The trouble with asymptotically safe inflation

Abstract

In this paper we investigate the perturbation theory of asymptotically safe inflation and we find that all modes of gravitational waves perturbation become ghosts in order to achieve a large enough number of e-folds. Formally we can calculate the power spectrum of gravitational waves perturbation, but we find that it is negative. It indicates that there is serious trouble with asymptotically safe inflation.

Appendix

We start with a general scalar perturbed metric about the flat FRW background as follows:

(A.1)

The gauge transformations are

(A.2)

(A.3)

(A.4)

(A.5)

In this paper we will work on the longitudinal gauge which corresponds to the gauge choice $\hat{\beta}=0$ and $\hat{\gamma}=0$ by setting the $\delta t=a(\beta+a\dot{\gamma})$ and δx=γ. In this gauge, the line element becomes

(A.6)

where we omitted the hat for perturbed quantities and τ is conformal time which is related to time t by

(A.7)

The equations of motion for α and ζ are governed by the action in Eq. (2.1). In the Longitudinal gauge, after a lengthy but straightforward calculation, we obtain the action for perturbations as follows:

(A.8)

where

(A.9)

(A.10)

and

(A.11)

From the above action, the equations of motion for α and ζ become

(A.12)

and

(A.13)

Similar to Sect. 3, ignoring the terms with ϵ and κ, the action for α and ζ becomes

(A.14)

We see that ghosts also emerge in the above action.