  We study a contracting universe composed of cold dark matter and radiation,
and with a positive cosmological constant. As is well known from standard
cosmological perturbation theory, under the assumption of initial quantum
vacuum fluctuations the Fourier modes of the comoving curvature perturbation
that exit the (sound) Hubble radius in such a contracting universe at a time of
matter-domination will be nearly scale-invariant. Furthermore, the modes that
exit the (sound) Hubble radius when the effective equation of state is slightly
negative due to the cosmological constant will have a slight red tilt, in
agreement with observations. We assume that loop quantum cosmology captures the
correct high-curvature dynamics of the space-time, and this ensures that the
big-bang singularity is resolved and is replaced by a bounce. We calculate the
evolution of the perturbations through the bounce and find that they remain
nearly scale-invariant. We also show that the amplitude of the scalar
perturbations in this cosmology depends on a combination of the sound speed of
cold dark matter, the Hubble rate in the contracting branch at the time of
equality of the energy densities of cold dark matter and radiation, and the
curvature scale that the loop quantum cosmology bounce occurs at. Importantly,
as this scenario predicts a positive running of the scalar index, observations
can potentially differentiate between it and inflationary models. Finally, for
a small sound speed of cold dark matter, this scenario predicts a small
tensor-to-scalar ratio.
