Decaying dark matter, the $H_0$ tension, and the lithium problem
  It has long been known that the sharpened tension between the observed and
inferred values of the Hubble constant $H_0$ can be alleviated if a fraction of
dark matter particles of type $\chi$ were produced non-thermally in association
with photons $\gamma$ through the decays of a heavy and relatively long-lived
state, viz. $X \to \chi + \gamma$. It was recently proposed that this model can
also resolve the longstanding lithium problem if $M= 4$ MeV and $m = 0.04$ keV,
where $M$ and $m$ are respectively the masses of $X$ and $\chi$. We confront
this proposal with experiment and demonstrate that cold dark matter decaying
before recombination cannot resolve the $H_0$ problem. Moreover, we show that
the best case scenario for alleviating the $H_0$ tension within the context of
cold dark matter decaying before recombination arises when the particles decay
exclusively into dark radiation, while leaving completely unmodified the
production of light elements. To this end we calculate the general functional
form describing the number of equivalent light neutrino species $\Delta
N_{eff}$ carried by $\chi$. We show that to resolve the $H_0$ tension at the
$1\sigma$ level a 55% correction in $m$ is needed and that the required $\Delta
N_{eff}$ is excluded at 95% CL by Planck data. We argue in favor of a more
complex model of dynamical dark matter to relax the $H_0$ tension.
