No Smooth Beginning for Spacetime.

We identify a fundamental obstruction to any theory of the beginning of the Universe, formulated as a semiclassical path integral. The Hartle-Hawking 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 mathematically meaningful in this approach, but the Euclidean version is not. The Lorentzian-Picard-Lefschetz formulation yields unambiguous predictions. 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.