Stepping out of Homogeneity in Loop Quantum Cosmology
Carlo Rovelli, Francesca Vidotto
TL;DR
The paper investigates stepping beyond homogeneous loop quantum cosmology by introducing a finite triangulation truncation Δ_n to capture inhomogeneous quantum geometry near the initial singularity. Using a dipole example Δ_2, it demonstrates a closed constraint algebra and derives LQC-like dynamics as the zeroth-order Born-Oppenheimer approximation, while providing a direct derivation of the LQC μ-parameter from LQG. This work establishes a principled bridge between full loop quantum gravity and LQC, enabling controlled study of quantum fluctuations and bounce behavior in a finite, tractable setting. It also outlines avenues to connect with Regge calculus and spinfoam evolution, offering a framework to examine structure formation and inflationary implications in an inhomogeneous quantum cosmology context.
Abstract
We explore the extension of quantum cosmology outside the homogeneous approximation, using the formalism of loop quantum gravity. We introduce a model where some of the inhomogeneous degrees of freedom are present, providing a tool for describing general fluctuations of quantum geometry near the initial singularity. We show that the dynamical structure of the model reduces to that of loop quantum cosmology in the Born-Oppenheimer approximation. This result corroborates the assumptions that ground loop cosmology, sheds light on the physical and mathematical relation between loop cosmology and full loop quantum gravity, and on the nature of the cosmological approximation. Finally, we show that the non-graph-changing Hamiltonian constraint considered in the context of algebraic quantum gravity provides a viable effective dynamics within this approximation.