Assessing Universal Relations for Rapidly Rotating Neutron Stars: Insights from an Interpretable Deep Learning Perspective

TL;DR

This study revisits established universal relations, introduces new ones, and reassess them using a feed-forward neural network as a regression model, and proposes ``deep''EoS-insensitive hypersurface relations for rapidly rotating compact objects between several of the star's global parameters.

Abstract

Relations between stellar properties independent of the nuclear equation of state offer profound insights into neutron star physics and have practical applications in data analysis. Commonly, these relations are derived from utilizing various realistic nuclear cold hadronic, hyperonic, and hybrid EoS models, each of which should obey the current constraints and cover a wide range of stiffnesses. Concurrently, the field of multimessenger astronomy has been significantly enhanced by the advent of gravitational wave astronomy, which increasingly incorporates deep learning techniques and algorithms. At the same time, X-ray spectral data from NICER based on known pulsars are available, and additional observations are expected from upcoming missions. In this study, we revisit established universal relations, introduce new ones, and reassess them using a feed-forward neural network as a regression model. More specifically, we mainly propose ``deep'' EoS-insensitive hypersurface relations for rapidly rotating compact objects between several of the star's global parameters, which achieve an accuracy of within $1\%$ in most cases, with only a small fraction of investigated models exceeding this threshold. While analytical expressions can be used to represent some of these relations, the neural network approach demonstrates superior performance, particularly in complex regions of the parameter space. Furthermore, we use the SHapley Additive exPlanations (SHAP) method to interpret the suggested network's predictions, since is based on a strong theoretical framework inspired by the field of cooperative Game Theory. Most importantly, these highly accurate universal relations empowered with the interpretability description could be used in efforts to constrain the high-density equation of state in neutron stars, with the potential to enhance our understanding as new observables emerge.

Figures (32)

• Figure 1: Illustration of the SHAP values contribution concerning four features $f_1,\cdots, f_4$. Each SHAP value $\phi_i$ associated with the corresponding feature $f_i$ contributes positively (green-colored vector) or negatively (red-colored vector) to the estimator's result $\hat{\mathcal{F}}(x_i)$ relatively to $\mathbb{E}_{X}(\hat{\mathcal{F}}(X))$. The associated inference is consistent with Eq. (\ref{['eq:SHAP']}), considering as input vector $x$ the specific vector $x_i$.
• Figure 2: Distributions of the $\chi-C$, $\chi-\log\bar{I}$, $\chi-\log\bar{Q}$, and $\chi-\log\bar{S}_3$ parameter spaces, encompassing a wide range of rotation rates and degree of stiffness. Each color represents the EOS mapping as highlighted in Fig. \ref{['fig:color_band']} of Appendix \ref{['app:EoS_list']}.
• Figure 3: Reduced moment of inertia $\bar{I}$ as a function of the dimensionless parameters $\chi$, and $\bar{Q}$. In this illustration, the colored black grid is associated with the ANN wireframe produced by Eq. (\ref{['eq:Ibar_fit']}). In addition, the colored variation of the presented data corresponds to the star's stellar compactness $C$, as highlighted in the accompanying vertical colored bar.
• Figure 4: Absolute relative error distribution ($100\% \times ({\bar{I}}_{\mathrm{model}} - \bar{I})/ \bar{I})$ derived evaluating the suggested regression model (\ref{['eq:Ibar_fit']}) on the test set. The vertical axis corresponds to the probability density function (PDF) of the test dataset, shown on a logarithmic scale. In addition, relative deviations for fitting functions proposed in the literature are shown in different colors, providing a basis for comparative analysis.
• Figure 5: $\bar{I} = \bar{I}(\bar{Q})$ theoretical curves for a discrete sample of spin parameter values $\chi \in [0.01,0.09]$ associated with slowly rotating NS configurations. The regression model (\ref{['eq:Ibar_fit']}) compared against the fitting function proposed in yagi2017approximate satisfactorily reproduces the $\bar{I}-\bar{Q}$ universal behavior in the slowly rotating limit.