Revisiting the 150 MHz Radio Luminosity Function of Star-Forming Galaxies with LOFAR Deep Fields through a Refined Statistical Framework

TL;DR

This work tackles the evolution of the 150 MHz luminosity function $\Phi(z,L)$ for star-forming galaxies by pairing non-parametric adaptive kernel density estimation with a global parametric maximum-likelihood approach. Guided by KDE, it tests three LF evolution models—PLE (Model A) and two LADE variants (Models B and C)—and constrains them using completeness corrections, the local LRLF, and Euclidean-normalized source counts via Markov Chain Monte Carlo. The results favor LADE over PLE, with Model C providing the best fit for the deep ELAIS-N1 field and Model B offering the best balance of fit quality and simplicity when all three LOFAR fields are combined. A mild bright-end excess, likely due to residual AGN contamination, persists, underscoring the need for improved AGN/SFG separation in future low-frequency surveys and highlighting the value of the KDE-plus-MLE framework for SKA-era data.

Abstract

We present a comprehensive analysis of the 150~MHz radio luminosity function (LF) of star-forming galaxies (SFGs) using deep observations from the LOFAR Two-metre Sky Survey in the ELAIS-N1, Boötes, and Lockman Hole fields. Our sample comprises $\sim$56,000 SFGs over $0 < z < 5.7$. We first analyze the deepest field (ELAIS-N1), then jointly model all three fields while accounting for their distinct flux limits and selection functions. Using adaptive kernel density estimation (KDE), we reconstruct the LF continuously across redshift and luminosity without binning or parametric assumptions. The KDE results reveal clear signatures of joint luminosity and density evolution (LADE). Motivated by this, we construct and fit three parametric models--pure luminosity evolution (PLE) and two LADE variants--using a full maximum-likelihood method that includes completeness corrections and constraints from the local radio LF and Euclidean-normalized source counts (SCs). Model selection using Akaike and Bayesian Information Criteria strongly favors LADE over PLE. For ELAIS-N1, the more flexible LADE model (Model C) provides the best fit, while for the combined fields, the simpler Model B balances fit quality and complexity more effectively. Both LADE models reproduce the observed LFs and SCs across luminosity and flux density ranges, whereas PLE underperforms. We also identify a mild excess at the bright end of the LF, likely due to residual AGN contamination. This study demonstrates that combining KDE with parametric modeling offers a robust framework for quantifying the evolving radio LF of SFGs, paving the way for future work with next-generation surveys like the SKA.

Figures (14)

• Figure 1: Redshift distribution ($top$) and scatter plot ($bottom$) of our SFG sample for three fields: ELAIS-N1 ($left$), Boötes ($middle$), and Lockman Hole ($right$). The red dashed lines indicate the flux limits of $F_{150\,\rm{MHz}}=100~\mu\rm{Jy}$ for ELAIS-N1, $160~\mu\rm{Jy}$ for Boötes, and $110~\mu\rm{Jy}$ for Lockman Hole. Note that a small number of sources fall below these flux limit lines and are excluded from our subsequent analysis.
• Figure 2: LFs estimated at a series of redshift grid points using the adaptive KDE method. The resulting curves are color-coded according to redshift. Solid circles indicate the flattest regions of the LF at each redshift.
• Figure 3: Redshift evolution of the luminosity (left) and comoving number density (right) of the reference points identified along the KDE-estimated LFs in the ELAIS-N1 field, shown as green hexagons. Assuming a redshift-invariant LF shape, the evolution of these reference points closely traces the underlying LADE trends. Colored curves represent the LE and DE functions derived from our three parametric LF models, with light shaded regions indicating their corresponding $3\sigma$ uncertainty intervals. Purple circles denote the estimates reported by 2023MNRAS.523.6082C.
• Figure 4: Radio LFs of SFGs in the ELAIS-N1 field at various redshifts. Blue, orange, and green solid lines show the best-fit LFs from Models A, B, and C, respectively. The light shaded area shows the 3$\sigma$ confidence interval. The vertical red dashed line in each panel indicates the luminosity threshold corresponding to the survey flux limit at the given redshift. The solid purple lines indicate adaptive KDE LFs and shaded areas indicate 3$\sigma$ confidence intervals. Circles with error bars denote the binned LF from 2023MNRAS.523.6082C.
• Figure 5: Similar to Figure \ref{['fig:en1plf']}, but for the combined analysis of all fields. The three dashed vertical lines in each panel indicate the luminosity thresholds corresponding to the survey flux limits of the ELAIS-N1, Boötes, and Lockman Hole fields at the given redshift.