On the cosmological variation of the fine structure constant

A phenomenological model is proposed to explain the recent observed cosmological variation of the fine structure constant as an effect of the quantum vacuum, assuming a flat universe with cosmological constant $\Lambda$ in the cases ($\Omega_M, \Omega_\Lambda$) equal to (0.3,0.7) and (1,0). Because of the fourth Heisenberg relation, the lifetime of the virtual pairs of the zero-point radiation must depend on the gravitational potential, so that the quantum vacuum changes its density and acquires a relative permittivity different from one. Since the matter was more concentrated in the past, the gravitational potential of all the universe was stronger and the optical density of the vacuum higher, the electron charge being then more renormalized and smaller than now. The model is based on a first order Newtonian approximation that is valid for the range of the observations, but not for very high redshift, its prediction being that $\Delta \alpha /\alpha$ is proportional to $\Omega_M[a(t)^{-1}-1]-2\Omega_\Lambda [a(t)^2-1]$, $a(t)$ being the scale factor. This agrees with the observations.