  One proposal for deriving effective cosmological models from theories of
quantum gravity is to view the former as a mean-field (hydrodynamic)
description of the latter, which describes a universe formed by a 'condensate'
of quanta of geometry. This idea has been successfully applied within the
setting of group field theory (GFT), a quantum field theory of 'atoms of space'
which can form such a condensate. We further clarify the interpretation of this
mean-field approximation, and show how it can be used to obtain a semiclassical
description of the GFT, in which the mean field encodes a classical statistical
distribution of geometric data. In this sense, GFT condensates are quantum
homogeneous geometries that also contain statistical information about
cosmological inhomogeneities. We show in the isotropic case how this
information can be extracted from geometric GFT observables and mapped to
quantities of observational interest. Basic uncertainty relations of
(non-commutative) Fourier transforms imply that this statistical description
can only be compatible with the observed near-homogeneity of the Universe if
the typical length scale associated to the distribution is much larger than the
fundamental 'Planck' scale. As an example of effective cosmological equations
derived from the GFT dynamics, we then use a simple approximation in one class
of GFT models to derive the 'improved dynamics' prescription of holonomy
corrections in loop quantum cosmology.
