[Paper Review] Reconstructing the galaxy redshift distribution from angular cross power spectra

Lei Sun, H. Zhan+1 more|arXiv (Cornell University)|Dec 2, 2015

Galaxies: Formation, Evolution, Phenomena3 references3 citations

TL;DR

This paper proposes using angular cross power spectra between tomographic photometric redshift bins to self-calibrate the true galaxy redshift distribution $n_i(z)$, crucial for weak lensing cosmology. Using Fisher matrix and MCMC analyses in an LSST-like survey with 10 bins, it demonstrates that mean redshift and width of $n_i(z)$ can be reconstructed with 1% and 10% fractional precision, respectively, even with catastrophic photo-z errors.

ABSTRACT

The control of photometric redshift (photo-$z$) errors is a crucial and challenging task for precision weak lensing cosmology. The spacial cross-correlations (equivalently, the angular cross power spectra) of galaxies between tomographic photo-$z$ bins are sensitive to the true redshift distribution $n_i(z)$ of each bin and hence can help calibrate the photo-$z$ error distribution for weak lensing surveys. Using Fisher matrix analysis, we investigate the contributions of various components of the angular power spectra to the constraints of $n_i(z)$ parameters and demonstrate the importance of the cross power spectra therein, especially when catastrophic photo-$z$ errors are present. We further study the feasibility of reconstructing $n_i(z)$ from galaxy angular power spectra using Markov Chain Monte Carlo estimation. Considering an LSST-like survey with $10$ photo-$z$ bins, we find that the underlying redshift distribution can be determined with a fractional precision ($σ(θ)/θ$ for parameter $θ$) of roughly $1\%$ and $10\%$ for the mean redshift and width of $n_i(z)$, respectively.

Motivation & Objective

• To address the critical challenge of photometric redshift (photo-z) errors in future weak lensing surveys, particularly their bias and scatter.
• To investigate whether angular cross power spectra between tomographic photo-z bins can self-calibrate the true redshift distribution $n_i(z)$ without relying solely on spectroscopic redshifts.
• To quantify the constraining power of different components of the angular power spectra—especially cross spectra—on $n_i(z)$ parameters.
• To assess the feasibility of reconstructing $n_i(z)$, including catastrophic outlier sub-samples, using MCMC estimation in realistic survey conditions.
• To evaluate the potential of combining this method with spectroscopic calibration to meet the stringent photo-z error requirements of surveys like LSST.

Proposed method

• Employs Fisher matrix analysis to decompose the contributions of auto- and cross-power spectra to the constraints on $n_i(z)$ parameters such as mean redshift $z_{mi}$ and width $\sigma_i$.
• Models the redshift distribution using a double-Gaussian component: a main component and a catastrophic outlier component with parameters $z^c$, $\sigma^c$, and $f^c$.
• Uses angular cross power spectra as the primary probe, since they are sensitive to the overlap of true redshift distributions between bins.
• Applies MCMC sampling to simulate realistic data analysis, testing reconstruction performance under realistic survey conditions including BAO features and observational systematics.
• Fixes cosmological parameters at fiducial values to isolate the performance of redshift distribution reconstruction, focusing on the self-calibration potential of clustering and lensing cross-correlations.
• Compares constraints in scenarios with and without catastrophic photo-z errors to assess robustness and degradation in precision.

Experimental results

Research questions

• RQ1 Can angular cross power spectra between tomographic photo-z bins effectively constrain the true redshift distribution $n_i(z)$ of galaxies in weak lensing surveys?
• RQ2 How do the contributions of auto-power spectra, cross-power spectra, and BAO features differ in constraining $n_i(z)$ parameters?
• RQ3 What is the achievable precision in reconstructing the mean redshift and width of $n_i(z)$ using angular power spectra, especially when catastrophic photo-z errors are present?
• RQ4 How well can the parameters of the catastrophic outlier sub-sample ($z^c$, $\sigma^c$, $f^c$) be constrained using cross spectra?
• RQ5 To what extent can this method reduce the need for external spectroscopic calibration in LSST-like surveys?

Key findings

• The angular cross power spectra are the dominant source of information for constraining $n_i(z)$, especially when catastrophic photo-z errors are present.
• For an LSST-like survey with 10 photo-z bins, the mean redshift $z_{mi}$ and width $\sigma_i$ of $n_i(z)$ can be reconstructed with fractional precisions of approximately 1% and 10%, respectively.
• The catastrophic outlier sub-sample parameters ($z^c$, $\sigma^c$, $f^c$) are less well constrained, with fractional precisions of 5–20% for $z^c$, 40–50% for $\sigma^c$, and 40% for $f^c$, due to degeneracy with galaxy bias.
• The BAO feature contributes relatively little to constraining $n_i(z)$, and its exclusion (via high-$\ell$ cut) leads to a 30–40% loss in precision, indicating it is not essential for $n_i(z)$ calibration.
• The method remains feasible in practical MCMC analysis, demonstrating that $n_i(z)$ can be reconstructed with sufficient accuracy to meet the $\sim 0.003$ bias requirement for LSST.
• The reconstructed uncertainty in $z_{mi}$, when expressed as $\sigma(z_{mi})/(1+z_{mi})$, corresponds to a photo-z bias of $\sim 0.005-0.008$, falling short of the $<0.003$ target, indicating the need for complementary calibration methods.