twin universe cosmology Astrophysics of ghost-matter. 1. 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 coordinate systems of their own folds. k and k* are the curvature indices.
Introduce dimensionless proper times :
(19) s = cT s s* = - cT s
and dimensionless scale factors :
(20)
R = cT R
R* = cT R*
The metrics become :
(21)
(22)
where the spatial part is :
(23)
dh² = du² + u² ( dq² + sin² q dj²)
Similarly, we can express the field equation system in a dimensionless form :
(24)
(25)
that is :
(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 characteristic mass densities and pressures. In this paper, we deal with the matter era. We assume that the matter densities and pressures are equal at 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 dimensionless forms :
(31)
where (w , w*) are dimensionless matter densities and (p , p*) dimensionless 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 light, the pressures can be neglected (dust universes). In a first step, just after decoupling, we have w = w*, the system becomes: