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

TL;DR

This work establishes a universal upper bound on the central EOS parameter $X=P_c/\varepsilon_c$ in neutron stars by combining the causality constraint from Special Relativity with a mass-sphere stability condition within the IPAD-TOV framework. The approach yields $X\lesssim 0.385$, modestly above the causality-only bound $X_+\approx 0.374$, and is shown to be EOS-independent across 284 realistic EOSs, including phase transitions and deconfined quark cores. The authors derive near-center NS compactness scaling relations, relate the bound to the dimensionless trace anomaly via $\phi\equiv P/\varepsilon \lesssim 0.385$ and $\Delta\equiv 1/3-\phi \gtrsim -0.051$, and demonstrate robustness under multiple correction forms (linear and quadratic) with an optimal effective parameter $\Theta$ around 0.93. The results provide a new, gravity–microphysics benchmark for superdense matter and offer a practical EOS-independent window into the behavior of matter under extreme compression in General Relativity.

Abstract

The equation-of-state (EOS) parameter $φ\equiv P/\varepsilon$, 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), $\x \equiv φ_{\rm c} = P_{\rm c}/\varepsilon_{\rm c}$, 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.

Figures (7)

• Figure 1: (Color Online). Analogy between many-body and gravitational effects. In a unitary Fermi gas, strong interactions substantially reduced the energy per particle $E_{\rm{UFG}}$ relative to that of a free Fermi gas $E$, namely $E_{\rm{UFG}}/E$ is much smaller than 1; analogously, strong-field gravity in GR tightens the upper bound on the central EOS-parameter $\mathrm{X}$ in NSs to be much smaller than 1 by the principle of SR.
• Figure 2: (Color Online). A Gedankenexperiment: Exerting an external force (pressure) on an NS while keeping its central energy density $\varepsilon_{\rm c}$ fixed (therefore the $\widehat{\varepsilon}(\widehat{r})$ profile changes). The pressure cannot increase without bound; once the configuration turns unstable, the transition defines the upper limit of $\mathrm{X} = P_{\rm c}/\varepsilon_{\rm c}$.
• Figure 3: (Color Online). Information encoded in the coefficient $A(\mathrm{X})$. For small $\mathrm{X}$, the mass-sphere $\widehat{M}(\widehat{r}_{\rm f})$ with $\widehat{r}_{\rm f}$ fixed increases with $\mathrm{X}$, implying $\mathrm{d}A/\mathrm{d}\mathrm{X} < 0$ (panel (a)). As $\mathrm{X}$ grows, further increase of $\widehat{M}(\widehat{r}_{\rm f})$ (i.e., further compression at fixed $\widehat{r}_{\rm f}$ and $\varepsilon_{\rm c}$) becomes progressively more difficult. When $\mathrm{X}$ approaches a critical value $\overline{\mathrm{X}}$, the increase of $\widehat{M}(\widehat{r}_{\rm f})$ also reaches a corresponding critical value. For $\mathrm{X} > \overline{\mathrm{X}}$, the increase of $\widehat{M}(\widehat{r}_{\rm f})$ starts to accelerate again, meaning that compression becomes easier as $\mathrm{X}$ increases, indicating instability. The second-order derivative of $A(\mathrm{X})$ with respect to $\mathrm{X}$ changes sign from positive to negative (panel (b)). If $\mathrm{X}$ is even large then $A(\mathrm{X})$ (or equivalently $a_2(\mathrm{X})$) may become negative (positive) which is naturally unphysical.
• Figure 4: (Color Online). Dependence of $\mathrm{X}_+$ and $\overline{\mathrm{X}}$ on the coefficient $\sigma$. The intersection of the two curves gives the final estimate of the upper bound on $\mathrm{X}$. The inset shows the effective corrections to $s_{\rm c}^2$ and $A_{\sigma}(\mathrm{X})$, and the two vertical dashed lines mark the positions at which the effective corrections in $A_{\sigma}(\mathrm{X})$ and $s_{\rm c}^2$ vanish, respectively. See the text for details.
• Figure 5: (Color Online). The numerical calculation corresponds to FIG. \ref{['fig_Ask']}. Without the $\sigma$-correction, $\overline{\mathrm{X}}$ and $\mathrm{X}_+$ are found to be about 0.368 and 0.381, respectively. Including the $\sigma$-correction shifts both values to $\approx 0.385$, inducing effects of about 4.5% and 1.0%, respectively. For $\mathrm{X} \ge 1/\sqrt{3} \approx 0.577$, the coefficient $A(\mathrm{X})$ becomes positive, corresponding to an unphysical state.