  The separable analytical solution in standard perturbation theory for an
Einstein de Sitter (EdS) universe can be generalized to the wider class of such
cosmologies (``generalized EdS'', or gEdS) in which a fraction of the
pressure-less fluid does not cluster. We derive the corresponding kernels in
both Eulerian perturbation theory (EPT) and Lagrangian perturbation theory,
generalizing the canonical EdS expressions to a one-parameter family where the
parameter can be taken to be the exponent $\alpha$ of the growing mode linear
amplification $D(a) \propto a^{\alpha}$. For the power spectrum (PS) at one
loop in EPT, the contribution additional to standard EdS is given, for each of
the `13' and `22' terms, as a function of two infra-red safe integrals. In the
second part of the paper we show that the calculation of cosmology-dependent
corrections in perturbation theory in standard (e.g. LCDM-like) models can be
simplified, and their magnitude and parameter dependence better understood, by
relating them to our analytic results for gEdS models. At second order the time
dependent kernels are equivalent to the analytic kernels of the gEdS model with
$\alpha$ replaced by a single redshift dependent effective growth rate
$\alpha_2(z)$. At third order the time evolution can be conveniently
parametrized in terms of two additional such effective growth rates. For the PS
calculated at one loop order, the correction to the PS relative to the EdS
limit can be expressed in terms of just $\alpha_2(z)$, one additional effective
growth rate function and the four infra-red safe integrals of the gEdS limit.
This is much simplified compared to expressions in the literature that use six
or eight red-shift dependent functions and are not explicitly infra-red safe.
Using the analytic gEdS expression for the PS with $\alpha=\alpha_2(z)$ gives a
good approximation (to $\sim 25 \%$) for the exact result.
