**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 the case of a so-called
cosmological constant, which corresponds to an equation of state
*p* = - . The
fluid equation then gives
= 0 and hence
=
_{0}, leading to

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
=
*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.