**Authors:** Carlos Castro

We review firstly why 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. 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. Next, a Jordan-Brans-Dicke gravity
model within the context of ordinary Riemannian geometry, yields also
the observed vacuum energy density (cosmological constant) to very high
precision. One finds that the temporal flow of the scalar field φ(t) in
ordinary Riemannian geometry, from t = 0 to t = to, has the same numerical
effects (as far as the vacuum energy density is concerned) as if
there were Weyl scalings from the field configuration φ(t), to the constant
field configuration φ_{o}, in Weyl geometry. Hence, Weyl scalings in Weyl
geometry can recapture the flow of time which is consistent with Segal's
Conformal Cosmology, in such a fashion that an expanding universe may
be visualized as Weyl scalings of a static universe. The main novel result
of this work is that one is able to reproduce the observed vacuum energy
density to such a degree of precision 10^{-123}M^{4}_{Planck}, while still having a
Big-Bang singularity at t = 0 when the vacuum energy density blows up.
This temporal flow of the vacuum energy density, from very high values
in the past, to very small values today, is not a numerical coincidence but
is the signal of an underlying Weyl geometry (conformal invariance) operating
in cosmology, combined with the dynamics of a Brans-Dicke-Jordan
scalar field.

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[v1] 3 Jan 2009

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