From Cosmicflows distance moduli to unbiased distances and peculiar velocities

Surveys of galaxy distances and radial peculiar velocities can be used to reconstruct the large scale structure. Other than systematic errors in the zero-point calibration of the galaxy distances the main source of uncertainties of such data are errors on the distance moduli, assumed here to be Gaussian and thus turn into lognormal errors on distances and velocities. Naively treated, it leads to spurious nearby outflow and strong infall at larger distances. The lognormal bias is corrected here and tested against mock data extracted from a $Lambda$CDM simulation, designed to statistically follow the grouped Cosmicflows-3 (CF3) data. Considering a subsample of data points, all of which have the same true distances or same redshifts, the lognormal bias arises because the means of the distributions of observed distances and velocities are skewed off the means of the true distances and velocities. Yet, the medians are invariant under the lognormal transformation. That invariance allows the Gaussianization of the distances and velocities and the removal of the lognormal bias. This Bias Gaussianization correction (BGc) algorithm is tested against mock CF3 catalogs. The test consists of a comparison of the BGC estimated with the simulated distances and velocities and of an examination of the Wiener filter reconstruction from the BGc data. Indeed, the BGc eliminates the lognormal bias. The estimation of Hubbles ($H_{0}$) constant is also tested. The residual of the BGc estimated $H_{0}$ from the simulated values is $0.6 pm 0.7 {rm kms}^{-1}{rm Mpc}^{-1}$ and is dominated by the cosmic variance. The BGc correction of the actual CF3 data yields $H_{0} = 75.8 pm 1.1 {rm kms}^{-1}{rm Mpc}^{-1}$ .