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cosmology of the twin universe astrophysics of ghost matter-matter. 3 : The radiative era : The problem of the "origin" of the universe. The problem of the homogeneity of the early universe.(p2)

**Astrophysics of ghost (twin) matter
3 : The radiative era : **

The problem of the "origin" of the Universe
The problem of the homogeneity of the early Universe

J.P.Petit & P.Midy Observatory of France - Centre de calcul d'Orsay France

Abstract :

We take the system of two coupled field equations and focuss on radiative era. We assume that R = R* . In order to avoid the trivial solution R » R* » t we apply a the variable constant model, presented in former papers. Then we get a model in which the constants of physics vary during the radiative era, then tend to absolute constants over the matter era. During the radiative era the entropy per baryon is no longer constant. The horizon varies like R, so that the homogeneity of the Universe is ensured at any time in the past : Inflation Theory is no longer necessary. We introduce a basic clock, composed by two masses orbiting around their common centre of gravity. Time is identified to the number of turns. We find that our clock made an infinite number of turns in the past so that the so-called "origin of the Universe, and t = 0 point" become questionable.

1. Introduction

In former papers ( [1] & [2] ) we have introduced a cosmological model based on a two-folds cover of a manifold (or on a two-points bundle of a M4 manifold, which is equivalent). We assumed it was governed by the following coupled field equations system :

(1)

S = c ( T - T* )

(2)

S* = c ( T* - T )

with :

(3)

T = Tr + Tm

(4)

T* = Tr* + Tm*

Obviously : (5)

S* = - S

where S and S* are geometrical tensors. The index m refers to matter while the index r refers to radiation.

Fig.1 : **The joint evolution of matter and ghost (twin) matter. **

On figure 1 we see that the two scales parameters depart from linear evolution, due to gravitational instability. The expansion of the ghost (twin) universe becomes slower and it pshed ours, whose expansion accelerates, so that the twin Universe behaves like a "cosmological constant". We assume discouplings between matter and radiation occur at the same moment in both Universes. In addition we assume that, during the radiative era :

(8)

R = R*.............. p = p*.............. r = r*

In references ( [4] ,[5] , and [6] ) we developed a model with"variable constants", applying both to radiative and matter eras, but this model introduced different gauge processes for gravitation and electromagnetism. Fora example, the mass was found to follow :

(8)

m » R

while the electric charge follows :

(9)

The Rydberg constant (ionization energy of the hydrogen atom) obeys :

(10)

Ei » R

which gives the redshift. The Jeans and Schwarzschild lengths vary like R while the Bohr radius was found to obey :

(11)

which, as noticed later by colleagues, would arise a severe problem for electron anti-electron pairs creation-annihilation. In the following we reconsider this model, applying this concept of the variable constants to radiative era only. Then, during the matter era the constants behave like absolute constants. We have no redshift on photons emitted before the radiative era, which is not a problem, for we cannot evidence it. Before discoupling the Universe is optically thick.

2. A model with variable constants.

The so-called constants of physics are :

(12) c : light velocity

(13) G : constant of gravity

(14) m : masses (neutral and charged particles)

(15) h : Planck constant

...Plus other constants, from electromagnetism :

e : electric charge

eo : dielectric constant of vacuum.

...G and c are linked through Einstein constant :

(16)

...As shown in reference [4] G and c may vary in time if :

(17)

Instead writing :

(18) x° = co t

where co is an absolute constant, we may write :

(19) x° = c(t) t

...A solution of the Einstein equation is an hypersurface. A solution of our field equations system is an hypersurface composed by two folds (the involutive mapping was described in [1] and [3]). In both cases we "read" these solution through an arbitrary choice of coordinates, where r is identified to a radial distance and t to cosmic time. The choice (19) must fit the matter dominated era solution (from the former paper [2]). It is possible if our "variable constants"c(t) , G(t) , h(t), m (t), e(t) , eo(t) tend rapidly to their today’s values, immediately after radiative era :

(20) Go (gravity) , co (light velocity) , mo (masses), ho ,(Planck)

(21) mo, eo (electromagnetic constants)

3) How to determine the time evolution of "variable constants" set.

G(t) and c(t) are coupled through (17) to fit the zero divergence condition. Physics depends on a certain set of basic equations (which are not all independant). We assume that the variations of the "constants" of physics, during the radiative era keeps all these equations invariant.

Schrödinger equation :

(22)

Boltzmann equation :

(23)

where f is the distribution function of the velocity v , of the position r = (x,y,z), t the time, (g, a, w) the classical impact parameters of a binary collision.

(new) Poisson equation for gravitation [1] :

(24) D f = 4 p G ( r - r*)

Maxwell equations :

(25)

(26)

(27) ї . B = 0

(28)

(29)

where re is the electric charge density and Q the cross-section :

(30)

is the mean thermal electron velocity.

...We put all these equations into a generalized adimensional form, considering that the constants can vary. We introduce length scale factor R and time scale factor T .

(31)

...In Schödinger equation, we can write :

(32)

Schrödinger equation becomes :

(34)

Its invariance will be ensured if :

(35)

where h , m , R , T are treated as variable quantities.

...For the Boltzmann equation, we write :

(36) v = c z..... r = R x..... g = c g .....a = R a

and :

(37)

In Boltzmann equation there is a force term, defined as the gradient of a potential f. Writing :

(38)

(we assume that the number of species is conserved)

...Boltzmann equation becomes :

(39)

Its invariance is be ensured if :

(40)

which mixes the space scale factor R, the time scale factor T and the "variable constants" G , m and c . We get :

(41) R » c T

and

(42)