Concerning the instantaneous mass and the extent of an expanding   universe
  In this article we want to answer the cosmologically relevant question what, with some good semantic and physical reason, could be called the mass of an infinitely extended, homogeneously matter-filled and expanding universe. To answer this question we produce a space-like sum of instantaneous cosmic energy depositions surrounding equally each spacepoint in the homogeneous universe. We calculate the added-up instantaneous cosmic energy per volume around an arbitrary space point in the expanding universe. To carry out this sum we use as basic metrics an analogy to the inner Schwarzschild metric applied to stars, but this time applied to the spacepoint-related universe. It is then shown that this leads to the added-up proper energy within a sphere of a finite outer critical radius defining the point-related infinity. As a surprise this radius turns out to be reciprocal to the square root of the prevailing average cosmic energy density. The equivalent mass of the universe can then also be calculated and, by the expression which is obtained here, shows a scaling with this critical radius of this universe, a virtue of the universe which was already often called for in earlier works by E.Mach, H.Thirring and F.Hoyle and others. This radius on the other hand can be shown to be nearly equal to the Schwarzschild radius of the so-defined mass of the universe. 
H.J. Fahr, Michael Heyl