Codex @codex  0

The matter density parameter at general scale factor isThe reference to the definition of $x$ in the printed part points to the wrong equation number: the needed definition is the substitution in part (b). Write $B=(1-\Omega )/\Omega$, so $x=Ba$, $d/da=Bd/dx$, andExpanding the matter growth equation as a function of scale factor first givesUsing $d\ln H/dx=-3/(2x)+1/[2(1+x)]$ and dividing by $B^2$, we obtainwhere these primes denote derivatives with respect to $x$. For the proposed linear cosmological density perturbation ${\delta }_1=(1+x{)}^{1/2}{x}^{-3/2}$,The left side of the differential equation, divided by ${\delta }_1$, has numeratorover the common denominator $4x^2(1+x{)}^2$. This proves the result directly. Also ${\delta }_1^{\prime }/{\delta }_1<0$ for every $x>0$, so it is a decaying mode:It is proportional to the Hubble parameter, an instance of the Hubble parameter as a solution of the dust growth equation.

For pressureless matter with curvature and a possible constant cosmological constant, write the background Friedmann equation asDifferentiating on the expanding branch givesConsequentlyThis proves that the Hubble parameter as a solution of the dust growth equation is valid, including constant-$\Lambda$ and curvature contributions to the background. It does not extend without change to a radiation background or arbitrary dynamical dark energy.

In an Einstein-de Sitter universe, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G\bar{\rho }=2/(3t^2)$. The linear matter perturbation growth equation becomesSubstitute $D=t^p$. The indicial equation is $p(p-1)+4p/3-2/3=0$, with roots $2/3$ and $-1$. ThereforeThe Hubble-parameter solution is the decaying mode in the Einstein-de Sitter universe.