Abstract

In  this  paper  we  start  from  astronomical  observations  confirming  the  fact  that  cosmic matter in form of stars and galaxies, at least in more recent cosmological times, is not homogeneously, but hierarchically distributed with respect to our cosmic vantage point and typically is described by two-point correlation functions. As we show here, with these correlations also a hierarchically structured cosmic mass distribution is associated. This stellar matter distribution enables to derive a law according to which the average cosmic mass density systematically falls off with cosmic distance. At larger distances comparable with  the  scale  R of the universe also the cosmic space-time geometry hereby has to be taken into account and the results strongly depend on the curvature of the universe. We show solutions for the average mass density for positively and negatively curved (k= ±1) and for Euclidean (k=0) universes. The interesting result is that only for positively curved universes (k= +1) one obtains finite values for the asymptotic mass density, while for other geometries the average mass density values monotonously fall off with cosmic distance not allowing for a reasonable input into the energy-momentum tensor Tik of Einstein‘s GTR field equations. We discuss the cosmologically essential question upcoming in this article, whether or not a positively curved universe in view of such results needs to be accepted.

Keywords

cosmic matter, stars-galaxies, structured cosmic mass distribution, curvature of the Universe, Einstein‘s GTR field equations