Abstract

Observations reveal the cosmos to be astonishingly simple, and yet deeply puzzling, on the largest observable scales. Why is the mean spatial geometry so close to flat? Why is the universe so uniform and isotropic on large scales? Why is there a cosmological constant and what fixes its value? How did everything we see emerge from a singular point in the past, and how did time emerge? Why do the density fluctuations take such a simple form? Many lines of evidence point to the need for a quantum theory of cosmology, and Feynman’s path integral for quantum gravity provides a sensible starting point. One attractive idea is the ``no boundary” hypothesis, that the past consisted of a compact four-geometry. I shall show how Picard-Lefschetz theory allows one to make sense of the Lorentzian (but not the Euclidean) path integral in this context, revealing the ``no boundary” and ``tunneling” proposals to be identical. One can then prove a theorem to the effect that no smooth, semi-classical quantum beginning of spacetime of the kind envisaged by Hartle and Hawking or Vilenkin is possible. I shall present a new proposal for the probability in quantum cosmology, which ``explains”the big bang singularity and offers new approaches to the above-mentioned puzzles.