Not all those missing are lost: leveraging galaxy clustering in incomplete catalogs to unleash dark sirens cosmology

Abstract: Gravitational waves offer a unique opportunity to solve the Hubble tension. In order to do so, we have to extract as much information as possible from cross correlating gravitational wave events (used as "dark sirens") with incomplete galaxy catalogs. Traditional methods assume a uniform in comoving volume distribution for galaxies missing from the catalogs, which neglects the fact that galaxies tend to cluster, leading to less precise and possibly biased posteriors. In this paper, we introduce a new method for accounting for galaxy clustering when dealing with an incomplete galaxy catalog, by adding back galaxies to the incomplete catalog and distributing them in pixels according to the correlation function $ξ(r)$. We find that our method drastically improves on traditional ones when the galaxy catalog is only $1\%-10\%$ complete, leading to posteriors that are between 2 and 4 times as precise, depending on the completeness fraction, without introducing any biases. Our method produces results comparable to traditional methods at extremely low catalog completeness fractions (<0.5\%) or for very high uncertainties in the recovered luminosity distance and sky localization of the gravitational wave events.