The linearization of the Kodama state

We study the question of whether the linearization of the Kodama state around classical de Sitter spacetime is normalizable in the inner product of the theory of linearized gravitons on de Sitter spacetime. We find the answer is no in the Lorentzian theory. However, in the Euclidean theory the corresponding linearized Kodama state is delta-functional normalizable. We discuss whether this result invalidates the conjecture that the full Kodama state is a good physical state for quantum gravity with positive cosmological constant.