Curvature Perturbations in the Effective Field Theory of Inflation

We discuss the difference between various gauge-invariant quantities typically used in single-field inflation, namely synchronous $zeta_s$, comoving $zeta_c$, and unitary $zeta_u$ curvatures. We show that conservation of $zeta_c$ outside the horizon is quite restrictive on models as it leads to conservation of $zeta_s$ and $zeta_u$, whereas the reverse does not hold. We illustrate the consequence of these differences with two inflationary models: ultra-slow-roll (USR) and braiding-ultra-slow-roll (BUSR). In USR, we show that out of the three curvatures, only $zeta_s$ is conserved outside the horizon, and we connect this result to the concepts of separate universe and the usage of the $delta N$ formalism. We find that even though $zeta_s$ is conserved, there is still a mild violation of the separate universe approximation in the continuity equation. Nevertheless, the $delta N$ formalism can still be applied to calculate the primordial power spectrum of some gauge-invariant quantities such as $zeta_u$, although it breaks down for others such as the uniform-density curvature. In BUSR, we show that both $zeta_u$ and $zeta_s$ are conserved outside the horizon, but take different values. Additionally, since $zeta_u ot=zeta_c$ we find that the prediction for observable curvature fluctuations after inflation does not reflect $zeta_c$ at horizon crossing during inflation and moreover involves not just $zeta_u$ at that epoch but also the manner in which the braiding phase ends.