  It is widely believed that string theory easily allows for a QCD axion in the
cosmologically favoured mass range. The required small decay constant, $f_a\ll
M_P$, can be implemented by using a large compactification volume. This points
to the Large Volume Scenario which in turn makes certain cosmological
predictions: First, the closed string axion behaves similarly to a
field-theoretic axion in the pre-inflationary scenario, i.e. the initial value
can be tuned but one is constrained by isocurvature fluctuations. In addition,
the volume represents a long-lived modulus that may lead to an early
matter-dominated phase. Finally, the decay of the volume modulus to its own
axion tends to overproduce dark radiation. In this paper we aim to carefully
analyze the cosmology by studying models that not only allow for a QCD axion
but also include inflation. Quite generally, limits on isocurvature
fluctuations restrict us to relatively low-scale inflation, which in the
present stringy context points to K\"ahler moduli inflation. As a novel feature
we find that the lightest (volume) modulus couples strongly to the Higgs. It
hence quickly decays to the SM, thus resolving the original dark radiation
problem. This decay is much faster than that of the inflaton, implying that
reheating is determined by the inflaton decay. The inflaton could potentially
reintroduce a dark radiation problem since it decays to lighter moduli and
their axions with equal rates. However, due its mixing with the QCD-saxion, the
inflaton has also a direct decay rate to the SM, enhanced by the number of SM
gauge bosons. This results in an amount of dark radiation that is consistent
with present limits but potentially detectable in future measurements.
