Cosmological perturbation theory using generalized Einstein de Sitter cosmologies

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.