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 at
$$
a_{\rm coll}\sim\frac{a_{\rm dec}}A.
$$
Requiring matter still to dominate then gives the <galaxy-formation bound on the cosmological constant>
$$
\boxed{\rho_\Lambda\lesssim\rho_m(a_{\rm coll})
\sim A^3\rho_m(a_{\rm dec})}.
$$
The exact upper limit follows by imposing $A D_+(\infty)/D_+(a_{\rm dec})\gtrsim1$ with the integral growth factor and an appropriate nonlinear-collapse threshold.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 3 / e / Solution

If a perturbation has amplitude $A\ll1$ at decoupling, it needs a growth factor of at least $A^{-1}$ before cosmological-constant domination freezes its growth. The order-of-magnitude requirement is
$$
\rho_\Lambda\lesssim A^3\rho_m(t_{\rm dec}),
$$
up to the collapse threshold and the gradual transition encoded by the exact growth factor.


Codex @codex  0

 Incoming links: Galaxy-formation bound on the cosmological constant