When FIMPs Decay into Neutrinos: The $N_\mathrm{eff}$ Story

The existence of feebly interacting massive particles (FIMPs) could have significant implications on the effective number of relativistic species $N_\mathrm{eff}$ in the early Universe. In this work, we investigate in detail how short-lived FIMPs that can decay into neutrinos affect $N_\mathrm{eff}$ and highlight the relevant effects that govern its evolution. We show that even if unstable FIMPs inject most of their energy into neutrinos, they may still decrease $N_{\mathrm{eff}}$, and identify neutrino spectral distortions as the driving power behind this effect. As a case study, we consider Heavy Neutral Leptons (HNLs) and indicate which regions of their parameter space increase or decrease $N_{\mathrm{eff}}$. Moreover, we derive bounds on the HNL lifetime from the Cosmic Microwave Background and comment on the possible role that HNLs could play in alleviating the Hubble tension.