The multipole expansion of the local expansion rate

We design a new observable, the expansion rate fluctuation $\eta$, to characterize deviations from the linear relation between redshift and distance in the local universe. We also show how to compress the resulting signal into spherical harmonic coefficients in order to better decipher the structure and symmetries of the anisotropies in the local expansion rate. We apply this analysis scheme to several public catalogs of redshift-independent distances, the Cosmicflows-3 and Pantheon data sets, covering the redshift range $0.01<z<0.05$. The leading anisotropic signal is stored in the dipole. Within the standard cosmological model, it is interpreted as a bulk motion ($307 \pm 23$ km/s) of the entire local volume in a direction aligned at better than $4$ degrees with the bulk component of the Local Group velocity with respect to the CMB. This term alone, however, provides an overly simplistic and inaccurate description of the angular anisotropies of the expansion rate. We find that the quadrupole contribution is non-negligible ($\sim 50\%$ of the anisotropic signal), in fact, statistically significant, and signaling a substantial shearing of gravity in the volume covered by the data. In addition, the 3D structure of the quadrupole is axisymmetric, with the expansion axis aligned along the axis of the dipole. Implications for the determination of the $H_0$ parameter are also discussed.