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Role of the $δ$ Meson in the Equation of State an Direct Urca Cooling of Neutron Stars

Luigi Scurto, Helena Pais, Marco Antonelli, Francesca Gulminelli

TL;DR

This study quantifies how adding the scalar isovector δ meson to relativistic mean-field neutron-star EoS models alters the proton fraction and the density at which direct Urca cooling becomes active, and it investigates the interplay with the $^1S_0$ proton pairing gap. Using a Bayesian framework with two model sets (without and with δ), constrained by astrophysical observations, nuclear experiments, and chiral EFT neutron-matter results, the authors demonstrate that δ inclusion broadens isovector nuclear-matter parameters and increases flexibility in NS predictions, notably shifting the dUrca threshold to higher masses and allowing very low proton fractions at high density. They implement a simplified pairing-gap model and explore how imposing a hypothetical dUrca threshold affects the allowed gap parameters and, consequently, cooling histories, finding strong constraints on the density interval of the superconducting phase under certain mass-threshold scenarios. The results highlight a meaningful link between nuclear microphysics (δ-induced mass splitting and symmetry energy derivatives) and macroscopic NS cooling, offering a pathway to constrain in-medium pairing and proton content via observed NS temperatures and cooling trajectories.

Abstract

The direct Urca (dUrca) process is a key mechanism driving rapid neutrino cooling in neutron stars, with its baryon density activation threshold determined by the microscopic model for nuclear matter. Understanding how nuclear interactions shape the dUrca threshold is essential for interpreting neutron star thermal evolution, particularly in light of recent studies on exceptionally cold objects. We investigate the impact of incorporating the scalar isovector $δ$ meson into the neutron star equation of state, which alters the internal proton fraction and consequently affects the dUrca cooling threshold. Since proton superfluidity is known to suppress dUrca rates, we also examine the interplay between the nuclear interaction mediated by the $δ$ meson and the $^1S_0$ proton pairing gap. We perform a Bayesian analysis using models built within a relativistic mean-field approximation, incorporating constraints from astrophysical observations, nuclear experiments, and known results of \textit{ab initio} calculations of pure neutron matter. We then impose a constraint on the dUrca threshold based on studies of fast-cooling neutron stars. The inclusion of $δ$ meson expands the range of possible internal compositions, directly influencing the stellar mass required for the central density to reach the dUrca threshold. Furthermore, we observe that the observation of relatively young and cold neutron stars provides insights into $^1S_0$ proton superfluidity in the core of neutron stars.

Role of the $δ$ Meson in the Equation of State an Direct Urca Cooling of Neutron Stars

TL;DR

This study quantifies how adding the scalar isovector δ meson to relativistic mean-field neutron-star EoS models alters the proton fraction and the density at which direct Urca cooling becomes active, and it investigates the interplay with the proton pairing gap. Using a Bayesian framework with two model sets (without and with δ), constrained by astrophysical observations, nuclear experiments, and chiral EFT neutron-matter results, the authors demonstrate that δ inclusion broadens isovector nuclear-matter parameters and increases flexibility in NS predictions, notably shifting the dUrca threshold to higher masses and allowing very low proton fractions at high density. They implement a simplified pairing-gap model and explore how imposing a hypothetical dUrca threshold affects the allowed gap parameters and, consequently, cooling histories, finding strong constraints on the density interval of the superconducting phase under certain mass-threshold scenarios. The results highlight a meaningful link between nuclear microphysics (δ-induced mass splitting and symmetry energy derivatives) and macroscopic NS cooling, offering a pathway to constrain in-medium pairing and proton content via observed NS temperatures and cooling trajectories.

Abstract

The direct Urca (dUrca) process is a key mechanism driving rapid neutrino cooling in neutron stars, with its baryon density activation threshold determined by the microscopic model for nuclear matter. Understanding how nuclear interactions shape the dUrca threshold is essential for interpreting neutron star thermal evolution, particularly in light of recent studies on exceptionally cold objects. We investigate the impact of incorporating the scalar isovector meson into the neutron star equation of state, which alters the internal proton fraction and consequently affects the dUrca cooling threshold. Since proton superfluidity is known to suppress dUrca rates, we also examine the interplay between the nuclear interaction mediated by the meson and the proton pairing gap. We perform a Bayesian analysis using models built within a relativistic mean-field approximation, incorporating constraints from astrophysical observations, nuclear experiments, and known results of \textit{ab initio} calculations of pure neutron matter. We then impose a constraint on the dUrca threshold based on studies of fast-cooling neutron stars. The inclusion of meson expands the range of possible internal compositions, directly influencing the stellar mass required for the central density to reach the dUrca threshold. Furthermore, we observe that the observation of relatively young and cold neutron stars provides insights into proton superfluidity in the core of neutron stars.

Paper Structure

This paper contains 15 sections, 44 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Construction of the Equation of State
  3. Bayesian Setup
  4. Construction of the Prior
  5. Posterior Distribution
  6. Astrophysical Constraints
  7. Nuclear Experimental Constraints
  8. Chiral EFT Constraints
  9. Comparison between Set A and Set B
  10. Direct Urca Constraint
  11. Introduction of the 1S0 Proton Pairing Gap
  12. Conclusions
  13. Acknowledgements
  14. Relativistic Mean Field Formalism Details
  15. Analytical expressions for the NMPs

Figures (12)

  • Figure 1: Energy per particle of pure neutron matter as a function of the density for the $\chi$-EFT constraint used in this work (blue band), of the calculations shown in Fig. 1 of Huth_2021 (grey band), and of the unitary gas limit (red solid line).
  • Figure 2: Marginalized posterior of the NMPs for the prior (dotted) and posterior (solid) distributions for Set A (blue) (without $\delta$ meson) and Set B (green) (with $\delta$ meson).
  • Figure 3: Marginalized posterior of the maximum mass (first panel), the central density of the most massive star (second panel) and the radius (third panel) and tidal deformability (fourth panel) of a $1.4M_\odot$ NS for the prior (dotted) and posterior (solid) distributions for Set A (blue) and Set B (green).
  • Figure 4: 90% quantiles for the distribution of the mass as a function of the radius (top panels) and as a function of the tidal deformability (bottom panels), for both the prior (left) and posterior (right) distributions, for Set A (blue) and Set B (green). In the $M$-$R$ plots we also show the 68% and 90% quantiles for the three NICER constraints.
  • Figure 5: 68%, 95% and 99% quantiles for the distribution of the proton fraction as a function of the baryonic density for the prior (left) and posterior (right) distributions for Set A (top) and Set B (bottom).
  • ...and 7 more figures