  The early universe provides an opportunity for quantum gravity to connect to
observation by explaining the large-scale structure of the Universe. In the
group field theory (GFT) approach, a macroscopic universe is described as a GFT
condensate; this idea has already been shown to reproduce a semiclassical large
universe under generic conditions, and to replace the cosmological singularity
by a quantum bounce. Here we extend the GFT formalism by introducing additional
scalar degrees of freedom that can be used as a physical reference frame for
space and time. This allows, for the first time, the extraction of correlation
functions of inhomogeneities in GFT condensates: in a way conceptually similar
to inflation, but within a quantum field theory of both geometry and matter,
quantum fluctuations of a homogeneous background geometry become the seeds of
cosmological inhomogeneities. We find approximately scale-invariant initial
quantum fluctuations in the local volume, with naturally small amplitude; this
behaviour extends to other quantities such as the matter density. These results
confirm the potential of GFT condensate cosmology to provide a purely quantum
gravitational foundation for the understanding of the early universe.
