A Universal Upper Bound on the Pressure-to-Energy Density Ratio in Neutron Stars

TL;DR

This paper establishes a refined, universal upper bound on the pressure-to-energy density ratio at neutron star centers, integrating causality and stability constraints within General Relativity, verified across numerous realistic equations of state.

Contribution

It introduces a self-consistent method to tighten the upper bound on the EOS parameter in neutron stars, incorporating stability conditions and validating universality across diverse models.

Findings

Derived a new upper bound $oxed{ ext{x} \, \lesssim \, 0.385}$ for the EOS parameter.

Verified the universality of the bound across 284 realistic neutron star EOSs.

Provided a scaling relation for neutron star compactness consistent with the bound.

Abstract

The equation-of-state (EOS) parameter $\phi \equiv P/\varepsilon$, defined as the ratio of pressure to energy density, encapsulates the fundamental response of matter under extreme compression. Its value at the center of the most massive neutron star (NS), $\x \equiv \phi_{\rm c} = P_{\rm c}/\varepsilon_{\rm c}$, sets a universal upper bound on the maximum denseness attainable by any form of visible matter anywhere in the Universe. Remarkably, owing to the intrinsically nonlinear structure of the EOS in General Relativity (GR), this bound is forced to lie far below the naive Special Relativity (SR) limit of unity. In this work, we refine the theoretical upper bound on $\x$ in a self-consistent manner by incorporating, in addition to the causality constraint from SR, the mass-sphere stability condition associated with the mass evolution pattern in the vicinity of the NS center. This condition is formulated within the intrinsic-and-perturbative analysis of the dimensionless Tolman--Oppenheimer--Volkoff equations (IPAD-TOV) framework. The combined constraints yield an improved bound, $\x \lesssim 0.385$, which is slightly above but fully consistent with the previously derived causal-only limit, $\x \lesssim 0.374$. We further derive an improved scaling relation for NS compactness and verify its universality across a broad set of 284 realistic EOSs, including models with first-order phase transitions, exotic degrees of freedom, continuous crossover behavior, and deconfined quark cores. The resulting bound on $\x$ thus provides a new, EOS-independent window into the microphysics of cold superdense matter compressed by strong-field gravity in GR.