  We derive a minimal basis of kernels furnishing the perturbative expansion of
the density contrast and velocity divergence in powers of the initial density
field that is applicable to cosmological models with arbitrary expansion
history, thereby relaxing the commonly adopted Einstein-de-Sitter (EdS)
approximation. For this class of cosmological models, the non-linear kernels
are at every order given by a sum of terms, each of which factorizes into a
time-dependent growth factor and a wavenumber-dependent basis function. We show
how to reduce the set of basis functions to a minimal amount, and give explicit
expressions up to order $n=5$. We find that for this minimal basis choice, each
basis function individually displays the expected scaling behaviour due to
momentum conservation, being non-trivial at $n\geq 4$. This is a highly
desirable property for numerical evaluation of loop corrections. In addition,
it allows us to match the density field to an effective field theory (EFT)
description for cosmologies with an arbitrary expansion history, which we
explicitly derive at order four. We evaluate the differences to the EdS
approximation for $\Lambda$CDM and $w_0w_a$CDM, paying special attention to the
irreducible cosmology dependence that cannot be absorbed into EFT terms for the
one-loop bispectrum. Finally, we provide algebraic recursion relations for a
special generalization of the EdS approximation that retains its simplicity and
is relevant for mixed hot and cold dark matter models.
