Repulsive dark matter (p3)
...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 (zero-pressure
dust universes model). In a first step, when decoupling has just occured, we
assume r = r*
and the system becomes :
(12-a)
(12- b)
(12- c)
(12- d)
...From
(12-b) and (12-d) the curvature indices k and k* must be negative. The evolution
laws are simply linear : R = R* » t
...After
that short period (the very begining of the matter dominated era), we return
to the system (11-a) to (11-d), with curvature indixes k = k* = -1 :
(13-a)
(13- b)
(13- c)
(13- d)
from which we get immediately
:
(14-a)
(14-b)
Introducing the mass-conservation in both folds :
(15)
r R3
= r*
R*3 = ro
Ro3 = constant
ro
and Ro being characteristic quantities. Let :
(16)
(17)
Then the system becomes
:
(18-a)
(18-b)
...Notice
that R = R* gives R" = R*" = 0. On another hand, if the two universes were
"fully coupled", i.e. R*/R = constant, then they would correspond to Friedmann
models, with "parallel evolutions". But we consider that they are coupled
by the gravitational field, through (18-a) and (18-b) that show that the linear
expansion is unstable. If, for example R > R* then R" > 0 and R* " < 0
. The system can be solved numerically. The typical solution corresponds to
figure 5.
.
Fig.
5 :The
evolution of the scale parameters
of the universe and
dark universe.
...With
respect to the classical Friedmann model, this new model enlarges the age
of the universe. Depends on the retained value of the Hubble constant Ho.
In 1993 measurements from Hubble telescope enlarged the constant from 45 to
70 km/s/Mpc which, based on the Einstein de Sitter model with zero cosmological
constant, gave an age of the universe lying between 8 and 9 billions yrs,
and arose a sever problem with respect to the age of the oldest stars of our
galaxy. But more recent calibration of cepheids, from Hipparcos satellite
measurements, lowered Ho . At the same time some people tended to low the
estimated age of the oldest stars of our galaxy.
...Notice
that the repulsive dark matter plays the role of a "cosmological constant",
for it gives a positive acceleration R" in our fold.
Anyway, the discussion
about the age of the universe is not ended.
