Past exam of the astrophysics course of the University of Oxford / 2018 / 2

Expand the density contrast in spatial Fourier modes using fixed comoving coordinates:The spatial Laplacian multiplies a mode by $-k^2$, while its comoving wavenumber $k=|\mathbf{k}|$ is time-independent. Writing $H=\dot{a}/a$, the linear cosmological density perturbation equation becomesThe proper wavenumber is $k^{(p)}=k/a$, since the proper wavelength is $2\pi a/k$. The proper Jeans wavenumber balances the pressure and gravity terms:

For $k^{(p)}\ll k_J^{(p)}$, neglect pressure support. The growing and decaying Jeans instability modes obeyIn an Einstein-de Sitter universe, $a\propto t^{2/3}$ gives $H=2/(3t)$, and the matter-density relation $8\pi G\bar{\rho }=3H^2$ gives $4\pi G\bar{\rho }=2/(3t^2)$. ThusTry a power law ${\delta }_{\mathbf{k}}\propto t^p$. Its characteristic equation isTherefore the Einstein-de Sitter density-growth modes areThis is a linear result while $|\delta |\ll 1$; a growing mode eventually leaves that regime.

In the formal static-background limit, $H=0$ and $\bar{\rho }$ is constant. The gravity-dominated linear cosmological density perturbation equation reduces toThe Static-universe Jeans modes areThe unstable perturbation grows exponentially rather than as $t^{2/3}$. In the expanding case, the $2H\dot{\delta }$ term resists peculiar motion and the falling background density also lengthens the gravitational growth timescale. A globally homogeneous static self-gravitating fluid is not obtained merely by turning off expansion; this calculation is the conventional local Jeans-background idealization, with the uniform background force removed or otherwise supported.

When $k^{(p)}\gg k_J^{(p)}$, omit the gravitational term and define $\omega (t)=c_s(t)k/a(t)$. ThenThis is an acoustic damped harmonic oscillator with time-dependent coefficients. Freezing the background locally at $t_0$, its amplitude damping rate is $H_0$, soThe oscillation-averaged quadratic oscillator energy decays twice as fast in this frozen-frequency model, with time $1/(2H_0)$. Stating which quantity is damped avoids a factor-of-two convention error. Locally the solution isFor evolution over many expansion times, retain the varying frequency. Put ${\delta }_{\mathbf{k}}=\chi /a$. Direct substitution givesWhen $\omega \gg H$ and $|\dot{\omega }|\ll {\omega }^2$, the WKB approximation yieldsThese are rapidly oscillating sound waves in comoving density, diluted by the expanding background. For constant $c_s$ their envelope scales as $a^{-1/2}$, not universally as $a^{-1}$, because $\omega \propto a^{-1}$. In the Einstein-de Sitter universe, introducing conformal time $d\eta =dt/a$ gives the exact constant-$c_s$ pressure-only solution ${\delta }_{\mathbf{k}}=[C\sin (k{c}_s\eta )+D\cos (k{c}_s\eta )]/\eta$, consistent with $a\propto {\eta }^2$. For that background the frozen-coefficient damping time is $3t/2$, while the slowly varying envelope has instantaneous decay time $2/H$.