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

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 $\Delta \varphi$ 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 $\varphi$ 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 $\frac{\Delta \varphi }{M_{\mathrm{Pl}}}<0.2\times \sqrt{10}\simeq 0.632$. For $n_s=0.9649$, $N_e=55$, and $\frac{\Delta \varphi }{M_{\mathrm{Pl}}}<0.632$, we predict that the tensor-to-scalar ratio is smaller than 0.0012 for such polynomial inflation to be consistent with QG/EFT.

arXiv comment12 pages

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