Tunneling probability for the birth of an asymptotically de Sitter universe

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\"odinger 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.