Silver Dark Sirens in Cosmology

• Silver dark sirens are gravitational-wave events with intermediate localization that allow statistical cosmological inference by matching with galaxy catalogs.
• They rely on targeted spectroscopic surveys, like HETDEX/VIRUS, to achieve nearly complete redshift coverage and minimize systematic errors.
• These events offer competitive Hubble constant constraints and pave the way for percent-level precision in mapping the universe’s expansion.

Silver dark sirens are a class of gravitational-wave (GW) events, specifically compact binary coalescences, whose electromagnetic (EM) counterparts are not identified, but which possess sky localization and distance information precise enough to enable meaningful cosmological inference by statistical correlation with external galaxy catalogues. They fill an intermediate regime between “bright sirens” (unique EM identification) and “golden dark sirens” (volume so small only one plausible host), offering rates and localization volumes compatible with current and near-future cosmological applications, notably direct, independent measurements of the Hubble constant ($H_0$) and the expansion history.

1. Definitions and Classification of Silver Dark Sirens

The principal taxonomy for standard siren cosmology includes:

• Bright Siren: A GW event with a unique EM counterpart, yielding a direct redshift ($z$) and precise $H_0$ posterior (e.g., GW170817).
• Dark Siren: A GW detection with no EM counterpart; its host redshift is inferred statistically by cross-matching the three-dimensional GW localization posterior with galaxy catalogs.
• Golden Dark Siren: A dark-siren event with 90% credible sky area $\Delta\Omega_{90} \le 0.1\,\mathrm{deg}^2$, effectively enclosing only a single plausible galaxy host within the localization volume.
• Silver Dark Siren: (editor's term) A dark-siren event with $$0.1\,\mathrm{deg}^2 < \Delta\Omega_{90} \le 1\,\mathrm{deg}^2$$ containing multiple plausible hosts, but few enough ($\lesssim$ tens–hundreds) that targeted spectroscopic follow-up (e.g., with IFU surveys like HETDEX/VIRUS) can capture all candidates efficiently (Dang et al., 25 Dec 2025).

This boundary is observationally motivated. For $\Delta\Omega_{90} \le 1\,\mathrm{deg}^2$, integral-field spectrographs can rapidly acquire complete redshift coverage down to $g\lesssim22$, eliminating major systematics from incomplete host identification (Dang et al., 25 Dec 2025). Silver dark sirens are distinguished from “well-localized dark sirens” ($\Delta\Omega_{68} \lesssim 30\,\mathrm{deg}^2$) in earlier O3 analyses in that only the former enable practical, near-complete, spectroscopic host-galaxy surveying (Mukherjee et al., 2022).

2. Methodology for Cosmological Inference with Silver Dark Sirens

Cosmological inference with silver dark sirens relies on matching the GW localization posterior in sky position and luminosity distance $p(\Omega, d_L)$ with the redshift distribution of candidate host galaxies. Two principal frameworks are deployed:

A. Direct Host Association (Silver/Golden Regime):

For each GW event, the Hubble constant posterior is

$$p(H_0 | \{x^{(i)}\},\{z^{(i)}_j,M^{(i)}_j\}) \propto \pi(H_0) \prod_{i=1}^{N_{\mathrm{GW}}} \frac{1}{\beta(H_0)} \sum_{j=1}^{N^{(i)}} \mathcal{L}^{(i)}_\mathrm{GW}(x^{(i)} \mid \Omega^{(i)}_j, d_L(z^{(i)}_j, H_0))\,p_{\mathrm{Host}}(M^{(i)}_j|H_0)$$

where $\mathcal{L}_\mathrm{GW}$ is the GW event likelihood, $p_{\mathrm{Host}}$ is the host prior (uniform or host-luminosity weighted), and $\beta(H_0)$ corrects for selection effects (Dang et al., 25 Dec 2025).

B. Clustering Redshift/Cross-Correlation (Statistical Host Regime):

Here, both GW and galaxy samples are treated as tracers of large-scale structure. The cross-angular power spectrum $C_\ell^{\rm GW,g}$ between GWs and galaxies provides a tomographic measurement. The full likelihood marginalizes over redshift position, GW and galaxy bias parameters, with the GW event selection function embedded in the redshift prior (Mukherjee et al., 2022).

$$P(\Theta_c|d_\mathrm{GW},d_g) \propto \int dz\, d\Theta_n\, \prod L(\hat C|z,\Theta_n,\Theta_c)\, \Pi_\mathrm{GW}(d_\mathrm{GW}|z,\Theta_c)\,P_g(d_g|C_\ell^{gg}(z))$$

Silver dark sirens allow the direct-summation approach, boosting precision per event and minimizing dependence on population and selection-function modeling.

3. Catalog Completion and Galaxy Selection in the Silver Regime

A critical source of error and bias in silver dark-siren cosmology is galaxy catalog incompleteness. Three main schemes have been analyzed for interpolating the host distribution:

• Homogeneous Completion: Missing galaxies are distributed uniformly, damping the true clustering signal and underestimating structure variance. This leads to over-smoothing of the redshift prior $p(z)$, degrading $H_0$ constraints (Dalang et al., 2023).
• Multiplicative Completion: Observed galaxy counts are up-weighted multiplicatively. This inflates variance and can bias $H_0$ posteriors via over-confident, peaked $p(z)$ when completeness is low.
• Variance (Clustering-Based) Completion: The missing galaxy distribution is optimized to exactly preserve both the mean $\bar n_g$ and variance $\sigma_g^2$, thereby optimally reconstructing the true large-scale structure. This is achieved by minimizing a quadratic cost function (Lagrangian) over local voxels, subject to known clustering statistics. Variance completion outperforms the other schemes at all but extremely low completeness, producing the lowest bias and variance in the $H_0$ posterior (Dalang et al., 2023).

For silver dark sirens, where integral-field surveys like HETDEX/VIRUS deliver $>98\%$ completeness down to $M_g<-17.6$ at $z<0.2$, the clustering-based scheme ensures negligible additional bias in host redshift assignment (Dang et al., 25 Dec 2025).

4. Observational Strategies and Practical Implementation

The transition from O3-era analyses (well-localized but not “silver-grade” dark sirens) to true silver dark sirens is driven by advances in GW detector sensitivity and survey capabilities:

• Localization Volumes and Expected Rates: With $\Delta\Omega_{90}\lesssim1\,\mathrm{deg}^2$, typical localization volumes are $10^3$–$10^4\,\mathrm{Mpc^3}$ and contain $\mathcal{O}(10)$–$\mathcal{O}(100)$ galaxies (Dang et al., 25 Dec 2025). Simulation forecasts indicate $7-17$ silver dark sirens per year for A$^+$ (HLV/HLI+), and up to $\sim130$ at A$\sharp$ sensitivity with the HLV networks (Dang et al., 25 Dec 2025).
• Spectroscopic Follow-up: The VIRUS instrument on HETDEX covers $1\,\mathrm{deg}^2$ in $\sim$60 pointings at standard depth, providing $>98\%$ spectroscopic completeness with individual galaxy redshifts to $30$–$100\,\mathrm{km\,s}^{-1}$ accuracy (Dang et al., 25 Dec 2025). This depth is sufficient that the missing-galaxy correction is negligible for $H_0$ inference at $z<0.2$.
• Event Selection: Silver dark sirens are prioritized based on $\Delta\Omega_{90}$, SNR ($>$12), and volume intersection with spectroscopic survey coverage. For O3 events, “silver” quality corresponded to $\Delta\Omega_{68}\lesssim30\,\mathrm{deg}^2$ and SNR$\geq11$ (Mukherjee et al., 2022). With current and future GW networks, the number of eligible silver events is expected to increase by over an order of magnitude annually.

5. Impact on Hubble Constant Determination

Silver dark sirens offer several key advantages in standard siren cosmology:

• Precision Constraints: In the combined silver+golden sample ($\sim$25 events), $\sigma_{H_0}$ can reach $\sim$1–2%, with silver-only samples ($\sim$22 events at $z<0.2$) achieving $7–8\%$ uncertainty on $H_0$ (Dang et al., 25 Dec 2025). This matches the systematics-limited precision of other cosmological probes and is competitive with, but independent from, EM distance ladder results.
• Statistical Robustness: Silver dark sirens are $\gtrsim10\times$ more common than golden events, offering a substantial statistical sample for network-level cosmological analyses (Dang et al., 25 Dec 2025).
• Systematics Control: Host redshift uncertainties, catalog incompleteness, and cosmic-variance biases are minimized via complete IFU observing strategies and clustering-based catalog completion schemes. Spectroscopic redshifts improve $H_0$ precision by $\sim15\%$ relative to photometric catalogs (Dang et al., 25 Dec 2025, Dalang et al., 2023). For O3 dark sirens alone, the $H_0$ posterior was uninformative, but in combination with GW170817, the constraint $H_0=75.4^{+11}_{-6}\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ (68.3% ETI) was obtained, driven almost entirely by the bright siren (Mukherjee et al., 2022).

6. Outlook and Future Directions

Precision cosmology with silver dark sirens is poised for rapid progress. The scaling of the cross-correlation methodology and direct host-marginalization approaches is assured as GW event rates and localizations improve, and as IFU spectroscopic coverage expands on both hemispheres. Forecasts anticipate that $\mathcal{O}(100)$ silver sirens per year, cross-matched against deep and nearly complete spectroscopic catalogs, will deliver percent‐level $H_0$ measurements and constrain the expansion rate $H(z)$ at $z>0.5$, with negligible dependence on host mass/luminosity priors or population models (Mukherjee et al., 2022, Dang et al., 25 Dec 2025). The methodology is also extensible to tests of GW propagation physics, lensing-induced perturbations, and potential deviations from standard $\Lambda$CDM cosmology.

In summary, silver dark sirens—GW events with $\Delta\Omega_{90}\lesssim1\,\mathrm{deg}^2$ and no EM counterpart—are emerging as the workhorse standard sirens for precision cosmology. Their high event rate, practical follow-up requirements, and minimized systematics make them central to future efforts resolving the Hubble tension and mapping the late-time expansion of the Universe (Dang et al., 25 Dec 2025, Mukherjee et al., 2022, Dalang et al., 2023).