Multipole expansion of the local expansion rate

Abstract

We design a new observable, the expansion rate fluctuation $\ensuremath{\eta}$, 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 datasets, 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 ($307\ifmmode\pm\else\textpm\fi{}23\text{ }\text{ }\mathrm{km}/\mathrm{s}$) of the entire local volume in a direction aligned at better than 4 degrees with the bulk component of the Local Group (LG) velocity with respect to the cosmic microwave background (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 ($\ensuremath{\sim}50%$ 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 discussed. We find that Hubble constant estimates may show variation as high as $\mathrm{\ensuremath{\Delta}}{H}_{0}=(4.1\ifmmode\pm\else\textpm\fi{}1.1)\text{ }\text{ }\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ between antipodal directions along the dipole axis. In the case of the Pantheon sample, this systematic difference is reduced to $\mathrm{\ensuremath{\Delta}}{H}_{0}=(2.4\ifmmode\pm\else\textpm\fi{}1.1)\text{ }\text{ }\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ once model-dependent correction for peculiar velocity flows are implemented. Notwithstanding, the axial anisotropy in the general direction of the CMB dipole is still detected. We thus show how to optimally subtract redshift anisotropies from Pantheon data in a fully model-independent way by exploiting the $\ensuremath{\eta}$ observable. As a result, the value of the best fitting ${H}_{0}$ is systematically revised upwards by nearly $0.7\text{ }\text{ }\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ (about $2\ensuremath{\sigma}$) compared to the value deduced from the Hubble diagram using the uncorrected observed redshift. The goodness of fit is also improved.