The constancy of ζin single-clock Inflation at all loops

TL;DR

The paper investigates infrared corrections to inflationary perturbations and proves that the curvature perturbation $\zeta$ is constant at large scales to all loop orders in single-clock inflation. By expressing the $n$-th order $\dot{\zeta}$ correlators as time integrals of Green's functions multiplied by local sources, and introducing a local frame that absorbs a long-wavelength mode, the authors show that late-time time dependence can only arise from the latest interactions and is suppressed. An inductive argument, supported by detailed analysis of SC and LC diagrams, demonstrates that $\dot{\zeta}$ vanishes at late times to all orders, ensuring the constancy of $\zeta$ and the robustness of inflationary predictions against infrared loop effects. This reinforces the theoretical consistency of single-clock inflation and underpins the slow-roll eternal inflation volume bound.

Abstract

Studying loop corrections to inflationary perturbations, with particular emphasis on infrared factors, is important to understand the consistency of the inflationary theory, its predictivity and to establish the existence of the slow-roll eternal inflation phenomena and its recently found volume bound. In this paper we show that ζ-correlators are time-independent at large distances at all-loop level in single clock inflation. We write the n-th order correlators of \dotζ as the time-integral of Green's functions times the correlators of local sources that are function of the lower order fluctuations. The Green's functions are such that only non-vanishing correlators of the sources at late times can lead to non-vanishing correlators for \dotζ at long distances. When the sources are connected by high wavenumber modes, the correlator is peaked at short distances, and these diagrams cannot lead to a time-dependence by simple diff. invariance arguments. When the sources are connected by long wavenumber modes one can use similar arguments once the constancy of ζ at lower orders was established. Therefore the conservation of ζ at a given order follows from the conservation of ζ at the lower orders. Since at tree-level ζ is constant, this implies constancy at all-loops by induction.

Figures (3)

• Figure 1: Example of a $SC$ (short-connected) diagram. Red-dashed lines ended by a red-cross represent free modes, and two crosses encircled by a blue-dashed ellipse represent a correlation function between those two free modes. The dashed-filled ellipse represents possible additional correlation functions. This is a $SC$ diagram as, among the various correlation functions present in the diagram, there are two involving two short modes at different locations $x_1$ and $x_2$ that are contracted among each other. If the distance between $x_1$ and $x_2$ is much longer than the wavelength and frequency of the two modes, the contribution becomes vanishingly small. The points $x_1$ and $x_2$ have therefore to be very close to each other. This diagram contributes to the first term of (\ref{['eq:short_average']}).
• Figure 2: Example of a $LC$ (long-connected) diagram. Green line represent Green's function. The dashed-filled ellipse contains possible additional correlations that connect $x_1$ and $x_2$ only though long modes. This is a $LC$ diagram as $x_1$ and $x_2$ are connected only by low momentum lines. Notice that $x_3$ must be very close to $x_1$ because it is connected to $x_1$ by a correlation of short modes. This diagram contributes to the second term of (\ref{['eq:short_average']}).
• Figure 3: Example of a diagram where the $\zeta_L$ dependence of the functions $F_A$ and $F_B$ appearing in the average over short modes of $\langle\sl(x_1) \sl(x_2) \rangle_{\rm short}$ (top shaded region) is generated by some interaction in the past (shaded regions). If the ellipses enclosing $x_1$ and $x_2$ contains a correlation function made of short modes that connect $x_1$ and $x_2$ we obtain a $SC$ diagram, and it contributes to the first term of (\ref{['eq:short_average']}). If instead $x_1$ and $x_2$ are connected only by long modes, we have a $LC$ diagram, and a contribution to the second term of (\ref{['eq:short_average']}). After taking expectation value over the long modes, we obtain a contribution respectively to the first and the second line of (\ref{['eq:long_short_average']}).