Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 66 / 1 / ii

Table of contents
- Solution ii

Solution

With constant equation-of-state parameter

w

, the cosmological perfect-fluid continuity equation becomes

\overset{ρ}{˙} + 3 (1 + w) H ρ = 0

. Consequently

\frac{d}{d t} (ρ a^{3 (1 + w)}) = a^{3 (1 + w)} [\overset{ρ}{˙} + 3 (1 + w) H ρ] = 0, ρ a^{3 (1 + w)} = C .

The constant-equation-of-state density scaling includes

ρ \propto a^{- 3}

for pressureless matter and

ρ \propto a^{- 4}

for radiation in cosmology.

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