On the Bias of the Distance-Redshift Relation from Gravitational Lensing

A long standing question in cosmology is whether gravitational lensing changes the distance-redshift relation $D(z)$ or the mean flux density of sources. Interest in this has been rekindled by recent studies in non-linear relativistic perturbation theory that find biases in both the area of a surface of constant redshift and in the mean distance to this surface, with a fractional bias in both cases on the order of the mean squared convergence $\langle \kappa^2 \rangle$. Any such area bias could alter CMB cosmology, and the corresponding bias in mean flux density could affect supernova cosmology. Here we show that, in an ensemble averaged sense, the perturbation to the area of a surface of constant redshift is in reality much smaller, being on the order of the cumulative bending angle squared, or roughly a part-in-a-million effect. This validates the arguments of Weinberg (1976) that the mean magnification $\mu$ of sources is unity and of Kibble \& Lieu (2005) that the mean direction-averaged inverse magnification is unity. It also validates the conventional treatment of lensing in analysis of CMB anisotropies. But the existence of a scatter in magnification will cause any non-linear function of these conserved quantities to be statistically biased. The distance $D$, for example, is proportional to $\mu^{-1/2}$ so lensing will bias $\langle D\rangle$ even if $\langle \mu \rangle=1$. The fractional bias in such quantities is generally of order $\langle \kappa^2 \rangle$, which is orders of magnitude larger than the area perturbation. Claims for large bias in area or flux density of sources appear to have resulted from misinterpretation of such effects: they do not represent a new non-Newtonian effect, nor do they invalidate standard cosmological analyses.