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4.13 Anisotropy: Phenomenological higher curvature

As in isotropic models, the discreteness of geometry also provides corrections of higher powers of extrinsic curvature components, and these can take different forms depending on how the discreteness of an underlying state changes during evolution. Due to the larger number of degrees of freedom, there are now more options for the behavior and a larger variety of potential phenomena. Viable classes of discretization effects are, however, rather strongly restricted by the requirement of showing the correct semiclassical behavior in the right regimes [75Jump To The Next Citation Point128Jump To The Next Citation Point]. Thus, even without developing a precise relation between a full Hamiltonian constraint and the effects it implies for the refinement in a model, one can self-consistently test symmetric models, although only initial steps [127126Jump To The Next Citation Point125] toward a comprehensive analysis have been undertaken.

Anisotropic solutions as analogs for Kasner solutions in the presence of higher powers of extrinsic curvature have first been studied in [142], although without considering refinements of the discreteness scale. These solutions have already shown that anisotropic models also exhibit bounces when such corrections are included phenomenologically, i.e., disregarding quantum backreaction. The issue has been revisited in [12842Jump To The Next Citation Point] for models including refinement (and a free, massless scalar as internal time) with the same qualitative conclusions. Similar calculations have been applied to Kantowski–Sachs models, which classically provide the Schwarzschild interior metric, without refinement in [231232] and with two different versions of refinement in [42]. They can thus be used to obtain indications for the behavior of quantum black holes, at least in the vacuum case. Then, however, one cannot use the arguments, which in isotropic cosmological models allowed one to conclude that higher-power corrections are dominant given a large matter content. Correspondingly, bounces of the higher-power phenomenological equations happen at much smaller scales than in massive isotropic models. They are thus less reliable because other corrections can become strong in those regimes.


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