\overset{ρ}{˙}_{m} + 3 H ρ_{m} = 0

. Integrating gives

ρ_{m} = ρ_{m, 0} (a_{0} / a)^{3}

ρ_{crit, 0} = \frac{3 H _{0}^{2}}{8 π G}, Ω_{m, 0} = \frac{ρ _{m, 0}}{ρ _{crit, 0}}, Ω_{Λ, 0} = \frac{Λ}{3 H _{0}^{2}}, Ω_{k, 0} = - \frac{k}{H _{0}^{2} a _{0}^{2}} .

Here

c = 1

and the curvature contribution is an effective density parameter, not an additional matter fluid. Substitution into the Friedmann equation gives

H^{2} = H_{0}^{2} [Ω_{m, 0} (\frac{a _{0}}{a})^{3} + Ω_{k, 0} (\frac{a _{0}}{a})^{2} + Ω_{Λ, 0}] .

H = \overset{a}{˙} / a

therefore yields

\overset{a}{˙}^{2} = H_{0}^{2} (\frac{Ω _{m, 0} a _{0}^{3}}{a} + Ω_{k, 0} a_{0}^{2} + Ω_{Λ, 0} a^{2}) .

a = a_{0}

the same equation implies

Ω_{m, 0} + Ω_{k, 0} + Ω_{Λ, 0} = 1

. The power in the matter term is

a_{0}^{3}

; the converted TeX's

a_{0}^{2}

is an error not present in the original PDF.

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