Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 / 4 / i

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- Solution i

Solution

Linearizing the cosmological Euler equation gives

v^{'} + H v = - \nabla ϕ .

Taking the curl removes the gradient force, so the vorticity

w = \nabla \times v

obeys

w^{'} + H w = 0 .

Since

H = a^{'} / a

, its solution is

w \propto a^{- 1} .

Thus expansion dilutes linear vorticity; in an Einstein-de Sitter universe,

a \propto τ^{2}

and

w \propto τ^{- 2}

.

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