Separable sequences in Bianchi I loop quantum cosmology

In this paper, we discuss the properties of one-parameter sequences that arise when solving the Hamiltonian constraint in Bianchi I loop quantum cosmology using a separation of variables method. In particular, we focus on finding an expression for the sequence for all real values of the parameter, and discuss the preclassicality of this function. We find that the behavior of these preclassical sequences imply time asymmetry on either side of the classical singularity in Bianchi I cosmology.

Figure 1

Graph of the exact solution $a_{\lambda }(m)$ given in terms of hypergeometric functions by (7) for $\lambda =3/2$. For positive $m$, the sequence increases without limit. For negative $m$, there is an oscillatory decay in the wave function, representing a region where there is a small probability of finding the Universe, compared to the positive side of the classical singularity.

Figure 2

Graph of the numerically evolved solution $a_{\lambda }(m)$ for $\lambda =3/2$ and using the recursion relation (3b) with the function $d(m)$ and discrete parameters $m=k+3/4$, $k$ an integer. Initial data are chosen for large positive $m$ using the exact solution (7), and evolved using the recursion relation in the negative direction. Note the large difference in slope between the two sides of the graph for both sequences.

Figure 3

Graph of the numerically evolved solution $a_{\lambda }(m)$ for $\lambda =-1$, $a(0)=1$ and using the recursion relation (3a) with the function $d_2(m)$ and discrete parameters $m=k+3/4$, $k$ an integer. Initial data are chosen for large positive $m$ using the exact solution (8), and evolved using the recursion relation in the negative direction. Unlike the previous graphs, the sequence becomes oscillatory with greatly increasing magnitude when $m<0$. For positive $m$, the sequence is gradually decreasing, but much smaller magnitude (so it is not visible).