Optimizing Bayesian model selection for equation of state of cold neutron stars

TL;DR

This work presents BEOMS, a Bayesian evidence framework to discriminate among cold neutron star EOSs using EOS-agnostic posterior samples from binary neutron star GW signals. By transforming and marginalizing posterior results into reduced spaces such as $(m,\Lambda)$ and $(\tilde{\Lambda},\eta)$, BEOMS computes EOS evidences $Z_k$ for multiple EOS hypotheses and combines them across a population of events for robust discrimination. The main finding is that evidence aggregation in the two-dimensional $m$-$\Lambda$ space yields higher discriminative power with fewer events than full 4D treatments, and next-generation detectors like ECC significantly enhance EOS distinguishability. The method provides a practical path to constrain dense-matter EOSs with GW data and is adaptable to other GW sources, with implications for multimessenger constraints and EOS modeling.

Abstract

We introduce a computational framework, Bayesian Evidence calculation fOr Model Selection (BEOMS) to evaluate multiple Bayesian model selection methods in the context of determining the equation of state (EOS) for cold neutron star (NS), focusing on their performance with current and next-generation gravitational wave (GW) observatories. We conduct a systematic comparison of various EOS models by using posterior distributions obtained from EOS-agnostic Bayesian inference of binary parameters applied to GWs from a population of binary neutron star (BNS) mergers. The cumulative evidence for each model is calculated in a multi-dimensional parameter space characterized by neutron star masses and tidal deformabilities. Our findings indicate that Bayesian model selection is most effective when performed in the two-dimensional subspace of component mass and tidal deformability, requiring fewer events to distinguish between EOS models with high confidence. Furthermore, we establish a relationship between the precision of tidal deformability measurements and the accuracy of model selection, taking into account the evolving sensitivities of current and planned GW observatories. BEOMS offers computational efficiency and can be adapted to execute model selection for gravitational wave data from other sources.

Figures (13)

• Figure 1: Here we show the flow chart of our calculation of evidence in different spaces and their relations as discussed in sec \ref{['sec:method']}. We start with the marginalized 4D posterior, $(m_1,m_2,\Lambda_1,\Lambda_2)$ from the Bilby LALSuite2020PE results and marginalize it to obtain two 2D posterior in $m_1-\Lambda_1$ and $m_2-\Lambda_2$. The 3D posterior, $\mathcal{M},\eta,\tilde{\Lambda}$ is obtained by transforming the 4D posterior to $(\mathcal{M},\eta,\tilde{\Lambda},\delta\tilde{\Lambda})$ and ignoring $\delta\tilde{\Lambda}$ for computational efficiency. In some cases, it is also useful to take a mean value of $\mathcal{M}$ reducing the 3D posterior to just 2D, $(\eta,\tilde{\Lambda})$ with negligible loss of the evidence.
• Figure 2: Mass-Radius-Tidal deformability curves for our choice of EOSs used in the current paper. We have chosen EOSs varying in stiffness, compactness, maximum mass and the formalism used for construction. EOS surves span wide variety of physical phenomena in dense matter in order to test the robustness of the method presented here.
• Figure 3: In this figure, we show the expected distributions and improvements in the measurement of three key quantities relevant for neutron star EOS problem. We plot the 1$\sigma$ full-width (on $\log_{10}$ scale) of the posteriors obtained from the PE runs of 100 events where injected EOS is APR4. The A$\#$ will bring only slight improvement in the measurement of $\mathcal{M}$, $\eta$ and $\tilde{\Lambda}$ while the proposed ECC configuration is expected to increase the accuracy of the measurement by 1-2 orders of magnitude or higher compared to our current accuracy. As the detector network sensitivity increases we also expect to see few events with exceptional measurement leading to golden binaries that may allow us to perform precision of tests of gravity. The runs with other EOSs have similar distributions where exactly same set of parameters have been used above.
• Figure 4: Here we show the distribution of evidence using the three methods described in section \ref{['sec:method']} and for our collection of 11 EOSs considered candidate models. Each ridgeline plot corresponding to a model is smoothened distribution of evidence of 100 events. The three panels show the results for three detector configurations where the injected EOS is DD2. We represent the median value and 1-$\sigma$ width with a filled circle and an error bar on the line, respectively. It is evident from the top two panels that A$\# ~$ (As in plot legend) does not bring much improvement in disinguishing EOSs while the ECC clearly picks up the correct EOS where the evidence for the DD2 EOS is large for most of the events. Please also note that errors reduce from the present to future detector configurations (top to bottom). The calculations in the subspace are highlighted by both distinguishing properties -- the median values being higher for the injected EOS and the width of the distribution being small, while the 4D space calculations are not informative even for our most promising detector confifurations. Similar plots for other EOS injection show similar behaviors with respect to detector networks and the model EOSs shown here.
• Figure 5: Here we plot the cumulative evidence on log scale of various EOS models as a function of the number of events with the three injection EOSs for detectors with A$\# ~$ sensitivity. The model with the highest evidence is the preferred model however confusion may arise based on the type of events selected. We show the uncertainties as color bands arising from the sampling errors from the set of events.