Comparison of Strong Gravitational Lens Model Software III. Direct and indirect semi-independent lens model comparisons of COSMOS J095930+023427, SDSS J1320+1644, SDSSJ1430+4105 and J1000+0021
DDraft version August 26, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
COMPARISON OF STRONG GRAVITATIONAL LENS MODEL SOFTWARE III. DIRECT AND INDIRECTSEMI-INDEPENDENT LENS MODEL COMPARISONS OF COSMOS J095930+023427, SDSS J1320+1644,SDSSJ1430+4105 AND J1000+0021
Alan T. Lefor and Toshifumi Futamase Draft version August 26, 2018
ABSTRACTAnalysis of strong gravitational lensing data is important in this era of precision cosmology. Theobjective of the present study is to directly compare the analysis of strong gravitational lens systemsusing different lens model software and similarly parameterized models to understand the differencesand limitations of the resulting models. The software lens model translation tool, HydraLens, was usedto generate multiple models for four strong lens systems including COSMOS J095930+023427, SDSSJ1320+1644, SDSSJ1430+4105 and J1000+0021. All four lens systems were modeled with PixeLens,Lenstool, glafic, and Lensmodel. The input data and parameterization of each lens model was similarfor the four model programs used to highlight differences in the output results. The calculation ofthe Einstein radius and enclosed mass for each lens model was comparable. The results were moredissimilar if the masses of more than one lens potential were free-parameters. The image tracingalgorithms of the software are different, resulting in different output image positions and differencesin time delay and magnification calculations, as well as ellipticity and position angle of the resultinglens model. In a comparison of different software versions using identical model input files, resultsdiffered significantly when using two versions of the same software. These results further support theneed for future lensing studies to include multiple lens models, use of open software, availability oflens model files use in studies, and computer challenges to develop new approaches. Future studiesneed a standard nomenclature and specification of the software used to allow improved interpretation,reproducibility and transparency of results.
Subject headings: strong gravitational lens models, direct comparison studies, indirect comparisonstudies, lens model software INTRODUCTION
The present time has been referred to as the ”GoldenAge” of Precision Cosmology. Strong gravitational lens-ing data is a rich source of information about the struc-ture and dynamics of the universe, and these data arecontributing significantly to this notion of precision cos-mology. Strong gravitational lens studies are highly de-pendent on the software used to create the models andanalyze the components such as lens mass, Einstein ra-dius, time delays etc. A comprehensive review of avail-able software has been conducted by Lefor et al. (2013).While many such software packages are available, moststudies utilize only a single software package for analy-sis. Furthermore, most authors of strong gravitationallensing studies use their own software only.More recently, the status of comparison studies ofstrong gravitational lens models has been reviewed byLefor and Futamase (2013). This study demonstratedthat changes in redshift affect time delay and mass cal-culations in a model dependent fashion, with variableresults with small changes in redshift for the same mod-els.An important resource for the conduct of comparisonstudies is the Orphan Lens Project, a compendium ofinformation about strong lens systems that as of May2014 contained data for 656 lens systems (Moustakas andBrownstein 2013). There are a number of barriers to the [email protected] Astronomical Institute, Faculty of Science, Tohoku Univer-sity, Sendai Japan conduct of lens model comparisons. Ideally, a compari-son study of a previously studied lens would include theoriginal model for comparison, but this is sometimes im-possible because the lens model code is not made publiclyavailable. Another barrier to performing comparativestudies is the complexity of the lens model files, sincethere are major differences among the commonly usedmodel software available. In order to facilitate this stepof the process the HydraLens program was developed togenerate model files for multiple strong gravitational lensmodel packages (Lefor 2014).To date, the largest comparison study of strong grav-itational lens models was an analysis of MACSJ1206.2-0847 as part of the CLASH survey conducted by Umetsuet al. (2012). This study included four different stronggravitational lens models including Lenstool ((Jullo et al.2007)), PixeLens ((Saha et al. 2006)), LensPerfect (Coeet al. (2008)) and SaWLens (Merten et al. (2009)). Theauthors conducted five lens model analyses using thesame data, and is thus categorized as a direct and semi-independent study. This type of study has great advan-tages in that all data and all models are available fordirect comparison in a single study.The Hubble Space Telescope (HST) Frontier Fieldsproject is reporting preliminary results (Laporte et al.2014). This important deep field observing program com-bines the power of the HST with gravitational lenses. a r X i v : . [ a s t r o - ph . I M ] M a y Lefor and FutamaseLensing analysis in the Frontier Fields project includesmodels from a number of software codes including ZB,GRALE, Lenstool, and two other non-LTM lens modelsoftware codes which facilitate direct comparison of re-sults from a number of lens models rather than dependingon a single model from which to draw conclusions. TheHubble Frontier Fields analysis uses models that are in-dependently developed and optimized by each group ofinvestigators for each code used. The power of this ap-proach has been reported, with more results surely tofollow (Coe et al. 2015).The goal of this study is to directly compare the re-sults of calculations among four model software codes inthe evaluation of four lens systems. The present studyhas several unique features. This study is the first touse computer-aided lens model design, using HydraLenssoftware to facilitate lens model generation.There are noprevious single studies which compare the results for mul-tiple lens systems using multiple lens model software.This study was designed to further evaluate comparativelens model analyses and includes both direct and indirectsemi-independent studies of four lens systems using fourdifferent software models. Other studies have includedindirect comparisons to previous lens model analyses, ordirect comparisons of several lens models of a single lenssystem. This is the first study to also include combinedindirect and direct analyses where previously publishedlens models were used for direct comparisons.The nomenclature of lens model comparison studies,lens systems studied, previous lens model studies of thesesystems and the lens model software used are describedin section §
2. The results of the lens model studies foreach of the four systems studied are presented in section § §
4. Conclusionsand suggestions for future lens model studies are in sec-tion § METHODS
Nomenclature
The use of standardized nomenclature to describe lens-ing studies is useful to evaluate multiple studies. Inthis paper we follow the nomenclature previously de-scribed (Lefor and Futamase 2013). Lens model com-parison studies are referred to as direct when the com-parison is made based on calculations using two soft-ware models in the same paper, and indirect when com-parison is made to previously published data. In thisstudy, we also use the actual models from published stud-ies (kindly supplied by the investigators) so these areconsidered combined indirect/direct comparisons. Lensmodel comparisons using the same data are referred to assemi-independent, and when different data is used, thecomparison is independent. Lastly, software is classifiedas Light Traces Mass (LTM, formerly known as para-metric), or non Light Traces Mass (non-LTM, formerlyknown as non-parametric).
Lens Model Preparation
Each lens model software package uses a different inputdata format to describe the lens model. All of themuse simple text files as input, but the format of the textfiles, available functionality and command structures are dependent on the particular software. Some lens modelsoftware uses multiple accessory files to provide otherdata. Each of them has a unique list of commands, withgreat variability. HydraLens (Lefor 2014) was writtento simplify the process of creating lens model input filesto facilitate direct comparison studies, and to assist thosestarting in the field.The four lens systems were evaluated using fourlens model codes, necessitating 16 different models.The Lenstool model for COSMOS J095930+023427 waskindly provided by Cao (Cao et al. 2013). The glaficmodel for SDSS J1320+1644 was kindly provided byRusu (Rusu et al. 2013). The remaining 14 models werewritten for this study using HydraLens. In the case ofCOSMOS J095930+023427 and SDSS J1320+1644, thetwo lens models we received from other investigators wereused as input to HydraLens which generated the modelsfor the other three software packages used in this study.In the case of SDSS J1430+4105 and J1000+0021, mod-els were first written for PixelLens . HydraLens wasthen used to translate the PixeLens model into the for-mat for the other strong gravitational lens model soft-ware, including Lenstool , Lensmodel and glafic .The translated files output from HydraLens were editedto assure that parameters were fixed or free as appro-priate, and that optimization parameters were correctlyset. The lens model files were then used as input to therespective lens model software. Gravitational Lenses Studied
The parameters used for the four lens systems wasobtained from previous studies. The geometry foreach system was identical in all four models evalu-ated, and therefore all studies conducted are classi-fied as semi-independent lens analyses. Three of thelens systems studied are listed in the Orphan LensDatabase (Moustakas and Brownstein 2013) includ-ing COSMOS J095930+023427, SDSS J1320+1644 andSDSSJ1430+4105.
COSMOS J095930+023427
The lens COSMOS J095930 was first described byJackson (Jackson 2008). COSMOS J095930 is an early-type galaxy with four bright images of a distant back-ground source. It is located at z lens =0.892, and thebackground source is estimated at z source =2.00. Whilethe exact z source is unknown, the value used by previousinvestigators is 2.00.Models of this system were described by Faure usingLenstool (Faure et al. 2011). This model used a SingularIsothermal Ellipsoid (SIE) with external shear (+ γ ) andfound an Einstein radius of 0.79” and σ V =255 km s − .More recently, an extensive multi-wavelength study ofthis system was reported by Cao and colleagues (Caoet al. 2013), also using Lenstool. This analysis used fourdifferent models, an SIE with two Singular IsothermalSpheres (SIS) as well as a Pseudo-Isothermal EllipticalMass Distribution (PIEMD) model with two SIS, both http://ascl.net/1402.023 http://ascl.net/1102.007 http://ascl.net/1102.004 http://ascl.net/1102.003 http://ascl.net/1010.012 trong Lens Model Comparisons 3with and without external shear (Kassiola and Kovner1993). We selected the SIE+SIS+SIS model used by Caoas the basis of the present indirect comparison with theirwork as well as the direct comparisons with the four lensmodels studied here.The Lenstool model developed by Cao and coworkerswas kindly supplied for this study and used as a baselinemodel which was then translated into input files for theother software by HydraLens. The Lenstool model usedby Cao included priors for the values of ellipticity ( (cid:15) =[0 . , . σ = [100 , m = 0 . Λ = 0 . H = 70 km s − cosmology, as was used byCao et al. (2013). SDSS J1320+1644
SDSSJ1320+1644 was initially described by Oguriet al. (2012) and Inada et al. (2012), and is a large separa-tion lensed quasar candidate identified in the SDSS, witha separation of 8 (cid:48)(cid:48) . ± (cid:48)(cid:48) .002 at z source =1.487 (Rusuet al. 2013). Both an elliptical and disk-like galaxy wereidentified almost symmetrically between the quasars atredshift z lens =0.899.A detailed lens model analysis of this system was con-ducted by Rusu et al. (2013), using glafic software. Basedon their analysis, they conclude that SDSSJ1320+1644 isa probable gravitationally lensed quasar, and if it is, thiswould be the largest separation two-imaged lensed quasarknown. They show that the gravitational lens hypothesisimplies that the galaxies are not isolated, but are embed-ded in a dark matter halo, using an NFW model and anSIS model. The SIS model has a σ V =645 ±
25 km s − .We use the ’SIS free’ model as the basis of the compari-son study, as defined by Rusu et al. (2013), which modelsthe three galaxies (referred to as G1, G2 and G4) as SISpotentials and leaves the position of the dark matter halo(also modeled as a SIS) as a free parameter. The modelused by Rusu includes priors for the velocity dispersionof the dark matter halo ( σ = [400 , χ <<
1. The ellip-ticity and position angle are used when the position of thedark matter halo is fixed. The models developed for thisstudy were similarly parameterized using the position ofthe dark matter halo as a free parameter (”SIS-free”) andfixed to introduce ellipticity and position angle.A number of glafic models developed by Rusu andcoworkers were kindly supplied for this comparative anal-ysis and used as a baseline model which was then trans-lated by HydraLens into models for the other software.The present study includes an indirect comparison withthe analysis of Rusu (Rusu et al. 2013) as well as a di-rect comparison of the four software lens models stud-ied. Since we were provided a model used by Rusu,this is a combined indirect/direct comparative analysisof SDSSJ1320+1644. All four models of this system useda Ω m = 0 .
27, Ω Λ = 0 . H = 70 km s − cosmology, aswas used by Rusu et al. (2013). SDSS J1430+4105
SDSS1430+4105 was first described by Bolton et al.(2008) as part of the SLACS survey. This system isat redshift z lens =0.285 with z source =0.575, and has acomplex morphology with several subcomponents as de-scribed by Eichner et al. (2012). Bolton reported aneffective radius of 2.55” and a σ SDSS =322 km s − .A very detailed lens model analysis of this system wasthen conducted by Eichner et al. (2012). This analy-sis was a direct, semi-independent comparative analysisusing both Gravlens (LTM) and Lensview (non-LTM)software. The authors studied five different models us-ing Gravlens/Lensmodel, including an SIE and a PowerLaw (PL) model as well as three two-component de Vau-couleurs plus dark matter models. Similar results werefound with the two different lens model analyses. Theyalso studied four models using Lensview (Wayth andWebster 2006) including an SIE and PL models with andwithout external shear. We use the Gravlens/LensmodelSIE model as the basis of the indirect comparison withtheir work. The plane of optimization used in the Eich-ner model is not explicitly stated in the report (Eichneret al. 2012).The models developed in the previous study were notavailable, and thus all models used were written forthis study. The results referred to as Model I by Eich-ner did not use any priors in the lens model for SDSSJ1430+4105, although priors were used in the develop-ment of the model with results within the error limitsreported. Similarly, priors were not used in the mod-els in this study. The free parameters used by Eichneret al included the lensing strength b, the ellipticity andthe orientation of the single-component SIE lens. Thesesame free parameters were used in the models developedfor this study. The positions of the multiple images ofthis system were taken from Table 2 in Eichner et al.(2012) .This is both an indirect comparison (compared withthe SIE model in the published study of (Eichner et al.2012)) and direct comparisons of the four lens modelsstudied here. All four models of this system used a Ω m =0 .
3, Ω Λ = 0 . H = 70 km s − cosmology, as was usedby Eichner et al. (2012). J1000+0021
Lefor and FutamaseUsing imaging data from CANDELS and the largebinocular telescope, van der Wel and colleaguesrecently reported the quadruple galaxy-galaxy lensJ100018.47+022138.74 (J1000+0221), which is the firststrong galaxy lens at z lens > z lens =1.53 and a z source =3.417.van der Wel et al. (2013), analyzed this system in themanner described previously by van de Ven et al. (2010),and reported an Einstein radius of R E = 0 .
35” with anenclosed mass of M E = (7 . ± . × M (cid:12) with anupper limit on the dark matter fraction of 60%. Thehighly magnified (40 × ) source galaxy has a very smallstellar mass ( ∼ M (cid:12) ). The z = 1 .
53 lens is a flattened,quiescent galaxy with a stellar mass of ∼ × M (cid:12) .There have been no other lens model analyses of thissystem using software models and therefore all modelswere developed for this study using data from van derWel et al. (2013), and is thus is a direct comparison of thefour lens software models studied. There were no priorsused in the lens models of J1000+0021 in this study. Thefree parameter in the SIS models was only the velocitydispersion. In the SIE models, free parameters includedthe velocity dispersion, orientation and ellipticity. Theimage positions in all models in this study for this systemwere taken from Table 2 in (van der Wel et al. 2013).All four models of this system used a Ω m = 0 .
3, Ω Λ =0 . H = 70 km s − cosmology. Lens Models
The analyses in this study were performed with fourstrong gravitational lens model software packages thathave been used extensively in the literature. All foursystems were modeled with all four lens model softwarepackages. Lenstool and Lensmodel were executed underScientific Linux version 6.4 (except as noted for Lens-model in section § σ con-fidence interval for velocity dispersions.The fit of the models is assessed by χ optimizationand the RMS uncertainty. The RMS is calculated by: RM S images = (cid:88) i (( x (cid:48) i − x i ) +( y (cid:48) i − y i ) ) / N images , (1)where x (cid:48) i and y (cid:48) i are the locations given by the model,and x i and y i are the real images location, and the sumis over all N images images. The χ results are calculatedfor the models by Lenstool, Lensmodel and glafic, andare reported in the data tables. The RMS value is re-ported by Lenstool directly, while a manual calculationwas necessary for models using Lensmodel and glafic. PixeLens
PixeLens is a non-LTM strong gravitational lens modelsoftware that is available for download as a Java pro-gram which runs in a browser window (Saha et al. 2006).Version 2.7 was used in these studies. PixeLens is accom-panied by a manual (Read 2012a) and a tutorial (Read2012b). PixeLens reconstructs a pixelated mass map forthe lens in terms of the arrival time surface and has beenused in several studies (Saha et al. 2006).PixeLens employs a built-in MCMC approach and cre-ates an ensemble of 100 lens models per given image con-figuration. The pixelated mass map offers the advantageof being linear in the unknown. Since all equations arelinear in the unknowns, the best-fitting model and its un-certainties are obtained by averaging over the ensemble(Saha et al. 2006; K¨ohlinger and Schmidt 2013).The pix-elated mass map differentiates PixeLens from the othersoftware used in this study which fit parametric func-tional forms. Lenstool
Lenstool has been used in many different studies andis available for download (Jullo et al. 2007). Version6.7.1 was used in these studies. Lenstool has features ofboth LTM and non-LTM modeling and uses a Bayesianapproach to strong lens modeling and has been well-described in the literature (Jullo et al. 2007; Jullo andKneib 2009). There are several resources available forwriting lens models for Lenstool (Kneib 2012; McCourt2006).Lenstool can optimize most of the parameters in amodel. Models produced by HydraLens for Lenstool weremodified slightly to add appropriate optimization param-eters and then used with Lenstool. Lenstool optimizationis performed with MCMC. Lenstool uses the geometryof the images given and then finds counter-images. Theimage positions are recomputed and the time delays de-termined. Gravlens
The Gravlens package includes two codes, Gravlensand Lensmodel (Keeton 2001b) accompanied by a usermanual (Keeton 2004). Version 1.99o was used in thesestudies, under the Linux operating system, downloadedfrom the Astrophysics Source Code Library . However,the Darwin (Macintosh) executable file provided for ver-sion 1.99o will only run on the now obsolete PowerPC ar-chitecture. A newer version to run on the Macintosh plat-form under OS/X 10.9 (Gravlens version dated Novem-ber 2012) was kindly provided by Professor Keeton, forthese studies. Lensmodel is an extension of Gravlens andwas used for all analyses here. It is fully described in twopublications by Keeton (Keeton 2001b,a), and has beenused extensively.Lensmodel is an LTM lens model software, which opti-mizes the selected lens parameters and uses a tiling algo-rithm and a simplex method with a polar grid centeredon the main galaxy. The tiles are used to determine theimage positions, and then uses a recursive sub-griddingalgorithm to more accurately determine image positions. glafic Glafic is an LTM lens model software, and includescomputation of lensed images for both point and ex-trong Lens Model Comparisons 5tended sources, handling of multiple sources, a wide va-riety of lens potentials and a technique for mass model-ing (Oguri 2010) with multiple component mass models.Version 1.1.5 was used on the OS/X platform and version1.1.6 was used with Linux in these studies .Each lens is defined by the lens model and seven pa-rameters. A large catalog of lens models is available (in-cluding point mass, Hernquist, NFW, Einsato, Sersic,etc.). After defining the parameters and the lens mod-els, parameters to be varied in the χ minimizations arespecified. Following this, the desired commands are is-sued such as computing various lensing properties, Ein-stein radius, write lensing properties to a FITS file, etc(Oguri 2013). Glafic has been used in a large number oflens model studies, including SDSSJ1004 (Oguri 2010),and performs lens model optimization.Glafic uses a downhill simplex method of optimization.The image plane is divided using square grids by an adap-tive meshing algorithm. The level of adaptive meshing isset as an optional parameter. RESULTS
Each of the four lens systems was modeled with all fourlens model software codes including PixeLens, Lenstool,Lensmodel, and glafic. Best-fit lens model parametersfrom previous studies are presented along with the re-sults from this study for each system. The results re-ported for each lens were intended to follow the formatof the data for best-fit lens parameters as reported in pre-vious studies, and therefore there are some differences inthe data presented for the four lens systems. Lenstooland glafic directly calculate the velocity dispersion andthen calculate the Einstein radius and mass within theEinstein radius. Lensmodel directly calculates the Ein-stein radius, from which the other values were deduced.PixeLens calculates mass at various distances from thelens mass. The figures shown are the output from each ofthe software packages used, and represent the graphicalcapabilities of that software.
COSMOS J095930+023427
Best-fit lens model parameters for COSMOSJ095930+023427 are shown in Table 1. The datareported by Cao et al. (2013) are at the upper portionof the table, and show the results of the Lenstool model.The results in this study using the Lenstool model aresomewhat different because the model in this study usedoptimization in the image plane, rather than the sourceplane optimization used by Cao. The glafic model wasalso conducted with optimization in the image plane,while the Lensmodel model is conducted with sourceplane optimization because image plane optimizationdid not yield a satisfactory model. Direct comparisonsof the four software models evaluated are shown next.The models used here were based on the SIE+SIS+SISmodel used by Cao et al. (2013). The Lenstool modelincludes an SIE potential at z lens =0.892, and two SISpotentials at z lens =0.7, as described by Cao et al.(2013).The PixeLens model used image coordinates from Caoet al. (2013), and calculated an enclosed mass insidethe Einstein radius very close to that calculated by theLenstool model. The Lenstool model optimized the ellip- ticity, position angle and velocity dispersion for the sin-gle SIE potential, and only the velocity dispersion for thetwo SIS potentials, as done by Cao et al. (2013) as freeparameters. Lensmodel sets all three lens potentials at z lens =0.892 because the software does not permit mul-tiple lens planes. The ellipticities and position anglesoptimized by each of the three codes are quite different.The Einstein radius of the SIE potentials are similarwhile there is some difference in the optimized velocitydispersions calculated by the three codes, particularly inthe values calculated by glafic for the second potential.In an effort to understand this, the velocity dispersionsof the first and second potentials were fixed at the valuescalculated by Lenstool at 234 and 412 km s − respec-tively and the velocity dispersion of the third potentialallowed to optimize, using glafic. This resulted in a ve-locity dispersion of 632 km s − for the third potential.When the first and third values were fixed at 238 and603 km s − (as found by Cao), the second potential wasoptimized at 57 km s − . Magnifications and time delaysfor this model are shown in Table 2. Both time delaysand magnifications calculated by all four models showgreat variability.The velocity dispersions shown in Table 1 as calculatedhere are slightly different from those reported by Caoet al. (2013), because of the different optimization tech-nique. The velocity dispersion values shown for the Lens-model and glafic models are somewhat different. TheLenstool model used by Cao (Cao et al. 2013) definedpotentials at z lens =0.892 and 0.7, although Lenstool al-lows only a single lens plane (Kneib 2012). When theresults were re-calculated defining all lenses in the sameplane ( z lens = 0 . SDSS J1320+1644
Best-fit lens model parameters for SDSS J1320+1644are shown in Table 3 with an indirect/direct comparisonto the study of Rusu et al. (2013) and the four directcomparisons in this study. Rusu et al. (2013) utilized aglafic model that modeled the potentials of G1, G2 andG4 which were boosted by an embedding dark matterhalo. One of the published models used four SIS poten- Lefor and Futamase
TABLE 1Best-fit lens model parameters for COSMOS J095930+023427
Software RA Dec χ e θ R E M ( < R E ) σ ( (cid:48)(cid:48) ) ( (cid:48)(cid:48) ) RMS( (cid:48)(cid:48) ) (deg) ( (cid:48)(cid:48) ) (10 M (cid:12) ) (km s − )Results from Cao et al. (2013)Lenstool 1.7SIE [0.0] [0.0] 0.28 -10 0.79 3 . +0 . − . · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Lenstool 1.2SIE [0.0] [0.0] 0.06 0 . ± . − ± . ± .
03 3 . ± . ± · · · · · · ± · · · · · · ± . ± .
002 84 ±
18 0 . ± .
06 1 . ± . ± · · · · · · ± · · · · · · ± . ± .
08 65 ± . ± .
02 2 . ± . ± · · · · · · . ± . · · · · · · ± Note . — Values shown in square brackets are fixed in the models. Values without brackets are theoptimized/calculated values from the model. *Calculated mass at 6.10E+00 M (cid:12) TABLE 2Magnification and Time Delays for Four Images inCOSMOS J095930+023427
Software A B C DPixeLensTime Delay 0 0.7 3.4 0.07LenstoolMagnification 7 . ± . . ±
10 8 . ± . . ± . ±
22 31 ±
18 32 ± . ± . . ± . . ± . . ± . ±
12 9 . ± . . ± . − . ± . . ± . − ± . ± ± . ± . . ± . Note . — Time delay is shown in days tials and fixed the locations of the first three, allowingthe position of the fourth (the dark matter halo) to op-timize (”SIS free”). Furthermore, they concluded thatany reasonable mass model reproduced the observed im-age configuration. The values shown in Table 3 are thoseas presented in the paper, as the ’SIS free’ model (Rusuet al. 2013). In this study, the values calculated by Rusuet al. (2013) and shown here were reproduced exactlyusing their model, and the ± values are at 1 σ .The PixeLens model has a much lower calculated timedelay than the other models, and an enclosed mass within 1 σ of the value reported by Rusu et al. (2013). As per-formed by Rusu et al. (2013), the positions of the sourceswere kept fixed for the first three SIS potentials. The ve-locity dispersion and position of the last potential (thedark matter halo) were optimized. The optimized po-sition of the fourth potential calculated in the Lenstoolmodel is quite different, and the velocity dispersion issimilar to other models. Lensmodel uses the Einsteinradius, rather than velocity dispersion so the Einsteinradii for the first three SIS potentials were fixed, andthe fourth was a free parameter. The mass of the fourthtrong Lens Model Comparisons 7 Fig. 1.—
Image plane for COSMOS J095930+023427 calculatedby Lenstool. The critical line is shown in red. Each image positionis shown as a cross with a label. The center of the mass distributionis shown in gray at the center potential calculated by Lensmodel is nearly identical tothe values calculated using glafic by Rusu et al. (2013) aswell as the Lenstool and glafic models reported here. Thetime delays and magnification values show more variabil-ity.The Lenstool, glafic and Lensmodel models conductedin this study use image plane optimization, similar to theglafic analysis conducted by Rusu. The calculated mod-els of SDSS J1320+1644 show similar optimization forthe mass of the fourth SIS potential, with fairly similarpositions calculated by Lensmodel and glafic, while thepositions calculated by Lenstool show greater variabil-ity. There is great variability among the calculated timedelays and magnifications.The calculations performed in this study using glaficare the same as the glafic SIS-free model reported byRusu et al. (2013). Table 3 shows that the mass calcu-lated for the fourth SIS potential, which was a free pa-rameter, optimized to the same value for Lenstool, Lens-model and glafic. The optimized geometry was slightlydifferent for Lenstool compared to the others. The Ein-stein radius calculated by all four models was almost thesame for the first SIS potential. The fact that the veloc-ity dispersion for the fourth lens potential was optimizedto the same value in all of the models may reflect the factthat there was only a single free parameter in each model.This is different from the results above with COSMOSJ095930+023427, which optimized three lens potentialsas free parameters, with varying results among the mod-els tested.The model of SDSS J1320+1644 was straightforwardincluding four SIS potentials which was reproduced in allsoftware models without difficulty. The model used byRusu (Rusu et al. 2013) had 0 degrees of freedom andwith a resulting χ <<
1, due in part to the design ofthe model with 14 nominal constraints and 14 param-eters. The similarity of the potentials used to modelthe system may have contributed to the close results foroptimization of the mass. Despite this, position, magni-fication and time delay showed great variability amongthe four models. The velocity dispersion for only thefourth lens potential was left as a free parameter, withthe other three fixed, which is likely a major factor inthe close agreement found among the various models inthe calculation of the velocity dispersion.The image plane of the glafic model is shown in Figure2, which is the same as shown in Figure 6 of Rusu et al.
Fig. 2.—
Image plane for SDSS J1320+1644 calculated by glafic.The blue line is the critical line. Image positions are shown as redtriangles. The centers of the masses are shown as black crosses. (2013). The image positions in the image plane are thesame as the input positions in all models. Despite this,there is variability in the time delay and magnificationcalculations.
SDSSJ1430+4105
The indirect comparison to the work of Eichneret al. (2012) and the results of the four direct com-parisons in this study are shown in Table 4. InEichner et al. (2012) there are five different modelstested for SDSSJ1430+4105. The models were testedwith Gravlens/Lensmodel (LTM) (Keeton 2001b) andLensview (LTM) (Wayth and Webster 2006), and theresults compared in a direct comparison.The model used in the current study is based on ModelI, as described in Eichner et al. (2012), which models thelens as an SIE, ignoring the environment of the lens. Thebest fitting parameters reported by Eichner et al. (2012)are shown in Table 4. The results of Eichner are in goodagreement with those by Bolton et al. (2008). In theSIE model using Lensview as reported by Eichner et al.(2012), their results were very similar to those with theLensmodel model. The input files for the model used byEichner et al. (2012) were not available for this study,making this study both an indirect and direct compari-son.The Lenstool, glafic and Lensmodel models conductedin this study use image plane optimization. The enclosedmass calculated by PixeLens inside the Einstein radius,is slightly higher than the result published by Eichneret al. (2012). The Einstein radii calculated by all themodels are very close to each other as well as close tothe result of Eichner et al. (2012). As shown in otherlens systems in this study, there is considerable variationin magnification and time delay calculations among thefour models studied as shown in Table 5. The optimizedellipticities among the four models are all quite close,but there is significant variability in the optimal positionangles calculated.The models used in this study (results shown in Ta-bles 4 and 5) were written without detailed knowledgeof the model used by Eichner et al. (2012). Despite this,the models all had similar results, especially in regardto Einstein radius, enclosed mass and velocity dispersioncalculations.The image plane of the glafic model of this system isshown in Figure 3. The glafic (Figure 3) model resultedin just 4 images in the output image plane. In contrast, Lefor and Futamase
TABLE 3Best-fit lens model parameters for SDSS J1320+1644
Software RA Dec χ µ ∆t R E M ( < R E ) σ ( (cid:48)(cid:48) ) ( (cid:48)(cid:48) ) RMS( (cid:48)(cid:48) ) (days) ( (cid:48)(cid:48) ) (10 M (cid:12) ) (km s − )Results from Rusu et al. (2013)glafic χ << ± − ± . ± . . ± . · · · · · · [163]SIS [-9.169] [5.173] · · · · · · · · · SIS -4.687 1.149 · · · · · · ± · · · · · · · · · Lenstool 11.5 1 . ± . − ± ± . ± . ± ± ± − ± . ± . . ± . ± Note . — Values shown in square brackets are fixed in the models. Values without brackets are theoptimized/calculated values from the model. + -2 -4 -6 ._._.._.__.__._......_.___.__._.._._._...__._...._._....___.___.___.__.__.'--'---J �-6 -4 -2 0 Fig. 3.—
Image plane for SDSS J1430+4105 calculated by glafic.The blue line is the critical line. Image positions are shown as redtriangles. The center of the mass density is shown as a black cross.
Lenstool identified a total of 28 images.The position angles were somewhat different but therewas good agreement among the models for ellipticity cal-culations. As with other models in this study, there wasvariation in the calculation of time delays and magnifi-cations.One of the reasons for such close agreement among themodels is that the models all used a single SIE potential,which allowed for comparable potentials among the fourlens model codes tested. There was a single lens planein all of the models.
J1000+0021
An analysis of this lens system was performed by vander Wel et al. (2013) with a calculated Einstein radius of R E = 0 .
35” (or 3.0 kpc) with an enclosed mass of M E =7 . ± . × M (cid:12) . There have been no extensive lensmodel analyses of this system published to date. This isthe first strong galaxy lens at z lens >
1. In all models, theposition (both RA and Dec) of the lens galaxy was keptconstant, and the mass was a free parameter optimizedby the software. Further details of the model used werenot provided, such as the model software used or the χ calculation.Results of the four direct comparisons done in thisstudy are shown in Table 6. This lens system was mod-eled both using an SIS and an SIE, with all lens modelsoftware tested. The Lenstool, glafic and Lensmodelmodels conducted in this study use image plane optimiza-tion. The PixeLens model calculated the enclosed massthe same as reported by van der Wel et al. (2013). Usingan SIS potential, the Einstein radius, enclosed mass andvelocity dispersion calculations were nearly the same forLenstool, Lensmodel and glafic. The Einstein radii andvelocity dispersions were very close to that reported byvan der Wel et al. (2013). Calculations of magnificationand time delay showed quite a bit of variability in thesemodels.The results of the models shown in Table 6 show verysimilar results for the SIS and the SIE models. Theenclosed mass within the Einstein radius is somewhatlower than that reported by van der Wel et al. (2013) forLenstool, Lensmodel and glafic although the PixeLenstrong Lens Model Comparisons 9 TABLE 4Best-fit lens model parameters for SDSSJ1430+4105
Software RA Dec χ e θ R E M ( < R E ) σ ( (cid:48)(cid:48) ) ( (cid:48)(cid:48) ) RMS( (cid:48)(cid:48) ) (degrees) ( (cid:48)(cid:48) ) (10 M (cid:12) ) (km s − )Results from Eichner et al. (2012)Lensmodel 11.5SIE [0.0] [0.0] 0 . +0 . − . − . +2 . − . . +0 . − . . +0 . − . ± · · · · · · · · · · · · Lenstool 4.9SIE [0.0] [0.0] 0.25 0 . ± .
03 82 ±
22 1 . ± .
02 3 . ± .
05 317 ± . ± .
32 22 ±
32 1 . ± .
02 3 . ± .
05 309 ± . ± . − ± . . ± .
02 3 . ± .
11 334 ± Note . — Values shown in square brackets are fixed in the models
TABLE 5Magnification and Time Delays for Five Images in SDSS J1430+4105
Software A B C D EPixeLensTime Delay 0 0 0 0 0LenstoolMagnification 1 . ± . . ± . . ± . . ± . . ± . ±
18 66 ±
22 90 ± − ± . ± . . ± . . ± .
15 0 . ± .
15 0 . ± . ± . ± . ± . ± . . ± . − . ± . − . ± .
02 1 . ± . · · · Time Delay 0 34 ± ± · · · Note . — Time delay is shown in days model reproduced the enclosed mass calculation verywell. Similar to the models used for SDSSJ1430+4105,these models were all quite straightforward with a sin-gle potential located at the origin, which may have con-tributed to the concordance of results.Comparing the results of the SIE models, the resultswith an SIE model using the four software packages werealso nearly identical, although among the SIE models,there was some variability in the calculations of ellipticityand position angle.The image plane of the glafic model of this system isshown in Figure 4. This system is particularly interestingas the image positions in the Lensmodel and glafic mod-els have an almost identical geometry, while the imagepositions in the Lenstool model are different. The timedelays and magnifications in the Lensmodel and glaficmodels are very similar, while the Lenstool model valuesare different.
Fig. 4.—
Image plane for J1000+0021 calculated by glafic. Theblue line is the critical line. Image positions are shown as redtriangles. The center of the mass density is shown as a black cross.
Comparison of Results
There are some generalizations that can be made com-paring the results calculated from the models for eachof the four lens systems studied. The Einstein radii and0 Lefor and Futamase
TABLE 6Best-fit lens model parameters for J1000+0021
Software RA Dec χ µ ∆t R E M ( < R E ) σ ( (cid:48)(cid:48) ) ( (cid:48)(cid:48) ) RMS( (cid:48)(cid:48) ) (days) ( (cid:48)(cid:48) ) (10 M (cid:12) ) (km s − )Results from van der Wel et al. (2013)40 ± . ± . ± · · · · · · · · · LenstoolSIS [0.0] [0.0] 2.9 1 . ± . − ±
22 0 . ± . . ± . ± . ± . . ± . . ± . . ± . ± . ±
12 3 . ± . . ± .
05 0 . ± .
25 192 ± χ e θ R E M ( < R E ) σ ( (cid:48)(cid:48) ) ( (cid:48)(cid:48) ) RMS( (cid:48)(cid:48) ) ( (cid:48)(cid:48) ) (10 M (cid:12) ) (km s − )LenstoolSIE [0.0] [0.0] 1.7 0 . ± . ±
18 0 . ± .
05 0 . ± .
07 190 ± . ± . − ±
35 0 . ± . . ± .
05 190 ± . ± . − ±
19 0 . ± .
05 0 . ± .
22 189 ± Note . — Values shown in square brackets are fixed in the models mass within the Einstein radii are quite close for the fourmodels of each system. The Einstein radius is calculatedfrom the average distance between the lens center andmultiple images, and is insensitive to the radial densityprofile (Oguri et al. 2013). The conversion from the Ein-stein radius to the enclosed mass within the Einstein ra-dius is dependent only on the lens and source redshifts,and is therefore model independent (Oguri et al. 2013).Thus, the similar results for Einstein radii and masswithin the Einstein radii are expected since all modelshad the same system geometry of z lens and z source .There is variation among the calculated time delaysand magnifications comparing the models generated byeach of the four lens model software programs. The im-age positions input to each model were identical. Theimage positions in the models studied change due to theray-tracing algorithms in each software model. Thesedifferences explain some of the variation seen in time de-lay and magnification. In some cases, the use of a simi-larly parameterized model leads to a model that has notconverged appropriately, which illustrates some of thedifferences in the software. This is evident in the RMSvalues calculated for the Lenstool and Lensmodel modelsof J1320+1644There is also little agreement among calculations ofellipticity and position angle. The variation in resultsfor calculated ellipticity and position angle may be a re-sult of differences in the optimization algorithms used byLenstool, Lensmodel and glafic. The complexity of the model also has an impacton agreement among the calculated values for veloc-ity dispersion. In the models for SDSS J1430+4105,J1000+0021and SDSS J1320+1644, there was only onepotential with the velocity dispersion as a free-parameterfor optimization. In all three of these systems, therewas close agreement among the calculated values. In themodel of COSMOS J095930+023427, there were threelens potentials which were optimized, with quite a bitof variation among the results from the three softwareprograms used. Comparison of Lens Model Software by Version
In order to evaluate the effect of software versionand/or operating system / hardware platform, the modelof SDSS J1320+1644 was evaluated with glafic and Lens-model on two different hardware platforms. Glafic is dis-tributed as an executable file with version 1.1.5 for theOS/X platform and version 1.1.6 for Linux. Lensmodelis available as an executable file only for download as ver-sion 1.99o for the Linux platform, and we were provideda version to run on OS/X.Input files for the models of SDSS J1320+1644 wereused unchanged. In the first test, the model was testedwith the two versions of glafic. The mass of the firstthree SIS potentials were held as fixed parameters andthe mass of the fourth potential, as well as its position,were free parameters to be optimized. Identical resultswere reported using either version of glafic, on both plat-trong Lens Model Comparisons 11forms. The results were identical including the numbersof models used for optimization in each run and the cal-culation of all parameters evaluated. The content of alloutput files produced by both versions was identical. Themodels for SDSS J1320+1644 were then tested with eachof the two versions of Lensmodel. In this same test, opti-mizing the fourth SIS potential, results with Lensmodelwere slightly different comparing the two versions. Theoptimized Einstein radius of the fourth potential usingthe Linux version is reported as 3.622605, and the OS/Xversion reports 3.622528. There are similarly small dif-ferences in the optimized position of the fourth potential.In the next test, the mass of all four potentials wasoptimized. The results with glafic, on both hardwareplatforms, were again identical in regard to all parame-ters evaluated, to the accuracy of the last decimal placereported. The contents of all output files produced byglafic were identical with the Linux and OS/X versions.However, the two versions of Lensmodel reported widelydisparate results with the two versions tested. The Ein-stein radii of the four optimized SIS potentials using theLinux version are 1.851, 1.004, 0.3161 and 1.660. Usingthe OS/X version, the four potentials are optimized at2.234, 1.818, 0.3139 and 2.006.Among the various studies reported in Tables 7 and 8,the software version used is reported in only one study.The hardware platform and/or operating system usedin the calculations is not reported in any of the studiesshown in these tables. DISCUSSION
Small changes in redshift have different effects on thecalculation of time delays and mass by different lensmodel software codes (Lefor and Futamase 2013). Inthat study, a mock model with a single potential andfour images as well as a model of SDSS J1004+4112 wereevaluated and the effect of changes in redshift on changesin calculations of time delay and mass were determined.The study showed that changes in time delay and masscalculations are not always proportional to changes in D d D s /D ds , as would be predicted. The image positionschange expectedly as a result of ray-tracing tracing al-gorithms which are not the same for all of the softwareused. This is partly responsible for the differences in thevalues of time delay and mass in both systems when com-paring the models from four different lens model softwarepackages.The present study was designed to specifically comparethe results using the same models with different software,rather than changes in the results, to compare resultsfrom different codes. The present study is the largeststrong gravitational lens software comparison study per-formed to date, evaluating four different lens systemswith four different lens model software codes in a sin-gle study, and is the first study to use HydraLens for thepreparation of multiple models. Indirect Comparison Studies
Table 7 shows a review of the existing literature whereparameters have been calculated using strong gravi-tational lens models and compared with other pub-lished results, and as such are referred to as ”indi-rect comparison studies”. In the indirect comparison ofCOSMOSJ095930 performed by Cao et al. (2013) and Faure et al. (2011), both analyses were conducted withLenstool, and had very similar results for Einstein radius,mass enclosed within the Einstein radius, and other pa-rameters. It is difficult to discern the details of the modelused by Faure et al. (2011) with regard to number, typeand geometry of the lens potentials used. Indirect com-parisons are further complicated by a lack of availabledetail of the model used, making it difficult to reproduceprevious results.
Direct Comparison Studies
Table 8 shows previous studies where different lensmodels were compared in the same study, as well as theevaluations performed in the present study, all of whichconstitute ”direct comparison studies”. The direct com-parisons performed of Abell 1703, MS1358, MACSJ1206and SDSS120602 have been described in detail in Leforand Futamase (2013) . The information in these di-rect studies was complementary in nature, leading toa greater understanding of the lens system. The lensSDSSJ1430 was investigated by Eichner et al. (2012)who compared the results using Lensview and Lens-model. The Lensmodel analysis assumes point sourceswhile Lensview uses the two-dimensional surface bright-ness distribution of the same system. Both analyses ledto the same conclusions regarding the mass distributionof the galaxy. The two lens model techniques were indeedcomplementary and led to similar results. In a compar-ative analysis of RX J1347.5-1145 using glafic and Pix-eLens, the authors note a 13 percent difference in thecalculation of mass enclosed within the Einstein radius(K¨ohlinger and Schmidt 2013). They suggest that theLTM model used by glafic may not be assigning suffi-cient mass to the profiles in the models used. We ob-served a similar underestimation of enclosed mass bynon-LTM models as compared to PixeLens in the analy-sis of J1000+0021. CONCLUSIONS
Indirect comparison studies are of value, but as someof the comparisons conducted in this study show, it maybe difficult to reproduce the results of previous studieswithout previous model files available to create the mod-els for other software, thus limiting the nature of thecomparisons performed. In the analyses of COSMOSJ095930+023427 and SDSS J1320+1644, being able touse the same models as used in the original studies, qual-ifies these as direct comparisons. This supports the im-portance of sharing lens model files in future studies.Even in direct comparisons, the results with one modelmay not be exactly the same as with another because ofthe difficulty in translating some of the features of onemodel to another because of the differences in featuresof the available software. For example, it is not possi-ble to parameterize a PixeLens model exactly the sameas a Lenstool model because of inherent differences inthe software. These differences may explain the observa-tions of (K¨ohlinger and Schmidt 2013) as well as someof the results in this study. Despite best efforts to sim-ilarly parameterize two models, there still may be smalldifferences. This suggests that using several models tounderstand a system may lead to improved understand-ing.2 Lefor and Futamase
TABLE 7Indirect Comparison Studies of Strong Gravitational Lens Models
PublicationsLens System Software/Reference Software/Reference Software/ReferenceAbell1689** LensPerfect ZB PixeLensCoe et al. (2010) Broadhurst et al. (2005) Saha et al. (2006)SDSSJ1004** glafic GRALE PixeLensOguri (2010) Liesenborgs et al. (2009) Saha et al. (2006)COSMOSJ095930 ∗ Lenstool Lenstool ∗ Cao et al. (2013) Faure et al. (2011)SDSSJ1430 ∗ Lensmodel / Lensview** ∗ Eichner et al. (2012)SDSSJ1320 ∗ glafic ∗ Rusu et al. (2013) ∗ indicates present study with PixeLens, Lenstool, Lensmodel, and glafic, ∗∗ indicates thatother studies of this lens have been published but are not listed. **Lensview is describedin Wayth and Webster (2006) TABLE 8Direct Comparison Studies of Strong Gravitational Lens Models
Lens System SoftwareReference 1 2 3 4 5SDSSJ1430 Eichner et al. (2012) Lensview** LensmodelAbell1703 Zitrin et al. (2010) ZB GRALEMS1358 Zitrin et al. (2011) ZB GRALEMACSJ1206 Umetsu et al. (2012) ZB Lenstool LensPerfect PixeLens SaWLens***SDSS120602 Lin et al. (2009) Lensmodel Lensview**RXJ1347.5 K¨ohlinger and Schmidt (2013) glafic PixeLensJ1000+0221 ∗ PixeLens Lenstool glafic LensmodelSDSSJ1430 ∗ PixeLens Lenstool glafic LensmodelSDSSJ1320 ∗ PixeLens Lenstool glafic LensmodelCOSMOSJ095930 ∗ PixeLens Lenstool glafic Lensmodel ∗ indicates the present study. **Lensview is described in Wayth and Webster (2006) In seeking agreement among various models, the num-ber of free parameters for the lens potentials is animportant factor. While there was reasonable agree-ment among the calculated values for Einstein radiusin single potential models, such as SDSS J1430+4105and J1000+0021 in this study, there was less agree-ment in a more complicated model such as COSMOSJ095930+023427, which may be a reflection of using morelens potentials to describe the system.Differences noted in time delay and magnification cal-culations may be due to differences in the image tracingalgorithms used by each of the software models. Theinput image positions are the same in all models. Thesoftware calculates new positions based on the softwarespecific ray-tracing algorithm used going from the sourceplane back to the image plane, resulting in differences intime delay results. The differences in optimization algo-rithms used also leads to some of the observed differencesamong the software models, with great variation in thecalculation of ellipticity and position angle.These results demonstrate that there are significantdifferences in results using lens models prepared with dif-ferent software, and are consistent with a previous studyof differences in lens models (Lefor and Futamase 2013).There is no intention to suggest that a particular group of models are necessarily more correct, but only to suggestthat future lensing studies should evaluate lens modelsusing several approaches to understand the system morethoroughly, as already being conducted in the HubbleFrontier Fields project.Based on the results of this study, in order to al-low comparisons across studies, it will be important touse a consistent nomenclature for lensing studies, spec-ifying indirect vs. direct comparisons, independent vs.semi-independent comparisons and the type of modelbeing used as LTM vs. non-LTM, as we have previ-ously described (Lefor and Futamase 2013). Further-more, this study has shown at least in one situation thatthe software version used can significantly affect the re-sults which stresses the importance of specifying the soft-ware version number being used in all future studies, inaddition to the hardware/operating system platform. Itis also suggested that more detail is provided in futurestudies to allow reproducibility of the models such as thenumber and types of potentials used along with the nameof the potential used in the various software packages.One of the most important aspects of any scientific ex-periment is reproducibility. In gravitational lens modelstudies, this is impossible in many cases because the soft-ware is not available to other investigators, or the lenstrong Lens Model Comparisons 13model files are not available. Code-sharing of software inastrophysics is essential, as emphasized by Shamir et al.(2013). Based on the studies reported here, the shar-ing of lens model files in gravitational lens studies is alsoessential to assure reproducibility and increased trans-parency in future gravitational lensing studies. Anotherapproach in lensing studies that has been successfullyapplied in weak lensing is computer challenges. The useof multiple approaches including comparative studies oflens models, open software, open lens model files, andcomputer challenges will help to assure increased trans-parency in future studies and enhance the results.4 Lefor and Futamase
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