ON THE CONTRIBUTION OF LARGE-SCALE STRUCTURE TO STRONG GRAVITATIONAL LENSING

In order to identify strong galaxy–galaxy lenses, Faure et al. (2008) first selected a sample 9452 of galaxies in the COSMOS-ACS survey, which will henceforth be called the "parent population." This parent population consists of all galaxies in the survey with photometric redshifts 0.2 ⩽ z_phot ⩽ 1.0, absolute V-band magnitude M_V < −20, and which are classified as early-type galaxies according to their color by the photometric-redshift software (Mobasher et al. 2007).

A 10'' × 10'' region around each galaxy in the parent population was visually inspected for multiple images and arc-like features. We also fitted and subtracted the stellar light of the lensing galaxy, and built ground-based color images to aid the detection of genuine strong lenses. This procedure yielded to 66 galaxy–galaxy lens candidates, which will henceforth be referred to as simply "lenses." Among these 66 galaxies, 19 systems are very convincing, showing multiple images of similar color or elongated curved arcs. They are called the "best systems" throughout this paper. The other 47 systems have generally a single arc, bluer than the lensing galaxy, and are referred to as "single arc systems."

The WL convergence map displayed in Figure 1 (top) is a measurement of the projected mass distribution. The convergence κ has been calculated from the image shapes of about half a million galaxies detected in the HST/ACS observation (Leauthaud et al. 2007, Massey et al. 2007), which have a density ∼70 arcmin⁻² and a median redshift z ∼1.17. The convergence map has been smoothed using a multiscale filtering method (Starck et al. 2006) with a maximum resolution of 12 FWHM. The measurement is sensitive to mass in a broad range of redshift 0.2 ⩽ z ⩽ 1.0, with a peak in the effective sensitivity at z ≈ 0.7 (Massey et al. 2007).

Zoom In Zoom Out Reset image size

Strongly lensed galaxies can typically lie at slightly higher reshifts (z ∼ 2) due to their additional magnification. However, the breadth of the lensing sensitivity kernel ensures that the WL mass reconstruction spans almost the same redshift range as the strong lens sample.

Finoguenov et al. (2007) and A. Finoguenov et al. (2009, in preparation) have discovered 202 galaxy groups and clusters in the COSMOS field, via X-ray emission from hot gas. This techniques probes virialised structures with a mass detection threshold lower than that of the WL mass map, but with a different sensitivity as a function of redshift. The galaxy clusters are characterized by a size r₅₀₀ (typically ∼001), the radius inside which the matter density is 500 times the critical density, and where most of the cluster's X-ray flux is generated (Markevitch 1998). Figure 1 (bottom) displays the X-ray galaxy clusters and groups detected in the COSMOS field. Each X-ray source is represented by a two-dimensional Gaussian, with FWHM equal to twice the radius r₅₀₀, and amplitude scaled to the lensing efficiency of the cluster, assuming background sources at z = 2. The lensing efficiency is defined by the weighted surface mass-density, $\Sigma \propto \frac{M_{500}}{\pi \times r_{500}^2}\times \frac{D_{LS}}{D_{S} }$, where M₅₀₀ is the mass encompassed in the radius r₅₀₀.

A comparison of Figure 1 (top and bottom) reveals some discrepancies in the mass distribution as traced by WL and X-ray emission. As noted by Massey et al. (2007), this is mainly due to the additional sensitivity of the X-ray analysis to compact objects. However, the WL map is also noisy and potentially subject to localized defects that are not revealed by systematic tests statistically averaged over the entire field.

To test whether our observational results are compatible with theoretical predictions, we have also created mock COSMOS data by ray-tracing through the MS (Springel et al. 2005). This is an N-body simulation of cosmic structure formation, which employed a TreePM version of Gadget-2 with 10¹⁰ particles of mass m_p = 8.6 × 10⁸h⁻¹ M_☉ to follow the structure formation in a cubic region of L = 500 h⁻¹ Mpc comoving on a side from redshift z = 127 and 0.

The ray-tracing through the MS is based on a multiple-lens-plane algorithm described in Hilbert et al. (2007, 2008a, 2008b). The dark-matter distribution in the observer's backward light-cone is generated directly from the particle data of the of the MS, and projected onto a series of lens planes. The stellar matter in galaxies is inferred from semianalytic galaxy-formation models implemented within the evolving dark-matter distribution of the MS (De Lucia & Blaizot 2007). Light rays are traced back from an observer to their source, assuming that the rays propagate unperturbed between lens planes, but are deflected when passing through a plane. The image distortions and amplifications resulting from differential deflections of the light rays at the lens planes are calculated from the projected matter distribution on the planes.

We randomly choose 20 fields of 0.5 × 0.5 deg². In each field, a grid of 8192 × 8192 rays is traced through the MS by the multiple-lens-plane algorithm. We calculate the convergence κ in each field from the ray distortions to sources at redshift z = 1.17 (i.e., the median redshift of the galaxies used for the WL map of the COSMOS field). We then add Gaussian noise to each pixel of the resulting convergence maps, with a variance appropriate for a source galaxy density of 70 arcmin⁻² and galaxy ellipticity variance ∼0.3. The noisy convergence maps are then smoothed using the multiscale, wavelet filtering method (Starck et al. 2006) that was applied to the real COSMOS map. This COSMOS-like WL simulation will then be processed through the same analysis as the real data, to investigate the correlation between the strong lenses and large scale structures.

In Figure 2, we display the projected mass map in nine of the 20 simulated fields. They contain noticeably less structure than the real COSMOS data. It is already known from a study of the angular correlation of the galaxies in the COSMOS field (McCracken et al. 2007) that this field is at the upper end of the expected cosmic variance scatter. Hence we expect from the numerical simulations to give us qualitative results on the behavior of strong lenses rather than direct quantitative results that can be compared to the COSMOS field.

Zoom In Zoom Out Reset image size

From the semianalytic galaxy catalog of De Lucia & Blaizot (2007), we select all galaxies with redshift 0.2 ⩽ z ⩽ 1.0, absolute V-band magnitude M_V < −20, and bulge-to-total B-band luminosity ratio L_B,bulge/L_B,total ⩾ 0.4 (Croton 2006), in order to select similar early-type galaxies to those visual inspected in the real data. We calculate the image positions of this "parent population" of simulated galaxies by linear interpolation between ray positions.

The identification of strong galaxy–galaxy lenses in the COSMOS field involves the human eye, whose "selection function" is difficult to model. In order to model the magnification bias, we assume an apparent magnitude threshold for detecting strongly lensed galaxy images, as well as a redshift and luminosity distribution for sources in the respective filter band. From the faintest arc detected in the Faure et al. (2008) sample, we deduce an apparent magnitude threshold of m_lim ∼ 26 [mag] in the F814W filter band (i.e., close to the COSMOS survey F814W magnitude limit; Leauthaud et al. 2007). We assume that the number density n_s(z_s, m_lim) of sources at redshift z_s brighter than the apparent magnitude m_min is simply given by a product

$$\begin{equation} n_{\rm s}(z_\mathrm{s},m_{\rm lim})=p_{\rm s}(z_\mathrm{s},m_{\rm lim}) n_{\rm s}(m_{\rm lim}) \end{equation} \tag{ 1 }$$

of a source redshift distribution

$$\begin{equation} p_{\rm s}(z_\mathrm{s},m_{\rm lim}) = \frac{\beta z_\mathrm{s}^2\exp \big[-\frac{z_\mathrm{s}}{z_0(m_{\rm lim})}\big]}{\Gamma (3/\beta) z_{0}^3(m_{\rm lim})} \end{equation} \tag{ 2 }$$

with parameters β = 3/2 and z₀(m_lim) = 0.13 m_lim − 2.2, and a cumulative source magnitude distribution

$$\begin{equation} n_{\rm s}(m_{\rm lim})=\!\!\int _{-\infty }^{m_{\rm lim}}\!\!\frac{n_{0}\,\mathrm{d}m}{\sqrt{10^{2a(m_1-m)}+10^{2b(m_1-m)}}} \end{equation} \tag{ 3 }$$

with parameters a = 0.30, b = 0.56, m₁ = 20, and n₀ = 3 × 10³ deg⁻² (Smail et al. 1994, Casertano et al. 2000, Leauthaud et al. 2007, Fedeli et al. 2008).¹⁶

Since most strong galaxy–galaxy lens candidates were selected due to arc-like features, we consider the optical depth τ for images with length-to-width ratio r > 7 for small circular sources (see Hilbert et al. 2008a, for details). Furthermore, we only consider strong-lensing regions in annuli between 02 and 5'' around a parent-sample galaxy, since closer images could not be distinguished from the center of the lens galaxy, and more distant images are outside the visually inspected regions (Faure et al. 2008).

We split the WL mass map of the COSMOS field in three regions according to their convergence values: a low-density region with κ < 0.4%, an intermediate-density region with 0.4% ⩽ κ < 1.4%, and the high-density region with κ ⩾ 1.4% (see Figure 1). The limits between the regions are set arbitrarily, and are only aimed at studying a possible transition in the strong lenses behavior of galaxies between regions of low and high convergence. The maximum value measured in the WL map is κ ∼   4.5%. Such a low maximum convergence is due to the finite spatial resolution (⩾2') of the WL mass map reconstruction, which washes out the stronger signal from the core of centrally concentrated matter structures. For the same reason, the convergence measured in the WL mass map cannot be accounted for the convergence that directly affects the strong lenses, as what is of matter for strong lensing is the fluctuation that is coherent over the angular scale of the lens (i.e., ∼1''). For each of the three defined convergence regions, Table 1 lists the density of elliptical galaxies and strong lenses, as well as the fraction of galaxies from the parent population that are strong lenses, best systems or single arc systems. The error bars correspond to simple Poisson statistics.

The results are summarized in Figure 3. We find a comparable density of strong lenses in the low mass density environment (all lenses: 35 ± 6 deg⁻², best systems: 12 ± 3 deg⁻²) and in the high mass density environment (all lenses: 45 ± 22 deg⁻², best systems: 22 ± 16 deg⁻²). If we take into account the fact that the density of elliptical galaxies is about twice as high in the densest regions of the field than in the low-density region, we also find a similar fraction of elliptical galaxies that are lensing galaxies in high-density regions (all lenses: 5.4 ± 2.7%, best systems: 2.7 ± 1.9%) and in the low density regions (all lenses: 7.9 ± 1.2%, best systems: 2.6 ± 0.7%).

Zoom In Zoom Out Reset image size

Surprisingly, we thus do not observe any particular correlation between the environment and the frequency of strong lenses. Strong lenses appear to be equally distributed throughout the different WL convergence regions of the field.

We now analyze our simulated data, to understand whether the absence of observed correlation between strong galaxy–galaxy lensing and WL convergence is compatible with the standard model of structure formation and galaxy evolution. The lensing simulations may also help to interpret the COSMOS observations.

The results of our lensing simulation are summarized in Table 2. We use the same threshold values for the convergence (i.e., κ = 0.004 and 0.014) in the smoothed convergence maps to divide our simulated lensing fields into low-, intermediate-, and high-convergence regions as for the observed field. The resulting area fraction covered by the low-(75% ± 9%), intermediate-(19% ± 7%), and high-convergence regions (7% ± 3%) in the simulated fields is very similar to the fractions measured in the COSMOS field (except that the fraction of the simulated survey area with high WL convergence is somewhat smaller than that in the COSMOS field).

The galaxy densities n of the simulated parent population in the low- and intermediate-convergence regions are slightly smaller than the densities observed in the COSMOS field. However, except for the low-convergence region, the observed densities are within the broad range spanned by the values calculated in individual ray-tracing fields. We are therefore confident that our lensing simulations reproduce the properties of the lensing observations reasonably well for a quantitative comparison of the correlation between strong galaxy–galaxy lensing and WL convergence.

As a simple approximation, we assume that each lens galaxy induces one strongly distorted image (a "single arc") of a source galaxy. Thus, the density n_L of strong lens galaxies is given by an integral

$$\begin{equation} n_\mathrm{L}=\int \tau (z_\mathrm{s},m_{\rm lim}) n_{\rm s}(z_\mathrm{s},m_{\rm lim}) \,dz_\mathrm{s}, \end{equation} \tag{ 4 }$$

of the optical depth τ(z_s, m_lim) and the source density n_s(z_s, m_lim) for given image-detection limit magnitude m_lim over all source redshifts z_s. In the following, we assume m_lim = 26.

We are primarily interested in the relative number of expected strong galaxy–galaxy lenses in the different convergence regimes. Therefore, we first discuss the optical depth and take the source redshift distribution and density into account later. The optical depth τ(z_s) in the low, intermediate and high WL convergence regions is shown in Figure 4(a) for source redshifts z_s = 1, 2, and 3. There is a clear trend of an increasing strong-lensing optical depth with increasing WL convergence for all considered source redshifts. On average, the optical depth in the high-convergence region is 9–15 times larger (depending on the source redshift considered) than in the low-convergence region. This strong increase is in contrast to the weak increase of the density of strong-lensing candidates seen in the COSMOS field.

Zoom In Zoom Out Reset image size

The increased strong-lensing optical depth τ in regions of higher WL convergence is partly due to the increased galaxy density n of the parent population. However, an increase in the strong-lensing probability remains even if the effect of the larger galaxy density is taken out. This can be seen in Figure 4(b), where the average strong-lensing cross section σ = τ/n of galaxies in the parent population is shown. The average cross section in high-convergence regions is three to seven times larger than in low-convergence regions. Thus, the expected fraction of strong lenses within the parent population in regions with high convergence is roughly three to seven times larger on average than in regions of low convergence. Among individual ray-tracing fields, the ratio of cross sections in the low- and high-convergence region ranges from 2 to 10. Hence, the simulations suggest that one should expect the fraction of strong lenses among the parent population to be at least 50% higher in the high-convergence regions than in the low-convergence regions (and even much larger ratios of 3–7 are expected on average).

The expected number of strong lenses in the different WL convergence regions is given by the product N_L = An_L of the area A and lens density n_L in the considered region. The expected fraction of galaxies in the parent population that are strong lenses is given by the ratio n_L/n_gal. The resulting strong lens numbers, assuming a total survey area of 1.8 deg², and the strong lens fraction are shown in Table 2. For the high-convergence region, the predicted number and fraction of strong lenses are consistent with the values observed in the COSMOS field. However, the simulation predicts far fewer galaxy–galaxy lenses in the intermediate- and low-convergence region than there are lens candidates in the COSMOS field.

The reasons for the different discrepancies between observations and simulations are discussed in Section 4.3.

From Tables 1 and 2, we see that the surface covered by the three convergence regions as well as the distribution of parent population galaxies is comparable in the simulated fields and in the observations. However, the increasing fraction of strong lenses in regions of increasing convergence expected from the simulations is not matched by the constant fraction of observed strong lenses in the COSMOS field.

The discrepancy between the simulations and observations may be due to a number of reasons. Some of the possible reasons related to the observations are as follows.

• 1.  
  The sample of strong lenses might be incomplete in a way that biases the results.
• 2.  
  A significant fraction of the strong-lensing candidates are not strong galaxy–galaxy lenses.
• 3.  
  The measured WL mass map does not represent the actual mass distribution in the COSMOS field very accurately.
• 4.  
  Our arbitrary cuts in WL convergence might not encompass important transitions between low- and high-density regimes.
• 5.  
  The COSMOS field might be a peculiar case due to cosmic variance.
• 6.  
  The number of strong lenses might be too low to reach the necessary statistical precision, and therefore we are missing the correlation.
• 7.  
  There may be actually no correlation between the strong lenses and the environment (at least for the used galaxy sample selection criteria).

Explanation (1) cannot yet be completely ruled out. Figure 5 (upper plot) shows the distribution of arc radii in our strong lenses. Note the conspicuous absence of observed systems with arc radii around ∼1''  or greater than 3.5 A much smoother distribution is seen in the simulations: we would have expected to find systems with arc radii up to 5'', which is still well within the 10'' × 10'' visually inspected regions. The gap at 1'' corresponds suspiciously to the mean seeing size of the ground based imaging. It is possible that genuine strong lenses were excluded during the transition from monochromatic, space-based images used to detect potential candidates, to ground-based, color images used to verify them. However, a second independent search (Jackson 2008) did not discover any new strong lens systems amongst the Faure et al. (2008) elliptical galaxies, so we are confident that our sample is close to complete. Confirming the statistical significance of the apparent lack of lenses with arc radii above 35  will require a larger sample. Overall, if the properties of strong lenses were strongly influenced by the environment of the main lensing potential, we would have expected a much smoother transition between systems with a galaxy-like (r_arc≲ 4'') and cluster-like (r_arc≳10'') mass deflectors, as seen in the simulations (Figure 5, bottom plot).

Zoom In Zoom Out Reset image size

Explanation (2) is currently being addressed with follow-up observations (C. Faure et al. 2009, in preparation). However, explaining this discrepancy would require a strong bias in the selection process as a function of environment—for example, favoring the spurious detection of false positives in low density environments. Explanation (3) is also subject to ongoing methodological improvements. The insignificant reconstructed B-modes (Massey et al. 2007) demonstrate the overall reliability of the WL map. However, isolated imperfections in the analysis (e.g., imperfect PSF modeling in one or two HST/ACS pointings) can go undetected when averaged over the entire field. Such artifacts (as well as noise) can manifest as spurious convergence peaks, which would dilute the fraction of strong lenses measured in high density environments. It is difficult for systematics to remove real lensing convergence and thus produce the opposite effect; the much larger area of low density than high density also means that noise acts in the same sense. This bias doubtless contributes to the observed results, but is not at a sufficient level to explain them alone. Explanation (4) can be excluded if the lensing simulations are not terribly unrealistic, since they show a clear change in the strong lens fraction. Explanation (5) can also be excluded, the measured scatter between the 20 simulated fields showing that cosmic variance alone is insufficient to explain the discrepancy. Future observations of larger fields will also test this. They will also be required to test explanations (6) and (7), which cannot be ruled out with existing data.

Apart from possible reasons related to the observations, the discrepancy may be caused by the simple assumptions and approximations used in the lensing simulations. These include the assumption of circularly symmetric projected stellar-mass profiles of the lens galaxies (i.e., lensing cross section), the neglect of the response of their dark-matter halo to baryon cooling (Gnedin et al. 2004), the assumption of point-like sources (instead of extended elliptical sources), and a simple source magnitude distribution (which neglects any evolution with redshift). In principle, we also cannot exclude the possibility that the discrepancy is due to shortcomings of the underlying dark-matter structure formation model. Moreover, the model for the strong lens selection function of the COSMOS sample that we use in the simulations might be unrealistic. First, the simulated parent population relies on semianalytic models to predict the galaxy properties; the number of small galaxies in high-density regions could have been overestimated. Then the detection criteria for strongly distorted images may be too simplistic. For example, the uniform magnitude detection threshold does not depend on the distance to the lens galaxy (apart from a cutoff for images closer than 0.2''), or on the lens galaxy's apparent size and brightness. In the real data, faint lensed images at small projected distances from a large and bright lensing galaxy may have been missed. In addition, there is considerable scatter in the average cross sections measured in different simulated fields (see Table 2). A different value for the normalization σ₈ might change the total number of strong lenses, but not their distribution among the different density region.

In summary, there remain many questions (which we intend to address in future work) before we can firmly conclude whether the observed lack of correlation between strong lenses and their environment is real, or whether predictions from current models are right. In particular, a sample of strong lenses identified via an automated detection algorithm would have a better constrained selection function. This would allow further discussion of the potential inadequacies of the simulations.

The semianalytic galaxy catalog (De Lucia & Blaizot 2007) used for the lensing simulations allow us to investigate the relation of various galaxy properties to observed trends in the strong lensing probabilities. Figure 6(a) shows the distribution n⁻¹dn/dlog M_* of the stellar mass M_* in galaxies of the parent population, for the three convergence regions. The distributions are very similar apart from a small shift towards higher stellar masses for regions with larger WL convergence. This shift, however, implies a higher number of galaxies with stellar masses M_* > 10¹¹h⁻¹ M_☉. Although a small fraction of the entire parent populations (∼4% of the parent population in low-convergence regions, ∼9% in high-convergence regions) these galaxies contribute significantly to the average lensing cross section of galaxies, as seen in Figure 6(b). These massive galaxies not only have higher stellar masses (and thus higher luminosities), but are often associated with significantly more massive dark-matter halos associated with them, resulting in very large total-to-stellar-mass ratios M_total/M_stellar > 100 (and large mass-to-light ratios), as can be seen in Figure 7. There are only a few of them, so they do not significantly affect the luminosity distribution of the parent population, but they dominate the lensing statistics.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

This result of the simulation is not in contradiction with the observations in the COSMOS field: the brightest galaxies do not dominate the population of galaxies in the high-convergence region (Figure 8). To verify the existence of these high total-to-stellar mass galaxies, galaxy–galaxy WL analysis of the COSMOS galaxies will be necessary (such as Mandelbaum et al. 2005), as well as a detail mass model of the strong lenses of the sample. Nevertheless, the low correlation between the strong lenses and the mass map measured in the COSMOS field suggests that there are fewer of theses massive galaxies in the observed field rather than in the simulated fields. This result, if verified, could also pinpoint a problem in the semianalytic model used to represent the galaxy luminosity function in the simulations.

Zoom In Zoom Out Reset image size

The higher cross section in the high-convergence regions predicted by the simulations is not only due to the higher number of massive galaxies. In Figure 9, the (average) strong-lensing cross section σ(M_*) = (dτ/dM_*)/(dn/dM_*) as a function of stellar mass M_* is shown for the different WL convergence regions. Even if we compare galaxies at fixed stellar mass M_*, the cross section σ(M_*) is consistently larger in the high-convergence regions than in the low-convergence regions. Using the simulation data, we can identify at least one cause: For given stellar mass, galaxies in high-convergence regions tend to have larger dark-matter halos than galaxies in low-convergence regions. However, other possible causes, e.g., additional matter from group/cluster halos or other structures along the line of sight, cannot be excluded so far (and should be investigated in more detail in future work).

Zoom In Zoom Out Reset image size

Here we want to verify whether the distribution of strong galaxy–galaxy lenses arc radii in the COSMOS field is a function of the environmental convergence. For that purpose, we split the lens sample into two (similarly sized) subsamples according to their arc radii: r^small_arc if r_arc <  15  and r^large_arc if r_arc ⩾ 15. The largest angular separation in our sample is r_arc = 354. The number and density of strong lenses as a function of environment are listed for the two subsamples in Table 1, and their densities are plotted in Figure 10. We find that the densities of r^small_arc and r^large_arc systems are similar in low- and intermediate-convergence regions. However, only r^large_arc systems are found in high-convergence environments (48 ± 24 deg⁻²) and their density is up to four times higher than in low-convergence regions (15 ± 3 deg⁻²).

Zoom In Zoom Out Reset image size

To estimate the density of strong lens systems with small and large arc radii, we separately calculate the strong-lensing optical depth for systems with arc radii r_arc < 15 and for systems with r_arc ⩾ 15. The resulting optical depths are shown in Figure 11 and given in Table 2 for different source redshifts. The optical depth for r^large_arc systems increases very strongly with increasing WL convergence (∼20 times higher in high-convergence regions). This matches the observed data, and is due to a combination of external shear/convergence plus an increased fraction of high mass-to-light ratio galaxies in dense environments.

Zoom In Zoom Out Reset image size

The optical depth for r^small_arc systems also increases (approximately 5 times higher in high-convergence regions) in the simulations. An equivalent increase in the number of observed r^small_arc systems is not seen in the COSMOS data. However, as already mentioned in Section 4.3, faint lensed images at small projected distances from a large and bright lens galaxy may not be visible in real images, thereby leading to an underestimate of the number of small arc radius systems.

High-density environments enhance the production of strong lens systems with large image separations with respect to low-density environments. This effect is observed both in the simulations and in the COSMOS field. The large image separations are due to the presence of high mass-to-light ratio galaxies in the dense regions and the strong external shear/convergence generated by the lens-galaxy environment. The latter can also be induced by massive structures along the same line of sight.

As discussed in Section 4.4, we do not find substantial evidence from the galaxy luminosity function (Figure 8) that the brightest galaxies (also likely to be the most massive) are preferentially found in high density environments. Nor does the low (constant) fraction of strong lenses in high-convergence regions of the COSMOS field hint at a excess of massive galaxies. However, the presence of only strong lenses with large image separations implies that this is indeed the case; or simply that it is only external shear that increases the image separation, not the presence of massive galaxies.

Seven lensing galaxies (two "best system" and five "single arc" systems) are members of an X-ray galaxy cluster or group (lying inside a projected radius r₅₀₀ and within redshift Δz = ±0.05). These represent 10 ± 4% of the strong lenses (10 ± 7% of the "best systems").

Two lensing galaxies (one "best system" and one "single arc;" both r^large_arc systems) are within only 0.02r₅₀₀ of the cluster core. The other five lensing galaxies (of which two are r^large_arc systems) lie at projected distances >0.3r₅₀₀ from the cluster center.

We find that 14 systems (six "best systems" and eight "single arc" systems) have an impact parameter with a galaxy cluster smaller than r₅₀₀, but have a different redshift than the cluster. By adding the seven systems embedded in a galaxy cluster, we find that 31% ± 7% of strong lenses (and 42% ± 15% of the best systems) are within or aligned with a galaxy cluster, with an impact parameter inside r₅₀₀. In comparison, Williams et al. (2006), in a study of the environment of 12 strongly lensed quasars with similar arc radii to our sample, found that 50% ± 20% to 67% ± 23% of strong lenses are aligned with a galaxy cluster/group. Within uncertainties, we are thus in rough agreement with their results.

If we assume that galaxy clusters in the COSMOS field are circular, do not overlap and have a projected size of πr²₅₀₀, they cover a projected area of 0.190 deg². We find a density of 110 ± 24 deg⁻² strong lenses inside this area, and a density of 28 ± 4 deg⁻² outside it (see Figure 12). The density of strong lenses is thus approximately four times higher in the projected region covered by galaxy clusters. However, the density of galaxies from the parent population (Figure 12) is also approximately four times higher in this region. The fraction of elliptical galaxies that are strong lenses is therefore approximately constant, regardless of whether the lensing galaxy is enclosed within (6% ± 1%) a projected radius r₅₀₀ from the center of a galaxy cluster or groups or not (7% ± 1%). This recovers the result of Section 4, that the population of strong lenses follows the distribution of bright elliptical galaxies, and that additional matter structures along the line of sight do not significantly increase the strong lensing cross section.

Zoom In Zoom Out Reset image size

The density of r^small_arc and r^large_arc systems is similar in the regions away from lines of sight to clusters (14 ± 3 deg⁻² and 11 ± 3 deg⁻², respectively). However, the density of r^large_arc systems is slightly higher (63 ± 18 deg⁻² for r^large_arc and 47 ± 16 deg⁻² for r^small_arc systems) when the projected distance of a lensing galaxy to a galaxy cluster center is within r₅₀₀. As in Section 5, two explanations are possible: additional matter along the line of sight increases the angular separation between the images, and/or more massive galaxies are found in clusters, which boost the lensing cross section.

Figure 13 shows the distribution of elliptical galaxies in the parent population, according to their brightness and as a function of their location. We find a similar distribution of galaxies inside and outside the region covered by clusters. An excess of bright, massive galaxies is therefore not a plausible explanation for the excessive fraction of r^large_arc in the region covered by the clusters, unless the galaxies in this region have an intrinsically higher mass-to-light ratio than those outside.

Zoom In Zoom Out Reset image size