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4.15 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.

4.15.1 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 modifications of 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 [4577Jump 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 [130]. 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 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.

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.

4.15.2 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, also here with a particular gradient term. In this model, cancellations have been observed that imply that corrections appear only at higher orders of the perturbation series and are too weak to be observable [126].

A common problem of both analyses is that the robustness of the observed effects has not yet been studied. This is in particular a pressing problem since one modification of the gradient term has been chosen without further motivation. Moreover, the modifications in both examples were different. Without a more direct derivation of the modifications from inhomogeneous models or the full theory one can only rely on a robustness analysis to show that the effects can be trusted. In particular the cancellation in the second example must be shown to be realized for a larger class of modifications.

4.15.3 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 modified 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 2 k for a mode of wave number k, which also depends on the metric in a way determined by the gradient term modification. 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 cut-off, possibly realizing older expectations [165166].

4.15.4 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 modified behavior of homogeneous modes has led to the suspicion that loop cosmology is not stable [9495] 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 modifications 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 modified differently in such a manner that the total behavior is stable [129]. Most importantly, this is an example where an inhomogeneous matter Hamiltonian with its modifications must be used and the qualitative result of stability can be shown to be robust under possible changes of the effective modification. This shows that reliable conclusions can be drawn for important issues without a precise definition of the effective inhomogeneous behavior.

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