Bright and Dark Standard Sirens

A compact-binary waveform provides an absolute luminosity-distance scale through its amplitude and phase evolution, but it does not generally provide an independent redshift. A standard siren combines the gravitational-wave distance likelihood with redshift information: one identified host for a bright siren, or a probability distribution over possible hosts for a dark siren. The cosmological likelihood must include distance–inclination uncertainty, source and host populations, catalog incompleteness, and the selection that placed an event in the sample.

Helpful background. Gravitational-Wave Astronomy supplies detector likelihoods, distance–inclination posteriors, and detection selection. Relativistic Cosmology supplies luminosity-distance and redshift relations. Black-Hole Astrophysics supplies source populations and host-rate hypotheses.

Distance from the waveform

For a nonprecessing circular binary, the leading polarization amplitudes scale as

$$\widetilde h_+\propto \frac{1+\cos^2\iota}{2D_L}, \qquad \widetilde h_\times\propto \frac{\cos\iota}{D_L},$$

where $\iota$ is the inclination of the orbital angular momentum to the line of sight. A detector network measures combinations of these polarizations. Near face-on or face-off orientation, changing inclination can mimic changing $D_L$, producing the familiar distance–inclination degeneracy. Calibration amplitude, polarization coverage, precession, and higher modes can also affect the distance likelihood.

Masses inferred from phasing are detector-frame masses,

$$m_{\rm det}=(1+z)m_{\rm src}.$$

Without an external redshift or a source-population scale, mass and redshift remain entangled. Standard-siren cosmology therefore adds information rather than extracting redshift from amplitude alone.

For a spatially flat background with cosmological parameters $\Lambda$,

$$D_L(z;\Lambda) =(1+z)c\int_0^z\frac{dz'}{H(z';\Lambda)}.$$

Perturbations, velocities, and lensing modify this ideal relation; their canonical treatment belongs to Perturbed Distance–Redshift Relations.

The detected-event likelihood

Let $\xi$ contain source redshift, direction, inclination, masses, and other parameters. For an event selected by a declared detection rule, a population-consistent likelihood has the form

$$\mathcal L_i(\Lambda) =\frac{ \displaystyle\int p(d_i\mid\xi,\Lambda)\, \pi(\xi\mid\Lambda)\,d\xi }{ \alpha(\Lambda) },$$

where

$$\alpha(\Lambda) =\int p_{\rm det}(\xi,\Lambda)\pi(\xi\mid\Lambda)\,d\xi$$

is the selection efficiency. Here $\pi$ is a normalized source-population density, and the sample count is conditioned on. At fixed source-frame masses and redshift, changing cosmology changes distance and hence detection probability; the explicit $\Lambda$ argument prevents that dependence from being lost. If the rate and observed count are modeled rather than conditioned away, use the associated Poisson count likelihood instead of adding a second selection correction.

To see the denominator, integrate the population predictive density $p(d\mid\Lambda)=\int p(d\mid\xi,\Lambda)\pi(\xi\mid\Lambda)d\xi$ over all data that pass the detection rule. That integral is $\alpha(\Lambda)$. Dividing by it makes $p(d\mid\Lambda,\mathrm{det})$ integrate to one on the selected data space. This is inference about cosmology and a population from a selected sample. It does not instruct an analyst to multiply an already valid single-source parameter posterior by $1/p_{\rm det}(\xi)$.

The same generative model must define numerator and denominator. Simulating sources with one population and evaluating $\alpha$ with another can manufacture cosmological bias even if the event likelihood is exact.

Bright sirens

Suppose a confidently associated host has direction $\Omega_h$ and redshift data $d_z$. A schematic bright-siren likelihood is

$$\begin{aligned} \mathcal L_i^{\rm bright}(\Lambda) \propto\frac{1}{\alpha_{\rm bright}(\Lambda)} \int dz\,d\iota\;& \mathcal L_{\rm GW} \bigl[D_L(z;\Lambda),\Omega_h,\iota\bigr]\\ &\times p(d_z\mid z) \pi(z,\iota\mid\Lambda). \end{aligned}$$

The host association is itself evidence and should be included rather than assumed when uncertain. The redshift likelihood must state reference frame, measurement uncertainty, and peculiar-velocity model. The selection denominator corresponds to the actual sample definition: gravitational-wave detection, counterpart detection and identification, and any follow-up cuts.

GW170817 with its identified host NGC 4993 supplied the first gravitational-wave standard-siren measurement of the Hubble constant. Its published likelihood explicitly combined gravitational-wave distance and inclination with host recessional and peculiar-velocity information LIGO/Virgo and partners 2017, Methods. The numerical posterior belongs to that event, calibration, prior, and velocity model; it is not a universal value reproduced here.

Dark sirens

The statistical-host construction was proposed before the first direct detections Schutz 1986, pp. 310–311. Without a unique host, let a catalog contain galaxies $g$ with sky positions $\Omega_g$, normalized redshift densities $p_g(z)$ conditional on their survey data, and normalized host weights $w_g$. Each $p_g$ must use the adopted redshift prior: a photometric-redshift posterior cannot silently be treated as a likelihood under a different prior. Suppressing other source parameters, the in-catalog contribution is

$$\mathcal L_i^{\rm cat}(\Lambda) \propto \sum_g w_g \int dz\,d\iota\, p_g(z)\, \mathcal L_{\rm GW} \bigl[D_L(z;\Lambda),\Omega_g,\iota\bigr] \pi(\iota).$$

A complete likelihood also includes hosts missing from the catalog:

$$\mathcal L_i^{\rm dark} =\frac{ P_{\rm cat}\,\mathcal L_i^{\rm cat} +(1-P_{\rm cat})\mathcal L_i^{\rm out} }{ \alpha_{\rm dark}(\Lambda) },$$

where $P_{\rm cat}$ is the scalar probability that a host belongs to the usable catalog under the specified population model. If survey completeness is a function $C(x)$ of position, redshift, and galaxy properties, then $P_{\rm cat}=\int C(x)p_0(x)\,dx$. The two branch likelihoods use separately normalized host densities. Putting $C(x)$ outside the integrals as though it were a scalar would give a different, generally unnormalized model. The host-systematics page derives these branch densities. The out-of-catalog term is drawn from the same merger and galaxy population that defines selection.

The sum must not be restricted to galaxies inside a plotted contour without accounting for the omitted probability. Renormalizing the retained part alone conditions on an additional host-location assumption; it does not restore missing support. Galaxy weights encode a physical host-rate hypothesis—perhaps uniform per galaxy or proportional to a measured property—and are not observational facts about which galaxy merged. The consistent galaxy-catalog framework, including selection and mock-data failure modes, is developed in Gair et al. 2023, §§II–IV, PDF.

Combining events and checking normalization

For independent detected events under fixed hyperparameters,

$$p(\Lambda\mid\{d_i\},{\rm det}) \propto \pi(\Lambda)\prod_{i=1}^{N}\mathcal L_i(\Lambda).$$

The events share source-population, host-weight, and selection uncertainties. Treating those as independently redrawn for every event can understate uncertainty. Posterior samples produced under an interim distance prior may be reused only with explicit prior division and support checks.

A minimum synthetic validation should:

1. draw sources and galaxies from a declared cosmology and host model;
2. apply gravitational-wave and counterpart or catalog selection to simulated data, not true parameters unless the likelihood matches that choice;
3. recover the injected cosmology with a complete catalog;
4. add catalog incompleteness, redshift error, peculiar velocities, lensing, calibration, and distance–inclination uncertainty one at a time;
5. refine event count, injection count, galaxy integration, sky resolution, and posterior-sample effective size.

An omitted selection denominator should visibly produce a controlled bias in an appropriate mock. A correct method need not recover the injected value in every finite realization. Nominal Bayesian coverage is expected for an ensemble drawn from the full analysis prior and generative model; fixed-cosmology simulations instead measure frequentist coverage, which need not be nominal. Both tests are useful, but they answer different questions.

What the results establish

As of 2026-08-18, the robust conclusion is methodological: bright sirens use an association-conditioned host redshift, while dark sirens marginalize over possible hosts and missing-catalog support. Both are valid when their generative and selection models match the data. GW170817 is the bounded bright-siren example above; Gair et al. 2023, PDF demonstrates unbiased dark-siren recovery under internally consistent mocks and shows how inconsistent selection or data generation can create spurious bias.

Every empirical result must pin event and catalog versions, strain and calibration, sky map, waveform, PSD, priors, host catalog and mask, redshift and velocity products, completeness, selection injections, software, and random seeds. Refresh after a superseding official event/catalog product, corrected selection set, material host-catalog update, or independent mock counterexample. No cosmological posterior was recomputed for this page.

Common pitfalls

Calling luminosity distance a redshift. The waveform measures distance conditional on source orientation, calibration, and gravity. Cosmology or a population model connects that distance to redshift.

Dropping the out-of-catalog term. Missing galaxies do not have zero host probability. Omitting them can force the cosmology toward catalog overdensities.

Correcting selection after fitting. Selection belongs inside the normalized likelihood. An after-the-fact scalar correction generally cannot repair a parameter-dependent denominator.

Exercises

1. A low-redshift estimate. Approximate $D_L=v_H/H_0$. If $D_L=40\pm8\,\mathrm{Mpc}$ and the peculiar-velocity-corrected Hubble-flow velocity is $v_H=3000\pm150\,\mathrm{km\,s^{-1}}$, estimate $H_0$ and its uncertainty by linear propagation.

Solution

The central value is

$$H_0=\frac{v_H}{D_L} =75\,\mathrm{km\,s^{-1}\,Mpc^{-1}}.$$

Assuming independent Gaussian errors,

$$\left(\frac{\sigma_{H_0}}{H_0}\right)^2 \simeq \left(\frac{150}{3000}\right)^2 +\left(\frac{8}{40}\right)^2 =0.0425.$$

Thus $\sigma_{H_0}/H_0\simeq0.206$ and $\sigma_{H_0}\simeq15\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$. A real posterior is asymmetric because distance and inclination are correlated and because priors and selection matter.

2. Include missing hosts. A catalog has two normalized host weights $(0.25,0.75)$ and gravitational-wave likelihood values $(0.8,0.2)$ in common arbitrary density units. The host catalog-membership probability is $P_{\rm cat}=0.8$, and the likelihood averaged over the normalized out-of-catalog host density is $0.4$ in the same units. Evaluate the numerator of the dark-siren mixture.

Solution

The catalog term is

$$\mathcal L^{\rm cat}=0.25(0.8)+0.75(0.2)=0.35.$$

Therefore

$$P_{\rm cat}\mathcal L^{\rm cat}+(1-P_{\rm cat})\mathcal L^{\rm out} =0.8(0.35)+0.2(0.4)=0.36.$$

This is only the event numerator. A cosmological likelihood still divides by the matching selection efficiency and evaluates all quantities as functions of $\Lambda$.

References

• Gair, Jonathan R., et al. “The Hitchhiker’s Guide to the Galaxy Catalog Approach for Dark Siren Gravitational-Wave Cosmology.” Astronomical Journal 166 (2023): 22. DOI. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and partners. “A Gravitational-Wave Standard Siren Measurement of the Hubble Constant.” Nature 551 (2017): 85–88. DOI. Open PDF.
• Schutz, Bernard F. “Determining the Hubble Constant from Gravitational Wave Observations.” Nature 323 (1986): 310–311. DOI.