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Loop quantum cosmology yields two kinds of quantum corrections to the effective equations of motion for cosmological perturbations. Here we focus on the holonomy kind and we study the problem of the closure of the resulting algebra of constraints when a scalar field is considered in the matter Hamiltonian. Up to now, tensor, vector and scalar perturbations were studied independently, leading to different algebras of constraints. The structures of the related algebras were imposed by the requirement of anomaly freedom. In this article we show that the algebra can be modified by a very simple quantum correction, holding for all types of perturbations. The Mukhanov-Sasaki equations of motion are similarly modified by a simple term.
Cailleteau et al. (Tue,) studied this question.