  Coherent state functional integrals for the minisuperspace models of quantum
cosmology are studied. By the well-established canonical theories, the
transition amplitudes in the path-integral representations of Wheeler-DeWitt
quantum cosmology and loop quantum cosmology can be formulated through group
averaging. The effective action and Hamiltonian with higher-order quantum
corrections are thus obtained in both models. It turns out that for a
non-symmetric Hamiltonian constraint operator, the Moyal (star)-product emerges
naturally in the effective Hamiltonian. This reveals the intrinsic relation
among coherent state functional integral, effective theory and star-product.
Moreover, both the resulted effective theories imply a possible quantum
cosmological effect in large scale limit.
