  Decaying dark matter models provide a physically motivated way of channeling
energy between the matter and radiation sectors. In principle, this could
affect the predicted value of the Hubble constant in such a way as to
accommodate the discrepancies between CMB inferences and local measurements of
the same. Here, we revisit the model of warm dark matter decaying
non-relativistically to invisible radiation. In particular, we rederive the
background and perturbation equations starting from a decaying neutrino model
and describe a new, computationally efficient method of computing the decay
product perturbations up to large multipoles. We conduct MCMC analyses to
constrain all three model parameters, for the first time including the mass of
the decaying species, and assess the ability of the model to alleviate the
Hubble and $\sigma_8$ tensions, the latter being the discrepancy between the
CMB and weak gravitational lensing constraints on the amplitude of matter
fluctuations on an $8 h^{-1}$ Mpc$^{-1}$ scale. We find that the model reduces
the $H_0$ tension from $\sim 4 \sigma$ to $\sim 3 \sigma$ and neither
alleviates nor worsens the $S_8 \equiv \sigma_8 (\Omega_m/0.3)^{0.5}$ tension,
ultimately showing only mild improvements with respect to $\Lambda$CDM.
However, the values of the model-specific parameters favoured by data is found
to be well within the regime of relativistic decays where inverse processes are
important, rendering a conclusive evaluation of the decaying warm dark matter
model open to future work.
