Scalar Cosmological Perturbations from Quantum Entanglement within Lorentzian Quantum Gravity

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 (GFT) 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 agreement with those of classical general relativity in the long-wavelength, super-horizon limit. In general, deviations become important for sub-horizon modes, which seem to be naturally associated with a trans-Planckian regime in our physical reference frame. We argue that these trans-Planckian corrections are quantum gravitational in nature. However, we explicitly show that for some physically interesting solutions these quantum gravity effects can be quite small, leading to a very good agreement with the classical GR behavior.