Galaxy density profiles and shapes – II. Selection biases in strong lensing surveys

Many current and future astronomical surveys will rely on samples of strong gravitational lens systems to draw conclusions about galaxy mass distributions. We use a new strong lensing pipeline (presented in Paper I of this series) to explore selection biases that may cause the population of strong lensing systems to differ from the general galaxy population. Our focus is on point-source lensing by early-type galaxies with two mass components (stellar and dark matter) that have a variety of density profiles and shapes motivated by observational and theoretical studies of galaxy properties. We seek not only to quantify but also to understand the physics behind selection biases related to: galaxy mass, orientation and shape; dark matter profile parameters such as inner slope and concentration; and adiabatic contraction. We study how all of these properties affect the lensing Einstein radius, total cross-section, quad/double ratio and image separation distribution, with a flexible treatment of magnification bias to mimic different survey strategies. We present our results for two families of density profiles: cusped and deprojected Sérsic models. While we use fixed lens and source redshifts for most of the analysis, we show that the results are applicable to other redshift combinations, and we also explore the physics of how our results change for very different redshifts. We find significant (factors of several) selection biases with mass; orientation, for a given galaxy shape at fixed mass; cusped dark matter profile inner slope and concentration; concentration of the stellar and dark matter deprojected Sérsic models. Interestingly, the intrinsic shape of a galaxy does not strongly influence its lensing cross-section when we average over viewing angles. Our results are an important first step towards understanding how strong lens systems relate to the general galaxy population.     

The image separation distribution of strong lenses: halo versus subhalo populations

We present a halo model prediction of the image separation distribution of strong lenses. Our model takes into account the subhalo population, which has been ignored in previous studies, as well as the conventional halo population. Haloes and subhaloes are linked to central and satellite galaxies by adopting a universal scaling relation between masses of (sub)haloes and luminosities of galaxies. Our model predicts that 10–20 per cent of lenses should be caused by the subhalo population. The fraction of lensing by satellite galaxies (subhaloes) peaks at ∼1 arcsec and decreases rapidly with increasing image separations. We compute fractions of lenses which lie in groups and clusters and find them to be ∼14 and ∼4 per cent, respectively; nearly half of such lenses are expected to be produced by satellite galaxies, rather than central parts of haloes. We also study mass distributions of lensing haloes and find that, even at image separations of ∼3 arcsec, the deviation of lens mass distributions from isothermal profiles is large; at or beyond ∼3 arcsec, image separations are enhanced significantly by surrounding haloes. Our model prediction agrees reasonably well with observed image separation distributions from galaxy to cluster scales.     

The Milky Way Galaxy as a strong gravitational lens

We study the gravitational lensing effects of spiral galaxies by taking a model of the Milky Way and computing its lensing properties. The model is composed of a spherical Hernquist bulge, a Miyamoto–Nagai disc and an isothermal halo. As a strong lens, a spiral galaxy like the Milky Way can give rise to four different imaging geometries. They are (i) three images on one side of the galaxy centre ('disc triplets'), (ii) three images with one close to the centre ('core triplets'), (iii) five images and (iv) seven images. Neglecting magnification bias, we show that the core triplets, disc triplets and fivefold imaging are roughly equally likely. Even though our models contain edge-on discs, their image multiplicities are not dominated by disc triplets. The halo is included for completeness, but it has a small effect on the caustic structure, the time delays and brightnesses of the images.
The Milky Way model has a maximum disc (i.e. the halo is not dynamically important in the inner parts). Strong lensing by nearly edge-on disc galaxies breaks the degeneracy between the relative contributions of the disc and halo to the overall rotation curve. If a spiral galaxy has a submaximum disc, then the astroid caustic shrinks dramatically in size, whilst the radial caustic shrinks more modestly. This causes changes in the relative likelihood of the image geometries, specifically (i) core triplets are now ∼9/2 times more likely than disc triplets, (ii) the cross-section for threefold imaging is reduced by a factor of ∼2/3, whilst (iii) the cross-section for fivefold imaging is reduced by ∼1/2. Although multiple imaging is less likely (the cross-sections are smaller), the average total magnification is greater. The time delays are smaller, as the total projected lensing mass is reduced.     

Weighing a galaxy bar in the lens Q2237 +=0305

In the gravitational lens system Q2237+0305 the cruciform quasar image geometry is twisted by 10c by the lens effect of a bar in the lensing galaxy. This effect can be used to measure the mass of the bar. We construct a new lensing model for this system with a power-law elliptical bulge and a Ferrers bar. The observed ellipticity of the optical isophotes of the galaxy leads to a nearly isothermal elliptical profile for the bulge, with a total quasar magnification of 16⁺⁵₋₄. We measure a bar mass of (7.5 ∼ 1.5) −10⁸ h ⁻¹₇₅ M⊙ in the region inside the quasar images.     

The Hubble constant from galaxy lenses: impacts of triaxiality and model degeneracies

The Hubble constant can be constrained using the time delays between multiple images of gravitationally lensed sources. In some notable cases, typical lensing analyses assuming isothermal galaxy density profiles produce low values for the Hubble constant, inconsistent with the result of the HST Key Project  (72 ± 8 km s⁻¹ Mpc⁻¹)  . Possible systematics in the values of the Hubble constant derived from galaxy lensing systems can result from a number of factors, for example, neglect of environmental effects, assumption of isothermality, or contamination by line-of-sight structures. One additional potentially important factor is the triaxial structure of the lensing galaxy halo; most lens models account for halo shape simply by perturbing the projected spherical lensing potential, an approximation that is often necessary but that is inadequate at the levels of triaxiality predicted in the cold dark matter paradigm. To quantify the potential error introduced by this assumption in estimates of the Hubble parameter, we strongly lens a distant galaxy through a sample of triaxial softened isothermal haloes and use an Markov Chain Monte Carlo method to constrain the lensing halo profile and the Hubble parameter from the resulting multiple image systems. We explore the major degeneracies between the Hubble parameter and several parameters of the lensing model, finding that without a way to accurately break these degeneracies accurate estimates of the Hubble parameter are not possible. Crucially, we find that triaxiality does not significantly bias estimates of the Hubble constant, and offer an analytic explanation for this behaviour in the case of isothermal profiles. Neglected triaxial halo shape cannot contribute to the low Hubble constant values derived in a number of galaxy lens systems.     

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.     

A comparison of different cluster mass estimates: consistency or discrepancy?

Rich and massive clusters of galaxies at intermediate redshift are capable of magnifying and distorting the images of background galaxies. A comparison of different mass estimators among these clusters can provide useful information about the distribution and composition of cluster matter and its dynamical evolution. Using the hitherto largest sample of lensing clusters drawn from the literature, we compare the gravitating masses of clusters derived from the strong/weak gravitational lensing phenomena, from the X-ray measurements based on the assumption of hydrostatic equilibrium, and from the conventional isothermal sphere model for the dark matter profile characterized by the velocity dispersion and core radius of galaxy distributions in clusters. While there is excellent agreement between the weak lensing, X-ray and isothermal sphere model-determined cluster masses, these methods are likely to underestimate the gravitating masses enclosed within the central cores of clusters by a factor of 2–4 as compared with the strong lensing results. Such a mass discrepancy has probably arisen from the inappropriate applications of the weak lensing technique and the hydrostatic equilibrium hypothesis to the central regions of clusters, as well as from assuming an unreasonably large core radius for both luminous and dark matter profiles. Nevertheless, it is pointed out that these cluster mass estimators may be safely applied on scales greater than the core sizes. Namely, the overall clusters of galaxies at intermediate redshift can still be regarded as the dynamically relaxed systems, in which the velocity dispersion of galaxies and the temperature of X-ray emitting gas are good indicators of the underlying gravitational potentials of clusters.     

Undistorted lensed images in galaxy clusters

To date, the study of high-magnification gravitational lensing effects of galaxy clusters has focused upon the grossly distorted, luminous arc-like features formed in massive, centrally condensed clusters. We investigate the formation of a different type of image, highly magnified yet undistorted, in two widely employed cluster mass density profiles, namely an isothermal sphere with a core, and a universal dark matter halo profile derived from the numerical simulations of Navarro et al. We examine the properties of images of extended sources produced by these two cluster profiles, paying particular attention to the undistorted images. Using simple assumptions about the source and lens population, we estimate the relative frequency of the occurrence of highly magnified, undistorted images and the more commonly known giant arcs.     

Vertical distribution of Galactic disc stars and gas constrained by a molecular cloud complex

We investigate the dynamical effects of a molecular cloud complex with a mass ∼ 10⁷ M_⊙ and a size ∼ a few 100 pc on the vertical distribution of stars and atomic hydrogen gas in a spiral galactic disc. Such massive complexes have now been observed in a number of spiral galaxies. The extended mass distribution in a complex, with an average mass density 6 times higher than the Oort limit, is shown to dominate the local gravitational field. This results in a significant redistribution or clustering of the surrounding disc components towards the mid-plane, with a resulting decrease in their vertical scaleheights.
The modified, self-consistent stellar density distribution is obtained by solving the combined Poisson equation and the force equation along the z -direction for an isothermal stellar disc on which the complex is imposed. The effect of the complex is strongest at its centre, where the stellar mid-plane density increases by a factor of 2.6 and the vertical scaleheight decreases by a factor of 3.4 compared with the undisturbed stellar disc. A surprising result is the large radial distance of ∼ 500 pc from the complex centre over which the complex influences the disc; this is due to the extended mass distribution in a complex. The complex has a comparable effect on the vertical distribution of the atomic hydrogen gas in the galactic disc. This 'pinching' or constraining effect should be detectable in the nearby spiral galaxies, as for example has been done for NGC 2403 by Sicking. Thus the gravitational field of a complex results in local corrugations of the stellar and H  i vertical scaleheights, and the galactic disc potential is highly non-uniform on scales of the intercomplex separation of ∼ 1 kpc.     

Lens models with density cusps

Lens models appropriate for representing cusped galaxies and clusters are developed. The analogue of the odd-number theorem for cusped density distributions is given. Density cusps are classified into strong, isothermal or weak, according to their lensing properties. Strong cusps cause multiple imaging for any source position, whereas isothermal and weak cusps give rise to only one image for distant sources. Isothermal cusps always possess a pseudo-caustic. When the source crosses the pseudo-caustic, the number of images changes by unity.   Two families of cusped galaxy and cluster models are examined in detail. The double power-law family has an inner cusp, followed by a transition region and an outer envelope. One member of this family — the isothermal double power-law model — possesses an exceedingly scarce property, namely the lens equation is exactly solvable for any source position. This means that the magnifications, the time delay and the lensing cross-sections are all readily available. The model has a three-dimensional density that is cusped like r ⁻² at small radii and falls off like r ⁻⁴ asymptotically. Thus, it provides a very useful representation of the lensing properties of a galaxy or cluster of finite total mass with a flat rotation curve. The second set of models studied is the single power-law family. These are single density cusps of infinite extent. The properties of the critical curves and caustics and the behaviour of the lenses in the presence of external shear are all discussed in some detail.     

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.     

Discs in early-type lensing galaxies: effects on magnification ratios and measurements of H0

Observations of early-type galaxies, both in the local Universe and in clusters at medium redshifts, suggest that these galaxies often contain discs or disc-like structures. Using the results of Kelson et al. for the incidence of disc-components among the galaxies in the redshift   z = 0.33  cluster CL 1358+62, we investigate the effect of disc structures on the lensing properties of early-type galaxies. Statistical properties, like magnification cross-sections and the expected number of quad (four-image) lens systems, are not affected greatly by the inclusion of discs that contain less than ∼10 per cent of the total stellar mass. However, the properties of individual lens systems are affected. We estimate that 10–30 per cent of all quad-lens systems, with early-type deflector galaxies, would be affected measurably by the presence of disc components. Intriguingly, the image magnification ratios are altered significantly. The amplitude of the predicted change is sufficient to explain the observed magnification ratios in systems like B1422+231 without requiring compact substructure. Furthermore, time-delays between images also change; fitting a bulge-only model to early-type lenses that in fact contain a disc would yield a value of the Hubble constant H ₀ that is systematically too low by ∼25 per cent.     

Submillimetre-wave gravitational lenses and cosmology

One of the most direct routes for investigating the geometry of the Universe is provided by the numbers of strongly magnified gravitationally lensed galaxies as compared with those that are either weakly magnified or de-magnified. In the submillimetre waveband the relative abundance of strongly lensed galaxies is expected to be larger as compared with the optical or radio wavebands, both in the field and in clusters of galaxies. The predicted numbers depend on the properties of the population of faint galaxies in the submillimetre waveband, which was formerly very uncertain; however, recent observations of lensing clusters have reduced this uncertainty significantly and confirm that a large sample of galaxy–galaxy lenses could be detected and investigated using forthcoming facilities, including the FIRST and Planck Surveyor space missions and a large ground-based millimetre/submillimetre-wave interferometer array (MIA). We discuss how this sample could be used to impose limits on the values of cosmological parameters and the total density and form of evolution of the mass distribution of bound structures, even in the absence of detailed lens modelling for individual members of the sample. The effects of different world models on the form of the magnification bias expected in sensitive submillimetre-wave observations of clusters are also discussed, because an MIA could resolve and investigate images in clusters in detail.     

B0712+472: a new radio four-image gravitational lens

A new four-image gravitational lens system, B0712+472, has been discovered during the Cosmic Lens All-Sky Survey. This system consists of four flat-spectrum radio images that are also seen on a Hubble Space Telescope ( HST ) image, together with the lensing galaxy. We present MERLIN, VLA and VLBA maps and WHT spectra of the system as well as the HST images. The light distribution of the lensing galaxy is highly elongated and so too is the mass distribution deduced from modelling. We suggest a redshift of ∼1.33 for the lensed object; the lens redshift will require further investigation. The discovery of this new system further increases the ratio of four-image to two-image lens systems currently known, exacerbating problems of required ellipticity of matter distributions in lensing galaxies.     

The influence of central black holes on gravitational lenses

Recent observations indicate that many if not all galaxies host massive central black holes. In this paper we explore the influence of black holes on the lensing properties. We model the lens as an isothermal ellipsoid with a finite core radius plus a central black hole. We show that the presence of the black hole substantially changes the critical curves and caustics. If the black hole mass is above a critical value, then it will completely suppress the central images for all source positions. Realistic central black holes are likely to have masses below this critical value. Even in such subcritical cases, the black hole can suppress the central image when the source is inside a zone of influence, which depends on the core radius and black hole mass. In the subcritical cases, an additional image may be created by the black hole in some regions, which for some radio lenses may be detectable with high-resolution and large dynamic range VLBI maps. The presence of central black holes should also be taken into account when one constrains the core radius from the lack of central images in gravitational lenses.     

A search for the third lensed image in JVAS B1030+074

Central gravitational image detection is very important for the study of the mass distribution of the inner parts (∼100 pc) of lens galaxies. However, the detection of such images is extremely rare and difficult. We present a 1.7-GHz High Sensitivity Array (HSA) observation of the double-image radio lens system B1030+074. The data are combined with archive Very Long Baseline Array and global very long baseline interferometry (VLBI) observations, and careful consideration is given to the effects of noise, clean ing and self-calibration. An upper limit is derived for the strength of the central image of 180 μJy (90 per cent confidence level), considerably greater than would have been expected on the basis of a simple analysis. This gives a lower limit of ∼10³ for the ratio of the brightest image to the central image. For cusped models of lens mass distributions, we have made use of this non-detection to constrain the relation between inner power-law slope β of the lensing galaxy mass profile, and its break radius r _b. For   r _b > 130 pc  the power-law slope is required to be close to isothermal  (β > 1.8)  . A flatter inner slope is allowed if a massive black hole is present at the centre of the lensing galaxy, but the effect of the black hole is small unless it is ∼10 times more massive than that implied by the relation between black hole mass and stellar velocity dispersion. By comparing four epochs of VLBI observations, we also detected possible superluminal motion in the jet in the brighter image A. The B jet remains unresolved, as expected from a simple lens model of the system.     

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.     

The edge-on spiral gravitational lens B1600+434

We present new observations of the gravitational lens (GL) system B1600+434, strongly suggesting that the lens is an edge-on spiral galaxy. These observations are used to constrain the mass model of the system, in particular the oblateness and velocity dispersion of the dark matter halo around the lensing galaxy. From an analytical model we find a lower limit on the halo oblateness q _halo=( c/a )_ρ≳0.4; more detailed numerical models give a lower limit of q _halo≳0.5. We determine an average halo velocity dispersion of σ_halo=190±15 km s⁻¹ over all non-singular isothermal elliptical (NIE) halo models. Constraining the models to larger and more massive discs decreases this average by only 10 km s⁻¹. A lower limit of σ_halo≳150 km s⁻¹ is found, even for disc masses larger than the mass inside the Einstein radius. This lower limit indicates the need for a massive dark matter halo, contributing at least half of the mass inside the Einstein radius. Time-delay calculations give (54±3)/ h ₅₀ d for the NIE halo model and (70±4)/ h ₅₀ d for the modified Hubble profile (MHP) halo model. Although the time delay for both NIE and MHP halo models is well constrained on our parameter grid, it strongly depends on the halo surface density profile. We furthermore find that the presence of a flat luminous mass distribution can severely alter the statistical properties of the lens.     

NICMOS images of JVAS/CLASS gravitational lens systems

We present Hubble Space Telescope ( HST ) infrared images of four gravitational lens systems from the JVAS/CLASS gravitational lens survey and compare the new infrared HST pictures with previously published WFPC2 HST optical images and radio maps. Apart from the wealth of information that we get from the flux ratios and accurate positions and separations of the components of the lens systems, which we can use as inputs for better constraints on the lens models, we are able to discriminate between reddening and optical/radio microlensing as the possible cause of differences observed in the flux ratios of the components across the three wavelength bands. Substantial reddening has been known to be present in the lens system B1600+434 and has been further confirmed by the present infrared data. In the two systems B0712+472 and B1030+074 microlensing has been pinpointed as the main cause of the flux ratio discrepancy both in the optical/infrared and in the radio, the radio possibly caused by the substructure revealed in the lensing galaxies. In B0218+357, however, the results are still not conclusive. If we are actually seeing the two 'true' components of the lens system then the flux ratio differences are attributed to a combination of microlensing and reddening or are alternatively the result of some variability in at least one of the images. Otherwise the second 'true' component of B0218+357 may be completely absorbed by a molecular cloud and the anomalous flux density ratios and large difference in separation between the optical/infrared and radio that we see can be explained by emission either from a foreground object or from part of the lensing galaxy.