Source–lens clustering effects on the skewness of the lensing convergence

The correlation between source galaxies and lensing potentials causes a systematic effect on measurements of cosmic shear statistics, known as the source–lens clustering (SLC) effect. The SLC effect on the skewness of lensing convergence, S ₃, is examined using a non-linear semi-analytic approach and is checked against numerical simulations. The semi-analytic calculations have been performed in a wide variety of generic models for the redshift distribution of source galaxies and power-law models for the bias parameter between the galaxy and dark matter distributions. The semi-analytic predictions are tested successfully against numerical simulations. We find the relative amplitude of the SLC effect on S ₃ to be of the order of  5–40  per cent. It depends significantly on the redshift distribution of sources and on the way in which the bias parameter evolves. We discuss possible measurement strategies to minimize the SLC effects.     

Sources of contamination to weak lensing three-point statistics: constraints from N-body simulations

We investigate the impact of the observed correlation between a galaxy's shape and its surrounding density field on the measurement of third-order weak lensing shear statistics. Using numerical simulations, we estimate the systematic error contribution to a measurement of the third-order moment of the aperture mass statistic (GGG) from three-point intrinsic ellipticity correlations (III), and the three-point coupling between the weak lensing shear experienced by distant galaxies and the shape of foreground galaxies (GGI and GII). We find that third-order weak lensing statistics are typically more strongly contaminated by these physical systematics compared to second-order shear measurements, contaminating the measured three-point signal for moderately deep surveys with a median redshift   z _m∼ 0.7  by ∼15 per cent. It has been shown that accurate photometric redshifts will be crucial to correct for this effect, once a model and the redshift dependence of the effect can be accurately constrained. To this end we provide redshift-dependent fitting functions to our results and propose a new tool for the observational study of intrinsic galaxy alignments. For a shallow survey with   z _m∼ 0.4  we find III to be an order of magnitude larger than the expected cosmological GGG shear signal. Compared to the two-point intrinsic ellipticity correlation which is similar in amplitude to the two-point shear signal at these survey depths, third-order statistics therefore offer a promising new way to constrain models of intrinsic galaxy alignments. Early shallow data from the next generation of very wide weak lensing surveys will be optimal for this type of study.     

Weak gravitational flexion

Flexion is the significant third-order weak gravitational lensing effect responsible for the weakly skewed and arc-like appearance of lensed galaxies. Here we demonstrate how flexion measurements can be used to measure galaxy halo density profiles and large-scale structure on non-linear scales, via galaxy–galaxy lensing, dark matter mapping and cosmic flexion correlation functions. We describe the origin of gravitational flexion, and discuss its four components, two of which are first described here. We also introduce an efficient complex formalism for all orders of lensing distortion. We proceed to examine the flexion predictions for galaxy–galaxy lensing, examining isothermal sphere and Navarro–Frenk–White (NFW) profiles and both circularly symmetric and elliptical cases. We show that in combination with shear we can precisely measure galaxy masses and NFW halo concentrations. We also show how flexion measurements can be used to reconstruct mass maps in two-dimensional projection on the sky, and in three dimensions in combination with redshift data. Finally, we examine the predictions for cosmic flexion, including convergence–flexion cross-correlations, and we find that the signal is an effective probe of structure on non-linear scales.     

Galaxy density profiles and shapes – I. Simulation pipeline for lensing by realistic galaxy models

Studies of strong gravitational lensing in current and upcoming wide and deep photometric surveys, and of stellar kinematics from (integral-field) spectroscopy at increasing redshifts, promise to provide valuable constraints on galaxy density profiles and shapes. However, both methods are affected by various selection and modelling biases, which we aim to investigate in a consistent way. In this first paper in a series, we develop a flexible but efficient pipeline to simulate lensing by realistic galaxy models. These galaxy models have separate stellar and dark matter components, each with a range of density profiles and shapes representative of early-type, central galaxies without significant contributions from other nearby galaxies. We use Fourier methods to calculate the lensing properties of galaxies with arbitrary surface density distributions, and Monte Carlo methods to compute lensing statistics such as point-source lensing cross-sections. Incorporating a variety of magnification bias modes lets us examine different survey limitations in image resolution and flux. We rigorously test the numerical methods for systematic errors and sensitivity to basic assumptions. We also determine the minimum number of viewing angles that must be sampled in order to recover accurate orientation-averaged lensing quantities. We find that for a range of non-isothermal stellar and dark matter density profiles typical of elliptical galaxies, the combined density profile and corresponding lensing properties are surprisingly close to isothermal around the Einstein radius. The converse implication is that constraints from strong lensing and/or stellar kinematics, which are indeed consistent with isothermal models near the Einstein radius, cannot trivially be extrapolated to smaller and larger radii.     

Self-consistent gravitational lens reconstruction

We present a new method for directly determining accurate, self-consistent cluster lens mass and shear maps in the strong lensing regime from the magnification bias of background galaxies. The method relies upon pixellization of the surface mass density distribution which allows us to write down a simple, solvable set of equations. We also show how pixellization can be applied to methods of mass determination from measurements of shear and present a simplified method of application. The method is demonstrated with cluster models and applied to magnification data from the lensing cluster Abell 1689.     

Weak lensing magnification reconstruction with the modified internal linear combination method

Measuring weak lensing cosmic magnification signal is very challenging due to the overwhelming intrinsic clustering in the observed galaxy distribution.In this paper,we modify the Internal Linear Combination (ILC) method to reconstruct the lensing signal with an extra constraint to suppress the intrinsic clustering.To quantify the performance,we construct a realistic galaxy catalogue for the LSST-like photometric survey,covering 20 000 deg~2 with mean source redshift at z_s~1.We find that the reconstruction performance depends on the width of the photo-z bin we choose.Due to the correlation between the lensing signal and the source galaxy distribution,the derived signal has smaller systematic bias but larger statistical uncertainty for a narrower photo-z bin.We conclude that the lensing signal reconstruction with the Modified ILC method is unbiased with a statistical uncertainty5%for bin width?z~P=0.2.     

Gravitational lensing of extended high-redshift sources by dark matter haloes

High-redshift galaxies and quasi-stellar objects (QSOs) are most likely to be strongly lensed by intervening haloes between the source and the observer. In addition, a large fraction of lensed sources is expected to be seen in the submillimetre region, as a result of the enhanced magnification bias on the steep intrinsic number counts. We extend in three directions Blain's earlier study of this effect.
First, we use a modification of the Press–Schechter mass function and detailed lens models to compute the magnification probability distribution. We compare the magnification cross-sections of populations of singular isothermal spheres and Navarro, Frenk & White (NFW) haloes and find that they are very similar, in contrast to the image-splitting statistics which were recently investigated in other studies. The distinction between the two types of density profile is therefore irrelevant for our purposes.
Secondly, we discuss quantitatively the maximum magnification, μ _max, that can be achieved for extended sources (galaxies) with realistic luminosity profiles, taking into account the possible ellipticity of the lensing potential. We find that μ _max plausibly falls into the range     for sources of     effective radius at redshifts within     .
Thirdly, we apply our model for the lensing magnification to a class of sources following the luminosity evolution typical for a unified scheme of QSO formation. As a result of the peculiar steepness of their intrinsic number counts, we find that the lensed source counts at a fiducial wave length of 850 μm can exceed the unlensed counts by several orders of magnitude at flux densities ≳100 mJy, even with a conservative choice of the maximum magnification.     

QSO lensing magnification associated with galaxy groups

We simulated both the matter and light (galaxy) distributions in a wedge of the Universe and calculated the gravitational lensing magnification caused by the mass along the line-of-sight of galaxies and galaxy groups identified in sky surveys. A large volume redshift cone containing cold dark matter particles mimics the expected cosmological matter distribution in a flat universe with low matter density and a cosmological constant. We generate a mock galaxy catalogue from the matter distribution and identify thousands of galaxy groups in the luminous sky projection. We calculate the expected magnification around galaxies and galaxy groups and then the induced quasi-stellar object (QSO)–lens angular correlation due to magnification bias. This correlation is observable and can be used both to estimate the average mass of the lens population and to make cosmological inferences. We also use analytical calculations and various analyses to compare the observational results with theoretical expectations for the cross-correlation between faint QSOs from the 2dF Survey and nearby galaxies and groups from the Automated Plate Measurement and Sloan Digital Sky Survey Early Data Release. The observed QSO–lens anticorrelations are stronger than the predictions for the cosmological model used. This suggests that there could be unknown systematic errors in the observations and data reduction, or that the model used is not adequate. If the observed signal is assumed to be solely due to gravitational lensing, then the lensing is stronger than expected, due to more massive galactic structures or more efficient lensing than simulated.     

Search for gravitational lenses near the extragalactic double object CSL-1

We consider additional arguments in favor of the first observable cosmic string. We discuss candidates for gravitational lensing events near the extragalactic double source CSL-1 (Capodimonte-Sternberg-Lens Candidate no. 1) discovered in the Osservatorio Astronomico di Capodimonte Deep Field (OACDF). The detected excess of candidates for such events cannot be explained in terms of the theory of gravitational lensing by standard extragalactic objects (galaxies, groups of galaxies, etc.) and is in close agreement with the proposed model of gravitational lensing by a cosmic string.     

Intrinsic correlation of galaxy shapes: implications for weak lensing measurements

Weak gravitational lensing is now established as a powerful method to measure mass fluctuations in the universe. It relies on the measurement of small coherent distortions of the images of background galaxies. Even low-level correlations in the intrinsic shapes of galaxies could however produce a significant spurious lensing signal. These correlations are also interesting in their own right, since their detection would constrain models of galaxy formation. Using     haloes found in N -body simulations, we compute the correlation functions of the intrinsic ellipticity of spiral galaxies assuming that the disc is perpendicular to the angular momentum of the dark matter halo. We also consider a simple model for elliptical galaxies, in which the shape of the dark matter halo is assumed to be the same as that of the light. For deep lensing surveys with median redshifts ∼1, we find that intrinsic correlations of ∼10⁻⁴ on angular scales     are generally below the expected lensing signal, and contribute only a small fraction of the excess signals reported on these scales. On larger scales we find limits to the intrinsic correlation function at a level ∼10⁻⁵, which gives a (model-dependent) range of separations for which the intrinsic signal is about an order of magnitude below the ellipticity correlation function expected from weak lensing. Intrinsic correlations are thus negligible on these scales for dedicated weak lensing surveys. For wider but shallower surveys such as SuperCOSMOS, APM and SDSS, we cannot exclude the possibility that intrinsic correlations could dominate the lensing signal. We discuss how such surveys could be used to calibrate the importance of this effect, as well as study spin–spin correlations of spiral galaxies.     

Angular cross-correlation of galaxies: a probe of gravitational lensing by large-scale structure

The angular cross-correlation between two galaxy samples separated in redshift is shown to be a useful measure of weak lensing by large-scale structure. Angular correlations in faint galaxies arise as a result of spatial clustering of the galaxies as well as gravitational lensing by dark matter along the line of sight. The lensing contribution to the two-point autocorrelation function is typically small compared with the gravitational clustering. However, the cross-correlation between two galaxy samples is almost unaffected by gravitational clustering provided that their redshift distributions do not overlap. The cross-correlation is then induced by magnification bias resulting from lensing by large-scale structure. We compute the expected amplitude of the cross-correlation for popular theoretical models of structure formation. For two populations with mean redshifts of ≃0.3 and 1, we find a cross-correlation signal of ≃1 per cent on arcmin scales and ≃3 per cent on scales of a few arcsec. The dependence on the cosmological parameters Ω and Λ, the dark matter power spectrum and the bias factor of the foreground galaxy population is explored.     

lemomaf: Lensed Mock Map Facility

We present the Lensed Mock Map Facility ( lemomaf ), a tool designed to perform mock weak-lensing measurements on numerically simulated chunks of the Universe. Coupling N -body simulations to a semi-analytical model of galaxy formation, lemomaf can create realistic lensed images and mock catalogues of galaxies, at wavelengths ranging from the ultraviolet to the submillimetre. To demonstrate the power of such a tool, we compute predictions of the source–lens clustering (SLC) effect on the convergence statistics, and quantify the impact of weak lensing on galaxy counts in two different filters. We find that the SLC effect skews the probability density function of the convergence towards low values, with an intensity which strongly depends on the redshift distribution of galaxies. On the other hand, the degree of enhancement or depletion in galaxy counts due to weak lensing is independent of the SLC effect. We discuss the impact on the two-point shear statistics to be measured by future missions like SNAP and LSST . The SLC effect would bias the estimation of σ₈ from two-point statistics up to 5 per cent for a narrow redshift distribution of mean   z ∼ 0.5  , and up to 2 per cent in small angular scales for a redshift distribution of mean   z ∼ 1.5  . We conclude that accurate photometric redshifts for individual galaxies are necessary in order to quantify and isolate the SLC effect.     

Designing weak lensing surveys: a generalized eigenmode analysis

We study the estimators of various second-order weak lensing statistics such as the shear correlation functions  ξ_±  and the aperture mass dispersion  〈 M ²_ap〉  which can directly be constructed from weak lensing shear maps. We compare the efficiency with which these estimators can be used to constrain cosmological parameters. To this end we introduce the Karhunen–Loève (KL) eigenmode analysis techniques for weak lensing surveys. These tools are shown to be very effective as a diagnostics for optimizing survey strategies. The usefulness of these tools to study the effect of angular binning, the depth and width of the survey and noise contributions due to intrinsic ellipticities and number density of source galaxies on the estimation of cosmological parameters is demonstrated. Results from independent analysis of various parameters and joint estimations are compared. We also study how degeneracies among various cosmological and survey parameters affect the eigenmodes associated with these parameters.     

Was Supernova 1997ff at z∼ 1.7 magnified by gravitational lensing?

The quest for the cosmological parameters has come to fruition with the identification of a number of supernovae at a redshift of     . Analyses of the brightness of these standard candles reveal that the Universe is dominated by a large cosmological constant. The recent identification of the     SN 1997ff in the northern Hubble Deep Field has provided further evidence for this cosmology. Here we examine the case for gravitational lensing of SN 1997ff owing to the presence of galaxies lying along our line of sight. We find that, while the alignment of SN 1997ff with foreground masses is not favourable for it to be multiply imaged and strongly magnified, two galaxies do lie close enough to result in significant magnification:     for the case where these elliptical galaxies have a velocity dispersion of 200 km s⁻¹. Given the small difference between supernova brightnesses in different cosmologies, detailed modelling of the gravitational lensing properties of the intervening matter is therefore required before the true cosmological significance of SN 1997ff can be deduced.     

The effect of weak lensing on the angular correlation function of faint galaxies

The angular correlation function ο(θ) of faint galaxies is affected both by non-linear gravitational evolution and by magnification bias resulting from gravitational lensing. We compute the resulting ο(θ) for different cosmological models and show how its shape and redshift evolution depend on Ω and Λ. For galaxies at redshift greater than 1 ( R magnitude fainter than about 24), magnification bias can significantly enhance or suppress ο(θ), depending on the slope of the number–magnitude relation. We show, for example, how it changes the ratio of ο(θ) for two galaxy samples with different number count slopes.     

Effects of cluster galaxies on arc statistics

We present the results of a set of numerical simulations evaluating the effect of cluster galaxies on arc statistics.
We perform a first set of gravitational lensing simulations using three independent projections for each of nine different galaxy clusters obtained from N -body simulations. The simulated clusters consist of dark matter only. We add a population of galaxies to each cluster, mimicking the observed luminosity function and the spatial galaxy distribution, and repeat the lensing simulations including the effects of cluster galaxies, which themselves act as individual lenses. Each galaxy is represented by a spherical Navarro, Frenk & White density profile.
We consider the statistical distributions of the properties of the gravitational arcs produced by our clusters with and without galaxies. We find that the cluster galaxies do not introduce perturbations strong enough to significantly change the number of arcs and the distributions of lengths, widths, curvature radii and length-to-width ratios of long arcs. We find some changes to the distribution of short-arc properties in the presence of cluster galaxies. The differences appear in the distribution of curvature radii for arc lengths smaller than 12 arcsec, while the distributions of lengths, widths and length-to-width ratios are significantly changed only for arcs shorter than 4 arcsec.     

Evolution of the dark matter distribution with three-dimensional weak lensing

We present a direct detection of the growth of large-scale structure, using weak gravitational lensing and photometric redshift data from the COMBO-17 survey. We use deep R -band imaging of two  0.5 × 0.5 deg²  fields, affording shear estimates for over 52 000 galaxies; we combine these with photometric redshift estimates from our 17-band survey, in order to obtain a 3D shear field. We find theoretical models for evolving matter power spectra and correlation functions, and fit the corresponding shear correlation functions to the data as a function of redshift. We detect the evolution of the power at the 4.7σ level given reasonable priors, and measure the rate of evolution for  0 < z < 1  . We also fit correlation functions to our 3D data as a function of cosmological parameters σ₈ and  Ω_Λ  . We find joint constraints on  Ω_Λ  and σ₈, demonstrating an improvement in accuracy by ≃40 per cent over that available from 2D weak lensing for the same area.     

A maximum-entropy method for reconstructing the projected mass distribution of gravitational lenses

The maximum-entropy method is applied to the problem of reconstructing the projected mass density of a galaxy cluster using its gravitational lensing effects on background galaxies. We demonstrate the method by reconstructing the mass distribution in a model cluster using simulated shear and magnification data to which Gaussian noise is added. The mass distribution is reconstructed directly and the inversion is regularized using the entropic prior for this positive additive distribution. For realistic noise levels, we find that the method faithfully reproduces the main features of the cluster mass distribution not only within the observed field but also slightly beyond it. We estimate the uncertainties in the reconstruction by calculating an analytic approximation to the covariance matrix of the reconstruction values of each pixel. This result is compared with error estimates derived from Monte Carlo simulations for different noise realizations and found to be in good agreement.