On using angular cross-correlations to determine source redshift distributions

TL;DR

This paper presents an improved quadratic estimator for reconstructing the redshift distribution of extragalactic objects using angular cross-correlations, demonstrating high sensitivity and robustness for cosmological surveys.

Contribution

It introduces a minimum variance quadratic estimator that simplifies application and enhances sensitivity over previous methods for determining redshift distributions.

Findings

The estimator constrains the product of linear bias and number of objects with fractional errors of (10^2 n/N)^1/2.

Sub-percent accuracy in measuring dN/dz is theoretically achievable, accounting for cosmic magnification.

Cross-correlations can effectively self-calibrate photometric redshift samples, improving survey strategies.

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.