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 ofsources. Interest in this has been rekindled by recent studies innon-linear relativistic perturbation theory that find biases in both thearea of a surface of constant redshift and in the mean distance to thissurface, with a fractional bias in both cases of the order of the meansquared convergence . Any such area biascould alter cosmic microwave background (CMB) cosmology, and thecorresponding bias in mean flux density could affect supernovacosmology. We show that the perturbation to the area of a surface ofconstant redshift is in reality much smaller, being of the order of thecumulative bending angle squared, or roughly a part-in-a-million effect. This validates the arguments of Weinberg that the mean magnification ofsources is unity and of Kibble & Lieu that the meandirection-averaged inverse magnification is unity. It also validates theconventional treatment of CMB lensing. But the existence of a scatter inmagnification will cause any non-linear function of these conservedquantities to be statistically biased. The fractional bias in suchquantities is generally of order , which isorders of magnitude larger than the area perturbation. Claims for largebias in area or flux density of sources appear to have resulted frommisinterpretation of such effects: they do not represent a newnon-Newtonian effect, nor do they invalidate standard cosmological analyses.