The cosmological constant and Pioneer anomaly from Weyl spacetimes and Mach's principle

It is shown how Weyl's geometry and Mach's holographic principle furnishes both the magnitude and sign (towards the sun) of the Pioneer anomalous acceleration aP∼−c2/RHubble firstly observed by Anderson et al. Weyl's geometry 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. A nonvanishing value of the vacuum energy density of the order of 10−123MPlanck4 is found consistent with observations. Weyl's geometry accounts also for the phantom scalar field in modern Cosmology in a very natural fashion.