Likelihood of the sky. Bayesian treatment of the geometric two-catalog matching problem

Abstract

Catalogs with celestial positions and uncertainties are among the primary data products of observational astronomy. Consequently, the (cross-)matching problem, namely, the question of how many (and which) pairs of positions pertain to the same celestial source, remains a central part of this scientific exploitation. We address the matching problem for two catalogs from a purely geometric point of view by adopting the concept of point processes and nearest-neighbor distributions. Suitable expressions for the nearest-neighbor distributions were derived and used to model the distribution of spatial offsets in the sky. Thus, we are able to estimate a total number of matching pairs, along with individual matching probabilities between sources in any two catalogs of positions and uncertainties. We demonstrate the workings of the model using mock data and apply it to cross-matching the X-ray sources of the eROSITA Final Equatorial-Depth survey with optical Gaia counterparts.