On the Bias of the Distance-Redshift Relation from Gravitational Lensing [CEA]

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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…

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