  We investigate the production of primordial Gravitational Waves (GWs) arising
from First Order Phase Transitions (FOPTs) associated to neutrino mass
generation in the context of type-I and inverse seesaw schemes. We examine both
"high-scale" as well as "low-scale" variants, with either explicit or
spontaneously broken lepton number symmetry $U(1)_L$ in the neutrino sector. In
the latter case, a pseudo-Goldstone majoron-like boson may provide a candidate
for cosmological dark matter. We find that schemes with softly-broken $U(1)_L$
and with single Higgs-doublet scalar sector lead to either no FOPTs or too weak
FOPTs, precluding the detectability of GWs in present or near future
measurements. Nevertheless, we found that, in the majoron-like seesaw scheme
with spontaneously broken $U(1)_L$ at finite temperatures, one can have strong
FOPTs and non-trivial primordial GW spectra which can fall well within the
frequency and amplitude sensitivity of upcoming experiments, including LISA,
BBO and u-DECIGO. However, GWs observability clashes with invisible Higgs decay
constraints from the LHC. A simple and consistent fix is to assume the
majoron-like mass to lie above the Higgs-decay kinematical threshold. We also
found that the majoron-like variant of the low-scale seesaw mechanism implies a
different GW spectrum than the one expected in the high-scale seesaw. This
feature will be testable in future experiments. Our analysis shows that GWs can
provide a new and complementary portal to test the neutrino mass generation
mechanism.
