paired: A Statistical Framework for Detecting Stellar Binarity with Gaia RVs. I. Sensitivity to Unresolved Binaries

Data Release 3 (DR3) from the Gaia Mission includes radial velocity measurements of over 33 million targets. Among many scientific applications, the overlap of this stellar sample with targeted exoplanet transit survey stars presents an opportunity to understand planet occurrence in the context of stellar multiplicity on a large scale. Yet, any interpretation of occurrence relies upon an understanding of survey sensitivity. While the sensitivity to planets in transit surveys is well understood, a characterization of the sensitivity of Gaia to unresolved binaries is also critical. We describe here a statistical framework called paired, which we developed to enable the forward modeling of Gaia radial velocity observables for large samples of stars. The paired machinery links the reported radial velocity noise for a given star from Gaia to the probability of a spatially unresolved stellar companion. We demonstrate how this enables the user, given an observed distribution of individual binary ``probabilities" for a set of stars, to understand this distribution within the sensitivity limits of Gaia. For the subset of stars with the highest probability of excess radial velocity noise, we describe the ability of paired to constrain the semi-amplitude of the stellar binary. Where possible, we benchmark our inferred radial velocity semi-amplitudes against those from ground-based radial velocity surveys, and the subset published by Gaia DR3 itself. We aim for paired to be a community tool for the exploration of the effects of binarity on planets at a population level, but also for any user interested in stellar populations.