Wheeler–DeWitt universe wave function in the presence of stiff matter

We study the Wheeler–DeWitt (WDW) equation close to the Big Bang. We argue that an interaction-dominated fluid (speed of sound equal to the speed of light), if present, would dominate during such an early phase. Such a fluid with p = ρ ∝ 1/a6 generates a term in the potential of the wave function of the WDW equation proportional to −1/a2. This very peculiar potential, which embodies a spontaneous breaking of dilatation invariance, has some very remarkable consequences for the wave function of the Universe: Ψ(a) vanishes at the Big Bang: Ψ(0) = 0; the wave function Ψ(a) is always real; a superselection rule assures that the system is confined to a ≥ 0 without the need of imposing any additional artificial barrier for unphysical negative a. These results are valid for a continuous class of choices of the operator ordering of the WDW equation.