Breaking the Degeneracy: Optimal Use of Three-point Weak Lensing Statistics

We study the optimal use of third order statistics in the analysis of weak lensing by large-scale structure. These higher order statistics have long been advocated as a powerful tool to break measured degeneracies between cosmological parameters. Using ray-tracing simulations, incorporating important survey features such as a realistic depth-dependent redshift distribution, we find that a joint two- and three-point correlation function analysis is a much stronger probe of cosmology than the skewness statistic. We compare different observing strategies, showing that for a limited survey time there is an optimal depth for the measurement of third-order statistics, which balances statistical noise and cosmic variance against signal amplitude. We find that the chosen CFHTLS observing strategy was optimal and forecast that a joint two- and three-point analysis of the completed CFHTLS-Wide will constrain the amplitude of the matter power spectrum $\sigma_8$ to 10% and the matter density parameter $\Omega_m$ to 17%, a factor of ~2.5 improvement on the two-point analysis alone. Our error analysis includes all non-Gaussian terms, finding that the coupling between cosmic variance and shot noise is a non-negligible contribution which should be included in any future analytical error calculations.Comment: 27 pages, 13 figures, 3 table

Cosmic variance of weak lensing surveys in the non-Gaussian regime

The results from weak gravitational lensing analyses are subject to a cosmic variance error term that has previously been estimated assuming Gaussian statistics. In this letter we address the issue of estimating cosmic variance errors for weak lensing surveys in the non-Gaussian regime. Using standard cold dark matter model ray-tracing simulations characterized by Omega_m=0.3, Omega_Lambda=0.7, h=0.7, sigma_8=1.0 for different survey redshifts z_s, we determine the variance of the two-point shear correlation function measured across 64 independent lines of sight. We compare the measured variance to the variance expected from a random Gaussian field and derive a redshift-dependent non-Gaussian calibration relation. We find that the ratio can be as high as ~30 for a survey with source redshift z_s ~ 0.5 and ~10 for z_s ~ 1. The transition scale theta_c above which the ratio is consistent with unity, is found to be theta_c ~ 20 arcmin for z_s ~ 0.5 and theta_c ~ 10 arcmin for z_s ~ 1. We provide fitting formula to our results permitting the estimation of non-Gaussian cosmic variance errors for any weak lensing analysis, and discuss the impact on current and future surveys. A more extensive set of simulations will however be required to investigate the dependence of our results on cosmology, specifically on the amplitude of clustering.Comment: 6 pages, 7 figures. MNRAS Accepted versio

Weak lensing from space: first cosmological constraints from three-point shear statistics

We use weak lensing data from the Hubble Space Telescope COSMOS survey to measure the second- and third-moments of the cosmic shear field, estimated from about 450,000 galaxies with average redshift ~ 1.3. We measure two- and three-point shear statistics using a tree-code, dividing the signal in E, B and mixed components. We present a detection of the third-order moment of the aperture mass statistic and verify that the measurement is robust against systematic errors caused by point spread function (PSF) residuals and by the intrinsic alignments between galaxies. The amplitude of the measured three-point cosmic shear signal is in very good agreement with the predictions for a WMAP7 best-fit model, whereas the amplitudes of potential systematics are consistent with zero. We make use of three sets of large Lambda CDM simulations to test the accuracy of the cosmological predictions and to estimate the influence of the cosmology-dependent covariance. We perform a likelihood analysis using the measurement and find that the Omega_m-sigma_8 degeneracy direction is well fitted by the relation: sigma_8 (Omega_m/0.30)^(0.49)=0.78+0.11/-0.26. We present the first measurement of a more generalised three-point shear statistic and find a very good agreement with the WMAP7 best-fit cosmology. The cosmological interpretation of this measurement gives sigma_8 (Omega_m/0.30)^(0.46)=0.69 +0.08/-0.14. Furthermore, the combined likelihood analysis of this measurement with the measurement of the second order moment of the aperture mass improves the accuracy of the cosmological constraints, showing the high potential of this combination of measurements to infer cosmological constraints.Comment: 17 pages, 11 figures. MNRAS submitte

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

CFHTLenS: Co-evolution of galaxies and their dark matter haloes

Galaxy-galaxy weak lensing is a direct probe of the mean matter distribution around galaxies. The depth and sky coverage of the CFHT Legacy Survey yield statistically significant galaxy halo mass measurements over a much wider range of stellar masses ($10^{8.75}$ to $10^{11.3} M_{\odot}$) and redshifts ($0.2 < z < 0.8$) than previous weak lensing studies. At redshift $z \sim 0.5$, the stellar-to-halo mass ratio (SHMR) reaches a maximum of $4.0\pm0.2$ percent as a function of halo mass at $\sim 10^{12.25} M_{\odot}$. We find, for the first time from weak lensing alone, evidence for significant evolution in the SHMR: the peak ratio falls as a function of cosmic time from $4.5 \pm 0.3$ percent at $z \sim 0.7$ to $3.4 \pm 0.2$ percent at $z \sim 0.3$, and shifts to lower stellar mass haloes. These evolutionary trends are dominated by red galaxies, and are consistent with a model in which the stellar mass above which star formation is quenched "downsizes" with cosmic time. In contrast, the SHMR of blue, star-forming galaxies is well-fit by a power law that does not evolve with time. This suggests that blue galaxies form stars at a rate that is balanced with their dark matter accretion in such a way that they evolve along the SHMR locus. The redshift dependence of the SHMR can be used to constrain the evolution of the galaxy population over cosmic time.Comment: 18 pages, MNRAS, in pres

CFHTLenS tomographic weak lensing: Quantifying accurate redshift distributions

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.Comment: 17 pages, 12 figures, submitted to MNRA

Cosmological Constraints From the 100 Square Degree Weak Lensing Survey

We present a cosmic shear analysis of the 100 square degree weak lensing survey, combining data from the CFHTLS-Wide, RCS, VIRMOS-DESCART and GaBoDS surveys. Spanning ~100 square degrees, with a median source redshift z~0.78, this combined survey allows us to place tight joint constraints on the matter density parameter Omega_m, and the amplitude of the matter power spectrum sigma_8, finding sigma_8*(Omega_m/0.24)^0.59 = 0.84+/-0.05. Tables of the measured shear correlation function and the calculated covariance matrix for each survey are included. The accuracy of our results is a marked improvement on previous work owing to three important differences in our analysis; we correctly account for cosmic variance errors by including a non-Gaussian contribution estimated from numerical simulations; we correct the measured shear for a calibration bias as estimated from simulated data; we model the redshift distribution, n(z), of each survey from the largest deep photometric redshift catalogue currently available from the CFHTLS-Deep. This catalogue is randomly sampled to reproduce the magnitude distribution of each survey with the resulting survey dependent n(z) parametrised using two different models. While our results are consistent for the n(z) models tested, we find that our cosmological parameter constraints depend weakly (at the 5% level) on the inclusion or exclusion of galaxies with low confidence photometric redshift estimates (z>1.5). These high redshift galaxies are relatively few in number but contribute a significant weak lensing signal. It will therefore be important for future weak lensing surveys to obtain near-infra-red data to reliably determine the number of high redshift galaxies in cosmic shear analyses.Comment: 14 pages, 6 figures, accepted for publication by MNRA