Impacts of Source Properties on Strong Lensing by Rich Galaxy Clusters
aa r X i v : . [ a s t r o - ph . C O ] O c t Draft version November 12, 2018
Preprint typeset using L A TEX style emulateapj v. 08/22/09
IMPACTS OF SOURCE PROPERTIES ON STRONG LENSING BY RICH GALAXY CLUSTERS
G. J. Gao , , Y. P. Jing , S. Mao , G. L. Li , , X. Kong Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80Nandan Road, Shanghai, 200030, China Graduate School of the Chinese Academy of Sciences, Beijing, 100039, China Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK Argelander-Institut f¨ur Astronomie, Auf dem H¨ugel 71, Bonn D-53121, Germany and Key Laboratory for Research in Galaxies and Cosmology, University of Science and Technology of China,Chinese Academy of Sciences, Hefei, 230026 China
Draft version November 12, 2018
ABSTRACTWe use a high-resolution N -body simulation to investigate the influence of background galaxyproperties, including redshift, size, shape and clustering, on the efficiency of forming giant arcs bygravitational lensing of rich galaxy clusters. Two large sets of ray-tracing simulations are carried outfor 10 massive clusters at two redshifts, i.e. z l ∼ . .
3. The virial mass ( M vir ) of the simulatedlens clusters at z ∼ . . × h − M ⊙ to 1 . × h − M ⊙ . The information ofbackground galaxies brighter than 25 magnitude in the I -band is taken from Cosmological EvolutionSurvey (COSMOS) imaging data. Around 1 . × strong lensing realizations with these images asbackground galaxies have been performed for each set. We find that the efficiency for forming giantarcs for z l = 0 . z l = 1 . z l = 1 .
5, respectively. We find that the efficiencyof producing giant arcs by rich clusters is weakly dependent on the source size and clustering. Ourprincipal finding is that a small proportion ( ∼ /
3) of galaxies with elongated shapes (e.g. ellipticity ǫ = 1 − b/a > .
5) can boost the number of giant arcs substantially. Compared with recent studieswhere a uniform ellipticity distribution from 0 to 0.5 is used for the sources, the adoption of directlyobserved shape distribution increases the number of giant arcs by a factor of ∼
2. Our results indicatethat it is necessary to account for source information and survey parameters (such as point-spread-function, seeing) to make correct predictions of giant arcs and further to constrain the cosmologicalparameters.
Subject headings: gravitational lensing – galaxies: clusters: general – dark matter – methods: dataanalysis INTRODUCTION
Giant arcs are spectacular examples of strong gravitational lensing found in rich galaxy clusters. Backgroundgalaxies are stretched into long, thin arcs by the intense foreground gravitational field. Hundreds of giant arcs havebeen found in both optically-selected and X-ray selected clusters (e.g. Luppino et al. 1999; Zaritsky & Gonzalez 2003;Gladders et al. 2003; Sand et al. 2005).Massive clusters are efficient producer of giant arcs and the number of giant arcs is a good indicator of the abun-dance of massive clusters. The halo mass function is very sensitive to the cosmological parameters, especially at themassive end. Moreover, the internal structures of massive clusters, such as substructures and ellipticity, also dependon the cosmological parameters. They all affect the lensing probability (optical depth, e.g., Meneghetti et al. 2003a,b;Wambsganss et al. 2004; Torri et al. 2004; Dalal et al. 2004; Li et al. 2005; Oguri et al. 2003, 2008; Hilbert et al. 2007;Puchwein et al. 2005, 2009) in various degree. Therefore, the observation of giant arcs is a useful probe of the cosmo-logical model, in particular the matter power-spectrum normalization, σ (Li et al. 2006b), the matter density, Ω m , ,and to a less extent the cosmological constant, Ω Λ , .However, in order to use the observations of giant arcs to constrain the cosmological models, one has to thoroughlyunderstand how the lensing probability is affected by the distribution of background source galaxies, as well as theintrinsic properties of lens population. It has been shown that the lensing probability increases significantly withthe increase of the source redshift, thus it is necessary to quantify the redshift distribution of source galaxies inlensing studies (Wambsganss et al. 2004). The lensing efficiency of massive clusters does not depend on the source sizesignificantly (Li et al. 2005), although not all the real source information has been used, in particular their shapes.Horesh et al. (2005) first adopted realistic galaxy images with known photometric redshifts from the Hubble DeepField (HDF) as background sources. They selected clusters from a cosmological N -body simulation in a ∼ h − Mpcbox at z l ∼ .
2. The mass of the simulated lens clusters (0 . . × h − M ⊙ ) is similar to that of 10 X-ray–selected Electronic address: [email protected]
Gao, Jing, Mao, Li, & Kongclusters (Smith et al. 2005), the (mean) mass range of which is around 6 . × h − M ⊙ ≤ M ≤ . × h − M ⊙ .Note that the mass is calculated based on the X-ray luminosity by using the L x - M relation (Reiprich & B¨ohringer2002). They argued that the probability of producing giant arcs is ∼ m , = 0 . , Ω Λ , = 0 . σ = 0 .
9) is consistent with the one observedin the massive clusters at redshift 0 . < z c < . §
2, we discuss COSMOS and background source population. In §
3, we discussthe simulation we use and our lensing methodology. In §
4, we present our main results and compare with recentstudies. We finish in § COSMOS AND BACKGROUND SOURCE POPULATION
COSMOS is a deep survey covering a 2 square degrees of equatorial field containing over 2 million galaxies. It is anideal sample of background galaxies for our lensing simulations due to the excellent resolution and large survey area (toreduce the cosmic variance). The image is centered on RA=10:00:28.6 and DEC=+02:12:21.0 from the Hubble SpaceTelescope Advanced Camera for Surveys (HST/ACS) (Koekemoer et al. 2007). Its calibration is relatively reliable dueto the absence of atmospheric absorption. The I -band image data we use is the second public release of COSMOSobservations. The camera has two 2048 × . ′′ . All the drizzled image datahave been flux calibrated and astrometrically registered. The whole image in the I -band has been cut into 575 edge-overlapped tiles of 4960 × ∼
10 degrees to thetile edge and embedded at the center of a rectangular box of 5600 × σ which is about 25mag / arcsec in surface brightness. Finally,a total of ∼ × galaxies are identified. The output of SExtractor contains the relative positions of sources inthe tile and positions in the celestial equator coordinate system. The relative position is useful for identifying whichgalaxies are strongly lensed in the lensing process (see § I -band AB magnitudeof 25 to obtain the redshift information. Only the galaxies successfully matched, thus having correct photometricredshifts, are added in our lensing simulations. Using the COSMOS mask catalog, we exclude all star-like sources,galaxies close to star-like sources, and those possibly contaminated by spikes due to the overflow of CCD pixels. Inparticular, we use the same shape analysis program as in search for giant-arcs (see § ≥
10 in all the tiles and exclude them to avoid “giant-arc like” artifactsources (e.g. imperfectly masked spikes due to CCD pixel overflow and asteroid trails) or real sources which couldhave been lensed by foreground objects in COSMOS field.All the cleaned COSMOS tiles are added behind the lens clusters as source patches. The sky coverage of stronglensing simulation is the same as the original COSMOS field. Due to the magnitude limit of the photometric catalog,the density of matched galaxies with photometric redshift is about 500 per tile of 5600 × ∼ . × . × − in the CCD imaging area. Tiles of 5600 × ∼
23 galaxies arcmin − in our lensing analysis.The scatter plots of source size (in pixels) vs. redshift, the ellipticity ( ǫ ) vs. redshift and ellipticity vs. source sizewith the median (as red filled circle) and the 68% value range (as red bars) for 20 randomly selected source tiles areshown in Figs. 1-3. The ellipticity ǫ is quantified by 1 − b/a , where b and a are semi-minor and semi-major axes of agalaxy image. The axes are obtained by fitting the image with an ellipse using the same method as that for quantifyingthe shape of giant arcs (see § ∼ z = 3 .
0. However, the ellipticity of galaxies is almost independent of redshift (seeFig. 2). The source size and ellipticity are somewhat correlated (especially for the sources of the effective diametermpacts of Source Properties on Strong Cluster Lensing 3smaller than ∼ . ′′ , as can be seen in Fig. 3 – larger sources appear to be somewhat more elliptical.The photometric redshift distribution of COSMOS background galaxies is plotted (as the black solid line) in Fig. 4for all the galaxies in 575 tiles of COSMOS field. As can be seen the redshift distribution peaks around z = 0 .
8, butextends out to z = 3 . p ( z s ) = βz Γ(3 /β ) z s2 exp h − (cid:18) z s z (cid:19) β i , (1)where z ∼ . β ∼ .
2; the best-fit line is indicated as the black dashed line in Fig. 4.With the photometric redshifts, we can construct an approximately three-dimensional distribution of backgroundgalaxies in order to study the effect of source redshift distribution on strong lensing efficiency. A series of source planesperpendicular to the line of sight is set between redshift z l ∼ . ∼ .
3) to z s = 3 .
0. The redshift separation betweentwo adjacent source planes δz is set to δz/z = 0 . . < z ≤ . δz/z = 0 .
05 for 0 . < z ≤ .
5. The galaxiesare projected onto the planes according to which redshift bin they fall in. In each source plane, we mark all the galaxypixels, which are used to identify the lensed sources in subsequent ray-tracing. There are 48 source planes in total forthe ray-tracing program (see § − b/a distributionhas a median of 0.4, which is similar to that presented in Ferguson et al. (2004). Compared to the studies of giant arcsusing a uniform distribution of the ellipticity with 0 ≤ − b/a ≤ . § D eff defined as the diameter of a circular-shaped galaxywith the same number of the image pixels. The size distribution of galaxies in these tiles is shown (as the blacksolid line) in Fig. 6. The size distribution of galaxies has a peak at D eff ∼ . ′′ , but ranges from effective diameter D eff < . ′′ to D eff > ′′ . The median diameter of background galaxies is about 0 . ′′ , smaller than that assumed tobe D eff = 1 ′′ (Li et al. 2005) since a relatively deeper galaxy survey is adopted in this work. NUMERICAL SIMULATION AND LENSING METHODOLOGY
Simulated Clusters
The lens clusters are selected from a N -body simulation generated with a upgraded version of vectorized-parallel P Mcode of Jing & Suto (2002) (Jing, Suto & Mo 2007). The underlying cosmological model is a ΛCDM (Ω m , = 0 . Λ , = 0 . n = 1) model. The simulation uses 1024 dark matter particles in a box with a side length of 600 h − Mpc(comoving). The mass resolution is 1 . × h − M ⊙ , thus massive clusters are reasonably resolved (with more than4 × particles in our samples). In this simulation (refer to Jing, Suto & Mo (2007) for more details), the Hubbleconstant h in unit of 100 km s − Mpc − is taken to be 0.71, while the amplitude of the linear density power spectrum, σ , is 0.85, which is obtained from CMBFAST (Seljak & Zaldarriaga 1996) directly. Particle pairwise interactions aresoftened on scales smaller than 30 h − kpc. The clusters are identified by the Friends-Of-Friends (FOF) method witha linking length of 0.2 times the mean particle separation, and the cluster mass M vir is defined as the mass enclosedwithin the virial radius according to the spherical collapse model (Kitayama & Suto 1996; Bryan & Norman 1998;Jing & Suto 2002).We primarily use a sample of 10 massive clusters at redshift z ∼ . z ∼ . . × h − M ⊙ to 1 . × h − M ⊙ , roughly consistent with theobserved range of mass (6 . × ≤ M ≤ . × h − M ⊙ of clusters at redshift 0 . < z c < . We shallalso use 10 most massive clusters at z ∼ . Ray-tracing and Mapping
We use the thin lens plane approximation for the lensing calculation; the line of sight contributions may be important(Wambsganss et al. 2004; Puchwein et al. 2009). The surface density is calculated on a mesh of 1024 × κ ) map is obtained by dividing the surface density by the critical value,Σ cr = c πG D s D l D ls , (2)where D s , D l , and D ls are the angular diameter distances between the source and the observer, the lens and theobserver, and the source and the lens. The lensing potential on the grids of the coarse mesh, φ , is calculated by the Notice that M is defined as the mass enclosed within in r , the radius within which the average density is equal to 200 times thecritical cosmological density. The difference between M and M vir are within 10% for 8 of 10 clusters. It is within 30% for the other twoclusters. Gao, Jing, Mao, Li, & Kong
Fig. 1.— : The scatter plot of size against redshift for galaxies randomly selected from 20 COSMOS image tiles witha bin width of ∆ z = 0 .
3. Each tile has 4096 × ∇ φ = 2 κ .The shear ( γ ) is then obtained by the second-order derivatives of the potential, and the (signed) magnification is givenby µ − = (1 − κ ) − γ .To better determine the geometry of an image and to perform an efficient lensing simulation, we refine the cell size ofthe central high magnification area with | µ | ≥ . . ′′ , identical to the intrinsic observationalresolution in the source planes. High resolution is necessary for resolving the lensed image well, especially in thedirection of minor axis, which is much more sensitive to the resolution. The lensing potential on the refined mesh isobtained by cubic spline interpolations of the neighboring 14 ×
14 coarse grids in the 1024 × ~α = ∇ φ . The lens mapping from the lens plane to each sourceplane at different redshifts can be constructed using the lens equation (i.e. ray-tracing), ~y = ~x − ~α , where ~y , ~x aredimensionless source position in source plane and image position in lens plane. The ray-tracing procedure is repeatedfor each of the ∼
50 source planes, as we discussed in § Fig. 2.— : The scatter plot of the ellipticity, ǫ = 1 − b/a , against redshift for galaxies randomly selected from 20 tileswith a bin width of ∆ z = 0 .
3. The median (red filled circle) and the 68% ellipticity range (red bar) are indicated in10 bins of redshift. As can be seen, the source ellipticity distribution is randomly distributed at all the redshifts. Themean value (blue filled circle) of ellipticity in each redshift bin is also plotted for comparison.corresponding point in the source planes falling into a galaxy. If so, we identify this grid point in the image planeas a lensed image pixel of that source galaxy; if not, it is a trivial grid. All the highly magnified lensed image pixels(e.g., | µ | ≥ .
5) can be located. Finally, we can obtain the whole image(s) of a background galaxy by collecting all theimage pixels of the same source using Friends-of-Friends (FOF) method.For source slices at different redshifts, the high-magnification areas will be mapped into areas of different sizes inthe sky. We choose a rectangular conjunctive angular area inclusive of every caustic area and put this angular boxrandomly in the source tile to select different background galaxies for the ray-tracing simulation. The box size isusually on the level of a few square arcmins (depending on projection and also mass profile). Since each added sourcetile is about 4 . ′ × . ′ , around 10 random lensing realizations are needed to cover a tile effectively. To avoid boundaryeffects, the source tile is periodically expanded in both dimensions which are then randomly sampled.The same process is carried out for all the three orthogonal projections of the massive clusters at the expectedredshifts. 10 ray-tracing simulations are carried out with all source galaxy samples in the COSMOS field for eachof 3 orthogonal projections. The lensing images found are stored and subject to more detailed analysis to find theirlength-to-width (L/W) ratios, sizes, the redshifts. Of particular importance is the L/W ratio. We determine thisquantity following Bartelmann et al. (1998) and Li et al. (2005). Briefly, we first identify the center of the image pixelsfor a given source ( ~x c ), and then locate the point ~x that is the furthest from the centre, and finally the point ~x thatis the furthest from ~x . We then fit a circular arc that passes through these three points ( ~x , ~x , ~x c ). The half length Gao, Jing, Mao, Li, & Kong Fig. 3.— : The scatter plot of the ellipticity, ǫ = 1 − b/a , against size of galaxies in 20 randomly selected tiles witha bin width of ∆ D eff = 0 .
3. The median (red filled circle) and the 68% ellipticity range (red bars) are indicated in10 bins of effective diameter D eff . The galaxies are more elliptical as the size increases for sources with the effectivediameter smaller than ∼ . ′′ , while the median ellipticity keeps around 0 . a , and the length of the semi-minor axis is taken tobe b = S image / ( πa ), where S image is the area covered by the image. The length-to-width ratio is then simply a/b . Asmentioned before, the same procedure is used to determine the ellipticity, 1 − b/a , for unlensed galaxies. Simulating test images
We carried out a test by putting a source around the tangential caustic curve (as the green dotted line) of a massivesimulated cluster. The simulated images (shown in red) are illustrated in Fig. 7 together with the source (shown inblack) positions. As expected, it produces one, two and three images when the mocked source is outside, on and insidethe caustic curve. Fig. 8 shows a simulated image of a massive lens cluster of a realization based on a real source tile.Clearly, the sources (shown in black) have been systematically stretched around the cluster center. In particular, thereis one giant arc at redshift z = 2 .
52 (labeled as “A1”) with
L/W ∼
22 among the lensed images produced from source“a”, which has an additional image labeled as “A2”. Notice that the positions of the images and background galaxiesin different planes are transformed into the same coordinate system centered at the cluster centre. RESULTS
Many recent studies have shown that the efficiency of forming giant arcs depends on the background source popu-lation, in particular, their sizes and redshifts (e.g., Wambsganss et al. 2004; Li et al. 2005; Horesh et al. 2005). Sincempacts of Source Properties on Strong Cluster Lensing 7
Fig. 4.— : Comparison plot of redshift probability distributions between the subpopulation of original galaxies whichhave been strongly lensed to be arcs of L/W ≥
10 ( z l = 0 .
2) and the parent population of COSMOS galaxies. Thenumber fractions per redshift bin of width ∆ z = 0 . Fig. 5.— : Comparison plot of shape probability distributions between the subpopulation of original galaxies whichhave been strongly lensed to be arcs of L/W ≥
10 ( z l = 0 .
2) and the parent population of COSMOS galaxies. Thenumber fractions per ellipticity bin of width ∆ ǫ = 0 . Lensing efficiency of COSMOS sources
Using the empirical COSMOS background source information, and the method outlined in the last section, weimplement a mock lensing survey for each projection of the massive clusters at redshift z l = 0 . z l = 0 . L/W ratio of Li et al. (2005). In total, 12816giant arcs of
L/W ≥
10 are found out of 172500 cases at z l = 0 .
2, while 9704 giant arcs for clusters at z l = 0 . z l = 0 . z l = 0 .
3. We illustrate the lens redshift ( z l ) dependence and the fluctuation of number counting in Fig. 9 for 20randomly selected tiles. All the giant arcs ( L/W ≥
10) produced by the 10 massive lens clusters from the galaxies ineach tile are taken into account. The lensing probability appears to decrease from z l = 0 . z l = 0 . Fig. 6.— : Comparison plot of size probability distributions between the subpopulation of original galaxies which havebeen strongly lensed to be arcs of L/W ≥
10 ( z l = 0 .
2) and parent population of COSMOS galaxies. The numberfractions per effective diameter bin of width ∆ D eff = 0 .
25 arcsec are plotted. There is little difference between thetwo distributions, which indicates they share the almost same population. The size distribution of the fiducial controlsample consisting of 100 randomly selected tiles (used in Table 1) is plotted in magenta. There is no big differencefrom the mean in most D eff range, except the places around the peak. For comparison, the distribution used in Li et al.(2005) is also indicated. Redshift influence
In recent works (Bartelmann et al. 1998; Li et al. 2005), the background sources are taken to be at the same redshift,e.g., z = 1, to estimate the theoretical overall efficiency of forming giant arcs in our universe. As shown in Fig. 4 thebackground sources in fact have a rather broad distribution, which we take into account by binning them into ∼ z s = 1, we find that in this case the numberof giant arcs ( L/W ≥
10) increases by a factor of ∼ .
05 compared with Case 1 in Table 1, since the equivalentsource redshift plane, in which we put all the source galaxies to generate similar number of giant arcs, for the 100randomly selected COSMOS tiles (i.e., “fiducial control sample” tiles which are considered as the “seeds” to generatemock samples for cases in Table 1) is about 0.9 which is slightly lower than z s = 1 .
0. Notice that the shape, sizedistributions and surface density follow the original distributions of COSMOS. Therefore, z s plane is coincidently agood approximation to the broad redshift distribution. However, according to Wambsganss et al. (2004) and Li et al.(2005), the lensing probability becomes larger when the sources are put at a higher redshift, which is also seen in theredshift distribution of lensed sources in Fig. 4 where the lensed galaxies, stretched to become giant arcs of L/W > (a) outside (b) across (c) inside
Fig. 7.— : Simulated images for a mocked circular galaxy located outside, on and inside the tangential caustic ofa merging galaxy cluster, which is selected to illustrate for easy identification. The black dots indicate the sources,while the red ones are the lensed images. The green and blue dotted lines indicate the caustics and critical curvesrespectively. As can be seen, the lens produces single, double and triple strongly lensed images.
TABLE 1 : Results of 7 different sets of simulations. “z”, “s”and “e”indicate the redshift , size and ellipticity ofthe background sources, while “1”and “0”represent following COSMOS and mocked. For example, “e0”implies theellipticity (1-b/a) is randomly chosen in the range between 0 to 0.5. We run 10 lensing simulations in each projectionof the 5 (middle) massive clusters at z l = 0 . z l = 0 . Case Name z s e N1 z1s1e1 COSMOS COSMOS COSMOS 12042 z0s1e1 z=1.0 COSMOS COSMOS 12643 z0s0e0 z=1.0 D eff = 1 . ′′ RANDOM 3214 z1s0e1 COSMOS D eff = 1 . ′′ COSMOS 8525 z1s0e0 COSMOS D eff = 1 . ′′ RANDOM 3696 z1s1e0 COSMOS COSMOS RANDOM 4937 z0s1e1 z=1.5 COSMOS COSMOS 1935
TABLE 2 : Ray-tracing simulations are carried out for the massive clusters selected from two snapshots at redshift ∼ . ∼ .
3. The cosmological model is a ΛCDM model with Ω m , = 0 . Λ , = 0 . σ = 0 . N R ) for each projection and each source tile. The whole simulationcovers the full sample of 575 source tiles (marked as N T ). Set z l M vir (10 h − M ⊙ ) N C N P N R N T redshift distribution in Fedeli et al. (2008) is also shown in Fig. 4. If we adopt a source redshift distribution following adeeper galaxy survey like in Fedeli et al. (2008) (keeping other properties unchanged), the giant-arc number is increasedby a factor ∼
2, compared with the number of giant arcs when the COSMOS redshift distribution is used (Case 1 inTable 1).However, the equivalent source redshift plane changes with source shape distributions. For example, considering thecomparison cases, Cases 3 and 5 in Table 1, we find a higher equivalent source redshift (e.g., z s ∼ .
6) for galaxies ofsimulated shapes as in Case 3 than those of the intrinsic shape distribution in Case 1, which share the same intrinsicredshift distribution. This can also be explained by Fig. 10. Galaxies of simulated ellipticities at higher redshift havea relatively higher weight than those of intrinsic ellipticities in producing giant arcs. It causes the equivalent sourceredshift for Case 5 to be larger than the mocked redshift in Case 3, e.g. z s = 1 .
0, thus a higher lensing efficiency.Also, the redshift of equivalent source plane may shift to a higher redshift for a deeper survey. In this case, the lensingprobability may increase substantially and a single source plane approximation (at z s = 1 .
0) can become violated. Formpacts of Source Properties on Strong Cluster Lensing 11
A1 A2b cde f a CB DE F
Fig. 8.— : Simulated images by a massive lens cluster ( z l = 0 .
3) for galaxies in a COSMOS tile. The sources and theirlensed images are shown in black and red respectively. The other unlensed galaxies in the tile are not plotted togetherfor a clear view . The blue and green curves are critical lines and their corresponding caustics for sources at redshift z s = 3 .
0, while the cyan cross indicates the center of lens. A strongly stretched arc “A1” is clearly seen correspondingto the source “a”; an additional image “A2” is highlighted (in magenta) for the same source.example, we find an increase of a factor ∼ Source shape influence
Previous studies (Li et al. 2005; Puchwein et al. 2009) often assumed the sources follow a uniform distribution inthe ellipticity ( ǫ ≡ − b/a ) between 0 and 0.5. However, as can be seen in Fig. 5, while it is true that the ellipticity1 − b/a of most sources is less than 0.5, there is still a high fraction of sources, about 1/3, with ellipticity larger than0.5. The sources with large ellipticities, such as edge-on spiral galaxies, could be stretched much more easily to formgiant arcs of large L/W values when the tangential direction of the lens is approximately along the direction of themajor axis of the background galaxy. Therefore, the small portion of high ellipticity sources may change the giant-arcstatistics substantially.For all the COSMOS sources in the fiducial control tiles, we change the shape of sources with random ellipticities (be-tween 0 and 0.5) and orientations (between 0 and π ) while keeping the source positions, sizes and redshifts unchanged.We perform a lensing simulation for this background galaxy sample. This case (Case 6) is labeled as “z1s1e0”. Wefind that the number of giant arcs is only ∼ / Fig. 9.— : The giant-arc statistics for three test runs at different lens redshifts, e.g., z l = 0 .
2, 0.3 and 0.5 are illustrated.The 20 test source tiles are randomly selected. For each source tile, all the giant arcs produced in 300 realizations of10 massive clusters are counted (see Table 2). As can be seen, the fluctuation is large for a single source tile in themock survey due to the limited realizations. The lensing efficiency decreases when the lens redshift ranges from 0.2 to0.5, which is consistent with the result in § L/W >
10 shows a clear shift tothe higher ellipticity tail.It is important to explore whether the effect of source shape changes with the source redshift. Although the overallcross-section becomes larger for a higher source redshift, the change of the effective lensing area, in which elongatedsources can be lensed to giant arcs while the simulated ones of 0 ≤ ǫ ≤ . z s = 0 . , . , . , .
5. As clearly seen in Fig. 10, while the number of giant arcs increases with the sourceredshift, the shape impact on lensing efficiency decreases from ∼ ∼
2, when source redshift ranges from z s =0.8to 3.5. The boost effect of source shape is expected to be larger at z s lower than 1.5 since the lensing cross-section issmaller at lower redshift and thus the enhancement due to the elliptical sources is relatively more important. Source size influence
Instead of assuming a same effective diameter for all sources (Li et al. 2005; Puchwein et al. 2009), we have adoptedthe COSMOS galaxy sizes in our fiducial case (Case 1). To see the size effect of background galaxies, we first checkthe size distribution for the lensed galaxies which are stretched to be giant arcs of
L/W >
10. The comparison resultsmpacts of Source Properties on Strong Cluster Lensing 13
Fig. 10.— : The shape impact on cross-section dependence of z s is illustrated. It is clearly seen that the lensingincidence is boosted after the real galaxy shape is adopted (top panel). The Poisson fluctuations are indicated as errorbars in the panel. As a comparison, we put the sources of random ellipticity (shown in red) or real ellipticity (shownin black) in source planes at redshift z = 0 . z = 1 . z = 1 . z = 3 .
5, keeping source size of D eff = 1 ′′ . Whileconfirming the shape impact on lensing cross-section is about a factor of 2 for z s > .
5, with a larger factor at z . D eff = 1 . ′′ and D eff = 0 . ′′ . We find a similar number of giant arcs (188 and 166 out of 3000 realizations), which is consistentwith Fig. 6. Source size thus has a moderate effect on producing giant arcs when massive clusters are taken as lenses(also see Fig. 3 and Fig. 11). The reason is that the typical source size is much smaller than the strong lensing causticsize, and thus a moderate change in the source size has a small impact on the number of giant arcs. Combination of Morphology and Redshift Distribution
To see the overall impact of the simple assumptions for the shape, size, and redshift of background galaxies, wecombine these effects to do lensing simulations (for random position effect, see § π ) are simulated as above, while theeffective diameter is assigned to be 1 ′′ . We find that the lensing efficiency for the Case 3 (“z0s0e0”) is a factor of 3.75lower than that for Case 1 (“z1s1e1”). It is roughly consistent with the results from the combination of the individualeffects, i.e. redshift effect (a factor of 1.15 which is got from comparing Case 3 and 5), shape effect (a factor of 2.314 Gao, Jing, Mao, Li, & Kong eff - b / a Fig. 11.— : The color-weighted distribution plot of the ellipticity against size for sources lensed to be images with
L/W ≥
10. The lensed sample is obtained from tracing back all the giant arcs produced by 10 massive clusters atredshift z ∼ .
2. The pattern of the distribution is similar to that shown in Fig. 3, although the elongated sourceshave a higher weight, which is consistent with that implied in Fig. 5.from Case 4 and 5) and size effect (a factor of 1.34 from Case 5 and 6), which is 1 . × . × .
34 = 3 .
56. It thusindicates that the galaxy shape, redshift and size are affecting strong lensing efficiency independently as well as thesethree properties are independent of each other intrinsically as shown in Fig. 1, 2 and 3.
Effect of Source Clustering
Theoretical calculations often assume that the background sources are randomly distributed in space. However,background galaxies are clearly clustered. It is therefore interesting to check the effect of source clustering on theincidence of giant arcs.The influence of the clustering effect is investigated by carrying out a ray-tracing simulation on a control sample ofgalaxies where the positions of original COSMOS galaxies in a fiducial tile are randomly shuffled while keeping thesource surface density, size and ellipticity unchanged. We find that the efficiency of producing giant arcs (166 giantarcs out of 3000 realizations for 10 clusters at z l = 0 .
3) is similar to what obtained in the COSMOS case (163 giantarcs) which implies the clustering effect of background sources is negligible on strong lensing. The likely reason is thatthe intrinsic source density is too low (43 per square arcmin) and close pairs of sources are too few to form connectedgiant arcs or to be strongly lensed simultaneously. Our results suggest that sources can be assumed to be random inthe prediction of giant arcs, at least to the depth of COSMOS (in deeper images with a larger source density, clusteringmay become more important).mpacts of Source Properties on Strong Cluster Lensing 15
Effects of Seeing in Ground-based Observations
For our fiducial simulation (Case 1), the observational data are taken by HST (as in the Hubble Deep Field used inHoresh et al. 2005). However, many giant arcs were discovered from the ground-based telescopes where the effect ofseeing may be important. A proper evaluation would require a detailed simulation of seeing effects. Here we discussbriefly the approximate seeing effect on the predicted number of giant arcs.In COSMOS, there is a significant portion ( ∼ D eff ≤ ′′ (see Fig. 6).The width of lensed giant arcs produced from these small sources probably remains less than 1 ′′ . In such cases, seeing(such as ∼ ′′ ) will have a much larger effect on the width of giant arcs than on the length (Meneghetti et al. 2007),and in general the L/W ratio will be substantially reduced. Assume the seeing is ∼ . ′′ , the width of arcs intrinsicallysmaller than 0 . ′′ will be blurred to ∼ . ′′ which we adopt as the “observed” width. We find that in this case thefraction of giant arcs of L/W ≥
10 will be reduced to 80% of the COSMOS value. If seeing is as bad as 1 ′′ , the lensingprobability could be reduced to about 40% of the original value. Although this is only an approximate estimation, it isclear that the impact of seeing should be carefully evaluated in a detailed comparison between theoretical predictionsand ground-based giant-arc surveys. Comparison with Observations and Horesh et al. (2005)
The giant arcs in this work are selected by quantifying the ratio of length to width (
L/W ≥ z l = 0 . z l = 0 . ∼ . z l = 0 . ∼ . z l = 0 . ∼ − in HDF, vs. 23 arcmin − in our case). The lensing efficiency with thesurface density normalized is 0.840 arcs and 0.636 per realization, respectively. Horesh et al. (2005) used a differentdefinition of L/W ratio which is smaller than ours by a factor of 4 /π (e.g., their L/W ≥
10 is equivalent to our
L/W ≥ . π ≈ . L/W ≥
10) will be boosted by a factor of (4 /π ) if we adopttheir length-to-width ratio definition. Taken these two factors into account, our clusters produce 0 . × (4 /π ) = 1 . . × (4 /π ) = 1 .
03 giant arcs per realization for z l = 0 . p ( > | µ | ) ∝ µ − , | µ | ≫ z l ∼ .
2) and observational samples (0 . < z c < . SUMMARY AND DISCUSSION
In this paper, aiming to study the impacts of background sources on strong lensing statistics, we use the I -bandgalaxy image data of HST/ACS in the COSMOS to quantify the distributions of background source size, shape andredshift. Each galaxy image is extracted by SExtractor limiting a surface brightness down to ∼
25 mag / arcsec . Theredshift of each galaxy image is obtained by matching its celestial position with the COSMOS photometric catalog. Theselected sources are then lensed by 10 massive clusters of the mass range 6 . × h − M ⊙ ≤ M vir ≤ . × h − M ⊙ at z l ∼ . . × h − M ⊙ ≤ M vir ≤ . × h − M ⊙ at z l ∼ . m , = 0 . , Ω Λ , = 0 . are fully used for the statistic study. 172500 ray-tracing lensing simulations are carried out for10 lens clusters to reduce the statistical fluctuation at the expected lens redshifts (i.e. z l = 0 . σ .There are a number of limitations in the present work. When scaling our results with the source number density,we implicitly assume that other source properties (like ellipticity and redshift) are the same in COSMOS as in adeeper observation (like HDF). However, ellipticity distribution of galaxies in a deeper survey may turn out to bemore elliptical (Vincent et al. 2005; see also the discussion on the surface brightness limit), which would increase theboosting factor of source ellipticity on strong lensing efficiency. We would also have a larger fraction of galaxies at thehigh redshift tail end in a galaxy survey like HDF, which increases the mean strong lensing cross section. Therefore,6 Gao, Jing, Mao, Li, & Kongboth effects will produce more giant arcs. For direct comparison with observation, the observational effects, such asspecific instrumental point-spread-function and observational seeing, are not fully included in our lensing simulations.As mentioned in § ′′ , then there may be only around 40% of giant arcs observed by a typicalground-based telescope compared with that by space-based one, such as HST.The same shape measurement method is used for quantifying the ellipticity of the original galaxies and lensed images.It could bring in some inaccuracy for measuring the ellipticity of very round images, since it is mainly designed forquantifying arcs. Nevertheless, this effect is rather small and negligible to our results, since we mainly focus on theelliptical galaxies. Besides, the intrinsic pixelization will affect the ellipticity quantification of small sources, especiallyfor sources of D eff < . ′′ (see Fig. 3). The influence of pixelization would probably shift down the measured ellipticitybecause of its relative larger effect on width measurement. Since the number of such tiny sources is small ( ∼ h − kpc, the softening mainly takes effect inthe center of major merging clumps, while it would also smooth away the small substructures of a similar scale ina minor merger case or those remaining in a main halo. The change of the lensing cross section due to this kind ofsmoothing would not be significant in both cases. Besides, according to a comparison plot of optical depth in Fig. 4of Li et al. (2005), the result agrees within 25% of Wambsganss et al. (2004), in which they adopt a much smallersoftening length of 3 . h − kpc than 30 h − kpc in our case, for a source redshift of z s = 1 .
0. Therefore, although therelatively large value of softening length could reduce the lensing efficiency, it would not affect our results significantly.Moreover, lensing clusters are selected from a dark matter only simulation, therefore, we cannot investigate the stronglensing dependence on source properties for clusters with realistic baryon distributions.We have only used one surface brightness limit (e.g., a detection threshold of 1.5 σ ) to extract the background galaxyimages. Since the redshift of galaxy is mainly constrained by the matching with photometric catalog, it will not changethe distribution by using a higher detection threshold in our case. A variation of detection threshold would changethe galaxy size, but the dependence of lensing efficiency on source size is weak (see § h − M ⊙ to ∼ h − M ⊙ in Fig. 7 ofHennawi et al. (2007a). However, the mean cross section ratio is less than 1.5 between the observed and our simulatedclusters, according to the plot. Moreover, they use a larger mass density parameter Ω m , = 0 . σ = 0 .
95 in their simulation. Besides, the simulated clusters are put at a much higherredshift z l = 0 .
41. All these effects help to make the mass dependence of lensing probability stronger. Therefore,the difference of mean cross section in two comparing samples would be much less than a factor of 1.5 and becomesnegligible in this work. Nevertheless, it is still necessary and interesting to select more comparable (between the simu-lated and X-ray selected) lens cluster samples in mass and redshift range for a fair comparison of giant-arc producingefficiency in the future.
ACKNOWLEDGMENT
We would like to thank the anonymous referee for his/her useful suggestions. We also thank W. P. Lin for managingthe large amount of the data and helpful discussions. This work is supported by NSFC (10533030, 10821302, 10878001),by the Knowledge Innovation Program of CAS (No. KJCX2-YW-T05), and by 973 Program (No. 2007CB815402).GL is supported by the Humboldt Foundation. SM acknowledges the Chinese Academy of Sciences and the ChineseNational Science Foundation for travel support. This work was also partly supported by the visitor’s grant at JodrellBank, a joint research grant from the NSFC and the Royal Society, the Department of Energy contract DE-AC02-76SF00515 and by the European Community’s Sixth Framework Marie Curie Research Training Network Programme,Contract No. MRTN-CT-2004-505183 “ANGLES”.
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