Scale-Invariance and the Strong Coupling Problem

TL;DR

The paper analyzes whether scale-invariant curvature fluctuations with weak interactions can arise around generic FRW backgrounds in a single-clock EFT framework. It shows that scale invariance of the two-point function does not guarantee weak coupling of interactions, and only backgrounds near de Sitter space maintain perturbative control over the observable range; non-attractor and strongly time-dependent setups typically hit strong coupling within ~10 e-folds. Through a nonlinear Emden–Fowler formulation, it demonstrates that the attractor case reduces to a background equation for which exact and approximate solutions reproduce inflation and ekpyrosis as limits, but generally lead to strong coupling away from quasi-de Sitter. Extensions to varying sound speed and higher-derivative terms do not provide robust non-de Sitter weakly-coupled scale-invariant solutions, strengthening the case that inflation uniquely realizes the observed scale invariance with weak coupling.

Abstract

The effective theory of adiabatic fluctuations around arbitrary Friedmann-Robertson-Walker backgrounds - both expanding and contracting - allows for more than one way to obtain scale-invariant two-point correlations. However, as we show in this paper, it is challenging to produce scale-invariant fluctuations that are weakly coupled over the range of wavelengths accessible to cosmological observations. In particular, requiring the background to be a dynamical attractor, the curvature fluctuations are scale-invariant and weakly coupled for at least 10 e-folds only if the background is close to de Sitter space. In this case, the time-translation invariance of the background guarantees time-independent n-point functions. For non-attractor solutions, any predictions depend on assumptions about the evolution of the background even when the perturbations are outside of the horizon. For the simplest such scenario we identify the regions of the parameter space that avoid both classical and quantum mechanical strong coupling problems. Finally, we present extensions of our results to backgrounds in which higher-derivative terms play a significant role.

Figures (4)

• Figure 1: Slow contraction: plot of $\log \epsilon$ and $\log \eta$ vs. $N$ for $\epsilon_i = 10^{-8}$ and $z_i = 0.01$.
• Figure 2: The value of the $\eta$ parameter 10 $e$-folds after the beginning of the scaling phase illustrates the strong coupling problem arising in backgrounds that deviate too much from quasi-de Sitter ($z_i = - 1 - \epsilon_i$).
• Figure 3: The value of the $\eta$ parameter 10 $e$-folds after the beginning of the scaling phase illustrates the strong coupling problem arising in backgrounds that deviate too much from quasi-de Sitter ($z_i = - 1 - \epsilon_i$).
• Figure 4: Plot of $\widehat{f}_{\rm NL}$ vs. $\beta$ for $y_i = e^{-10} y_{\rm end}$ (blue) and $y_i = e^{-50} y_{\rm end}$ (red). We show two cases: $f_X^{(0)} = 1$ (DBI) ($\Rightarrow \widehat{f}_{\rm NL} > 0$) and $(f_X^{(0)} - 1)= 1$ ($\Rightarrow \widehat{f}_{\rm NL} < 0$). The plot implies an upper limit on $\beta$ but no lower limit. This difference from the result of Justin arises from corrections to their result for the bispectrum (see Appendix \ref{['sec:bispectra']}).