Inflaton perturbations through an Ultra-Slow Roll transition and Hamilton-Jacobi attractors

We examine the behaviour of the gauge invariant scalar field perturbations in an analytic inflationary model that transitions from slow-roll to an ultra-slow-roll (USR) phase. We find that the numerical solution of the Mukhanov-Sasaki equation is well described by Hamilton-Jacobi (HJ) theory, as long as the appropriate branches of the Hamilton-Jacobi solutions are invoked: Modes that exit the horizon during the slow-roll phase evolve into the USR as described by the first HJ branch, up to a subdominant $\mathcal{O}(k^2/H^2)$ correction to the Hamilton-Jacobi prediction for their final amplitude that we compute, indicating the influence of neglected gradient terms. Modes that exit during the USR phase are described by a separate HJ branch once they become sufficiently superhorizon, obtained by the shift $\left(\epsilon_1,\epsilon_2\right) \simeq \left(0,-6+\Delta \right) \rightarrow \left(\tilde{\epsilon}_1,\tilde{\epsilon}_2\right)\simeq (0,-\Delta)$ and corresponding to a slow-roll solution (very close to de Sitter) supported by the same potential. This transition is similar to the conveyor belt concept put forward in our previous work [1] and suggests that the limit $\epsilon_2\rightarrow -6$ is unphysical as an asymptotic value for the background/long wavelength solution. We further discuss implications for the validity of the stochastic equations arising from the Hamilton-Jacobi formulation. Our work suggests that if Hamilton-Jacobi attractors are appropriately used, they can successfully describe the dynamics of long wavelength inflationary inhomogeneities for potentials with USR regions.