Effective dynamics in Bianchi type II loop quantum cosmology

We numerically investigate the solutions to the effective equations of the Bianchi II model within the “improved” loop quantum cosmology dynamics. The matter source is a massless scalar field. We perform a systematic study of the space of solutions, and focus on the behavior of several geometrical observables. We show that the big bang singularity is replaced by a bounce and the pointlike singularities do not saturate the energy density bound. There are up to three directional bounces in the scale factors, one global bounce in the expansion, the shear presents up to four local maxima and can be zero at the bounce. This allows for solutions with density larger than the maximal density for the isotropic and Bianchi I cases. The asymptotic behavior is shown to behave like that of a Bianchi I model, and the effective solutions connect anisotropic solutions even when the shear is zero at the bounce. All known facts of Bianchi I are reproduced. In the “vacuum limit,” solutions are such that almost all the dynamics is due to the anisotropies. Since Bianchi II plays an important role in the Bianchi IX model and the Belinskii, Khalatnikov, Lifshitz conjecture, our results can provide an intuitive understanding of the behavior in the vicinity of general spacelike singularities, when loop-geometric corrections are present.

Figure 1

Comparison between classical and effective solutions. Here it can be appreciated that near the big bang the classical solutions have an infinite density because the scale factors go to zero.

Figure 2

Isotropic limit. The density at the bounce is $\rho /{\rho }_{\mathrm{crit}}=1$; the shear is close to zero; the Kasner exponents are equal and change from $1/3$ to $1/3-2/3=-1/3$ when they are evolved back in time; and the Ricci scalar strongly changes its behavior near to the bounce and has its maximal value at the bounce.

Figure 3

Bianchi I limit. The density and shear have a dynamical contribution ($\Omega \ne 0$, ${\Sigma }^2\ne 0$); the shear reaches its maximal value (for Bianchi I) at the bounce; the Kasner exponents change from $k_i$ to $k_i-2/3$ when they are evolved back in time, as was shown by 10; one direction is expanding ($a_3$), one contracting ($a_2$), and one bounces ($a_1$); the expansion has one zero (bounce) and it is finite all the time; and the Ricci scalar presents a slower changes in its behavior in comparison to the isotropic limit and it is finite too.

Figure 4

LRS solution. The density reaches its maximal value when the shear is zero at the bounce, but the shear is different from zero in the rest of the evolution; the dynamical observables $a_i$, $H_i$, $k_i$ for directions 2 and 3 are equal (this shows that it is a LRS solution); the directional scale factor $a_1$ bounces one time and has two turn-around points; and the curvature parameter $K$ is different from zero and asymptotically it satisfies $\Omega +{\Sigma }^2+K=1$.

Figure 5

Vacuum limit. The shear reaches its maximal value at one side of the bounce and it is zero on the other side (the zero is not at the bounce); the density is not zero but has a small value (even at the bounce); ${\overline{\mu }}_i{c}_i$ evolve from one classical solution (${\overline{\mu }}_i{c}_i\approx 0$) to another one (${\overline{\mu }}_i{c}_i\approx \pi$); ${\dot{p}}_2$ and ${\dot{p}}_3$ are zero 3 times; all the dynamical contributions come from the shear ($\Omega \approx 0$, $K\approx 0$, ${\Sigma }^2\approx 1$); and the Ricci scalar reaches its maximal value at one side of the bounce and a local minimum on the other side.

Figure 6

Convergence of the constraint. The first plot shows the convergence of the solutions and that the constraint is conserved. The second plot shows the error functions versus the bigger resolution used to calculate their.