Lyman-$\alpha$ as a tracer of cosmic reionisation in the SPHINX
  radiation-hydrodynamics cosmological simulation
  The Ly$\alpha$ emission line is one of the most promising probes of cosmic
reionisation but isolating the signature of a change in the ionisation state of
the IGM is challenging because of intrinsic evolution and internal radiation
transfer effects. We present the first study of the evolution of Ly$\alpha$
emitters (LAE) during the epoch of reionisation based on a full
radiation-hydrodynamics cosmological simulation that is able to capture both
the large-scale process of reionisation and the small-scale properties of
galaxies. We predict the Ly$\alpha$ emission of galaxies in the $10^3$ cMpc$^3$
SPHINX simulation at $6\leq z\leq9$ by computing the full Ly$\alpha$ radiation
transfer from ISM to IGM scales. SPHINX is able to reproduce many observational
constraints such as the UV/Ly$\alpha$ luminosity functions and stellar mass
functions at z $\geq$ 6 for the dynamical range probed by our simulation
($M_{\rm 1500}\gtrsim-18$, $L_{\rm Ly\alpha}\lesssim10^{42}$ erg/s,
$M_{\star}\lesssim10^9$ M$_{\odot}$). As intrinsic Ly$\alpha$ emission and
internal Ly$\alpha$ escape fractions barely evolve from $z=6$ to 9, the
observed suppression of Ly$\alpha$ luminosities with increasing redshift is
fully attributed to IGM absorption. For most observable galaxies ($M_{\rm
1500}\lesssim-16$), the Ly$\alpha$ line profiles are slightly shifted to the
red due to internal radiative transfer effects which mitigates the effect of
IGM absorption. Overall, the enhanced Ly$\alpha$ suppression during
reionisation traces the IGM neutral fraction $x_{\rm HI}$ well but the
predicted amplitude of this reduction is a strong function of the Ly$\alpha$
peak shift, which is set at ISM/CGM scales. We find that a large number of LAEs
could be detectable in very deep surveys during reionisation when $x_{\rm HI}$
is still $\approx 50\%$.
