Foundations of Physics 37 (3):366-409 (2007)

Two different derivations of the observed vacuum energy density are presented. One is based on a class of proper and novel generalizations of the de Sitter solutions in terms of a family of radial functions R that provides an explicit formula for the cosmological constant along with a natural explanation of the ultraviolet/infrared entanglement required to solve this problem. A nonvanishing value of the vacuum energy density of the order of ${10^{- 123} M_{\rm Planck}^4}$ is derived in agreement with the experimental observations. A correct lower estimate of the mass of the observable universe related to the Dirac–Eddington–Weyl’s large number N = 1080 is also obtained. The presence of the radial function R is instrumental to understand why the cosmological constant is not zero and why it is so tiny. Finally, we rigorously prove why the proper use of Weyl’s Geometry within the context of Friedman–Lemaitre–Robertson–Walker cosmological models can account for both the origins and the value of the observed vacuum energy density. The source of dark energy is just the dilaton-like Jordan–Brans–Dicke scalar field that is required to implement Weyl invariance of the most simple of all possible actions. The full theory involving the dynamics of Weyl’s gauge field Aμ is very rich and may explain also the anomalous Pioneer acceleration and the temporal variations of the fundamental constants resulting from the expansion of the Universe. This is consistent with Dirac’s old idea of the plausible variation of the physical constants but with the advantage that it is not necessary to invoke extra dimensions
Keywords dark energy  cosmological constant  Weyl geometry  Jordan–Brans–Dicke gravity
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DOI 10.1007/s10701-007-9106-z
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