The magnification of distant light sources by isothermal spheres in an expanding Universe

The discovery that the Universe has a vanishingly small global space curvature is usually interpreted as implying a large scale mean density of space close to the critical value. Since at low redshifts a substantial fraction of the matter is not homogeneously distributed, a frequently asked question is whether the propagation of light through this region of space still reflects the Euclidean geometry, even on average. By considering the mass clumps as singular isothermal spheres, we demonstrate using two independent mathematical proofs that as long as the light intercepts a sufficient number of these spheres the average convergence of the beam by them will in the weak lensing limit exactly cancel the divergence within the homogeneous subcritical density Universe-the result is statistically identical to propagation in flat space. However, the notion of an average convergence applies only to the mean amplification among numerous remote point sources, but not to the angular size of large emitters like the primary acoustic peaks of the microwave background. The reason is because most (by mass) of the isothermal spheres are galaxies and their halos, which typical light paths stay well away from, so that the picture is one in which the acoustic peaks all lie behind the giant bubbles (or 'concave' lenses) of the ambient Universe, with some tiny galactic 'convex' lenses which are too small to remagnify a large source. In a conservative approach where galaxies encompass 50 % of the total matter out to z = 1, the acoustic peaks demagnifies by ≈ 3 % relative to Euclidean geometry. This expectation is in conflict with observed data, i.e. the standard cosmological model cannot yet be deemed to have provided a clear and consistent picture of the Universe.