This paper studies how a nonstandard early expansion of the universe can both help generate the matter–antimatter imbalance and change the pattern of primordial gravitational waves. The authors consider brane‑inspired scenarios in which extra scalar fields make the universe expand faster than in the usual radiation era. That faster, “stiff” expansion can alter the creation of leptons (leptogenesis) and imprint a clear change in the spectrum of primary gravitational waves (PGWs) produced during inflation. The idea is that future gravitational‑wave measurements could provide an indirect window on high‑scale leptogenesis.

Concretely, the authors model one or more short epochs after inflation when the universe is dominated by a scalar field with a stiff equation of state (meaning pressure is large compared to energy density; in one extreme called kination the equation of state is w = 1). They feed this modified expansion history into the Boltzmann equations that control how a lepton asymmetry builds up. The leptogenesis mechanism they study is the standard high‑scale, unflavored scenario with heavy right‑handed neutrinos from the type‑I seesaw (masses of order 10^9–10^11 GeV). Because the universe expands faster during stiff phases, the time and temperature evolution that set the lepton asymmetry change, and this can open up regions of parameter space where leptogenesis succeeds but would fail under the usual radiation‑dominated history.

They also follow how these stiff phases change the spectral energy density (SED) of PGWs. Under simple inflation models the SED is nearly flat over a range of scales. But modes that re‑enter the Hubble radius during a stiff epoch get their SED reshaped. For example, a kination era (w = 1) produces a linear rise in SED at small scales just below the scale that left the Hubble radius at the end of inflation. The paper gives formulas for how the Hubble rate depends on temperature in single‑ and two‑field stiff scenarios and explains how those changes map onto features in the GW spectrum. The authors focus only on PGWs from inflation and note that the raw inflationary spectrum must be regularized to avoid an unphysical steep rise at sub‑Hubble scales.