Equation-of-state-informed pulse profile modeling

TL;DR

The study tackles the inefficiency and potential biases of performing pulse-profile modeling (PPM) and EOS inference separately with EOS-agnostic priors by introducing an EOS-informed mass–radius ($M$–$R$) prior learned via normalizing flows. This intermediate approach constrains PPM to EOS-consistent regions of parameter space, using χEFT-based priors (PP and CS) and high-density extensions, and is validated on PSR J0740+6620 and PSR J0437–4715. Results show tighter $M$–$R$ posteriors and radius shifts consistent with the underlying EOS: CS favors softer, PP favors stiffer EOSs; a new geometric mode emerges for J0437–4715 under EOS-informed priors but raises questions about physical plausibility. Including the PPM posteriors in subsequent EOS inference further tightens high-density constraints, though sensitivity to high-density parametrizations remains, motivating future fully hierarchical modeling and the use of more agnostic EOS representations.

Abstract

NICER has enabled mass-radius inferences for pulsars using pulse profile modeling (PPM), providing constraints on the equation of state (EOS) of cold, dense matter. To date, PPM and EOS inference have been carried out as two separate steps, with the former using EOS-agnostic priors. This approach has several drawbacks. Ideally, one would perform a fully hierarchical Bayesian inference where the pulse profile and EOS model parameters are jointly fit, but implementing such a framework is complex and computationally demanding. Here, we present an intermediate solution introducing an EOS-informed prior on mass-radius into the existing PPM pipeline using normalizing flows. By focusing on the parameter space consistent with certain EOSs, this approach both tightens constraints on neutron star parameters while reducing computational costs and requiring minimal additional implementation effort. We test this approach on two pulsars, PSR J0740+6620 and PSR J0437-4715, and with two EOS model families: a model based on the speed of sound inside the neutron star interior (CS) and a piecewise-polytropic (PP) model. Both EOS models implement constraints from chiral effective field theory calculations of dense matter. For both pulsar datasets, the inferred radius credible intervals are narrower than in the EOS-agnostic case, with CS favoring smaller radii and PP favoring larger radii. For PSR J0437-4715, the EOS-informed priors reveal a new, more extreme geometric mode that is statistically favored but physically questionable. Including the PPM posteriors in the subsequent EOS inference further tightens the mass-radius posteriors through the chiral effective field theory constraints. However, there is also a sensitivity to the high-density extensions, where the PP (CS) model produces a shift towards larger (smaller) radii and corresponding stiffening (softening) of the pressure-energy density relation.

Figures (10)

• Figure 1: Probabilistic graphical model illustrating the hierarchical Bayesian framework for the joint inference of neutron star properties and EOS parameters. Arrows denote statistical dependences between variables. The shaded gray circle represents observed variables, while double circles indicate deterministic variables. The box represents a summation that applies to all parameters contained within it. See Table \ref{['tab:params hierarical model']} for a detailed description of all variables.
• Figure 2: Mass-radius prior distributions for the PP EOS model (left) and CS EOS model (right) when using the N$^3$LO $\chi$EFT calculations up to $1.5 n_0$. The blue contours show the 68%, 95%, and 99.7% credible regions of the EOS prior distributions. Black dotted lines show the corresponding contours from the normalizing flow distributions.
• Figure 3: Training (orange) and validation (blue) loss curves for the PP EOS model (top) and CS EOS model (bottom).
• Figure 4: Radius, compactness, and mass posterior distributions using the PSR J0740+6620 joint NICER and XMM-Newton dataset conditional on the ST-U model. Three posterior distributions are shown: the results from Salmi2024-J0740 without an EOS-informed prior (red), the results obtained with the CS-informed $M$–$R$ prior (orange), and the ones obtained with the PP-informed $M$–$R$ prior (blue). All inference results are obtained using MultiNest with $4\times10^4$ live points and a sampling efficiency of 0.01. The marginal prior probability distribution functions for each parameter are displayed as dashed-dotted lines. The shaded regions in the diagonal panels contain the 68.3% credible interval for each parameter symmetric around the median. The contours in the off-diagonal panels contain the 68.3%, 95.4%, and 99.7% credible regions.
• Figure 5: Radius, compactness, and mass posterior distributions using the PSR J0437-4715 NICER dataset conditional on the CST+PDT model. Three posterior distributions are shown: the headline result from Choudhury2024-J0437 using an EOS-agnostic approach (red), the results obtained with the CS-informed $M$–$R$ prior (orange), and the ones obtained with the PP-informed $M$–$R$ prior (blue). The EOS-agnostic analysis used MultiNest with $2\times10^4$ live points and a sampling efficiency of 0.3, while the EOS-informed runs used MultiNest with $4\times10^3$ live points and a sampling efficiency of 0.1. See Fig. \ref{['fig:corner_J0740']} for more details about the figure elements.