Data-driven exploration of the neutron $^3\text{P}_2$ pairing gap using Cassiopeia A neutron star observational data: Direct $\chi^2$ minimization

The rapid cooling observed in the Cassiopeia~A neutron star (Cas~A NS) is one of the most stringent tests for neutron-star cooling theory. While Cooper-pair breaking and formation (PBF) neutrino emission is a leading candidate, uncertainties remain regarding the PBF efficiency factor $q$ and the neutron ${}^{3}\mathrm{P}_{2}$ pairing gap. This work explores in a data-driven manner how the optimized gap shape responds to variations of the PBF emissivity parameter $q$ within a fixed cooling setup. We introduce a novel gap parametrization, in which each parameter carries direct physical meaning and controls the gap amplitude, peak location, width, and asymmetry. Using a Fortran-based cooling code and the BSk24 equation of state, we perform parameter-space exploration guided by the Cas~A NS data. Global optimization is carried out with Optuna's tree-structured Parzen estimator, followed by local refinement using the Nelder--Mead method. The optimized solutions yield physically reasonable gaps with peak amplitudes $\Delta_{\max}\approx0.5$--$0.6~\mathrm{MeV}$. Although the multi-objective formulation explores the parameter space more broadly, the single-objective $\chi^{2}$-only optimization achieves the lowest $\chi^{2}$. For $M_{\mathrm{NS}}=1.4\,M_{\odot}$, increasing $q$ drives the optimized gap and critical-temperature profiles toward smoother and more localized shapes, improving consistency with the observed trend. Models with $q\gtrsim0.4$ reproduce the decline rate within the $1\sigma$ confidence interval, whereas the baseline case $q\simeq0.19$ lies near the $3\sigma$ level. Our results suggest larger effective PBF emissivities than the baseline estimate, although robust constraints on $q$ require future Bayesian inference including uncertainties in mass, envelope composition, equation of state, pairing microphysics, and age offset. (Shortened due to the arXiv abstract length limit.)