Formulation and constraints on decaying dark matter with finite mass
  daughter particles
  Decaying dark matter cosmological models have been proposed to remedy the
overproduction problem at small scales in the standard cold dark matter
paradigm. We consider a decaying dark matter model in which one CDM mother
particle decays into two daughter particles, with arbitrary masses. A complete
set of Boltzmann equations of dark matter particles is derived which is
necessary to calculate the evolutions of their energy densities and their
density perturbations. By comparing the expansion history of the universe in
this model and the free-streaming scale of daughter particles with astronomical
observational data, we give constraints on the lifetime of the mother particle,
$\Gamma^{-1}$, and the mass ratio between the daughter and the mother particles
$m_{\rm D}/m_{\rm M}$. From the distance to the last scattering surface of the
cosmic microwave background, we obtain $\Gamma^{-1}>$ 30 Gyr in the massless
limit of daughter particles and, on the other hand, we obtain $m_{\rm D} >$
0.97$m_{\rm M}$ in the limit $\Gamma^{-1}\to 0$. The free-streaming constraint
tightens the bound on the mass ratio as $(\Gamma^{-1}/10^{-2}{\rm Gyr})
\lesssim ((1-m_{\rm D1}/m_{\rm M})/10^{-2})^{-3/2}$ for $\Gamma^{-1} <
H^{-1}(z=3)$.
