In the context of the metric-affine Chern-Simons gravity endowed with
projective invariance, we derive analytical solutions for torsion and
nonmetricity in the homogeneous and isotropic cosmological case, described by a
flat Friedmann-Robertson-Walker metric. We describe in some details the general
properties of the cosmological solutions in the presence of a perfect fluid,
such as dynamical stability and the settling of big bounce points, and we
discuss the structure of some specific solutions reproducing de Sitter and
power law behaviours for the scale factor. Then, we focus on first-order
perturbations in the de Sitter scenario, and we study the propagation of
gravitational waves in the adiabatic limit, looking at tensor and scalar
polarizations. In particular, we find that metric tensor modes couple to
torsion tensor components, leading to the appearance, as in the metric version
of Chern-Simons gravity, of birefringence, described by different dispersion
relations for the left and right circularized polarization states. As a result,
the purely tensor part of torsion propagates like a wave, while nonmetricity
decouples and behaves like a harmonic oscillator. Finally, we discuss scalar
modes, outlining as they decay exponentially in time and do not propagate.
