Calibrating Photometric Redshifts of Luminous Red Galaxies

TL;DR

This work shows how to build a robust photometric redshift catalogue for LRGs by combining careful photometric selection, empirical calibration of redshift errors, and a regularized deconvolution to recover the true redshift distribution $dN/dz$ from the observed photometric redshift distribution. It compares a simple template-fitting approach with a hybrid method that refines templates using calibration data, finding similar overall accuracy with $\sigma$ around $0.03$ for $z \lesssim 0.55$ and larger errors at higher redshift due to photometric scatter and template/zeropoint systematics. The core contribution is a discretized Fredholm-inversion framework that models the convolution relationship between true and photometric redshifts and stabilizes the reconstruction via a smoothness prior, with a data-driven merit function to set the regularization strength. The approach is demonstrated on SDSS and SDSS-2dF data and is positioned as broadly applicable to any multi-band photometric survey, enabling reliable three-dimensional mapping for studies of large-scale structure and weak gravitational lensing.

Abstract

We discuss the construction of a photometric redshift catalogue of Luminous Red Galaxies (LRGs) from the Sloan Digital Sky Survey (SDSS), emphasizing the principal steps necessary for constructing such a catalogue -- (i) photometrically selecting the sample, (ii) measuring photometric redshifts and their error distributions, (iii) and estimating the true redshift distribution. We compare two photometric redshift algorithms for these data and find that they give comparable results. Calibrating against the SDSS and SDSS-2dF spectroscopic surveys, we find that the photometric redshift accuracy is $σ\sim 0.03$ for redshifts less than 0.55 and worsens at higher redshift ($\sim 0.06$). These errors are caused by photometric scatter, as well as systematic errors in the templates, filter curves, and photometric zeropoints. We also parametrize the photometric redshift error distribution with a sum of Gaussians, and use this model to deconvolve the errors from the measured photometric redshift distribution to estimate the true redshift distribution. We pay special attention to the stability of this deconvolution, regularizing the method with a prior on the smoothness of the true redshift distribution. The methods we develop are applicable to general photometric redshift surveys.

Figures (12)

• Figure 1: A model spectrum of an early type galaxy from Bruzual & Charlot (2003MNRAS.344.1000B). The model was formed from a single burst of star formation 11 Gyr ago, and assumes a solar metallicity. Note the prominent break in the spectrum at 4000 Å. Also overplotted are the response functions (including atmospheric absorption) for the SDSS filters.
• Figure 2: The top panel shows simulated $g-r$ and $r-i$ colours of an early-type galaxy as a function of redshift. The spectrum used to generate the track is the same as in Fig. \ref{['fig:modelspec']}, but evolved in redshift. Also shown are the colour cuts for Cut I (dashed, black) and Cut II galaxies (solid, blue). The points show the stellar locus as determined by a sample of stars with $r$-band magnitudes less than 19.5. The lower panel shows the colours $c_{||}$ (diamonds, black) and $d_{\perp}$ (triangles, red), as a function of redshift. Also shown are fiducial redshift boundaries for Cut I (0.2 -- 0.4) and Cut II (0.4 -- 0.6). Note that the range in $g-r$ is identical to the range in $1+z$.
• Figure 3: The top panel shows the spectroscopic redshift distribution, $dN/dz$ of the SDSS (solid, black) and the SDSS-2dF (dashed, red) samples trimmed using the selection criteria of Sec.\ref{['sec:select']}. Note that the SDSS sample is dominated by the low redshift MAIN sample, accounting for the low normalization at high redshift. The lower panel shows the $i$ band absolute magnitude distribution for the two samples, demonstrating that our absolute magnitude cuts are selecting a sample with $M_{i} \sim -22$ as desired. Both $dN/dz$ and $dN/dM$ are normalized so that they integrate to unity.
• Figure 4: Scatter plot showing the photometric redshift versus the spectroscopic redshift for a random 10000 galaxies from our calibration sample. The upper panel shows the results for the simple template fitting code of Sec.\ref{['sec:single']}, and the lower panel are the results for the hybrid code of Sec.\ref{['sec:hybrid']}. The solid (red) line has slope 1, while the dashed line marks the fiducial lower redshift limit of any photometric LRG sample. The difficulty of estimating redshifts at $z\sim 0.4$ is evident from the increased scatter.
• Figure 5: Simulations showing the effect of magnitude errors on the accuracy of the photometric redshifts. The upper left plot shows the reconstructed photometric redshifts for a magnitude error, $\sigma_{m} = 0.03$ in all 5 bands, while the upper right panel has no S/N in the $u$ and $z$ bands and $\sigma_{m} = 0.03$ in the remaining bands. The lower panel shows the redshift error induced by magnitude errors; the solid line has constant error across the bands, while the dashed line has constant error in $g,r,i$ and zero S/N in $u$ and $z$. Since the magnitude errors are independant of redshift, the redshift errors are simply computed over the entire redshift range.