 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 3

Since $P=P(\rho)$ and $\rho=\bar\rho(1+\delta)$,
$$
\frac{c^2}{\rho}\nabla P
=\frac{c^2}{\rho}\frac{dP}{d\rho}\nabla\rho
\simeq c_s^2\nabla\delta.
$$
To first order, continuity and the divergence of Euler give
$$
\dot\delta+\theta=0,\qquad
\dot\theta+2\frac{\dot a}{a}\theta
=-\frac{4\pi G}{c^2}\bar\rho\,\delta-c_s^2\nabla^2\delta.
$$
Eliminating $\theta$ and using $\nabla\mapsto i\mathbf k/a$ yields
$$
\ddot\delta+2\frac{\dot a}{a}\dot\delta
-\left(\frac{4\pi G}{c^2}\bar\rho-\frac{c_s^2k^2}{a^2}\right)\delta=0.
$$
The second term is Hubble damping, gravity drives growth, and <pressure> produces oscillatory restoration. Their balance defines
$$
k_J^2=\frac{4\pi G\bar\rho\,a^2}{c^2c_s^2},
\qquad \lambda_J=\frac{2\pi a}{k_J}.
$$
Modes longer than the Jeans length are gravitationally unstable; shorter modes are <pressure> supported.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 14D / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 14D / a / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 14D / a / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)