The convergence fluctuation of light by clusters of galaxies - a cosmological test using the second acoustic peak of the microwave background

The weak lensing magnification of distant light by clusters of galaxies, assumed to be virialized systems with an NFW density profile, was calculated analytically as a statistical average over a random non-evolving distribution within z = 1. The effect is exactly cancelled by the Dyer-Roeder beam in between clusters this conclusion is valid also for an ensemble of spherical clumps of arbitrary density profile. An analytical expression for the variance in the mean magnification is provided, using the general formula from our previous work. The all sky fluctuation in the angular size of the microwave background second acoustic peak, due to the random placement of cluster lenses in the intervening space, is then derived to be ≈ 33 %. Moreover, this standard deviation corresponds to that of a highly skewed magnification distribution, with a long tail extending towards the side of positive magnification, since any negative excursion from the mean cannot cross the hard limit imposed by the Dyer-Roeder empty beam. The absence of a left tail (in ℓ space) of this magnitude in the WMAP TT cross power spectrum is a result to be reckoned with. It indicates that either the NFW profile is not an accurate description of clusters, or our interpretation of the WMAP data is oversimplistic.