  The effective description of loop quantum cosmology (LQC) has proved to be a
convenient platform to study phenomenological implications of the quantum
bounce that resolves the classical big-bang singularity. Originally, this
description was derived using Gaussian quantum states with small dispersions.
In this paper we present a generalization to incorporate states with large
dispersions. Specifically, we derive the \emph{generalized} effective Friedmann
and Raychaudhuri equations and propose a generalized effective Hamiltonian
which are being used in an ongoing study of the phenomenological consequences
of a broad class of quantum geometries. We also discuss an interesting
interplay between the physics of states with larger dispersions in standard
LQC, and of sharply peaked states in (hypothetical) LQC theories with larger
area gap.
