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In loop quantum cosmology, one has to make a choice of SU(2) irreduciblerepresentation in which to compute holonomies and regularize the curvature ofthe connection. The systematic choice made in the literature is to work in thefundamental representation, and very little is known about the physicsassociated with higher spin labels. This constitutes an ambiguity whoseunderstanding, we believe, is fundamental for connecting loop quantum cosmologyto full theories of quantum gravity like loop quantum gravity, its spin foamformulation, or cosmological group field theory. We take a step in thisdirection by providing here a new closed formula for the Hamiltonian of flatFLRW models regularized in a representation of arbitrary spin. This expressionis furthermore polynomial in the basic variables which correspond towell-defined operators in the quantum theory, takes into account the so-calledinverse-volume corrections, and treats in a unified way two differentregularization schemes for the curvature. After studying the effectiveclassical dynamics corresponding to single and multiple spin Hamiltonians, westudy the behavior of the critical density when the number of representationsis increased, and the stability of the difference equations in the quantumtheory.

New Hamiltonians for loop quantum cosmology with arbitrary spin representations