الكونيات الكونية التوأمية

cosmology of the twin universe Astrophysics of matter ghost-matter. 1. The geometrical framework. The matter era and the newtonian approximation. (p2)
2) Matter dominated cosmological model.

...Assuming the two universes to be isotropic and homogeneous, the metrics, in spherical coordinates, are :

(17)

(18)

...These two metrics are expressed in the coordinates systems of their own fold. k and k* are the curvature indices.
Introduce adimensional proper times :
(19) s = cT s s* = - cT s

and adimensional scale factors :
(20)

R = cT R

R* = cT R*

the metrics become :

(21)

(22)

where the spatial parts is :
(23)

dh2 = du2 + u2 ( dq2 + sin2 q dj2)

Similarly we can put the field equation system into an adimensional form :
(24)

(25)

i.e :

(26)

with

(27)

where the subscript r refers to radiation and the subscript m to matter.

c (Einstein constant) --> - 8 π

R ---> R

R* ---> R*

r = ro w

r* = ro w

p = po p

p* = po p

{ ro , ro , po , po } being characteritic mass densities and pressures. .In this paper we deal with matter era. We assume that the matter densities and pressures are equal at the the end of the radiative era and write :
(27b)

ro = ro ; po = po

In the matter era we have :
(28)

and the field equations system becomes :
(29)

(30)

The tensors are written in their adimensional forms :

(31)

were (w , w*) are adimensional matter-densities and (p , p*) adimensional matter pressures, all positive. We get the following system of four differential equations :

(32-a)

(33-b)

(32-c)

(32-d)

...If we assume the thermal velocities, in both folds, to be negligible with respect to the velocity of the light, the pressures can be neglected (dust universes). In a first step, when discoupling has just occured, we have w = w* the system becomes: