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 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 mode magnification , which corresponds to the acoustic 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 \Omega _ { m } = 0.278 \pm 0.040 and \Omega _ { \Lambda } = 0.782 \pm 0.040 .