Evolution of perturbations and cosmological constraints in decaying dark matter models with arbitrary decay mass products

Abstract: Decaying dark matter (DDM) is a candidate which can solve the discrepancies between predictions of the concordance $\Lambda$CDM model and observations at small scales such as the number counts of companion galaxies of the Milky Way and the density profile at the center of galaxies. Previous studies are limited to the cases where the decay particles are massless and/or have almost degenerate masses with that of mother particles. Here we expand the DDM models so that one can consider the DDM with arbitrary lifetime and the decay products with arbitrary masses. We calculate the time evolutions of perturbed phase-space distribution functions of decay products for the first time and study effects of DDM on the temperature anisotropy in the cosmic microwave background and the matter power spectrum at present. From a recent observational estimate of $\sigma_{8}$, we derive constraints on the lifetime of DDM and the mass ratio between the decay products and DDM. We also discuss implications of the DDM model for the discrepancy in the measurements of $\sigma_8$ recently claimed by the Planck satellite collaboration.