Effective dynamics, big bounces, and scaling symmetry in Bianchi type I loop quantum cosmology

The detailed formulation for loop quantum cosmology (LQC) in the Bianchi I model with a scalar massless field has been constructed. In this paper, its effective dynamics is studied in two improved strategies for implementing the LQC discreteness corrections. Both schemes show that the big bang is replaced by the big bounces, which take place up to 3 times, once in each diagonal direction, when the area or volume scale factor approaches the critical values in the Planck regime measured by the reference of the scalar field momentum. These two strategies give different evolutions: In one scheme, the effective dynamics is independent of the choice of the finite sized cell prescribed to make Hamiltonian finite; in the other, the effective dynamics reacts to the macroscopic scales introduced by the boundary conditions. Both schemes reveal interesting symmetries of scaling, which are reminiscent of the relational interpretation of quantum mechanics and also suggest that the fundamental spatial scale (area gap) may give rise to a temporal scale.

Figure 1

The Kasner-unlike solution in the $\overline{\mu }$-scheme effective dynamics. ${\kappa }_1=1/5$, ${\kappa }_2=1/4$, ${\kappa }_3=11/20$, ${\kappa }_{\varphi }=\sqrt{119/200}$; $p_1({\varphi }_0)=1.\times {10}^5{\ell }_{\mathrm{Pl}}^2$, $p_2({\varphi }_0)=2.\times {10}^5{\ell }_{\mathrm{Pl}}^2$, $p_3({\varphi }_0)=3.\times {10}^5{\ell }_{\mathrm{Pl}}^2$; and $p_{\varphi }=2.\times {10}^3\hslash \sqrt{8\pi G}$ (i.e., $\mathcal{K}{\kappa }_{\varphi }=2.\times {10}^3$). The values of ${\varrho }_{I,\mathrm{crit}}$ given by (3.18) are pointed by the arrows in (c). (The Barbero-Immirzi parameter is set to $\gamma =1$.)

Figure 2

The Kasner-like solution in the $\overline{\mu }$-scheme effective dynamics. ${\kappa }_1=-1/4$, ${\kappa }_2=3/4$, ${\kappa }_3=1/2$, ${\kappa }_{\varphi }=1/\sqrt{8}$; $p_1({\varphi }_0)=1.\times {10}^5{\ell }_{\mathrm{Pl}}^2$, $p_2({\varphi }_0)=2.\times {10}^5{\ell }_{\mathrm{Pl}}^2$, $p_3({\varphi }_0)=3.\times {10}^5{\ell }_{\mathrm{Pl}}^2$; and $p_{\varphi }=2.\times {10}^3\hslash \sqrt{8\pi G}$ (i.e., $\mathcal{K}{\kappa }_{\varphi }=2.\times {10}^3$).

Figure 3

The Kasner-unlike solution in the ${\overline{\mu }}^{\prime }$-scheme effective dynamics. ${\kappa }_1=1/5$, ${\kappa }_2=1/4$, ${\kappa }_3=11/20$, ${\kappa }_{\varphi }=\sqrt{119/200}$; $p_1({\varphi }_0)=1.\times {10}^5{\ell }_{\mathrm{Pl}}^2$, $p_2({\varphi }_0)=2.\times {10}^5{\ell }_{\mathrm{Pl}}^2$, $p_3({\varphi }_0)=3.\times {10}^5{\ell }_{\mathrm{Pl}}^2$; and $p_{\varphi }=2.\times {10}^3\hslash \sqrt{8\pi G}$ (i.e., $\mathcal{K}{\kappa }_{\varphi }=2.\times {10}^3$). The values of ${\rho }_{I,\mathrm{crit}}$ given by (3.43) are pointed by the arrows in (c). The epochs of big bounces are indicated by dotted lines in (a), which are very close to the moments corresponding to ${\rho }_{I,\mathrm{crit}}$, shown in (c).

Figure 4

The Kasner-like solution in the ${\overline{\mu }}^{\prime }$-scheme effective dynamics. ${\kappa }_1=-1/4$, ${\kappa }_2=3/4$, ${\kappa }_3=1/2$, ${\kappa }_{\varphi }=1/\sqrt{8}$; $p_1({\varphi }_0)=1.\times {10}^4{\ell }_{\mathrm{Pl}}^2$, $p_2({\varphi }_0)=2.\times {10}^4{\ell }_{\mathrm{Pl}}^2$, $p_3({\varphi }_0)=3.\times {10}^4{\ell }_{\mathrm{Pl}}^2$; and $p_{\varphi }=2.\times {10}^3\hslash \sqrt{8\pi G}$ (i.e., $\mathcal{K}{\kappa }_{\varphi }=2.\times {10}^3$). The epochs of big bounces are only slightly different from the moments corresponding to ${\rho }_{I,\mathrm{crit}}$.

Figure 5

(a) The shaded surface (in pink) is ${\square }_{JK}$, the physical area of which is to be shrunk to $\Delta$ in the ${\overline{\mu }}^{\prime }$-scheme. (b) The shaded surfaces (in blue) are ${⊡}_J$ and ${⊡}_K$, the physical areas of which are to be shrunk to $\Delta$ in the $\overline{\mu }$-scheme.

Figure 6

(a)–(b) The action of the Hamiltonian acting on a spin network state. (c) The resulting state is equivalent to the original one superposed with a triangular loop. (d) The idea of the ${\overline{\mu }}^{\prime }$-scheme. (e) The idea of the $\overline{\mu }$-scheme.