The Cosmological Constant From Planckian Fluctuations and the Averaging Procedure

Abstract

In this paper I continue the investigation in Viaggiu (Class Quantum Gravity 35:215011, 2018), Viaggiu (Phys Scr 94:125014, 2019) 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 (Buchert in Gen Relativ Gravit 33:1381, 2001) 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 (Doplicher et al. in Commun Math Phys 172:187, 1995; Doplicher in Space-time and fields: a quantum texture, in Karpacz, new developments in fundamental interaction theories, arXiv:hep-th/0105251, 2001; Bahns et al. in Advances in algebraic quantum field theory, Springer, Cham; Tomassini and Viaggiu in Class Quantum Gravity 28:075001, 2011) and in a cosmological context (Tomassini and Viaggiu in Class Quantum Gravity 31:185001, 2014). 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.