Cosmological Singularity and Power-Law Solutions in Modified Gravity

TL;DR

The paper addresses the issue of cosmological singularities by testing modified gravity theories against power-law cosmologies using the Raychaudhuri focusing theorem. It employs three models—Stelle gravity, $f(R)$ gravity, and Brans-Dicke theory—to derive conditions under which focusing can be violated, allowing non-singular evolution such as bounces or accelerated expansion. The key finding is that Stelle gravity does not resolve the singularity, while $f(R)$ gravity can support accelerating solutions and Brans-Dicke theory can realize both acceleration and ekpyrosis under specific energy conditions; ekpyrosis remains a credible alternative in several cases. These results hint at a possible unification of singularity-resolution approaches within modified gravity and provide concrete criteria for future observational tests.

Abstract

A bouncing Universe avoids the big-bang singularity. Using the time-like and null Raychaudhhuri equations, we explore whether the bounce near the big-bang, within a broad spectrum of modified theories of gravity, allows for cosmologically relevant power-law solutions under reasonable physical conditions. Our study shows that certain modified theories of gravity, such as Stelle gravity, do not demonstrate singularity resolution under any reasonable conditions, while others including $f(R)$ gravity and Brans-Dicke theory can demonstrate singularity resolution under suitable conditions. For these theories, we show that the accelerating solution is slightly favoured over ekypyrosis.

Figures (13)

• Figure 1: Displays the left hand side of the inequality Eq.(\ref{['fqa']}) for $\alpha=0.323$ (red), $\alpha=0.313$ (blue) and $\alpha=0.303$ (green).
• Figure 2: Displays the left hand side of the inequality Eq.(\ref{['fqa']}) for $\alpha=1.1$ (red), $\alpha=1.2$ (blue) and $\alpha=1.3$ (green).
• Figure 3: Displays the left hand side (red) and the right hand side (blue) of the inequality Eq.(\ref{['ast']}).
• Figure 4: Displays the left hand side of the inequality Eq.(\ref{['full0']}) for a fixed value of $\omega = -3.5/2$ (consistent with Eq.(\ref{['awq']}) and for various values of $\alpha = 1.2, 2, 3$ that is larger than 1.
• Figure 5: The left hand side (red) and right hand side (cyan) of Eq.(\ref{['zow2']}) for the values of $\gamma$ that is consistent with Eq.(\ref{['full0']}) for a fixed value of $\omega = -3.5/2$ that satisfies the inequality $\omega <-3/2$.