On the Absence of Gravitational Lensing of the Cosmic Microwave Background

The magnification of distant sources by mass clumps at lower (z ≤ 1) redshifts is calculated analytically. The clumps are initially assumed to be galaxy group isothermal spheres with properties inferred from an extensive survey. The average effect, which includes strong lensing, is exactly counteracted by the beam divergence in between clumps (more precisely, the average reciprocal magnification cancels the inverse Dyer-Roeder demagnification). This conclusion is independent of the matter density function within each clump, and remains valid for arbitrary values of Ωm and ΩΛ. When tested against the cosmic microwave background data, a rather large lensing-induced dispersion in the angular size of the primary acoustic peaks of the temperature-temperature (TT) power spectrum is inconsistent with WMAP observations. The situation is unchanged by the use of Navarro-Frenk-White (NFW) profiles for the density distribution of groups, which lead in fact to slightly larger fluctuations. Finally, our formulae are applied to an ensemble of NFW mass clumps or isothermal spheres having the properties of galaxy clusters. The acoustic peak size dispersion remains unobservably large and is also excluded by WMAP. For galaxy groups, two possible ways of reconciling with the data are proposed, both exploiting maximally the uncertainties in our knowledge of group properties. The same escape routes are not available in the case of clusters, however, because their properties are well understood. Here we have a more robust conclusion: neither the NFW nor isothermal sphere profiles are accurate descriptions of clusters, or important elements of physics responsible for shaping zero-curvature space are missing from the standard cosmological model. When all the effects are accrued, it is difficult to understand how WMAP could reveal no evidence whatsoever of lensing by groups and clusters.