Effect upon universal order of Hubble expansion

The level of order R in a spherical system of radius r 0 with a probability amplitude function ψ ( x ) , x = r , θ , ϕ obeys R = ( 1 / 2 ) r 0 2 I , where I = 4 ∫ d x | ∇ ψ | 2 is its Fisher information level. We show that a flat space universe obeying the Robertson–Walker metric has an invariant value of the order as it undergoes either uniform Hubble expansion or contraction. This means that Hubble expansion per se does not cause a loss of universal order as time progresses. Instead, coarse graining processes characterizing decoherence and friction might cause a loss of order. Alternatively, looking backward in time, i.e. under Hubble contraction, as the big bang is approached and the Hubble radius r 0 approaches small values, the structure in the amplitude function ψ ( x ) becomes ever more densely packed, increasing all local slopes ∇ ψ and causing the Fisher information I to approach unboundedly large values. As a speculation, this ever-well locates the initial position of the universe in a larger, multiverse.