The faint end of the $z \sim 3 - 7$ Luminosity Function of Lyman-alpha emitters behind lensing clusters observed with MUSE.

We present the results obtained with VLT/MUSE on the faint-end of the Lyman-alpha luminosity function(LF) based on deep observations of four lensing clusters. The precise aim of the present study is to further constrain the abundance of Lyman-alpha emitters(LAEs) by taking advantage of the magnificationprovided by lensing clusters. We blindly selected a sample of 156 LAEs, with redshiftsbetween $2.9 \le z \le 6.7$ and magnification-corrected luminosities in the range $ 39 \lesssim \log L_{Ly_{\alpha}}$ [erg s$^{-1}$] $\lesssim 43$. The price to pay to benefit from magnificationis a reduction of the effective volume of the survey, together with a more complex analysisprocedure. To properly take into account the individual differences in detection conditions (including lensing configurations, spatial and spectral morphologies) when computing the LF, a new method based on the 1/Vmax approach was implemented. The LAE LF has been obtained in four different redshiftbins with constraints down to $\log L_{Ly_{\alpha}} = 40.5$. From our data only, no significant evolution of LF mean slope can be found. When performing a Schechter analysis including data from the literature to complete the present sample a steep faint-end slope was measured varying from $\alpha = -1.69^{+0.08}_{-0.08}$ to $\alpha = -1.87^{+0.12}_{-0.12}$ between the lowest and the highest redshiftbins. The contribution of the LAE population to the star formation rate density at $z \sim 6$ is $\lesssim 50$% depending on the luminosity limit considered, which is of the same order as the Lyman-break galaxy(LBG) contribution. The evolution of the LAE contribution with redshiftdepends on the assumed escape fraction of Lyman-alpha photons, and appears to slightly increase with increasing redshiftwhen this fraction is conservatively set to one. ( abridged)