Evolution to a smooth universe in an ekpyrotic contracting phase with w > 1

TL;DR

The paper addresses whether ekpyrotic smoothing during a contracting phase with equation of state $w>1$ remains effective when initial conditions are highly inhomogeneous. It employs a scale-invariant tetrad formulation of the Einstein–scalar field equations with a negative exponential potential $V(\phi)=-V_0 e^{-k\phi}$ and adaptive mesh refinement to evolve a 1D inhomogeneity setup toward a possible isotropic singularity. The main finding is that regions with $w>1$ grow in proper volume relative to $w=1$ anisotropic regions, with the volume ratio $R$ obeying $R\propto e^{-t(1-6/k^2)}$, diverging for $k>\sqrt{6}$ as $t\to -\infty$, and that asymptotically $W\to k$, $\bar{V}\to 3-k^2/2$, $N\to 2/k^2$, and $w\to k^2/3-1$ in smooth sectors while anisotropic sectors maintain $w=1$. These results support ekpyrotic and cyclic cosmologies by demonstrating robustness to non-linear inhomogeneities and suggesting the possibility of isotropic singularities in the smooth regions. The work implies that a pre-ekpyrotic dark-energy epoch is not a necessary prerequisite for smoothing and motivates further study of bouncing scenarios with controlled anisotropy growth.

Abstract

A period of slow contraction with equation of state w > 1, known as an ekpyrotic phase, has been shown to flatten and smooth the universe if it begins the phase with small perturbations. In this paper, we explore how robust and powerful the ekpyrotic smoothing mechanism is by beginning with highly inhomogeneous and anisotropic initial conditions and numerically solving for the subsequent evolution of the universe. Our studies, based on a universe with gravity plus a scalar field with a negative exponential potential, show that some regions become homogeneous and isotropic while others exhibit inhomogeneous and anisotropic behavior in which the scalar field behaves like a fluid with w=1. We find that the ekpyrotic smoothing mechanism is robust in the sense that the ratio of the proper volume of the smooth to non-smooth region grows exponentially fast along time slices of constant mean curvature.

Figures (5)

• Figure 1: $t={\rm const.}$ snapshots of the normalized energy density in matter $\Omega_m$ (solid line), curvature $\Omega_k$ (dot-dash line) and shear $\Omega_s$ (dashed line) for $0\le x \le 2 \pi$ at several times during the evolution of the initial data described in Sec.\ref{['sec_results']}. Time runs from left to right along the top row and continues along the bottom row. The shaded slit (dotted outline) in the last panel ($t = -150$) indicates the range of x shown in the blow-up in Fig.\ref{['omega_inset']}.
• Figure 2: Zooming in on one of the spike structures that formed in the anisotropic region at $t=-150$, as shown in Fig.\ref{['omega_panel']}.
• Figure 3: The effective equation of state parameter $w$ (\ref{['weffdef']}) for the simulation described in Sec.\ref{['sec_results']}, for $0 \le x \le 2 \pi$ at the same times as in Fig.\ref{['omega_panel']}. At late times $w\rightarrow k^2/3-1$ in the matter dominated region, and $w\rightarrow 1$ in the anisotropic region (in this simulation $k=10$ for the potential (\ref{['vdef']})).
• Figure 4: 3 times the scale invariant lapse ${\cal N}$ for the simulation described in Sec.\ref{['sec_results']}, for $0 \le x \le 2 \pi$ at the same output times as Figs.\ref{['omega_panel']} and \ref{['w_panel']}.
• Figure 5: The state space orbits for worldlines at $x=0, 3.0, 3.9, 4.0, 4.4$. Towards the singularity, the first three orbits (solid lines) approach the origin of the plane, indicating isotropization. The fourth (dotted) and the fifth (dashed) do not isotropize.