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.