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96 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
Even if these last two equations do not contain the scale factor
that with other fluids is always present. This fact makes it unnecessary
recourse to a third order differential equation in order to eliminate the
aforementioned term. For Eq.(25), and Eq.(26) obtain
with while while remains free.
The discriminant is
In all, three different cosmic solutions for this FRW flat space
model exist distinguished by the discriminant whose sign depends on the
sign of The solutions have been analyzed in [7]. For Eq.(25)
and Eq.(26) have the following solutions
the first is for and with the second
is for and and for the third, the last is for
and Moreover, when depending on the sign of
for each type of determinant one can get two solutions.
Other particular solutions for are found when
with so that
This last expression can be given in terms of the “cosmic time The
result is a linear expansion
because, from the constriction one gets