Reconciling the observed all-sky CMB flux with its expected value from an inhomogeneous Universe

An analytical expression is given for the total flux amplification of a large distant source due to the presence of a small foreground gravitational lens. In the weak lensing limit, the result is independent of the internal density profile of the lensing mass. By comparing the above with the behavior (deflection) of light rays from the outer boundary of the souce, which do not pass through the lens, it is shown that to first order in the gravitational potential weak lensing conserves surface brightness of emission. When the exercise is repeated with extra flux amplification from strong lensing of the very central light rays included, it is found that the movement of the outermost source boundary remains the same as before, i.e. there is no further solid angle magnification of the source to accompany (and accomodate) the excess flux. The inevitable conclusion is that strong lensing of a large source leads to higher surface brightness relative to the unlensed configuration. The physical reason is related to backtracking of the source when a caustic is encountered a phenemenon not exhibited by weak lensing. While photon number is strictly conserved, there is the consequence of an inequality between flux and angular size distance. For this reason, the total flux from the entire microwave background sky is enhanced from the value of 4πσsT 4 s , by ≈ 1.5 % when strong lensing by galaxies is taken into account. The only way of reconciling this with the COBE/FIRAS measurement is to envisage a galaxy number density or velocity disperion grossly at odds with their observed values. The evidence brought forth here represents another formidable inconsistency between the standard cosmological model and reality.