Core States of Neutron Stars from Anatomizing their Scaled Structure Equations

Given an Equation of State (EOS) for neutron star (NS) matter, there is a unique mass-radius sequence characterized by a maximum mass $M_{\rm{NS}}^{\max}$ at radius $R_{\max}$. We first show analytically that the $M_{\rm{NS}}^{\max}$ and $R_{\max}$ scale linearly with two different combinations of NS central pressure $P_{\rm{c}}$ and energy density $\varepsilon_{\rm{c}}$ by dissecting perturbatively the dimensionless Tolman-Oppenheimer-Volkoff (TOV) equations governing NS internal variables. The scaling relations are then verified via 87 widely used and rather diverse phenomenological as well as 17 microscopic NS EOSs with/without considering hadron-quark phase transitions and hyperons by solving numerically the original TOV equations. The EOS of densest NS matter allowed before it collapses into a black hole (BH) is then obtained. Using the universal $M_{\rm{NS}}^{\max}$ and $R_{\max}$ scalings and NICER (Neutron Star Interior Composition Explorer) and XMM-Newton mass-radius observational data for PSR J0740+6620, a very narrow constraining band on the NS central EOS is extracted directly from the data for the first time without using any specific input EOS model.