Non-singular big-bounces and evolution of linear fluctuations

TL;DR

This work analyzes how linear scalar curvature perturbations evolve as a closed FLRW universe passes through a smooth, non-singular bounce within General Relativity. It identifies a gauge-invariant quantity $Φ$ and a corresponding $C$-mode that remains constant through the bounce, setting the amplitude of the growing mode in the expanding phase. Across several bounce models (pressureless, fluid with exotic contributions, and a massive scalar field), it shows that the $C$-mode is robust when large-scale conditions hold, whereas curvature effects during the bounce can disrupt mode tracing and induce instabilities or oscillations. The results have implications for bouncing and cyclic cosmologies, informing the viability of generating scale-invariant spectra in ekpyrotic/branelike scenarios and clarifying the role of linear perturbation theory near bounces.

Abstract

We consider evolutions of linear fluctuations as the background Friedmann world model goes from contracting to expanding phases through smooth and non-singular bouncing phases. As long as the gravity dominates over the pressure gradient in the perturbation equation the growing-mode in the expanding phase is characterized by a conserved amplitude, we call it a C-mode. In the spherical geometry with a pressureless medium, we show that there exists a special gauge-invariant combination Φwhich stays constant throughout the evolution from the big-bang to the big-crunch with the same value even after the bounce: it characterizes the coefficient of the C-mode. We show this result by using a bounce model where the pressure gradient term is negligible during the bounce; this requires additional presence of an exotic matter. In such a bounce, even in more general situations of the equation of states before and after the bounce, the C-mode in the expanding phase is affected only by the C-mode in the contracting phase, thus the growing mode in the contracting phase decays away as the world model enters expanding phase. In the case the background curvature has significant role during the bounce, the pressure gradient term becomes important and we cannot trace C-mode in the expanding phase to the one before the bounce. In such situations, perturbations in a fluid bounce model show exponential instability, whereas the ones in a scalar field bounce model show oscillatory behaviors.

Figures (4)

• Figure 1: Right: Evolutions of $\varphi_{+}$ (blue, line), $\varphi_{-}$ (cyan, dashed line), $\varphi_{d}$ (blue, dotted line), and $\Phi$ (black, horizontal line).
• Figure 2: Evolution of the scale factor (line). The dotted lines indicate the extensions of the collapsing, bouncing and expanding phases without matchings. The bouncing phase lasts for $|t_B| <5$ with $\Lambda = 1$ and $K = 1$. As an example, we take $w_{I} = 0$ and $w_{II} = {1 \over 3}$.
• Figure 3: Evolutions of $\Phi$ for the $C$-mode (blue, line) and $d$-mode (red, long-dashed line) initial conditions, and $\varphi_\chi$ for the $C$-mode (cyan, dotted line) and $d$-mode (magenta, dot-short-dashed line) initial conditions. We take $n=10$. The pressure terms become important in $|t|<.4$ for $\varphi_\chi$ and in $|t|<.2$ for $\Phi$. Notice that the $C$-mode of $\Phi$ changes sign twice, whereas the $d$-mode changes once. The sign changes in the expanding phase occur at the same time.
• Figure 4: Evolutions of $\varphi_\chi$ (red, long-dashed line), $\Phi$ (blue, line), $\varphi_v$ (cyan, dotted line), and $\varphi_\kappa$ (green, dot-short-dashed line). We omit $\varphi_\delta$ which has a larger amplitude and approaches $\Phi$ in the large-scale. We take $n = 10$. $t = 0$ corresponds to the minimum of the bounce, and the pressure dominates over the gravity till $t \sim 1$. We can show that $\Phi$, $\varphi_v$, $\varphi_\kappa$ and $\varphi_\delta$ stay constant in the large-scale, whereas $\varphi_\chi$ still adjusting its value even in the large-scale as the background equation of state effectively changes from $w_\phi \simeq -1$ while $\phi$ is rolling to $w_\phi = 0$ as $\phi$ starts oscillating which occurs for $t>11$.