On using angular cross-correlations to determine source redshift distributions

TL;DR

The paper tackles reconstructing a population's redshift distribution dN/dz from angular cross-correlations with a well-characterized spectroscopic sample. It develops a minimum-variance quadratic estimator and analyzes Limber, Schur-Limber, abundant- and rare-sample limits to yield simple scaling relations for the precision on N_i^(p) and b_i^(p)N_i^(p). The approach is demonstrated to yield percent-level constraints for realistic survey combinations (e.g., LSST-like photometry with BigBOSS-like spectroscopy) while accounting for biases such as magnification and RSDs, and it provides practical guidance for photometric redshift calibration and diffuse-background mapping. The results offer a versatile, implementable framework for planning surveys and for improving redshift calibration in large-scale structure and weak-lensing analyses.

Abstract

We investigate how well the redshift distribution of a population of extragalactic objects can be reconstructed using angular cross-correlations with a sample whose redshifts are known. We derive the minimum variance quadratic estimator, which has simple analytic representations in very applicable limits and is significantly more sensitive than earlier proposed estimation procedures. This estimator is straightforward to apply to observations, it robustly finds the likelihood maximum, and it conveniently selects angular scales at which fluctuations are well approximated as independent between redshift bins and at which linear theory applies. We find that the linear bias times number of objects in a redshift bin generally can be constrained with cross-correlations to fractional error (10^2 n/N)^1/2, where N is the total number of spectra per dz and n is the number of redshift bins spanned by the bulk of the unknown population. The error is often independent of the sky area and sampling fraction. Furthermore, we find that sub-percent measurements of the angular source density per unit redshift, dN/dz, are in principle possible, although cosmic magnification needs to be accounted for at fractional errors of <~ 10 per cent. We discuss how the sensitivity to dN/dz changes as a function of photometric and spectroscopic depth and how to optimize the survey strategy to constrain dN/dz. We also quantify how well cross-correlations of photometric redshift bins can be used to self-calibrate a photometric redshift sample. Simple formulae that can be quickly applied to gauge the utility of cross correlating different samples are given.

Figures (15)

• Figure 1: Shown are the $dN/dz$ of different galaxy populations. The dashed curves are for surveys complete to i-band magnitude limits of $21,\; 23,$ and $25.3$, calculated via Eq. (\ref{['eqn:pziband']}). Also shown are the density of SDSS+BOSS spectroscopic quasars and estimates for the future combined density of luminous red galaxies, emission line galaxies, and quasars with BigBOSS. The solid curves represent the critical densities for whether a sample is in the rare galaxy limit (Section \ref{['ss:rare']}).
• Figure 2: The fractional error on the photometric number density for different spectroscopic and photometric samples. The contours represent $\log_{10}$ of the fractional error on $N_{i}^{(p)}$ with $i = N_{\rm bin}/2$. They consider an idealized survey in which the $N_i^{(x)}$ are equal and span $z= 0-1$ with $10$ redshift bins of the same width, covering 1 per cent of the sky ($400\,$deg$^2$). Contours are labelled for the solid curves, and the corresponding contour for the other curves is the adjacent curve at higher number densities. The calculations assume our fiducial parameters except $f_{\rm over} = 0$. (For $f_{\rm over} = 1$, the curves buckle outwards when the number densities become equal.) The black solid curves are the sensitivity of the optimal estimator. The purple dotted curves show the approximation that sets to zero terms in ${\mathbfss{F}}$ in which the derivatives hit $A_{00}$. The short dashed green is the diagonal approximation to the remaining Fisher matrix, a limit that also works excellently. The long dashed blue is the error on the estimator in the Schur-Limber limit (Section \ref{['sec:schurlimber']} and Eq. \ref{['eqn:fishC00large']}).
• Figure 3: Contours showing $\log_{10}$ of the fractional error in $b_{i}^{(p)}N_{i}^{(p)}$, where $i = N_{\rm bin}/2$ in the Limber approximation (black solid curves) and the full calculation without this approximation (blue dashed curves; which for the same fractional error fall immediately upwards of the solid curves). The contours are calculated for a survey that spans $z= 0-1$ with $10$ (left panel) and $100$ (right panel) redshift bins of equal width over 1 per cent of the sky. Roughly, the errors are $\sqrt{10}$ larger in the right panel than in the left panel. This figure illustrates that the Limber approximation works well for the $\Delta z = 0.1$ case, but is starting to break down at $\Delta z = 0.01$. While making the Limber approximation leads to errors in the uncertainty estimate, we find in Section \ref{['sec:bias']} that the bias on $N^{(p)}_i$ is always quite small.
• Figure 4: An illustration of the scales that contribute to the constraint on $N^{(p)}_i$ in different limits. The areas under these curves, which are of $d[1/{\mathbfss{F}}^{-1}_{ii}]/d\log \ell$, are proportional to the information that contributes to the estimate in the $i=6$ bin for a measurement in $10$ redshift bins with $\Delta z = 0.1$ and spanning $0 <z<1$. For illustrative purposes, we have assumed constant $dN^{(p)}/dz$ and $dN^{(s)}/dz$. The first adjective for each curve's label in the key describes the spectroscopic sample (rare=$10\,$deg$^{-2}$ and many=$10^{5}\,$deg$^{-2}$), and the second describes the photometric sample (rare=$100\,$deg$^{-2}$ and many=$10^{6}\,$deg$^{-2}$). However, the curves are not significantly impacted at linear scales by the assumed densities as long as 'many' equates to $\gtrsim 10^4\,$deg$^{-2}$ and 'rare' to $\lesssim 10^3\,$deg$^{-2}$, with the exception being the many-many case. In the text we describe why these limits select the scales that they do. The vertical lines denote significant scales discussed in the text. The thin red dot-dashed curve does not assume the Limber approximation whereas the corresponding thick curve assumes it.
• Figure 5: The source clustering angular power spectrum under different approximations and for different source number densities. Shown is the clustered component of the power, $C_{ii}$, for $z_i=1$, $\Delta z_i = 0.1$, and our fiducial bias model. The $C_{ii}$ are calculated under various approximations -- linear theory (dashed curve) and the Limber approximation (solid curves) -- and for the full peacockdodds nonlinear power spectrum (dotted curve). Also depicted are the stochastic component of the power for two characteristic number densities and $f_{\rm sat} = 0$ (horizontal dashed lines). The auto-power of spectroscopic bin $i$, $\langle s_i^2\rangle$, equals $C_{ii}$ plus the stochastic component. The optimal quadratic estimator selects information that roughly falls in the range of the two vertical dotted lines (Section \ref{['sec:approx']}), between where $P(k)$ roughly scales as $k^{-1}$ and $k^{-2}$. Conveniently, both linear theory and the Limber approximation apply around these scales.