# No smooth beginning for spacetime

## TL;DR

This paper introduces Picard-Lefschetz theory to define the Lorentzian path integral in quantum cosmology, revealing an obstruction to smooth beginnings of the universe and showing that primordial gravitational waves are unsuppressed.

## Contribution

It presents a novel mathematical approach to quantum gravity path integrals and demonstrates a fundamental obstruction to smooth universe origins.

## Key findings

- Lorentzian path integral is well-defined with positive cosmological constant
- Euclidean path integral is not well-defined in this framework
- Primordial tensor fluctuations are not suppressed, implying observable gravitational waves

## 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.