Von Neumann stability of modified loop quantum cosmologies

Abstract

Von Neumann stability analysis of quantum difference equations in loop quantized spacetimes has often proved useful to understand viability of quantizations and whether general relativistic description is recovered at small spacetime curvatures. We use this technique to analyze the infra-red behavior of quantum Hamiltonian constraint in recently explored modifications of loop quantum cosmology: mLQC-I and mLQC-II, for the spatially flat FLRW model. We investigate the behavior for $μ_o$ scheme, where minimum area of loops in quantization procedure does not take physical metric in to account, and the $\bar μ$ scheme where quantization procedure uses physical metric. The fate of stability of quantum difference equations is tested for massless scalar field as well as with inclusion of a positive cosmological constant. We show that for mLQC-I and mLQC-II, difference equation fails to be von Neumann stable for the $μ_o$ scheme if cosmological constant is included signaling problematic behavior at large volumes. Both of the modified loop quantum cosmologies are von Neumann stable for $\bar μ$ scheme. In contrast to standard loop quantum cosmology, properties of roots turn out to be richer and intricate. Our results demonstrate the robustness of $\bar μ$ scheme (or `improved dynamics') in loop quantization of cosmological spacetimes even when non-trivial quantization ambiguities of Hamiltonian are considered, and show that $μ_o$ scheme is non-viable in this setting.