Combining simulation-based inference and universal relations for precise and accurate neutron star science

TL;DR

The paper tackles EOS-induced uncertainties in neutron star bulk properties by combining simulation-based inference (SBI) with a structured search over high-dimensional bulk-property combinations to identify robust universal relations. SBI treats EOS variability as intrinsic noise, yielding predictive posteriors and explicit EOS-noise uncertainties, and guides the traditional construction of a new universal relation $R = R(M,f,p_1)$. The results show SBI can match or surpass the predictive power of the UR with quantified systematic errors, while calibrated URs provide reliable uncertainty estimates and highlight the value of EOS-noise calibration. The framework offers EOS-insensitive constraints with calibrated uncertainties and is extendable to slowly rotating stars and future gravitational-wave and oscillation measurements.

Abstract

In this work, we propose a novel approach for identifying, constructing, and validating precise and accurate universal relations for neutron star bulk quantities. A central element is simulation-based inference (SBI), which we adopt to treat uncertainties due to the unknown nuclear equation of state (EOS) as intrinsic non-trivial noise. By assembling a large set of bulk properties of non-rotating neutron stars across multiple state-of-the-art EOS models, we are able to systematically explore universal relations in high-dimensional parameter spaces. Our framework further identifies the most promising parameter combinations, enabling a more focused and traditional construction of explicit universal relations. At the same time, SBI does not rely on explicit relations; instead, it directly provides predictive distributions together with a quantitative measure of systematic uncertainties, which are not captured by conventional approaches. As an example, we report a new universal relation that allows us to obtain the radius as a function of mass, fundamental mode, and one pressure mode. Our analysis shows that SBI can surpass the predictive power of this universal relation while also mitigating systematic errors. Finally, we demonstrate how universal relations can be further calibrated to mitigate systematic errors accurately.

Figures (7)

• Figure 1: Posterior distributions obtained from applying SBI (blue), calibrated universal relation (orange), and uncalibrated universal relation (green) to the triple of $(M, f, p_1) = (2.33\,M_\odot, 1.87\,\textrm{kHz}, 7.08\,\textrm{kHz})$. The true value of the radius ($12.7\,\textrm{km}$) is shown in black for comparison. The shaded areas represent the 68 % HDIs of each distribution.
• Figure 2: Histograms of the deviations of the mean values of the network's posteriors (SBI) and the calibrated universal relation (UR) from the true value for all data in the test data set.
• Figure 3: Histograms of the widths of the 68% HDIs that are returned by the trained network (SBI) and the universal relation (UR) for all data in the test data set. See the main text on how we sample posteriors from the universal relation.
• Figure 4: Calibration plot visualizing the reliability of the posteriors of SBI and the universal relation (uncalibrated and calibrated). The $x-$axis shows the HDI probability, the $y-$axis shows the coverage probability, i.e., the observed fraction of true values within each HDI. The ideal case corresponds to a diagonal (shown as a black, dashed line) which is very well achieved by the calibrated universal relation; due do the finite accuracy in the SBI network and the finite number of test data, one expects small deviations captured by the binomial proportion confidence interval (shown as blue, solid lines for a 95 % confidence level). The uncalibrated universal relation severely underestimates the true posterior uncertainty.
• Figure 5: Empirical cumulative density function of the posterior ranks for the true neutron star radius. The grey area shows the 95% confidence interval for a uniform distribution of ranks.