Hamiltonian constraint of loop quantum cosmology

In this paper we construct the Hamiltonian constraint operator of loop quantum cosmology using holonomies defined for arbitrary irreducible SU(2) representations labeled by spin $J$. We show that modifications to the effective semiclassical equations of motion arise both in the gravitational part of the constraint as well as matter terms. The modifications are important for phenomenological investigations of the cosmological imprints of loop quantum cosmology. We discuss the implications for the early universe evolution.

Figure 1

Log plot of the difference equation solution (dots) compared with Wheeler-DeWitt solution (solid line) for a massless scalar field at large volume. The presence of spurious solutions to the higher order difference equation eventually dominate to solution.

Figure 2

LQC modified function $s_J(a)$ (27) compared with the classical expression for the function $\mathrm{sgn}(p)\sqrt{|p|}=a$. The modified $s_J(a)$ function behaves quadratically for $a<a_{*}=\sqrt{(\gamma J{\mu }_0)/3}{l}_p$. The values of $a_{*}$ in the graph are $1.290,2.89,4.08$ for $J=10,50,100$ respectively.