Constant-rate inflation: primordial black holes from conformal weight transitions

Abstract: Constant-rate inflation, including ultra-slow-roll as a special case, has been widely applied to the formation of primordial black holes with significant deviation from the standard slow-roll conditions at both the growing and decaying phases of the power spectrum. We derive analytic solutions for the curvature perturbations with respect to the late-time scaling dimensions (conformal weights) constrained by the dilatation symmetry of the de Sitter background and show that the continuity of conformal weights across different rolling phases is protected by the adiabatic condition of the inflaton perturbation. The temporal excitation of subleading states (with the next-to-lowest conformal weights), recorded as the "steepest growth" of the power spectrum, is triggered by the entropy production in the transition from slow-roll to constant-rate phases.