Strong restriction on inflationary vacua from the local gauge invariance III: Infrared regularity of graviton loops

T Tanaka,Y Urakawa

doi:10.1093/ptep/ptu071

Abstract

It has been claimed that the super-Hubble modes of the graviton generated during inflation can make loop corrections diverge. Even if we introduce an infrared (IR) cutoff at a comoving scale as an ad hoc but practical method of regularization, we encounter secular growth, which may lead to the breakdown of perturbative expansion for a sufficiently long-lasting inflation. In this paper, we show that the IR pathology concerning the graviton can be attributed to the presence of residual gauge degrees of freedom in the local observable universe, as in the case of the adiabatic curvature perturbation. We will show that choosing the Euclidean vacuum as the initial state ensures invariance under the above-mentioned residual gauge transformations. We will also show that, as long as we consider a gauge invariant quantity in the local universe, we encounter neither IR divergence nor secular growth. The argument in this paper applies to general single-field models of inflation up to a sufficiently high order in perturbation.

Highlights

The inflationary spacetime leads to the generation of gravitational waves
We will show that, when we choose the Euclidean vacuum as the initial state, the npoint functions of Rx g ζ and Rx δgγij no longer suffer from the IR divergence (IRdiv), IR secular growth (IRsec), and Secular growth (SG)
We have addressed the regularity of the graviton loops

Summary

Introduction

The inflationary spacetime leads to the generation of gravitational waves. Even though the amplitude of gravitational waves is smaller than the amplitude of the adiabatic curvature perturbation, detection of the primordial gravitational waves generated during inflation is expected to be within our reach. Tsamis and Woodard [106] pointed out that using the geodesic normal coordinates can introduce an additional origin of UV divergence, yielding contributions that may not be renormalized by local counter terms [107] This is presumably because specifying the spatial distance precisely in the presence of the gravitational perturbation requires taking account of all short-wavelength modes. We will show that, when we choose the Euclidean vacuum as the initial state, the n-point functions for g ζ (x) and δgγij (x) can be expanded only in terms of the interaction picture fields ζ I (x) and δ γi j I (x) with the IR-suppressing operators Rx. While the interaction Hamiltonian density. We should stress that the SG never appears in slow-roll inflation, unless the order of perturbative expansion N takes an extremely large value, such as 1/ε1 O(102 )

Concluding remarks

Solving the equations of motion

Computation of g ζ

One-loop corrections

IR regularity condition on the mode function