[Paper Review] Elliptic Inflation: Interpolating from natural inflation to $R^2$-inflation

Tetsutaro Higaki, Fuminobu Takahashi|arXiv (Cornell University)|Jan 10, 2015

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proposes elliptic inflation, a model where the inflaton potential is derived from Jacobi elliptic functions, theta functions, or the Dedekind eta function—functions that naturally arise in string compactifications. It shows that such models interpolate between natural inflation and $R^2$-inflation, predicting a spectral index $n_s \approx 0.967$ for $N=60$ e-foldings, with a potential for sizable running via the Dedekind eta function.

ABSTRACT

We propose an extension of natural inflation, where the inflaton potential is a general periodic function. Specifically, we study elliptic inflation where the inflaton potential is given by Jacobi elliptic functions, Jacobi theta functions or the Dedekind eta function, which appear in gauge and Yukawa couplings in the string theories compactified on toroidal backgrounds. We show that in the first two cases the predicted values of the spectral index and the tensor-to-scalar ratio interpolate from natural inflation to exponential inflation such as $R^2$- and Higgs inflation and brane inflation, where the spectral index asymptotes to $n_s = 1-2/N \simeq 0.967$ for the e-folding number $N = 60$. We also show that a model with the Dedekind eta function gives a sizable running of the spectral index due to modulations in the inflaton potential. Such elliptic inflation can be thought of as a specific realization of multi-natural inflation, where the inflaton potential consists of multiple sinusoidal functions. We also discuss examples in string theory where Jacobi theta functions and the Dedekind eta function appear in the inflaton potential.

Motivation & Objective

• To extend natural inflation by replacing the single cosine potential with more general periodic functions arising in string compactifications.
• To explore whether periodic potentials based on Jacobi elliptic functions and theta functions can realize inflation models that interpolate between natural inflation and $R^2$-inflation.
• To investigate the cosmological predictions—specifically the spectral index $n_s$ and tensor-to-scalar ratio $r$—of such models in the slow-roll regime.
• To examine the possibility of generating a sizable running of the spectral index through modulations in the inflaton potential using the Dedekind eta function.
• To provide a UV-completion of multi-natural inflation by identifying the inflaton with open string moduli in type IIB orientifolds with flux compactification.

Proposed method

• Modeling the inflaton potential using Jacobi elliptic functions, such as $ V(\phi) = 2\Lambda^4(1 - \text{cn}^2(\phi/(2F), k)) $, where $k$ is the elliptic modulus.
• Using Jacobi theta functions $\vartheta_3$ and the Dedekind eta function $\eta(\tau)$ as inflaton potentials, motivated by their appearance in gauge and Yukawa couplings in toroidally compactified string theories.
• Analyzing the slow-roll parameters and cosmological observables ($n_s$, $r$) in the large-$N$ limit, showing interpolation between natural inflation and exponential inflation.
• Deriving the inflaton potential from the real part of light open string moduli in type IIB orientifold compactifications with flux stabilization.
• Applying the alignment mechanism to enhance the decay constant $F$ beyond the Planck scale using multiple light moduli.
• Evaluating reheating via operators involving the inflaton coupling to SM gauge bosons, top quarks, and right-handed neutrinos to assess thermal leptogenesis viability.

Experimental results

Research questions

• RQ1 Can periodic inflaton potentials based on elliptic functions reproduce the observed cosmological parameters, particularly $n_s \approx 0.967$?
• RQ2 How do the predictions of the spectral index $n_s$ and tensor-to-scalar ratio $r$ evolve as the elliptic modulus $k$ varies from 0 to 1?
• RQ3 Can the Dedekind eta function in the inflaton potential generate a significant running of the spectral index $dn_s/d\ln k$?
• RQ4 What is the UV origin of such periodic potentials in string theory, particularly in toroidal compactifications?
• RQ5 Is it possible to achieve a super-Planckian decay constant $F$ in this framework through moduli alignment?

Key findings

• The model with Jacobi elliptic functions interpolates between natural inflation and $R^2$-inflation, with the spectral index asymptoting to $n_s = 1 - 2/N \approx 0.967$ for $N=60$ e-foldings.
• The tensor-to-scalar ratio $r$ decreases from the natural inflation regime toward the $R^2$-inflation regime, consistent with Planck constraints.
• The model based on the Dedekind eta function generates a sizable running of the spectral index, with $dn_s/d\ln k \sim -0.01$ achievable over CMB scales.
• The inflaton potential can be naturally realized in string compactifications, particularly in type IIB orientifolds with flux compactification, where the inflaton corresponds to the real part of the lightest open string modulus.
• A super-Planckian decay constant $F$ can be achieved via the alignment mechanism in the presence of multiple light moduli, with enhancement factors up to $I/|J_1 - J_2|$.
• Reheating via inflaton couplings to SM fields is viable, with the potential to support thermal leptogenesis at reheating temperatures above $10^9$ GeV.