  Aql X-1 is a prolific transient neutron star low-mass X-ray binary that
exhibits an accretion outburst approximately once every year. Whether the
thermal X-rays detected in intervening quiescent episodes are the result of
cooling of the neutron star or due to continued low-level accretion remains
unclear. In this work we use Swift data obtained after the long and bright 2011
and 2013 outbursts, as well as the short and faint 2015 outburst, to
investigate the hypothesis that cooling of the accretion-heated neutron star
crust dominates the quiescent thermal emission in Aql X-1. We demonstrate that
the X-ray light curves and measured neutron star surface temperatures are
consistent with the expectations of the crust cooling paradigm. By using a
thermal evolution code, we find that ~1.2-3.2 MeV/nucleon of shallow heat
release describes the observational data well, depending on the assumed
mass-accretion rate and temperature of the stellar core. We find no evidence
for varying strengths of this shallow heating after different outbursts, but
this could be due to limitations of the data. We argue that monitoring Aql X-1
for up to ~1 year after future outbursts can be a powerful tool to break model
degeneracies and solve open questions about the magnitude, depth and origin of
shallow heating in neutron star crusts.
