Numerical techniques for solving the quantum constraint equation of generic lattice-refined models in loop quantum cosmology
arXiv:0803.4483 · doi:10.1103/PhysRevD.78.024030
Abstract
To avoid instabilities in the continuum semi-classical limit of loop quantum cosmology models, refinement of the underlying lattice is necessary. The lattice refinement leads to new dynamical difference equations which, in general, do not have a uniform step-size, implying complications in their analysis and solutions. We propose a numerical method based on Taylor expansions, which can give us the necessary information to calculate the wave-function at any given lattice point. The method we propose can be applied in any lattice-refined model, while in addition the accuracy of the method can be estimated. Moreover, we confirm numerically the stability criterion which was earlier found following a von Neumann analysis. Finally, the `motion' of the wave-function due to the underlying discreteness of the space-time is investigated, for both a constant lattice, as well as lattice refinement models.
17 pages, 14 figures
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- Chimera: A hybrid approach to numerical loop quantum cosmology
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- Unstable Anisotropic Loop Quantum Cosmology
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- Loop Quantum Gravity and Cosmology: A dynamical introduction
- Lattice refinement in loop quantum cosmology
- Discretisation parameter and operator ordering in loop quantum cosmology with the cosmological constant
- Phenomenology of loop quantum cosmology
- Quantum gravity effects on space-time