The PAU Survey: The $i$-band galaxy luminosity function from the present-day to $z = 2$

TL;DR

The paper develops a framework to measure the rest-frame $i$-band LF from $z=0$ to $z=2$ using PAUS, leveraging 40 narrow-band filters for precise photometric redshifts and a GALFORM-based lightcone mock to quantify selection and redshift-error systematics. A random forest regression model predicts $k$-corrections from $u,g,r,i,z$ photometry, enabling robust rest-frame magnitude estimates up to $z\lesssim 1$, while high-$z$ performance shows biases from photometric redshift outliers. The LF is estimated with the $1/V_{ m max}$ method in thin redshift slices, revealing good agreement with the mock at $z<1$ and highlighting the impact of photo-$z$ outliers and selection incompleteness at higher redshift; red and blue populations evolve distinctly, with red galaxies dominating the bright end and blue galaxies driving the faint end. The results demonstrate PAUS’s capability to characterize the galaxy population across a wide redshift baseline with photometric redshifts, providing a benchmark for galaxy evolution models and informing future wide-field, high-precision photometric surveys.

Abstract

We present a measurement of the $i$-band galaxy luminosity function from the present-day to $z = 2$, using over 1.1 million galaxies from the Physics of the Accelerating Universe Survey (PAUS). PAUS combines broad-band imaging from the Canada-France-Hawaii Telescope Lensing Survey with narrow-band photometry from PAUCam, enabling high-precision photometric redshifts with an accuracy of $σ_{68} (Δz) = 0.019$ down to $i_{\textrm{AB}} = 23$. A synthetic lightcone mock catalogue built using the \texttt{GALFORM} semi-analytic model is used to simulate PAUS selection effects and photometric uncertainties, and to derive a machine-learning based estimate of the $k$-correction. We recover rest-frame $i$-band luminosities using a random forest regressor trained on simulated $ugriz$ photometry and redshifts. Luminosity functions are estimated using the $1/V_{\textrm{max}}$ method, accounting for photometric redshift and magnitude errors, and validated against mock data. We find good agreement between observations and models at $z < 1$, with increasing discrepancies at higher redshifts due to photometric redshift outliers. The bright-end of the luminosity function becomes flatter at high redshift, primarily driven by redshift errors. We show that the faint-end of the luminosity function becomes more incomplete with increasing redshift, but is still useful for constraining models. We analyze the red and blue galaxy populations separately, observing distinct evolutionary trends. The model overpredicts the number of both faint red and blue galaxies. Our study highlights the importance of accurate redshift estimation and selection modeling for robust luminosity function recovery, and demonstrates that PAUS can characterise the galaxy population with photometric redshifts across a wide redshift baseline.

Figures (17)

• Figure 1: Redshift distribution (normalised to the number of galaxies per unit redshift and per square degree) of the GALFORM lightcone mock catalogue (black), PAUS W1 (green), and PAUS W3 (red) to $i_{\rm AB} = 23$. The vertical grey bands show the redshift bins used for calculating the luminosity function.
• Figure 2: The photometric redshift estimated from the random sample of galaxies drawn from the lightcone mock catalogue to $i_{\rm AB}=23$ using the BCNZ2 code. The bottom panel shows the true (specz) redshifts from the mock catalogue on the $y$-axis compared to the BCNZ2 estimated redshifts on the $x$-axis. The top panel shows the parameter $\eta$, which is the number of galaxies with photometric redshifts in a particular redshift bin (indicated by the shaded vertical bands) divided by the original number of galaxies in the bin (i.e. using the spectroscopic or true redshift value). The horizontal dashed and dotted lines in the upper panel indicate errors on this number ratio of $10$ per cent and $25$ per cent, respectively. The green and red regions show the redshift bins with $|\eta - 1|$ lower and greater than 10 per cent, respectively
• Figure 3: The exact $i-$band $k$-correction as a function of redshift predicted by GALFORM from the lightcone mock catalogue. The pixel shading shows the galaxy number count for a sample with $i_{\rm AB}=23$, with black representing 1 galaxy and the brightest colour pixel having $\sim1\,300$ galaxies as shown in the colour bar on the right.
• Figure 4: The performance of the RFR machine learning estimate of the $k$-correction, expressed in terms of the difference between the predicted and true absolute magnitude for each galaxy. The true magnitude is predicted using the exact $k$-correction from GALFORM. The solid lines show the median difference between the true and predicted magnitudes. The dotted lines show the 16-84 th percentile interval, a centralised version of the $1$-$\sigma$ scatter which is not affected by outliers. The short vertical lines at the bottom of the panel show the absolute magnitude limits per redshift bin. Different colours indicate different redshift ranges, as shown by the legend.
• Figure 5: The impact of selection effects on the estimated luminosity function, shown at $z \sim 1$ for illustration. In the upper panel the blue curve shows the GALFORM LF without any selection effects. The red points show the LF after applying the observed $i$-band limit of $i_{\rm AB}=23$, using a weighted average of the LFs for the snapshots that fall within the redshift range. The weighting is given by ${\rm d}N/{\rm d}z$. The lower panel shows the observed and rest $i$-band absolute magnitudes of the model galaxies; points coloured red pass the sample selection. The green curve in the upper panel shows the LF estimated from the lightcone mock, assuming the exact $k$-correction predicted by GALFORM. The orange line shows the LF recovered using the $k$-correction obtained using the random forest and the $ugriz$ photometry.