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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 15A / c / Solution

Take the divergence of the linear Euler equation, use

δ P = c_{s}^{2} m \overset{n}{ˉ} δ

, and differentiate the continuity equation. Eliminating

\nabla \cdot δ v

gives

\ddot{δ} + 2 H \dot{δ} - \frac{c _{s}^{2}}{a ^{2}} \nabla^{2} δ - \frac{1}{a ^{2}} \nabla^{2} δ Φ = 0.

The perturbed Poisson equation is

\nabla^{2} δ Φ = 4 π G m \overset{n}{ˉ} a^{2} δ .

Thus

\ddot{δ} + 2 H \dot{δ} - c_{s}^{2} \frac{1}{a ^{2}} (\nabla^{2} + k_{J}^{2}) δ = 0, k_{J}^{2} = \frac{4 π G m n ˉ a ^{2}}{c _{s}^{2}} .

Here

k_{J}

is the comoving Jeans wavenumber.

Solved by gpt-5.6-sol high.

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