RESUMO
Realizing the no-boundary proposal of Hartle and Hawking as a consistent gravitational path integral has been a long-standing puzzle. In particular, it was demonstrated by Feldbrugge, Lehners, and Turok that the sum over all universes starting from a zero size results in an unstable saddle point geometry. Here we show that, in the context of gravity with a positive cosmological constant, path integrals with a specific family of Robin boundary conditions overcome this problem. These path integrals are manifestly convergent and are approximated by stable Hartle-Hawking saddle point geometries. The price to pay is that the off-shell geometries do not start at a zero size. The Robin boundary conditions may be interpreted as an initial state with Euclidean momentum, with the quantum uncertainty shared between the initial size and momentum.
RESUMO
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.
RESUMO
In inflationary models, the predicted amplitude of primordial density perturbations Q is much larger than the observed value (â¼10(-5)) for natural choices of parameters. To explain the requisite exponential fine-tuning, anthropic selection is often invoked, especially in cases where microphysics is expected to produce a complex energy landscape. By contrast, we find examples of ekpyrotic models based on heterotic M theory for which dynamical selection naturally favors the observed value of Q.