The Upper Bound on the Tensor-to-Scalar Ratio Consistent with Quantum Gravity

TL;DR

This paper investigates the maximum tensor-to-scalar ratio compatible with quantum gravity constraints in polynomial inflation, finding it to be less than 0.0012, which challenges the detectability of primordial gravitational waves.

Contribution

It introduces a practical bound on inflaton excursion and demonstrates that the tensor-to-scalar ratio must be below 0.0012 under quantum gravity and effective field theory constraints.

Findings

Tensor-to-scalar ratio < 0.0012 for polynomial inflation

Inflaton excursion bound: Δφ/Mpl < 0.632

Violates the traditional Lyth bound

Abstract

We consider the polynomial inflation with the tensor-to-scalar ratio as large as possible which can be consistent with the Quantum Gravity (QG) corrections and Effective Field Theory (EFT). To get a minimal field excursion $\Delta\phi$ for enough e-folding number $N$, the inflaton field traverses an extremely flat part of the scalar potential, which results in the Lyth bound to be violated. We get a CMB signal consistent with Planck data by numerically computing the equation of motion for inflaton $\phi$ and using Mukhanov-Sasaki formalism for primordial spectrum. Inflation ends at Hubble slow-roll parameter $\epsilon_1^H=1$ or $\ddot{a}=0$. Interestingly, we find an excellent practical bound on the inflaton excursion in the format $a+b{\sqrt r}$, where $a$ is a tiny real number and $b$ is at the order 1. To be consistent with QG/EFT and suppress the high-dimensional operators, we show that the concrete condition on inflaton excursion is $\frac{\Delta \phi}{M_{\rm Pl}} < 0.2 \times \sqrt{10}\simeq 0.632$. For $n_s=0.9649$, $N_e=55$, and $\frac{\Delta \phi}{M_{\rm Pl}} < 0.632$, we predict that the tensor-to-scalar ratio is smaller than 0.0012 for such polynomial inflation to be consistent with QG/EFT.