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

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.