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.
