Universality Class in Conformal Inflation

TL;DR

This work establishes a universality class of conformal and superconformal inflation models in which the inflationary potential is exponentially flattened by the stretching of moduli space when moving to the Einstein frame. The approach shows that a broad family of constructions, including the T-Model and variations that reproduce Starobinsky- and Higgs-type potentials, yield the same leading predictions $n_s = 1 - 2/N$ and $r = 12/N^2$, independent of detailed microphysics. By exploiting gauges related to rapidity, the authors connect conformal symmetry, Kähler geometry, and SUSY embeddings, revealing a robust mechanism for inflation even for steep original potentials. The results also frame a cosmological version of critical phenomena, with attractor points controlled by enhanced symmetry, and extend to superconformal theories with stable moduli and plateau-like potentials that align with Planck data.

Abstract

We develop a new class of chaotic inflation models with spontaneously broken conformal invariance. Observational consequences of a broad class of such models are stable with respect to strong deformations of the scalar potential. This universality is a critical phenomenon near the point of enhanced symmetry, SO(1,1), in case of conformal inflation. It appears because of the exponential stretching of the moduli space and the resulting exponential flattening of scalar potentials upon switching from the Jordan frame to the Einstein frame in this class of models. This result resembles stretching and flattening of inhomogeneities during inflationary expansion. It has a simple interpretation in terms of velocity versus rapidity near the Kahler cone in the moduli space, similar to the light cone of special theory of relativity. This effect makes inflation possible even in the models with very steep potentials. We describe conformal and superconformal versions of this cosmological attractor mechanism.

Figures (5)

• Figure 1: Potentials for the T-Model inflation ${\tanh}^{2n}(\varphi/\sqrt6)$ for $n = 1,2,3,4$ (blue, red, brown and green, corresponding to increasingly wider potentials). We took $\lambda_{n} = 1$ for each of the potential for convenience of comparison. As we see, 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 $V = 1-3/2N = 0.975$ correspond to the points where the perturbations are produced in these models on scale corresponding to $N = 60$. Asymptotic height of the potential is the same for all models of this class, see (\ref{['energylevel']}).
• Figure 2: Models of conformal inflation based on generalizations of the Starobinsky model, with $F({\phi/\chi}) \sim {\phi^{2n}\over \chi^{2n-2}(\phi+\chi)^{2}}$, $n = 1,2,3,4$.
• Figure 3: Flattening of the sinusoidal potential $V(\phi)$ near the boundary of the moduli space $\phi = \sqrt 6$ by boost in the moduli space, $V(\phi) \to V(\sqrt 6 \, \tanh {\varphi\over \sqrt{6}})$. Inflationary plateau of the function $V(\phi)$ appears because of the exponential stretching of the last growing part of the sinusoidal function $V(\phi)$.
• Figure 4: Flattening of a generic potential near the boundary of the moduli space by boost in the moduli space $V(\phi) \to V(\sqrt 6 \, \tanh {\varphi\over \sqrt{6}})$. In essence, what we see is 'inflation of the inflationary landscape' at the boundary of the moduli space, which solves the flatness problem of the inflationary potential required for inflation in the landscape.
• Figure 5: Supergravity version of the T-Model. The $S$-direction is steep, the system quickly reaches the minimum at $S=0$ and evolves in the inflaton direction $\varphi$ at stabilized $S$.