[Paper Review] Inferring the neutron star equation of state simultaneously with the population of merging neutron stars

D. M. Wysocki, R. O’Shaughnessy+2 more|arXiv (Cornell University)|Jan 6, 2020

Pulsars and Gravitational Waves Research1 references20 citations

TL;DR

This paper presents a hierarchical Bayesian inference framework that simultaneously constrains the neutron star equation of state (EOS), the population distribution of merging neutron stars (including mass, spin, and merger rate), and accounts for intrinsic diversity such as bimodal mass distributions. By using principal component analysis to reparameterize the EOS in a physically valid, high-efficiency sampling space, the method improves sampling efficiency by over 3 orders of magnitude, enabling robust EOS inference even with small event counts (N ≲ 10), and demonstrates that biased population inferences can significantly distort EOS recovery.

ABSTRACT

Observations of the properties of multiple coalescing neutron stars will simultaneously provide insight into neutron star mass and spin distribution, the neutron star merger rate, and the nuclear equation of state. Not all merging binaries containing neutron stars are expected to be identical. Plausible sources of diversity in these coalescing binaries can arise from a broad or multi-peaked NS mass distribution; the effect of different and extreme NS natal spins; the possibility of NS-BH mergers; or even the possibility of phase transitions, allowing for NS with similar mass but strongly divergent radius. In this work, we provide a concrete algorithm to combine all information obtained from GW measurements into a joint constraint on the NS merger rate, the distribution of NS properties, and the nuclear equation of state. Using a concrete example, we show how biased mass distribution inferences can significantly impact the recovered equation of state, even in the small-$N$ limit. With the same concrete example, we show how small-$N$ observations could identify a bimodal mass and spin distribution for merging NS simultaneously with the EOS. Our concordance approach can be immediately generalized to incorporate other observational constraints.

Motivation & Objective

• To develop a unified inference framework that simultaneously constrains the neutron star equation of state (EOS), the population distribution of merging neutron stars (including mass, spin, and merger rate), and intrinsic diversity such as bimodal distributions.
• To address the challenge of highly correlated, non-uniformly distributed EOS parameters in spectral representation, which leads to extremely low sampling efficiency in standard Monte Carlo methods.
• To improve sampling efficiency in EOS inference by transforming the parameter space using principal component analysis (PCA) on physical, valid EOS samples, reducing the volume of unphysical parameter space.
• To demonstrate that population-level biases—such as incorrect assumptions about mass or spin distributions—can significantly distort the inferred EOS, even with small numbers of observed events.
• To provide a general-purpose, extensible codebase that integrates population modeling with EOS inference, applicable to future gravitational wave observations of binary neutron star mergers.

Proposed method

• The method uses a hierarchical Bayesian approach to jointly infer the neutron star equation of state (EOS), the population distribution of merging neutron stars (mass, spin, merger rate), and intrinsic diversity.
• It employs Lindblom’s spectral EOS parameterization with basis functions (γ₀ to γ₃) to represent the EOS, but transforms the parameter space to improve sampling efficiency.
• A Monte Carlo sampling procedure identifies valid, physical EOS within the original hypercube, and PCA is applied to the resulting physical samples to find a new, rotated coordinate system (r′) that captures most variance in fewer dimensions.
• The method constructs a minimal hypercube 𝒞′ in the transformed r′ space that encloses the physical EOS subset, increasing sampling efficiency from ~0.005% to ~19% (a 3.2-order-of-magnitude improvement) with a 10% buffer to reduce risk of missing physical solutions.
• The transformation involves standardizing the physical samples (subtracting mean μᵣ, dividing by standard deviation σᵣ), followed by PCA to compute the rotation matrix S that maps the standardized coordinates to the principal component basis.
• The final sampling is performed uniformly within the optimized hypercube 𝒞′, which is significantly smaller and better aligned with the physical EOS subspace, enabling efficient and accurate joint inference.

Experimental results

Research questions

• RQ1 Can joint inference of the neutron star equation of state and the population distribution of merging neutron stars improve constraints on nuclear physics, even with small numbers of observed events?
• RQ2 How does bias in the assumed population distribution (e.g., unimodal vs. bimodal mass distribution) affect the accuracy of the inferred equation of state?
• RQ3 What is the optimal way to parameterize the neutron star equation of state to maximize sampling efficiency while ensuring physical validity?
• RQ4 Can a hierarchical inference framework detect intrinsic diversity in the neutron star population—such as bimodal mass or spin distributions—simultaneously with EOS constraints?
• RQ5 How can the parameter space of spectral EOS representations be restructured to reduce the computational cost of Markov Chain Monte Carlo sampling?

Key findings

• The method improves sampling efficiency for physical neutron star equations of state by a factor of approximately 19% (a 3.2-order-of-magnitude improvement) when using a 10% buffer around the optimized hypercube in the principal component-transformed space.
• The physical EOS subspace occupies only about 0.005% of the original 4D hypercube of spectral parameters, making uniform sampling in the original space highly inefficient.
• Biased assumptions about the neutron star population—such as assuming a unimodal mass distribution when the true distribution is bimodal—can lead to significant systematic errors in the inferred equation of state, even with small N.
• The framework successfully identifies a bimodal mass and spin distribution in the merging neutron star population simultaneously with EOS constraints, using only a small number of simulated events.
• The PCA-based reparameterization effectively captures the dominant variance in the physical EOS subspace, reducing the dimensionality of the inference problem while preserving physical consistency.
• The method is generalizable and can be extended to other astrophysical inference problems involving parametric dimensional reduction, such as subpopulations of black hole binaries with zero spin.