Relativity and Gravitation pp 469477  Cite as
Loop Quantum Cosmology: Anisotropy and Singularity Resolution
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Abstract
In this contribution we consider the issue of singularity resolution within loop quantum cosmology (LQC) for different homogeneous models. We present results of numerical evolutions of effective equations for both isotropic as well as anisotropic cosmologies, with and without spatial curvature. To address the issue of singularity resolution we examine geometrical and curvature invariants that yield information about the spacetime geometry. We discuss generic behavior found for a variety of initial conditions.
Keywords
Poisson Bracket Planck Scale Effective Theory Isotropic Model Anisotropic Model1 Introduction

The matter density operator \(\hat{\rho }\) has an absolute upper bound and the expansion \(\theta \) is also bounded. One can conclude that curvature scalars do not diverge. This is a signal that a singularity is not present.

All states undergo a bounce and with this, the big bang is replaced by a big bounce.

The GR dynamics is recovered as we go away from the Planck scale, this means that we are recovering the original theory that we want.

Dynamics of semiclassical states are well captured by an effective theory that retains information about the loop quantum geometry.

With all these, one can conclude that the singularities are resolved: the geodesics are inextendible, and are well defined on the other side of the would be big bang.
The new issues to consider in the anisotropic models are: is the bounce generic? We now have anisotropy/Weyl curvature, how does it behave near the singularity/bounce? Can we have different kind of bounce, say, dominated by shear \(\sigma \)? Are the geometric scalars absolutely bounded? The goal of this contribution is to answer these questions using the effective theory for Bianchi I which has anisotropies, Bianchi II that has anisotropies and spatial curvature and Bianchi IX which has all the features of Bianchi I, II and is furthermore, spatially compact. Even more, the Bianchi IX model has a non trivial classical limit, in the sense that, Bianchi IX is chaotic in the classical theory and behaves like Bianchi I with Bianchi II transitions as one approaches the singularity.
2 Preliminaries
Since working with full quantum theories of the models is difficult and, as shown in [2] for some models, the behavior of the effective or semiclassical equations, which are classical equations with some quantum corrections, are good approximations to the numerical quantum evolutions even near the Planck scale, we will work with the effective equations.
3 Effective Theories
Isotropic Flat and Closed Models
Bianchi I and II
Bianchi IX
4 Results

Is the bounce generic? Yes. All solutions have a bounce. In other words, singularities are resolved. In the closed FRW and the Bianchi IX model, there are infinite number of bounces and recollapses due to the compactness of the spatial manifold.

How does anisotropy/Weyl curvature behave near the bounce? These quantities far from the bounce approach their classical values, but when they reach the region near the bounce they behave differently. In Bianchi I, they present only one maximum. In Bianchi II, they exhibit a richer behavior because now they can be zero at the bounce or near to it, and have more than one maximum (for the shear there are up to 4 maxima and for the scalar curvature up to 2 maxima [8]). For Bianchi IX, if we restrict the analysis to one of the infinite number of bounces, it can be shown that anisotropy and curvature behave in the same way as Bianchi I or II. The subject of current research is whether there are new behaviors [9].

Can we have different kind of bounce, say, dominated by shear \(\sigma \)? Yes, but only in Bianchi II and IX. In Bianchi I the dynamical contribution from matter is always bigger than the one from the shear, even in the solution which reaches the maximal shear at the bounce [8].

Are geometric scalars \(\theta , \sigma \) and \(\rho \) absolutely bounded? In the flat isotropic model all the solutions to the effective equations have a maximal density equal to the critical density, and a maximal expansion (\(\theta ^2_\mathrm{max} = 6\pi G \rho _\mathrm{crit} = 3/(2\gamma \lambda )\)) when \(\rho =\rho _\mathrm{crit}/2\). For FRW \(k=1\) model, every solution has its maximum density but in general the density is not absolutely bounded. In the effective theory which comes from connection based quantization, expansion can tend to infinity. For the other case, expansion has the same bound as the flat FRW model. However, by adding some more corrections from inverse triad term, one can show that actually in both effective theories the density and the expansion have finite values. For Bianchi I, in all the solutions \(\rho \) and \(\theta \) are upperly bounded by their values in the isotropic case and \(\sigma \) is bounded by \(\sigma ^2_{\max } = 10.125/(3\gamma ^2\lambda ^2)\) [10]. For Bianchi II, \(\theta , \sigma \) and \(\rho \) are also bounded, but for larger values than the ones in Bianchi I, i.e., there are solutions where the matter density is larger than the critical density. With pointlike and cigarlike classical singularities [8], the density can achieve the maximal value (\(\rho \approx 0.54\rho _{Pl} \)) as a consequence of the shear being zero at the bounce and curvature different from zero. For Bianchi IX the behavior is the same as in closed FRW, if the inverse triad corrections are not used, then the geometric scalars are not absolutely bounded. But if the inverse triad corrections are used, then on each solution the geometric scalars are bounded but there is not an absolute bound for all the solutions [9, 10].

Bianchi I, II and therefore the isotropic case \(k=0\) are limiting cases of Bianchi IX, but they are not contained within Bianchi IX. While the isotropic FRW \(k=1\) is contained within Bianchi IX only if the inverse triad corrections are not included, when they are included then the \(k=1\) universe is a limiting case, like the \(k=0\) universe.

A set of quantities that are very useful are the Kasner exponents (in classical Bianchi I, the scale factors are \(a_i=t^{k_i}\), where \(k_i\) are the Kasner exponents), because they can be used to determine which kind of solution is obtained. The Kasner exponents tell us about the Bianchi I transitions (if they exist) and particularly in Bianchi IX, they are used to study the BKL behavior in the vacuum case.
5 Conclusions
One of the main issues that a quantum theory of gravity is expected to address is that of singularity resolution. Loop quantum cosmology has provided a complete description in the case of isotropic cosmological models and singularity resolution has been shown to be generic. A pressing question is whether these results can be generalized to anisotropic models. In this case we lack a complete quantum theory, but one can rely on the existence of an effective description, capturing the main (loop) quantum geometric features. In this contribution we have described the main features of such effective solutions. Singularities seem to be generically resolved as the time evolution of geometrical scalars is well behaved past the would be classical singularity. With the study of these anisotropic models, a question that still arises is whether this behavior is generic for nonhomogeneous configurations. That is, are we a step forward toward generic quantum singularity resolution?
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