Analysis of two-point statistics of cosmic shear: III. Covariances of shear measures made easy

In recent years cosmic shear, the weak gravitational lensing effect by the large-scale structure of the Universe, has proven to be one of the observational pillars on which the cosmological concordance model is founded. Several cosmic shear statistics have been developed in order to analyze data from surveys. For the covariances of the prevalent second-order measures we present simple and handy formulae, valid under the assumptions of Gaussian density fluctuations and a simple survey geometry. We also formulate these results in the context of shear tomography, i.e. the inclusion of redshift information, and generalize them to arbitrary data field geometries. We define estimators for the E- and B-mode projected power spectra and show them to be unbiased in the case of Gaussianity and a simple survey geometry. From the covariance of these estimators we demonstrate how to derive covariances of arbitrary combinations of second-order cosmic shear measures. We then recalculate the power spectrum covariance for general survey geometries and examine the bias thereby introduced on the estimators for exemplary configurations. Our results for the covariances are considerably simpler than and analytically shown to be equivalent to the real-space approach presented in the first paper of this series. We find good agreement with other numerical evaluations and confirm the general properties of the covariance matrices. The studies of the specific survey configurations suggest that our simplified covariances may be employed for realistic survey geometries to good approximation.Comment: 15 pages, including 4 figures (Fig. 3 reduced in quality); minor changes, Fig. 4 extended; published in A&

Simulations of Wide-Field Weak Lensing Surveys I: Basic Statistics and Non-Gaussian Effects

We study the lensing convergence power spectrum and its covariance for a standard LCDM cosmology. We run 400 cosmological N-body simulations and use the outputs to perform a total of 1000 independent ray-tracing simulations. We compare the simulation results with analytic model predictions. The semi-analytic model based on Smith et al.(2003) fitting formula underestimates the convergence power by ~30% at arc-minute angular scales. For the convergence power spectrum covariance, the halo model reproduces the simulation results remarkably well over a wide range of angular scales and source redshifts. The dominant contribution at small angular scales comes from the sample variance due to the number fluctuations of halos in a finite survey volume. The signal-to-noise ratio for the convergence power spectrum is degraded by the non-Gaussian covariances by up to a factor 5 for a weak lensing survey to z_s ~1. The probability distribution of the convergence power spectrum estimators, among the realizations, is well approximated by a chi-square distribution with broadened variance given by the non-Gaussian covariance, but has a larger positive tail. The skewness and kurtosis have non-negligible values especially for a shallow survey. We argue that a prior knowledge on the full distribution may be needed to obtain an unbiased estimate on the ensemble averaged band power at each angular scale from a finite volume survey.Comment: 11 pages, 11 figures. Accepted for publication in the Astrophysical Journal. Corrected typo in the equation of survey window function below Equation (18). The results unchange

A bias in cosmic shear from galaxy selection: results from ray-tracing simulations

We identify and study a previously unknown systematic effect on cosmic shear measurements, caused by the selection of galaxies used for shape measurement, in particular the rejection of close (blended) galaxy pairs. We use ray-tracing simulations based on the Millennium Simulation and a semi-analytical model of galaxy formation to create realistic galaxy catalogues. From these, we quantify the bias in the shear correlation functions by comparing measurements made from galaxy catalogues with and without removal of close pairs. A likelihood analysis is used to quantify the resulting shift in estimates of cosmological parameters. The filtering of objects with close neighbours (a) changes the redshift distribution of the galaxies used for correlation function measurements, and (b) correlates the number density of sources in the background with the density field in the foreground. This leads to a scale-dependent bias of the correlation function of several percent, translating into biases of cosmological parameters of similar amplitude. This makes this new systematic effect potentially harmful for upcoming and planned cosmic shear surveys. As a remedy, we propose and test a weighting scheme that can significantly reduce the bias.Comment: 9 pages, 9 figures, version accepted for publication in Astronomy & Astrophysic

Sources of contamination to weak lensing tomography: redshift-dependent shear measurement bias

The current methods available to estimate gravitational shear from astronomical images of galaxies introduce systematic errors which can affect the accuracy of weak lensing cosmological constraints. We study the impact of KSB shape measurement bias on the cosmological interpretation of tomographic two-point weak lensing shear statistics. We use a set of realistic image simulations produced by the STEP collaboration to derive shape measurement bias as a function of redshift. We define biased two-point weak lensing statistics and perform a likelihood analysis for two fiducial surveys. We present a derivation of the covariance matrix for tomography in real space and a fitting formula to calibrate it for non-Gaussianity. We find the biased aperture mass dispersion is reduced by ~20% at redshift ~1, and has a shallower scaling with redshift. This effect, if ignored in data analyses, biases sigma_8 and w_0 estimates by a few percent. The power of tomography is significantly reduced when marginalising over a range of realistic shape measurement biases. For a CFHTLS-Wide-like survey, [Omega_m, sigma_8] confidence regions are degraded by a factor of 2, whereas for a KIDS-like survey the factor is 3.5. Our results are strictly valid only for KSB methods but they demonstrate the need to marginalise over a redshift-dependent shape measurement bias in all future cosmological analyses.Comment: 13 pages, 8 figures. Submitted MNRA

The removal of shear-ellipticity correlations from the cosmic shear signal: Influence of photometric redshift errors on the nulling technique

Cosmic shear is regarded one of the most powerful probes to reveal the properties of dark matter and dark energy. To fully utilize its potential, one has to be able to control systematic effects down to below the level of the statistical parameter errors. Particularly worrisome in this respect is intrinsic alignment, causing considerable parameter biases via correlations between the intrinsic ellipticities of galaxies and the gravitational shear, which mimic lensing. In an earlier work we have proposed a nulling technique that downweights this systematic, only making use of its well-known redshift dependence. We assess the practicability of nulling, given realistic conditions on photometric redshift information. For several simplified intrinsic alignment models and a wide range of photometric redshift characteristics we calculate an average bias before and after nulling. Modifications of the technique are introduced to optimize the bias removal and minimize the information loss by nulling. We demonstrate that one of the presented versions is close to optimal in terms of bias removal, given high quality of photometric redshifts. For excellent photometric redshift information, i.e. at least 10 bins with a small dispersion, a negligible fraction of catastrophic outliers, and precise knowledge about the redshift distributions, one version of nulling is capable of reducing the shear-intrinsic ellipticity contamination by at least a factor of 100. Alternatively, we describe a robust nulling variant which suppresses the systematic signal by about 10 for a very broad range of photometric redshift configurations. Irrespective of the photometric redshift quality, a loss of statistical power is inherent to nulling, which amounts to a decrease of the order 50% in terms of our figure of merit.Comment: 26 pages, including 16 figures; minor changes to match accepted version; published in Astronomy and Astrophysic

Baryons, neutrinos, feedback and weak gravitational lensing

The effect of baryonic feedback on the dark matter mass distribution is generally considered to be a nuisance to weak gravitational lensing. Measurements of cosmological parameters are affected as feedback alters the cosmic shear signal on angular scales smaller than a few arcminutes. Recent progress on the numerical modelling of baryon physics has shown that this effect could be so large that, rather than being a nuisance, the effect can be constrained with current weak lensing surveys, hence providing an alternative astrophysical insight on one of the most challenging questions of galaxy formation. In order to perform our analysis, we construct an analytic fitting formula that describes the effect of the baryons on the mass power spectrum. This fitting formula is based on three scenarios of the OverWhelmingly Large hydrodynamical simulations. It is specifically calibrated for z < 1.5, where it models the simulations to an accuracy that is better than 2 per cent for scales k < 10 h Mpc−1 and better than 5 per cent for 10 < k < 100 h Mpc−1. Equipped with this precise tool, this paper presents the first constraint on baryonic feedback models using gravitational lensing data, from the Canada France Hawaii Telescope Lensing Survey (CFHTLenS). In this analysis, we show that the effect of neutrino mass on the mass power spectrum is degenerate with the baryonic feedback at small angular scales and cannot be ignored. Assuming a cosmology precision fixed by WMAP9, we find that a universe with massless neutrinos is rejected by the CFHTLenS lensing data with 85–98 per cent confidence, depending on the baryon feedback model. Some combinations of feedback and non-zero neutrino masses are also disfavoured by the data, although it is not yet possible to isolate a unique neutrino mass and feedback model. Our study shows that ongoing weak gravitational lensing surveys (KiDS, HSC and DES) will offer a unique opportunity to probe the physics of baryons at galactic scales, in addition to the expected constraints on the total neutrino mass

Cosmic Shear E/B-mode Estimation with Binned Correlation Function Data

In this work I study the problem of E/B-mode separation with binned cosmic shear two-point correlation function data. Motivated by previous work on E/B-mode separation with shear two-point correlation functions and the practical considerations of data analysis, I consider E/B-mode estimators which are linear combinations of the binned shear correlation function data points. I demonstrate that these estimators mix E- and B-modes generally. I then show how to define estimators which minimize this E/B-mode mixing and give practical recipes for their construction and use. Using these optimal estimators, I demonstrate that the vector space composed of the binned shear correlation function data points can be decomposed into approximately ambiguous, E- and B-mode subspaces. With simple Fisher information estimates, I show that a non-trivial amount of information on typical cosmological parameters is contained in the ambiguous mode subspace computed in this formalism. Next, I give two examples which apply these practical estimators and recipes to generic problems in cosmic shear data analysis: data compression and spatially locating B-mode contamination. In particular, by using wavelet-like estimators with the shear correlation functions directly, one can pinpoint B-mode contamination to specific angular scales and extract information on its shape. Finally, I discuss how these estimators can be used as part of blinded or closed-box cosmic shear data analyses in order to assess and find B-mode contamination at high-precision while avoiding observer biases.Comment: 15 pages, 5 figures, 3 appendices, MNRAS submitted, comments welcome

Phenotypic redshifts with self-organizing maps: A novel method to characterize redshift distributions of source galaxies for weak lensing

Wide-field imaging surveys such as the Dark Energy Survey (DES) rely on coarse measurements of spectral energy distributions in a few filters to estimate the redshift distribution of source galaxies. In this regime, sample variance, shot noise, and selection effects limit the attainable accuracy of redshift calibration and thus of cosmological constraints. We present a new method to combine wide-field, few-filter measurements with catalogs from deep fields with additional filters and sufficiently low photometric noise to break degeneracies in photometric redshifts. The multi-band deep field is used as an intermediary between wide-field observations and accurate redshifts, greatly reducing sample variance, shot noise, and selection effects. Our implementation of the method uses self-organizing maps to group galaxies into phenotypes based on their observed fluxes, and is tested using a mock DES catalog created from N-body simulations. It yields a typical uncertainty on the mean redshift in each of five tomographic bins for an idealized simulation of the DES Year 3 weak-lensing tomographic analysis of $\sigma_{\Delta z} = 0.007$, which is a 60% improvement compared to the Year 1 analysis. Although the implementation of the method is tailored to DES, its formalism can be applied to other large photometric surveys with a similar observing strategy.Comment: 24 pages, 11 figures; matches version accepted to MNRA

Simulations of Baryon Acoustic Oscillations II: Covariance matrix of the matter power spectrum

We use 5000 cosmological N-body simulations of 1(Gpc/h)^3 box for the concordance LCDM model in order to study the sampling variances of nonlinear matter power spectrum. We show that the non-Gaussian errors can be important even on large length scales relevant for baryon acoustic oscillations (BAO). Our findings are (1) the non-Gaussian errors degrade the cumulative signal-to-noise ratios (S/N) for the power spectrum amplitude by up to a factor of 2 and 4 for redshifts z=1 and 0, respectively. (2) There is little information on the power spectrum amplitudes in the quasi-nonlinear regime, confirming the previous results. (3) The distribution of power spectrum estimators at BAO scales, among the realizations, is well approximated by a Gaussian distribution with variance that is given by the diagonal covariance component. (4) For the redshift-space power spectrum, the degradation in S/N by non-Gaussian errors is mitigated due to nonlinear redshift distortions. (5) For an actual galaxy survey, the additional shot noise contamination compromises the cosmological information inherent in the galaxy power spectrum, but also mitigates the impact of non-Gaussian errors. The S/N is degraded by up to 30% for a WFMOS-type survey. (6) The finite survey volume causes additional non-Gaussian errors via the correlations of long-wavelength fluctuations with the fluctuations we want to measure, further degrading the S/N values by about 30% even at high redshift z=3.Comment: submitted to ApJ, 14 pages, 12 figures. The full halo model is included. Minor changes also made, and references adde

A Unified Analysis of Four Cosmic Shear Surveys

In the past few years, several independent collaborations have presented cosmological constraints from tomographic cosmic shear analyses. These analyses differ in many aspects: the datasets, the shear and photometric redshift estimation algorithms, the theory model assumptions, and the inference pipelines. To assess the robustness of the existing cosmic shear results, we present in this paper a unified analysis of four of the recent cosmic shear surveys: the Deep Lens Survey (DLS), the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS), the Science Verification data from the Dark Energy Survey (DES-SV), and the 450 deg$^{2}$ release of the Kilo-Degree Survey (KiDS-450). By using a unified pipeline, we show how the cosmological constraints are sensitive to the various details of the pipeline. We identify several analysis choices that can shift the cosmological constraints by a significant fraction of the uncertainties. For our fiducial analysis choice, considering a Gaussian covariance, conservative scale cuts, assuming no baryonic feedback contamination, identical cosmological parameter priors and intrinsic alignment treatments, we find the constraints (mean, 16% and 84% confidence intervals) on the parameter $S_{8}\equiv \sigma_{8}(\Omega_{\rm m}/0.3)^{0.5}$ to be $S_{8}=0.94_{-0.045}^{+0.046}$ (DLS), $0.66_{-0.071}^{+0.070}$ (CFHTLenS), $0.84_{-0.061}^{+0.062}$ (DES-SV) and $0.76_{-0.049}^{+0.048}$ (KiDS-450). From the goodness-of-fit and the Bayesian evidence ratio, we determine that amongst the four surveys, the two more recent surveys, DES-SV and KiDS-450, have acceptable goodness-of-fit and are consistent with each other. The combined constraints are $S_{8}=0.79^{+0.042}_{-0.041}$, which is in good agreement with the first year of DES cosmic shear results and recent CMB constraints from the Planck satellite.Comment: 22 pages, 15 figures, 7 tables; submitted to MNRA