Calibrating Redshift Distributions Beyond Spectroscopic Limits with Cross-Correlations

TL;DR

This work introduces a cross-correlation-based method to recover the true redshift distribution phi_p(z) of photometric samples by leveraging angular cross-correlations with overlapping spectroscopic surveys. The approach uses the cross-correlation w_sp and the auto-correlations of each sample under a simple bias model to reconstruct phi_p(z), with error budgets dominated by Poisson statistics and mitigated by survey design and sample variance corrections. Extensive Monte Carlo tests show that mean redshifts and dispersions can be recovered with precisions reaching the stringent requirements of upcoming dark energy experiments (order 10^{-3} level), even for faint photometric samples, provided sufficient spectroscopic coverage and sky overlap. The paper also analyzes potential systematics—bias evolution, autocorrelation errors, zero-point variations, and cosmology—and offers practical guidance for optimizing future surveys to maximize the efficacy of cross-correlation redshift calibration.

Abstract

We describe a new method for measuring the true redshift distribution of any set of objects studied only photometrically. The angular cross-correlation between objects in a photometric sample with objects in some spectroscopic sample as a function of the spectroscopic z, in combination with standard correlation measurements, provides sufficient information to reconstruct the true redshift distribution of the photometric sample. This technique enables the robust calibration of photometric redshifts even beyond spectroscopic limits. The spectroscopic sample need not resemble the photometric one in galaxy properties, but must overlap in sky coverage and redshift range. We test this new technique with Monte Carlo simulations using realistic error estimates. RMS errors in recovering both the mean and sigma of the true, Gaussian redshift distribution of a single photometric redshift bin are 1.4x10^(-3) (sigma_z/0.1) (Sigma_p/10)^(-0.3) (dN_s/dz / 25,000)^(-0.5), where sigma_z is the true sigma of the redshift distribution, Sigma_p is the surface density of the photometric sample in galaxies/arcmin^2, and dN_s/dz is the number of galaxies with a spectroscopic redshift per unit z. We test the impact of redshift outliers and of a variety of sources of systematic error; none dominate measurement uncertainties in reasonable scenarios. With this method, the true redshift distributions of even arbitrarily faint photometric redshift samples may be determined to the precision required by proposed dark energy experiments (errors in mean and sigma below 3x10^(-3) at z~1) using expected extensions of current spectroscopic samples.

Figures (8)

• Figure 1: Redshift distributions assumed for current spectroscopic samples (blue dashed line) and future samples (red solid line). The assumed characteristics of each sample are given in Table \ref{['tab:dndz']}. The differences are the addition of an intermediate-redshift survey, PRIMUS (Eisenstein et al. 2007, in prep.); a baryonic oscillation survey, WiggleZ (Glazebrook et al. 2007, in prep.); zCOSMOS 2005Msngr.121...42L; and larger samples at $z>2$ in the near-future scenario. These samples were used to produce the Monte Carlo realizations shown in Figure \ref{['fig:montecarlo']}. The black, dot-dashed line indicates the assumption used for our standard scaling scenario, which approximates current redshift samples at $z\sim 1$.
• Figure 2: Examples of individual Monte Carlo realizations for the recovery of $\phi_{p}(z)$ using the combinations of current spectroscopic datasets (left) or of current and future datasets (right) shown in Fig. \ref{['fig:dndz']}. Each realization was generated by randomly drawing from realistic error distributions for the recovery of $\phi_p(z)$ in bins of width $\Delta z=0.01$. Plotted in blue is the true, input redshift distribution, given by Equation \ref{['eq:gz']} with $\sigma_z=0.1$. The black histogram shows one realization for the distribution measured using cross-correlation techniques, with realistic errors determined as described in § \ref{['sec:errors']}. Shown in red is the distribution determined from a least-squares fit to the simulated data shown by the black histogram. The recovery is good enough in each case that the blue curve is essentially invisible.
• Figure 3: Errors in the recovery of the mean redshift, $\langle z \rangle$ (red) or the RMS dispersion of redshifts, $\sigma_z$ (blue) for objects in a photometric sample versus their surface density in galaxies per square arcminute, $\Sigma_p$, as measured in our Monte Carlo simulations. Note that for a single photometric redshift bin drawn from a larger sample, $\Sigma_p$ is the surface density only for objects in that bin, not for the overall sample. The black, dashed line indicates the estimated maximum error in $\langle z \rangle$ allowable for proposed dark energy surveys using the SNAP satellite or LSST. We assume a spectroscopic sample with $dN_s/dz = 25,000$ (roughly corresponding to current samples at $z \sim 1$) and a true $\phi_{p}(z)$ having $\sigma_z=0.1$. If sample variance is negligible, both errors scale as $\sigma \propto \Sigma_p^{-1/2}$; if it has maximal impact, their scaling is weaker, $\sigma \propto \Sigma_p^{-0.3}$. If the photometric sample has very low surface density, larger numbers of redshifts or a narrower redshift distribution than assumed in our standard scenario may be required to meet the requirements of future dark energy surveys.
• Figure 4: Errors in the recovery of $\langle z \rangle$ (red) or $\sigma_z$ (blue) versus the true value of $\sigma_z$, from our Monte Carlo tests. The black, dashed line indicates the estimated maximum error in $\langle z \rangle$ allowable for proposed dark energy surveys using LSST or the SNAP satellite. We assume here a spectroscopic sample with $dN_s/dz = 25,000$ (roughly corresponding to current samples at $z \sim 1$) and a photometric sample with a surface density of 10 galaxies per square arcminute. If sample variance is negligible, both errors scale as $\sigma_z^{3/2}$; if it has maximal impact, their $\sigma_z$--dependence is weaker, $\sigma \propto \sigma_z$. In all plotted cases, the errors in measuring the parameters of the redshift distribution are much smaller than required for future dark energy surveys.
• Figure 5: Errors in the recovery of $\langle z \rangle$ (red) and $\sigma_z$ (blue) versus the number of spectroscopic galaxies per unit redshift, $dN_s/dz$, from our Monte Carlo tests. The black, dashed line indicates the estimated maximum error allowable for proposed dark energy surveys using the SNAP satellite or LSST. We assume here that the photometric sample has a surface density of 10 galaxies per square arcminute and a true $\phi_{p}(z)$ having $\sigma_z=0.1$. Regardless of assumptions about sample variance, all uncertainties scale as $(dN_s/dz)^{-1/2}$. If $dN_s/dz$ is small, as at $z>2$ currently, meeting the tolerances of future dark energy surveys may be problematic. However, accuracy requirements at $z\sim 2$ are in general less restrictive than the $z \sim 1$ tolerance plotted here, as angular diameter distance and lookback time evolve more slowly with redshift at higher $z$.