  The discovery of gravitational waves opens new opportunities to test BSM
physics. In particular, the production of a stochastic background of primordial
gravitational waves could provide a signature of the generation of the
right-right Majorana neutrino mass term necessary, within type-I seesaw
mechanism, to explain lightness of neutrinos and their mixing parameters. I
will discuss the possibility that such a generation occurs during a strong
first order phase transition within Majoron models [1]. As well known, this can
indeed produce a stochastic background of gravitational waves. The scale of the
phase transition can or cannot coincide with the seesaw scale. In the latter
case a low scale phase transition, occurring in the pre-recombination era,
might be tested at very low frequencies ($10^{-9}$--$10^{-6}\,{\rm Hz}$). Even
though the signal can hardly reproduce the NANOGrac putative signal such new
physics at low scale might help ameliorating the tensions in the $\Lambda$CDM
cosmological model (e.g., the Hubble tension). I will also discuss how a phase
transition might be responsible for the generation of dark matter in the form
of dark neutrinos coupling to the seesaw neutrinos via Higgs induced right
handed-right handed neutrino mixing [2].
