The Energy of the Universe, the Scaling-Dependent Coupling Constant β and the Pion Mass

From the hypothesis of a quadratic dependence of the binding energy of the quarks on the length scale and from the idea of a universal running coupling constant, we get the closure mass of the Universe and the pion mass. Besides this, we propose a free energy for the pions which stationarity condition leads to a closed form for the pion mass as a function of the Planck mass and of the Hubble constant.