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

Abstract

Two different 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. 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 A µ 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.

On some toy quantum cosmology

“…Another way to work the quantum action (1.2) into a cosmological framework would be to consider conformally coupled QM (quantum matter) as in [41,45] where one coupled dark matter DM (cf. also [23]). First one could dismiss W µν terms as irrelevant for cosmology (here W µν = 0 automatically of course) and then, following pp.…”

The Unexpected Resurgence of Weyl Geometry in late 20th-Century Physics

“…When he became aware of the new attempts at using Weyl geometric methods in gravity and in high energy physics, he took up the Weylian thread again. His guiding questions were now how Weyl's scale geometry may be used for understanding dark energy and, perhaps, the Pioneer anomaly which at that time could still appear as a challenge for gravity theories Castro (2007Castro ( , 2009. 132 Castro speculated with grand visions for his newly detected interest in Weyl geometric methods, in contrast to Quiros' more sober perspective.…”

THE EXCEPTIONAL E8 GEOMETRY OF CLIFFORD (16) SUPERSPACE AND CONFORMAL GRAVITY YANG–MILLS GRAND UNIFICATION

“…Such scalar component is a dilaton-like Jordan-Brans-Dicke scalar field. In [40] we were able to show how Weyl-geometry solves the riddle of the cosmological constant within the context of a Robertson-Friedmann-Lemaitre-Walker cosmology by coupling the Weyl scalar curvature to the Jordan-Brans-Dicke scalar φ field with a self-interacting potential V (φ) and kinetic terms (D µ φ)(D µ φ). Upon eliminating the Weyl gauge field of dilations A µ from its algebraic (nonpropagating) equations of motion, and fixing the Weyl gauge scalings, by setting the scalar field to a constant φ o such that φ 2 o = 1 16 πG, where G is the present day observed Newtonian constant, we were able to prove that V (φ o ) = …”