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Take the usual smooth late dark energy, in particular a cosmological constant, and hold the primordial perturbations fixed. Increasing its physical density while holding the matter and radiation densities fixed raises $H_0$ in a flat universe. It lowers ${\Omega }_{m0}$, but leaves the physical product ${\Omega }_{m0}{H}_0^2=\kappa {\rho }_{m0}/3$ fixed. It also leaves the early equality ratio $a_{\mathrm{eq}}={\rho }_{r0}/{\rho }_{m0}$ and the equality wavenumber in ${\mathrm{Mpc}}^{-1}$ essentially unchanged, provided dark energy remains negligible at equality. The early cosmological transfer function, including the acoustic scale, is then unchanged.

More dark energy causes accelerated expansion to start earlier, reducing the time for matter growth. The suppression of matter growth by smooth accelerated expansion lowers $D(1)$ and hence the present spectrum. For the subhorizon scales shown,This is approximately a downward amplitude change with the same turnover and wiggle locations in physical inverse-megaparsec units. For matter plus a cosmological constant, an explicit early-normalized growth solution iswith radiation neglected in this late-time growth equation. A numerical factor needs the old and new dark-energy densities. A change to axes expressed using $h$ would additionally relabel wavenumbers and volumes; the printed axes use ${\mathrm{Mpc}}^{-1}$ and ${\mathrm{Mpc}}^3$.

Yes. A perturbation of amplitude $A$ at CMB decoupling must grow by at least $A^{-1}$ before becoming nonlinear. During matter domination ${\delta }_m\propto a$, but once ${\rho }_{\Lambda }$ dominates, the suppression of matter growth by smooth accelerated expansion makes the growth approach a finite limit. Increasing ${\rho }_{\Lambda }$ moves this freeze-out to an earlier scale factor and eventually prevents ${\delta }_m$ from reaching unity.

For an order-of-magnitude bound, matter growth would reach unity atRequiring matter still to dominate then gives the galaxy-formation bound on the cosmological constantThe exact upper limit follows by imposing $A{D}_{+}(\infty )/D_{+}(a_{\mathrm{dec}})\gtrsim 1$ with the integral growth factor and an appropriate nonlinear-collapse threshold.

The dominant component is smooth, so only the subdominant matter density clusters. The cold-matter perturbation equation is thereforeAt late times, $a\gg 1$, the source term is negligible compared with the expansion terms. Setting $\tau =t-t_{\ast }$ and using $H=2/\tau$ givesIt integrates toThe constant mode shows the suppression of matter growth by smooth accelerated expansion; the second mode decays.