  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.
