Are the WMAP angular magnification measurements consistent with an inhomogeneous critical density Universe?

Abstract

The propagation of light through a Universe of (a) isothermal mass spheres amidst (b) a homogeneous matter component, is considered. We demonstrate by an analytical proof that as long as a small light bundle passes {\it through} sufficient number of (a) at various impact parameters - a criterion of great importance - its average convergence will exactly compensate the divergence within (b). The net effect on the light is statistically the same as if all the matter in (a) is `fully homogenized'. When applying the above ideas towards understanding the angular size of the primary acoustic peaks of the microwave background, however, caution is needed. The reason is that most (by mass) of (a) are in galaxies - their full mass profiles are not sampled by passing light - at least the inner 20 kpc regions of these systems are missed by the majority of rays, while the rest of the rays would map back to unresolvable but magnified, randomly located spots to compensate for the loss in angular size. Therefore, a scanning pair of WMAP beams finds most frequently that the largest temperature difference occurs when each beam is placed at diametrically opposite points of the Dyer-Roeder collapsed sections. This is the {\it mode} magnification, which corresponds to the acoustic {\it peaks}, and is less than the mean (or the homogeneous pre-clumping angular size). Since space was seen to be Euclidean without taking the said adjustment into account, the true density of the Universe should be supercritical. Our analysis gives $Ω_m =$ 0.278 $\pm$ 0.040 and $Ω_Λ =$ 0.782 $\pm$ 0.040.