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 ζ_s, comoving ζ_c, and unitary ζ_u curvatures. We show that conservation of ζ_c outside the horizon is quite restrictive on models as it leads to conservation of ζ_s and ζ_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 ζ_s is conserved outside the horizon, and we connect this result to the concepts of a separate universe and the usage of the δ N formalism. We find that even though ζ_s is conserved, there is still a mild violation of the separate universe approximation in the continuity equation. Nevertheless, the δ N formalism can still be applied to calculate the primordial power spectrum of some gauge-invariant quantities such as ζ_u, although it breaks down for others such as the uniform-density curvature. In BUSR, we show that both ζ_u and ζ_s are conserved outside the horizon, but take different values. Additionally, since ζ_u≠ζ_c, we find that the prediction for observable curvature fluctuations after inflation does not reflect ζ_c at the horizon crossing during inflation and moreover involves not just ζ_u at that epoch but also the manner in which the braiding phase ends.