Non-minimal Inflationary Attractors

TL;DR

The paper investigates inflation with negative non-minimal coupling $\xi<0$ and shows that a broad class of models exhibits an attractor mechanism: as $|\xi|$ increases, predictions for the spectral index $n_s$ and tensor-to-scalar ratio $r$ rapidly converge to the universal values $1-n_s=2/N$ and $r=12/N^2$, matching the conformal coupling case $\xi=-1/6$ and the $\xi\to-\infty$ limit. It develops this result first in a conformal (T-Model) setup where $V(\varphi)=\lambda_n\tanh^{2n}(\varphi/\sqrt{6})$, then extends to general $\xi<0$ showing the flattening of potentials in the Einstein frame near the moduli-space boundary. It also discusses Higgs-type potentials and other attractors, illustrating the robustness of the predictions across a wide range of non-minimally coupled theories. These findings support a model-independent target for cosmological observations and bolster the case for tensor-mode searches by tying diverse models to a common attractor framework.

Abstract

Recently we identified a new class of (super)conformally invariant theories which allow inflation even if the scalar potential is very steep in terms of the original conformal variables. Observational predictions of a broad class of such theories are nearly model-independent. In this paper we consider generalized versions of these models where the inflaton has a non-minimal coupling to gravity with ξ<0 different from its conformal value ξ= -1/6. We show that these models exhibit attractor behavior. With even a slight increase of |ξ| from |ξ| = 0, predictions of these models for n_s and r rapidly converge to their universal model-independent values corresponding to conformal coupling ξ= -1/6. These values of n_s and r practically coincide with the corresponding values in the limit of infinitely large negative ξ.

Figures (6)

• Figure 1: Potentials of the T-Model $V(\varphi) \sim {\tanh}^{2n}(\varphi/\sqrt6)$ for $n = 1,2,3,4$ (blue, red, brown and green, corresponding to increasingly wider potentials). These potentials differ from each other quite considerably, especially at $\varphi \lesssim 1$: at small $\phi$ they behave as $\varphi^{2n}$. Nevertheless all of these models predict the same values $n_{s} =1-2/N$, $r = 12/N^{2}$, in the leading approximation in $1/N$, where $N\sim 60$ is the number of e-foldings. The points where each of these potentials cross the red dashed line $V = 1-3/2N = 0.96$ correspond to the points where the perturbations are produced in these models on scale corresponding to $N = 60$. Asymptotic height of the potential is determined by the required amplitude of perturbations of metric; it is the same for all models of this class.
• Figure 2: Basic mechanism which leads to inflation in the theories with generic functions $F(\phi/\sqrt6)$. The potential can have a nearly arbitrary shape in terms of variables $\phi$. If this potential is non-singular at the boundary of the moduli space, it looks exponentially stretched and flat at large values of the canonically normalized field $\varphi$. This stretching makes inflation very natural, and leads to universal observational predictions for a very broad class of such models Kallosh:2013hoa.
• Figure 3: The potential $V(\varphi)$ for $F(\sqrt{|\xi|}\phi) \sim \phi^{2}$. The upper curve corresponds to $\xi = 0$, other curves correspond to $\xi$ = -1/24, -1/12, -1/8 and -1/6.
• Figure 4: The potential $V(\varphi)$ for $F(\sqrt{|\xi|}\phi) \sim \phi^{4}$. The upper curve corresponds to $\xi = 0$, other curves correspond to $\xi$ = -1/24, -1/12, -1/8 and -1/6.
• Figure 5: Behavior of $n_{s}$ and $r$ in nonminimal T-Model with quadratic and quartic potentials as a function of the parameter $\xi$ in the interval from $0$ to $-1/6$, for the number of e-foldings $N = 60$. The dark violet curve beginning with yellow star ($\xi = 0$) corresponds to the theory $\lambda\phi^{4}/4$. The orange curve beginning with the green star corresponds to the theory $m^{2}\phi^{2}/2$. The red line corresponds to the more conventional chaotic inflation in the theory $\lambda\phi^{4}/4$ with $\xi > 0$. In all of these theories, the point $\xi =0$ is a strong repeller: Even a tiny $\xi$ dramatically changes predictions of the theory. One can see it by looking at the red stars, which correspond to $\xi = -0.01$ for the T-model, and $\xi = 0.01$ for the usual chaotic inflation with $\xi > 0$ (red line). Increase of $|\xi|$ beyond $1/6$ practically does not change predictions of the models. They reach at attractor shown by the blue star, which shows the unique prediction of a broad class of models of conformal chaotic inflation, including the T-model and the Starobinsky model Kallosh:2013hoa.