Cosmological Density Perturbation of Cold Dark Matter in Newtonian Gravity Scenario after Recombination: a Symmetry-Based Approach

We study the evolution of cosmological density perturbations of cold dark matter (CDM) in an expanding matter-dominated universe with respect to a homogeneous spatially flat Friedman-Lemaître-Robertson-Walker (FLRW) background under a Newtonian gravity scenario after recombination. In particular, the density contrast equation for CDM is constructed from a two-fluids model. In deriving the equation, we use the well-known solution for the density contrast of baryonic matter presented in one of our recent papers. We employ a symmetry-based approach to study the CDM density contrast equation and find eight-parameter Lie group symmetries. We use these Lie symmetries to find group-invariant solutions of the density contrast equation for CDM governed by a Newtonian force law from the invariant curve condition and visualize the growth of density perturbation after recombination in the presence of baryonic matter. We also solve numerically the CDM density contrast equation in the presence of a baryonic component and check the evolution of the density contrast from recombination to the present age. In addition, considering only growing mode solutions, we compare the evolution of density contrasts of CDM (WIMPs) in the presence of a baryonic component and that of only baryons. We also compare the evolution of density contrasts of CDM in the presence of baryons under Newtonian gravity, baryons under Newtonian gravity, and baryons under MONDian gravity for the time period from recombination to the present age.