Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 10D (Cosmology)

The physical Jeans length is the wavelength for which pressure and gravity balance:The physical wavelength of the comoving mode is $2\pi a/k$. Modes with wavelength larger than ${\lambda }_J$ have a gravitationally destabilizing coefficient and can grow; shorter wavelengths are pressure-supported and oscillate, with expansion also damping the motion. A time-dependent background makes the exact growth history depend on the epoch, not just an instantaneous sign. For pressureless matter the Jeans length vanishes.

For an Einstein-de Sitter universe, $H=\dot{a}/a=2/(3t)$. The flat matter-only Friedmann equation $H^2=8\pi G{\rho }_m/3$ gives ${\rho }_m=1/(6\pi G{t}^2)$. With zero sound speed the perturbation equation is thereforeSubstituting $t^s$ yields $s(s-1)+4s/3-2/3=0$, or $(3s-2)(s+1)=0$. Hencewith growing mode proportional to the scale factor and decaying mode proportional to $t^{-1}$.

For subhorizon pressureless matter during radiation domination, expansion is too rapid for significant self-gravitating growth. Neglecting the small matter-gravity term gives ${\ddot{\delta }}_m+t^{-1}{\dot{\delta }}_m\simeq 0$, hence a constant plus a logarithm of time; the exact coupled radiation problem supplies the detailed amplitudes. This is the Mészáros effect. Modes entering the horizon during radiation domination are suppressed relative to modes entering later, while after equality ordinary matter growth resumes.

The matter power spectrum consequently has a turnover near the equality horizon scale $k_{\mathrm{eq}}\sim a_{\mathrm{eq}}{H}_{\mathrm{eq}}$ in units with $c=1$. Shorter-scale modes retain the radiation-era suppression. This statement concerns cold pressureless matter; baryons coupled to photons also show acoustic effects before recombination.