Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 71 / 3 / b

Table of contents
- Solution b

Solution

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 .

This is the conformal Riccati equation for matter and a coasting fluid. The prime denotes conformal time, whereas

H_{0}

is the physical present Hubble parameter.

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