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.     

Models of the Cosmic Horseshoe gravitational lens J1004+4112

We model the extremely massive and luminous lens galaxy in the Cosmic Horseshoe Einstein ring system J1004+4112, recently discovered in the Sloan Digital Sky Survey. We use the semilinear method of Warren & Dye, which pixelizes the source surface brightness distribution, to invert the Einstein ring for sets of parametrized lens models. Here, the method is refined by exploiting Bayesian inference to optimise adaptive pixelization of the source plane and to choose between three differently parametrized models: a singular isothermal ellipsoid, a power-law model and a Navarro, Frenk & White (NFW) profile. The most probable lens model is the power law with a volume mass density  ρ∝ r ^{−1.96±0.02}  and an axis ratio of ∼0.8. The mass within the Einstein ring (i.e. within a cylinder with projected distance of ∼30 kpc from the centre of the lens galaxy) is  (5.02 ± 0.09) × 10¹² M _⊙  , and the mass-to-light ratio is ∼30. Even though the lens lies in a group of galaxies, the preferred value of the external shear is almost zero. This makes the Cosmic Horseshoe unique amongst large separation lenses, as almost all the deflection comes from a single, very massive galaxy with little boost from the environment.     

Non-parametric inversion of gravitational lensing systems with few images using a multi-objective genetic algorithm

Galaxies acting as gravitational lenses are surrounded by, at most, a handful of images. This apparent paucity of information forces one to make the best possible use of what information is available to invert the lens system. In this paper, we explore the use of a genetic algorithm to invert in a non-parametric way strong lensing systems containing only a small number of images. Perhaps the most important conclusion of this paper is that it is possible to infer the mass distribution of such gravitational lens systems using a non-parametric technique. We show that including information about the null space (i.e. the region where no images are found) is prerequisite to avoid the prediction of a large number of spurious images, and to reliably reconstruct the lens mass density. While the total mass of the lens is usually constrained within a few per cent, the fidelity of the reconstruction of the lens mass distribution depends on the number and position of the images. The technique employed to include null space information can be extended in a straightforward way to add additional constraints, such as weak-lensing data or time-delay information.     

A genetic algorithm for the non-parametric inversion of strong lensing systems

We present a non-parametric technique to infer the projected mass distribution of a gravitational lens system with multiple strong-lensed images. The technique involves a dynamic grid in the lens plane on which the mass distribution of the lens is approximated by a sum of basis functions, one per grid cell. We used the projected mass densities of Plummer spheres as basis functions. A genetic algorithm then determines the mass distribution of the lens by forcing images of a single source, projected back on to the source plane, to coincide as well as possible. Averaging several tens of solutions removes the random fluctuations that are introduced by the reproduction process of genomes in the genetic algorithm and highlights those features common to all solutions. Given the positions of the images and the redshifts of the sources and the lens, we show that the mass of a gravitational lens can be retrieved with an accuracy of a few percent and that, if the sources sufficiently cover the caustics, the mass distribution of the gravitational lens can also be reliably retrieved. A major advantage of the algorithm is that it makes full use of the information contained in the radial images, unlike methods that minimize the residuals of the lens equation, and is thus able to accurately reconstruct also the inner parts of the lens.     

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.     

SDSS J120923.7+264047: a new massive galaxy cluster with a bright giant arc

Highly magnified lensed galaxies allow us to probe the morphological and spectroscopic properties of high-redshift stellar systems in great detail. However, such objects are rare, and there are only a handful of lensed galaxies that are bright enough for a high-resolution spectroscopic study with current instrumentation. We report the discovery of a new massive lensing cluster, SDSS J120923.7+264047, at z = 0.558. Present around the cluster core, at angular distances of up to ∼40 arcsec, are many arcs and arc candidates, presumably due to lensing of background galaxies by the cluster gravitational potential. One of the arcs, 21 arcsec long, has an r -band magnitude of 20, making it one of the brightest known lensed galaxies. We obtained a low-resolution spectrum of this galaxy, using the Keck-I telescope, and found it is at redshift of z = 1.018.     

A photometric study of the field around the candidate recoiling/binary black hole SDSS J092712.65+294344.0

We present a photometric far-ultraviolet (FUV) to K _s-band study of the field around quasar SDSS J092712.65+294344.0. The SDSS spectrum of this object shows various emission lines with two distinct redshifts, at   z = 0.699  and 0.712. Because of this peculiar spectroscopic feature, this source has been proposed as a candidate recoiling or binary black hole. A third alternative model involves two galaxies moving in the centre of a rich galaxy cluster. Here, we present a study addressing the possible presence of such a rich cluster of galaxies in the SDSS J092712.65+294344.0 field. We observed the  3.6 × 2.6  arcmin² field in the K _s band and matched the near-infrared data with the FUV and near-ultraviolet images in the Galaxy Evolution Explorer archive and the ugriz observations in the SDSS. From various colour–colour diagrams, we were able to classify the nature of 32 sources, only 6–11 of which have colours consistent with galaxies at   z ≈ 0.7  . We compare these numbers with the surface density of galaxies, stars and quasars and the expectations for typical galaxy clusters both at low and high redshift. Our study shows that the galaxy cluster scenario is in clear disagreement with the new observations.     

New multiply-lensed galaxies identified in ACS/NIC3 observations of Cl0024+1654 using an improved mass model

We present an improved strong-lensing analysis of Cl0024+1654  ( z = 0.39)  using deep Hubble Space Telescope ( HST )/Advanced Camera for Surveys (ACS)/NIC3 images, based on 33 multiply-lensed images of 11 background galaxies. These are found with a model that assumes mass approximately traces light, with a low-order expansion to allow for flexibility on large scales. The model is constrained initially by the well-known five-image system  ( z = 1.675)  and refined as new multiply-lensed systems are identified using the model. Photometric redshifts of these new systems are then used to constrain better the mass profile by adopting the standard cosmological relation between redshift and lensing distance. Our model requires only six free parameters to describe well all positional and redshift data. The resulting inner mass profile has a slope of  d log  M /d log  r ≃−0.55  , consistent with new weak-lensing measurements where the data overlap, at   r ≃ 200 kpc/ h ₇₀  . The combined profile is well fitted by a high-concentration Navarro, Frenk & White (NFW) mass profile,   C _vir∼ 8.6 ± 1.6  , similar to other well-studied clusters, but larger than predicted with standard Λ cold dark matter (ΛCDM). A well-defined radial critical curve is generated by the model and is clearly observed at   r ≃ 12  arcsec, outlined by elongated images pointing towards the centre of mass. The relative fluxes of the multiply-lensed images are found to agree well with the modelled magnifications, providing an independent consistency check.     

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.     

A new gravitational lens from the MUSCLES survey: ULAS J082016.1+081216

We present observations of a new double-image gravitational lens system, ULAS J082016.1+081216, of image separation 2.3 arcsec and high (∼6) flux ratio. The system is selected from the Sloan Digital Sky Survey (SDSS) spectroscopic quasar list using new high-quality images from the UKIRT (United Kingdom Infrared Telescope) Deep Sky Survey (UKIDSS). The lensed quasar has a source redshift of 2.024, and we identify the lens galaxy as a faint red object of redshift  0.803 ± 0.001  . Three other objects from the UKIDSS survey, selected in the same way, were found not to be lens systems. Together with the earlier lens found using this method, the SDSS–UKIDSS lenses have the potential to significantly increase the number of quasar lenses found in SDSS, to extend the survey to higher flux ratios and lower separations, and to give greater completeness which is important for statistical purposes.     

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.     

The rapidly pulsating sdO star, SDSS J160043.6+074802.9

A spectroscopic analysis of Sloan Digital Sky Survey (SDSS) J160043.6+074802.9, a binary system containing a pulsating subdwarf-O (sdO) star with a late-type companion, yields   T _eff= 70 000 ± 5000 K  and  log  g = 5.25 ± 0.30  , together with a most likely type of K3 V for the secondary star. We compare our results with atmospheric parameters derived by Fontaine et al. and in the context of existing evolution models for sdO stars. New and more extensive photometry is also presented which recovers most, but not all, frequencies found in an earlier paper. Therefore, it seems probable that some pulsation modes have variable amplitudes. A non-adiabatic pulsation analysis of uniform metallicity sdO models show those having  log  g > 5.3  to be more likely to be unstable and capable of driving pulsation in the observed frequency range.