Je abstrakter die Wahrheit ist, die du lehren willst, um so mehr mußt du noch die Sinne zu ihr verführen.

(The more abstract the truth you want to teach is, the more you have to seduce to it the senses.)

Friedrich Nietzsche

Beyond Good and Evil

The gravitational field equations, for instance in the case of cosmology where one can assume homogeneity and isotropy, involve components of curvature as well as the inverse metric. (Computational methods to derive information from these equations are described in [5].) Since singularities occur, these curvature components become large in certain regimes, but the equations have been tested only in small curvature regimes. On small length scales, such as close to the Big Bang, modifications to the classical equations are not ruled out by observations and can be expected from candidates of quantum gravity. Quantum cosmology describes the evolution of a universe by a constraint equation for a wave function, but some effects can be included already at the level of effective classical equations. In loop quantum gravity one characteristic quantum correction occurs through inverse metric components, which, e.g., appear in the kinematic term of matter Hamiltonians; see Section 4.4. In addition, holonomies provide higher powers of connection components and thus additional effects in the dynamics as described in Section 4.7. While the latter dominate in many homogeneous models, which have been analyzed in detail, this is a consequence of the homogeneity assumption. More generally, effects from inverse metric components play equally important roles and are currently under better control, e.g., regarding the anomaly issue (see Section 6.5.4).

4.1 Isotropy

4.2 Isotropy: Connection variables

4.3 Isotropy: Implications of a loop quantization

4.4 Isotropy: Effective densities in phenomenological equations

4.5 Isotropy: Properties and intuitive meaning of effective densities

4.6 Isotropy: Applications of effective densities

4.6.1 Collapsing phase

4.6.2 Expansion

4.6.3 Model building

4.6.4 Stability

4.7 Isotropy: Phenomenological higher curvature corrections

4.8 Isotropy: Intuitive meaning of higher power corrections

4.9 Isotropy: Applications of higher-power corrections

4.10 Anisotropies

4.11 Anisotropy: Connection variables

4.12 Anisotropy: Applications

4.12.1 Isotropization

4.12.2 Bianchi IX

4.12.3 Isotropic curvature suppression

4.13 Anisotropy: Phenomenological higher curvature

4.14 Anisotropy: Implications for inhomogeneities

4.15 Inhomogeneities

4.16 Inhomogeneous matter with isotropic quantum geometry

4.17 Inhomogeneity: Perturbations

4.18 Inhomogeneous models

4.19 Inhomogeneity: Results

4.19.1 Matter gradient terms and small-a effects

4.19.2 Matter gradient terms and large-a effects

4.19.3 Non-inflationary structure formation

4.19.4 Stability

4.19.5 Cosmological perturbation theory

4.19.6 Realistic equations of state

4.19.7 Big Bang nucleosynthesis

4.20 Summary

4.2 Isotropy: Connection variables

4.3 Isotropy: Implications of a loop quantization

4.4 Isotropy: Effective densities in phenomenological equations

4.5 Isotropy: Properties and intuitive meaning of effective densities

4.6 Isotropy: Applications of effective densities

4.6.1 Collapsing phase

4.6.2 Expansion

4.6.3 Model building

4.6.4 Stability

4.7 Isotropy: Phenomenological higher curvature corrections

4.8 Isotropy: Intuitive meaning of higher power corrections

4.9 Isotropy: Applications of higher-power corrections

4.10 Anisotropies

4.11 Anisotropy: Connection variables

4.12 Anisotropy: Applications

4.12.1 Isotropization

4.12.2 Bianchi IX

4.12.3 Isotropic curvature suppression

4.13 Anisotropy: Phenomenological higher curvature

4.14 Anisotropy: Implications for inhomogeneities

4.15 Inhomogeneities

4.16 Inhomogeneous matter with isotropic quantum geometry

4.17 Inhomogeneity: Perturbations

4.18 Inhomogeneous models

4.19 Inhomogeneity: Results

4.19.1 Matter gradient terms and small-a effects

4.19.2 Matter gradient terms and large-a effects

4.19.3 Non-inflationary structure formation

4.19.4 Stability

4.19.5 Cosmological perturbation theory

4.19.6 Realistic equations of state

4.19.7 Big Bang nucleosynthesis

4.20 Summary

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