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A singlet majoron can arise from the seesaw framework as a pseudo-Goldstoneboson when the heavy Majorana neutrinos acquire masses via the spontaneousbreaking of global ${\rm U}(1)_L$ symmetry. The resulting cosmological impactsare usually derived from the effective majoron-neutrino interaction, and themajoron abundance is accumulated through the freeze-in neutrino coalescence.However, a primordial majoron abundance can be predicted in a minimal setup andlead to distinctive cosmological effects. In this work, we consider such aprimordial majoron abundance from relativistic freeze-out and calculate themodification to the effective neutrino number $N_{\rm eff}$. We demonstratethat the measurements of $N_{\rm eff}$ will constrain the parameter space froma primordial majoron abundance in an opposite direction to that from neutrinocoalescence. When the contributions from both the primordial abundance and thefreeze-in production coexist, the ${\rm U}(1)_L$-breaking scale (seesaw scale)$f$ will be pushed into a ''sandwiched window''. Remarkably, for majoron massesbelow 1 MeV and above the eV scale, the future CMB-S4 experiment willcompletely 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 thatcouples to SM particles may lead to a similar sandwiched window, and such ageneral phenomenon deserves careful investigation.

A cosmological sandwiched window for seesaw with primordial majoron abundance