User:Pao.christ/sandbox4

Fundamental cosmological information has come from the Hubble parameter whose present value $$H_{0}$$ can be used to determine the approximate age of the Universe. Indeed, since $$[1/H_{0} ] = [t] $$ the first obvious candidate is R/c  where R stands for the dimension of the visible Universe  and c for the velocity of light. Hence

$$  t_{U} = \frac{1}{H_{0}}  = \frac{R}{c} $$

Numerically, with the value $$R\simeq 10^{26} $$ it yields $$ t_{U} \simeq .3 \times 10^{18} $$ But, interestingly there is another quantity which, to the best of our knowledge, has not been considered so far with the same dimensions $$ t'_{U} = \frac{GM}{c^3} $$ which yields the same result for $$M= N m_{N}$$, where $$m_{N}$$ stands for the nucleon mass  and $$N\simeq 10^{80}$$  for the nucleon number. These two figures are  taken from . The two values  coincide as they should. In that case one immediately obtains

$$ t_{U} = \frac{R}{c} =  t'_{U}  = \frac{GM}{c^3} $$

i. e.

$$\epsilon = \frac{GM}{c^2R }=1 $$

the well known black hole (b.h.) condition which supports the suggestion by. The preceding equation embodies the  striking relation between the age of the Universe and its mass content. Now tautologically in the past  the age must have been smaller and this implies that the (visible) mass must have been less or in other words that  there has been matter creation. Restraining for simplicity to the matter dominated regime i.e. the post CMB times (where $$R \simeq 10^{23})$$ we have the situation represented in Fig.1)

Thus the linear relation between mass and radius can be regarded as almost model independent. Indeed it is also valid for Planck quantities

$$ \frac{GM_{P}}{c^2R_{P} }=1 $$

so that $$\epsilon$$ must be constant in times as proved in. The description of the Universe evolution as a gigantic and evolving black hole (b.h.), has encountered many objections. In particular how is it possible the development to the present unconventional one with a big mass in a large volume obeying however the same relation $$ M/R = c^2/G$$ of a conventional tiny and very dense one i.e. the Planck where gravitation and Q.M. succesfully combine. In fact this condition which essentially corresponds to energy conservation does not correspond to a minimum in energy. Indeed if we allow a perturbation in R at the Planck era

$$ R \to R +dR $$

we cannot have shrinking with a radius smaller than the Planck one and a bigger one naturally entails a correspondingly increase in the mass. The same argument also holds true also for later times even if a smaller radius cannot be discarded in principle. However the same observation remains valid : no restoring force ! Consider indeed the radiation dominated era where the mass is given by {xrist3} $$ M \simeq (kT)^4 R^3 $$

Of course in principle both possibilities exist i.e. increase and decrease in R with constant $$M/R \simeq (kT)^4 R^2$$. In the first case M/R remains constant at the price of a decreasing temperature ($$(KT)^2 \simeq 1/R $$ ) which is what is actually observed. In the second case the opposite should happen in contrast to actuality.

The possibility of perturbing to a smaller radius (at constant mass) would result in energy violation since the negative self energy would overcompensate  the mass. In other words  again only a bigger radius is possible (smaller self energy) and mass creation is demanded to restore energy balance.

In the case of the matter dominated era even if photons are a very small fraction of nucleons the above argument remains true. A contraction would decrease the photon wavelength (anti CMB) and this implies that also for nucleons expansion is the only possibility. Thus the particle mass content simply increases. Thus expansion in the radiation dominated era would correspond, loosely speaking, to the Boltzmann thermal death whereas in the matter dominated era (where nucleons are non relativistic) the negative heat capacity would allow the birth of new structures. In the

In the b.h. model where  $$\epsilon$$must be constant in time as proved in  \cite{xrist2}    the mass variation required by Eq. (2) has therefore another fundamental effect in the equations of motion

$$ d\epsilon = 0 = - \frac{GM}{R^2}+ \frac{G dM}{RdR} $$

where the first term represents the well known Newtonian acceleration counterbalanced by the second one, due to mass variation. So '''self energy  is seen to provide the repulsive  force  since it increases  the total energy when particles move away and thus demands matter creation. This is the missing dark energy at present represented by the cosmological constant.'''

Consider now the density given in the b.h. model by the density

\rho_{b.h.} = \frac{3}{4 \pi G} \frac{c^2}{r^2}

which,in line with the previous arguments, reads

$$\rho_{b.h.} = \frac{3 H^2}{4 \pi  G}$$

This has to be compared with the critical density of the standard GR treatment in flat space

$$ \rho_{cr} = \frac{3 H^2}{8 \pi  G}$$

$$ rho_{b.h.} = 1/ 2 \rho_{cr} $$

This represents probably a rather unexpected result in the sense, first, that it seems to suggest that a sort of black hole description is contained also in a particular GR formulation, but with a numerical difference. This point  can be understood by remembering that (probably inspired  by a non relativistic origin) $$H^2$$ appears in GR with the coefficient 1/2 and that in the given case (without the cosmological constant)  also GR describes the same situation of the black hole model i.e. indefinite expansion consistent with energy conservation determined only by the density. Of course the difference between the two theories lies in the acceleration equation where  the mass variation, necessary to account mainly for the time dependent age of the Universe provides the repulsive agent. The halved $$\rho$$ has implications for the amount of the presumed dark energy in that it proves how this quantity  be model dependent. In the b.h. model there is no critical density. The given one, smaller than the GR's, is just of the right amount predicted by the model and the expansion, accompanied by matter creation and density decrease in time,  happens independent of the Universe curvature. That must have evolved becoming flatter and flatter from the Planck radius to the present one ($$ 1/R_{c}^2 \simeq G\rho/c^2, R_{C}$$  standing for the curvature radius) i.e. a Universe in the matter era essentially flat. A totally different scenario than the GR one, which as regards curvature probably suffers from being an essentially static, matter conserving, one. This also determines the fate of the Universe : no big crunch, nor big bounce but a density decrease which might anyway foresee the possibility of more diluted structure formation due to the negative heat capacity of gravitation. So an innocent looking, unassuming relation turns out to produce two equations which reproduce and correct the cherished GR ones in the Friedman's metric without the epicycle add-ons criticized by Perlmutter \cite {perl}

The possibility of perturbing to a smaller radius (at constant mass) would result in energy violation since the negative self energy would overcompensate the mass. In other words again only a bigger radius is possible (smaller self energy) and mass creation is demanded to restore energy balance.

In the case of the matter dominated era even if photons are a very small fraction of nucleons the above argument remains true. A contraction would decrease the photon wavelength (anti CMB) and this implies that also for nucleons expansion is the only possibility. Thus the particle mass content simply increases. The expansion in the radiation dominated era would correspond, loosely speaking, to the Boltzmann thermal death whereas in the matter dominated era (where nucleons are non relativistic}) the negative heat capacity would allow the birth of structures. THe mass variation variation required by Eq.[3] has another fundamental effect in the equations of motion

$$ d\epsilon = 0 = - \frac{GM}{R^2}+ \frac{G dM}{RdR} $$

where the first term represents the well known Newtonian acceleration counterbalanced by the second one, due to mass variation, the same mass variation which determines  the Hubble time  in  Eq. [3]. This term, additionally justified because the cosmological potential is not a state function,  which predicts  a steady expansion  represents the "misterious force"  (dark energy) which balances gravitational attraction. So self energy is seen to provide the repulsive  force since it increases  the total energy when particles move away. This  shows the misleading parallelism with Newtonian treatment which is seen to represent correctly only  a local description of gravitation and the inadequacy of GR and therefore of  the Lambda CDM model to account for reality. In this connection it is worth stressing that the reported supernovae acceleration is a model dependent effect existing only in the standard treatment and disappearing in the present approach. The (im)possibility of detecting matter non conservation in present times has already been considered in. The new term  in Eq.[6] may be related at present to the GR "cosmological constant" as a vacuum (v) density of the order of $$ \rho_{v} \simeq \frac{M}{R^3} \simeq 10^{-25}  $$ to be compared to the same quantity at Planck (P) times $$\rho^P_{v} \simeq 10^{97} $$ with the notorious ratio of $$\simeq 10^{120}$$,. Of course at earlier times we would have different "cosmological constants".

Of course in principle both possibilities exist i.e. increase and decrease in R with constant $$M/R  \simeq (kT)^4 R^2$$. In the first case M/R remains constant at the price of a decreasing temperature ( $$($KT)^2 \simeq 1/R  $$  ) which is what is actually observed. In the second case the opposite should happen in contrast to actuality.

The possibility of perturbing to a smaller radius (at constant mass) would result in energy violation since the negative self energy would overcompensate  the mass. In other words  again only a bigger radius is possible (smaller self energy) and mass creation is demanded to restore energy balance.

In the case of the matter dominated era even if photons are a very small fraction of nucleons the above argument remains true. A contraction would decrease the photon wavelength (anti CMB) and this implies that also for nucleons expansion is the only possibility. Thus the particle mass content simply increases. Thus expansion in the radiation dominated era would correspond, loosely speaking, to the Boltzmann thermal death whereas in the matter dominated era (where ( nucleons are non relativistic ) the negative heat capacity would allow the birth of structures.

Consider now the density given in the b.h. model by $$ \rho_{b.h.} = \frac{3}{4 \pi G} \frac{c^2}{r^2}$$

which, in line with the previous arguments, reads

$$\rho_{b.h.} = \frac{3 H^2}{4 \pi  G}$$

This has to be compared with the critical density of the standard GR treatment in flat space

$$ \rho_{cr} = \frac{3 H^2}{8 \pi  G} $$ $$ \rho_{b.h.} = 1/ 2 \rho_{cr} $$ This represents probably a rather unexpected result in the sense, first, that it seems to suggest that a sort of black hole description is contained also in a particular GR formulation, but with a numerical difference. This point  can be understood by remembering that (probably inspired  by a non relativistic origin) $$H^2$$ appears in GR with the coefficient $1/2$ and that in the given case (without the cosmological constant)  also GR describes the same situation of the black hole model i.e. indefinite expansion consistent with energy conservation determined only by the density. Of course the difference between the two theories lies in the acceleration equation where  the mass variation, necessary to account mainly for the time dependent age of the Universe provides the repulsive agent. The halved $\rho$ has implications for the amount of the presumed dark energy in that it proves how this quantity  be model dependent. In the b.h. model there is no critical density. The given one, smaller than the GR's, is just of the right amount predicted by the model and the expansion, accompanied by matter creation and density decrease in time,  happens independent of the Universe curvature. That must have evolved becoming flatter and flatter from the Planck radius to the present one ($$ 1/R_{c}^2 \simeq G\rho/c^2, R_{C}$$  standing for the curvature radius) i.e. a Universe in the matter era essentially flat. A totally different scenario than the GR one, which as regards curvature probably suffers from being an essentially static, matter conserving, one. This also determines the fate of the Universe : no big crunch, nor big bounce but a density decrease which might anyway foresee the possibility of more diluted structure formation due to the negative heat capacity of gravitation. So an innocent looking, unassuming relation turns out to produce two equations which reproduce and correct the cherished GR ones in the Friedman's metric without the epicycle add-ons criticized by Perlmutter \cite {perl}

\