Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 74 / 3 / i

A superhorizon density contrast is gauge dependent. The stated growth laws describe the growing adiabatic mode in the comoving-gauge density contrast (and the corresponding usual synchronous growing mode), not an arbitrary coordinate density perturbation. Work with a flat, single-fluid background with constant $w=p/\rho$, negligible anisotropic stress, $c=1$, and a comoving Fourier wavenumber $K$.

Here is a derivation from the linear gravitational constraints and evolution equation. For an adiabatic mode, the scalar gravitational potential $\Phi$ obeysPrimes denote conformal time derivatives. The Friedmann equation gives $a\propto {\tau }^{2/(1+3w)}$ and $\mathcal{H}=2/[(1+3w)\tau ]$, so the bracket vanishes. On superhorizon scales, $K\tau \ll 1$, the leading solutions are a constant potential and a decaying solution proportional to ${\tau }^{-(5+3w)/(1+3w)}$. Keeping the growing adiabatic branch therefore makes $\Phi$ time independent to leading order.

Combining the time-time and time-space linear Einstein constraints eliminates the coordinate velocity and gives the comoving density constraintThis is the relativistic comoving constraint, not a subhorizon Newtonian approximation. Since $\rho \propto a^{-3(1+w)}$, it implies $\Delta \propto (a^2{H}^2{)}^{-1}\propto a^{1+3w}$. Also $a\propto t^{2/[3(1+w)]}$ in cosmic time, whenceBoth epochs here mean their respective leading growing solution; matching through matter-radiation equality requires the full two-component evolution. The constant superhorizon curvature or potential is consistent with this growth because the comoving density perturbation carries the additional factor $(K/aH{)}^2$.