Loop quantum cosmology in Bianchi type I models: Analytical investigation

The comprehensive formulation for loop quantum cosmology in the spatially flat, isotropic model was recently constructed. In this paper, the methods are extended to the anisotropic Bianchi I cosmology. Both the precursor and the improved strategies are applied and the expected results are established: (i) the scalar field again serves as an internal clock and is treated as emergent time; (ii) the total Hamiltonian constraint is derived by imposing the fundamental discreteness and gives the evolution as a difference equation; and (iii) the physical Hilbert space, Dirac observables, and semiclassical states are constructed rigorously. It is also shown that the state in the kinematical Hilbert space associated with the classical singularity is decoupled in the difference evolution equation, indicating that the big bounce may take place when any of the area scales undergoes the vanishing behavior. The investigation affirms the robustness of the framework used in the isotropic model by enlarging its domain of validity and provides foundations to conduct the detailed numerical analysis.

Figure 1

(a): Given the values of $e_{{\lambda }_{12}}$ at the lattice points marked •, the value at the point marked $\times$ can be uniquely determined via the recursion relation (7.14). (b): If the values of $e_{{\lambda }_{12}}$ are specified on the boundary points (marked •), all the values at the points marked $\times$ can be obtained by iterating the scheme depicted in (a).

Figure 2

(a) The lattice ${\mathcal{L}}_{{ϵ}_1=0}\otimes {\mathcal{L}}_{{ϵ}_2\ne 0}$. (b) The lattice ${\mathcal{L}}_{{ϵ}_1=0}\otimes {\mathcal{L}}_{{ϵ}_2=0}$. The points marked • are the boundary lines with given values; in particular, ○ denotes the boundary points at which the value of $e_{{\lambda }_{12}}$ vanishes. The points marked $\times$ are the places whose values can be obtained easily, while those with ⋄, □, and $\otimes$ require more careful consideration. (The largeness of ${\mu }_1^{\star }$ and ${\mu }_2^{\star }$ is assumed but not properly presented in this figure.)