The MUSE Hubble Ultra Deep Field Survey X. Ly$\alpha$ Equivalent Widths at $2.9 < z < 6.6$

We present rest-frame Ly$\alpha$ equivalent widths (EW) of 417 Ly$\alpha$ emitters (LAEs) detected with Multi Unit Spectroscopic Explorer (MUSE) on the Very Large Telescope (VLT) at $2.9 < z < 6.6$ in the Hubble Ultra Deep Field. Based on the deep MUSE spectroscopy and ancillary Hubble Space Telescope (HST) photometry data, we carefully measured EW values taking into account extended Ly$\alpha$ emission and UV continuum slopes ($\beta$). Our LAEs reach unprecedented depths, both in Ly$\alpha$ luminosities and UV absolute magnitudes, from log($L_{\rm Ly\alpha}$/erg s$^{-1}$) $\sim$41.0 to 43.0 and from Muv $\sim$ -16 to -21 (0.01-1.0 $L^{*}_{\rm z=3}$). The EW values span the range of $\sim$ 5 to 240 \AA\ or larger, and their distribution can be well fitted by an exponential law $N = N_{\rm 0}$ exp($-$EW/$w_{\rm 0}$). Owing to the high dynamic range in Muv, we find that the scale factor, $w_{\rm 0}$, depends on Muv in the sense that including fainter Muv objects increases $w_{\rm 0}$, i.e., the Ando effect. The results indicate that selection functions affect the EW scale factor. Taking these effects into account, we find that our $w_{\rm 0}$ values are consistent with those in the literature within $1\sigma$ uncertainties at $2.9 < z < 6.6$ at a given threshold of Muv and $L_{\rm Ly\alpha}$. Interestingly, we find 12 objects with EW $>200$ \AA\ above $1\sigma$ uncertainties. Two of these 12 LAEs show signatures of merger or AGN activity: the weak CIV $\lambda 1549$ emission line. For the remaining 10 very large EW LAEs, we find that the EW values can be reproduced by young stellar ages ($< 100$ Myr) and low metallicities ($\lesssim 0.02$ $Z_{\rm \odot}$). Otherwise, at least part of the Ly$\alpha$ emission in these LAEs needs to arise from anisotropic radiative transfer effects, fluorescence by hidden AGN or quasi-stellar object activity, or gravitational cooling.