Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 14D / i

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- Solution i

Solution

Expand the density contrast in Fourier modes,

δ (x, t) = \int δ_{k} (t) e^{i k \cdot x} d^{3} k

. In the static universe each mode satisfies

\ddot{δ}_{k} + c_{s}^{2} (k^{2} - k_{J}^{2}) δ_{k} = 0.

For

k > k_{J}

there are sound waves

δ_{k} = A_{k} e^{i ω_{k} t} + B_{k} e^{- i ω_{k} t}, ω_{k} = c_{s} k^{2} - k_{J}^{2} .

For

k < k_{J}

, Jeans instability gives growing and decaying exponentials with rate

c_{s} k_{J}^{2} - k^{2}

. At the Jeans wavenumber the two solutions are constant and linear in time.

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