Setting the energy density ε equal to zero is one way of simplifying the
Friedmann equation. Another way is to set κ = 0 and to demand that the
universe contain only a single component, with a single value of w. In such a
spatially flat, single-component universe, the Friedmann equation takes the
simple form
˙a 2 =
8πGε 0
3c 2
a −(1+3w) . (5.39)
To solve this equation, we first make the educated guess that the scale factor
has the power law form a ∝ t q . The left hand side of equation (5.39) is then
∝ t 2q−2 , and the right hand side is ∝ t −(1+3w)q , yielding the solution
q =
2
3 + 3w
,