A study on the Belinski-Khalatnikov-Lifshitz scenario through quadrics of kinetic energy

TL;DR

The paper analyzes asymptotic dynamics near the cosmological singularity in the BKL scenario for a diagonal Bianchi IX reduction by recasting the equations in a Lagrangian framework and introducing a cone of kinetic energy in the velocity space. It introduces the $u$-variables ($u_1= frac{1}{3}\ln(abc)$, $u_2=\tfrac{1}{2}\ln(c/a)$, $u_3=\tfrac{1}{6}\ln(b^2/ac)$) to diagonalize the kinetic term, yielding a kinetic-energy cone $E_k=3\dot{u}_1^2-\dot{u}_2^2-3\dot{u}_3^2=\epsilon$ and a total-energy constraint $\mathcal{H}=0$ with an exponential potential. The main results show that the exact apex-ending solution, given by $u_1=\tfrac{1}{3}\ln\frac{10800}{|t-t_0|^9}$, $u_2=\tfrac{1}{2}\ln\frac{40}{|t-t_0|^4}$, $u_3=\tfrac{1}{6}\ln\frac{5}{2}$, is the unique differentiable path to all-direction collapse and is unstable, implying chaotic collapse; Kasner-like end-states do not strictly satisfy the BKL equations, though quasi-Kasner epochs arise as the trajectory reflects off hyperboloidal walls. The cone-and-wall framework provides a geometric, rigorous account of BKL chaos and a potentially generalizable approach to other Lagrangian systems with indefinite kinetic energy.

Abstract

A detailed description of the asymptotic behaviour in the Belinski-Khalatnikov-Lifshitz (BKL) scenario is presented through a simple geometric picture illustrating the geometry of their ordinary differential equations (ODE), which describe a neighbourhood of the cosmic singularity. The Lagrangian version of the dynamics governed by these equations is described in terms of trajectories inside a conical subset of the corresponding space of the generalised velocities. The calculations confirm that the initial conditions of decreasing volume inevitably result in eventual total collapse, while oscillations along paths reflecting from a hyperboloid, similar to those predicted by Kasner's solutions, occur on the way. The exact solution, found in our previous work, proves to be the only one that shrinks to a point along a differentiable path. Therefore, its instability means that the collapse is always chaotic. It is also shown that the BKL equations are not satisfied by the Kasner solutions exactly, even in the asymptotic regime, although the precision of their approximation may be high.

Figures (3)

• Figure 1: The lower half (=shrinking universe) of the cone $3\dot{u}_1^2-\dot{u}_2^2-3\dot{u}_3^2 > 0$. The dynamics of the system takes place inside the cone. The line with the arrow shows the exact solution; the arrow indicates its direction of evolution. For $t\to\infty$, the line tends to the apex of the cone. A position in the cone, together with the tangent to the trajectory, provide complete information on $u_1, u_2, u_3$, and their derivatives.
• Figure 2: Three components of the velocity $\dot{u}_1,\,\dot{u}_2$ and $\dot{u}_3$ as functions of time parameter $t$. Each of them has time intervals of apparently constant values and there are intervals in which all three seem to be constant. Revealing their variability requires a logarithmic scale, as seen in the next figure.
• Figure 3: The kinetic energy as a function of the time parameter $t$. In the upper graph, apparently, $E_k$ systematically reaches zero corresponding to the surface of the cone, and stays at this level for a long time, but the logarithmic scale in the lower graph reveals its oscillatory behaviour with reflections from the hyperboloidal surfaces \ref{['hyper']}.

Theorems & Definitions (27)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Corollary 1
• Corollary 2