A unified harmonic framework for dark siren cosmology

Abstract

The galaxy catalog dark siren method aims to infer cosmological parameters from gravitational waves (GWs) without an electromagnetic counterpart by statistically marginalizing over possible host galaxies. The cross-correlation of GW sources and galaxies is a promising avenue for cosmological inference without requiring observed host galaxies, by leveraging 2-point statistics. We provide a detailed guide to the cross-correlation method, clarifying its relationship to standard dark siren techniques as well as the assumptions necessary to be able to use this formalism on GW data. We show that the cross-correlation method is an extension of the angular part of the galaxy catalog method in which we effectively marginalize over all possible realizations of the unknown galaxy field, jointly adding information from galaxy--galaxy clustering. Combined with the spectral sirens method, which encodes information from the GW rate evolution, mass distribution, and selection effects, one can perform an inference that leverages the joint constraining power of all dark siren methods. We also present a strategy to rigorously fold GW measurement errors into the likelihood. Using this method, we show that with a 2 Einstein Telescope + 1 Cosmic Explorer setup, the GW--galaxy cross-correlation part alone can jointly measure $H_0$ and $Ω_{m,0}$ to 1\% and 5\% precision with just 2 years of data, demonstrating its potential as a precise and scalable inference technique in the next generation of GW and galaxy surveys. This is in contrast with canonical population inference techniques, which are known to scale poorly with the precision and catalog size expected of next-generation GW experiments. Contrary to some previous projections, we remain pessimistic about the cross-correlation method until these next generation detectors are online, due to its implicit requirement of large-number statistics.

Figures (9)

• Figure 1: A flowchart summarizing our cross-correlation formalism. Pre-processing steps for the data (\ref{['sec:noise']}) are shown in the left box, which are used to compute the theoretical cross-correlation and covariance matrix during inference (right box). Boxes are colored roughly as follows: data (blue), estimated noise data products used during inference (teal), theory-level calculations (purple), likelihoods (orange), and the spectral sirens method (red). The cross-correlation and spectral sirens methods are complementary, so they can be done separately and/or combined.
• Figure 2: A slice of the mock galaxy catalog, with measured redshifts $1.17 < \hat{z} < 1.22$. The galaxy sky mask, which we have taken from the WISE x SuperCosmos analysis, is clearly visible.
• Figure 3: 90% localization sky area and relative luminosity distance error $\Delta(\ln \hat{d}_L) = \Delta d_L / \hat{d}_L$ of the BBH in the mock catalog. Colors correspond to the GW radial bins in the top panel of \ref{['fig:window']}, i.e., the observed source distance $\hat{d}_L$.
• Figure 4: Galaxy and BBH effective window functions for our equally populated bins evaluated in the fiducial cosmology, normalized to the number count per bin. Dotted lines show bins that were removed due to being too close to the edge of the simulation volume at $\chi_\text{max}=6$ Gpc.
• Figure 5: Beam window functions for the 1st, 6th, and last of our 11 GW bins. The solid blue line shows the average of $B_\ell$ over all events, while the dotted black line shows $B_\ell$ computed from the median angular RMS localization error of each bin.