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Decaying dark matter (DDM) is a candidate which can solve the discrepanciesbetween predictions of the concordance $\Lambda$CDM model and observations atsmall scales such as the number counts of companion galaxies of the Milky Wayand the density profile at the center of galaxies. Previous studies are limitedto the cases where the decay particles are massless and/or have almostdegenerate masses with that of mother particles. Here we expand the DDM modelsso that one can consider the DDM with arbitrary lifetime and the decay productswith arbitrary masses. We calculate the time evolutions of perturbedphase-space distribution functions of decay products for the first time andstudy effects of DDM on the temperature anisotropy in the cosmic microwavebackground and the matter power spectrum at present. From a recentobservational estimate of $\sigma_{8}$, we derive constraints on the lifetimeof DDM and the mass ratio between the decay products and DDM. We also discussimplications of the DDM model for the discrepancy in the measurements of$\sigma_8$ recently claimed by the Planck satellite collaboration.

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