Intimate Relationship Between Sterile Neutrino Dark Matter and $\Delta
  N_{\rm eff}$
  The self-interacting neutrino hypothesis is well motivated for addressing the
tension between the origin of sterile neutrino dark matter and indirect
detection constraints. It can also result in a number of testable signals from
the laboratories to the cosmos. We explore a model of neutrino self-interaction
mediated by a Majoron-like scalar with sub-MeV mass, and show that explaining
the relic density of sterile neutrino dark matter implies a lower bound on the
amount of extra radiation in early universe, in particular $\Delta N_{\rm
eff}>0.12$ at the CMB epoch. This lower bound will be further strengthened with
an improved $X$-ray search at the Athena observatory. Such an intimate
relationship will be unambiguously tested by the upcoming CMB-S4 project.
