Dynamical eigenfunctions and critical density in loop quantum cosmology

Abstract

We offer a new, physically transparent argument for the existence of the critical, universal maximum matter density in loop quantum cosmology for the case of a flat Friedmann-Lemaitre-Robertson-Walker cosmology with scalar matter. The argument is based on the existence of a sharp exponential ultraviolet cutoff in momentum space on the eigenfunctions of the quantum cosmological dynamical evolution operator (the gravitational part of the Hamiltonian constraint), attributable to the fundamental discreteness of spatial volume in loop quantum cosmology. The existence of the cutoff is proved directly from recently found exact solutions for the eigenfunctions for this model. As a consequence, the operators corresponding to the momentum of the scalar field and the spatial volume approximately commute. The ultraviolet cutoff then implies that the scalar momentum, though not a bounded operator, is in effect bounded on subspaces of constant volume, leading to the upper bound on the expectation value of the matter density. The maximum matter density is universal (i.e. independent of the quantum state) because of the linear scaling of the cutoff with volume. These heuristic arguments are supplemented by a new proof in the volume representation of the existence of the maximum matter density. The techniques employed to demonstrate the existence of the cutoff also allow us to extract the large volume limit of the exact eigenfunctions, confirming earlier numerical and analytical work showing that the eigenfunctions approach superpositions of the eigenfunctions of the Wheeler-DeWitt quantization of the same model. We argue that generic (not just semiclassical) quantum states approach symmetric superpositions of expanding and contracting universes.

23 pages, 8 figures. Minor corrections throughout. Significant improvement to key figure illustrating behavior of the eigenfunctions. Version to appear in Classical and Quantum Gravity

References in corpus (15)

• Background Independent Quantum Gravity: A Status Report
• Loop Quantum Cosmology: A Status Report
• Quantum Nature of the Big Bang: Improved dynamics
• Quantum Nature of the Big Bang: An Analytical and Numerical Investigation
• Robustness of key features of loop quantum cosmology
• Loop quantum cosmology of k=1 FRW models
• Loop quantum cosmology of Bianchi I models
• Anti-deSitter universe dynamics in LQC
• Quantum bounce and cosmic recall
• Further Improvements in the Understanding of Isotropic Loop Quantum Cosmology
• Casting Loop Quantum Cosmology in the Spin Foam Paradigm
• Cosmic recall and the scattering picture of Loop Quantum Cosmology
• Path Integrals and the WKB approximation in Loop Quantum Cosmology
• Consistent Probabilities in Wheeler-DeWitt Quantum Cosmology
• Group theoretical Quantization of Isotropic Loop Cosmology

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