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

In this paper, we study Raychaudhuri’s equation in the background of $R + \beta R^2$gravitywithaphenomenologicalmatter($\rho \propto a(t)^{-n}$). We conclude that even though the Strong Energy Condition (S.E.C.) for Einstein’s gravity, which guarantees singularity, is $n\ge 2$ for $\rho \propto a(t)^{-n}$, a perturbative analysis of Raychaudhuri’s equation in the background of $R+\beta R^2$ gravity reveals that the big bang singularity may not be guaranteed for $n>4$. We derive the following Strong Energy Conditions for $R+\beta R^2$ ($\beta ≠0$):

1. For $k<0$ FRW metric, S.E.C. is ($0\le n\le 4$) i.e., $-{\rho }_n\le p_n\le \frac{1}{3}{\rho }_n$.
2. For $k=0$ FRW metric, S.E.C. is ( $1\le n\le 4$) i.e., $-\frac{2}{3}{\rho }_n\le p_n\le \frac{1}{3}{\rho }_n$.
3. For $k>0$ FRW metric, S.E.C. is ($2\le n\le 4$) i.e., $-\frac{1}{3}{\rho }_n\le p_n\le \frac{1}{3}{\rho }_n$.