Dark Siren Method in GW Cosmology

• Dark siren method is a statistical technique that infers source redshifts for gravitational-wave events lacking electromagnetic counterparts using galaxy catalog data.
• It applies Bayesian marginalization over potential host galaxies to derive key cosmological parameters, including the Hubble constant (H0), even with broad and multimodal posteriors.
• The method leverages both spectroscopic and photometric surveys, with improved precision from enhanced localization and clustering techniques to refine future cosmic measurements.

The dark siren method is a statistical standard-siren technique in gravitational-wave cosmology for sources without an identified electromagnetic counterpart. In this setting, the gravitational-wave signal provides an absolute luminosity distance from the waveform amplitude, but the source redshift is not known from a uniquely identified host galaxy. The missing redshift information is therefore supplied statistically by a galaxy catalog covering the gravitational-wave localization volume, and cosmological parameters are inferred by marginalizing over all plausible host galaxies. The method was introduced observationally for binary-black-hole events such as GW170814 and has since been applied with photometric and spectroscopic galaxy surveys, hierarchical Bayesian population analyses, and large-scale-structure formalisms (Collaboration et al., 2019, Ballard et al., 2023).

1. Definition and physical basis

A standard siren is a compact-binary gravitational-wave source whose signal directly yields a luminosity distance. A bright siren has an electromagnetic counterpart or an unambiguous host galaxy, so the redshift is measured directly. A dark siren has no such counterpart, so the host galaxy is unknown and the redshift must be inferred statistically from galaxies inside the three-dimensional localization region. Binary black hole mergers are the canonical dark-siren sources because they are not expected to have detectable electromagnetic counterparts in the standard picture (Borhanian et al., 2020).

The basic observable pair is therefore asymmetric. The gravitational-wave data constrain $d_L$, while the galaxy catalog constrains a discrete or continuous prior over possible $z$. At low redshift some analyses use the approximation

$D_L=\frac{cz}{H_0},$

whereas other analyses use the full flat $\Lambda$CDM luminosity-distance relation with fixed $\Omega_m$. The operational distinction from bright-siren cosmology is not in the distance measurement but in the treatment of redshift uncertainty: object-by-object host identification is replaced by host-galaxy marginalization.

The method also admits an intermediate regime. Forecast work has argued that upgraded detectors plus higher-order spherical harmonic modes can make some dark sirens so well localized that a single host galaxy can be identified within $z=0.1$, using a sky-area criterion $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$. In that regime the event behaves effectively like a bright siren, even though no electromagnetic counterpart is observed. The same study reports that higher modes improve sky localization by about a factor of $\sim 2$ and distance estimation by as much as a factor of $\sim 6$ for the populations considered (Borhanian et al., 2020).

2. Statistical formulation

The core dark-siren posterior is written in Bayesian form as

$$p(H_0\mid d_{\rm GW},d_{\rm EM}) \propto p(d_{\rm GW},d_{\rm EM}\mid H_0)\,p(H_0),$$

with the gravitational-wave and catalog data treated as independent. The essential step is then a marginalization over all potential host galaxies. In the galaxy-catalog formulation, each galaxy $z$0 contributes a term weighted by its host prior $z$1, and the gravitational-wave likelihood is evaluated at the luminosity distance implied by that galaxy’s redshift and the trial value of $z$2 (Collaboration et al., 2019).

A representative flat-$z$3CDM relation used in early catalog-based analyses is

$z$4

with $z$5 fixed and $z$6 inferred. In the first GW170814 analysis, the host-galaxy weights were taken mostly to be uniform because the host-galaxy properties were unknown, and a selection factor $z$7 was included to account for gravitational-wave and galaxy-survey selection effects (Collaboration et al., 2019).

Later implementations made the same structure more explicit in the presence of photometric-redshift uncertainty and photo-$z$8 bias. One form used for catalog-based inference is

$z$9

in a flat $D_L=\frac{cz}{H_0},$0CDM background with $D_L=\frac{cz}{H_0},$1 and a flat prior $D_L=\frac{cz}{H_0},$2. For multiple independent events, the combined posterior is the product of the single-event likelihoods, so broad and multimodal single-event posteriors can sharpen rapidly when many events are accumulated (Bom et al., 2024).

This likelihood structure makes clear what the method assumes. The event occurred in one of the catalog galaxies, the gravitational-wave distance posterior must be compatible with the redshift–distance relation of that galaxy, and the final $D_L=\frac{cz}{H_0},$3 posterior is a weighted mixture over all such host hypotheses. A plausible implication is that most methodological disputes in dark-siren cosmology reduce to disagreements about the host prior, the treatment of missing galaxies, and the fidelity of the galaxy redshift model.

3. Galaxy catalogs, redshift information, and survey construction

The redshift side of the method has been implemented with both photometric and spectroscopic catalogs. The first observational application used the Dark Energy Survey Year 3 catalog as a list of possible hosts for GW170814. In the analysis region, that study quotes $D_L=\frac{cz}{H_0},$4 galaxies inside the 90% localization volume, $D_L=\frac{cz}{H_0},$5 galaxies when using 99.7% of the distance posterior, and about $D_L=\frac{cz}{H_0},$6 galaxies with spectroscopic redshifts. The catalog was made volume-limited by an absolute-luminosity cut so that it remained complete over the redshift interval relevant for the gravitational-wave prior (Collaboration et al., 2019).

The first dark-siren analysis with DESI used the Bright Galaxy Sample from DESI Iron internal data release plus daily reductions for GW190412. After cuts to the 90% localization region, redshift consistency, and reliable DESI redshifts, the fiducial Iron sample contained 6039 galaxies, while a larger daily-reductions sample contained 8442 galaxies. That work emphasized that DESI spectroscopic redshifts are about two orders of magnitude more precise than photometric redshifts, and it applied FKP, WEIGHT_SYS, and WEIGHT_COMP large-scale-structure weights to account for survey nonuniformities (Ballard et al., 2023).

Photometric implementations have focused on full redshift probability density functions rather than point estimates. DELVE DR2 was used for two new O3 dark sirens, with photometric redshifts estimated by a Mixture Density Network using griz magnitudes and colors and producing a mixture-of-Gaussians photo-$D_L=\frac{cz}{H_0},$7 PDF. The analysis explicitly marginalized over a photo-$D_L=\frac{cz}{H_0},$8 bias $D_L=\frac{cz}{H_0},$9. A related O4a analysis used Legacy Survey catalogues with a deep-learning Mixture Density Network built on a Legendre Memory Unit, with

$\Lambda$0

to retain asymmetric or multi-modal redshift uncertainty for each galaxy (Alfradique et al., 2023, Bom et al., 2024).

These survey choices matter because the dark-siren likelihood is sensitive not only to the central redshift of each candidate host but also to the detailed structure of the line-of-sight redshift distribution. Spectroscopic catalogs reduce redshift uncertainty directly; photometric catalogs trade precision for sky coverage and depth; full-PDF photo-$\Lambda$1 treatments preserve non-Gaussian features that would be erased by Gaussian approximations.

4. Observational measurements of $\Lambda$2

Representative measurements illustrate both the viability and the current breadth of the method. The first binary-black-hole dark siren, GW170814 combined with DES Y3 galaxies, yielded

$\Lambda$3

for a flat prior $\Lambda$4. With a broader prior $\Lambda$5, the same analysis found

$\Lambda$6

showing explicit prior dependence for a single event (Collaboration et al., 2019).

GW190412, analyzed as a dark siren with DESI galaxies, produced

$\Lambda$7

with a 68% credible interval and a posterior shape described as consistent with redshift overdensities. That event was localized to 12 deg$\Lambda$8 at 90% credible level, and the posterior exhibited multiple peaks corresponding to redshift overdensities in the DESI galaxy distribution along the line of sight (Ballard et al., 2023).

A DELVE-based analysis of ten well-localized dark sirens from the first three LIGO/Virgo observing runs reported

$\Lambda$9

while the combination of GW190924_021846, GW200202_154313, and the bright siren GW170817 gave

$\Omega_m$0

The paper states that adding the two new dark sirens reduced the 68% confidence interval by 7% relative to GW170817 alone (Alfradique et al., 2023).

A later catalogue-only analysis combining five new O4a events, three updated O3 events, and the previous ten-event sample reported

$\Omega_m$1

from 15 dark sirens, with the paper describing this as an improvement of $\Omega_m$2 over the previous ten-dark-siren constraint. When combined with GW170817 and recent jet constraints, that study quoted

$\Omega_m$3

corresponding to a $\Omega_m$4 precision from standard sirens (Bom et al., 2024).

Measurement	Galaxy data	Reported $\Omega_m$5
GW170814	DES Y3	$\Omega_m$6
GW190412	DESI	$\Omega_m$7
10 dark sirens	DELVE	$\Omega_m$8
15 dark sirens	Legacy Survey	$\Omega_m$9

These results show a recurrent pattern. Single-event posteriors are broad and often multimodal; better-localized events produce narrower structure; combinations with bright sirens tighten constraints substantially; and the overall precision improves primarily by accumulating events with sufficiently informative galaxy coverage.

5. Systematics, weighting, clustering, and completeness

Redshift precision is a leading systematic. A 2025 study found that spectroscopic-like redshifts always improve the $z=0.1$0 constraint relative to photometric-like redshifts, but that the gain depends strongly on localization area and gravitational-wave distance precision. When the localization area is reduced from $z=0.1$1 to $z=0.1$2 deg$z=0.1$3, the mean uncertainty in $z=0.1$4 falls by 43% for spectroscopic-like redshifts and 36% for photometric-like redshifts. Comparing spectroscopic-like with photometric-like redshifts directly, the improvement is 4% at $z=0.1$5 deg$z=0.1$6 and 15% at $z=0.1$7 deg$z=0.1$8. The same study found no significant bias in $z=0.1$9 from redshift outliers in a redMaGiC-like photometric sample, but showed that at 50% completeness the degradation from incompleteness can outweigh the precision advantage of spectroscopy (Cross-Parkin et al., 25 Feb 2025).

Host-galaxy weighting is another nontrivial prior choice. Equal weighting, stellar-mass weighting, and star-formation-rate weighting have all been studied. Mock-catalog analyses found that incorrect weighting schemes can bias $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$0 through two mechanisms: assuming an incorrect galaxy redshift distribution and preferentially weighting incorrect host galaxies during inference. The severity of the bias depends on the number of galaxies in the localization volume, the gravitational-wave distance uncertainty, and correlations among galaxy properties. A separate 2025 study concluded that unbiased estimates are obtained when the corrected weighting scheme is applied to a complete or volume-limited catalog, and reported that 100 binary-black-hole detections with $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$1 can reach approximately 3% precision with stellar-mass weighting and approximately 6% with equal weighting (Hanselman et al., 2024, Alfradique et al., 24 Mar 2025).

Catalog incompleteness has motivated a distinct methodological literature. Simulation studies have shown that clustering in the galaxy catalog can materially improve dark-siren inference. One study found clear recovery of the input $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$2 as early as $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$3 events for clustered catalogs, while uniform catalogs remained largely unconstrained even with $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$4 events; the same analysis reported that completeness levels of $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$5 perform statistically similar to complete catalogs with as few as $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$6 events (Kalomenopoulos et al., 15 Nov 2025). Another mock-catalog study found that for well-localized events, if hosts are found in galaxies with $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$7, catalogs need only be complete down to the 1% brightest $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$8-band galaxies, $$\Omega_{90}\lesssim 4.4\times 10^{-2}\,\mathrm{deg}^2$$9, to yield an unbiased and informative posterior, and traced this to the clustering of faint galaxies around brighter and more massive galaxies (VanWyngarden et al., 6 Nov 2025).

Completeness estimation itself can change the posterior. A robust test implemented in gwcosmo used the Rauzy statistic $\sim 2$0 with threshold $\sim 2$1 to estimate the apparent-magnitude completeness limit of a magnitude-redshift sample. Applied to GWTC-1 with GLADE and GLADE+, the method produced a 3.4% improvement for dark sirens only with GLADE and an 8.6% improvement with GLADE+; for GWTC-3 with GLADE+ $\sim 2$2-band, the paper reported no improvement because the catalog provided little or no $\sim 2$3-band coverage for those events (Datrier et al., 20 Feb 2025).

A common misconception is that incompleteness can be repaired adequately by a uniform-in-comoving-volume background. Multiple recent works reject that as a physically faithful description because galaxies trace large-scale structure, not a homogeneous random field. This suggests that completion schemes informed by clustering are not optional refinements but structural components of the host prior.

6. Generalizations, alternative formalisms, and future directions

Several recent methods replace homogeneous completion with explicit large-scale-structure modeling. Variance completion splits the line-of-sight prior into catalogued and missing contributions and rescales the missing term through

$\sim 2$4

Implemented with GLADE+ and gwcosmo, this framework produced results consistent with homogeneous completion for O3, while a GW190814 proof of concept gave a slightly tighter posterior under variance completion. The same paper argued that the payoff should grow as localization volumes shrink (Dalang et al., 2024).

A more ambitious development is Bayesian reconstruction of the true galaxy field from incomplete catalogs. The “Cosmic Cartography” framework models the latent dark-matter overdensity as a log-normal field, infers the galaxy magnitude distribution jointly with the spatial galaxy field, and reconstructs a three-dimensional host prior in $\sim 2$5 rather than appending a homogeneous background. The follow-up “Cosmic Cartography II” further infers the detection probability and the true magnitude distribution jointly with the spatial field, with validation on Millennium-based simulated data (Leyde et al., 2024, Leyde et al., 16 Jul 2025).

An alternative formalism treats dark-siren cosmology as a clustering problem rather than a host-association problem. In the cross-correlation method, galaxy catalogs binned in redshift are cross-correlated with gravitational-wave events binned in luminosity distance. One 2026 study emphasized that catalog incompleteness can be incorporated directly into the theoretical prediction through the radial window function, without explicitly modeling the missing population, and found that with appropriate modeling choices and a sufficiently large sample of precise events the resulting systematic biases can be mitigated (Cross-Parkin et al., 7 May 2026). A unified harmonic framework then showed that the cross-correlation method is the angular extension of the galaxy-catalog method, effectively marginalizing over all realizations of the unknown galaxy field, and forecast that with a 2 Einstein Telescope + 1 Cosmic Explorer setup the GW–galaxy cross-correlation part alone can jointly measure $\sim 2$6 and $\sim 2$7 to 1% and 5% precision with 2 years of data, while remaining pessimistic about the method before next-generation detectors because of its implicit large-number requirement (Cheng et al., 13 Mar 2026).

Detector forecasts remain correspondingly optimistic. One study argued that upgraded 2G+ detectors could deliver better than a few percent precision, with a headline forecast of about 2% precision in about five years using uniquely hosted “golden” dark sirens (Borhanian et al., 2020). A third-generation forecast using ET and CE found that a network made of ET and two CEs can constrain $\sim 2$8 to 0.8% and $\sim 2$9 to 10.0% at 90% confidence interval within one year, assuming a catalog complete up to $\sim 6$0 and events with network signal-to-noise ratio greater than 300 (Muttoni et al., 2023). At the same time, a Fisher-information study warned that to achieve a total error budget of 1% in $\sim 6$1, the galaxy mass-function redshift evolution should be known to $\sim 6$2, underscoring that astrophysical model uncertainty can become dominant in the third-generation regime (Wang et al., 2024).

The method has also expanded beyond $\sim 6$3. A GWTC-4 dark-siren analysis of higher-dimensional gravitational-wave propagation combined 141 compact binary coalescences with GLADE+ galaxies and obtained

$\sim 6$4

for a baseline prior, with the result described as fully consistent with four-dimensional General Relativity (Chen et al., 12 Jun 2026). This indicates that the dark siren method is not restricted to local-expansion measurements: it is a general framework for combining gravitational-wave distance information with statistical redshift information from large-scale structure.