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 $6leq zleq9$ 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 ($M_{rm 1500}gtrsim-18$, $L_{rm Lyalpha}lesssim10^{42}$ 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 $approx 50%$.
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