Propagation Speed, Dispersion, and Damping

Propagation tests separate four effects that are often conflated: a frequency-independent wave speed, frequency-dependent dispersion, amplitude damping, and polarization-dependent birefringence. They act between source and detector, whereas altered binary dynamics acts at emission. The distinction matters because the same measured phase or amplitude can be explained by source physics, detector calibration, cosmology, or propagation unless its characteristic frequency, distance, redshift, and polarization dependence is retained.

Helpful background. Weak-Field Gravity supplies the wave limit of metric perturbations; Probability and Statistical Inference supplies nuisance marginalization; Gravitational-Wave Data Analysis and Inference supplies strain, response, and distance posteriors; Tests of Gravity supplies the null-test logic; and Black-Hole Perturbation Theory distinguishes source-mode structure from effects accumulated during travel.

Speed and dispersion are different observables

With time dependence $e^{-i\omega t}$, a useful phenomenological dispersion relation is

$$E^2=p^2+A_\alpha p^\alpha,$$

Here $E$ and $p$ are the locally measured wave energy and momentum magnitude. The coefficient $A_\alpha$ has dimension $[\text{energy}]^{2-\alpha}$, and the GR null is $A_\alpha=0$. A dispersion relation must be accompanied by its propagation prescription and the corresponding waveform phase; one should not transfer a bound between different prescriptions by copying the symbol $A_\alpha$.

A transparent example is $\alpha=0$, $A_0=m_g^2$, for which a freely propagating massive mode has

$$v_g=\frac{dE}{dp}=\frac{p}{E} =\sqrt{1-\frac{m_g^2}{E^2}} \simeq1-\frac{m_g^2}{2E^2}.$$

The expansion assumes $m_g/E\ll1$. Over a fixed nearby distance $L$, the excess travel time is $L(1/v_g-1)\simeq Lm_g^2/(2E^2)$: lower frequencies arrive later. At cosmological distance, $E$ changes along the path and this expression must be integrated on the expanding background. More general phase parameterizations have special powers that are degenerate with coalescence time or phase, and sometimes logarithmic limits; those cases require the exact implementation used by the reported analysis. A massive-mode propagation bound alone does not establish the consistency of a complete massive-gravity theory.

A counterpart arrival time instead constrains

$$\Delta t_\mathrm{obs} =(1+z)\Delta t_\mathrm{em}^{\rm source}+\Delta t_\mathrm{prop},$$

where positive lag means the gravitational wave arrives after the electromagnetic signal, and $\Delta t_\mathrm{em}^{\rm source}$ uses the same sign at emission. The factor $1+z$ converts that source-frame interval to an observer-frame interval; it can be omitted only when the lag has already been redshifted. The intrinsic lag is not known from the detector clocks. A narrow observed lag constrains a frequency-independent speed difference only after the source mechanism bounds that term. By contrast, dispersion can be inferred from phase accumulated within one gravitational-wave signal, but it remains covariant with intrinsic phasing and waveform error. The GWTC-3 treatment makes this separation explicit in its modified-dispersion analysis LIGO–Virgo–KAGRA 2023, §VI and Table VII.

Amplitude friction and polarization transport

On a homogeneous background, parameterize tensor propagation in conformal time by

$$h_A''+(2+\nu)\mathcal H h_A' +\left(c_T^2 k^2+a^2\mu_T^2\right)h_A=0,$$

where $A$ labels polarization, primes denote conformal-time derivatives, $k$ is comoving wavenumber, $\mathcal H=a'/a$, $\nu$ modifies friction, $c_T$ the speed, and $\mu_T$ an effective mass. If only friction changes ($c_T=1$, $\mu_T=0$), wavelengths are short compared with the background evolution scale, and source emission and detector coupling use the same normalization as in GR, the gravitational-wave luminosity distance satisfies

$$\frac{d_L^\mathrm{GW}(z)}{d_L^\mathrm{EM}(z)} =\exp\!\left[ \frac12\int_0^z\frac{\nu(z')}{1+z'}\,dz' \right].$$

This is an amplitude effect, derived in the standard-siren treatment and in Belgacem et al. 2018, Eqs. (8)–(13). It is degenerate with detector amplitude calibration, binary inclination, weak lensing, host redshift, population selection, and the electromagnetic cosmology used for $d_L^\mathrm{EM}$. Parity-violating propagation may give right- and left-handed modes different phase or amplitude evolution. That signature must be projected through the network antenna response; a single effective amplitude shift is not automatically birefringence.

The review by Yunes, Siemens, and Yagi develops the required separation between direct and generic tests, generation and propagation, detector response, and parameterized deformations Yunes, Siemens, and Yagi 2025, §§3.1–3.5 and §§4.1–4.4. A bottom-up bound constrains the declared dispersion or damping coordinate. Translating it into a graviton mass, Lorentz-violating coefficient, or modified-gravity coupling requires that theory’s field content, stability domain, and source-generation corrections.

Likelihood, release, and selection

The data vector is versioned calibrated strain, sometimes augmented by a counterpart time and host redshift. Collect the source, redshift, intrinsic-lag, calibration, waveform, and host nuisances into $\Xi$. A normalized marginal likelihood takes the form

$$p(d_\mathrm{GW},d_\mathrm{EM}\mid\lambda) =\int p(d_\mathrm{GW}\mid\theta,\lambda,C,\mathcal M) \,p(d_\mathrm{EM}\mid z,\Delta t_\mathrm{em},H) \,p(\Xi\mid\lambda)\,d\Xi,$$

with propagation parameters $\lambda$, calibration $C$, waveform model $\mathcal M$, and host model $H$ marginalized rather than fixed. Event cuts on signal-to-noise ratio, distance information, counterpart association, frequency bandwidth, and data quality define the population to which the result applies. Cross-event products require a selection normalization and shared calibration or cosmological covariance.

At the evidence cutoff 2026-08-19, the official current strain/posterior baseline is GWTC-5/GWOSC stable parameter-estimation release 9, with file identity and calibration pinned through the official documentation GWOSC 2026, “Parameter Estimation” and “Notes on Strain Data”. The 16 kHz and decimated 4 kHz products are not interchangeable, and data below 10 Hz are uncalibrated. P2600410-v1 is the separate release of result products for the tests manuscript LIGO–Virgo–KAGRA 2026, official tests data release P2600410-v1. Crucially, the submitted GWTC-5 tests manuscript defers dispersive propagation; its absence must not be reported as a current constraint or null result LIGO–Virgo–KAGRA 2026, submitted P2500781-v17, §§2 and 8.

This deferral does not erase newer-than-GWTC-3 propagation work. The submitted GWTC-4.0 parameterized-tests paper includes a modified-dispersion analysis and a polarization-dependent propagation analysis LIGO–Virgo–KAGRA 2026, arXiv:2603.19020v2, §§3.1–3.2. Its O4a selection, likelihood, priors, and result products must be used together; replacing its strain or event list with GWTC-5 products would be a new analysis, not reproduction of that result.

A reproducible rerun also records the software and sampler versions, random seeds, frequency cutoffs, posterior convergence criteria, and the machine-readable injection set; a plotted marginal bound without those data products is not independently checkable.

Results and limitations through 2026-08-19

Evidence or result	Physical position	Limitation or scope
Multimessenger arrival-time comparisons	Bound a propagation delay once an intrinsic emission-lag model is supplied	Source lag, association, clocks, and line-of-sight effects remain nuisance terms
Peer-reviewed GWTC-3 dispersion analysis	Found no established frequency-dependent propagation anomaly in its selected sample	Bound is tied to O3 calibration, waveform families, priors, and chosen dispersion powers
Submitted GWTC-4.0 propagation analyses	Extend dispersion and birefringence tests with an O4a selected sample	These are separate results with their own event cuts and modeling assumptions
GWTC-5 submitted test suite	Provides current generation and other tests but no updated dispersive result	Deferral is neither a new bound nor evidence that dispersion is absent

The conservative interpretation is that tested propagation coordinates remain compatible with GR in their stated domains. Competing modified-propagation interpretations remain possible but must predict the joint frequency, redshift, distance, and polarization scaling. An adversarial independent check injects speed, dispersion, and damping one at a time over a redshift distribution, recovers each expected scaling, and demonstrates that source phasing and calibration injections do not masquerade as propagation. Without that separation, the conclusion remains unresolved and model-dependent.

What could change the conclusion. The interpretation could change when an official propagation analysis using a superseding catalog appears, a counterpart changes the speed baseline, or a calibrated injection campaign materially changes degeneracies among emission, dispersion, damping, and detector response.

Exercises

For constant $\nu=0.1$, evaluate $d_L^\mathrm{GW}/d_L^\mathrm{EM}$ at $z=1$ in the friction-only model above, and identify two nuisance effects that could imitate the change.

Solution

The integral is $\nu\ln(1+z)=0.1\ln2$, so the ratio is $\exp(0.05\ln2)=2^{0.05}\simeq1.035$. Thus the gravitational-wave amplitude would be interpreted as a distance about $3.5\%$ larger than the electromagnetic distance. The comparison is not decisive because an amplitude-calibration shift or an inclination bias can produce a similar change; weak lensing and host-redshift errors add further covariance. Therefore a population likelihood and calibration marginalization are required.

References

• Enis Belgacem, Yves Dirian, Stefano Foffa, and Michele Maggiore, “Gravitational-Wave Luminosity Distance in Modified Gravity Theories,” Physical Review D 97, 104066 (2018), DOI.
• Gravitational Wave Open Science Center (2026), “GWTC-5.0 Data Release Documentation,” stable release 9 parameter-estimation, strain, and data-quality products, official release, DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “Data Release for GWTC-5.0: Tests of General Relativity,” version P2600410-v1 (2026), official release.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “GWTC-4.0: Tests of General Relativity. II. Parameterized Tests,” submitted manuscript, arXiv:2603.19020v2 (2026), Preprint.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “GWTC-5.0: Tests of General Relativity,” submitted manuscript P2500781-v17 (2026), official DCC record.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “Tests of General Relativity with GWTC-3,” Physical Review D 107, 022005 (2023), DOI.
• Nicolás Yunes, Xavier Siemens, and Kent Yagi, “Gravitational-Wave Tests of General Relativity with Ground-Based Detectors and Pulsar-Timing Arrays,” Living Reviews in Relativity 28, article 3 (2025), DOI.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Gravitational-Wave Speed, Dispersion, and Damping