  A fully consistent linear perturbation theory for cosmology is derived in the
presence of quantum corrections as they are suggested by properties of inverse
volume operators in loop quantum gravity. The underlying constraints present a
consistent deformation of the classical system, which shows that the
discreteness in loop quantum gravity can be implemented in effective equations
without spoiling space-time covariance. Nevertheless, non-trivial quantum
corrections do arise in the constraint algebra. Since correction terms must
appear in tightly controlled forms to avoid anomalies, detailed insights for
the correct implementation of constraint operators can be gained. The
procedures of this article thus provide a clear link between fundamental
quantum gravity and phenomenology.
