QFT in curved spacetime from quantum gravity: Proper WKB decomposition of the gravitational component

Starting from a reanalysis of previous work, we construct the proper low-energy quantum field theory (QFT) limit of a full quantum gravity theory in the Born-Oppenheimer approach. We separate the gravitational sector into a classical background, given by a vacuum diagonal Bianchi I cosmology, and its quantum perturbations represented by the two graviton degrees of freedom; we further include quantum matter in the form of a test scalar field. We then implement a Born-Oppenheimer separation, where the gravitons and matter play the roles of "slow" and "fast" quantum components, respectively, and perform a WKB expansion in a Planckian parameter. The functional Schrodinger evolution for matter is recovered after averaging over quantum-gravitational effects, provided that a condition is imposed on the gravitons' wave functional. Such a condition fixes the graviton dynamics and is equivalent to the purely gravitational Wheeler-DeWitt constraint imposed in previous approaches. The main accomplishment of the present work is to clarify that QFT in curved spacetime can be recovered in the low-energy limit of quantum gravity only after averaging over the graviton degrees of freedom, in the spirit of effective field theory. Furthermore, it justifies a posteriori the implementation of the gravitational Wheeler-DeWitt equation on the "slow" gravitons' wave functional rather than assuming its validity a priori.