  If a fraction of the dark matter is unstable and decays into dark radiation
at around the time of matter-radiation equality, it could impact the expansion
history of the universe in a way that helps to ameliorate the long-standing
tension between the locally measured value of the Hubble constant and the value
inferred from measurements of the cosmic microwave background and baryon
acoustic oscillations (assuming standard $\Lambda$CDM cosmology). If this
component of decaying dark matter is cold, however, it will modify the
evolution of the gravitational potentials, leading to inconsistencies with
these same data sets. With this in mind, we consider here a component of
decaying warm dark matter, with a free-streaming length that is long enough to
remain consistent with existing data. We study the background and perturbation
evolution of warm decaying dark matter, and use cosmological data to constrain
the mass, abundance and decay rate of such a particle. We find that a component
of warm decaying dark matter can significantly reduce the tension between local
and cosmological determinations of the Hubble constant.
