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Wave function of the Universe as a sum over eventually inflating universes
Journal
Physical Review D
ISSN
24700010
Date Issued
2022-07-15
Author(s)
Rajeev, Karthik
Abstract
We consider a proposal to define the wave function of the Universe as a sum over spacetimes that eventually inflate. In the minisuperspace model, we explicitly show that a simple family of initial conditions, parametrized by a positive real number a0, can be imposed to realize this prescription. The resulting wave function is found to be proportional to the Hartle-Hawking wave function and its dependence on a0 is only through an overall phase factor. Motivated by this observation, we ask whether it is possible to analytically extend a0 to an extended region D¯ in complex a0 plane, while retaining the Hartle-Hawking form of the wave function. We use the condition for convergence of path integral and a recent theorem due to Kontsevich and Segal, further extended by Witten, to explicitly find D¯. Interestingly, a special point on the boundary of D¯ recovers the exact no-boundary wave function. Following that, we show that our prescription leads to a family of quantum states for the perturbations, which give rise to scale-invariant power spectra. If we demand, as an extra ingredient to our prescription, a matching condition at the "no-boundary point"in D¯, we converge on a unique quantum state for the perturbations.