Constraining the Self-Interacting Neutrino Interpretation of the Hubble
  Tension
  Large, non-standard neutrino self-interactions have been shown to resolve the
$\sim 4\sigma$ tension in Hubble constant measurements and a milder tension in
the amplitude of matter fluctuations. We demonstrate that interactions of the
necessary size imply the existence of a force-carrier with a large neutrino
coupling ($> 10^{-4}$) and mass in the keV -- 100 MeV range. This mediator is
subject to stringent cosmological and laboratory bounds, and we find that
nearly all realizations of such a particle are excluded by existing data unless
it carries spin 0 and couples almost exclusively to $\tau$-flavored neutrinos.
Furthermore, we find that the light neutrinos must be Majorana, and that a
UV-complete model requires a non-minimal mechanism to simultaneously generate
neutrino masses and appreciable self-interactions.
