Darboux Parameter for Frw “quantum” Universes

I perform the factorization(s) of the Wheeler-DeWitt equation for the FRW minisuperspace universes of arbitrary Hartle-Hawking factor ordering introducing by this means a new, Darboux-type, free parameter in “quantum” cosmology. This parameter can be thought of as giving the time scale of a force due to a cosmological field. Mathematically speaking, the parameter is a Riccati integration constant. PACS number(s): 98.80.Hw, 11.30.Pb In cosmology one can encounter arbitrary parameters that are difficult to determine by experiment and theory. One famous example is Einstein’s cosmological constant [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 FriedmannRobertson-Walker (FRW) universes, I introduce an arbitrary parameter of Darboux type for this class of universes. Recalling that factorizations of second order linear one-dimensional differential operators are common tools in Witten’s supersymmetric quantum mechanics [3] and imply particular solutions of Riccati equations known as superpotentials, I would like to explore here a more general factoring of the WDW-FRW equation for arbitrary Hartle-Hawking factor ordering Q [6] by means of the general solution of the Riccati equation, a procedure which has been first used in physics by Mielnik [4] for the quantum harmonic oscillator. Stated differently, I shall exploit the non-uniqueness of the factorization of second-order linear differential operators on the simple example of the WDW-FRW differential equation with Hartle-Hawking factor ordering,