DARBOUX PARAMETER FOR EMPTY FRW QUANTUM UNIVERSES AND QUANTUM COSMOLOGICAL SINGULARITIES

I present the factorization(s) of the Wheeler-DeWitt equation for vacuum FRW minisuperspace universes of arbitrary Hartle-Hawking factor ordering, including the so-called strictly isospectral supersymmetric method. By the latter means, one can introduce an infinite class of singular FRW minisuperspace wavefunctions characterized by a Darboux parameter that mathematically speaking is a Riccati integration constant, while physically determines the position of these strictly isospectral singularities on the Misner time axis. In cosmology one can encounter arbitrary parameters that are difficult to determine by experiment and theory. One famous example is Einstein’s cosmological constant, which does set the time scale in the vacuum dominated universe. 1 Another, very recent example is Immirzi’s parameter in quantum general relativity. 2 In this work, using the factorization(s) of the Wheeler-DeWitt (WDW) minisuperspace equation for the vacuum Friedmann-Robertson-Walker (FRW) universes, an infinite class of singular FRW wavefunctions are introduced containing an arbitrary parameter of Darboux type3 determining the position of the singularities during