The mass-radius relation for neutron stars in $f(R)=R+\alpha R^2$ gravity: a comparison between purely metric and torsion formulations

Within the framework of $f(R)=R+\alpha R^2$ gravity, we study realistic models of neutron stars, using equations of state compatible with the LIGO constraints. i.e. APR4, MPA1, SLy, and WW1. By numerically solving modified Tolman-Oppenheimer-Volkoff equations, we investigate the Mass--Radius relation in both metric and torsional $f(R)=R+\alpha R^2$ gravity models. In particular, we observe that torsion effects decrease the compactness and total mass of neutron star with respect to the General Relativity predictions, therefore mimicking the effects of a repulsive massive field. The opposite occurs in the metric theory, where mass and compactness increase with $\alpha$, thus inducing an excess of mass that overtakes the standard General Relativity limit. We also find that the sign of $\alpha$ must be reversed whether one considers the metric theory (positive) or torsion (negative) to avoid blowing up solutions. This could draw an easy test to either confirm or discard one or the other theory by determining the sign of parameter $\alpha$.