Behavior of nonlinear anisotropies in bouncing Bianchi I models of loop quantum cosmology

In homogeneous and isotropic loop quantum cosmology, gravity can behave repulsively at Planckian energy densities leading to the replacement of the big bang singularity with a big bounce. Yet in any bouncing scenario it is important to include nonlinear effects from anisotropies which typically grow during the collapsing phase. We investigate the dynamics of a Bianchi I anisotropic model within the framework of loop quantum cosmology. Using effective semiclassical equations of motion to study the dynamics, we show that the big bounce is still predicted with only differences in detail arising from the inclusion of anisotropies. We show that the anisotropic shear term grows during the collapsing phase, but remains finite through the bounce. Immediately following the bounce, the anisotropies decay and with the inclusion of matter with equation of state $w<+1$, the universe isotropizes in the expanding phase.

Figure 1

Mean scale factor $a(t)$ and directional scale factors $a_I(t)$ for the vacuum ${\rho }_M=0$ case. The initial conditions are chosen corresponding to an initially contracting universe.

Figure 2

Shear term for vacuum case. Classically, the value is a constant, though quantum mechanically it is not constant near the bounce. The postbounce value approaches the prebounce value, and nowhere does it blow up.

Figure 3

Mean scale factor $a(t)$ and directional scale factors $a_I(t)$ for a massless scalar field with momentum $P_{\varphi }=10l_p^2$. Initially, all directions are contracting and a bounce occurs leading to expansion in all directions.

Figure 4

Shear term for massless scalar field. As in the vacuum case, the prebounce value is conserved after the bounce.

Figure 5

Mean scale factor $a(t)$ and directional scale factors $a_I(t)$ for radiation.

Figure 6

Shear term for radiation. The postbounce value does not equal the prebounce values although the value is finite through the entire evolution.

Figure 7

Ratios of directional Hubble rates for radiation. At late times, the ratios approach unitary indicative of equal expansion rates in all directions as the universe isotropizes.

Figure 8

Case (i): Quantum corrections take effect in the Kasner phase. With $w=1/3$ (radiation filled); ${\kappa }_1=-2/7$, ${\kappa }_2=3/7$, ${\kappa }_3=6/7$; $\mathcal{K}=1.\times {10}^3$; $A=1.\times {10}^2\hslash {\ell }_{\mathrm{Pl}}^{3w-1}$; and $p_1(t_0)=3.\times {10}^4{\ell }_{\mathrm{Pl}}^2$, $p_2(t_0)=2.\times {10}^4{\ell }_{\mathrm{Pl}}^2$, $p_3(t_0)=1.\times {10}^4{\ell }_{\mathrm{Pl}}^2$. (Also set $\gamma =1$.) The solid lines represent the effective loop quantum evolution for the triad components $p_I$ (left plot) and scale factors $a_I$ (right plot). The dashed lines represent the classical behavior. The proper time $t$ is offset such that the classical singularity is at the origin of $t$. The arrows at the right denote the values of $p_{I,\mathrm{crit}}$ with the bottom most arrow giving $p_{\mathrm{crit}}$. In this case, the bouncing point of each $p_I$ matches $p_{I,\mathrm{crit}}$ very precisely and we have $p_{I,\mathrm{crit}}\gg p_{\mathrm{crit}}$ (or say ${\rho }_M\ll {\varrho }_I\sim {\rho }_{\mathrm{Pl}}$ near the bounces). [Note that the bounce of $p_3$ is out of the shown range. The isotropized phases on both sides of the bounces are also out of plot.]

Figure 9

Case (ii): Quantum corrections take effect in the isotropized phase. With $w=1/3$ (radiation filled); ${\kappa }_1=-2/7$, ${\kappa }_2=3/7$, ${\kappa }_3=6/7$; $\mathcal{K}=1.\times {10}^3$; $A=1.\times {10}^6\hslash {\ell }_{\mathrm{Pl}}^{3w-1}$; and $p_1(t_0)=9.\times {10}^4{\ell }_{\mathrm{Pl}}^2$, $p_2(t_0)=6.\times {10}^4{\ell }_{\mathrm{Pl}}^2$, $p_3(t_0)=3.\times {10}^4{\ell }_{\mathrm{Pl}}^2$. In this case, the bouncing point of each $p_I$ roughly matches $p_{\mathrm{crit}}$ and we have $p_{I,\mathrm{crit}}\ll p_{\mathrm{crit}}$ (or say ${\varrho }_I\ll {\rho }_M\sim {\rho }_{\mathrm{Pl}}$ near the bounces). [Note that in the backward evolution the contracting curve of the classical $a_1$ eventually becomes expanding at the epoch extremely close to the singularity, indicating that the Kasner phase does occur classically although almost invisible in the plot.]

Figure 10

Case (iii): Quantum corrections take effect in the transition phase. With $w=1/3$ (radiation filled); ${\kappa }_1=-2/7$, ${\kappa }_2=3/7$, ${\kappa }_3=6/7$; $\mathcal{K}=1.\times {10}^3$; $A=2.\times {10}^4\hslash {\ell }_{\mathrm{Pl}}^{3w-1}$; and $p_1(t_0)=3.\times {10}^4{\ell }_{\mathrm{Pl}}^2$, $p_2(t_0)=2.\times {10}^4{\ell }_{\mathrm{Pl}}^2$, $p_3(t_0)=1.\times {10}^4{\ell }_{\mathrm{Pl}}^2$. In this case, we have $p_{I,\mathrm{crit}}\sim p_{\mathrm{crit}}$, roughly around which all $p_I$ are bounced (or say ${\varrho }_I\sim {\rho }_M\sim {\rho }_{\mathrm{Pl}}$ near the bounces).