The validity of separate-universe approach in transient ultra-slow-roll inflation

We investigate the breakdown of the separate-universe approximation during transitions in transient ultra-slow-roll inflation by analyzing the evolution of the comoving curvature perturbation ${\cal R}$ and its conjugate momentum $\Pi$. It is well known that spatial gradient terms lead to a failure of this approximation, particularly at the transition from slow-roll to ultra-slow-roll phase. We show that a similar breakdown also occurs during the second transition back to slow-roll when considering the evolution of $\Pi$. Interestingly, while the homogeneous solution for $\Pi$ accurately captures the dynamics across the first transition, it is the homogeneous solution for ${\cal R}$ that becomes valid across the second. Furthermore, we demonstrate that the spatial curvature term introduced in the extended $\delta N$ formalism of \cite{Artigas:2024ajh} can be interpreted as arising from the contribution of $\Pi$ to the energy density perturbation. Importantly, this modification of the local Hubble parameter is valid only when the first slow-roll parameter is both small and strictly constant.