The cosmological constant from Planckian fluctuations and the averaging procedure

In this paper I continue the investigation in \cite{1,1b} concerning my proposal on the nature of the cosmological constant. In particular, I study both mathematically and physically the quantum Planckian context and I provide, in order to depict quantum fluctuations and in absence of a complete quantum gravity theory, a semiclassical solution where an effective inhomogeneous metric at Planckian scales or above is averaged. In such a framework, a generalization of the well known Buchert formalism \cite{2} is obtained with the foliation in terms of the mean value $s(\hat{t})$ of the time operator $\hat{t}$ in a maximally localizing state $\{s\}$ of a quantum spacetime \cite{3,4,5,6} and in a cosmological context \cite{7}. As a result, after introducing a decoherence length scale $L_D$ where quantum fluctuations are averaged on, a classical de Sitter universe emerges with a small cosmological constant depending on $L_D$ and frozen in a true vacuum state (lowest energy), provided that the kinematical backreaction is negligible at that scale $L_D$. Finally, I analyse the case with a non-vanishing initial spatial curvature $\mathcal{R}$ showing that, for a reasonable large class of models, spatial curvature and kinematical backreation $\mathcal{Q}$ are suppressed by the dynamical evolution of the spacetime.