Curvature Perturbations in the Effective Field Theory of Inflation

TL;DR

This paper examines different gauge-invariant curvature perturbations in single-field inflation, analyzing their conservation properties and implications for the $ ext{delta}N$ formalism in various inflationary models.

Contribution

It clarifies the conditions under which different curvature perturbations are conserved and explores their impact on the $ ext{delta}N$ formalism in ultra-slow-roll and braiding-ultra-slow-roll models.

Findings

Only $ ext{zeta}_s$ is conserved in USR, affecting the $ ext{delta}N$ formalism.

Both $ ext{zeta}_u$ and $ ext{zeta}_s$ are conserved in BUSR but differ in value.

Prediction of observable curvature fluctuations depends on the end of the braiding phase.

Abstract

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.