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eISSN: 2576-4543

Physics & Astronomy International Journal

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Received: January 01, 1970 | Published: ,

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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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfaaaa@36BA@ 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 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaabaaeaa aaaaaaa8qacaWGRbGaeyypa0JaaeiiaiabgglaXkaaigdaa8aacaGL OaGaayzkaaaaaa@3CDD@ and for Euclidean ( k= 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaabaaeaa aaaaaaa8qacaWGRbGaeyypa0Jaaeiiaiaaicdaa8aacaGLOaGaayzk aaaaaa@3AEE@ universes. The interesting result is that only for positively curved universes ( k= +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaabaaeaa aaaaaaa8qacaWGRbGaeyypa0JaaeiiaiabgUcaRiaaigdaa8aacaGL OaGaayzkaaaaaa@3BD1@ 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 T ik MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaamivamaaBaaabaWaaSbaaeaacaWGPbGaam4Aaaqabaaabeaaaaa@38FC@ 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

Introduction

It is generally known that Einstein´s general relativistic field equations (see Einstein,1915,1917, or later presented in books e.g. by Rindler, 1977, or Tolman, 1987)1,2 describe the 4-dim space time geometry through the gravitational geometry source, i.e. through the cosmic energy-momentum tensor T ik MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaamivamaaBaaabaWaaSbaaeaacaWGPbGaam4Aaaqabaaabeaaaaa@38FC@ . Oppositely to what is commonly thought, this source tensor is not an easy-to-handle quantity, since the tensor ingredients are dependent on cosmic time in a non-trivial, but fairly complicated, and in a physically not evident or straightforward way. Even though all cosmological models start from the basis of the cosmological principle requiring that the universe at identical cosmic times looks the same from all space points in the universe, this does not make it evident what that means in terms of these T ik MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaamivamaaBaaabaWaaSbaaeaacaWGPbGaam4Aaaqabaaabeaaaaa@38FC@ -tensor-ingredient data, even not in such a homogeneous and isotropic universe. In case the matter content of the universe can be described as a homogeneously distributed baryonic gas, then the mass density and the scalar pressure of this gas may count as space-averaged quantities, but in later, closer to the present phases of the universe, when matter is structured in stars, galaxies and galaxy clusters what in replace of these former quantities should be used then? In terms of gravity sources baryons imprisoned in the body of a star are not like the same number of baryons freely distributed as a cosmic baryon gas. Stellar baryons are much hotter and in the stellar interiors their pressure may strongly count in terms of T ik MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaamivamaaBaaabaWaaSbaaeaacaWGPbGaam4Aaaqabaaabeaaaaa@38FC@ -ingredients. So the question arises how to make spatially averaged quantities out of them under such conditions.

While already this aspect makes the T ik MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaamivamaaBaaabaWaaSbaaeaacaWGPbGaam4Aaaqabaaabeaaaaa@38FC@ -ingredients a deeply problematic quantity, there is still another aspect which makes them an even much more problematic quantity. Namely the observationally confirmed fact that stellar and galactic matter is not homogeneously distributed in space but in a hierarchical structure seen in the visible light, making it an even more problematic question how under these conditions spatial averages of the T ik MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaamivamaaBaaabaWaaSbaaeaacaWGPbGaam4Aaaqabaaabeaaaaa@38FC@ - ingredients as they are nevertheless needed for Einstein‘s GRT field equations can be reasonably well constructed. In the following part of the paper we shall attempt to give an answer to this delicate question. While in earlier papers3–5 we considered the effect of gravitational binding energy formation connected with hierarchical clusterings, we here shall especially study what average cosmic mass densities under these conditions could mean.

The hierarchically structured cosmic mass density

We start from the astronomical observations carried out by Bahcall et al.6 and Bahcall et al.,7and or equally well in more recent times by Sylos-Labini et al.,8,9 and take their two-point correlation function ξ( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabe67a4naabm aabaGaaGymaaGaayjkaiaawMcaaaaa@39EA@  denoting the probability to find other stellar objects at a distance l from any other arbitrarily taken stellar object. The quantity l has to be considered as the so-called distance parameter; astronomers take it to be identical with the visual or redshift distance, making it evident that it necessarily is a cosmologically biased quantity. From this function ξ( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabe67a4naabm aabaGaaGymaaGaayjkaiaawMcaaaaa@39EA@ we create the underlying model of cosmic matter distribution. This correlation function has been confirmed and supported by astronomical observations of the visible star and galaxy formation structure which is at present surrounding us at our cosmic standpoint, and, based on the generally respected cosmological principle, also should surround every other cosmic space point in an analogous and equivalent manner, unless the cosmological principle would turn out to be violated. This two-point correlation function ξ( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabe67a4naabm aabaGaaGymaaGaayjkaiaawMcaaaaa@39EA@  defines the probability to find another star (or galaxy) at a distance l from our arbitrary standpoint and, based on astronomical observations, is expressed in the form:

ξ( 1 )= ξ 0 . ( l 0 l ) α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabe67a4naabm aabaGaaGymaaGaayjkaiaawMcaaiabg2da9maavababeqaaiaaicda aeqabaGaeqOVdGhaaiaac6cadaqadaqaamaalaaabaWaaubeaeqaba GaaGimaaqabeaacaWGSbaaaaqaaiaadYgaaaaacaGLOaGaayzkaaWa aWbaaeqabaGaeqySdegaaaaa@4471@ (1)

where ξ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaavababeqaai aaicdaaeqabaGaeqOVdGhaaaaa@388E@ is a reference value valid at the reference distance l0. The correlation index α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHbaa@3782@ has been determined by Bahcall et al.6 as α=1.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHjabg2 da9iaaigdacaGGUaGaaGioaaaa@3AB7@ . This first of all has an interesting consequence which we want to mention first here: If we may assume here that this form of a stellar or galactic clustering continues to prevail to larger and larger cosmic distances (i.e.: scale-invariant clustering!), then first of all this fact perhaps could give an evident solution of the Olbers paradox (H.W.M. Olbers,1826), namely the fact that the sky during night is dark. Other solutions meanwhile have been offered, all perhaps worth a discussion, but none convincing up to the present, only one has been overlooked up to now. Because under these above mentioned conditions of a scale-invariant stellar clustering one would simply obtain the following growth O( l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+eadaqada qaamaavababeqaaiabg6HiLcqabeaacaWGSbaaaaGaayjkaiaawMca aaaa@3AD0@ of the illuminated part of the sky in any arbitrary direction with a view cone d 2 Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgadaahaa qabeaacaaIYaaaaiabfM6axbaa@3938@

O( l )= 1 l 2 d 2 Ω l 0 l l 2 d 2 Ω.ξ( 1 ) π r s 2 l 2 dl= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+eadaqada qaamaavababeqaaiabg6HiLcqabeaacaWGSbaaaaGaayjkaiaawMca aiabg2da9maalaaabaGaaGymaaqaamaavadabeqaaiabg6HiLcqaai aaikdaaeaacaWGSbaaaiaadsgadaahaaqabeaacaaIYaaaaiabfM6a xbaadaWdXaqaamaavacabeqabeaacaaIYaaabaGaamiBaaaadaqfGa qabeqabaGaaGOmaaqaaiaadsgaaaaabaWaaubeaeqabaGaaGimaaqa beaacaWGSbaaaaqaaiaadYgacqGHEisPaiabgUIiYdGaeuyQdCLaai Olaiabe67a4naabmaabaGaaGymaaGaayjkaiaawMcaamaalaaabaGa eqiWda3aaubmaeqabaGaam4CaaqaaiaaikdaaeaacaWGYbaaaaqaai aadYgadaahaaqabeaacaaIYaaaaaaacaWGKbGaamiBaiabg2da9aaa @5D36@

1 l 2 l0 l l 2 .ξ( 1 ) π r s 2 l 2 dl= π r s 2 l 0 2 ξ 0 . l 0 α l0 l 1 l dl= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaabaGaaG ymaaqaamaavadabeqaaiabg6HiLcqaaiaaikdaaeaacaWGSbaaaaaa daWdXaqaaiaadYgadaahaaqabeaacaaIYaaaaiaac6cacqaH+oaEda qadaqaaiaaigdaaiaawIcacaGLPaaaaeaacaWGSbGaaGimaaqaaiaa dYgacqGHEisPaiabgUIiYdWaaSaaaeaacqaHapaCdaqfWaqabeaaca WGZbaabaGaaGOmaaqaaiaadkhaaaaabaGaamiBamaaCaaabeqaaiaa ikdaaaaaaiaadsgacaWGSbGaeyypa0ZaaSaaaeaacqaHapaCdaqfWa qabeaacaWGZbaabaGaaGOmaaqaaiaadkhaaaaabaWaaubmaeqabaGa aGimaaqaaiaaikdaaeaacaWGSbaaaaaacqaH+oaEdaWgaaqaaiaaic daaeqaaiaac6cadaqfWaqabeaacaaIWaaabaGaeqySdegabaGaamiB aaaadaWdXaqaamaalaaabaGaaGymaaqaaiaadYgacqGHEisPaaaaba GaamiBaiaaicdaaeaacaWGSbGaeyOhIukacqGHRiI8aiaadsgacaWG SbGaeyypa0daaa@6ABE@

π r s 2 ξ 0 l 0 α2 l 0 α2 1α 1α [ ( l l0 ) 1α 1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWnaava dabeqaaiaadohaaeaacaaIYaaabaGaamOCaaaacqaH+oaEdaWgaaqa aiaaicdaaeqaamaavadabeqaaiaaicdaaeaacqaHXoqycqGHsislca aIYaaabaGaamiBaaaadaWcaaqaamaavadabeqaaiaaicdaaeaacqaH XoqycqGHsislcaaIYaaabaGaamiBaaaaaeaacaaIXaGaeyOeI0Iaeq ySdegaamaaCaaabeqaaiaaigdacqGHsislcqaHXoqyaaGaai4wamaa bmaabaWaaSaaaeaacaWGSbGaeyOhIukabaGaamiBaiaaicdaaaaaca GLOaGaayzkaaWaaWbaaeqabaGaaGymaiabgkHiTiabeg7aHbaacqGH sislcaaIXaGaaiyxaaaa@5AED@  (2)

First here one can see that for α=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHjabg2 da9iaaicdaaaa@3942@ (i.e. no structuring; homogeneous matter distribution!) one would have the following result

O( l )=π r s 2 ξ 0 l 0 1 [( l l0 )1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+eadaqada qaamaavababeqaaiabg6HiLcqabeaacaWGSbaaaaGaayjkaiaawMca aiabg2da9iabec8aWnaavadabeqaaiaadohaaeaacaaIYaaabaGaam OCaaaacqaH+oaEdaWgaaqaaiaaicdaaeqaaiaadYgadaqhaaqaaiaa icdaaeaacqGHsislcaaIXaaaaiaacUfadaqadaqaamaalaaabaGaam iBaiabg6HiLcqaaiaadYgacaaIWaaaaaGaayjkaiaawMcaaiabgkHi TiaaigdacaGGDbaaaa@4FAC@ (3)

clearly showing that for increasing values of ( l/ l 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaabaGaam iBaiabg6HiLkaac+cacaWGSbWaaSbaaeaacaaIWaaabeaaaiaawIca caGLPaaaaaa@3C4D@ the sky coverage would grow to O( l )>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+eadaqada qaamaavababeqaaiabg6HiLcqabeaacaWGSbaaaaGaayjkaiaawMca aiabg6da+iaaigdaaaa@3C93@ (i.e. illuminated sky = Olbers paradox!).

To the contrast, however, for values as observationally confirmed, namely alpha=1.8, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggacaWGSb GaamiCaiaadIgacaWGHbGaeyypa0JaaGymaiaac6cacaaI4aGaaiil aaaa@3E67@ one can see that O( l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+eadaqada qaamaavababeqaaiabg6HiLcqabeaacaWGSbaaaaGaayjkaiaawMca aaaa@3AD0@ always leads to values O( l )>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+eadaqada qaamaavababeqaaiabg6HiLcqabeaacaWGSbaaaaGaayjkaiaawMca aiabg6da+iaaigdaaaa@3C93@ (i.e. non-illuminated sky = no Olbers paradox!).

To continue now with another interesting problem connected with the above mentioned clustering we here want to emphasize the cosmologically important point, that the existence of the above correlation function can evidently also be interpreted as an expression for the structured stellar density or stellar mass density distribution of surrounding stars or galaxies in our cosmic environment, and consequently thereby also expresses a standpoint-oriented mass density distribution p=p( l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadchacqGH9a qpcaWGWbWaaeWaaeaacaWGSbaacaGLOaGaayzkaaaaaa@3B4D@ when recognizing that the mass is closely associated with the number of stars with a typical stellar or galactic mass m 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2gadaWgaa qaaiaaicdaaeqaaaaa@37B0@ on a spherical shell at distance l , based at a first approximation here, Euklidean geometry conditions (i.e. flat universe: k = 0), is then given by

dM( l )=4π l 2 m 0 ξ 0 ( l 0 l ) α dl=4π ρ 2 . ( l 0 l ) α dl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgacaWGnb WaaeWaaeaacaWGSbaacaGLOaGaayzkaaGaeyypa0JaaGinaiabec8a WjaadYgadaahaaqabeaacaaIYaaaaiaad2gadaWgaaqaaiaaicdaae qaaiabe67a4naaBaaabaGaaGimaaqabaWaaeWaaeaadaWcaaqaaiaa dYgadaWgaaqaaiaaicdaaeqaaaqaaiaadYgaaaaacaGLOaGaayzkaa WaaWbaaeqabaGaeqySdegaaiaadsgacaWGSbGaeyypa0JaaGinaiab ec8aWjabeg8aYnaaCaaabeqaaiaaikdaaaGaaiOlamaabmaabaWaaS aaaeaacaWGSbWaaSbaaeaacaaIWaaabeaaaeaacaWGSbaaaaGaayjk aiaawMcaamaaCaaabeqaaiabeg7aHbaacaWGKbGaamiBaaaa@5A86@ (4)

with ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaaGimaaqabaaaaa@387E@ denoting a reference value of the hierarchy-typical mass density. Bahcall7and Chokski7 point out furthermore the astonishing fact that the general type of the above mentioned two-point correlation function ξ( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabe67a4naabm aabaGaaGymaaGaayjkaiaawMcaaaaa@39EA@ interestingly enough is observationally confirmed as well for galaxy correlations, as for cluster correlations, as also for super-cluster correlations, with the difference that only the reference scale l0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgacaaIWa aaaa@378E@ and the reference probability ξ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabe67a4naaBa aabaGaaGimaaqabaaaaa@3881@ or mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaaGimaaqabaaaaa@387E@ have to be adapted from the galaxy-case up to the super-cluster case, while however as a surprise the same correlation index of α=1.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHjabg2 da9iaaigdacaGGUaGaaGioaaaa@3AB7@ is reappearing as a common number for all these hierarchies.

Taking the largest hierarchy, i.e. super-clusters, in order to cover the largest achievable distances of the order 100Mpc and more, we thus would use the corresponding SC-SC correlation function, and the corresponding mass increment with distance l would then - in accordance with Eq.(4) - be given by

dM( l )=4π l 2 m SC , 0 ξ SC , 0 . ( l SC , 0 l ) α dl= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgacaWGnb WaaeWaaeaacaWGSbaacaGLOaGaayzkaaGaeyypa0JaaGinaiabec8a WjaadYgadaahaaqabeaadaahaaqabeaacaaIYaaaaaaacaWGTbWaaS baaeaacaWGtbGaam4qaaqabaGaaiilamaaBaaabaGaaGimaaqabaGa eqOVdG3aaSbaaeaacaWGtbGaam4qaaqabaGaaiilamaaBaaabaGaaG imaaqabaGaaiOlamaabmaabaWaaSaaaeaacaWGSbWaaSbaaeaacaWG tbGaam4qaaqabaGaaiilamaaBaaabaGaaGimaaqabaaabaGaamiBaa aaaiaawIcacaGLPaaadaahaaqabeaacqaHXoqyaaGaamizaiaadYga cqGH9aqpaaa@54F1@

4π l 2 ρ SC ,0. ( l SC , 0 l ) α dl= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaisdacqaHap aCcaWGSbWaaWbaaeqabaWaaWbaaeqabaGaaGOmaaaaaaGaeqyWdi3a aSbaaeaacaWGtbGaam4qaaqabaGaaiilaiaaicdacaGGUaWaaeWaae aadaWcaaqaaiaadYgadaWgaaqaaiaadofacaWGdbaabeaacaGGSaWa aSbaaeaacaaIWaaabeaaaeaacaWGSbaaaaGaayjkaiaawMcaamaaCa aabeqaaiabeg7aHbaacaWGKbGaamiBaiabg2da9aaa@4B54@ (5)

where ρ SC ,0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaam4uaiaadoeaaeqaaiaacYcacaaIWaaaaa@3ACE@ is a typical reference density for the super cluster scale l SC ,0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgadaWgaa qaaiaadofacaWGdbaabeaacaGGSaGaaGimaaaa@39FF@ . In order to address the largest achievable cosmic distances lR,         R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgacqWIdj YocaWGsbGaaiilauaabaqaceaaaeaaqaaaaaaaaaWdbiaacckaa8aa baWdbiaacckaaaWdauaabaqaceaaaeaapeGaaiiOaaWdaeaapeGaai iOaaaapaGaamOuaaaa@3F97@ being the scale of the universe, where the prevailing cosmic geometry conditions become important, we should then also take furthermore into account that most likely we are sitting in a non-Euklidean universe with a curved general-relativistic pace time geometry. One therefore would have to pay attention to the fact that the radial distance parameter l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgaaaa@36D4@ , in the Robertson-Walker approximation of the cosmic geometry, is transformed into a geometrical distance r(l) given by (see e.g. Goenner10, or Fliessbach11):

r( l )=l. ( 1+k l 2 ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkhadaqada qaaiaadYgaaiaawIcacaGLPaaacqGH9aqpcaWGSbGaaiOlamaabmaa baGaaGymaiabgUcaRiaadUgacaWGSbWaaWbaaeqabaGaaGOmaaaaai aawIcacaGLPaaadaahaaqabeaacqGHsislcaaIXaaaaaaa@43AC@ (6)

i.e. the spherical area associated to the distance parameter l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgaaaa@36D4@ is Φ( 1 )=4 π 2 ( l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agnaabm aabaGaaGymaaGaayjkaiaawMcaaiabg2da9iaaisdacqaHapaCdaah aaqabeaacaaIYaaaamaabmaabaGaamiBaaGaayjkaiaawMcaaaaa@407A@ where k denotes the cosmic curvature parameter which needs to be determined by looking for the best fitting cosmological FLRW-model (e.g. see Bennet et al.12 yielding k0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqWIdj YocaaIWaaaaa@38BE@ ).

This latter cosmological model interestingly enough needs, however, spatially averaged quantities as T ik MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaamivamaaBaaabaWaaSbaaeaacaWGPbGaam4Aaaqabaaabeaaaaa@38FC@ -ingredients which latter, under the perspectives given above, are highly problematic quantities. Anticipating the value for k as input from this model one can include this geometric distance transformation and would bring the above Eq.(5) for dM (l) into the following form:

dM( l )=4π l 2 +         . ( 1+k l 2 ) 2     . ρ SC ,0.     ( l SC , 0 l ) α .     ( 1     k l 2 ) ( 1+     k l 2 ) 2 d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgacaWGnb WaaeWaaeaacaWGSbaacaGLOaGaayzkaaGaeyypa0JaaGinaiabec8a WjaadYgadaahaaqabeaadaahaaqabeaacaaIYaaaaaaacqGHRaWkfa qaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaaa8aa faqaaeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdaiaac6cada qadaqaaiaaigdacqGHRaWkcaWGRbGaamiBamaaCaaabeqaaiaaikda aaaacaGLOaGaayzkaaWaaWbaaeqabaGaeyOeI0IaaGOmaaaafaqaae GabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdaiaac6cacqaHbpGC daWgaaqaaiaadofacaWGdbaabeaacaGGSaGaaGimaiaac6cafaqaae GabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdamaabmaabaWaaSaa aeaacaWGSbWaaSbaaeaacaWGtbGaam4qaaqabaGaaiilamaaBaaaba GaaGimaaqabaaabaGaamiBaaaaaiaawIcacaGLPaaadaahaaqabeaa cqaHXoqyaaGaaiOlauaabaqaceaaaeaapeGaaiiOaaWdaeaapeGaai iOaaaapaWaaSaaaeaadaqadaqaaiaaigdacqGHsislfaqaaeGabaaa baWdbiaacckaa8aabaWdbiaacckaaaWdaiaadUgacaWGSbWaaWbaae qabaGaaGOmaaaaaiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGH RaWkfaqaaeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdaiaadU gacaWGSbWaaWbaaeqabaGaaGOmaaaaaiaawIcacaGLPaaadaahaaqa beaacaaIYaaaaaaacaWGKbaaaa@7811@ (7)

or rearranging things and putting them into a better logic order then gives

dM( l )=4π ρ SC ,0.         ( l SC , 0 l ) α .     ( 1     k l 2 ) ( 1+     k l 2 ) 4 l 2 dl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgacaWGnb WaaeWaaeaacaWGSbaacaGLOaGaayzkaaGaeyypa0JaaGinaiabec8a Wjabeg8aYnaaBaaabaGaam4uaiaadoeaaeqaaiaacYcacaaIWaGaai OlauaabaqaceaaaeaaqaaaaaaaaaWdbiaacckaa8aabaWdbiaaccka aaWdauaabaqaceaaaeaapeGaaiiOaaWdaeaapeGaaiiOaaaapaWaae WaaeaadaWcaaqaaiaadYgadaWgaaqaaiaadofacaWGdbaabeaacaGG SaWaaSbaaeaacaaIWaaabeaaaeaacaWGSbaaaaGaayjkaiaawMcaam aaCaaabeqaaiabeg7aHbaacaGGUaqbaeaabiqaaaqaa8qacaGGGcaa paqaa8qacaGGGcaaa8aadaWcaaqaamaabmaabaGaaGymaiabgkHiTu aabaqaceaaaeaapeGaaiiOaaWdaeaapeGaaiiOaaaapaGaam4Aaiaa dYgadaahaaqabeaacaaIYaaaaaGaayjkaiaawMcaaaqaamaabmaaba GaaGymaiabgUcaRuaabaqaceaaaeaapeGaaiiOaaWdaeaapeGaaiiO aaaapaGaam4AaiaadYgadaahaaqabeaacaaIYaaaaaGaayjkaiaawM caamaaCaaabeqaaiaaisdaaaaaaiaadYgadaahaaqabeaacaaIYaaa aiaadsgacaWGSbaaaa@69C8@ (8)

The cosmic curvature parameter k can be restricted to values of k=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcaaIWaaaaa@3893@ ; k=         ±1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaa a8aafaqaaeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdaiabgg laXkaaigdaaaa@3FB6@ if l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgaaaa@36D4@ is scaled with the cosmic scale parameter R     =     R( t )         by         k=K         .     R 2 ,         K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfafaqaae Gabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaaa8aacqGH 9aqpfaqaaeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdaiaadk fadaqadaqaaiaadshaaiaawIcacaGLPaaafaqaaeGabaaabaWdbiaa cckaa8aabaWdbiaacckaaaWdauaabaqaceaaaeaapeGaaiiOaaWdae aapeGaaiiOaaaapaGaamOyaiaadMhafaqaaeGabaaabaWdbiaaccka a8aabaWdbiaacckaaaWdauaabaqaceaaaeaapeGaaiiOaaWdaeaape GaaiiOaaaapaGaam4Aaiabg2da9iaadUeafaqaaeGabaaabaWdbiaa cckaa8aabaWdbiaacckaaaWdauaabaqaceaaaeaapeGaaiiOaaWdae aapeGaaiiOaaaapaGaaiOlauaabaqaceaaaeaapeGaaiiOaaWdaeaa peGaaiiOaaaapaGaamOuamaaCaaabeqaaiaaikdaaaGaaiilauaaba qaceaaaeaapeGaaiiOaaWdaeaapeGaaiiOaaaapaqbaeaabiqaaaqa a8qacaGGGcaapaqaa8qacaGGGcaaa8aacaWGlbaaaa@6001@ being the cosmic curvature scalar or the contracted Ricci tensor K     = R i i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUeafaqaae Gabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaaa8aacqGH 9aqpcaWGsbWaa0baaeaacaWGPbaabaGaamyAaaaaaaa@3D30@ (i.e. Ricci scalar). Therefore, besides the Euklidean case k = 0, favoured by Bennet et al.12 one obtains the following two more options:

dM( l )=4π ρ SC ,0 l SC,o α .             ( 1 l 2 R 2 ) ( 1±     l 2 R 2 ) 4 l 2 α dl= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgacaWGnb WaaeWaaeaacaWGSbaacaGLOaGaayzkaaGaeyypa0JaaGinaiabec8a Wjabeg8aYnaaBaaabaGaam4uaiaadoeaaeqaaiaacYcacaaIWaGaam iBamaaDaaabaGaam4uaiaadoeacaGGSaGaam4Baaqaaiabeg7aHbaa caGGUaqbaeaabiqaaaqaaabaaaaaaaaapeGaaiiOaaWdaeaapeGaai iOaaaapaqbaeaabiqaaaqaa8qacaGGGcaapaqaa8qacaGGGcaaa8aa faqaaeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdamaalaaaba WaaeWaaeaacaaIXaGaeS4eI02aaSaaaeaacaWGSbWaaWbaaeqabaGa aGOmaaaaaeaacaWGsbWaaWbaaeqabaGaaGOmaaaaaaaacaGLOaGaay zkaaaabaWaaeWaaeaacaaIXaGaeyySaeBbaeaabiqaaaqaa8qacaGG Gcaapaqaa8qacaGGGcaaa8aadaWcaaqaaiaadYgadaahaaqabeaaca aIYaaaaaqaaiaadkfadaahaaqabeaacaaIYaaaaaaaaiaawIcacaGL PaaadaahaaqabeaacaaI0aaaaaaacaWGSbWaaWbaaeqabaGaaGOmaa aadaahaaqabeaacqGHsislcqaHXoqyaaGaamizaiaadYgacqGH9aqp aaa@6AA2@ (9)

4π ρ SC ,0 l SC,o α .             g( l )dl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaisdacqaHap aCcqaHbpGCdaWgaaqaamaaBaaabaGaam4uaiaadoeaaeqaaaqabaGa aiilaiaaicdacaWGSbWaa0baaeaacaWGtbGaam4qaiaacYcacaWGVb aabaGaeqySdegaaiaac6cafaqaaeGabaaabaaeaaaaaaaaa8qacaGG Gcaapaqaa8qacaGGGcaaa8aafaqaaeGabaaabaWdbiaacckaa8aaba WdbiaacckaaaWdauaabaqaceaaaeaapeGaaiiOaaWdaeaapeGaaiiO aaaapaGaam4zamaabmaabaGaamiBaaGaayjkaiaawMcaaiaadsgaca WGSbaaaa@5118@

The average cosmic mass density

Considering here the case of k=         ±1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaa a8aafaqaaeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdaiabgg laXkaaigdaaaa@3FB6@ (as the most promising with respect to asymptotically reach a constant mass density ρ     =     ρ( l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaeyOhIukabeaafaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaa paqaa8qacaGGGcaaa8aacqGH9aqpfaqaaeGabaaabaWdbiaacckaa8 aabaWdbiaacckaaaWdaiabeg8aYnaabmaabaGaamiBaiabgkziUkab g6HiLcGaayjkaiaawMcaaaaa@4707@ with the over-Euklidean growth of the sphere surface at large distances lR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgacqWIdj YocaWGsbaaaa@38DC@ leads to the following remaining geometrical expression:13

g( l )dl[ ( 1+ l 2 R 2 ) ( 1     l 2 R 2 ) 4 l 2 α dl] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgadaqada qaaiaadYgaaiaawIcacaGLPaaacaWGKbGaamiBaiaacUfadaWcaaqa amaabmaabaGaaGymaiabgUcaRmaalaaabaGaamiBamaaCaaabeqaai aaikdaaaaabaGaamOuamaaCaaabeqaaiaaikdaaaaaaaGaayjkaiaa wMcaaaqaamaabmaabaGaaGymaiabgkHiTuaabaqaceaaaeaaqaaaaa aaaaWdbiaacckaa8aabaWdbiaacckaaaWdamaalaaabaGaamiBamaa CaaabeqaaiaaikdaaaaabaGaamOuamaaCaaabeqaaiaaikdaaaaaaa GaayjkaiaawMcaamaaCaaabeqaaiaaisdaaaaaaiaadYgadaahaaqa beaacaaIYaaaamaaCaaabeqaaiabgkHiTiabeg7aHbaacaWGKbGaam iBaiaac2faaaa@544B@ (10)

which at large distances lR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgacqWIdj YocaWGsbaaaa@38DC@ leads to the following asymptotic behaviour

g( l     R ) = lR lim [ ( 1+ l 2 R 2 ) ( 1     l 2 R 2 ) 4 l 2 α dl]= 2 1δ l 2α dl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgadaqada qaaiaadYgacqGHsgIRfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaa paqaa8qacaGGGcaaa8aacaWGsbaacaGLOaGaayzkaaGaeyypa0Zaa0 baaeaacaWGSbGaeyOKH4QaamOuaaqaaiGacYgacaGGPbGaaiyBaaaa caGGBbWaaSaaaeaadaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaadY gadaahaaqabeaacaaIYaaaaaqaaiaadkfadaahaaqabeaacaaIYaaa aaaaaiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGHsislfaqaae GabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdamaalaaabaGaamiB amaaCaaabeqaaiaaikdaaaaabaGaamOuamaaCaaabeqaaiaaikdaaa aaaaGaayjkaiaawMcaamaaCaaabeqaaiaaisdaaaaaaiaadYgadaah aaqabeaacaaIYaaaamaaCaaabeqaaiabgkHiTiabeg7aHbaacaWGKb GaamiBaiaac2facqGH9aqpdaWcaaqaaiaaikdaaeaacaaIXaGaeyOe I0IaeqiTdqgaaiaadYgadaahaaqabeaadaahaaqabeaacaaIYaGaey OeI0IaeqySdegaaaaacaWGKbGaamiBaaaa@6AEA@ (11)

For small distances ( l/R )1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaabaGaam iBaiaac+cacaWGsbaacaGLOaGaayzkaaGaeSOAI0JaaGymaaaa@3BFC@ one would instead obtain the following behaviour

g( lR )=[ ( 1+ l 2 R 2 ) ( 1     l 2 R 2 ) 4 l 2 α dl]lR= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgadaqada qaaiaadYgacqWIQjspcaWGsbaacaGLOaGaayzkaaGaeyypa0Jaai4w amaalaaabaWaaeWaaeaacaaIXaGaey4kaSYaaSaaaeaacaWGSbWaaW baaeqabaGaaGOmaaaaaeaacaWGsbWaaWbaaeqabaGaaGOmaaaaaaaa caGLOaGaayzkaaaabaWaaeWaaeaacaaIXaGaeyOeI0sbaeaabiqaaa qaaabaaaaaaaaapeGaaiiOaaWdaeaapeGaaiiOaaaapaWaaSaaaeaa caWGSbWaaWbaaeqabaGaaGOmaaaaaeaacaWGsbWaaWbaaeqabaGaaG OmaaaaaaaacaGLOaGaayzkaaWaaWbaaeqabaGaaGinaaaaaaGaamiB amaaCaaabeqaaiaaikdaaaWaaWbaaeqabaGaeyOeI0IaeqySdegaai aadsgacaWGSbGaaiyxaiaadYgacqWIQjspcaWGsbGaeyypa0daaa@59D0@

( 1+ l 2 R 2 )( 1+4 l 2 R 2 ) l 2 α dl( 1+5 l 2 R 2 ) l 2 α dl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaabaGaaG ymaiabgUcaRmaalaaabaGaamiBamaaCaaabeqaaiaaikdaaaaabaGa amOuamaaCaaabeqaaiaaikdaaaaaaaGaayjkaiaawMcaamaabmaaba GaaGymaiabgUcaRiaaisdadaWcaaqaaiaadYgadaahaaqabeaacaaI YaaaaaqaaiaadkfadaahaaqabeaacaaIYaaaaaaaaiaawIcacaGLPa aacaWGSbWaaWbaaeqabaGaaGOmaaaadaahaaqabeaacqGHsislcqaH XoqyaaGaamizaiaadYgacqWIdjYodaqadaqaaiaaigdacqGHRaWkca aI1aWaaSaaaeaacaWGSbWaaWbaaeqabaGaaGOmaaaaaeaacaWGsbWa aWbaaeqabaGaaGOmaaaaaaaacaGLOaGaayzkaaGaamiBamaaCaaabe qaaiaaikdaaaWaaWbaaeqabaGaeyOeI0IaeqySdegaaiaadsgacaWG Sbaaaa@596D@ (12)

After this inspection one can then state that the expression for the average cosmic density in such a hierarchically structured universe for example with k=         1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaa a8aafaqaaeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdaiabgk HiTiaaigdaaaa@3EB5@ finally takes the following form:

 ρ( l )   = M( l ) V( l ) =     m SC ,0 ξ SC ,0 l SC ,0 α 0 1 [ ( 1+ l 2 R 2 ) ( 1 l 2 R 2 ) 4 l 2α dl] 0 1 [ ( 1+ l 2 R 2 ) ( 1 l 2 R 2 ) 4 l 2 dl] = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbawaabaqaceaaae aaqaaaaaaaaaWdbiaacckacqaHbpGCdaqadaqaaiaadYgaaiaawIca caGLPaaaa8aabaWdbiaacckaaaWdaiabg2da9maalaaabaGaamytam aabmaabaGaamiBaaGaayjkaiaawMcaaaqaaiaadAfadaqadaqaaiaa dYgaaiaawIcacaGLPaaaaaGaeyypa0tbaeaabiqaaaqaa8qacaGGGc aapaqaa8qacaGGGcaaa8aacaWGTbWaaSbaaeaadaWgaaqaaiaadofa caWGdbaabeaaaeqaaiaacYcacaaIWaGaeqOVdG3aaSbaaeaadaWgaa qaaiaadofacaWGdbaabeaaaeqaaiaacYcacaaIWaGaamiBamaaDaaa baWaaSbaaeaadaWgaaqaaiaadofacaWGdbaabeaaaeqaaiaacYcaca aIWaaabaGaeqySdegaamaalaaabaWaa8qmaeaacaGGBbWaaSaaaeaa daqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaadYgadaahaaqabeaaca aIYaaaaaqaaiaadkfadaahaaqabeaacaaIYaaaaaaaaiaawIcacaGL PaaaaeaadaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadYgadaahaa qabeaacaaIYaaaaaqaaiaadkfadaahaaqabeaacaaIYaaaaaaaaiaa wIcacaGLPaaadaahaaqabeaacaaI0aaaaaaacaWGSbWaaSbaaeaaca aIYaGaeyOeI0IaeqySdegabeaacaWGKbGaamiBaiaac2faaeaacaaI WaaabaGaaGymaaGaey4kIipaaeaadaWdXaqaaiaacUfadaWcaaqaam aabmaabaGaaGymaiabgUcaRmaalaaabaGaamiBamaaCaaabeqaaiaa ikdaaaaabaGaamOuamaaCaaabeqaaiaaikdaaaaaaaGaayjkaiaawM caaaqaamaabmaabaGaaGymaiabgkHiTmaalaaabaGaamiBamaaCaaa beqaaiaaikdaaaaabaGaamOuamaaCaaabeqaaiaaikdaaaaaaaGaay jkaiaawMcaamaaCaaabeqaaiaaisdaaaaaaiaadYgadaWgaaqaaiaa ikdaaeqaaiaadsgacaWGSbGaaiyxaaqaaiaaicdaaeaacaaIXaaacq GHRiI8aaaacqGH9aqpaaa@89B5@

m SC ,0 ξ SC ,0 l SC ,0 α R α 0 X [ ( 1+ x 2 ) ( 1 x 2 ) 4 x 2α dx] 0 X [ ( 1+ x 2 ) ( 1 x 2 ) 4 x 2 dx] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2gadaWgaa qaamaaBaaabaGaam4uaiaadoeaaeqaaaqabaGaaiilaiaaicdacqaH +oaEdaWgaaqaamaaBaaabaGaam4uaiaadoeaaeqaaaqabaGaaiilai aaicdadaWcaaqaaiaadYgadaqhaaqaamaaBaaabaWaaSbaaeaacaWG tbGaam4qaaqabaaabeaacaGGSaGaaGimaaqaaiabeg7aHbaaaeaaca WGsbWaaWbaaeqabaGaeqySdegaaaaadaWcaaqaamaapedabaGaai4w amaalaaabaWaaeWaaeaacaaIXaGaey4kaSIaamiEamaaCaaabeqaai aaikdaaaaacaGLOaGaayzkaaaabaWaaeWaaeaacaaIXaGaeyOeI0Ia amiEamaaCaaabeqaaiaaikdaaaaacaGLOaGaayzkaaWaaWbaaeqaba GaaGinaaaaaaGaamiEamaaBaaabaGaaGOmaiabgkHiTiabeg7aHbqa baGaamizaiaadIhacaGGDbaabaGaaGimaaqaaiaadIfaaiabgUIiYd aabaWaa8qmaeaacaGGBbWaaSaaaeaadaqadaqaaiaaigdacqGHRaWk caWG4bWaaWbaaeqabaGaaGOmaaaaaiaawIcacaGLPaaaaeaadaqada qaaiaaigdacqGHsislcaWG4bWaaWbaaeqabaGaaGOmaaaaaiaawIca caGLPaaadaahaaqabeaacaaI0aaaaaaacaWG4bWaaSbaaeaacaaIYa aabeaacaWGKbGaamiEaiaac2faaeaacaaIWaaabaGaamiwaaGaey4k Iipaaaaaaa@72BC@ (13)

In the presented Figures 1,2,3 we show the quantities M(l), V (l) and ρ( l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaeqyWdi3aaeWaaeaacaWGSbaacaGLOaGaayzkaaaaaa@3A3D@ as functions of l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgaaaa@36D4@ , for k=     +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaa a8aacqGHRaWkcaaIXaaaaa@3C18@ , k=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcaaIWaaaaa@3893@  and k=     1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaa a8aacqGHsislcaaIXaaaaa@3C23@ , respectively.14

Figure 1 This shows the increase of the cosmic mass as a function of l in arbitrary units for curvature k=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcqGHsislcaaIXaaaaa@3981@ , k=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcaaIWaaaaa@3893@ and k=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcaaIXaaaaa@3894@ , respectively.

Figure 2 This shows the increase of the cosmic volume as a function of l in arbitrary units for curvature k=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcqGHsislcaaIXaaaaa@3981@ , k=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcaaIWaaaaa@3893@ and k=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcaaIXaaaaa@3894@ , respectively.

Figure 3 This shows the increase of the cosmic density as a function of l in arbitrary units for curvature k=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcaaIXaaaaa@3894@ , k=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcaaIWaaaaa@3893@ and k=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcqGHsislcaaIXaaaaa@3981@ , respectively.

Obtained results

In the preceding section of this paper we have taken serious the astronomically confirmed fact that we are surrounded by a hierarchically structured stellar universe. In order to be cosmologically based on this important fact, we have first derived a point-related spatial mass configuration of surrounding stellar mass sources from the astonomically confirmed two-point correlation functions ξ( l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabe67a4naabm aabaGaamiBaaGaayjkaiaawMcaaaaa@3A20@ describing the spatial configuration of radiating stellar or galactic sources. As we have derived in this paper there exists the following connection between this correlation function and a hierarchy typical mass density

0     =     m 0 ξ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaBaaabaGaaG imaaqabaqbaeaabiqaaaqaaabaaaaaaaaapeGaaiiOaaWdaeaapeGa aiiOaaaapaGaeyypa0tbaeaabiqaaaqaa8qacaGGGcaapaqaa8qaca GGGcaaa8aacaWGTbWaaSbaaeaacaaIWaaabeaacqaH+oaEdaWgaaqa aiaaicdaaeqaaaaa@4163@ (14)

where m 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2gadaWgaa qaaiaaicdaaeqaaaaa@37B0@  denotes the average mass of the hierarchy-typical source, i.e. either stars, or galaxies, or clusters of galaxies. For the spatially largest, observed hierarchies, super clusters, this would give a mass density typical for super clusters given by

ρ SC,0     =         m SC,0 ξ SC,0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaam4uaiaadoeacaGGSaGaaGimaaqabaqbaeaabiqaaaqaaaba aaaaaaaapeGaaiiOaaWdaeaapeGaaiiOaaaapaGaeyypa0tbaeaabi qaaaqaa8qacaGGGcaapaqaa8qacaGGGcaaa8aafaqaaeGabaaabaWd biaacckaa8aabaWdbiaacckaaaWdaiaad2gadaWgaaqaaiaadofaca WGdbGaaiilaiaaicdaaeqaaiabe67a4naaBaaabaGaam4uaiaadoea caGGSaGaaGimaaqabaaaaa@4CA5@ (15)

Since astronomers (Bahcall,6 Bahcall7 and Chokski7) have given there correlation function in absolute numbers, one thus from their results can determine absolute values of hierarchy-typical mass densities, e.g. like the value ρ SC,0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaam4uaiaadoeacaGGSaGaaGimaaqabaaaaa@3ACE@  Figure 1 & Figure 2.

Studying therefore our hierarchical density distributions in Figure 3 we find that only for the case of a positively curved universe, i.e. k=     +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaa a8aacqGHRaWkcaaIXaaaaa@3C18@ , a constant value for the cosmic density at lR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgacqGHsg IRcaWGsbaaaa@3998@ can be expected, namely just the value ρ SC,0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaam4uaiaadoeacaGGSaGaaGimaaqabaaaaa@3ACE@ .In contrast to that result, for Euclidean and negatively curved universes, i.e. k=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpcaaIWaaaaa@3893@ or k=         1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacqGH9a qpfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qacaGGGcaa a8aafaqaaeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdaiabgk HiTiaaigdaaaa@3EB5@ , no finite density is to be expected, but density values that permanently decrease with increasing values of l. But now one has to clearly see the critical point from that outcome - namely that without a finite density value for the density of the universe, one cannot enter and solve the FLRW-field equations.

Conclusion

In the frame of our deductions only one cosmic solution can be envisioned - the one with a positively curved universe and a finite cosmic density of ρ= ρ SC,0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYjabg2 da9iabeg8aYnaaBaaabaGaam4uaiaadoeacaGGSaGaaGimaaqabaaa aa@3D94@ .But in order to logically and scientifically bring things together one has to make sure that a universe with this density ρ SC,0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaam4uaiaadoeacaGGSaGaaGimaaqabaaaaa@3ACE@ is a positively curved one. In order to be in fact positively curved ( k=+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaabaGaam 4Aaiabg2da9iabgUcaRiaaigdaaiaawIcacaGLPaaaaaa@3AFF@ , the density of such a universe has to be larger than it scritical density

ρ c =3 H 0 2 /8πG ρ SC,0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaam4yaaqabaGaeyypa0JaaG4maiaadIeadaqhaaqaaiaaicda aeaacaaIYaaaaiaac+cacaaI4aGaeqiWdaNaam4raiabgsMiJkabeg 8aYnaaBaaabaGaam4uaiaadoeacaGGSaGaaGimaaqabaaaaa@4772@ (16)

In order to decide upon that, one needs to know the Hubble constant Ho MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeacaWGVb aaaa@37A4@ of the present universe and the density value ρ SC,0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaam4uaiaadoeacaGGSaGaaGimaaqabaaaaa@3ACE@ . At this point we have to recognize a special problem for the continuation of the above presented chain of ideas, namely the question concerning the actual value of the Hubble constant Ho MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeacaWGVb aaaa@37A4@ which is extremely essential for our above ideas. The most modern value of this constant, Ho     =73km/s/ M pc MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeacaWGVb qbaeaabiqaaaqaaabaaaaaaaaapeGaaiiOaaWdaeaapeGaaiiOaaaa paGaeyypa0JaaG4naiaaiodacaWGRbGaamyBaiaac+cacaWGZbGaai 4laiaad2eadaWgaaqaaiaadchacaWGJbaabeaaaaa@43DA@ , is derived with the help of the standard _CDM- cosmological model on the basis of a Robertson-Walker spacetime geometry, trying to fit into the cosmological expansion model the cosmic matter evolution from the times of matter recombination (Bennet et al., 2003) to the present times of visible galactic structurings (millennium simulation!, see Springel et al., 2005). The essential finding from these attempts is that the energetic ingredients of this best-fitting universe are: Ω=0.72;ΩD=0.23;ΩB=0.04;Ω=0!, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axjabgE Iizlabg2da9iaaicdacaGGUaGaaG4naiaaikdacaGG7aGaeuyQdCLa amiraiabg2da9iaaicdacaGGUaGaaGOmaiaaiodacaGG7aGaeuyQdC LaamOqaiabg2da9iaaicdacaGGUaGaaGimaiaaisdacaGG7aGaeuyQ dCLaeyikIOTaeyypa0JaaGimaiaacgcacaGGSaaaaa@521F@ where the different Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axbaa@3771@ values are defined by

Ω i = ρ i ρc = ρ i ( 3 H 0 2 8πG ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axnaaBa aabaGaamyAaaqabaGaeyypa0ZaaSaaaeaacqaHbpGCdaWgaaqaaiaa dMgaaeqaaaqaaiabeg8aYjaadogaaaGaeyypa0ZaaSaaaeaacqaHbp GCdaWgaaqaaiaadMgaaeqaaaqaamaabmaabaWaaSaaaeaacaaIZaGa amisamaaDaaabaGaaGimaaqaaiaaikdaaaaabaGaaGioaiabec8aWj aadEeaaaaacaGLOaGaayzkaaaaaaaa@4AF8@ (17)

with i =         ;i=         D;i=         B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaBaaabaGaam yAaaqabaGaeyypa0tbaeaabiqaaaqaaabaaaaaaaaapeGaaiiOaaWd aeaapeGaaiiOaaaapaqbaeaabiqaaaqaa8qacaGGGcaapaqaa8qaca GGGcaaa8aacqGHNis2caGG7aGaaiyAaiabg2da9uaabaqaceaaaeaa peGaaiiOaaWdaeaapeGaaiiOaaaapaqbaeaabiqaaaqaa8qacaGGGc aapaqaa8qacaGGGcaaa8aacaGGebGaai4oaiaacMgacqGH9aqpfaqa aeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdauaabaqaceaaae aapeGaaiiOaaWdaeaapeGaaiiOaaaapaGaaiOqaaaa@5014@ determining the critical density of the vacuum, of the dark matter, and of the baryonic matter, respectively, and ρ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYnaaBa aabaGaam4yaaqabaaaaa@38AC@ denoting the so-called critical cosmic density , critical with respect to the actually prevailing curvature of the universe, and with the validity:

Ω+     ΩD+     ΩB+     Ω+     Ωk=     1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axjabgE IizlabgUcaRuaabaqaceaaaeaaqaaaaaaaaaWdbiaacckaa8aabaWd biaacckaaaWdaiabfM6axjaadseacqGHRaWkfaqaaeGabaaabaWdbi aacckaa8aabaWdbiaacckaaaWdaiabfM6axjaadkeacqGHRaWkfaqa aeGabaaabaWdbiaacckaa8aabaWdbiaacckaaaWdaiabfM6axjabgI IiAlabgUcaRuaabaqaceaaaeaapeGaaiiOaaWdaeaapeGaaiiOaaaa paGaeuyQdCLaam4Aaiabg2da9uaabaqaceaaaeaapeGaaiiOaaWdae aapeGaaiiOaaaapaGaaGymaaaa@55BA@ (18)

Hereby the cosmic curvature energy Ωk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axjaadU gaaaa@3861@ is defined by

Ωk=     1 H 0 2 k c 2 R 0 2 =k c 2 R 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axjaadU gacqGH9aqpfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qa caGGGcaaa8aacqGHsisldaWcaaqaaiaaigdaaeaacaWGibWaa0baae aacaaIWaaabaGaaGOmaaaaaaWaaSaaaeaacaWGRbGaam4yamaaCaaa beqaaiaaikdaaaaabaGaamOuamaaDaaabaGaaGimaaqaaiaaikdaaa aaaiabg2da9iabgkHiTiaadUgadaWcaaqaaiaadogadaahaaqabeaa caaIYaaaaaqaaiaadkfadaqhaaqaaiaaicdaaeaacaaIYaaaaaaaaa a@4C83@ (19)

and, in the above mentioned simulation model, with k = 0, has been set to Ω k =0! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0=Mr0xXdbba91rFfpec8EKe9Vze9Vze9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqqHPoWvdaWgaa qaaiaadUgaaeqaaiabg2da9iaaicdacaGGHaaaaa@3C21@ Also the contribution of cosmic photons for the cosmic times after recombination has been neglected setting Ω =0! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0=Mr0xXdbba91rFfpec8EKe9Vze9Vze9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqqHPoWvdaWgaa qaaiabgIIiAdqabaGaeyypa0JaaGimaiaacgcaaaa@3CE1@ . If on the other hand, however, one would have to admit a positive curvature with k = +1, as e.g. favoured in the argumentation of the upper part of this paper, one would obtain a new cosmic energy balance given by ( see e.g. Goenner, 1996, Fahr, 2017):

Ω+     ΩD+     ΩB+     Ωk=     Ω+     ΩD+     ΩB c 2 R 0 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0=Mr0xXdbba91rFfpec8EKe9Vze9Vze9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqqHPoWvcqGHNi s2cqGHRaWkfaqaaeGabaaabaaeaaaaaaaaa8qacaGGGcaapaqaa8qa caGGGcaaa8aacqqHPoWvcaWGebGaey4kaSsbaeaabiqaaaqaa8qaca GGGcaapaqaa8qacaGGGcaaa8aacqqHPoWvcaWGcbGaey4kaSsbaeaa biqaaaqaa8qacaGGGcaapaqaa8qacaGGGcaaa8aacqqHPoWvcaWGRb Gaeyypa0tbaeaabiqaaaqaa8qacaGGGcaapaqaa8qacaGGGcaaa8aa cqqHPoWvcqGHNis2cqGHRaWkfaqaaeGabaaabaWdbiaacckaa8aaba WdbiaacckaaaWdaiabfM6axjaadseacqGHRaWkfaqaaeGabaaabaWd biaacckaa8aabaWdbiaacckaaaWdaiabfM6axjaadkeacqGHsislda WcaaqaaiaadogadaahaaqabeaacaaIYaaaaaqaaiaadkfadaqhaaqa aiaaicdaaeaacaaIYaaaaaaacqGH9aqpcaaIXaaaaa@654A@ (20)

With the values Ωi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0=Mr0xXdbba91rFfpec8EKe9Vze9Vze9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqqHPoWvcaWGPb aaaa@3999@ found in the millennium fit (Springel et al., 2005) the above requirement evidently would not be fulfilled. Instead one would have to enhance the sum Ωi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaqaeabaGaeu yQdCLaamyAaaqabeqacqGHris5aaaa@3A5B@ of the present values Ωi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axjaadM gaaaa@385F@ by an amount of ( c/ R ˙ 0 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaabaGaam 4yaiaac+caceWGsbGbaiaadaWgaaqaaiaaicdaaeqaaaGaayjkaiaa wMcaamaaCaaabeqaaiaaikdaaaaaaa@3BA0@ , which by the way could be achieved, when keeping the relative contributions of the Ωi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axjaadM gaaaa@385F@ -terms, by lowering the value of the actual Hubble constant H0. This means that , admitting a positive (instead of a vanishing) curvature k, one would either need a completely new fit to the cosmic data, evidently also leading to a new value of the best-fitting Hubble constant H0, or perhaps try the following way: One could perhaps argue that asymptotically ( e.g.         for         Ω1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaabaGaam yzaiaac6cacaWGNbGaaiOlauaabaqaceaaaeaaqaaaaaaaaaWdbiaa cckaa8aabaWdbiaacckaaaWdauaabaqaceaaaeaapeGaaiiOaaWdae aapeGaaiiOaaaapaGaamOzaiaad+gacaWGYbqbaeaabiqaaaqaa8qa caGGGcaapaqaa8qacaGGGcaaa8aafaqaaeGabaaabaWdbiaacckaa8 aabaWdbiaacckaaaWdaiabfM6axjabgEIizlabgkziUkaaigdaaiaa wIcacaGLPaaaaaa@4DB8@ the CDM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0=Mr0xXdbba91rFfpec8EKe9Vze9Vze9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqGHNis2caWGdb Gaamiraiaad2eaaaa@3B2E@ -universe will approach an expansion rate with a "constant" Hubble constant H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaaiisaiabgEIizdaa@387D@ given by

H 2 = 8πGρ 3 =   H 0 2         . Ω 0 =0.72.        H 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbawaabaqaceaaae aaqaaaaaaaaaWdbiaacIeadaqhaaqaaiabgEIizdqaaiaaikdaaaGa eyypa0ZaaSaaaeaacaaI4aGaeqiWdaNaam4raiabeg8aYjabgEIizd qaaiaaiodaaaGaeyypa0dapaqaa8qacaGGGcaaa8aafaqaaeGabaaa baWdbiaacIeadaqhaaqaaiaaicdaaeaacaaIYaaaa8aafaqaaeGaba aabaWdbiaacckaa8aabaWdbiaacckaaaWdauaabaqaceaaaeaapeGa aiiOaaWdaeaapeGaaiiOaaaapaGaaiOlaiabfM6axnaaBaaabaGaey 4jIKTaaGimaaqabaGaeyypa0JaaGimaiaac6cacaaI3aGaaGOmaiaa c6capeGaaiiOaaWdaeaapeGaaiiOaaaapaqbaeaabiqaaaqaa8qaca GGGcaapaqaa8qacaGGGcaaaiaacIeadaqhaaqaaiaaicdaaeaacaaI Yaaaaaaa@5CD5@ (21)

meaning that the asymptotic Hubble constant would be smaller than the presently determined Hubble constant by the factor 0.72, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaakaaabaGaaG imaiaac6cacaaI3aGaaGOmaiaacYcaaeqaaaaa@398C@ i.e. yielding H=0.85 H 0 =62,5.km/s/ M pc MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rs0=Mr0=Mr0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape GaaiisaiabgEIizlabg2da9iaaicdacaGGUaGaaGioaiaaiwdacaWG ibWaaSbaaeaacaaIWaaabeaacqGH9aqpcaaI2aGaaGOmaiaacYcaca aI1aGaaiOlaiaadUgacaWGTbGaai4laiaadohacaGGVaGaamytamaa BaaabaGaamiCaiaadogaaeqaaaaa@49CB@ .

Acknowledgements

None.

Conflict of interest

The author declares there is no conflict of interest.

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