Validity of separate-universe approach in transient ultraslow-roll inflation

We investigate the breakdown of the separate-universe approximation during transitions in transient ultraslow-roll inflation by analyzing the evolution of the comoving curvature perturbation $\mathcal{R}$ and its conjugate momentum $\mathrm{\ensuremath{\Pi}}$. It is well known that spatial gradient terms lead to a failure of this approximation, particularly at the transition from slow- to ultraslow-roll phase. We show that a similar breakdown also occurs during the second transition back to slow roll when considering the evolution of $\mathrm{\ensuremath{\Pi}}$. Interestingly, while the homogeneous solution for $\mathrm{\ensuremath{\Pi}}$ accurately captures the dynamics across the first transition, it is the homogeneous solution for $\mathcal{R}$ that becomes valid across the second. Furthermore, we demonstrate that the spatial curvature term introduced in the extended $\ensuremath{\delta}N$ formalism of Artigas et al. [Phys. Rev. Lett. 134, 221001 (2025).] can be interpreted as arising from the contribution of $\mathrm{\ensuremath{\Pi}}$ 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.