  We investigate a concrete scenario of a light scalar with a mass around 1 MeV
which can be connected to the origin of neutrino masses and simultaneously
survive current bounds on relativistic degrees of freedom in the early
universe. In particular we show that a feeble coupling to the Standard Model
neutrinos can relax the stringent bounds on the decays to photons inferred from
the measured value of $N_{\rm eff}$. Interestingly, we find that such a scalar
whose diphoton coupling is generated by a tree-level coupling to the muon of
similar strength as that of the Standard Model Higgs boson can simultaneously
explain the long-standing discrepancy in the measured value of the muon
magnetic moment. We present a possible ultraviolet completion of this scenario
providing a link between new physics in the early universe and the generation
of neutrino masses.
