**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
_{0} at time
*t*_{0}, where here and throughout the subscript zero
indicates present value.

A more intriguing solution appears for domination by the cosmological constant, namely

This is equivalent to the solution for a fluid with equation of
state *p*_{}
= -_{}.
The fluid equation then gives
_{} = 0
and hence _{}
/
8*G*.

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 using conformal time
=
*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.