Non-parametric strong lens inversion of Cl 0024+1654: illustrating the monopole degeneracy

The cluster lens Cl 0024+1654 is undoubtedly one of the most beautiful examples of strong gravitational lensing, providing five large images of a single source with well-resolved substructure. Using the information contained in the positions and the shapes of the images, combined with the null space information, a non-parametric technique is used to infer the strong lensing mass map of the central region of this cluster. This yields a strong lensing mass of  1.60 × 10¹⁴ M_⊙  within a 0.5  arcmin radius around the cluster centre. This mass distribution is then used as a case study of the monopole degeneracy, which may be one of the most important degeneracies in gravitational lensing studies and which is extremely hard to break. We illustrate the monopole degeneracy by adding circularly symmetric density distributions with zero total mass to the original mass map of Cl 0024+1654. These redistribute mass in certain areas of the mass map without affecting the observed images in any way. We show that the monopole degeneracy and the mass-sheet degeneracy together lie at the heart of the discrepancies between different gravitational lens reconstructions that can be found in the literature for a given object, and that many images/sources, with an overall high image density in the lens plane, are required to construct an accurate, high-resolution mass map based on strong lensing data.     

Depletion of background galaxies owing to the cluster lens CL0024+1654: U- and R-band observations

We have obtained U - and R -band observations of the depletion of background galaxies resulting from the gravitational lensing of the galaxy cluster CL0024+1654 ( z =0.39). The radial depletion curves show a significant depletion in both bands within a radius of 40–70 arcsec from the cluster centre. This is the first time that depletion is detected in the U band. This gives independent evidence for a break in the slope of the U -band luminosity function at faint magnitudes. The radially averaged R -band depletion curve is broader and deeper than in the U band. The differences can be attributed to the wavelength dependence of the slope of the luminosity function and to the different redshift distribution of the objects probed in the two bands. We estimate the Einstein radius, r _E, of a singular isothermal sphere lens model using maximum-likelihood analysis. Adopting a slope of the number counts of α =0.2 and using the background density found beyond r =150 arcsec, we find r _E=17±3 and 25±3 arcsec in the U and R bands, respectively. When combined with the redshift of the single background galaxy at z =1.675 seen as four giant arcs around 30 arcsec from the cluster centre, these values indicate a median redshift in the range 〈 z _S〉≈0.7 to 1.1 for the U _AB≥24 mag and R _AB≥24 mag populations.     

Quasar–galaxy and galaxy–galaxy cross-correlations: model predictions with realistic galaxies

Several measurements of quasi-stellar object (QSO)–galaxy correlations have reported signals much larger than predictions of magnification by large-scale structure. We find that the expected signal depends strongly on the properties of the foreground galaxy population. On arcmin scales, it can be either larger or smaller by a factor of 2 for different galaxy types in comparison with a linearly biased version of the mass distribution. Thus the resolution of some of the excess measurements may lie in examining the halo occupation properties of the galaxy population sampled by a given survey; this is also the primary information such measurements will provide.
We use the halo model of clustering and simulations to predict the magnification-induced cross-correlations and errors for forthcoming surveys. With the full Sloan Digital Sky Survey, the statistical errors will be below 1 per cent for the galaxy–galaxy correlations and significantly larger for QSO–galaxy correlations. Thus accurate constraints on parameters of the galaxy halo occupation distribution can be obtained from small-scale measurements and on the bias parameter from large scales. Since the lensing-induced cross-correlation measures the first moment of the halo occupation number of galaxies, these measurements can provide the basis for interpreting galaxy clustering measurements that measure the second- and higher-order moments.     

Noise properties of gravitational lens mass reconstruction

Gravitational lensing is potentially able to observe mass-selected haloes, and to measure the projected cluster mass function. An optimal mass selection requires a quantitative understanding of the noise behaviour in mass maps. This paper is an analysis of the noise properties in mass maps reconstructed from a maximum-likelihood method.
The first part of this work is the derivation of the noise power spectrum and the mass error bars as a straightforward extension of the Kaiser & Squires algorithm for the case of a correlated noise. Very good agreement is found between these calculations and the noise properties measured in the mass reconstructions limited to non-critical clusters of galaxies. It demonstrates that Kaiser & Squires and maximum-likelihood methods have similar noise properties and that the weak lensing approximation is valid for describing these properties .
In a second stage I show that the statistics of peaks in the noise follows accurately the peak statistics of a two-dimensional Gaussian random field (using the BBKS techniques) if the smoothing aperture contains enough galaxies. This analysis provides a full procedure for deriving the significance of any convergence peak as a function of its amplitude and profile.
I demonstrate that a detailed quantitative analysis of the structures in mass maps can be carried out, and that, to a very good approximation, a mass map is the sum of the lensing signal and known two-dimensional Gaussian random noise. A straightforward application is the measurement of the projected mass function in wide-field lensing surveys, down to small mass overdensities that are individually undetectable.     

Non-parametric reconstruction of cluster mass distribution from strong lensing: modelling Abell 370

We describe a new non-parametric technique for reconstructing the mass distribution in galaxy clusters with strong lensing, i.e. from multiple images of background galaxies. The observed positions and redshifts of the images are considered as rigid constraints, and through the lens (ray-trace) equation they provide us with linear constraint equations. These constraints confine the mass distribution to some allowed region, which is then found by linear programming. Within this allowed region we study in detail the mass distribution with minimum mass-to-light variation, and also some other distributions, such as the smoothest mass distribution.
The method is applied to the extensively studied cluster Abell 370, which hosts a giant luminous arc and several other multiply imaged background galaxies. Our mass maps are constrained by the observed positions and redshifts (spectroscopic or model-inferred by previous authors) of the giant arc and multiple-image systems. The reconstructed maps obtained for Abell 370 reveal a detailed mass distribution, with substructure quite different from the light distribution. The method predicts the bimodal nature of the cluster, and that the projected mass distribution is indeed elongated along the axis defined by the two dominant cD galaxies. However, the peaks in the mass distribution appear to be offset from the centres of the cDs.
We also present an estimate for the total mass of the central region of the cluster. This is in good agreement with previous mass determinations. The total mass of the central region is M =(2.0–2.7)×10¹⁴ M⊙ h ⁻¹₅₀, depending on the solution chosen.     

Detecting mass substructure in galaxy clusters: an aperture mass statistic for gravitational flexion

Gravitational flexion has been introduced as a technique by which one can map out and study substructure in clusters of galaxies. Previous analyses involving flexion have measured the individual galaxy–galaxy flexion signal, or used either parametric techniques or a Kaiser, Squires and Broadhurst (KSB)-type inversion to reconstruct the mass distribution in Abell 1689. In this paper, we present an aperture mass statistic for flexion, and apply it to the lensed images of background galaxies obtained by ray-tracing simulations through a simple analytic mass distribution and through a galaxy cluster from the Millennium Simulation. We show that this method is effective at detecting and accurately tracing structure within clusters of galaxies on subarcminute scales with high signal to noise even using a moderate background source number density and image resolution. In addition, the method provides much more information about both the overall shape and the small-scale structure of a cluster of galaxies than can be achieved through a weak lensing mass reconstruction using gravitational shear data. Lastly, we discuss how the zero-points of the aperture mass might be used to infer the masses of structures identified using this method.     

Infrared constraints on the dark mass concentration observed in the cluster Abell 1942

We present a deep H -band image of the region in the vicinity of the cluster Abell 1942 containing the puzzling dark matter concentration detected in an optical weak lensing study by Erben et al. We demonstrate that our limiting magnitude, H =22 , would be sufficient to detect clusters of appropriate mass out to redshifts comparable with the mean redshift of the background sources. Despite this, our infrared image reveals no obvious overdensity of sources at the location of the lensing mass peak, nor an excess of sources in the I − H versus H colour–magnitude diagram. We use this to constrain further the luminosity and mass-to-light ratio of the putative dark clump as a function of its redshift. We find that for spatially flat cosmologies, background lensing clusters with reasonable mass-to-light ratios lying in the redshift range 0< z <1 are strongly excluded, leaving open the possibility that the mass concentration is a new type of truly dark object.     

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.     

X-ray and strong lensing mass estimate of MS2137.3–2353

We present new mass estimates of the cluster of galaxies MS2137.3–2353, inferred from X-ray and strong lensing analyses. This cluster exhibits an outstanding strong lensing configuration and indicates a well-relaxed dynamical state, being most suitable for a mass reconstruction which combines both techniques. Despite this, several previous studies have claimed a significant discrepancy between the X-ray and the strong lensing mass estimates. The primary aim of this paper is to address and explain this mismatch. For this purpose, we have analysed Chandra observations to recover the profiles of the intracluster medium properties and, assuming a functional form for the matter density, the total mass distribution. The notable strong-lensing features of MS2137.3 allow us to reconstruct its projected mass in the central regions with good accuracy, by taking advantage of the lensing inversion code lenstool . We compare the results obtained for both methods. Our mass estimates for MS2137.3 are in agreement within errors, leading to a mean, extrapolated value of   M ₂₀₀≃ 4.4 ± 0.3 × 10¹⁴ M_⊙  , under the assumption of the Navarro–Frenk–White (NFW) mass profile. However, the strong lensing mass estimate is affected by the details of the brightest cluster galaxy mass modelling, since the radial arc is a very sensitive probe of the total mass derivative in the central region. In particular, we do not find evidence for a high concentration for the NFW density profile, as reported in some earlier works.     

Probing galactic dark matter in dense environments: on the strong lensing efficiency of galaxies in rich clusters

The recent detection by Limousin et al. of five new strong lensing events dominated by galaxy cluster members in Abell 1689, and outside the critical regime of the cluster itself, offers a way to obtain constraints on the cluster mass distribution in a region inaccessible to standard lensing analysis. In addition, modelling such systems will provide another window on the dark matter haloes of galaxies in very dense environments. Here, it is shown that the boost in image separation due to the external shear and convergence from a smooth cluster component means that more numerous, less massive galaxies have the potential to create multiple images with detectable separations, relative to isolated field galaxies. This comes in addition to a potential increase in their lensing (source plane) cross-section. To gain insight into the factors involved and as a precursor to a numerical study using N -body simulations, a simple analytic model of a cluster at   z = 0.3  lensing background galaxies at   z = 2  is considered here. The fiducial model has cluster members with isothermal density profiles and luminosities L , distributed in a Schechter function (faint-end slope  ν=−1.25  ), related to their velocity dispersions σ via the Faber–Jackson scaling L ∝σ⁴. Just outside the critical regime of the cluster, the scale of galaxy-dominated image separations is significantly increased. Folding in the fact that less massive galaxies present a lower lensing cross-section, and that the cross-section can itself be enhanced in an external field leads to a factor of a few times more detected events relative to field galaxies. These values will be higher closer to the critical curve. Given that the events in Abell 1689 were detected over a very small region of the cluster where ACS data were available, this motivates the search for such events in other clusters.     

The JVAS/CLASS search for 6-arcsec to 15-arcsec image separation lensing

The Jodrell Bank–VLA Astrometric Survey (JVAS) and the Cosmic Lens All Sky Survey (CLASS) have been systematically searched for multiple gravitational imaging of sources with image separations between 6 arcsec and 15 arcsec, associated with galaxy group and cluster lensing masses. The radio and optical follow-up observations of all candidates are presented. From a total of ∼15 000 sources only one weak candidate remains and this is not contained in the statistically complete sample of flat-spectrum JVAS/CLASS sources of 11 670 sources. A simple Press–Schechter analysis is performed. For singular isothermal sphere lenses the lack of multiple image systems is inconsistent with the currently favoured cosmologies with     at the 4.2 σ level. Cored isothermal lenses reduce the expected number of lens systems and we suggest that the most probable interpretation of our results is that the surface mass density of groups and clusters of galaxies is not high enough to cause multiple imaging and the presence of the mass concentrations associated with individual galaxies is required to produce image separations such as those in B0957+561.     

The relation between stellar mass and weak lensing signal around galaxies: implications for modified Newtonian dynamics

We study the amplitude of the weak gravitational lensing signal as a function of stellar mass around a sample of relatively isolated galaxies. This selection of lenses simplifies the interpretation of the observations, which consist of data from the Red-Sequence Cluster Survey and the Sloan Digital Sky Survey. We find that the amplitude of the lensing signal as a function of stellar mass is well described by a power law with a best-fitting slope  α= 0.74 ± 0.08  . This result is inconsistent with modified Newtonian dynamics (MOND), which predicts  α= 0.5  (we find  α > 0.5  with 99.7 per cent confidence). As a related test, we determine the MOND mass-to-light ratio as a function of luminosity. Our results require dark matter for the most luminous galaxies ( L ≳ 10¹¹ L_⊙). We rule out an extended halo of gas or active neutrinos as a way of reconciling our findings with MOND. Although we focus on a single alternative gravity model, we note that our results provide an important test for any alternative theory of gravity.