Abstract
The usual description of inflationary fluctuations uses the framework of quantum field theory (QFT) in curved spacetime, in which quantum fluctuations are superimposed on a classical background spacetime. Even for large fluctuations, such as those envisioned during a regime of eternal inflation, this framework is frequently used. In the present work we go one step beyond this description by quantising both the scalar field and the scale factor of the universe. Employing the Lorentzian path integral formulation of semi-classical gravity we restrict to a simplified minisuperspace setting by considering homogeneous transitions. This approach allows us to determine the dominant geometry and inflaton evolution contributing to such amplitudes. We find that for precisely specified initial scale factor and inflaton values (and uncertain momenta), two distinct saddle point geometries contribute to the amplitude, leading to interference effects. However, when the momenta of both scale factor and inflaton are specified with sufficient certainty, only a single saddle point is relevant and QFT in curved spacetime is applicable. In particular we find that for inflaton transitions up the potential, meaningful results are only obtained when the initial uncertainty in the inflaton value is large enough, allowing the dominant evolution to be a complexified slow-roll solution \emph{down} from a comparatively unlikely position higher up in the potential.
Highlights
A beautiful idea of modern cosmology is that the origin of the largest structures in the universe may lie in primordial quantum fluctuations [1,2]
The amplification itself is calculated within the framework of quantum field theory (QFT) in curved spacetime
It is important to gain an understanding of the quantization of the entire system in order to assess under what circumstances the approximation of QFT in curved spacetime breaks down, and to see what might replace it in such a regime
Summary
A beautiful idea of modern cosmology is that the origin of the largest structures in the universe may lie in primordial quantum fluctuations [1,2]. All of nature should be described by the same theory, and the background should be thought of as being just as much part of the quantum wave function as the fluctuations Beyond this conceptual consideration, it is important to gain an understanding of the quantization of the entire system in order to assess under what circumstances the approximation of QFT in curved spacetime breaks down, and to see what might replace it in such a regime. A general feature that we observe is that the transition amplitude is governed by contributions from two saddle points when the uncertainty in the initial value of the scalar field is small, but with large uncertainty in the inflaton velocity In this case a description in terms of QFT in curved spacetime breaks down, as two separate backgrounds contribute significantly.
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