Bayesian Inference of the Tip of the Red Giant Branch: I. One-Dimensional Analysis

Abstract

The tip of the red giant branch (TRGB) is a relative distance measure that can be used, as part of a cosmic distance ladder, to infer the Hubble constant. We present a Bayesian framework for TRGB inference that incorporates selection effects and propagates per-star flux uncertainties, modelling the observed catalogue as a realisation of an inhomogeneous Poisson point process which includes the RGB and contamination from the asymptotic giant branch (AGB). We derive expressions for the predicted TRGB uncertainty in terms of the photometric noise, stellar density fainter than the tip, and the ratio of the AGB to AGB+RGB number densities, $r$. We validate these results using synthetic data from a simplified luminosity function. Applying the full model to synthetic catalogues, we find that the dependence on $r$ is the strongest of the three, and that reliable inference requires $r\lesssim0.5$; otherwise, the posterior can be multimodal, and the TRGB cannot reliably be identified. We then fit data from 11 Hubble Space Telescope fields of the calibrator galaxy NGC 4258. We find intrinsic field-to-field scatter of $\tau=0.067^{+0.023}_{-0.017}$ mag (beyond the measurement noise), and a galaxy-level, extinction-corrected TRGB magnitude of $25.324 \pm 0.026$ in the F814W filter. Combining this with the maser distance modulus to NGC 4258 of $29.397 \pm 0.032$ mag, we determine the absolute magnitude of the TRGB to be $M_{\mathrm{F814W}}^{\mathrm{TRGB}} = -4.073 \pm 0.026$ (tip) $\pm 0.032$ (dist), consistent with, but brighter than, other recent analyses of the same data.