Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 10D / Solution

The Jeans wavenumber in physical coordinates is $k_J=\sqrt{4\pi G\rho }/c_s$. With wavelength defined as $2\pi /k$, the Jeans length isPressure opposes gravitational growth on physical wavelengths shorter than this; on longer wavelengths the gravitational term wins. Expansion also provides the damping term $2H\dot{\delta }$, so the actual growth is not generally the exponential growth of a static medium.

For the pressure-free, spatially flat matter solution, $H=2/(3t)$ and the Friedmann equation gives $\rho =1/(6\pi G{t}^2)$. The perturbation equation becomesSubstituting ${\delta }_k=t^{\nu }$ gives $3{\nu }^2+\nu -2=(3\nu -2)(\nu +1)=0$. ThusThe growing mode scales as $a(t)$ and permits the formation of structure; the decaying mode becomes negligible at late times. This is a linear result, valid while the density contrast remains small.