Counting eras near a cosmological singularity

A spatially homogeneous universe can contract at different rates along three spatial axes. In the vacuum Bianchi IX model the relative rates change repeatedly as the singularity is approached. A succession of curvature transitions forms an era. Its length is an integer: it counts transitions. The Belinski, Khalatnikov and Lifshitz description predicts both short eras and occasional very long ones. Can one read the predicted frequencies from a single solution of the Einstein evolution equations?

Two bodies of work give this question its shape. On the differential-equation side, Ringström proved attraction to the Mixmaster attractor, and Brehm obtained estimates on approach to this set. On the discrete side, Damour and Lecian described era statistics for cosmological billiards. The distinction between asymptotic attraction and statistics along individual solutions is discussed by Heinzle and Uggla.

What is observed along one solution

The evolution is the exact normalized ordinary differential equation for vacuum Bianchi IX with zero cosmological constant. The physical states have three positive curvature walls and satisfy the constraint with value one. Time increases toward the singularity. On an orbit, a complete era has an entrance, a finite succession of core passages and the connection to the next entrance. Permuting the spatial axes lets all entrances be described in the same coordinates.

Write $t_n$ for the entrance times and $R_n$ for the radius in those coordinates. The era parameter is $R_n/2$, the phase is $y_n=2/R_n$, and the length $K_n$ equals both the number of core passages and $\lfloor R_n/2\rfloor$. An observation gives each era one vote. In particular, an era with many transitions makes just one contribution to a frequency.

The target: one exceptional set for all the statistics

Theorem 2.4 gives an invariant Borel exceptional set. Its pullback has measure zero on every regular physical transverse section and in every regular four-dimensional constraint chart, also for measures with densities. Outside this one set the entrances, frames and complete eras can be chosen measurably. The times $t_n$ are positive and increase strictly to infinity in normalized time.

For every continuous observable $f$ on $[0,1]$ the phases satisfy

$$\lim_{N\to\infty}\frac1N\sum_{n=0}^{N-1}f(y_n) =\frac1{\log 2}\int_0^1\frac{f(y)}{1+y}\,dy .$$

Every finite word of positive era lengths occurs with the Gauss measure of its continued-fraction cylinder. A cylinder consists of the numbers in $(0,1)$ whose continued fractions start with that word. In particular, for each integer $k\geq1$,

$$\lim_{N\to\infty}\frac{\#\{0\leq n<N:K_n\geq k\}}{N} =\log_2(1+1/k).$$

The intermediate results concern success on entrance graphs (Theorem 8.20), two regions of a physical section distinguished by the relative size of shear and curvature (Theorems 12.15 and 13.13), their common section (Theorem 13.15), and a measurable choice of entrance from physical initial data (Theorem 14.21).

Frequent short eras and an infinite mean

The one-digit law gives a concrete example: the fraction of eras of length one tends to $\log_2(4/3)$, about $41.5\%$. Long eras are rare enough to have a limiting frequency at every threshold, yet frequent enough to make the mean length diverge. If the count in each era is capped at an integer $r\geq1$, its mean instead tends to the finite value $\log_2(r+1)$. These conclusions and all finite-word frequencies hold along the same trajectory. The Lean formalization identifies the digits to which these laws apply with the lengths measured at physical entrances.

Why the statistics require a flow theorem

The discrete era map acts on a real variable and has a known invariant probability measure. The exact flow acts on a four-dimensional constraint surface. A typical orbit for the first measure supplies no physical initial condition for the second. Moreover, near a Taub direction neighboring initial data can have very different era lengths. An estimate that locates the orbit near the attractor leaves the relative frequencies of those lengths undetermined. The measure statement must therefore apply in the coordinates in which physical initial data can be prescribed, while the counted events must belong to one exact orbit.

Formalization scope

The development starts from the normalized vector field. It includes its maximal trajectories, section flux, physical crossings and their measurable selection, the coding by positive integers, and the transfer of continuous and cylinder averages. The Lean statement has two explicit literature hypotheses: the real-valued pointwise ergodic theorem for an ergodic probability-preserving real map (Walkden, Corollary 10.1.2), and decay of the complementary curvature-wall quantities outside the Taub subspaces (Brehm, Theorems 2 and 3). The latter input asserts decay along specified trajectories; the null-set and frequency conclusions are proved in the development.

Selected references

• H. Ringström, The Bianchi IX attractor, Annales Henri Poincaré (2001).
• J. M. Heinzle and C. Uggla, Mixmaster: Fact and Belief, Classical and Quantum Gravity (2009).
• T. Damour and O. M. Lecian, Statistical Properties of Cosmological Billiards, Physical Review D (2011).
• B. Brehm, Bianchi VIII and IX vacuum cosmologies: Almost every solution forms particle horizons and converges to the Mixmaster attractor, arXiv:1606.08058v1 (2016), Section 6, Theorem 2, (6.2c), and Theorem 3, pp. 44-45.
• C. Walkden, Ergodic Theory, University of Manchester lecture notes, Corollary 10.1.2, p. 89.