General gravitational properties of neutron stars: curvature invariants, binding energy, and trace anomaly

Abstract: We investigate the behavior of curvature invariants for a large ensemble of neutron stars built with equations of state (EOSs) that satisfy constraints from nuclear theory and perturbative QCD, as well as measurements of neutron-star masses, radii, and gravitational waves from binary neutron-star mergers. Surprisingly, our analysis reveals that stars with negative Ricci scalar $\mathcal{R}$ are rather common and about $\sim 50\%$ of our EOSs produce one or more stars with Ricci curvature that is negative somewhere inside the star. The negative curvature is found mostly but not exclusively at the highest densities and pressures, and predominantly for stiff EOSs and for the most compact and most massive stars. Furthermore, we improve the quasi-universal relation between the stellar gravitational mass $M$ and the baryonic mass $M_\mathrm{b}$, which allows us to express analytically one in terms of the other with a maximum variance of only $\sim 3\%$. Finally, using the relation between the Ricci scalar and the trace anomaly $Δ$, we determine the conditions under which $Δ$ vanishes or becomes negative in neutron stars.