Binary-Neutron-Star Inspiral Inference

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Binary-neutron-star inspirals can constrain masses, aligned spin, tidal deformability, distance, and sky position, but not independently with equal precision. Chirp mass is sharply encoded in the long inspiral; mass ratio, spin, and tides enter at higher orders and are correlated. Equation-of-state information therefore depends on waveform completeness, spin and mass priors, high-frequency data, and the relation imposed between the two stars.

Helpful background. Tidal Signatures in the Inspiral defines $\widetilde\Lambda$ . Priors, Measures, and Reparameterization explains why a prior described in different coordinates changes the inference.

The hierarchy of phase information

For detector-frame total mass $M=M^{\rm det}$ , symmetric mass ratio $\eta$ , observed frequency $f$ , and $v=(\pi GMf)^{1/3}$ , the stationary-phase inspiral has schematic phase

\Psi(f)=2\pi ft_c-\phi_c-\frac{\pi}{4} +\frac{3}{128\eta v^5} \left[1+\sum_k a_k(\eta,\boldsymbol\chi)v^k -\frac{39}{2}\widetilde\Lambda v^{10}+\cdots\right].

The leading chirp term determines detector-frame $\mathcal M=M\eta^{3/5}$ . Mass ratio and aligned spin alter subleading point-particle phasing; tides grow late in the signal. A change in $q$ or spin can partially offset tidal phase over a finite band, so marginalizing them broadens $\widetilde\Lambda$ .

Neutron-star spins in known merging Galactic binaries motivate relatively small-spin priors, but a population expectation is not a direct constraint on every gravitational-wave source. A broad spin prior allows more mass–spin degeneracy. Reporting both can show how much apparent tidal precision is prior supplied.

One tidal combination, two stars

The leading combination

\widetilde\Lambda =\frac{16}{13} \frac{(m_1+12m_2)m_1^4\Lambda_1 +(m_2+12m_1)m_2^4\Lambda_2} {M^5}

does not normally determine $\Lambda_1$ and $\Lambda_2$ separately. Three common parameterizations add different information:

independent $\Lambda_1,\Lambda_2$ priors make the weakest shared-matter assumption;
binary-Love relations restrict the two deformabilities using approximate equation-of-state-insensitive structure;
a common equation of state maps both masses to one function $\Lambda(m)$ and enforces stability and causality conditions.

The latter approaches can greatly tighten radii but are not the same measurement. Phase transitions or nonstandard compact stars can weaken assumed universal relations.

High-frequency likelihood choices

Tidal information is concentrated where detector sensitivity worsens and waveform differences grow. The analysis must state low- and high-frequency cutoffs, PSD segment and estimator, calibration marginalization, glitch subtraction, waveform and tidal order, and whether merger information is included.

A useful cumulative test recomputes the posterior as the high-frequency cutoff increases. Genuine broadband tidal information should accumulate consistently; a sudden shift from a narrow band calls for detector and waveform diagnostics. Comparing post-Newtonian, effective-one-body, and phenomenological tidal models gauges model uncertainty only where each is calibrated.

Observational examples

GW170817 had a long in-band inspiral, a strong network detection, and an electromagnetic counterpart. Its official analyses constrained masses and tidal deformability under both low- and broader-spin priors, illustrating the mass–spin–tide hierarchy LVK 2019, §§III–VII.

GW190425 was consistent with a substantially heavier binary-neutron-star system but lacked a confirmed counterpart and had weaker network geometry. Its classification and source interpretation therefore rely more heavily on masses, priors, and population context LVK 2020, §§2–5.

At the cutoff 2026-08-18, GWTC-5.0 adds no new confidently neutron-star-containing event from O4b LVK 2026, observations §§1 and 3. Current combined matter constraints should name the exact earlier events and reanalysis products rather than imply that every catalog update adds equivalent tidal information.

Systematics and reproduction

A bounded reproduction pins the GWOSC strain release, detector segments, sample rate, calibration, PSD construction, data-quality mask, waveform version, priors, sampler, and random seeds. It should recover injections spanning plausible mass ratios, spins, and deformabilities in real off-source noise.

Independent checks include:

posterior calibration for $\mathcal M$ , $q$ , $\chi_{\rm eff}$ , and $\widetilde\Lambda$ ;
waveform comparisons at the same priors and frequency support;
high-frequency cutoff and detector-removal tests;
independent versus common-EOS tidal parameterizations;
residual checks and calibration perturbations;
prior predictive plots showing which waveform phases the prior permits.

The detector likelihood can constrain $\Lambda(m)$ . Translating it into pressure, composition, or phase transitions belongs to Relativistic Matter and Compact Objects and must retain the detector-layer uncertainty.

Exercises

1. Why chirp mass wins. Why is chirp mass usually measured more accurately than mass ratio in a long inspiral?

Solution

Chirp mass controls the leading $v^{-5}$ phase evolution and therefore accumulates coherently over many cycles. Mass ratio first changes higher-order coefficients and is correlated with spin. The likelihood is consequently much narrower along the chirp-mass direction than along the orthogonal component-mass direction.

2. A tight radius posterior. An analysis replaces independent $\Lambda_1,\Lambda_2$ with one causal spectral EOS shared by both stars. Why can the radius posterior narrow?