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 $\zeta_u\not=\zeta_c$ 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.
