Lyman Alpha line properties at $z \simeq 3.78$ and their environmental dependence: a case study around a massive proto-cluster

Ly$\alpha$-emitting galaxies (LAEs) are easily detectable in the high-redshift Universe and are potentially efficient tracers of large scale structure at early epochs, as long as their observed properties do not strongly depend on environment. We investigate the luminosity and equivalent width functions of LAEs in the overdense field of a protocluster at redshift $z \simeq 3.78$. Using a large sample of LAEs (many spectroscopically confirmed), we find that the Ly$\alpha$ luminosity distribution is well-represented by a Schechter (1976) function with $\log(L^{\ast}/{\rm erg s^{-1}}) = 43.26^{+0.20}_{-0.22}$ and $\log(\phi^{\ast}/{\rm Mpc^{-3}})=-3.40^{+0.03}_{-0.04}$ with $\alpha=-1.5$. Fitting the equivalent width distribution as an exponential, we find a scale factor of $\omega=79^{+15}_{-15}$ Angstroms. We also measured the Ly$\alpha$ luminosity and equivalent width functions using the subset of LAEs lying within the densest cores of the protocluster, finding similar values for $L^*$ and $\omega$. Hence, despite having a mean overdensity more than 2$\times$ that of the general field, the shape of the Ly$\alpha$ luminosity function and equivalent width distributions in the protocluster region are comparable to those measured in the field LAE population by other studies at similar redshift. While the observed Ly$\alpha$ luminosities and equivalent widths show correlations with the UV continuum luminosity in this LAE sample, we find that these are likely due to selection biases and are consistent with no intrinsic correlations within the sample. This protocluster sample supports the strong evolutionary trend observed in the Ly$\alpha$ escape fraction and suggest that lower redshift LAEs can be on average significantly more dusty that their counterparts at higher redshift