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We derive the dynamics of (isotropic) scalar perturbations from the mean-field hydrodynamics of full Lorentzian quantum gravity, as described by a two-sector (timelike and spacelike) Barrett-Crane group field theory model. The rich causal structure of this model allows us to consistently implement in the quantum theory the causal properties of a physical Lorentzian reference frame composed of four minimally coupled, massless, and free scalar fields. Using this frame, we are able to effectively construct relational observables that are used to recover macroscopic cosmological quantities. In particular, small isotropic scalar inhomogeneities emerge as a result of (relational) nearest-neighbor two-body entanglement between degrees of freedom of the underlying quantum gravity theory. The dynamical equations we obtain for geometric and matter perturbations show remarkable agreement with those of classical general relativity for sub-Planckian modes. Quantum gravity effects produce important deviations from the classical general relativistic dynamics for trans-Planckian modes, which we show to be associated to subhorizon scales in the physical reference frame we are employing.
Jercher et al. (Tue,) studied this question.