The effective gravitational decoupling between dark matter and the CMB
We present a detailed and self-contained analytical derivation of the evolution of sub-horizon cosmological perturbations before decoupling, based on previous work by S. Weinberg. These solutions are valid in the minimal Lambda CDM scenario, to first order in perturbation theory, in the tight-coupling limit and neglecting neutrino shear stress. We compare them to exact numerical solutions computed by a Boltzmann code, and we find the two to be in very good agreement. The analytic solutions show explicitly that CDM and the baryon-photon fluid effectively behave as separate self-gravitating fluids until the epoch of baryon drag. This in turn leads to the surprising conclusion that the CMB is much less sensitive to the clustering properties of minimally coupled Dark Matter models than what would be naively expected.