The MUSE Hubble Ultra Deep Field Survey VI: The Faint-End of the Lya
  Luminosity Function at 2.91 < z < 6.64 and Implications for Reionisation
  We present the deepest study to date of the Lya luminosity function (LF) in a
blank field using blind integral field spectroscopy from MUSE. We constructed a
sample of 604 Lya emitters (LAEs) across the redshift range 2.91 < z < 6.64
using automatic detection software in the Hubble Ultra Deep Field. We calculate
accurate total Lya fluxes capturing low surface brightness extended Lya
emission now known to be a generic property of high-redshift star-forming
galaxies. We simulated realistic extended LAEs to characterise the selection
function of our samples, and performed flux-recovery experiments to test and
correct for bias in our determination of total Lya fluxes. We find an accurate
completeness correction accounting for extended emission reveals a very steep
faint-end slope of the LF, alpha, down to luminosities of log10 L erg s^-1<
41.5, applying both the 1/Vmax and maximum likelihood estimators. Splitting the
sample into three broad redshift bins, we see the faint-end slope increasing
from -2.03+1.42-inf at z ~ 3.44 to -2.86+0.76-inf at z ~ 5.48, however no
strong evolution is seen between the 68% confidence regions in L*-alpha
parameter space. Using the Lya line flux as a proxy for star formation
activity, and integrating the observed LFs, we find that LAEs' contribution to
the cosmic SFRD rises with redshift until it is comparable to that from
continuum-selected samples by z ~ 6. This implies that LAEs may contribute more
to the star-formation activity of the early Universe than previously thought -
any additional interglactic medium correction would act to further boost the
Lya luminosities. Finally, assuming fiducial values for the escape of Lya and
LyC radiation, and the clumpiness of the IGM, we integrated the maximum
likelihood LF at 5.00 < z < 6.64 and find we require only a small extrapolation
beyond the data (< 1 dex in L) for LAEs alone to maintain an ionised IGM at z ~
6.
