Numerical solutions of the Mukhanov-Sasaki equation for inflaton perturbations across a slow-roll to ultra-slow-roll transition are well described by Hamilton-Jacobi theory when appropriate solution branches and a parameter shift are used for modes exiting in each phase.
Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation,
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Inflaton perturbations through an Ultra-Slow Roll transition and Hamilton-Jacobi attractors
Numerical solutions of the Mukhanov-Sasaki equation for inflaton perturbations across a slow-roll to ultra-slow-roll transition are well described by Hamilton-Jacobi theory when appropriate solution branches and a parameter shift are used for modes exiting in each phase.