Lattice refining loop quantum cosmology and the matter Hamiltonian

In the context of loop quantum cosmology, we parametrize the lattice refinement by a parameter, $A$, and the matter Hamiltonian by a parameter, $\delta$. We then solve the Hamiltonian constraint for both a self-adjoint and a non-self-adjoint Hamiltonian operator. Demanding that the solutions for the wave functions obey certain physical restrictions, we impose constraints on the two-dimensional, $(A,\delta )$, parameter space, thereby restricting the types of matter content that can be supported by a particular lattice refinement model.

Figure 1

Regions in which the solutions to the different vacuum Hamiltonian constraint equations are normalizable at large scales. Note that ${}^{\mathrm{self}\mathrm{\text{−}}\mathrm{adjoint},\mathrm{varying}\mathrm{lattice}}\Psi (\nu )$ is oscillatory only for $A<-1/12$.

Figure 2

The regions of parameter space in which the wave functions of the self-adjoint Hamiltonian constraint equation with lattice refinement are physically acceptable. Notice that there are regions (crosses) in which the Taylor expansions used to calculate the large scale behavior of the wave functions are no longer valid, and hence while we can say that these wave functions decay sufficiently quickly on large scales to be normalizable and are physical (i.e., oscillating), we cannot be sure that there is no new large scale behavior due to the underlying discreteness.

Figure 3

The full loop quantum gravity theory allows only the range $0<A<-1/2$. Within this range we can see that the acceptable types of matter content are significantly restricted. In addition, notice that for a varying lattice ($A\ne 0$) it is not always possible to treat the large scale behavior of the wave functions perturbatively (dashed line with crosses).