Power-Law Bounces in $f(R)$ Gravity: Analysis of the Ekpyrosis and Accelerating Regimes

TL;DR

This work addresses the big bang singularity problem by using a model-independent, compact dynamical-systems framework for $f(R)$ gravity with a power-law ansatz. It derives a four-dimensional autonomous system in cosmographic variables and identifies fixed points corresponding to non-singular bounces, notably at $H=0$ (i.e., $\bar{Q}=0$) for dust and radiation, without specifying a particular $f(R)$ form. Perturbation analysis shows these bounce points are saddles that mediate contraction to expansion, with $w_{\text{eff}}< -1$ after the bounce due to modified gravity terms. The results generalize bounce scenarios beyond $f(R) \simeq R^n$ and indicate a robust non-singular cosmology in $f(R)$ gravity, with potential extensions to Brans-Dicke and other scalar-tensor theories.

Abstract

We investigate the dynamics of the Friedmann-Lemaître-Robertson-Walker spacetime within the framework of $f(R)$ gravity using a compact, model-independent dynamical systems approach. By assuming a power-law scale factor, we explore ekpyrotic and accelerating solutions to address the big bang singularity. Our analysis demonstrates that a cosmological bounce, characterized by a transition from contraction to expansion, possibly avoids the singularity without directly using the Raychaudhuri equation, unlike previous approaches using specific $f(R) \simeq R^n$ forms. We identify a key fixed point in the phase space corresponding to the bounce, supported by perturbation analysis and qualitative description of trajectories in the phase space. The results suggest that $f(R)$ gravity provides a robust framework for non-singular cosmologies.

Figures (4)

• Figure 1: Phase space trajectories perturbed from the fixed point $F = (\bar{\Omega}, \bar{Q}) = (0.4, 0)$ are shown in the $\bar{\Omega}$--$\bar{Q}$ plane. Left plot illustrates the ekpyrotic regime ($\beta < \tfrac{1}{3}$, $w = 0$). The trajectories evolve from the minimum ($\bar{Q} = 0$) to expansion ($\bar{Q} > 0$), indicating a cosmological bounce. Right plot illustrates the accelerating regime ($\beta > 1$, $w = 0$). A bounce is again present, with the trajectories exhibiting pronounced post-bounce expansion characterized by $\bar{Q} > 0$.
• Figure 2: Phase space trajectories perturbed from the fixed point $F$ ($(\bar{x}, \bar{Q}) = (0, 0)$) are shown in the $\bar{x}$--$\bar{Q}$ plane. Left plot illustrates the ekpyrotic regime ($\beta < \tfrac{1}{3}$, $w = 0$). The trajectories cross the $\bar{Q} = 0$ axis, indicating a bounce from the minimum value of $\bar{Q}$ to expansion. Right plot illustrates the accelerating regime ($\beta > 1$, $w = 0$). The bounce is again evident, with trajectories extending into the expanding phase where $\bar{Q} > 0$.
• Figure 3: Phase space trajectories perturbed from the fixed point $F$ ($(\bar{K}, \bar{Q}) = (0.6, 0)$) are shown in the $\bar{K}$--$\bar{Q}$ plane. Left plot illustrates the ekpyrotic regime ($\beta < \tfrac{1}{3}$, $w = 0$). The trajectories exhibit a bounce, transitioning through $\bar{Q} = 0$ from contraction to expansion. Right plot illustrates the accelerating regime ($\beta > 1$, $w = 0$), the bounce is again confirmed, with trajectories displaying strong post-bounce expansion characterized by $\bar{Q} > 0$.
• Figure 4: Illustrative 2D phase portrait in the $(\bar{Q}, \bar{x})$ plane showing qualitative trajectories associated with a cosmological bounce. The vertical dashed line at $\bar{Q} = 0$ denotes the bounce surface where the Hubble parameter vanishes. Red points mark representative fixed points, with $V$ corresponding to a contracting source, $D$ to a post-bounce expanding saddle, and $C$ (not shown) expected near the bounce at $\bar{Q} = 0$. The flow pattern reflects the qualitative structure described in the analysis of Tables \ref{['table1a']} and \ref{['table2c']}.