Fisher order measure and Petri's universe

Abstract

Given a closed system described via an amplitude function ψ(x), what is its level of order? We consider here a quantity R defined by the property that it decreases (or stays constant) after the system is coarse grained. It was recently found that (i) this quantity exhibits a series of properties that make it a good order-quantifier and (ii) for a very simple model of the universe the Hubble expansion does not in itself lead to changes in the value of R. Here we determine the value of the concomitant invariant for a somewhat more involved universe-model recently advanced by Petri. The answer is simply R=2( rHr0-r0rH), where rH is a model's parameter and r0 is the Planck length. Thus, curiously, the Petri-order seems to be a geometric property and not one of its mass-energy levels. Numerically, R=26.0×1060. This is a colossal number, which approximates other important cosmological constants such as the ratio of the mass of a typical star to that of the electron ∼ 1060, and microlevel constants such as exp(1α), where α is the fine structure constant.