Slow-roll inflation produces nearly scale-invariant perturbations. Ultra-slow-roll inflation enhances perturbations dramatically, potentially producing primordial black holes. The transition between these regimes determines the shape of the power spectrum enhancement — and whether the enhancement is sharp enough to produce observable consequences. Previous models treated the transition as instantaneous, which simplifies the calculation but introduces artificial discontinuities.
This paper constructs the first fully analytical model of a smooth slow-roll to ultra-slow-roll transition. A minimal time dependence in the effective mass term of the Mukhanov-Sasaki equation keeps the second slow-roll parameter continuously varying throughout. Both the background evolution and curvature perturbations have closed-form solutions, making the parameter dependence of the power spectrum transparent.
The smooth transition differs from its sharp counterpart in the spectral shape: the enhancement is broader and the oscillatory features are damped. Nature doesn't make jumps — and the smooth model shows that the sharpness of the transition, usually treated as a free parameter, is physically constrained by requiring continuity of the slow-roll parameters. The analytical tractability means the relationship between model parameters and observational signatures can be read off directly rather than extracted numerically.