Central Speed of Sound, Trace Anomaly and Observables of Neutron Stars from Perturbative Analyses of Scaled TOV Equations

The central speed of sound (SS) measures the stiffness of the Equation of State (EOS) of superdense neutron star (NS) matter. Its variations with density and radial coordinate in NSs in conventional analyses often suffer from uncertainties of the specific nuclear EOSs used. Using the central SS and NS mass/radius scaling obtained from solving perturbatively the scaled Tolman-Oppenheimer-Volkoff (TOV) equations, we study the variations of SS, trace anomaly and several closely related properties of NSs in an EOS-model independent manner. We find that the SS increases with the reduced central pressure $\widehat{P}_{\rm{c}}\equiv P_{\rm{c}}/\varepsilon_{\rm{c}}$ (scaled by the central energy density $\varepsilon_{\rm{c}}$), and the conformal bound for SS is violated for NSs with masses higher than about 1.9M$_{\odot}$. The ratio $P/\varepsilon$ is upper bounded as $P/\varepsilon\lesssim0.374$ around the centers of stable NSs. We demonstrate that it is an intrinsic property of strong-field gravity and is more relevant than the perturbative QCD bound on it. While a sharp phase transition at high densities characterized by a sudden vanishing of SS in cores of massive NSs are basically excluded, the probability for a continuous crossover signaled by a peaked radial profile of SS is found to be enhanced as $\widehat{P}_{\rm{c}}$ decreases, implying it likely happens near the centers of massive NSs. Moreover, a new and more stringent causality boundary as $R_{\max}/\rm{km}\gtrsim 4.73M_{\rm{NS}}^{\max}/M_{\odot}+1.14$ for NS M-R curve is found to be excellently consistent with observational data on NS masses and radii. Furthermore, new constraints on the ultimate energy density and pressure allowed in NSs before collapsing into black holes are obtained and compared with earlier predictions in the literature.Comment: 23 page