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The cosmological fluid equations describe the early gravitational dynamics of cold dark matter (CDM), exposed to a uniform component of dark energy, the cosmological constant $Λ$. Perturbative predictions for the fluid equations typically assume that the impact of $Λ$ on CDM can be encapsulated by a refined growing mode $D$ of linear density fluctuations. Here we solve, to arbitrary high perturbative orders, the nonlinear fluid equations with an {\it Ansatz} for the fluid variables in increasing powers of $D$. We show that $Λ$ begins to populate the solutions starting at the fifth order in this strict $D$-expansion. By applying suitable resummation techniques, we recast these solutions to a standard perturbative series where not $D$, but essentially the initial gravitational potential serves as the bookkeeping parameter within the expansion. Then, by using the refined growth functions at second and third order in standard perturbation theory, we determine the matter power spectrum to one-loop accuracy as well as the leading-order contribution to the matter bispectrum. We find that employing our refined growth functions impacts the total power- and bispectra at a precision that is below one percent at late times. However, for the power spectrum, we find a characteristic scale-dependent suppression that is fairly similar to what is observed in massive neutrino cosmologies. Therefore, we recommend employing our refined growth functions in order to reduce theoretical uncertainties for analysing data in related pipelines.