Tidal Signatures in the Inspiral

A neutron star’s companion raises a quadrupolar tide that stores binding energy and modifies the late-inspiral phase. The leading measurable combination is the mass-weighted tidal deformability $\widetilde\Lambda$, not a radius or equation of state directly. Tidal information accumulates mainly at high frequency, where detector noise, point-particle waveform error, spin, calibration, and merger modeling are most consequential.

Helpful background. Tidal Deformability and Love Response develops stellar structure and the equation-of-state map. Parameter Degeneracies develops the point-particle likelihood.

Adiabatic tidal response

For a slowly varying external quadrupolar tidal field $\mathcal E_{ij}$, write

$$Q_{ij}=-\lambda\mathcal E_{ij}, \qquad \lambda=\frac{2}{3G}k_2R^5, \qquad \Lambda=\frac{\lambda}{G^4m^5} =\frac23 k_2\left(\frac{R}{Gm}\right)^5.$$

Here $Q_{ij}$ is the induced mass quadrupole, $k_2$ the quadrupolar Love number, and $R$ the areal radius of the isolated, approximately nonrotating star. These are source-frame stellar quantities. The dimensionless $\Lambda$ is extremely sensitive to compactness, but $k_2$ also depends on internal structure. A tidal measurement therefore does not convert to $R$ without a stellar model or relation. The response definition and leading phasing calculation originate in Flanagan and Hinderer 2008, Eqs. (4) and (10).

For a binary, the leading phase depends on

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

Equal masses and identical stars give $\widetilde\Lambda=\Lambda$. The mass weights are unchanged if all masses are multiplied by $1+z$, but an EOS relation $\Lambda(m)$ must be evaluated at the source-frame mass. If one body is a general-relativistic black hole with vanishing static Love number, its term is zero but the companion’s weighted response remains. This static statement does not remove dissipative effects such as horizon absorption.

Where the phase information enters

With detector-frame total mass $M=M^{\rm det}$, observed frequency $f$, and $v=(\pi G M f)^{1/3}$, the leading Fourier-domain phase schematically contains

$$\Psi(f)=\Psi_{\rm pp}(f) +\frac{3}{128\eta v^5} \left[-\frac{39}{2}\widetilde\Lambda v^{10} +\cdots\right].$$

Thus the leading adiabatic tidal term is a relative fifth-post-Newtonian correction. One can understand the power counting from the induced quadrupole: $\mathcal E\sim Gm/r^3$, so its interaction energy scales as $\lambda\mathcal E^2$. Relative to the orbital binding energy it is of order $(R/r)^5$, or $v^{10}$ at fixed stellar compactness. Both the modified binding energy and the extra radiation enter the phase coefficient; scaling alone does not determine $39/2$.

For the transform $\tilde h(f)=\int h(t)e^{+2\pi ift}dt$ and the phase convention above, the leading tidal contribution is

$$\Delta\Psi_{\rm tidal} =-\frac{117}{256\eta}\widetilde\Lambda v^5.$$

For an illustrative equal-mass binary with $M^{\rm det}=2.8M_\odot$ and $\widetilde\Lambda=400$, it is about $-0.084$, $-1.23$, and $-3.91$ radians at $100$, $500$, and $1000\,\mathrm{Hz}$, respectively, using $GM_\odot/c^3=4.92549\,\mu\mathrm{s}$. These are leading-order phase differences at fixed coalescence time and phase, not measured residuals or an accuracy forecast: a fit can absorb part of them into other parameters, and higher-order tidal physics becomes increasingly relevant at high frequency. The increasing effect explains both the promise of late-inspiral data and the need to control its modeling.

Subleading tidal terms include another mass-weighted deformability, higher multipoles, spin–tidal couplings, and dynamical tides. Near stellar mode resonances or merger, the adiabatic expansion can fail. The usable frequency ceiling is therefore part of the inference.

Separating tides from the baseline

Tidal inference is a small difference between two large phases. Bias can arise from point-particle phasing, spin-induced quadrupoles, residual eccentricity, calibration, PSD error, or a high-frequency glitch. Mass ratio and aligned spin correlate with $\widetilde\Lambda$; restrictive spin priors can sharpen tides by assumption.

The most discriminating checks are:

• recover zero tides from binary-black-hole injections and known tides from neutron-star injections;
• vary the high-frequency cutoff and verify that information accumulates coherently rather than in one contaminated band;
• compare waveform families with different point-particle and tidal resummations;
• propagate calibration and PSD uncertainty;
• inspect detector-by-detector residuals;
• test spin priors and, for unequal masses, the relation imposed between $\Lambda_1$ and $\Lambda_2$.

A binary-Love or common-equation-of-state relation can reduce dimensionality, but it adds an equation-of-state universality assumption. An “independent” measurement of both stars’ deformabilities is weaker and often much broader.

Observational anchor and cutoff

GW170817 remains the clearest public inspiral example with matter-sensitive phasing and a multimessenger classification. The official discovery and parameter analyses established a binary-neutron-star interpretation and constrained tidal combinations under specified waveform and spin priors LVK 2017, pp. 3–8 and supplemental material. Its GWOSC release fixes the strain and event products used for reproduction GWOSC, GW170817 event page.

At the evidence cutoff 2026-08-18, GWTC-5.0 adds no new confidently classified neutron-star-containing event in O4b; current matter claims therefore continue to depend strongly on earlier events and their reanalyses LVK 2026, observations §§1 and 3. This is a dated catalog statement, not a prediction about future detections.

Interpretation boundary

$\widetilde\Lambda$ is closest to the waveform. Individual $\Lambda(m)$ values add a binary relation and mass inference. Radius adds stellar-structure modeling. Pressure versus density adds an equation-of-state parameterization, phase-transition assumptions, causality and stability constraints, and often external pulsar or X-ray data. Each layer can be scientifically justified, but uncertainty must be propagated through it.

The detector-level analysis belongs here. Love numbers, stellar structure, and microphysical interpretation belong in Relativistic Matter and Compact Objects.

Exercises

1. Equal-star limit. Verify that $m_1=m_2=m$ and $\Lambda_1=\Lambda_2=\Lambda$ imply $\widetilde\Lambda=\Lambda$.

Solution

Each numerator term is $(m+12m)m^4\Lambda=13m^5\Lambda$, so the sum is $26m^5\Lambda$. The denominator is $(2m)^5=32m^5$. Hence

$$\widetilde\Lambda=\frac{16}{13}\frac{26}{32}\Lambda=\Lambda.$$

2. Compactness sensitivity. At fixed $k_2$ and mass, by what factor does $\Lambda$ change if the radius increases by $10\%$?

Solution

Since $\Lambda\propto R^5$, the factor is $1.1^5\approx1.61$. Realistic $k_2$ also changes with compactness, so this is a scaling illustration rather than a radius-conversion formula.

References

• Flanagan, Éanna É., and Tanja Hinderer. “Constraining Neutron-Star Tidal Love Numbers with Gravitational-Wave Detectors.” Physical Review D 77 (2008): 021502. DOI. Open PDF.
• Gravitational Wave Open Science Center. “GW170817 Event Data.” Data and documentation.
• LIGO Scientific Collaboration and Virgo Collaboration. “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral.” Physical Review Letters 119 (2017): 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.

Further reading

• Baiotti, Luca. “Gravitational Waves from Neutron Star Mergers and Their Relation to the Nuclear Equation of State.” Progress in Particle and Nuclear Physics 109 (2019): 103714. DOI. Open PDF.