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 Incoming links: Conformal Riccati equation for matter and a coasting fluid

Write

Ω = Ω_{M 0}

β^{2} = H_{0}^{2} (1 - Ω)

and

H = a^{'} / a = \overset{a}{˙}

. The Friedmann equation is

H^{2} = H_{0}^{2} (\frac{Ω}{a} + 1 - Ω) .

The Raychaudhuri equation has only the matter component on its active-gravity side, so

H^{'} = a \overset{a}{¨} = - \frac{4 π G}{3} a^{2} ρ_{M} = - \frac{H _{0}^{2} Ω}{2 a} .

Combining these expressions proves

2 H^{'} + H^{2} - β^{2} = 0 .

H_{0}

Put

Ω = Ω_{m, 0}

and

b = 1 - Ω

ρ_{m} = ρ_{m, 0} a^{- 3}, ρ_{X} = ρ_{X, 0} a^{- 2} .

In particular, $ρ_{X} a^{2}$ is constant. At the present epoch the flat Friedmann equation implies

Ω_{X, 0} = b

. Nonnegative fluid densities therefore require

0 \leq Ω \leq 1

d t = a d τ

H = a H

. Consequently

H^{2} = H_{0}^{2} (\frac{Ω}{a} + b), 2 H H^{'} = - H_{0}^{2} \frac{Ω a ^{'}}{a ^{2}} .

On the expanding branch, divide by

H

and use

a^{'} = a H

2 H^{'} = - \frac{H _{0}^{2} Ω}{a} .

2 H^{'} + H^{2} - α^{2} = 0, α = H_{0} 1 - Ω_{m, 0} .

The nonnegative square root fixes the convenient parameter convention; only

α^{2}