Suppression of matter growth by smooth accelerated expansion

If a smooth component dominates the expansion but does not cluster, pressureless matter still satisfiesAs ${\bar{\rho }}_m$ becomes negligible, Hubble friction leaves a constant mode and a decaying mode, so structure growth freezes.

When $H$ is constant and the matter source is negligible,so ${\delta }_m=C_1+C_2{a}^{-2}$. The growing matter-era mode therefore approaches a constant after cosmological-constant domination begins.

For constant positive Hubble parameter $\mu$ and a four-thirds polytropic equation of state, a linear cosmological density perturbation obeys $\ddot{\delta }+2\mu \dot{\delta }+{\eta }^2{e}^{-3\mu t}\delta =0$. Put $\delta =e^{-\mu t}y$ and $x=2|\eta |e^{-3\mu t/2}/(3\mu )$. The resulting Bessel differential equation has order $2/3$; use ordinary Bessel functions for ${\eta }^2>0$ and modified Bessel functions for ${\eta }^2<0$. Their small-$x$ expansions give one constant limiting mode and one $e^{-2\mu t}$ mode. The constant limit need not exceed the initial amplitude without an initial-velocity condition.

If ${\eta }^2=-s^2<0$, $\delta (0)>0$ and $\dot{\delta }(0)\ge 0$, the integrating factor identity $(e^{2\mu t}\dot{\delta }{)}^{\prime }=s^2{e}^{-\mu t}\delta$ proves that $\delta$ stays positive and increases. Its finite limit satisfies ${\delta }_{\infty }=\delta (0)+\dot{\delta }(0)/(2\mu )+s^2{\int }_0^{\infty }{e}^{-3\mu t}\delta (t)dt/(2\mu )>\delta (0)$. The initial sign condition is essential: the positive decaying modified Bessel function mode has limit zero.

In a matter-plus-cosmological-constant universe, changing the independent variable from cosmic time to $a$ gives

The growing solution of the matter perturbation equation in a spatially flat matter-plus-cosmological-constant universe isnormalized so that $D_{+}(a)\sim a$ during early matter domination.

When $H\to H_{\Lambda }>0$, the growth integral has an integrable $a^{-3}$ tail. Its product with $H$ tends to a constant, with leading correction proportional to $a^{-2}$ in a matter-plus-Lambda background. Both displayed basis functions can tend to constants; a linear combination cancels that constant and isolates the truly decaying late-time $a^{-2}$ behavior.

Changing the lower limit in $H(a){\int }_{a_i}^{a}d\tilde{a}/(\tilde{a}H{)}^3$ adds a constant multiple of $H(a)$, the independent matter-era decaying solution. Thus a finite lower limit defines a useful independent solution but can include a decaying admixture. A pure growing normalization is selected by an appropriate linear combination.

If a perturbation has amplitude $A\ll 1$ at decoupling, it needs a growth factor of at least $A^{-1}$ before cosmological-constant domination freezes its growth. The order-of-magnitude requirement isup to the collapse threshold and the gradual transition encoded by the exact growth factor.