Some exact results on the Belinski-Khalatnikov-Lifshitz scenario

Abstract

The well-known Bielinski-Khalatnikov-Lifshitz (BKL) scenario for the universe near the cosmological singularity is supplemented with a few exact results following from the BKL asymptotic of the Einstein equations: (1) The cosmological singularity is proved to be an inevitable beginning or end of the universe as described by these equations. (2) Attaining the singularity from shrinking initial conditions requires infinite time parameter $τ$; no singularity of any kind may occur in a finite $τ$. (3) The previously found exact solution [P.G. and W. Piechocki, Eur. Phys. J. C 82:216 (2022)] is the only asymptotic with well-defined proportions between the directional scale factors which have been appropriately compensated against indefinite growth of anisotropy. In all other cases, the universe undergoes oscillations of Kasner type, which reduce the length scales to nearly zero in some directions, while largely extending it in the others. Together with instability of the exact solution [op. cit.], it makes the approach to the singularity inevitably chaotic. (4) Reduced equations are proposed and explicitly solved to describe these oscillations near their turning points. In logarithmic variables, the oscillations are found to have sawtooth shapes. A by-product is a quadric of kinetic energy, a simple geometric tool for all this analysis.

Figures (4)

• Figure 1: (from GP). The lower half (= universe shrinking with $\tau$) 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 $\tau\to\infty$, 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 the time parameter $\tau$. The initial conditions correspond to $q(0)=0.225,\, r(0)=0.25,\,s(0)=0.1,\,\dot{q}(0)=-2,25,\,\dot{r}(0)=-2,5$. Each of the velocity components has time intervals of apparently constant value. Revealing their variability requires a logarithmic scale, as seen in the next figures.
• Figure 3: The kinetic energy as a function of the time parameter $\tau$ for $q(0)=0.225,\, r(0)=0.25,\,s(0)=0.1,\,\dot{q}(0)=-2,25,\,\dot{r}(0)=-2,5$. 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 sawtooth oscillations, with the teeth a little rounded (of shape $\ln\, \mathop{\mathrm{sech}}\nolimits^2$) at the reflections from the hyperboloidal surfaces \ref{['hyper']}.
• Figure 4: The logarithmic variables $y_1=\ln q,\,y_2=\ln r$ and $y_3=\ln s$ as functions of time parameter $\tau$ for $q(0)=0.3,\, r(0)=0.33,\,s(0)=0.12,\,\dot{q}(0)=-1,\,\dot{r}(0)=-1$. Variables $y_1$ and $y_2$ exhibit sawtooth behavior, exchanging their dominant roles, while $y_3$ is much smaller. The kinetic energy $E_k$ (dashed line) is the upper envelope of the $y_i$ variables. The tips of the saw teeth are more rounded than those in the previous graph (Fig. \ref{['kin-en']}) as the initial "velocities" are less than halves of those in Fig. \ref{['kin-en']}. Still the term "sawtooth" is justified by the rectilinear shape of the teeth outside their blades.

Theorems & Definitions (30)

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