  In loop quantum cosmology, one has to make a choice of SU(2) irreducible
representation in which to compute holonomies and regularize the curvature of
the connection. The systematic choice made in the literature is to work in the
fundamental representation, and very little is known about the physics
associated with higher spin labels. This constitutes an ambiguity whose
understanding, we believe, is fundamental for connecting loop quantum cosmology
to full theories of quantum gravity like loop quantum gravity, its spin foam
formulation, or cosmological group field theory. We take a step in this
direction by providing here a new closed formula for the Hamiltonian of flat
FLRW models regularized in a representation of arbitrary spin. This expression
is furthermore polynomial in the basic variables which correspond to
well-defined operators in the quantum theory, takes into account the so-called
inverse-volume corrections, and treats in a unified way two different
regularization schemes for the curvature. After studying the effective
classical dynamics corresponding to single and multiple spin Hamiltonians, we
study the behavior of the critical density when the number of representations
is increased, and the stability of the difference equations in the quantum
theory.
