Time-covariant Schr\"{o}dinger equation and invariant decay probability: The $\Lambda$-Kantowski-Sachs universe

Abstract

The system under study is the $\Lambda$-Kantowski-Sachs universe. Its canonical quantization is provided based on a recently developed method: the singular minisuperspace Lagrangian describing the system, is reduced to a regular (by inserting into the dynamical equations the lapse dictated by the quadratic constraint) possessing an explicit (though arbitrary) time dependence; thus a time-covariant Schr\"{o}dinger equation arises. Additionally, an invariant (under transformations $t=f(\tilde{t})$) decay probability is defined and thus ``observers'' which correspond to different gauge choices obtain, by default, the same results. The time of decay for a Gaussian wave packet localized around the point $a=0$ (where $a$ the radial scale factor) is calculated to be of the order $\sim 10^{-42}-10^{-41}\mathrm{s}$. The acquired value is near the end of the Planck era (when comparing to a FLRW universe), during which the quantum effects are most prominent. Some of the results are compared to those obtained by following the well known canonical quantization of cosmological systems, i.e. the solutions of the Wheeler-DeWitt equation.