For a <barotropic equation of state>, $\delta P=c_s^2\delta\rho=c_s^2\bar\rho\,\delta$. A spatial <Fourier mode> of comoving wavenumber $k$ therefore satisfies
$$
\ddot\delta_k+2H\dot\delta_k
+\left(\frac{c_s^2k^2}{a^2}-4\pi G\bar\rho\right)\delta_k=0.
$$
The <Jeans wavenumber> is defined by equality of the pressure and self-gravity terms:
$$
\boxed{k_J=\frac{a\sqrt{4\pi G\bar\rho}}{c_s}}.
$$
For $k\gg k_J$, pressure dominates and produces acoustic oscillations whose amplitude is affected by <Hubble friction>. For $k\ll k_J$, self-gravity dominates and a growing mode develops: this is <Jeans instability>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 4 / c / Solution

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Jeans instability is the gravitational growth of density perturbations whose self-gravity overcomes pressure support. For physical sound speed $c_s$, the comoving threshold is
$$
k_J=\frac{a\sqrt{4\pi G\bar\rho}}{c_s}.
$$
Modes with $k\ll k_J$ grow gravitationally, whereas modes with $k\gg k_J$ undergo pressure-supported acoustic oscillations.


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