Angular clustering and bias of photometric quasars in the Kilo-Degree Survey Data Release 4

TL;DR

This paper measures the angular clustering and scale-independent bias of photometrically selected quasars in KiDS DR4 by updating their photo-$z$s with Hybrid-$z$, a deep learning framework trained on DESI DR1 and SDSS DR17. Four tomographic redshift bins are used to infer $b(z)$ from the observed $\omega(\theta)$ via Limber projection and a dark-matter reference, revealing a quadratic growth $b(z) \approx b_0 + b_1 z + b_2 z^2$ with $b(z)$ rising from $\sim1.6$ at $z_{\rm eff}\approx0.6$ to $\sim4.0$ at $z_{\rm eff}\approx2.2$, and corresponding host halos of $\log_{10}(M_{\rm eff}/h^{-1}M_\odot) \sim 12.7$--$12.9$. The analysis demonstrates that the assumed redshift distribution $dN/dz$ strongly affects $b(z)$, while stellar contamination has little impact on the clustering results. The work provides a first cosmological application of KiDS-selected quasars and establishes a framework for future KiDS DR5, LSST, and DESI-era clustering studies, highlighting the need for robust $dN/dz$ calibration for photometric-quasar cosmology. The derived bias evolution is broadly consistent with previous quasar-bias measurements, supporting a picture in which higher-redshift quasars occupy more massive halos.

Abstract

We investigate the angular clustering and effective bias of photometrically selected quasars in the Kilo-Degree Survey Data Release 4 (KiDS DR4). We update the previous photometric redshifts (photo-$z$s) of the KiDS quasars using Hybrid-z, a deep learning framework combining four-band KiDS images and nine-band KiDS+VIKING magnitudes. Hybrid-z is trained on the latest Dark Energy Spectroscopic Instrument (DESI) DR1 and Sloan Digital Sky Survey (SDSS) DR17 quasars matching with KiDS, and achieves average bias $\langle δz \rangle < 0.01$ and scatter $\sim 0.04(1 + z)$ on a test sample. The updated catalog of $\sim 157k$ quasars over $777~\mathrm{deg}^2$ is divided into four tomographic bins spanning $0.1 \leq z_{\mathrm{phot}} \leq 2.7$. In each bin, we measure the angular two-point correlation function and compare it with theoretical predictions for dark matter clustering. We estimate the best-fit scale-independent quasar bias, which increases from $b \approx 1.6$ at $z \approx 0.6$ to $b \approx 4.0$ at $z \approx 2.2$, and is well matched by a quadratic relation in redshift. Our clustering analysis indicates that KiDS quasars reside in dark matter halos of mass $\log_{10}(M_{\mathrm{eff}}/h^{-1}M_\odot)$ in the range $\sim 12.7$--$12.9$ and effective peak heights $ν_{\mathrm{eff}}$ rising from $\sim 1.5$ to $2.9$ over our redshift span. We study two systematics that could affect the bias derivation: stellar contamination and the redshift distribution assumed in the theoretical modeling. The former has a negligible effect, whereas the latter significantly impacts the derived $b(z)$, emphasizing the importance of redshift calibration. Our work is the first cosmological application of quasars selected from KiDS and paves the way for future extensions in the final KiDS DR5, the Legacy Survey of Space and Time, or the 4-metre Multi-Object Spectroscopic Telescope.

Figures (7)

• Figure 1: Density plot comparing spectroscopic and photometric redshifts of KiDS DR4 quasars derived with our Hybrid-z method. The color bar indicates counts per hex-bin for a test sample not seen in the training. The red dashed line corresponds to the identity relation $z_{\mathrm{spec}} = z_{\mathrm{phot}}$.
• Figure 2: Comparison of redshift distributions for the KiDS DR4 safe quasar sample. The orange-filled histogram shows the spectroscopic training sample used in our Hybrid-z model. The green dotted line represents the photo-$z$s of KiDS DR4 safe quasars from Nakoneczny2021, while the blue line corresponds to the photo-$z$s derived in this work using the Hybrid-z framework.
• Figure 3: Angular auto-correlation function of KiDS DR4 quasars measured in four photometric redshift bins. Red data points with error bars are the quasar 2PCF measurements, while the dashed lines show the predicted matter 2PCF $\omega_m(\theta)$, obtained for Planck $\Lambda$CDM cosmology and quasar redshift distribution taken as $dN/dz_\mathrm{phot}$. The solid black line shows $b^2 \omega_m$, with the best-fit bias value indicated in the legend. $N$ and $z_{\rm eff}$ are the number of quasar objects and the effective redshift, respectively. The measurements shown here were corrected for stellar contamination using $p_\mathrm{star}=0.99$ as discussed in Sec. \ref{['sec:stellar_correction']}.
• Figure 4: Quasar redshift distributions for the four tomographic bins used in this study. Shaded histograms are the photo-$z$ distributions of the KiDS DR4 QSO sample, while the thick lines correspond to the spec-$z$ distributions obtained by direct cross-match of the quasars from a given photo-$z$ bin with overlapping DESI and SDSS spectroscopy. Each distribution is normalized to unity under the curve.
• Figure 5: Effective bias of KiDS DR4 quasars a function of redshift, estimated from angular clustering analysis. We present results from two modeling choices: blue circles were obtained by assuming $dN/dz_{\mathrm{phot}}$ directly as the underlying redshift distribution in the photo-$z$ bins, while and for green triangles we used per-bin cross-matches of KiDS photometric quasars with DESI and SDSS spectroscopy. Other model assumptions are the same in both cases. The dashed lines indicate the best-fit models of the form $b(z) = b_0 + b_1 z + b_2 z^2$. The corresponding best-fit parameters are shown in the legend, while their uncertainties are reported in Table \ref{['tab:quad_fit']}. The shaded regions represent the $\pm1\sigma$ uncertainty bands of the fitted models.