Discreteness corrections to the effective Hamiltonian of isotropic loop quantum cosmology

One of the qualitatively distinct and robust implications of loop quantum gravity is the underlying discrete structure. In the cosmological context elucidated by loop quantum cosmology, this is manifested by the Hamiltonian constraint equation being a (partial) difference equation. One obtains an effective Hamiltonian framework by making the continuum approximation followed by a WKB approximation. In the large volume regime, these lead to the usual classical Einstein equation which is independent of both the Barbero-Immirzi parameter gamma as well as h. In this work, we present an alternative derivation of the effective Hamiltonian by-passing the continuum approximation step. As a result, the effective Hamiltonian is obtained as a closed form expression in gamma. These corrections to the Einstein equation can be thought of as corrections due to the underlying discrete (spatial) geometry with gamma controlling the size of these corrections. These corrections imply a bound on the rate of change of the volume of the isotropic universe. In most cases these are perturbative in nature but for a cosmological constant dominated isotropic universe, there are significant deviations.