Standard Sirens and Propagation Friction

A compact-binary waveform supplies an amplitude distance without a cosmic distance ladder, while a counterpart or galaxy population supplies redshift information. Comparing that gravitational-wave distance with an electromagnetic distance–redshift relation can test anomalous propagation friction. The comparison is only as clean as its amplitude calibration, distance–inclination covariance, host association, redshift completeness, peculiar velocities, weak lensing, source population, detection selection, waveform model, and background cosmology.

Required background. Probability and Statistical Inference supplies hierarchical host marginalization, selection normalization, and the prior measures needed for bright and dark sirens.

Helpful background. Weak-Field Gravity supplies tensor propagation; Relativistic Cosmology supplies luminosity distance and redshift; and Tests of Gravity supplies the separation of propagation, source, and detector effects.

Gravitational and electromagnetic distances

For one tensor polarization on an FLRW background, write

$$h_A''+[2+\nu(\tau)]\mathcal H h_A' +k^2h_A=0,$$

where primes denote conformal time $\tau$, $a_0=1$, $k$ is comoving wavenumber, $\mathcal H=a'/a$, and $\nu=0$ in GR for the propagation effect considered here. Assume luminal, massless tensor propagation, a slowly evolving background relative to the wave period, and unchanged source emission and detector coupling. To obtain the amplitude law, define $q'/q=(1+\nu/2)\mathcal H$ and set $v_A=q h_A$. The wave equation becomes

$$v_A''+\left(k^2-\frac{q''}{q}\right)v_A=0.$$

When $k^2\gg\lvert q''/q\rvert$, $v_A$ has approximately constant oscillation amplitude. Hence $h_A\propto q^{-1}$, with $q\propto a\exp[\tfrac12\int\nu\,d\ln a]$. Comparing propagation from emission to observation with the GR $1/a$ damping yields

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

Positive $\nu$ gives extra damping and therefore a larger inferred distance for the same emitted waveform. The sign and factor depend on the friction definition: $\nu=-2\delta$ in the notation of Belgacem et al. 2018, Eqs. (12) and (20). The GR null is equality of the two distances after using the same redshift and cosmology. This propagation-only expression is not a complete prediction if the source, its local gravitational coupling, or the detector response also changes.

For tensor friction generated by a homogeneous effective Planck mass, $\nu=\alpha_M=d\ln M_*^2/d\ln a$, the same integral gives $d_L^{\rm GW}/d_L^{\rm EM}=M_*(0)/M_*(z)$. Endpoint normalization and screening must be included when translating this relation to a particular theory; one cannot infer a cosmological Planck-mass history from an arbitrary distance bias.

The waveform amplitude scales roughly as

$$\tilde h(f)\propto \frac{\mathcal M_z^{5/6}f^{-7/6}}{d_L^{\rm GW}} \mathcal A(\iota,\psi,\hat n),$$

where $\mathcal M_z=(1+z)\mathcal M$ is the redshifted chirp mass, $f$ the observed frequency, and $\mathcal A$ the angular detector response. Distance is covariant with inclination $\iota$, polarization, sky position, higher modes, and precession. Detector amplitude calibration multiplies the same observable. A bright siren uses an electromagnetic counterpart and host redshift, as in the GW170817 analysis LIGO Scientific Collaboration and Virgo Collaboration 2017, main result and Methods; a dark siren marginalizes over possible galaxies; a spectral siren infers redshift statistically from population features. These are different likelihoods, not interchangeable labels.

Host and population likelihood

For a dark siren with galaxy data $d_{\rm gal}$, let $p_{\rm host}(z,\hat n\mid d_{\rm gal},\Lambda)$ be a normalized host-redshift and sky density. It includes candidate galaxies with declared weights and a component for hosts missing from the catalog. For an analysis conditioned on the event count, a schematic event factor is

$$\begin{aligned} \mathcal L_i(\lambda,\Lambda) &\propto\frac{1}{\beta(\lambda,\Lambda)} \int dz\,d\hat n\,d\theta\, p(d_i^{\rm GW}\mid d_L^{\rm GW}(z;\lambda),\hat n,\theta)\\ &\qquad\times p(\theta\mid z,\Lambda) p_{\rm host}(z,\hat n\mid d_{\rm gal},\Lambda). \end{aligned}$$

where $\theta$ includes source-frame masses and orientation, $\lambda$ contains propagation and cosmology parameters, $\Lambda$ describes the source and galaxy population, and $\beta$ is the matching detection-selection normalization. The waveform uses detector-frame masses $(1+z)m$ inside the likelihood. If host weights or volume factors depend on cosmology, that dependence belongs in $p_{\rm host}$ as well. The denominator is essential: selection acts on detected data rather than merely truncating a source prior Mandel, Farr, and Gair 2019, §§2–4. For a bright siren the host mixture concentrates on the identified galaxy, but its redshift uncertainty, peculiar velocity, counterpart selection, and possible misidentification remain. A published distance posterior must first be divided by its interim prior before it is used as a likelihood in this integral.

The population distribution of masses and redshifts matters because detector-frame masses are redshifted and detection probability depends on amplitude, orientation, and waveform. Weak lensing broadens and skews high-redshift distances. Galaxy catalog masks, magnitude limits, photometric-redshift calibration, and stellar contamination enter completeness. A shared Hubble calibration or galaxy sample generates covariance across events.

Versioned inputs and controls

At the evidence cutoff 2026-08-19, the official gravitational-wave input baseline is GWTC-5/GWOSC stable parameter-estimation release 9, DOI 10.7935/bk00-6a89, with its calibration and file documentation GWOSC 2026, official GWTC-5.0 data release. The full catalog contains 390 detections, but any siren result applies only to the event, detector, host, and quality selection explicitly modeled. Frequencies below 10 Hz are uncalibrated, and the source-matched 16 kHz products must not be confused with the decimated 4 kHz products.

The electromagnetic background should be pinned to named public likelihoods, such as Planck Collaboration 2020, final-release papers V–VI and DESI DR2 BAO cosmology chains DESI Collaboration 2025–2026, official DR2 cosmology products. Euclid Consortium 2025, Q1 release overview identifies Q1 as a calibrated quick, non-cosmology release, not a mature standard-siren host or gravity likelihood. A current inference must state the galaxy catalog version and retrieval date rather than assuming a generic complete sky.

Reproduction records strain and posterior identifiers, calibration version, waveform and sampler, sky maps, galaxy/host catalog and mask, redshift likelihoods, completeness model, peculiar-velocity field, lensing prescription, population hyperpriors, cosmological priors, selection injections, software commit, and convergence. A normalized likelihood and its covariance are essential; stacking distance ratios after selecting well-localized events biases the population.

Results and limitations through 2026-08-19

Evidence or result	Position supported	Limitation or counterevidence
Bright sirens with secure hosts	Directly join a waveform distance to a measured redshift	Inclination, amplitude calibration, peculiar velocity, lensing, and counterpart selection remain
Dark or statistical sirens	Use many binary events without requiring unique counterparts	Galaxy completeness, host weights, population evolution, sky localization, and cosmology dominate
GWTC-5 release 9 plus electromagnetic distances	Supplies a current reproducible input basis for a friction hierarchy	No generic propagation-friction verdict follows from the catalog or background chains alone

The conservative position is that standard sirens provide a principled propagation test, but present conclusions remain tied to sparse hosts, broad distance posteriors, catalog completeness, and population assumptions. A competing apparent friction signal can arise from amplitude calibration, inclination selection, host/redshift error, weak lensing, population evolution, or the electromagnetic cosmology. The issue remains unresolved and model-dependent outside the analyzed hierarchy.

An adversarial independent check injects GR catalogs with $d_L^{\rm GW}=d_L^{\rm EM}$ through the actual detector and galaxy selection, then injects a redshift-dependent friction function. Recovery is repeated under alternate inclination, host, lensing, peculiar-velocity, population, and calibration models. A genuine propagation effect should follow its redshift scaling and survive those alternatives; a catalog bias generally will not.

What could change the conclusion. The interpretation could change when a superseding stable posterior, amplitude calibration, bright-siren host, galaxy catalog, selection-injection set, or peer-reviewed friction hierarchy changes the distance comparison or its coverage.

Exercises

For constant $\nu=0.2$, compute $d_L^{\rm GW}/d_L^{\rm EM}$ at $z=1$. Why could a face-on selection bias imitate the same sign of effect?

Solution

The integral is $\nu\ln(1+z)=0.2\ln2$, so the exponent is $0.1\ln2$ and the ratio is $2^{0.1}\simeq1.072$. Therefore the inferred gravitational-wave distance is about $7.2\%$ larger. Face-on binaries are louder and preferentially detected, while distance and inclination are covariant in amplitude. If the population or selection model overweights orientations that make sources appear farther away, it can produce the same sign of distance shift without modified propagation.

References

• DESI Collaboration, “DESI DR2 Cosmology Results and Supporting Data Products,” official publication list and cosmology-chain release (2025–2026), DESI Data.
• Enis Belgacem, Yves Dirian, Stefano Foffa, and Michele Maggiore, “Modified Gravitational-Wave Propagation and Standard Sirens,” Physical Review D 98, 023510 (2018), DOI.
• Euclid Consortium, “Euclid Data Release: Q1,” official release overview (2025), Release page.
• Ilya Mandel, Will M. Farr, and Jonathan R. Gair, “Extracting Distribution Parameters from Multiple Uncertain Observations with Selection Biases,” Monthly Notices of the Royal Astronomical Society 486, 1086–1093 (2019), DOI.
• LIGO Scientific Collaboration and Virgo Collaboration, “A Gravitational-Wave Standard Siren Measurement of the Hubble Constant,” Nature 551, 85–88 (2017), DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “GWTC-5.0: Gravitational-Wave Transient Catalog,” official catalog and stable parameter-estimation release 9 (2026), GWOSC.
• Planck Collaboration, “Planck 2018 Results. V and VI,” Astronomy & Astrophysics 641, A5 and A6 (2020), official publication list.