A cosmological sandwiched window for lepton-number breaking scale

A singlet majoron can arise from the seesaw framework as a pseudo-Goldstone boson when the heavy Majorana neutrinos acquire masses via the spontaneous breaking of global ${\rm U}(1)_L$ symmetry. The resulting cosmological impacts are usually derived from the effective majoron-neutrino interaction, and the majoron abundance is accumulated through the freeze-in neutrino coalescence. However, a primordial majoron abundance can be predicted in a minimal setup and lead to distinctive cosmological effects. In this work, we consider such a primordial majoron abundance from relativistic freeze-out and calculate the modification to the effective neutrino number $N_{\rm eff}$. We demonstrate that the measurements of $N_{\rm eff}$ will constrain the parameter space from a primordial majoron abundance in an opposite direction to that from neutrino coalescence. When the contributions from both the primordial abundance and the freeze-in production coexist, the ${\rm U}(1)_L$-breaking scale (seesaw scale) $f$ will be pushed into a ''sandwiched window''. Remarkably, for majoron masses below 1 MeV and above the eV scale, the future CMB-S4 experiment will completely close such a low-scale seesaw window for $f\in [1,10^5]~{\rm GeV}$. We highlight that any new light particle with a primordial abundance that couples to SM particles may lead to a similar sandwiched window, and such a general phenomenon deserves careful investigation.