Large scale structure prior knowledge in the dark siren method

Gravitational wave dark sirens are a powerful tool for cosmology and inference of compact object population hyperparameters. They allow for a measurement of the luminosity distance to the source, but not their redshift. Galaxy catalogues in the source localization volume can be used to infer the redshift of the source in a statistical manner. Catalogues are, however, limited by their incompleteness, which can be significant at redshifts corresponding to current GW events. In this work, we detail how to implement in practice variance completion, a novel galaxy completion method which uses knowledge of the large scale structure to optimize the potential of dark sirens analyses. We compress the prediction for the missing number of galaxies into a ratio between the predictions of variance completion and the standard homogeneous completion method. This ratio format can be easily incorporated into existing line of sight computations used in dark sirens software; we demonstrate this procedure using the GLADE+ galaxy catalogue and the gwcosmo software package. We discuss the robustness of the method, and apply it to well-localized event GW190814 as a proof of concept. Finally, we apply the method to data from the third observing run of LIGO-Virgo-KAGRA, finding that it yields results that are consistent with homogeneous completion. We also discuss the prospects for an improvement if the GW localization volume shrinks.

💡 Research Summary

This paper addresses a central challenge in the use of gravitational‑wave (GW) “dark sirens” for cosmology: the incompleteness of galaxy catalogs that are required to statistically associate a GW event with a host redshift. Traditional approaches, most notably homogeneous completion, assume that galaxies missing from a catalog are uniformly distributed in space. While simple, this assumption neglects the well‑known clustering of galaxies driven by large‑scale structure (LSS) and can bias the inferred Hubble constant (H₀).

The authors introduce and implement a “variance completion” technique that explicitly incorporates LSS information. The method proceeds as follows. First, the line‑of‑sight redshift prior p(z, n̂) – the probability density that a GW originated at redshift z in direction n̂ – is expressed in terms of the weighted galaxy number density. The galaxy field is discretized into three‑dimensional voxels defined by redshift bins and angular pixels (HEALPix with n_side = 32). Each galaxy contributes to one or several voxels depending on its redshift uncertainty, which is modeled as a Gaussian PDF.

To capture spatial variations in catalog completeness, the authors construct a magnitude‑threshold map m_th(n̂) based on the median apparent magnitude of galaxies in each sky pixel. Pixels are sorted by this threshold and grouped into twelve classes, each containing 1 024 voxels. Classes with large internal variations in completeness (the most and least complete) are excluded from variance completion and treated with homogeneous completion; the remaining seven classes are used for the LSS‑aware correction.

For each class the completeness fraction f̂_S is estimated using a Schechter luminosity function. The expected number of missing galaxies n_m is then predicted in two ways: (i) homogeneous completion, which simply scales the observed number by 1/f̂_S, and (ii) variance completion, which adds the contribution of LSS clustering by using the measured mean and variance of galaxy counts within the class. The key output is a ratio

R_vc(z, n̂) = n_m^vc / n_m^hom,

which quantifies how much the LSS‑based estimate deviates from the homogeneous assumption. This ratio can be multiplied into the existing line‑of‑sight prior in the publicly available gwcosmo pipeline, requiring only a small additional data file.

The authors validate the approach on the well‑localized binary‑neutron‑star/black‑hole merger GW190814. Both homogeneous and variance completions yield posterior distributions for H₀ that peak near 70 km s⁻¹ Mpc⁻¹, with variance completion modestly tightening the credible interval (≈ 3 % reduction). They then apply the method to all dark‑sirens from the third LIGO‑Virgo‑KAGRA observing run (O3). The combined H₀ posterior remains consistent with the homogeneous result (H₀ ≈ 68.5 km s⁻¹ Mpc⁻¹, uncertainties ≈ ± 10 km s⁻¹ Mpc⁻¹), indicating that, with current catalog depths and GW localization volumes, the LSS‑based correction does not produce a statistically significant shift.

The paper discusses limitations. Variance completion relies only on the first two moments (mean and variance) of the galaxy distribution, neglecting higher‑order clustering statistics that could become important in highly non‑linear regions. The Gaussian treatment of photometric redshift errors is a simplification that may not capture asymmetric or multi‑modal uncertainties. Moreover, the method’s benefit is strongest in regions where catalog completeness exceeds ~70 %; in low‑completeness zones the correction can add noise.

Looking forward, the authors argue that as GW detectors improve and localization volumes shrink, the number of voxels intersecting the true host galaxy will increase, making the precise modeling of LSS more valuable. In that regime, variance completion (or even more sophisticated Bayesian hierarchical models of the galaxy field) could substantially reduce systematic uncertainties in H₀ and other cosmological parameters derived from dark sirens.

In summary, the work provides a practical, easily integrable prescription for incorporating large‑scale‑structure knowledge into dark‑siren analyses. While current data show no dramatic change relative to homogeneous completion, the framework is ready for the next generation of GW observations where it may deliver a measurable improvement in cosmological inference.