No smooth beginning for spacetime

TL;DR

A new mathematical tool is introduced-Picard-Lefschetz theory-for defining the semiclassical path integral for gravity, and it is proved that primordial tensor (gravitational wave) fluctuations are unsuppressed.

Abstract

We identify a fundamental obstruction to any theory of the beginning of the universe, formulated as a semiclassical path integral. Hartle and Hawking's no boundary proposal and Vilenkin's tunneling proposal are examples of such theories. Each may be formulated as the quantum amplitude for obtaining a final 3-geometry by integrating over 4-geometries. We introduce a new mathematical tool - Picard-Lefschetz theory - for defining the semiclassical path integral for gravity. The Lorentzian path integral for quantum cosmology with a positive cosmological constant is meaningful in this approach, but the Euclidean version is not. Framed in this way, the resulting framework and predictions are unique. Unfortunately, the outcome is that primordial tensor (gravitational wave) fluctuations are unsuppressed. We prove a general theorem to this effect, in a wide class of theories.

Figures (2)

• Figure 1: Left: the smooth, regular picture of the no boundary background. Middle: the no boundary picture with hoped-for small fluctuations, in agreement with observations. Right: the fluctuations implied by the more rigorous, Lorentzian-Picard-Lefschetz approach developed here. Our analysis shows that, to leading semiclassical order, large fluctuations are preferred, leading to a breakdown of the theory.
• Figure 2: Left: Analytic continuation contours (red) in the Hartle-Hawking (H-H) and Picard-Lefschetz (P-L) descriptions, above and below the branch cut in the complex $q$-plane. Right: Corresponding contours for the conformal time.