Cosmological perturbations for two cold fluids in $\Lambda$CDM

The cosmic large-scale structure of our Universe is comprised of baryons and cold dark matter (CDM). Yet it is customary to treat these two components as a combined single-matter fluid with vanishing pressure, which is justified only for sufficiently large scales and late times. Here we go beyond the single-fluid approximation and develop the perturbation theory for two gravitationally coupled fluids while still assuming vanishing pressure. We mostly focus on perturbative expansions in powers of $D$ (or $D_+$), the linear structure growth of matter in a $\Lambda$CDM Universe with cosmological constant $\Lambda$. We derive in particular (1) explicit recursion relations for the two fluid densities, (2) complementary all-order results in the Lagrangian-coordinates approach, as well as (3) the associated component wavefunctions in a semi-classical approach to cosmic large-scale structure. In our companion paper (Hahn et al. 2020) we apply these new theoretical results to generate novel higher-order initial conditions for cosmological hydrodynamical simulations.