Median Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely Deep Field

We present the median surface brightness profiles of diffuse Ly$\alpha$ haloes (LAHs) around star-forming galaxies by stacking 155 spectroscopically confirmed Ly$\alpha$ emitters (LAEs) at 3<z<4 in the MUSE Extremely Deep Field (MXDF), with median Ly$\alpha$ luminosity $\mathrm{L_{Ly\alpha} \approx 10^{41.1} erg\,s^{-1}}$. After correcting for a systematic surface brightness offset we identified in the datacube, we detect extended Ly$\alpha$ emission out to a distance of 270 kpc. The median Ly$\alpha$ surface brightness profile shows a power-law decrease in the inner 20 kpc, and a possible flattening trend at larger distance. This shape is similar for LAEs with different Ly$\alpha$ luminosities, but the normalisation of the surface brightness profile increases with luminosity. At distances larger than 50 kpc, we observe strong overlap of adjacent LAHs, and the Ly$\alpha$ surface brightness is dominated by the LAHs of nearby LAEs. We find no clear evidence of redshift evolution of the observed Ly$\alpha$ profiles when comparing with samples at 4<z<5 and 5<z