Cosmological scalar perturbations for a metric reconstructed from group field theory

While homogeneous cosmologies have long been studied in the group field theory (GFT) approach to quantum gravity, including a quantum description of cosmological perturbations is highly non-trivial. Here we apply a recent proposal for reconstructing an effective spacetime metric in GFT to the case of a metric with small inhomogeneities over a homogeneous background. We detail the procedure and give general expressions for cosmological scalar perturbations defined in terms of the GFT energy-momentum tensor. These include all the scalar components of standard perturbation theory and hence can be used to define gauge-invariant quantities. This is a major advantage of the effective metric approach compared to previous GFT studies limited to volume perturbations. We compute these perturbations explicitly for a particular Fock coherent state. While it was previously shown that such a state can be interpreted as an approximately flat homogeneous cosmology at late times, here we find that, in a very simple example, inhomogeneities do not follow the dynamics of general relativity in the semiclassical regime. More specifically, restricting ourselves to a specific coherent state in a simple (free) GFT, we study two types of perturbative GFT modes, squeezed and oscillating modes. For squeezed modes we find perturbation equations with Euclidean signature and a late-time limit that differs from general relativistic perturbation equations. Oscillating modes satisfy different dynamical equations that also differ from those of general relativity, but show a Lorentzian signature. Considering that our results were obtained within a number of simplifying assumptions [...], we discuss how going beyond these assumptions could lead to a more desirable phenomenology. Overall, our analysis should be understood as a first step in understanding cosmological perturbations within the effective GFT metric.