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The detection of Ly$\alpha$ nebulae around $z\gtrsim 6$ quasars providesevidence for extended gas reservoirs around the first rapidly growingsupermassive black holes. Observations of $z > 6$ quasars can be explained bycosmological models provided that the black holes by which they are poweredevolve in rare, massive dark matter haloes. Whether these theoretical modelsalso explain the observed extended Ly$\alpha$ emission remains an openquestion. We post-process a suite of cosmological, radiation-hydrodynamicsimulations targeting a quasar host halo at $z>6$ with the Ly$\alpha$ radiativetransfer code RASCAS. A combination of recombination radiation fromphoto-ionised hydrogen and emission from collisionally excited gas powersLy$\alpha$ nebulae with a surface brightness profile in close agreement withobservations. We also find that, even on its own, resonant scattering of theLy$\alpha$ line associated to the quasar's broad line region can also generateLy$\alpha$ emission on $\sim 100 \, \rm kpc$ scales, resulting in comparableagreement with observed surface brightness profiles. Even if powered by a broadquasar Ly$\alpha$ line, Ly$\alpha$ nebulae can have narrow line-widths$\lesssim 1000 \, \rm km \, s^{-1}$, consistent with observational constraints.Even if there is no quasar, we find that halo gas cooling produces a faint,extended Ly$\alpha$ glow. However, to light-up extended Ly$\alpha$ nebulae withproperties in line with observations, our simulations unambiguously requirequasar-powered outflows to clear out the galactic nucleus and allow theLy$\alpha$ flux to escape and still remain resonant with halo gas. The closematch between observations and simulations with quasar outflows suggests thatAGN feedback already operates before $z \, = \, 6$ and confirms that high-$z$quasars reside in massive haloes tracing overdensities.

AGN-driven outflows and the formation of Ly$\alpha$ nebulae around high-z quasars