The MUSE-Wide Survey: A determination of the Lyman $\alpha$ emitter luminosity function at $3 < z < 6$

(Abridged) We investigate the Lyman $\alpha$ emitter luminosity function (LAE LF) within the redshift range $2.9 \leq z \leq 6$ from the first instalment of the blind integral field spectroscopic survey MUSE-Wide. This initial part of the survey probes a region of 22.2 arcmin$^2$ in the CANDELS/GOODS-S field. The dataset provided us with 237 LAEs from which we construct the LAE LF in the luminosity range $42.2 \leq \log L_\mathrm{Ly\alpha} [\mathrm{erg\,s}^{-1}]\leq 43.5$ within a volume of $2.3\times10^5$ Mpc$^3$. For the LF construction we utilise three different non-parametric estimators: The classical $1/V_\mathrm{max}$ method, the $C^{-}$ method, and an improved binned estimator for the differential LF. All three methods deliver consistent results, with the cumulative LAE LF being $\Phi(\log L_\mathrm{Ly\alpha} [\mathrm{erg\,s}^{-1}] = 43.5) \simeq 3\times 10^{-6}$ Mpc$^{-3}$ and $\Phi(\log L_\mathrm{Ly\alpha} [\mathrm{erg\,s}^{-1}] = 42.2) \simeq 2 \times 10^{-3}$ Mpc$^{-3}$ towards the bright- and faint-end of our survey, respectively. By employing a non-parametric statistical test, as well as by comparing the full sample to sub-samples in redshift bins, we find no supporting evidence for an evolving LAE LF over the probed redshift and luminosity range. We determine the best-fitting Schechter function parameters $\alpha = -1.84^{+0.42}_{-0.41}$ and $\log L^* [\mathrm{erg\,s}^{-1}] = 42.2^{+0.22}_{-0.16}$ with the corresponding normalisation $\log \phi^* [\mathrm{Mpc}^{-3}] = -2.71$. When correcting for completeness in the LAE LF determinations, we take into account that LAEs exhibit diffuse extended low surface-brightness haloes. We compare the resulting LF to one obtained where we apply a correction assuming compact point-like emission. We find that the standard correction underestimates the LAE LF at the faint end of our survey by a factor of 2.5.