Equation-of-State Inference

Equation-of-state inference treats all neutron stars as draws from one dense-matter relation and pushes that relation through stellar structure into each event likelihood. It can combine weak tidal measurements coherently, enforce stability and causality, and incorporate pulsar or X-ray data. Its precision is conditional on the EOS parameterization, stellar branches, population model, waveform systematics, and compatibility of the datasets.

Helpful background. Binary-Neutron-Star Inspiral Inference supplies event mass–spin–tide likelihoods. Hierarchical Equation-of-State Reconstruction develops pressure–density models and Gravitational-Wave Tidal Inference supplies the stellar-response interface.

From pressure to a waveform parameter

Let $\boldsymbol\lambda_{\rm EOS}$ denote EOS hyperparameters defining pressure $p(\epsilon\mid\boldsymbol\lambda_{\rm EOS})$ as a function of energy density $\epsilon$. For the inspiral application, this is usually a cold, beta-equilibrated relation, and the stellar sequence is approximately nonrotating; thermal postmerger matter and rapid rotation require extensions. Solving the Tolman–Oppenheimer–Volkoff equations and the tidal perturbation equation gives, along each stable branch,

$$M(\rho_c;\boldsymbol\lambda_{\rm EOS}), \qquad R(\rho_c;\boldsymbol\lambda_{\rm EOS}), \qquad \Lambda(\rho_c;\boldsymbol\lambda_{\rm EOS}).$$

Eliminating central density yields $R(M;\boldsymbol\lambda_{\rm EOS})$ and $\Lambda(M;\boldsymbol\lambda_{\rm EOS})$ where the map is single valued. The binary waveform then uses

$$\widetilde\Lambda =\widetilde\Lambda \bigl(m_1,m_2,\Lambda(m_1;\boldsymbol\lambda_{\rm EOS}), \Lambda(m_2;\boldsymbol\lambda_{\rm EOS})\bigr).$$

The detector does not observe pressure directly. It constrains the end of this forward chain.

Strong phase transitions can create disconnected stable branches or twins with similar mass and different radius. An analysis assuming one smooth branch can exclude that possibility by construction. The maximum supported mass, low-density crust matching, and central-density prior are also part of the model.

Hierarchical likelihood

Define the event likelihood restricted to an EOS curve by

$$\begin{aligned} \mathcal L_i^{\rm EOS}(m_1,m_2,\nu) &=p\bigl(d_i\mid m_1,m_2,\nu,\Lambda_1,\Lambda_2\bigr),\\ \Lambda_a&=\Lambda(m_a;\boldsymbol\lambda_{\rm EOS}). \end{aligned}$$

The masses in this stellar relation are source-frame masses; the waveform uses $(1+z)m_a$, with redshift supplied by the distance–cosmology part of the model. Nuisances $\nu$ include spins, distance, orientation, calibration, and noise parameters. For a catalog conditioned on $N$ detections, one schematic form is

$$\mathcal L(\{d_i\}\mid\boldsymbol\lambda_{\rm EOS},\Lambda_{\rm pop},N) \propto \alpha(\boldsymbol\lambda_{\rm EOS},\Lambda_{\rm pop})^{-N} \prod_i\mathcal I_i,$$ $$\mathcal I_i= \int dm_1\,dm_2\,d\nu\, \mathcal L_i^{\rm EOS}(m_1,m_2,\nu)\, p(m_1,m_2,\nu\mid\Lambda_{\rm pop},\boldsymbol\lambda_{\rm EOS}).$$

The population distribution must be normalized on the allowed stellar masses and branches. EOS-dependent maximum masses or detection efficiencies can therefore affect both the numerator and $\alpha$. Neglecting that dependence is an approximation to check, not a general property of tidal inference. A rate analysis instead uses the corresponding Poisson likelihood.

Posterior-sample recycling must remove each event’s interim prior. There is an extra numerical subtlety: a deterministic EOS restricts continuous $(m_1,m_2,\Lambda_1,\Lambda_2)$ space to a lower-dimensional surface. Almost no samples from independent tidal priors lie exactly on that surface. Ordinary finite importance weights cannot impose this singular constraint. One must sample the common-EOS model directly, evaluate a validated likelihood or density interpolation along the curve, or introduce a justified finite-width relation and verify convergence as its numerical representation is refined.

A common EOS ties events more strongly than merely multiplying independent $\widetilde\Lambda$ posteriors. It evaluates whether each event’s masses and tides can lie on the same stellar sequence.

Parameterization is a prior on functions

Piecewise polytropes, spectral expansions, sound-speed models, Gaussian processes, and direct pressure knots explore different function spaces. Uniform coefficients do not induce uniform pressure, radius, or deformability. Smooth spectral models can struggle with sharp phase transitions; extremely flexible models broaden constraints and need regularization.

Physical restrictions commonly include:

• thermodynamic stability, $dp/d\epsilon\ge0$;
• causal sound speed, $0\le c_s^2=dp/d\epsilon\le1$;
• matching to a stated low-density crust or nuclear calculation;
• support for observed massive pulsars;
• a declared treatment of self-bound matter and multiple branches.

These are informative physical priors. Their effect should be shown with prior predictive bands in $p(\epsilon)$, $M$–$R$, and $\Lambda(M)$ space.

Results at the stated cutoff

GW170817 provides the strongest public gravitational-wave tidal likelihood for this purpose, with GW190425 contributing differently because of its mass, signal quality, and weaker matter information. Official and independent reanalyses show that radii and EOS bands move with waveform family, high-frequency cutoff, spin prior, and EOS representation LVK 2018, pp. 3–8 and Narikawa et al. 2020, §§II–IV.

As of 2026-08-18, GWTC-5.0 contributes no new confidently neutron-star-containing O4b event LVK 2026, observations §§1 and 3. Exact radius, maximum-mass, pressure, and phase-transition odds remain dated and model conditional.

Evidence line	Supported inference	Main limitation
Inspiral likelihood disfavors extremely large tidal phase in GW170817	Very stiff stellar responses are restricted under the stated waveform and priors	High-frequency waveform and prior sensitivity
Shared-EOS analyses combine tides with massive-pulsar support	A common stable sequence occupies a narrower function band	Parameterization and branch assumptions
Flexible/nonparametric analyses widen some bands	Function-space uncertainty is material	Flexibility can be prior dominated where data are weak
Multimessenger combinations sharpen radii	Independent datasets can complement GW tides	Shared systematics and model consistency must be checked

Combining other observations

For conditionally independent data $D_{\rm GW}$, $D_X$, and $D_{\rm radio}$ under the same EOS,

$$p(\boldsymbol\lambda_{\rm EOS}\mid D) \propto p(\boldsymbol\lambda_{\rm EOS}) \mathcal L_{\rm GW}(\boldsymbol\lambda_{\rm EOS}) \mathcal L_X(\boldsymbol\lambda_{\rm EOS}) \mathcal L_{\rm radio}(\boldsymbol\lambda_{\rm EOS}).$$

Independence cannot be assumed if analyses reuse a prior posterior as a likelihood, share calibration or distance information, or impose inconsistent mass/radius definitions. Each likelihood must be recovered with its interim prior removed. Electromagnetic remnant arguments additionally depend on ejecta, jet, and merger simulations. For a concrete combination of radio, X-ray, gravitational-wave, and kilonova information, see Raaijmakers et al. 2021, §§ II–III and V.3; their likelihood approximations and parameterization comparisons are part of the inference.

Validation and resolution criteria

A strong analysis performs simulated-EOS recovery across parameterizations, injection recovery at the event-likelihood level, waveform and frequency-cutoff variation, population and selection stress tests, and leave-one-event-out comparisons. It reports whether the posterior reaches parameterization or maximum-mass boundaries.

A phase-transition claim requires evidence across models that can represent both smooth and discontinuous behavior, plus a data feature not equally explained by waveform or population assumptions. Review after a new matter event, a revised tidal likelihood, a stronger maximum-mass measurement, or a validated EOS model that changes the evidence ranking.

Exercises

1. Why two events help. Two binaries have different component masses but individually broad tidal likelihoods. How can a shared EOS add information?

Solution

The EOS requires all four stars to lie on one $\Lambda(M)$ curve. Parameter combinations that fit one event but predict incompatible deformabilities at the other event’s masses are removed. The gain is valid only if the common-EOS and branch assumptions hold.

2. A smooth-model null result. A spectral EOS analysis finds no phase transition. What has been established if the basis cannot represent a sharp transition well?

Solution

Only that no transition is required within the smooth spectral family. The analysis has low power against the excluded alternative. A fair conclusion needs a model that explicitly permits sharp transitions or multiple branches and tests recovery on such injections.

References

• LIGO Scientific Collaboration and Virgo Collaboration. “GW170817: Measurements of Neutron Star Radii and Equation of State.” Physical Review Letters 121 (2018): 161101. DOI. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. “GWTC-5.0: Observations from the Second Part of the Fourth LIGO-Virgo-KAGRA Observing Run and Updates to the Gravitational-Wave Transient Catalog.” arXiv:2605.27225 (2026). Open PDF.
• Narikawa, Tatsuya, et al. “Reanalysis of the Binary Neutron Star Mergers GW170817 and GW190425 Using Numerical-Relativity Calibrated Waveform Models.” Physical Review Research 2 (2020): 043039. DOI. Open PDF.
• Raaijmakers, Geert, et al. “Constraints on the Dense Matter Equation of State and Neutron Star Properties from NICER’s Mass–Radius Estimate of PSR J0740+6620 and Multimessenger Observations.” Astrophysical Journal Letters 918 (2021): L29. DOI. Open PDF.