The Mass-Radius relation for Neutron Stars in $f(R)$ gravity

We discuss the Mass -Radius diagram for static neutron star models obtained by the numerical solution of modified Tolman-Oppenheimer-Volkoff equations in $f(R)$ gravity where the Lagrangians $f(R)=R+\alpha R^2 (1+\gamma R)$ and $f(R)=R^{1+\epsilon }$ are adopted. Unlike the case of the perturbative approach previously reported, the solutions are constrained by the presence of an extra degree of freedom, coming from the trace of the field equations. In particular, the stiffness of the equation of state determines an upper limit on the central density $\rho_c$ above which the the positivity condition of energy-matter tensor trace $T^{\rm m}=\rho - 3 p$ holds. In the case of quadratic f(R)-gravity, we find higher masses and radii at lower central densities with an inversion of the behavior around a pivoting $\rho_c$ which depends on the choice of the equation of state. When considering the cubic corrections, we find solutions converging to the required asymptotic behavior of flat metric only for $\gamma < 0$. A similar analysis is performed for $f(R)=R^{1+\epsilon }$ considering $\epsilon$ as the leading parameter. We work strictly in the Jordan frame in order to consider matter minimally coupled with respect to geometry. This fact allows us to avoid ambiguities that could emerge in adopting the Einstein frame.