On the maximum compactness of neutron stars

TL;DR

This work addresses the question of how compact a neutron star can be given the uncertainty in the nuclear EOS. It builds a large, constrained ensemble of EOSs and tests the conjecture that, for a fixed EOS, the maximum compactness occurs at the maximum-mass nonrotating configuration, enabling a focus on $\mathcal{C}_{\rm TOV}$. Across EOSs that satisfy nuclear theory, pQCD, and astrophysical constraints, the study identifies a universal upper bound $\mathcal{C}_{\rm max} \approx 0.3329$, essentially $1/3$, set predominantly by pQCD constraints and largely mass-independent. The results provide a robust link between high-density QCD and observable neutron-star structure, offering a falsifiable target for future measurements of mass and radius that could probe the behavior of matter at extreme densities.

Abstract

The stellar compactness, that is, the dimensionless ratio between the mass and radius of a compact star, $\mathcal{C} := M/R$, plays a fundamental role in characterising the gravitational and nuclear-physics aspects of neutron stars. Yet, because the compactness depends sensitively on the unknown equation of state (EOS) of nuclear matter, the simple question: ``how compact can a neutron star be?'' remains unanswered. To address this question, we adopt a statistical approach and consider a large number of parameterised EOSs that satisfy all known constraints from nuclear theory, perturbative Quantum Chromodynamics (QCD), and astrophysical observations. Next, we conjecture that, for any given EOS, the maximum compactness is attained by the star with the maximum mass of the sequence of nonrotating configurations. While we can prove this conjecture for a rather large class of solutions, its general proof is still lacking. However, the evidence from all of the EOSs considered strongly indicates that it is true in general. Exploiting the conjecture, we can concentrate on the compactness of the maximum-mass stars and show that an upper limit appears for the maximum compactness and is given by $\mathcal{C}_{\rm max} = 1/3$. Importantly, this upper limit is essentially independent of the stellar mass and a direct consequence of perturbative-QCD constraints.

Figures (4)

• Figure 1: Left panel: behaviour in the $(M,R)$ space of the sequences of nonrotating stars with EOSs leading to maximum-mass stars with the largest (smallest) compactness $\mathcal{C}_{_{\rm TOV}, {\rm max}}$ ($\mathcal{C}_{_{\rm TOV}, {\rm min}}$). Thin red (blue) lines refer to EOSs near the maximum (minimum) compactness, which is instead indicated with a thick red (blue) line. A golden star (circle) marks the position in the $(M,R)$ space of the star with the maximum (minimum) compactness of $\mathcal{C}_{\rm max} \simeq 0.3329$ ($\mathcal{C}_{\rm min} \simeq 0.2294$) in the whole ensemble. Also shown with a solid grey (green dashed) line are the contours of the allowed ranges when considering $100\%$ ($90\%$) of the stars in the ensemble. Finally, shown with dark (light) grey shaded areas are the regions where $\mathcal{C} \geq 4/9$ ($\mathcal{C} \geq 1/3$). Right panel: the same as on the left but in the $(p,e)$ space.
• Figure 2: Distribution of the logarithmic derivative $d\ln R/d\ln M$ for all of the stellar models in our ensemble shown as a function of the stellar mass. Since $d\ln R/d\ln M \lesssim 0.18$, for each EOS the most massive star is also the most compact one.
• Figure 3: Distribution of the maximum compactness $\mathcal{C}_{_{\rm TOV}}$ as a function the maximum mass $M_{_{\rm TOV}}$. Shown as blue- and red-shaded areas are the constraints coming from the maximum mass and the binary tidal deformability, respectively. Also reported with a red dashed line is the analytic fit \ref{['eq:Cmin']} for the minimum of the maximum-mass compactness $\mathcal{C}_{_{\rm TOV}, {\rm min}}$, while the star and circle are the same models as in Fig. \ref{['fig:EOSs']}. The outer bounds without imposing the pQCD constraints is shown with the black solid contour, while the horizontal red solid line marks $\mathcal{C}=1/3$, highlighting that all stellar models are below this limit when the pQCD constraint is imposed.
• Figure 4: The same as in Fig. \ref{['fig:EOSs']} but for the behaviour of the sound speed (left panel) and of the conformal anomaly as a function of the energy density (right panel).