How Heavy Neutrinos Leave a Double Gravitational Trail express
In the first moments after the Big Bang, the Universe was a hot soup of particles. Among them floated superheavy neutrinos — thousands of times more massive than their light counterparts that permeate us today. Their masses were almost identical, like triplets rather than random passersby.
This picture was painted by a mathematical symmetry — the unusual modular group A₄. It rigidly ties particle masses together, leaving no freedom for fine-tuning. As a result, the superheavy neutrinos end up nearly degenerate in mass (differing by fractions of a percent). It's like three weights cast from the same mold. Their decays in the early Universe produced a tiny excess of matter over antimatter — the very excess that makes our existence possible.
But this elegant picture has a cosmological shadow. As the Universe cooled, it underwent a phase transition, like water freezing into ice. The field responsible for the symmetry 'froze' differently in different regions of space, and domain walls grew between them. Such a fence of energy would have come to dominate the Universe, yet we don't see it.
The authors found an elegant way out: the same superheavy neutrinos create a tiny pressure imbalance on the walls. The walls start to move, collide, and disappear, generating ripples in spacetime — gravitational waves. Their spectrum is unusual: two peaks separated by many octaves. One comes from the wall collisions, the other from the phase transition itself. Together, they resemble a double bell strike, from which one can reconstruct the mass of the invisible clapper.
Future detectors, the space observatory LISA and the ground-based Einstein Telescope, will be able to hear both peaks. This is a rare chance to peer into the epoch when particle masses were formed and to indirectly weigh what will never be trapped in an accelerator.
Original abstract
as written by the authors arXiv:2607.18803
We study a type-I seesaw framework based on non-holomorphic $$A_4$$ modular symmetry, where polyharmonic Maa\ss\ forms construct the Yukawa couplings and right-handed neutrino (RHN) Majorana mass matrix. The use of non-holomorphic modular forms yields highly constrained neutral lepton mass matrices with a more restrictive lepton-sector structure and naturally generates a quasi-degenerate RHN mass spectrum, enabling resonant leptogenesis at an intermediate scale with RHN masses of $$\mathcal{O}(10^3)$$ TeV without requiring an ad hoc mass degeneracy. We further extend the model by introducing a complex scalar field $$(\Phi)$$ charged under $$\mathbb{Z}_3$$ symmetry. The spontaneous breaking of the discrete symmetry after the phase transition associated with $$\Phi$$ leads to domain-wall (DW) formation. A radiatively induced bias term associated with the RHN sector triggers DW annihilation, resolving the cosmological DW problem, and producing a stochastic gravitational wave (GW) signal that indirectly probes the RHN mass scale. The accompanying first-order phase transition produces a second GW peak, yielding a characteristic double-peaked spectrum with frequencies separated by several orders of magnitude and potentially observable by complementary future GW detectors.
The domain walls in question are not physical barriers but regions of sharp energy field gradients. If such a wall passed through a room, it wouldn't hit you in the nose, but it would change the masses of elementary particles inside your body.
In Greg Egan's novel 'The Clockwork Rocket', physicists use gravitational waves from phase transitions to study alternative vacua. Here, the idea is similar: waves tell us about a symmetry that broke in the first fractions of a second of the Universe's life.