The early Universe resembles a boiling cauldron, where particles and antiparticles are born and die in equal numbers. But today, all around us is only matter — stars, planets, ourselves. How did a tiny excess of one part in a billion emerge from the initial equality? One answer hides in the oscillations of a heavy field, like a rock that the tide alternately submerges and exposes.

The authors returned to the simplest scenario of spontaneous leptogenesis. Here, the main character is the axion field (a hypothetical particle that solves physics puzzles), which slowly rolls toward its energy minimum. Its motion disrupts the balance between leptons and antileptons, and a special process — the Weinberg operator (a mechanism giving mass to neutrinos)—freezes in this imbalance. For the first time, flavors, the different types of leptons, were fully taken into account, and this dramatically changes the picture: as if you added not just one spice to the pot, but a whole bouquet, making the flavor more complex.

The scientists derived a simple algebraic formula that predicts the final matter excess for a wide class of models. It works like a kitchen recipe: knowing the speed of the 'rock' and its coupling to particles, you immediately get the result. But not all axions are equally good. A popular candidate — the majoron (a particle linked to neutrino mass)—is almost ruled out: its late decays dilute the asymmetry, like extra water spoils a broth. However, axions interacting with sphalerons (processes that convert leptons into baryons) survive at enormous temperatures around a thousand billion degrees and energy scales close to grand unification.

Thus, the cosmic rock scenario finds its conclusion: its rhythmic motions in the tidal zone of the early Universe leave behind the matter from which everything around us is woven. And the recipe-like formula now allows quick testing of new hypotheses without going through complex equations each time.