Self-interacting dark matter from late decays and the $H_0$ tension

We study a dark matter production mechanism based on decays of a messenger WIMP-like state into a pair of dark matter particles that are self-interacting via exchange of a light mediator. Its distinctive thermal history allows the mediator to be stable and therefore avoid strong limits from the cosmic microwave background and indirect detection. A natural by-product of this mechanism is a possibility of a late time, i.e., after recombination, transition to subdominant dark radiation component through three-body and one-loop decays to states containing the light mediator. We examine to what extent such a process can help to alleviate the $H_0$ tension. Additionally, the mechanism can provide a natural way of constructing dark matter models with ultra-strong self-interactions that may positively affect the supermassive black hole formation rate. We provide a simple realization of the mechanism in a Higgs portal dark matter model and find a significant region of the parameter space that leads to a mild relaxation of the Hubble tension while simultaneously having the potential of addressing small-scale structure problems of $\Lambda$CDM.