\overset{ρ}{˙} + 3 H (ρ + p) = 0.

w = p / ρ

, integration gives

ρ_{i} (a) = ρ_{i, 0} a^{- 3 (1 + w_{i})} = ρ_{i, 0} (1 + z)^{3 (1 + w_{i})},

where

a_{0} = 1

Ω_{i, 0} = \frac{ρ _{i, 0}}{ρ _{c, 0}}, ρ_{c, 0} = \frac{3 H _{0}^{2}}{8 π G} .

H (z) = H_{0} E (z), E (z) = [i \sum Ω_{i, 0} (1 + z)^{3 (1 + w_{i})}]^{1/2} .

Spatial curvature may be included as an effective

w = - 1/3

component. Once the parameters are specified, the scale factor follows from the first-order equation

\overset{a}{˙} = a H (a)

, or equivalently

t (a) - t (a_{i}) = \int_{a_{i}}^{a} \frac{d a ^{'}}{a ^{'} H ( a ^{'} )} .

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