CFHTLenS tomographic weak lensing: Quantifying accurate redshift distributions

TL;DR

CFHTLenS addresses the challenge of accurate source redshift distributions in weak lensing tomography by showing that summing the full photometric redshift PDFs provides an unbiased redshift distribution for $z_p<1.3$, validated against spectroscopic and COSMOS data. The study conducts a two-bin tomographic weak lensing analysis, measures $oldsymbol{\xi_\pm}$ over $1-40$ arcmin, and tests redshift-scaling against flat and curved ΛCDM models, using non-linear $P_\delta$ with halo-model corrections. When combined with external constraints from WMAP7, BOSS, and $H_0$ priors, CFHTLenS yields $oldsymbol{ ext{$ ext{Ω_m}$}}=0.2762±0.0074$ and $oldsymbol{ ext{$ ext{σ}_8$}}=0.802±0.013$, representing a significant precision improvement. The paper also demonstrates robustness to non-linear and baryonic uncertainties within the analyzed scales and discusses implications for applying PDF-based redshift distributions to future weak-lensing surveys.

Abstract

The Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) comprises deep multi-colour (u*g'r'i'z') photometry spanning 154 square degrees, with accurate photometric redshifts and shape measurements. We demonstrate that the redshift probability distribution function summed over galaxies provides an accurate representation of the galaxy redshift distribution accounting for random and catastrophic errors for galaxies with best fitting photometric redshifts z_p < 1.3. We present cosmological constraints using tomographic weak gravitational lensing by large-scale structure. We use two broad redshift bins 0.5 < z_p <= 0.85 and 0.85 < z_p <= 1.3 free of intrinsic alignment contamination, and measure the shear correlation function on angular scales in the range ~1-40 arcmin. We show that the problematic redshift scaling of the shear signal, found in previous CFHTLS data analyses, does not afflict the CFHTLenS data. For a flat Lambda-CDM model and a fixed matter density Omega_m=0.27, we find the normalisation of the matter power spectrum sigma_8=0.771 \pm 0.041. When combined with cosmic microwave background data (WMAP7), baryon acoustic oscillation data (BOSS), and a prior on the Hubble constant from the HST distance ladder, we find that CFHTLenS improves the precision of the fully marginalised parameter estimates by an average factor of 1.5-2. Combining our results with the above cosmological probes, we find Omega_m=0.2762 \pm 0.0074 and sigma_8=0.802 \pm 0.013.

Figures (12)

• Figure 1: Comparison of the predicted redshift distributions within each broad redshift bin, labelled $z_{\rm i}$. A magnitude cut of $i' < 23.0$ is used for comparison with spectroscopic redshifts. Solid lines (pink) show the summed PDFs for all galaxies within a given redshift bin. Dashed lines (green) show the spectroscopic redshift distribution. The listed P-values are the result of a two-sample Kolmogorov-Smirnov test of the distributions, we adopt a significance level of $\alpha=0.05$ rejecting the null hypothesis that the two distributions are drawn from the same population for the highest redshift bin.
• Figure 2: Comparison of the predicted redshift distributions with a magnitude cut of $i' < 24.7$. Solid lines (pink) show the summed PDFs for all galaxies within a given redshift bin. Dot-dashed histogram (cyan) shows the result of resampling the CFHTLenS redshifts using the constructed conditional probability $P(z_{\rm\mathsmaller{30}}|z_{\rm p})$. The P-values are the result of a KS test, we reject the null hypothesis for the highest redshift bin at $\alpha=0.05$.
• Figure 3: Comparison of the predicted true redshift distribution within each broad redshift bin, labelled $z_{\rm i}$. A magnitude cut of $i' < 23.0$ is used for comparison with spectroscopic redshifts. All horizontal error bars denote the width of the redshift bin and points are offset horizontally for clarity. Crosses with solid lines (pink) denote the summed PDFs when integrated within a given broad redshift bin, the error is calculated as the standard deviation from 1000 bootstrap samples. Filled circles with dotted lines (blue) show the result from our contamination analysis with 68 per cent confidence region. Filled squares with dashed lines (green) show the spectroscopic redshift data integrated within each broad redshift bin. The error is the standard deviation of 1000 bootstrap samples.
• Figure 4: Same as Figure \ref{['fig:contam-23.0']} except for the following differences. A magnitude cut of $i' < 24.7$ is used. Filled squares with dot-dashed line (cyan) show the resampled COSMOS-30 data integrated within each broad redshift bin. The error is given as the standard deviation of the 100 low-resolution reconstructions (see Section \ref{['sec:COSMOS']}).
• Figure 5: Redshift distributions used in the weak lensing analysis. Low and high redshift bins correspond to $z_{\rm p}=(0.5,0.85]$ and $z_{\rm p}=(0.85,1.3]$ respectively. Smooth curves show the result of summing the photometric redshift probability distribution functions (PDFs) of all galaxies within the respective redshift bin, and the solid and dashed curves are used in the tomographic analysis. The sum of the PDFs over the entire redshift range is given by the dot-dashed line which is used in the 2D lensing analysis. For comparison, the histograms show the redshift distribution obtained from the photometric redshifts.