi} note that can be rewritten as {{NumBlk|:|\xi_s = \ln \frac{\left ( k_s + x_s \right ) \delta_s} {x_s \left (1 - \delta_{s+1} \right )} = \ln \frac{\delta_s} {x_s x_{s-1} \left ( 1 - \delta_{s+1} \right )}|}} For the stationary distribution \overline{\ln x_s} = \overline{\ln x_{s-1}}, and in virtue of the symmetry of the function
P(δ) also \overline{\ln \delta_s} = \overline{\ln \left ( \delta_{s+1} \right )}. Hence \bar{\xi} = -2 \overline{\ln x} = -2 \int_0^1 w (x) \ln x\ dx = \frac{\pi^2}{6 \ln 2} = 2.37 (
w(
x) from ). Thus {{NumBlk|:|\overline{\tau_s} = 2.37s, |}} which determines the mean doubly-logarithmic time interval containing
s successive eras. For large
s the number of terms in the sum is large and according to general theorems of the ergodic theory the values of τs are distributed around \overline{\tau_s} according to
Gauss's law with the density {{NumBlk|:| \rho (\tau_s) = \left ( 2 \pi D_{\tau} \right )^{-1/2} \exp \left [ - \left ( \tau_s - \overline{\tau_s} \right )^2 / 2 D_{\tau} \right ] |}} Calculation of the variance
Dτ is more complicated since not only the knowledge of \bar{\xi} and \overline{\xi^2} are needed but also of the correlations \overline{\xi_p \xi_{p \prime}}. The calculation can be simplified by rearranging the terms in the sum . By using the sum can be rewritten as \sum_{p=1}^s \xi_p = \ln \prod_{p=1}^s \frac{\delta_p}{\left (1 - \delta_{p+1} \right ) x_p x_{p-1}} = \ln \prod_{p=1}^s \frac{\delta_p}{\left (1 - \delta_{p} \right ) x_{p-1}^2} + \ln \frac{x_0}{x_s} + \ln \frac{1 - \delta_1}{1 - \delta_{s+1}} The last two terms do not increase with increasing
s; these terms can be omitted as the limiting laws for large
s are dominating. Then {{NumBlk|:|\sum_{p=1}^s \xi_p = \sum_{p=1}^s \ln \frac{x_p^{-}}{x_p^{+}}|}} (the expression for δp is taken into account). To the same accuracy (i.e., up to the terms which do not increase with
s) the equality {{NumBlk|:|\sum_{p=1}^s \xi_p^{+} = \sum_{p=1}^s \xi_p^{-}|}} is valid. Indeed, in virtue of x_{p+1}^{+} + \frac{1}{x_{p+1}^{-}} = \frac{1}{x_p^{+}} + x_p^{-} and hence \ln \left ( 1 + x_{p+1}^{+} x_{p+1}^{-} \right ) - \ln x_{p+1}^{-} = \ln \left ( 1 + x_{p}^{+} x_{p}^{-} \right ) - \ln x_{p}^{+} By summing this identity over
p is obtained. Finally again with the same accuracy x_p^{+} is changed for
xp under the summation sign and thus represent τ
s as {{NumBlk|:|\tau_s = \sum_{p=1}^{\infty} \eta_p, \quad \eta_p = -2 \ln x_p|}} The variance of this sum in the limit of large
s is {{NumBlk|:|D_{{\tau}_s} = \overline{ \left (\tau_s - \overline{\tau_s} \right )^2} \approx s \left \{ \overline{\eta^2} - \bar{\eta}^2 + 2 \sum_{p=1}^{\infty} \left ( \overline{\eta_0 \eta_p} - \bar{\eta}^2 \right ) \right \}|}} It is taken into account that in virtue of the statistical homogeneity of the sequence
X the correlations \overline{\eta_p \eta_{p \prime}} depend only on the differences |
p −
p′|. The mean value \bar{\eta} = \bar{\xi}; the mean square \overline{\eta^2} = 4 \int_0^1 w(x) \ln^2 x\ dx = \frac{6 \xi (3)}{\ln 2} = 10.40 By taking into account also the values of correlations \overline{\eta_0 \eta_p} with
p = 1, 2, 3 (calculated numerically) the final result
Dτ
s = (3.5 ± 0.1)
s is obtained. At increasing
s the relative fluctuation D_{{\tau}_s} / \overline{\tau_s} tends to zero as
s−1/2. In other words, the statistical relation becomes almost certain at large
s. This makes it possible to invert the relation, i.e., to represent it as the dependence of the average number of the eras
sτ that are interchanged in a given interval τ of the double logarithmic time: {{NumBlk|:|\overline{s_{\tau}} = 0.47\tau.|}} The statistical distribution of the exact values of
sτ around its average is also Gaussian with the variance D_{s_{\tau}} = 3.5 \frac{\overline{s_{\tau}}^3}{\tau^2} = 0.26 \tau The respective statistical distribution is given by the same Gaussian distribution in which the random variable is now
sτ at a given τ: {{NumBlk|:|\rho(s_\tau) \propto \exp \left \{ - \left ( s_\tau - 0.47\tau \right )^2/0.43\tau \right \}.|}} From this point of view, the source of the statistical behavior is the arbitrariness in the choice of the starting point of the interval τ superimposed on the infinite sequence of the interchanging eras. Respective to matter density, can be re-written with account of in the form :\ln \ln \frac{\varepsilon^{(s+1)}}{\varepsilon^{(s)}} = \eta_s + \sum_{p=0}^{s-1} \xi_p, \quad \eta_s = \ln \left [ 2\delta^{(s)} \left ( k^{(s)} + x^{(s)} - 1 \right ) \Omega^{(0)} \right ] and then, for the total energy change during
s eras, {{NumBlk|:|\ln \ln \frac{\varepsilon^{(s)}}{\varepsilon^{(0)}} = \ln \sum_{p=0}^{s-1} \exp \left \{ \sum_{q=0}^p\xi_q + \eta_p \right \}.|}} 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 , one gets in its right hand side the expression s\bar{\xi} which coincides with ; 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, {{NumBlk|:|\overline{\ln \ln \left (\frac{\varepsilon^{(s)}}{\varepsilon^{(0)}} \right )} = \overline{\ln \left ( \frac{\Omega^{(s)}}{\Omega^{(0)}} \right )}.|}} By virtue of the almost certain character of the relation between τ
s and
s can be written as :\overline{\ln \ln \left ( \varepsilon_\tau/\varepsilon^{(0)} \right )} = \tau \quad \text{or} \quad \overline{\ln \ln \left ( \varepsilon^{(s)}/\varepsilon^{(0)} \right )} = 2.1 s, which determines the value of the double logarithm of density increase averaged by given double-logarithmic time intervals τ or by a given number of eras
s. These stable statistical relationships exist specifically for doubly-logarithmic time intervals and for the density increase. For other characteristics, e.g., ln (ε(
s)/ε(0)) or Ω(s) / Ω(0) = exp τs the relative fluctuation increase exponentially with the increase of the averaging range thereby voiding the term mean value of a stable meaning. The origin of the statistical relationship can be traced already from the initial law governing the variation of the density during the individual Kasner epochs. According to , during the entire evolution we have \ln \ln \varepsilon (t) = \text{const} + \ln \Omega + \ln 2 (1 - p_3 (t)), with 1 −
p3(
t) changing from epoch to epoch, running through values in the interval from 0 to 1. The term ln Ω = ln ln (1/
t) increases monotonically; on the other hand, the term ln2(1 −
p3) can assume large values (comparable with ln Ω) only when values of
p3 very close to unity appear (i.e., very small |
p1|). These are precisely the "dangerous" cases that disturb the regular course of evolution expressed by the recurrent relationships –. It remains to show that such cases actually do not arise in the asymptotic limiting regime. The spontaneous evolution of the model starts at a certain instant at which definite initial conditions are specified in an arbitrary manner. Accordingly, by "asymptotic" is meant a regime sufficiently far away from the chosen initial instant. Dangerous cases are those in which excessively small values of the parameter
u =
x (and hence also |
p1| ≈
x) appear at the end of an era. A criterion for selection of such cases is the inequality {{NumBlk|:|x^{(s)} \exp \left | \alpha^{(s)} \right | \lesssim 1,|}} where | α(
s) | is the initial minima depth of the functions that oscillate in era
s (it would be more appropriate to choose 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 δ
x(0) ~ exp ( − |α(0)| ), and also in intervals that could result in dangerous cases in the next eras. In order for
x(
s) to fall in the dangerous interval δ
x(
s) ~ exp ( − | α(
s) | ), the initial value
x(0) should lie into an interval of a width δ
x(0) ~ δ
x(
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: {{NumBlk|:|\lambda = \exp \left ( \left |-\alpha^{(s)} \right | \right ) + \sum_{s=1}^\infty \sum_k \frac{\exp \left ( \left |-\alpha^{(s)} \right | \right )}{k^{(1)^2} k^{(2)^2} \dots k^{(s)^2}}|}} (the inner sum is taken over all the values
k(1),
k(2), ... ,
k(
s) from 1 to ∞). It is easy to show that this era converges to the value λ \ll 1 whose order of magnitude is determined by the first term in . This can be shown by a strong majoration of the era 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 :\sum_k \frac{1}{k^{(1)^2} k^{(2)^2} \dots k^{(s)^2}} = \left ( \pi^2 / 6 \right )^s one obtains :\lambda = \exp \left ( \left |-\alpha^{(0)} \right | \right )\sum_{s=0}^\infty \left [ \left ( \pi^2 / 6 \right ) \exp \left ( \left |-\alpha^{(0)} \right | \right ) \right ]^s \approx \exp \left ( \left |-\alpha^{(0)} \right | \right ). 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, ... , asymptotically converging to zero. ==General solution with small oscillations==