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Connecting Faint End Slopes of the Lyman-$α$ emitter and Lyman-break Galaxy Luminosity Functions

We predict Lyman-$α$ (Ly$α$) luminosity functions (LFs) of Ly$α$-selected galaxies (Ly$α$ emitters, or LAEs) at $z=3-6$ using the phenomenological model of Dijkstra & Wyithe (2012). This model combines observed UV-LFs of Lyman-break galaxies (LBGs, or drop out galaxies), with constraints on their distribution of Ly$α$ line strengths as a function of UV-luminosity and redshift. Our analysis shows that while Ly$α$ LFs of LAEs are generally not Schechter functions, these provide a good description over the luminosity range of $\log_{10}( L_α/{\rm erg}\,{\rm s}^{-1})=41-44$. Motivated by this result, we predict Schechter function parameters at $z=3-6$. Our analysis further shows that (i) the faint end slope of the Ly$α$ LF is steeper than that of the UV-LF of Lyman-break galaxies, (with a median $α_{Lyα} < -2.0$ at $z\gtrsim 4$), and (ii) a turn-over in the Ly$α$ LF of LAEs at Ly$α$ luminosities $10^{40}$ erg s$^{-1}<L_α\lesssim 10^{41}$ erg s$^{-1}$ may signal a flattening of UV-LF of Lyman-break galaxies at $-12>M_{\rm UV}>-14$. We discuss the implications of these results - which can be tested directly with upcoming surveys - for the Epoch of Reionization.