A Cosmological Underdensity Does Not Solve the Hubble Tension

A potential solution to the Hubble tension is the hypothesis that the Milky Way is located near the center of a matter underdensity. We model this scenario through the Lema\^itre-Tolman-Bondi formalism with the inclusion of a cosmological constant ($\Lambda$LTB) and consider a generalized Gaussian parametrization for the matter density profile. We constrain the underdensity and the background cosmology with a combination of data sets: the Pantheon Sample of type Ia supernovae (both the full catalogue and a redshift-binned version of it), a collection of baryon acoustic oscillations data points and the distance priors extracted from the latest Planck data release. The analysis with the binned supernovae suggests a preference for a $-13 \%$ density drop with a size of approximately 300 Mpc, interestingly matching the prediction for the so-called KBC void already identified on the basis of independent analyses using galaxy distributions. The constraints obtained with the full Pantheon Sample are instead compatible with a homogeneous cosmology and we interpret this radically different result as a cautionary tale about the potential bias introduced by employing a binned supernova data set. We quantify the level of improvement on the Hubble tension by analyzing the constraints on the B-band absolute magnitude of the supernovae, which provides the calibration for the local measurements of $H_0$. Since no significant difference is observed with respect to an analogous fit performed with a standard $\Lambda$CDM cosmology, we conclude that the potential presence of a local underdensity does not resolve the tension and does not significantly degrade current supernova constraints on $H_0$.