Tunneling probability for the birth of an asymptotically DeSitter universe

TL;DR

This paper quantizes a cosmological model with a positive cosmological constant and radiation, solving the Wheeler-DeWitt equation numerically to analyze the tunneling probability for universe creation, considering boundary conditions at zero scale factor.

Contribution

It introduces a more general approach by including the boundary condition at zero scale factor, improving upon previous WKB-based methods.

Findings

Tunneling probability increases with the cosmological constant.

The boundary condition at zero scale factor affects tunneling outcomes.

Numerical solutions provide detailed tunneling probabilities as a function of initial energy.

Abstract

In the present work, we quantize a closed Friedmann-Robertson-Walker model in the presence of a positive cosmological constant and radiation. It gives rise to a Wheeler-DeWitt equation for the scale factor which has the form of a Schr\"{o}dinger equation for a potential with a barrier. We solve it numerically and determine the tunneling probability for the birth of a asymptotically DeSitter, inflationary universe, initially, as a function of the mean energy of the initial wave-function. Then, we verify that the tunneling probability increases with the cosmological constant, for a fixed value of the mean energy of the initial wave-function. Our treatment of the problem is more general than previous ones, based on the WKB approximation. That is the case because we take into account the fact that the scale factor ($a$) cannot be smaller than zero. It means that, one has to introduce an infinity potential wall at $a = 0$, which forces any wave-packet to be zero there. That condition introduces new results, in comparison with previous works.