Cut-off free finite zero-point vacuum energy and the cosmological missing mass problem

Abstract

As the mass-energy is universally self-gravitating, the gravitational binding energy must be subtracted self-consistently from its bare mass value so as to give the physical gravitational mass. Such a self-consistent gravitational self-energy correction can be made non-perturbatively by the use of a gravitational `charging' technique, where we calculate the incremental change $dm$ of the physical mass of the cosmological object, of size $r_o$ due to the accretion of a bare mass $dM$, corresponding to the gravitational coupling-in of the successive zero-point vacuum modes, i.e., of the Casimir energy, whose bare value $\Sigma_{\bf k} \hbar ck$ is infinite. Integrating the `charging' equation, $dm = dM - (3\alpha/5)Gm\Delta M/r_o c^2$, we get a gravitational mass for the cosmological object that remains finite even in the limit of the infinite zero-point vacuum energy, i.e., without any ultraviolet cut-off imposed. Here $\alpha$ is a geometrical factor of order unity. Also, setting $r_o = c/H$, the Hubble length, we get the corresponding cosmological density parameter $\Omega \simeq 1$, without any adjustable parameter. The cosmological significance of this finite and unique contribution of the otherwise infinite zero-point vacuum energy to the density parameter can hardly be overstated.