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

TL;DR

This work develops a hierarchical Bayesian framework to estimate the luminosity function (LF) of astrophysical objects from multi-depth surveys with complex selection effects. Modeling the LF with a Schechter form and treating normalization as a Poisson parameter, the method jointly analyzes data from diverse MUSE surveys, accounting for measurement uncertainties and survey completeness via selection functions, and uses inhomogeneous Poisson likelihoods across surveys. Validation with mock catalogues demonstrates that combining deep, small-area data with wide-area, shallow surveys significantly reduces biases and tightens constraints across the LF, particularly at the faint end, while preserving information about the bright end. Application to 1176 Lyman-Alpha Emitters at $3<z<5$ yields LF parameters $\log(\Phi^*/\mathrm{Mpc^{-3}})=-2.86^{+0.15}_{-0.17}$, $\log(L^*/\mathrm{erg\,s^{-1}})=42.72^{+0.10}_{-0.09}$, and $\alpha=-1.81^{+0.09}_{-0.09}$, with evolution across redshift bins and with results broadly consistent with lensing-based studies at the faint end; the study highlights how systematic errors in completeness corrections become important and how future large-volume surveys can better constrain the bright end.

Abstract

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 $\gtrsim 1.5$ dex below a characteristic luminosity ($\tilde{L}^\star$) are essential to reducing biases at the faint end ($\lesssim 0.1$ dex) and that wide-area data help constrain the bright end. As a proof of concept, we considered a combined sample of 1176 Lyman $α$ emitters at redshift $3 < z < 5$ drawn from several MUSE surveys, ranging from ultra-deep ($\gtrsim 90$ hr) and narrow ($\lesssim 1$ arcmin$^2$) fields to shallow ($\lesssim 5$ hr) and wide ($\gtrsim 20$ arcmin$^2$) fields. With this complete sample, we constrain the luminosity function parameters $\log(Φ^\star/\mathrm{Mpc^{-3}}) = -2.86^{+0.15}_{-0.17}$, $\log(L^\star/\mathrm{erg\,s^{-1}}) = 42.72^{+0.10}_{-0.09}$, and $α= -1.81^{+0.09}_{-0.09}$, where the uncertainties represent the $90\%$ credible intervals. These values are in agreement with the results of studies based on gravitational lensing that reach $\log(L/\mathrm{erg\,s^{-1}}) \approx 41$, 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 $α$ emitters [$\log(L/\mathrm{erg\,s^{-1}}) \gtrsim 43$], where statistical uncertainties still dominate.

Figures (4)

• Figure 1: Results of the Bayesian model applied to mock catalogues for both individual surveys and their combination. This test demonstrates how combining multiple surveys with varying depths helps constrain the LF with reduced statistical uncertainty across a wider dynamical range of luminosities. In the top panel the solid dark blue line indicates the true LF used to generate the mock catalogues. The light blue (magenta) diamonds represent the $1/V_\mathrm{max}$ estimator of the LF obtained from the MUDF 1- (MAGG)-like survey, along with the median of the posterior samples from the Bayesian model (dashed line) and the corresponding $90\%$ credible interval. The solid black line shows the median when combining the two surveys, with the associated $90\%$ credible interval. Horizontal bars on the diamonds represent the bin widths, while vertical bars are the Poisson errors associated with the $1/V_\mathrm{max}$ estimator. In the bottom panel the dashed lines (same colours as in the top panel) show the logarithmic difference between the median LF obtained from the Bayesian model and the true LF. The solid lines indicate the $5$th and $95$th percentiles relative to the true LF.
• Figure 2: LF obtained from the combination of the four MUSE surveys considered in this study (see Table \ref{['tab:samples_prop']}). Top panel: Median LF reconstructed using our Bayesian model applied to the full sample (solid black line), along with the corresponding $90\%$ credible interval. The blue colour-coded diamonds represent the $1/V_\mathrm{max}$ estimates from each individual survey, with vertical bars indicating the Poisson uncertainties and horizontal bars the bin widths. The dark blue circles show the weighted average of the $1/V_\mathrm{max}$ points across surveys in each bin. Arrows in the same colour scheme indicate the luminosity limit above which each survey reaches a completeness higher than $10\%$. For comparison, coloured lines with different line styles display the best-fit Schechter LFs from various studies in the literature Herenz2019Thai2023. Bottom panel: Statistical uncertainty on the LF as a function of luminosity. The solid black lines show the relative $5$th and $95$th percentiles of the Bayesian posterior distribution.
• Figure 3: Posterior distributions of the LF parameters $\alpha$, $\tilde{L}^\star$, and $\log(\Phi^\star)$ inferred from the Bayesian model for the overall sample (dark blue) and for two redshift bins: $3 < z < 4$ (light blue) and $4 < z < 5$ (purple). Rightmost panels: Marginalized one-dimensional posterior distributions for each parameter. Other panels: Joint two-dimensional posterior distributions, highlighting the correlations between parameters. The $5$th, $50$th, and $95$th percentiles are shown.
• Figure 4: Selection functions $P_\mathrm{det,k}(\tilde{L}, z)$ for LAEs analysed in each individual survey. The white and black contours indicate the $10\%$ and $90\%$ completeness limits, respectively. The MW and MAGG selection functions are taken from Herenz2019 and Fossati2021, respectively. The vertical stripe with a detection fraction of zero around redshift $z \sim 3.8$ in the MXDF and MUDF surveys is caused by the Ground Layer Adaptive Optics module, which uses an artificial laser guide star to improve image quality during the observations Fossati2019Bacon2023. This region therefore contains no data.