Strong Energy Condition in $R + R^2$ Gravity

Abstract

In this paper, we study Raychaudhuri&#39;s equation in the background of $R + βR^2$ gravity with a phenomenological matter ($ρ\propto a(t)^{-n}$). We conclude that even though the Strong Energy Condition (S.E.C.) for Einstein&#39;s gravity, which guarantees singularity, is $n\geq 2$ for $ρ\propto a(t)^{-n}$, a perturbative analysis of Raychaudhuri&#39;s equation in the background of $R + βR^2$ gravity reveals that the big bang singularity may not be guaranteed for $n > 4$. We derive the following Strong Energy Conditions for $R + βR^2$ ($β\not= 0$): 1) For $k<0$ FRW metric, S.E.C. is ($0\leq n\leq 4$) i.e., $-ρ_n \leq p_n \leq {1\over 3}ρ_n$. 2) For $k=0$ FRW metric, S.E.C. is ( $1\leq n\leq 4$) i.e., $-{2\over 3}ρ_n \leq p_n \leq {1\over 3}ρ_n$. 3) For $k>0$ FRW metric, S.E.C. is ($2\leq n\leq 4$) i.e., $-{1\over 3}ρ_n \leq p_n \leq {1\over 3}ρ_n$.