Bayesian luminosity function estimation in multi-depth datasets with selection effects: A case study for 3 < z < 5 Lyman α emitters

We present a hierarchical Bayesian framework designed to infer the luminosity function of any class of object by jointly modelling data from multiple surveys with varying depth, completeness, and sky coverage. Our method explicitly accounts for selection effects and measurement uncertainties (e.g. in luminosity) and can be generalized to any extensive quantity, such as mass. We validated the model using mock catalogues; from this we determined that deep data reaching greater than or similar to 1.5 dex below a characteristic luminosity (L-star) are essential to reducing biases at the faint end (less than or similar to 0.1 dex) and that wide-area data help constrain the bright end. As a proof of concept, we considered a combined sample of 1176 Lyman alpha emitters at redshift 3 &lt; z &lt; 5 drawn from several MUSE surveys, ranging from ultra-deep (greater than or similar to 90 h) and narrow (less than or similar to 1 arcmin(2)) fields to shallow (less than or similar to 5 h) and wide (greater than or similar to 20 arcmin(2)) fields. With this complete sample, we constrain the luminosity function parameters log(Phi(star)/Mpc(-3)) = -2.86(-0.17)(+0.15), log(L-star/erg s(-1)) = 42.72(-0.09)(+0.10), and alpha = -1.81(-0.09)(+0.09), where the uncertainties represent the 90% credible intervals. These values are in agreement with the results of studies based on gravitational lensing that reach log(L/erg s(-1))approximate to 41, although differences in the faint-end slope underscore how systematic errors are starting to dominate. In contrast, wide-area surveys represent the natural extension needed to constrain the brightest Lyman alpha emitters [log(L/erg s(-1))greater than or similar to 43], where statistical uncertainties still dominate.