Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 41 / 2

The term $2H\dot{\delta }$ is Hubble friction in the equation for the density contrast. During expansion, $H>0$, it damps the peculiar motion producing density growth. It therefore slows gravitational instability relative to the otherwise identical static model; it does not necessarily stop growth.

Multiplying by $a^2$ shows the mechanism without confusing it with microscopic dissipative friction:When the self-gravity term is negligible, the expansion-weighted derivative of a passive density perturbation is conserved, so $\dot{\delta }\propto a^{-2}$. Expansion also reduces ${\rho }_m\propto a^{-3}$ and hence weakens the source on the right. A contracting background has $H<0$ and reverses the sign of this effect.

For constant $a$ and constant ${\rho }_m>0$, Hubble friction vanishes. Put ${\omega }_J=\sqrt{4\pi G{\rho }_m}$. The constant-coefficient ordinary differential equation is $\ddot{\delta }-{\omega }_J^2\delta =0$, so its characteristic polynomial has the real roots of a polynomial $\pm {\omega }_J$:These are the pressureless special case of the Static-universe Jeans modes. A positive growing-mode coefficient gives exponential gravitational instability, with time scale $(4\pi G{\rho }_m{)}^{-1/2}$. For initial data $\delta (t_{\ast })={\delta }_{\ast }$ and $\dot{\delta }(t_{\ast })=v_{\ast }$, the equivalent answer is ${\delta }_{\ast }\cosh [{\omega }_J(t-t_{\ast })]+(v_{\ast }/{\omega }_J)\sinh [{\omega }_J(t-t_{\ast })]$. This solves the stated perturbation model; a globally static dust-only FRW metric would require additional background support.

In an Einstein-de Sitter universe, $a\propto t^{2/3}$ and $H=2/(3t)$. The Friedmann equation givesThis is an Euler differential equation. Substitution of $\delta =t^p$ gives $p(p-1)+(4/3)p-2/3=0$, namely $(p-2/3)(p+1)=0$. The Einstein-de Sitter density-growth modes are thereforeThe linear growth factor for the growing mode is proportional to $a$, rather than an exponential in cosmic time. These expressions apply while the density contrast is small enough for linear cosmological density perturbations.

After the decay products dominate the background, radiation domination gives ${\rho }_r\propto a^{-4}$. Neglecting curvature and vacuum terms, the Friedmann equation implies $\dot{a}/a=C/a^2$, so $a\propto (t-t_B{)}^{1/2}$ andThe displayed $1/(2t)$ form uses a cosmic-time origin with $t_B=0$. After a late transition the extrapolated radiation-era origin need not be the original Big Bang; this time shift does not change the growth law.

For the remaining pressureless matter, neglect its self-gravity to leading order in ${\rho }_m/{\rho }_r$. Its density contrast obeys $\ddot{\delta }+(t-t_B{)}^{-1}\dot{\delta }=0$. Using the conserved expansion-weighted derivative of a passive density perturbation givesThus logarithmic growth of matter perturbations during radiation domination replaces the static exponential growth and the matter-era $t^{2/3}$ growth. The expansion persists while its dominant radiation component supplies little sustained clustering in this pressureless test-component approximation. The residual matter can eventually become important again because ${\rho }_m/{\rho }_r\propto a$; the approximation is for the radiation-dominated interval. The equation is not a pressureless growth equation for the relativistic decay products themselves, whose pressure and, when a fluid description applies, radiation acoustic oscillations must be retained.