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158 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
where the are some unknown functions of that determine the
anisotropic character of the solutions. Furthermore, Bianchi models
obey the condition
to demand consistency with Eq. (7). For the Bianchi type V model one
has additionally that since as mentioned above. For
FRW models one has that
In order to analyze the anisotropic character of the solutions, we consider
the anisotropic shear,
is a necessary condition to obtain a FRW cosmology since it
implies cf. Ref. [7, 15]. If the sum of the squared
differences of the Hubble expansion rates tends to zero, it would mean
that the anisotropic scale factors tend to a single function of time which
is, certainly, the scale factor of a FRW solution.
The dimensionless, anisotropic shear parameter [16] becomes,
using Eqs. (7) and (8):
If the above equations admit solutions such that as
then one has time asymptotic isotropization solutions, similar
to solutions found for the Bianchi models in GR [4]. In fact, one does
not need to impose an asymptotic, infinity condition, but just that
where is yet some arbitrary value to warrant that can be
bounded from above.
3. SOLUTIONS AND ASYMPTOTIC
BEHAVIOR IN BD THEORY
The problem to find solutions of Bianchi and FRW models in the
scalar-tensor theories is, firstly, to solve Eqs. (4) and (6), and, secondly,
to solve for each scale factor through Eq. (1). This task, is very
complicated to achieve analytically, therefore this work have to be done
numerically. However, we observe from Eq. (6) that if then this
equation canbe once integrated to get that
where is an integration constant. This equation permit us to know the
average Hubble rate in terms of and substituting it into Eq. (7), gives
us each of the Hubble rates. The point that is left is to have solutions