2.3. Standard cosmological solutions
When k = 0 the Friedmann and fluid equations can readily be solved for the equations of state given earlier, leading to the classic cosmological solutions
In both cases the density falls as t-2. When k
= 0 we have the freedom
to rescale a and it is normally chosen to be unity at the present,
making physical and comoving scales coincide. The proportionality
constants are then fixed by setting the density to be
A more intriguing solution appears for the case of a so-called
cosmological constant, which corresponds to an equation of state
p = -
More complicated solutions can also be found for mixtures of components.
For example, if there is both matter and radiation the Friedmann
equation can be solved be using conformal time
0 at time
t0, where here and throughout the subscript zero
indicates present
value.
. The
fluid equation then gives
= 0 and hence
=
0, leading to
=
dt /
a, while
if there is matter and a non-zero curvature term the solution can be
given either in parametric form using normal time t, or in closed form
with conformal time.