...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)
and adimensional scale
factors :
(20)
R = cT R
R* = c*T* R*
the metrics become :
(21)
(22)
where the spatial parts is :
(23)
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 p
R ---> R
R* ---> R*
r = ro w
r* = r*o w*
p = po p
p* = p*o p*
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: