Twin universe cosmology Astrophysics of ghost-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 consider the system of two coupled field equations and focus on the radiative era. We assume that R = R*. In order to avoid the trivial solution R » R* » t, we apply a model with variable constants, presented in previous papers. Thus, we obtain a model in which the constants of physics vary during the radiative era, then tend to absolute constants during 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 fundamental clock composed of two masses orbiting around their common center of gravity. Time is identified to the number of turns. We find that our clock has made an infinite number of turns in the past, so that the so-called "origin of the Universe" and the t = 0 point become questionable.
- Introduction
In previous papers ([1] & [2]), we introduced a cosmological model based on a two-fold 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 system of coupled field equations:
(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 geometric 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 scale parameters depart from linear evolution, due to gravitational instability. The expansion of the ghost (twin) universe becomes slower and it lags behind ours, whose expansion accelerates, so that the twin Universe behaves like a "cosmological constant". We assume the decouplings 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", applied both to the radiative and matter eras, but this model introduced different gauge processes for gravitation and electromagnetism. For 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 raise a serious problem for electron-antielectron pair creation-annihilation. In the following, we reconsider this model, applying this concept of variable constants only to the radiative era. 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, as we cannot detect it. Before decoupling, the Universe is optically thick.
- A model with variable constants.
The so-called constants of physics are:
(12) c: light velocity
(13) G: gravitational constant
(14) m: masses (neutral and charged particles)
(15) h: Planck constant
...Other constants, from electromagnetism:
e: electric charge
eo: dielectric constant of vacuum.
...G and c are linked through the Einstein constant:
(16)
...As shown in reference [4], G and c may vary in time if:
(17)
Instead of 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 of two folds (the involutive mapping was described in [1] and [3]). In both cases, we "read" these solutions 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 previous 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 current values, immediately after the radiative era:
(20) Go (gravity), co (light velocity), mo (masses), ho (Planck)
(21) mo, eo (electromagnetic constants)
- How to determine the time evolution of the set of "variable constants".
G(t) and c(t) are coupled through (17) to satisfy the zero divergence condition. Physics depends on a certain set of basic equations (which are not all independent). We assume that the variations of the "constants" of physics, during the radiative era, keep 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.
(Poisson equation for gravity [1]):
(24) D f = 4 p G ( r - r* )
Maxwell equations:
(25)
(26)
(27) Ñ . 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 dimensionless form, considering that the constants can vary. We introduce a length scale factor R and a time scale factor T.
(31)
...In the Schrödinger equation, we can write:
(32)
The 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 the 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)
...The Boltzmann equation becomes:
(39)
Its invariance will 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)
