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Decaying dark matter cosmological models have been proposed to remedy theoverproduction problem at small scales in the standard cold dark matterparadigm. We consider a decaying dark matter model in which one CDM motherparticle decays into two daughter particles, with arbitrary masses. A completeset of Boltzmann equations of dark matter particles is derived which isnecessary to calculate the evolutions of their energy densities and theirdensity perturbations. By comparing the expansion history of the universe inthis model and the free-streaming scale of daughter particles with astronomicalobservational 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 thecosmic microwave background, we obtain $\Gamma^{-1}>$ 30 Gyr in the masslesslimit 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 constrainttightens 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 $\Gamma^{-1} <H^{-1}(z=3)$.

Formulation and constraints on decaying dark matter with finite mass daughter particles