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6.3 Isotropic cosmology

The use of ˆ ⟨C⟩ for effective constraints explains why loop cosmology receives corrections from quantizations of the inverse volume and higher power corrections from holonomies. Such corrections must be present since the corresponding quantizations are crucial features of the full theory, and they have given rise to many phenomenological effects. (The precise form of corrections depends on the suitable semiclassical states used, while terms in phenomenological equations using effective densities, as well as higher powers of connection components, have been read off from expressions in eigenstates of triads or holonomies. Qualitative features are insensitive to the difference, and so simply reading off terms is good enough for phenomenology.) But for a reliable analysis one must to derive a complete set of quantum corrections to any given order, which the phenomenological corrections described in Section 4 cannot provide because they ignore quantum backreaction, the main source of corrections in genuine quantum systems. A systematic analysis of possible coupling terms between expectation values and quantum variables is required.

Our cosmology, however, is not close to a harmonic oscillator, which makes this solvable system unsuitable as zeroth order for a perturbative analysis. There is, however, a solvable system available for quantum cosmology: a spatially-flat isotropic model sourced by a free scalar [70Jump To The Next Citation Point]. The Friedmann equation, written in isotropic Ashtekar variables, is

2 c2∘ |p-| = 4-πG--pφ- . (75 ) 3 |p|3∕2
This is certainly a constraint, and so we could use the framework of effective constraints. In this form, however, the system would not be recognized as solvable because the constraint is obviously not quadratic in the canonical variables. On the other hand, in this case, it is easy to parameterize the system by choosing the scalar φ as internal time. Then, the solution
3 |p φ| = ----|pc| = H (p,c) (76 ) 4πG
of (75View Equation) provides an ordinary Hamiltonian H (p,c) for φ-evolution, which, except for the absolute value, is quadratic. The absolute value can in fact be shown to be irrelevant for the most interesting aspects of these kinds of cosmologies [69Jump To The Next Citation Point], such that we obtain a solvable system for quantum cosmology.

More remarkably, this solvability behavior even extends to loop quantum cosmology. In this case, there is no operator for c and thus one has to refer to non-canonical variables involving holonomies or the exponentials eic. For this reason, the Hamiltonian operator as constructed in Section 5.4 receives higher-order corrections and takes the form Hˆ = ˆps^in c in a certain factor ordering. Since this is no longer quadratic in canonical variables and we are forced to use a non-canonical basic operator algebra, it seems difficult to relate this to the solvable free scalar cosmology. However, there is a change of basic variables, which provides an exact solvable model even for loop quantum cosmology: using ˆp and ˆJ := ˆpexp (ic) (in this ordering) as basic operators, we obtain a closed algebra of basic variables together with the Hamiltonian operator Hˆ = − 1i(Jˆ− Jˆ†) 2. This provides a solvable model for loop quantum cosmology as the starting point for a systematic derivation of effective equations.

While we keep calling the geometrical variable p, one should note that the solvable model is insensitive to the precise form of refinement used in the underlying quantization. Instead of starting with the canonical pair (c,p), one can use

1− x v := -p---- and P := pxc, (77 ) 1 − x
which is also canonical. For − 1∕2 < x < 0 we obtain precisely the refinement models discussed in Sections 5.5 and 6.4 with x δ(p) = p. For any x, we are led to the same algebra of quantum operators between ˆv, ˆJ = ˆv exp(iP) and a linear Hamiltonian ˆH = − 1i(Jˆ− Jˆ†) 2 [69Jump To The Next Citation Point]. Only the geometrical interpretation of the variables depends on x.

This model also provides an instructive example for the role of physical inner-product issues in effective theory. By our definition of ˆJ, we will have to deal with complex-valued expectation values of our basic operators, in particular ⟨Jˆ⟩. After solving the effective equations, we have to impose reality conditions, which, due to the form of Jˆ, are not linear but derived from the quadratic relation ˆ ˆ† 2 JJ = ˆp. Taking an expectation value of this relation provides the reality condition to be imposed, which not only involves expectation values but also fluctuations. Thus, fluctuations can in principle play a significant role for admissible solutions, but this does not happen for initially semiclassical states. Physical solutions for expectation values and higher moments can then be found more conveniently than it would be for states in a Hilbert-space representation.

Thanks to the solvability of this model it is clear that it does not receive quantum corrections from backreaction. Thus, it is strictly justified in this model to use higher-power corrections as they are implemented by sin c. But inverse volume corrections have been neglected by construction, since they would alter the Hamiltonian and destroy the solvability. In a complete treatment, such corrections would have to arise, too, and with them quantum backreaction would play a role in the absence of solvability. Nevertheless, for many applications, one can argue that the latter types of corrections are small in an isotropic model because one usually requires a large matter content and thus large pφ, which enhances higher-power corrections. However, the situation is not fully clarified since the original versions, where the magnitudes of higher power and inverse volume corrections were compared numerically [26Jump To The Next Citation Point27Jump To The Next Citation Point28], were based on formulations of isotropic loop quantum cosmology, where inverse volume corrections are artificially suppressed due to the form of the symmetry reduction [75]. Taking this into account makes inverse volume corrections important for a larger parameter range. They are also much more prominent in inhomogeneous models.

Independent of this question, quantum backreaction inevitably results in the presence of matter interactions, anisotropies or inhomogeneities. In those cases, taking higher-power corrections, as in the free isotropic model, is not as reliable, and a detailed analysis is required, which so far has been provided only for a perturbative treatment of scalar potentials [87Jump To The Next Citation Point]. As anticipated, the resulting effective equations do not seem to allow an adiabatic approximation since quantum variables in the underlying solvable models are not constant. One thus has to deal with higher-dimensional effective systems, where some of the quantum variables remain as independent variables. Effective equations have been derived to first order in the perturbative potential and in a semiclassical expansion, but are rather involved and have not been analyzed much yet.

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