Isolating a Minimum Radius of the Universe Consistent with the Production of at Least 1 Unit of Entropy, at the Start of Inflation

We begin by examining a general expression of entropy, and its links to a minimum radius of the universe. We derive an expression for the production of at least 1 unit of entropy, which translates to a value of Planck length in radii to 1000 times Planck radii, for the quantum bubble of space-time which dependent upon, of all things, the initial Hubble expansion rate value. If the Hubble parameter has the value of 10^ 10 GeV, we see a minimum radial length of the Universe of about 1 billion times Planck length. If the Hubble parameter is of 10^19 GeV, the minimum radial length of the universe would be about one Planck length, which is surprising, to put it mildly. The higher the initial temperate is, up to a point, the more likely the initial entropy is closer to the Causal barrier mentioned in an earlier publication by the author Keywords. Causal barrier, Inflaton, Minimum entropy (non-zero), initial radii (of universe)