Bayesian Nonparametric Inference of the Neutron Star Equation of State via a Neural Network

Abstract

We develop a new nonparametric method to reconstruct the equation of state (EoS) of a neutron star with multimessenger data. As a universal function approximator, the feed-forward neural network (FFNN) with one hidden layer and a sigmoidal activation function can approximately fit any continuous function. Thus, we are able to implement the nonparametric FFNN representation of the EoSs. This new representation is validated by its capability of fitting the theoretical EoSs and recovering the injected parameters. Then, we adopt this nonparametric method to analyze the real data, including the mass–tidal deformability measurement from the binary neutron star merger gravitational-wave event GW170817 and mass–radius measurement of PSR J0030+0451 by NICER. We take the publicly available samples to construct the likelihood and use the nested sampling to obtain the posteriors of the parameters of the FFNN according to the Bayesian theorem, which in turn can be translated to the posteriors of the EoS parameters. Combining all of these data for a canonical 1.4 M ⊙ neutron star, we get a radius R 1.4 = 11.83 − 1.08 + 1.25 km and tidal deformability Λ 1.4 = 323 − 165 + 334 (90% confidence interval). Furthermore, we find that in the high-density region (≥3ρ sat), the 90% lower limits of c s 2 / c 2 (where c s is the sound speed and c is the velocity of light in vacuum) are above 1/3, which means that the so-called conformal limit (i.e., c s 2 / c 2 < 1 / 3 ) is not always valid in the neutron stars.

Relativistic description of dense matter equation of state and compatibility with neutron star observables: a Bayesian approach

“…In order to account for all possible EOS compatible with these two constraints, the EOS at the two extreme densities are connected using a piecewise polytropic interpolation, a speed-of-sound interpolation or a spectral interpolation, and causality is imposed when necessary Lindblom & Indik (2012); Kurkela et al (2014); Most et al (2018); Lope Oter et al (2019); Annala et al (2020Annala et al ( , 2021. Of late, a nonparametric inference of the NS EOS has also been proposed based on Gaussian processes (GPs) Essick et al (2020) or using machine learning techniques Han et al (2021). However, such EOS models have strong limitations because they do not assume any kind of composition of matter in the intermediate density regime.…”

Relativistic Description of Dense Matter Equation of State and Compatibility with Neutron Star Observables: A Bayesian Approach

“…In order to account for all possible EOSs compatible with these two constraints, the EOS at the two extreme densities is connected using a piecewise polytropic interpolation, a speed of sound interpolation, or a spectral interpolation, and causality is imposed when necessary (Lindblom & Indik 2012;Kurkela et al 2014;Most et al 2018;Lope Oter et al 2019;Annala et al 2020Annala et al , 2022. Of late, a nonparametric inference of the NS EOS has also been proposed based on Gaussian processes (Essick et al 2020) or using machine learning techniques (Han et al 2021). However, such EOS models have strong limitations because they do not assume any kind of composition of matter in the intermediate density regime.…”

Nonparametric Representation of Neutron Star Equation of State Using Variational Auto-Encoder

“…Gaussian process (GP) has been used as the nonparametric method (Landry & Essick 2019;Essick et al 2020;Landry et al 2020), but such a method is hard to be incorporated by Bayesian inference with the Markov Chain Monte Carlo (MCMC) algorithm. In Han et al (2021), we have developed a nonparametric method via the feed-forward neural network (FFNN), and by using the sampling algorithm MultiNest we obtained the posterior distributions of EoS given the observations. To make the model nonparametric, we have to use 31 parameters in the FFNN.…”

“…However, both methods are deterministic, and they estimate the uncertainties simply by repeating the optimization procedure many times. As mentioned in the previous paragraph, the nonparametric method introduced in Han et al (2021) combines the nonparametric representation of EoS and the Bayesian inference, which can naturally handle the uncertainties. However, the high dimensionality of the parameters in such a method increases the difficulty of sampling.…”

“…Once a VAE is trained one can take the decoder of the VAE as a generator. We employ 100,000 EoSs generated with the nonparametric representation method in Han et al (2021) as the training set and try different settings of the neural network, then get an EoS generator (trained VAE's decoder) with 4 parameters. We use the mass-tidal-deformability data of binary neutron star (BNS) merger event GW170817, and the mass-radius data of PSR J0030+0451, PSR J0740-6620, PSR J0437-4715, and 4U 1702-429 to perform the joint Bayesian inference.…”