Tidal evolution and diffusive growth during high-eccentricity planet migration: revisiting the eccentricity distribution of hot Jupiters

High-eccentricity tidal migration is a potential formation channel for hot Jupiters. During this process, the planetary f-mode may experience a phase of diffusive growth, allowing its energy to quickly build up to large values. In Yu et al. (2021, ApJ, 917, 31), we demonstrated that nonlinear mode interactions between a parent f-mode and daughter f- and p-modes expand the parameter space over which the diffusive growth of the parent is triggered. We extend that study by incorporating (1) the angular momentum transfer between the orbit and the mode, and consequently the evolution of the pericenter distance, (2) a phenomenological correction to the nonlinear frequency shift at high parent mode energies, and (3) dissipation of the parent's energy due to both turbulent convective damping of the daughter modes and strongly nonlinear wave-breaking events. The new ingredients allow us to follow the coupled evolution of the mode and orbit over $\gtrsim 10^4$ years, covering the diffusive evolution from its onset to its termination. We find that the semi-major axis shrinks by a factor of nearly ten over $10^4$ years, corresponding to a tidal quality factor $\mathcal{Q}\sim10$. The f-mode's diffusive growth terminates while the eccentricity is still high, at around e=0.8-0.95. Using these results, we revisit the eccentricity distribution of proto-hot Jupiters. We estimate that less than 1 proto-HJ with eccentricity >0.9 should be expected in Kepler's data once the diffusive regime is accounted for, explaining the observed paucity of this population.