  One of the fundamental challenges for quantum cosmology is to explain the
emergence of our macroscopic Universe from physics at the Planck scale. In the
group field theory (GFT) approach to quantum gravity, such a macroscopic
universe results from the formation of a "condensate" of fundamentally discrete
degrees of freedom. It has been shown that the effective dynamics of such GFT
condensates follows the classical Friedmann dynamics at late times, while
avoiding the classical singularity by a bounce akin to the one of loop quantum
cosmology (LQC). It was also shown how quantum fluctuations in a GFT condensate
provide an initial power spectrum of volume fluctuations around exact
homogeneity. Here we connect the results for quantum fluctuations in GFT to the
usual formalism for cosmological perturbations within quantum field theory in
curved spacetime. We consider a bouncing universe filled with a massless scalar
field, in which perturbations are generated by vacuum fluctuations in the
contracting phase. Matching conditions at the bounce are provided by working
within LQC. We then compare the results to the GFT condensate scenario for
quantum gravity with massless scalar matter. Here, instead, an initial quantum
phase described by a GFT condensate generates initial scalar perturbations
through quantum fluctuations. We show general agreement in the predictions of
both approaches, suggesting that GFT condensates can provide a physical
mechanism for the emergence of a slightly inhomogeneous universe from full
quantum gravity.
