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6.4 Inhomogeneity

Inhomogeneities can only be treated in a field-theory context, which requires infinitely many classical and quantum variables. The discreteness of loop quantum gravity might help in avoiding some of the field theory subtleties, since finite regions contain only finitely many variables. But there are still many of them, and the dynamical equations are rather involved. At present, it is only possible to include inhomogeneities in the sense of classical cosmological perturbation theory around isotropic models, where the isotropic background for the perturbations is introduced by selecting specific states within the background-independent quantum theory [66Jump To The Next Citation Point].

Since, even for isotropic models, complete effective equations have been derived only in rare cases so far, it will take longer to obtain complete effective equations for inhomogeneities. Nevertheless, the corrections can be derived systematically, and several of the simpler terms have already been obtained along the lines of [86]. Of advantage in this perturbative setting is the fact that, to linear order in inhomogeneities, one can split the perturbations in different modes (scalar, vector and tensor) and compute their corrections separately. For each mode, gauge invariant variables can be determined. This is discussed for scalar modes in [8492Jump To The Next Citation Point], for vector modes in [90Jump To The Next Citation Point] and for tensor modes in [91Jump To The Next Citation Point]. Such a procedure would not be available at higher orders, where cosmological observables can be computed in a canonical scheme [157156], but are difficult to introduce into a loop quantization.

Once effective equations for inhomogeneities have been obtained, valuable insights can be derived for fundamental, as well as phenomenological, questions. As already mentioned, effective constraints allow a simpler discussion of anomalies than the full quantum theory provides. Anomalies are a pressing issue in the presence of inhomogeneities since the constraint algebra is quite non-trivial. By analyzing whether anomaly freedom is possible in the presence of quantum corrections, one can get hints as to whether such quantizations can be anomaly free in a full setting. This usually leads to additional conditions on the corrections, by which one can reduce quantization ambiguities. Since closed equations of motion for gauge invariant perturbations are possible only in the absence of anomalies, this issue also has direct implications for phenomenology. Anomaly-free effective constraints and the equations of motion they generate can be formulated in terms of gauge invariant quantities only, where gauge invariance refers to gauge transformations generated by the effective constraints including corrections. In this way, a complete analysis can be performed and evaluated for cosmological effects.

Effective equations for inhomogeneous models also provide the means to analyze refinements of the underlying discreteness as they are suggested by the full Hamiltonian constraint operators. In this context, inhomogeneous considerations help to elucidate the role of certain auxiliary structures in homogeneous models, which have often led to a considerable amount of confusion. The basic objects of inhomogeneous states are edge holonomies of the form &tidle; exp (iℓ0A ) where ℓ0 is the coordinate length of the edge and &tidle; A is some integrated connection component. (This may not directly be a component of Ai a due to the path ordering involved in non-Abelian holonomies. Nevertheless, we can think of matrix elements of holonomies as of this form for the present purpose.) Similarly, basic fluxes are given by surfaces transversal to a single edge in the graph. These are the elementary objects on which constructions of inhomogeneous operators are based, in contrast to homogeneous models, where only the total space or a chosen box of size V0, as in Section 4.2, is available to define edges for holonomies and surfaces for fluxes. Such a box is one of the auxiliary structures appearing in the definition of homogeneous models since there is no underlying graph to relate these objects to a state.

If a graph is being refined during evolution (in volume as internal time, say), the parameter ℓ0 can be thought of as being non-constant, but rather a function of volume. Note that the whole expression of a holonomy is coordinate-independent since the product &tidle; ℓ0A comes from a scalar quantity. The refinement can thus also be formulated in coordinate independent terms. For instance, if we are close to an isotropic configuration, we can, as in Equation (22View Equation), introduce the isotropic connection component 1∕3 c = V0 &tidle;A with coordinate size V0 of the above-mentioned box. Isotropic holonomies thus take the form exp (iℓ V −1∕3c) 0 0, where, for a nearly-regular graph with respect to the background geometry, 1∕3 − 1∕3 ℓ0∕V 0 =: 𝒩 is the inverse cubic root of the number of vertices within the box. For a refining model, 𝒩 increases with volume. For instance, if 𝒩 ∝ V with the total geometrical volume V = |p|3∕2 we have holonomies of the form exp(iδ(p)c) with ∘ --- δ(p) ∝ 1∕ |p|. This agrees with the suggestion of [27], or x = − 1∕2 in the notation of Section 5.5 [66Jump To The Next Citation Point], but is recognized here only as one special case of possible refinements. Moreover, one can easily convince oneself that a vertex number proportional to volume is not allowed by the dynamics of loop quantum gravity: this would require the Hamiltonian constraint to create only new vertices but not change the spin of edges. The other limiting case, where only spins change but no new vertices are created, corresponds to the non-refining model, where δ(p) = const. A realistic refinement must therefore lie between those behaviors, i.e., x δ(p) ∝ |p| with 0 < x < − 1∕2, if it follows a power law.

One can also see that V0 appears only when the exactly isotropic model is introduced (possibly as a background for perturbations), but not in inhomogeneous models. Basic corrections are thus independent of V 0 and the chosen box. They refer rather to sizes of the elementary discrete variables: local edge holonomies and fluxes. This characterization is independent of any auxiliary structures but directly refers to properties of the underlying state. One should certainly expect this, because it is the quantum state, which determines the quantum geometry and its corresponding corrections to classical behavior.

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