Cosmology of an Axion-Like Majoron

We propose a singlet majoron model that defines an inverse seesaw mechanism in the $\nu$ sector. The majoron $\phi$ has a mass $m_\phi\approx 0.5$ eV and a coupling to the $\tau$ lepton similar to the one to neutrinos. In the early universe it is initially in thermal equilibrium, then it decouples at $T\approx 500$ GeV and contributes with just $\Delta N_{\rm eff}=0.026$ during BBN. At $T=26$ keV (final stages of BBN) a primordial magnetic field induces resonant $\gamma \leftrightarrow \phi$ oscillations that transfer $6\%$ of the photon energy into majorons, implying $\Delta N_{\rm eff}=0.55$ and a $4\%$ increase in the baryon to photon ratio. At $T\approx m_\phi$ the majoron enters in thermal contact with the heaviest neutrino and it finally decays into $\nu \bar \nu$ pairs near recombination, setting $\Delta N_{\rm eff}=0.85$. This boost in the expansion rate at later times solves the Hubble tension, while the neutrino--majoron interactions suppress the $\nu$ free streaming and make the model consistent with large scale structure observations. Its lifetime and the fact that it decays into neutrinos instead of photons lets this axion-like majoron avoid the strong bounds that affect other axion-like particles of similar mass and coupling to photons.