Cosmological dynamics in $R^2$ gravity with logarithmic trace term

A novel function for modified gravity is proposed, $f(R, T)=R+\lambda R^2+2\beta\ln(T)$, with constants $\lambda$ and $\beta$, scalar curvature $R$, and the trace of stress energy tensor $T$, satisfying $T=\rho-3p>0$. Subsequently, two equations of state (EoS) parameters, namely $\omega$ and a parametric form of the Hubble parameter $H$, are employed in order to study the accelerated expansion and initial cosmological bounce of the corresponding universe. Hubble telescope experimental data for redshift $z$ within the range $0.07\leq z \leq 2.34$ are used to compare the theoretical and observational values of the Hubble parameter. Moreover, it is observed that all the energy conditions are fulfilled within a neighborhood of the bouncing point $t=0$, what shows that the necessary condition for violation of the null energy condition, within a neighborhood of the bouncing point in general relativity, could be avoided by modifying the theory in a reasonable way. Furthermore, a large amount of negative pressure is found, which helps to understand the late time accelerated expansion phase of the universe.