On Dark Energy, Weyl’s Geometry, Different Derivations of the Vacuum Energy Density and the Pioneer Anomaly

Abstract

Two dierent derivations of the observed vacuum energy density are presented. One is based on a class of proper and novel generalizations of the (Anti) de Sitter solutions in terms of a family of radial functions R(r) that provides an explicit formula for the cosmological constant along with a natural explanation of the ultraviolet/infrared ( UV/IR) entanglement required to solve this problem. A nonvanishing value of the vacuum energy density of the order of 10 123 M 4 Planck 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 = 10 80 is also obtained. The presence of the radial function R(r) is instrumental to understand why the cosmological constant is not zero and why it is so tiny and provides a natural derivation of the Pioneer’s anomaly. 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 ( dark energy ). 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 ( over cosmological scales ) 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. 1 Dedicated to the loving memory of Rachael Bowers