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Neutron stars provide a unique opportunity to study strongly interacting matter under extreme density conditions. The intricacies of matter inside neutron stars and their equation of state are not directly visible, but determine bulk properties, such as mass and radius, which affect the star's thermal X-ray emissions. However, the telescope spectra of these emissions are also affected by the stellar distance, hydrogen column, and effective surface temperature, which are not always well-constrained. Uncertainties on these nuisance parameters must be accounted for when making a robust estimation of the equation of state. In this study, we develop a novel methodology that, for the first time, can infer the full posterior distribution of both the equation of state and nuisance parameters directly from telescope observations. This method relies on the use of neural likelihood estimation, in which normalizing flows use samples of simulated telescope data to learn the likelihood of the neutron star spectra as a function of these parameters, coupled with Hamiltonian Monte Carlo methods to efficiently sample from the corresponding posterior distribution. Our approach surpasses the accuracy of previous methods, improves the interpretability of the results by providing access to the full posterior distribution, and naturally scales to a growing number of neutron star observations expected in the coming years.

The interiors of neutron stars reach densities and temperatures beyond the limits of terrestrial experiments, providing vital laboratories for probing nuclear physics. While the star's interior is not directly observable, its pressure and density determine the star's macroscopic structure which affects the spectra observed in telescopes. The relationship between the observations and the internal state is complex and partially intractable, presenting difficulties for inference. Previous work has focused on the regression from stellar spectra of parameters describing the internal state. We demonstrate a calculation of the full likelihood of the internal state parameters given observations, accomplished by replacing intractable elements with machine learning models trained on samples of simulated stars. Our machine-learning-derived likelihood allows us to perform maximum a posteriori estimation of the parameters of interest, as well as full scans. We demonstrate the technique by inferring stellar mass and radius from an individual stellar spectrum, as well as equation of state parameters from a set of spectra. Our results are more precise than pure regression models, reducing the width of the parameter residuals by 11.8% in the most realistic scenario. The neural networks will be released as a tool for fast simulation of neutron star properties and observed spectra.

Neutron stars provide a unique laboratory for studying matter at extreme pressures and densities. While there is no direct way to explore their interior structure, X-rays emitted from these stars can indirectly provide clues to the equation of state (EOS) of the superdense nuclear matter through the inference of the star's mass and radius. However, inference of EOS directly from a star's X-ray spectra is extremely challenging and is complicated by systematic uncertainties. The current state of the art is to use simulation-based likelihoods in a piece-wise method which relies on certain theoretical assumptions and simplifications about the uncertainties. It first infers the star's mass and radius to reduce the dimensionality of the problem, and from those quantities infer the EOS. We demonstrate a series of enhancements to the state of the art, in terms of realistic uncertainty quantification and a path towards circumventing the need for theoretical assumptions to infer physical properties with machine learning. We also demonstrate novel inference of the EOS directly from the high-dimensional spectra of observed stars, avoiding the intermediate mass-radius step. Our network is conditioned on the sources of uncertainty of each star, allowing for natural and complete propagation of uncertainties to the EOS.