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162 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
(4). For this Bianchi model it implies, by imposing the condition that
are constants, severe algebraic constraints on the scale factors, so it
seems more likely that are functions. This explains why no totally
anisotropic solution has been found yet. In Ref. [7] we
have analyzed the case when the polynomial solution for is valid. In
this case, unfortunately, we could not found explicitly the values of the
constants If this solution is valid, however, one has that
where D, F, G are constants. Accordingly,
Eq. (9) indicates that the solution must tend, as time evolves, to the
positive curvature FRW solution, i.e. one has again that as
However, a definitive answer will arrive by obtaining explicitly
the values of
4. CONCLUSIONS
We have presented a set of differential equations written in rescaled
variables that are valid for Bianchi type I, V and IX models, as well as for
FRW models within general scalar–tensor theories, including Dehnen’s
IG theory. It seems to be very difficult to solve analytically this system
given by Eqs. (1), (4) and (6). However, for the special case in which the
potential vanishes, or when matter terms dominate over the potential
[9], that is for the BD case, one is able to find solutions. Accordingly,
one can integrate Eq. (10) for the cases of FRW models, and Bianchi
type I and V models, because in these cases the anisotropic parameters
are constants.
The solutions discussed here are:
(i) A particular solution for non-flat FRW models. We have found a
new solution of Eq. (10) valid for curved FRW cosmologies, that
is, a solution with This is the general solution subject to some
constraint that permits one to have two important physical cases: the
case when having some interest in string cosmology [17], that
implies an equation of state of a quasi dust model and the case
when consistent with current BD local experimental constraints
[18], implying that The new solution is non-inflationary and
for asymptotic times is of power-law type.
(ii) The general solution of Bianchi type I model and a particular
solution of type V; both solutions are quadratic polynomials. These
solutions let the models isotropize as time evolves, however, this can
happens only for some parameter range. The polynomial solution
may also be valid for Bianchi type IX, but it is not proved yet. If it
were, isotropization would be also guaranteed.