Data-driven exploration of the neutron $^3\text{P}_2$ pairing gap using Cassiopeia A neutron star observational data

Abstract

This work aims to elucidate whether the PBF process alone (i.e., without invoking other processes like direct Urca) can explain the observed rapid cooling of Cas~A NS, by incorporating the significant uncertainties in both $q$ and the $^{3}\text{P}_{2}$ pairing gap function into an optimization of cooling models against the Cas~A NS data. To this end, we introduce a novel parametrization of the pairing gap, in which each parameter has a direct physical meaning, and perform systematic parameter-space exploration with the BSk24 equation of state (EoS). Using a newly-developed Fortran-based cooling code coupled to Optuna's TPE algorithm, we conduct both single-objective ($χ^2$ only) and multi-objective ($χ^2$ + slope difference) optimizations under identical conditions. By optimizing the neutron $^3\text{P}_2$ pairing gap parameters to best reproduce the Cas~A NS observational data during repeated neutron-star cooling simulations, we obtain reasonably-behaving neutron $^3\text{P}_2$ pairing gap functions with maximum values of $Δ_\text{max}\approx$\,0.5--0.6\,MeV. Fixing $M=1.4\,M_\odot$, increasing $q$ progressively drives the optimized gap and the critical temperature $T_\text{c}$ profiles toward smoother, more traditional shapes and improves agreement with the observational data; the PBF efficiency factor of $q\gtrsim0.4$ reproduces the Cas~A NS slope well, whereas $q\simeq0.19$ remains insufficient. Our results support previous indications that enhanced PBF efficiency or additional rapid-cooling channels may be required to fully explain the Cas~A NS observational data. The new parametrization not only improves interpretability but also provides a framework for future Bayesian inference and machine-learning applications. Extensions of this work will further advance the systematic study of dense-matter physics with neutron-star cooling. (Shortened due to the arXiv words limit.)

Figures (16)

• Figure 1: A figure that shows the problem of the conventional pairing gap function (\ref{['traditional_parametrization']}). Three pairing gap functions with distinct parameter sets are shown as functions of the Fermi wave number. Two parameters, $k_0=1$ and $k_2=3$, are fixed, which are the left and the right edges of the gap functions, respectively. Red solid, green dashed, blue dotted lines correspond to the cases with $(\Delta_0,k_1,k_3)=(10,1,1)$, $(50,3.45,3.45)$, and $(100,5.3,5.3)$, respectively. Despite significant difference in parameter space, the resulting functions exhibit nearly identical shapes, demonstrating the inherent difficulty for automated parameter optimization algorithms to distinguish between these parameter configurations.
• Figure 2: Comparison of eight commonly used neutron $^3\mathrm{P}_2$ pairing gap models, as listed in Table II of Ref. Ho_2015—AO, BEEHS, EEHO, EEHOr, SYHHP, T, TTav, and TToa—drawn with the traditional parametrization \ref{['traditional_parametrization']} (solid line) and the new parametrization \ref{['new_parametrization']} proposed in this work (dashed line). For each model the curve is shown only over its physical domain $k_0\le k_{\mathrm{Fn}}\le k_2$, and all panels share common axes. The new form reproduces the shape and peak location of the traditional curves with only minor deviations, which are negligible for practical neutron–star cooling calculations.
• Figure 3: Cooling curves obtained from 1,000 trials during the TPE optimization. Each line represents a cooling calculation for a distinct parameter set sampled by TPE, compared against the Cas A NS data (red data points). A colormap is applied for trials with $\chi^2 \le 100$, while those with $\chi^2 > 100$ are shown in gray. Lower $\chi^2$ values indicate better agreement with the Cas A NS observations. Note that the Cas A NS data are allowed to shift within $\pm19$ yr relative to each cooling curve when evaluating the fit; hence, the data points shown here are aligned to the best-fit curve (red dashed) for visualization.
• Figure 4: Parameter-space projections of $\Delta_{\max}$ versus each gap parameter for the single-objective (top row) and multi-objective (bottom row) optimizations. Points are colored by $\chi^{2}$ (brighter is better); red markers denote the top 1% in $\chi^{2}$ (100 out of 10,000). Note that a colormap is applied for trials with $\chi^2 \le 100$, while those with $\chi^2 > 100$ are shown in gray. The multi-objective run concentrates competitive solutions near $k_{\max}\!\approx\!2.0\,\mathrm{fm}^{-1}$ while allowing broader support in $k_{2}$, which is weakly constrained once it exceeds the $k_{\mathrm{Fn}}$ at the center of the star. Note that, because the admissible range of $w$ spans several orders of magnitude, we re-parameterize and plot $w^{-1}$ (the optimization was also carried out in $w^{-1}$) to stabilize the scale and improve readability.
• Figure 5: Evolution of the best (top-1) $\chi^{2}$ versus the number of valid iterations for the single-objective ($\chi^{2}$ only) and multi-objective ($\chi^{2}\,+$ slope-diff.) runs at fixed $q\simeq 0.19$ and $M_\text{NS}=1.4\,M_\odot$. The horizontal axis is in logarithmic scale. Because the multi-objective run must balance two targets, it converges more slowly; nevertheless, after $\sim\!7\times10^{3}$ valid trials it attains a lower top-1 $\chi^{2}$ than the single-objective run.