Dark-Siren Method in GW Cosmology

• Dark-siren method is a statistical technique in gravitational-wave cosmology that infers redshift from galaxy catalogues for events without electromagnetic counterparts.
• It utilizes Bayesian hierarchical inference to combine luminosity distance data from gravitational waves with host-galaxy priors, enabling meaningful H0 measurements.
• Recent implementations with spectroscopic catalogues and advanced weighting schemes demonstrate improved precision and reduced uncertainty in cosmological parameter estimation.

The dark-siren method is a statistical standard-siren technique in gravitational-wave cosmology for sources without an identified electromagnetic counterpart. A gravitational-wave signal provides an absolute luminosity distance, but not a direct redshift; the missing redshift is inferred statistically by combining the gravitational-wave localization volume with a galaxy catalogue and marginalizing over all plausible host galaxies. In current usage, the method is primarily applied to binary black hole mergers and other compact-binary coalescences without secure host identification, with the Hubble constant $H_0$ as the canonical target parameter, although the same framework has also been extended to joint population inference and tests of modified gravitational-wave propagation (Collaboration et al., 2019, Gair et al., 2022).

1. Definition, scope, and historical emergence

A standard siren is the gravitational-wave analogue of a standard candle: the waveform directly yields the luminosity distance $d_L$ without external calibration. A bright siren has an identified electromagnetic counterpart or secure host galaxy, so distance and redshift are known for the same source. A dark siren has no electromagnetic counterpart, so the host is unknown and the redshift must be inferred statistically from galaxies in the localization volume (Ballard et al., 2023).

This distinction became operationally important once binary-black-hole mergers were recognized as viable cosmological probes despite the absence of bright electromagnetic emission. The first measurement of the Hubble constant from a black-hole dark siren used GW170814 and the DES Year 3 galaxy catalogue, producing $$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$ with a photometric-redshift catalogue (Collaboration et al., 2019). That result established the feasibility of host-galaxy marginalization for a binary-black-hole event and made explicit that a one-event dark-siren posterior is broad and prior-sensitive, but still cosmologically meaningful (Collaboration et al., 2019).

Subsequent analyses moved from proof-of-principle to increasingly structured implementations. The GW190412 analysis with DESI used a well-localized asymmetric binary black hole merger inside the DESI footprint and emphasized that spectroscopic redshifts produce narrower posterior peaks than earlier photometric dark-siren analyses (Ballard et al., 2023). O1–O4a catalogue-based analyses then combined multiple well-covered events, showing that the method improves through posterior multiplication across events rather than through any single detection alone (Bom et al., 2024, Alfradique et al., 2023).

2. Bayesian and hierarchical structure

The dark-siren method is ordinarily formulated as Bayesian hierarchical inference in which cosmological parameters enter through the luminosity-distance–redshift relation and the host redshift is marginalized over a catalogue-based prior. In its standard event-level form, the single-event likelihood can be written as

$$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$

where the numerator marginalizes the gravitational-wave likelihood over source redshift and the denominator applies the selection correction (Gair et al., 2022).

When the galaxy redshifts are treated as exact, the same structure reduces to an explicit sum over candidate hosts,

$$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$

which is the operational galaxy-catalog form of the method (Gair et al., 2022). A compact weighted expression used in spectroscopic dark-siren work is

$$p(H_0 \mid \mathrm{GW}, \mathrm{gal}) \propto p(H_0)\sum_i w_i\, p(\mathrm{GW}\mid d_{L,i}, \Omega_i)\,p(z_i\mid H_0),$$

with $w_i$ carrying catalogue and selection information (Ballard et al., 2023).

Several analyses adopt a flat $\Lambda$CDM cosmology with fixed $\Omega_m = 0.3$ and a uniform prior on $H_0$ over $d_L$0, quoting 68% credible intervals (Ballard et al., 2023). In this setup, the gravitational-wave data contribute the distance posterior and sky localization, the galaxy catalogue contributes a redshift prior over possible hosts, and the final $d_L$1 posterior is obtained by marginalizing over thousands of galaxies rather than selecting a single host (Ballard et al., 2023).

The same hierarchical logic can be extended beyond a single cosmological parameter. Joint cosmological and compact-binary population inference allows $d_L$2, mass-distribution hyperparameters, and merger-rate evolution parameters to be varied simultaneously, thereby avoiding fixed-population assumptions in the selection function and redshift prior (Gray et al., 2023). This became a central methodological step once it was recognized that selection effects couple directly to the compact-binary population model (Gray et al., 2023).

3. Galaxy catalogues, redshift information, and catalogue completeness

The method is only as informative as its host-galaxy prior. In practice this has led to several catalogue regimes: photometric wide-field catalogues, spectroscopic catalogues, and hybrid strategies. Representative implementations are summarized below.

Catalogue	Role in dark-siren inference	Example
DES Year 3	Photometric host-galaxy prior	GW170814 (Collaboration et al., 2019)
DESI BGS	Spectroscopic host-galaxy prior	GW190412 (Ballard et al., 2023)
Legacy Survey / DELVE	Photometric-redshift PDFs from deep learning	O1–O4a multi-event analyses (Bom et al., 2024)
GLADE+	K-band catalogue with luminosity weights	GWTC-4 dark-siren analyses (Pierra et al., 6 Jan 2026)

The redshift-precision problem has been studied directly. A controlled analysis comparing spectroscopic-like redshifts $d_L$3 with photometric-like redshifts $d_L$4 found that spectroscopic-like redshifts always improve the $d_L$5 constraint, but the improvement is much larger when the GW localization area is small. The reported reduction in mean $d_L$6 uncertainty is about 15% for 1 deg$d_L$7 and only about 4% for 50 deg$d_L$8. The same study found that photometric redshift outliers do not bias the measurement of $d_L$9, while uniform sub-sampling of spectroscopic catalogues increases the uncertainty as completeness decreases; at 50% completeness, the gain from spectroscopic precision is outweighed by the degradation from incompleteness (Cross-Parkin et al., 25 Feb 2025).

This catalogue trade-off is evident in event-based analyses. The DESI GW190412 analysis used spectroscopic redshifts from the DESI Bright Galaxy Survey and reported that DESI redshifts are about two orders of magnitude more accurate than photometric redshifts, which makes the $$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$0 peaks narrower and the inference sharper. For GW190412, the analysis considered 6039 galaxies from the Iron release after cuts and 8442 galaxies in the daily reductions sample, both inside the gravitational-wave localization and compatible with the distance likelihood under the adopted $$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$1 prior (Ballard et al., 2023).

Photometric approaches have correspondingly emphasized full redshift PDFs rather than point estimates. Catalogue-only O4a and previous-run analyses used a Mixture Density Network with a Legendre Memory Unit architecture to obtain full photometric-redshift PDFs from Legacy Survey data, and DELVE-based O3 analyses used Mixture Density Networks with mixtures of 20 Gaussians for host-galaxy redshift PDFs (Bom et al., 2024, Alfradique et al., 2023). These studies treated photo-$$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$2 bias, PDF calibration, and footprint coverage as part of the cosmological inference rather than as auxiliary preprocessing (Bom et al., 2024, Alfradique et al., 2023).

A recurrent conclusion is that a competitive dark-siren programme will require a hybrid photometric/spectroscopic catalogue until highly complete spectroscopic catalogues become available (Cross-Parkin et al., 25 Feb 2025). This suggests that catalogue depth, completeness, and redshift precision are coupled design parameters rather than separable optimizations.

4. Selection effects, host weighting, and systematic structure

Selection effects enter the dark-siren likelihood through the denominator, and incorrect treatment can induce spurious bias. A technical review of the galaxy-catalog method showed that the method is unbiased when the data-generating process and the analysis model are consistent, and attributed reported failures primarily to inconsistent mock generation or incorrect handling of selection effects, latent variables, and likelihood normalization (Gair et al., 2022).

That review identified several specific failure modes: double counting galaxies, using a Heaviside step in true luminosity distance when the GW likelihood has finite width, ignoring the simulation redshift cutoff, mismodeling the distance-likelihood normalization, and using unrealistically sparse mock catalogues so that multiple GW events may share the same host (Gair et al., 2022). Its central claim was that no clustering does not imply bias; rather, it makes the inference less informative (Gair et al., 2022).

Host-galaxy weighting is a separate source of systematics. A study based on mock galaxy catalogues examined equal weights, stellar-mass weights, and star-formation-rate weights, and found that incorrect weighting schemes can lead to significant biases due to two effects: the assumption of an incorrect galaxy redshift distribution, and preferentially weighting incorrect host galaxies during the inference (Hanselman et al., 2024). The severity of the bias depends on the number of galaxies along the line of sight, the GW luminosity-distance uncertainty, and correlations in the parameter space of galaxies (Hanselman et al., 2024).

A complementary simulation study of O4/O5-like LVK dark sirens reached a related conclusion. It found that an unbiased estimate of $$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$3 can be obtained when the corrected weighting scheme is applied to a complete or volume-limited catalog, and that stellar-mass weighting yields tighter constraints than equal weighting. The same study showed that incomplete, magnitude-limited catalogues bias $$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$4 low, while wrong host-weighting prescriptions can produce large event-level shifts even when the combined posterior appears nearly unbiased because well-localized events dominate the stack (Alfradique et al., 24 Mar 2025).

Event-level analyses implement weighting and completeness corrections in catalogue-specific ways. The DESI GW190412 analysis used large-scale-structure weights to account for survey selection effects and incomplete targeting, including FKP weights, WEIGHT_SYS, and WEIGHT_COMP, while also using the daily reductions sample as a cross-check (Ballard et al., 2023). In practice, the dominant uncertainty in one-event dark-siren measurements still comes from the intrinsic GW distance uncertainty and marginalization over many possible hosts, but residual catalogue incompleteness and weighting uncertainty remain explicit components of the error budget (Ballard et al., 2023).

5. Extensions, variants, and large-scale-structure formulations

The dark-siren method has diversified into several technically distinct but related lines of development. One line combines galaxy catalogues with population inference. An enhanced version of the Python package GWCOSMO was introduced specifically to allow joint estimation of cosmological and compact-binary population parameters, motivated by the fact that informative priors on the mass distribution and merger-rate evolution directly affect the inferred $$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$5 posterior (Gray et al., 2023). In the same direction, a GWTC-4 analysis with 142 CBCs from GWTC-4 and the K-band GLADE+ galaxy catalogue performed joint Bayesian inference with the icarogw hierarchical Bayesian pipeline and reported that a new mass feature at $$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$6 is strongly anti-correlated with $$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$7 in the dark-siren posterior, tightening the cosmological constraint (Pierra et al., 6 Jan 2026).

A second line addresses catalogue incompleteness by reconstructing the missing galaxy field. Homogeneous completion assumes that missing galaxies are uniformly distributed in comoving volume; multiplicative completion assumes that missing galaxies trace the observed galaxies; both are simple but systematically limited at low completeness (Dalang et al., 2023, Dalang et al., 2024). Variance completion uses large-scale-structure information to redistribute missing galaxies where overdensities and voids are expected, and can be incorporated into existing line-of-sight computations through a ratio

$$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$8

so that $$H_0 = 75.2^{+39.5}_{-32.4}~{\rm km~s^{-1}~Mpc^{-1}}$$9 (Dalang et al., 2024).

A broader Bayesian reconstruction programme models the observed catalogue as a noisy, magnitude-limited realization of an underlying full galaxy field. This approach infers voxelized galaxy counts, cosmological parameters, power-spectrum parameters, galaxy-bias parameters, and absolute-magnitude-distribution parameters jointly, and is explicitly motivated by the inadequacy of the usual “missing galaxies are uniformly distributed” prescription (Leyde et al., 2024, Leyde et al., 16 Jul 2025). This suggests a shift from catalogue completion by ansatz to field-level host-prior inference.

A third line replaces host-galaxy marginalization by 2-point statistics. Cross-correlation methods infer cosmology from the angular cross-power spectrum of GW events and galaxies, treating them as tracers of the same large-scale structure. One study emphasized that selection effects can be incorporated directly into the theoretical prediction, without the need to model the missing population explicitly, which it identified as a key advantage over the standard galaxy-catalog approach (Cross-Parkin et al., 7 May 2026). A later unified harmonic treatment argued that the cross-correlation formalism is an extension of the angular part of the galaxy-catalog method in which one effectively marginalizes over all realizations of the unknown galaxy field, and forecast $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$0 precision on $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$1 from the GW–galaxy cross-correlation part alone with a 2 Einstein Telescope + 1 Cosmic Explorer setup and 2 years of data (Cheng et al., 13 Mar 2026).

The framework has also been generalized beyond $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$2. A GWTC-4 dark-siren analysis of higher-dimensional GW propagation used 141 compact binary coalescences and GLADE+ to constrain a modified-propagation model with hyperparameters $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$3, obtaining $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$4 for a narrow $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$5 prior and finding results consistent with four-dimensional General Relativity (Chen et al., 12 Jun 2026).

6. Empirical measurements and future regime

Measured dark-siren constraints on $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$6 remain broad for single events but are steadily tightening in stacked analyses. The first black-hole dark-siren measurement from GW170814 and DES obtained $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$7 (Collaboration et al., 2019). A spectroscopic analysis of GW190412 with DESI obtained $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$8 km/s/Mpc, with a multi-peaked posterior consistent with redshift overdensities, and reported $$\mathcal{L}(x_i|H_0) = \frac{\int dz~ \mathcal{L}_{\rm GW}(x_i|d_L(z,H_0))~p_{\rm CBC}(z)} {\int dz~ P^{\rm GW}_{\rm det}(z,H_0)~p_{\rm CBC}(z)} ,$$9 km/s/Mpc when combined with the bright-siren measurement from GW170817 in the abstract; the results section gives the fiducial combined value $$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$0 (Ballard et al., 2023).

Multi-event photometric analyses have now produced substantially tighter catalogue-only results. A DELVE-based study of 10 well-localized dark sirens from the first three observing runs reported $$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$1, while combining two new DELVE dark sirens with the bright siren GW170817 gave $$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$2 (Alfradique et al., 2023). An O4a and previous-runs catalogue-method analysis combining 15 dark sirens reported $$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$3, and the combination of those 15 dark sirens with GW170817 plus recent jet constraints yielded $$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$4, described as a $$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$5 precision from standard sirens (Bom et al., 2024). In a GWTC-4 hierarchical analysis with 142 CBCs and a new population model, the dark-siren result was $$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$6 (Pierra et al., 6 Jan 2026).

Future performance is expected to depend as much on methodology as on event counts. Forecast work argued that higher-order spherical-harmonic modes can shrink sky areas by roughly 20%–45% and improve distance errors by factors of about 3–6, enabling few-percent $$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$7 measurements from dark sirens in the upgraded-detector era (Borhanian et al., 2020). At the same time, a Fisher study for third-generation detectors found that, to achieve a total error budget of 1% in the Hubble constant, the galaxy mass-function redshift evolution should be known to $$\mathcal{L}(x_i|H_0) = \frac{\sum_i^{N_{\rm gal}} \mathcal{L}_{\rm GW}(x_i|d_L(\hat z_g^i,H_0))} {\sum_i^{N_{\rm gal}} P^{\rm GW}_{\rm det}(\hat z_g^i,H_0)} ,$$8, with galaxy redshift uncertainty, survey magnitude limit, and GW angular localization error as important factors (Wang et al., 2024).

The emerging picture is therefore technically specific. Dark sirens are not a single estimator but a family of hierarchical inference schemes whose performance is controlled by localization area, distance precision, redshift precision, catalogue completeness, host-weighting assumptions, and the degree to which large-scale structure is modeled rather than averaged away. This suggests that the method’s long-term precision will be set not only by detector sensitivity and event rate, but also by how completely the host-galaxy prior is embedded in the inference architecture.