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4.19 Inhomogeneity: Results

There are some results obtained for inhomogeneous systems. We have already discussed glimpses from the BKL picture, which used loop results only for anisotropic models. Methods described in this section have led to some preliminary insights into possible cosmological scenarios.

4.19.1 Matter gradient terms and small-a effects

When an inhomogeneous matter Hamiltonian is available it is possible to study its implications on the cosmic microwave background with standard techniques. With quantum-corrected densities there are then different regimes, since the part of the inflationary era responsible for the formation of currently visible structure can be in the small-a or large-a region of the effective density.

The small-a regime, below the peak of effective densities, has more dramatic effects since inflation can here be provided by quantum geometry effects alone and the matter behavior changes to be anti-frictional [53111Jump To The Next Citation Point]. Mode evolution in this regime has been investigated for a particular choice of gradient term and using a power-law approximation for the effective density at small a, with the result that there are characteristic signatures [187]. As in standard inflation models, the spectrum is nearly scale invariant, but its spectral index is slightly larger than one (blue tilt) as compared to slightly smaller than one (red tilt) for single-field inflaton models. Since small scale factors at early stages of inflation generate structure which today appears on the largest scales, this implies that low multipoles of the power spectrum should have a blue tilt. The running of the spectral index in this regime can also be computed but depends only weakly on ambiguity parameters.

The main parameter then is the duration of loop inflation. In the simplest scenario, one can assume only one inflationary phase, which would require huge values for the ambiguity parameter j. This is unnatural and would likely imply that the spectrum is blue on almost all scales, which is in conflict with present observations. Thus, not only conceptual arguments but also cosmological observations point to smaller values for j, which is quite remarkable. On the other hand, while one cannot achieve a large ratio of scale factors during a single phase of super-inflation with small j, the ratio of the Hubble parameter multiplied by the scale factor increases more strongly. This is in fact the number relevant for structure formation, and so even smaller j may provide the right expansion for a viable power spectrum [134Jump To The Next Citation Point].

However, cosmological perturbation theory in this regime of non-perturbative corrections is more subtle than on larger scales [86Jump To The Next Citation Point], and so a precise form of the power spectrum is not yet available. Several analyses have been performed based on the classical perturbation theory but with super-inflationary quantum corrections for the isotropic background [237121134], showing that scale-invariant spectra can be obtained from certain classes of matter potentials driving the background in a suitable way; see also [226227] for a preliminary analysis of tensor modes on a super-inflationary background.

In order to have sufficient inflation to make the universe big enough, one then needs additional stages provided by the behavior of matter fields. One still does not need an inflaton since now the details of the expansion after the structure generating phase are less important. Any matter field being driven away from its potential minimum during loop inflation and rolling down its potential thereafter suffices. Depending on the complexity of the model there can be several such phases.

4.19.2 Matter gradient terms and large-a effects

At larger scale factors above the peak of effective densities there are only perturbative corrections from loop effects. This has been investigated with the aim of finding trans-Planckian corrections to the microwave background, here also with a particular gradient term [182].

A common problem of both analyses is that the robustness of the observed effects has not yet been studied. This is particularly a pressing problem since one modification of the gradient term has been chosen without further motivation. Moreover, the modifications of several examples were different. Without a more direct derivation of the corrections from inhomogeneous models or the full theory, one can only rely on a robustness analysis to show that the effects can be trusted. In addition, reliable conclusions are only possible if corrections in matter as well as gravitational terms of the Hamiltonian are taken into account; for the latter see Section 4.19.5.

4.19.3 Non-inflationary structure formation

Given a modification of the gradient term, one obtains effective equations for the matter field, which for a scalar results in a corrected Klein–Gordon equation. After a mode decomposition, one can then easily see that all the modes behave differently at small scales with the classical friction replaced by anti-friction as in Section 4.5. Thus, not only the average value of the field is driven away from its potential minimum but also higher modes are being excited. The coupled dynamics of all the modes thus provides a scenario for structure formation, which does not rely on inflation but on the anti-friction effect of loop cosmology.

Even though all modes experience this effect, they do not all see it in the same way. The gradient term implies an additive contribution to the potential proportional to k2 for a mode of wave number k, which also depends on the metric in a way determined by the gradient term corrections. For larger scales, the additional term is not essential and their amplitudes will be pushed to similar magnitudes, suggesting scale invariance for them. The potential relevant for higher modes, however, becomes steeper and steeper such that they are less excited by anti-friction and retain a small initial amplitude. In this way, the structure formation scenario provides a dynamical mechanism for a small-scale cutoff, possibly realizing older expectations [246247].

In independent scenarios, phenomenological effects from loop cosmology have been seen to increase the viability of mechanisms to generate scale-invariant perturbations, such as in combination with thermal fluctuations [221].

4.19.4 Stability

As already noted, inhomogeneous matter Hamiltonians can be used to study the stability of cosmological equations in the sense that matter does not propagate faster than light. The behavior of homogeneous modes has led to the suspicion that loop cosmology is not stable [139140] since other cosmological models displaying super-inflation have this problem. A detailed analysis of the loop equations, however, shows that the equations as they arise from loop cosmology are automatically stable. While the homogeneous modes display super-inflationary and anti-frictional behavior, they are not relevant for matter propagation. Modes relevant for propagation, on the other hand, are corrected differently in such a manner that the total behavior is stable [186]. Most importantly, this is an example in which an inhomogeneous matter Hamiltonian with its corrections must be used and the qualitative result of stability can be shown to be robust under possible changes of the effective term. This shows that reliable conclusions can be drawn for important issues without a precise definition of the effective inhomogeneous behavior. Consistency has been confirmed for tensor modes based on a systematic analysis of anomaly cancellation in effective constraints [91Jump To The Next Citation Point].

4.19.5 Cosmological perturbation theory

For linear metric and matter perturbations the complete set of cosmological perturbation equations is available from a systematic analysis of scalar, vector and tensor modes [84Jump To The Next Citation Point92Jump To The Next Citation Point90Jump To The Next Citation Point91Jump To The Next Citation Point]. The main focus of those papers is on effective-density corrections both in the gravitational and matter terms. These equations have been formulated for gauge invariant perturbations (whose form also changes for quantum-corrected constraints) and thus provide physical results. By the very fact that such equations exist with non-trivial quantum corrections it has also been proven that loop quantum gravity can provide a consistent deformation of general relativity at least in linear regimes. Quantum corrections do allow anomaly-free effective constraints, which can then be written in a gauge-invariant form. The study of higher-power corrections has provided several consistency tests and restricted the class of allowed refinement models which affect coefficients of the correction terms [90Jump To The Next Citation Point91Jump To The Next Citation Point].

This has led so far to several indications, while a complete evaluation of the cosmological phenomenology remains to be done. It has been shown that quantum corrections can add up in long cosmological regimes, magnifying potentially visible effects [85]. The scalar mode equations can be combined with a quantum-corrected Poisson equation, whose classical solution would be the Newton potential. Thus, this line of research allows a derivation of the Newton potential in a classical limit of quantum gravity together with a series of corrections. This offers interesting comparisons with the spin-foam approach to the graviton propagator [257]. Although the formulations are quite different, there are some similarities, e.g., in the role of semiclassical states in both approaches; compare, e.g., [216].

4.19.6 Realistic equations of state

Due to quantum corrections to matter Hamiltonians, equations of state of matter change in loop cosmology, which by itself can have cosmological effects. The main example is that of a scalar field, whose effective density and pressure directly provide the equation of state. Similarly, one can easily derive equations of state for some perfect fluid models, such as dust (w = 0). In this case, the total energy is just a field-independent constant, which remains a simple constant as a contribution to the simple quantum Hamiltonian. The dust equation of state thus receives no quantum corrections. (Note that it is important that the Hamiltonian is quantized, not energy density, for which no analog in the full theory would exist. For the energy density of dust, one certainly has inverse powers of the scale factor and could thus wrongly expect corrections from loop cosmology; see also [277] for such expressions for other fluids.)

A derivation of effective equations of state is more indirect for other perfect fluids since a fundamental realization through a field is required for a loop quantization. This can be done for the radiation equation of state w = 1 3 since it is a result of the Maxwell Hamiltonian of the electromagnetic field. Here, one is required to use an inhomogeneous setting in order to have a non-trivial field. Techniques to derive quantum-corrected equations of state thus resemble those used for cosmological perturbation theory. The Hamiltonian derivation of the equation of state, including inverse volume corrections in loop quantum cosmology, can be found in [77Jump To The Next Citation Point]. A similar derivation for relativistic fermions, which obey the same equation of state, is given in [78Jump To The Next Citation Point].

4.19.7 Big Bang nucleosynthesis

Big Bang nucleosynthesis takes place at smaller energy densities than earlier stages of the Big Bang, but there are rather tight constraints on the behavior of different types of matter, radiation and fermions during this phase. Thus, this might provide another handle on quantum-gravity phenomenology, or at least present a phase where one has to make sure that quantum gravity already behaves classically enough to be consistent with observations.

For a detailed analysis, equations of state are required for both the Maxwell fields and fermions. The latter can be assumed to be massless at those energies and thus relativistic, so that all matter fields classically satisfy the equation of state P = 1ρ 3. However, their Hamiltonians receive quantum corrections in loop quantum gravity, which could lead to different equations of state for the two types of fields, or, in any case, to corrections that may be common to both fields. Any such correction may upset the fine balance required for successful Big Bang nucleosynthesis, and thus puts limits on parameters of loop quantum gravity.

Corrections for the equation of state of radiation have been computed in [77]. Fermions are more difficult to quantize, and the necessary steps are dispersed over several articles. Their kinematics can be found in [32289Jump To The Next Citation Point], where [289Jump To The Next Citation Point] introduces half-densitized fermions, which play a key role in consistent quantization. Dynamics, i.e., the definition of Hamiltonian constraint operators, is also discussed in [289] as well as in older work [235234236]; however, torsion, which inevitably occurs for connection theories of gravity in the presence of fermions, was not included. This is completed in the full treatment of Dirac fermions, starting from an analysis of the Holst-Dirac action [183], provided in [76]. One can take the resulting canonical quantization in the presence of torsion as a starting point for quantum corrections to the fermion equation of state [78Jump To The Next Citation Point]. Applications to Big Bang nucleosynthesis [78] have provided some bounds on parameters, which can play a role once more information about the precise dynamics of inhomogeneous quantum states has been obtained. This application has also highlighted the need for understanding the role of the thermodynamics of a system on a quantum corrected spacetime. From effective loop cosmology one can derive corrections to the energy density ρ(a) or equation of state, but additional input is needed for energy density ρ (T ) as a function of temperature as it often occurs. Such considerations of matter thermodynamics on a quantum spacetime have just started.

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