Solution (source code)

= Solution

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 $a_{\rm eq}=\rho_{r0}/\rho_{m0}$ 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,
$$
\boxed{\frac{P_m^{\rm new}(k,0)}{P_m^{\rm old}(k,0)}
\simeq\left[\frac{D_{\rm new}(1)}{D_{\rm old}(1)}\right]^2<1.}
$$
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 is
$$
D(a)=\frac{5\Omega_{m0}H_0^2}{2}H(a)
\int_0^a\frac{da'}{a'^3H(a')^3},
$$
with 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 $\mathrm{Mpc}^3$.