Non Singular Origin of the Universe and the Cosmological Constant Problem (CCP)

We consider a non singular origin for the Universe starting from an Einstein static Universe in the framework of a theory which uses two volume elements $\sqrt{-{g}}d^{4}x$ and $Φd^{4}x$, where $Φ$ is a metric independent density, also curvature, curvature square terms, first order formalism and for scale invariance a dilaton field $ϕ$ are considered in the action. In the Einstein frame we also add a cosmological term that parametrizes the zero point fluctuations. The resulting effective potential for the dilaton contains two flat regions, for $ϕ\rightarrow \infty$ relevant for the non singular origin of the Universe and $ϕ\rightarrow -\infty$, describing our present Universe. Surprisingly, avoidance of singularities and stability as $ϕ\rightarrow \infty$ imply a positive but small vacuum energy as $ϕ\rightarrow -\infty$. Zero vacuum energy density for the present universe is the "threshold" for universe creation.

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