Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 37E / a

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- Solution a

Solution

The cosmological perfect-fluid continuity equation and the constant barotropic equation of state

P = wρ

give

\overset{ρ}{˙} = - 3 (1 + w) \frac{a ˙}{a} ρ .

On every interval where

a

is monotone, separation of variables yields

\frac{d ρ}{ρ} = - 3 (1 + w) \frac{d a}{a} .

Hence the constant-equation-of-state density scaling is

ρ = \frac{3 μ}{8 π} a^{- 3 (1 + w)},

where

μ

is a constant. The same expression extends through a regular turning point by continuity.

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