A new strategy for matching observed and simulated lensing galaxies
MMNRAS , 1–10 () Preprint 23 February 2021 Compiled using MNRAS L A TEX style file v3.0
A new strategy for matching observed and simulatedlensing galaxies
Philipp Denzel, , (cid:63) Sampath Mukherjee, Prasenjit Saha , Institute for Computational Science, University of Zurich, CH-8057 Zurich, Switzerland Physics Institute, University of Zurich, CH-8057 Zurich, Switzerland STAR Institute, Quartier Agora - All´ee du six Ao ˆ u t, 19c B-4000 Li`ege, Belgium ABSTRACT
The study of strong-lensing systems conventionally involves constructing a mass distribution that can reproduce theobserved multiply-imaging properties. Such mass reconstructions are generically non-unique. Here, we present analternative strategy: instead of modelling the mass distribution, we search cosmological galaxy-formation simulationsfor plausible matches. In this paper we test the idea on seven well-studied lenses from the SLACS survey. For eachof these, we first pre-select a few hundred galaxies from the EAGLE simulations, using the expected Einstein radiusas an initial criterion. Then, for each of these pre-selected galaxies, we fit for the source light distribution, whileusing MCMC for the placement and orientation of the lensing galaxy, so as to reproduce the multiple images andarcs. The results indicate that the strategy is feasible, and even yields relative posterior probabilities of two differentgalaxy-formation scenarios, though these are not statistically significant yet. Extensions to other observables, suchas kinematics and colours of the stellar population in the lensing galaxy, is straightforward in principle, though wehave not attempted it yet. Scaling to arbitrarily large numbers of lenses also appears feasible. This will be especiallyrelevant for upcoming wide-field surveys, through which the number of galaxy lenses will rise possibly a hundredfold,which will overwhelm conventional modelling methods.
Key words: gravitational lensing: strong — galaxies: formation — galaxies: evolution — methods: numerical
Four decades after the first discovery by Walsh et al. (1979),galaxies exhibiting strong gravitational lensing seem almostcommonplace. The SLACS sample (Sloan Lens ACS; Boltonet al. 2006, 2008; Shu et al. 2017) alone has over a hun-dred strong lensing galaxies. The next generation of wide-field surveys (LSST/Rubin from the ground, and Euclid andWFIRST/RST in space) promise many many more. Extrap-olation from small fields that have been surveyed at differentresolutions indicate (see e.g., Collett 2015) that > (cid:48) strong-lensing galaxies will be discovered.Meanwhile, the past decade has seen significant progress onthe structure and formation of galaxies. Within the ΛCDMparadigm, there is general agreement regarding the growth ofdensity perturbations under gravity, from the level observedin the cosmic microwave background to the formation of dark-matter halos. The subsequent processes of star formation and (cid:63) Email: [email protected] the resultant feedback are less well understood and requiresub-grid models to simulate, but still the galaxies formedin simulations like Illustris (Vogelsberger et al. 2014), FIRE(Feedback In Realistic Environments; Hopkins et al. 2014),and EAGLE (Evolution and Assembly of GaLaxies and theirEnvironments; Crain et al. 2015) are much more crediblethan previous generations of simulated galaxies. The
SEAGLE pipeline (Simulating EAGLE LEnses; Mukherjee et al. 2018)producing simulated lenses from EAGLE is of particular in-terest in this work. In addition to galaxy-formation simula-tions, there are also distribution-function models for galax-ies, such as from
AGAMA (action-based galaxy modelling archi-tecture; Vasiliev 2018), which provide self-consistent phase-space distributions for dark matter, stars and gas.One would like to compare lensing observations with galaxysimulations. Let us first consider this task in a rather ab-stract way. Let F be some galaxy-formation scenario, and let D represent the observational data. In Bayesian terms, theposterior probability of F after comparison with D would be P ( F | D ) = P ( D | F ) P ( F ) P ( D ) (1)where P ( F ) represents the probability of F before the data, © The Authors a r X i v : . [ a s t r o - ph . GA ] F e b Denzel et al. and P ( D ) is the probability of the data marginalized over allpossible F . The factors P ( F ) and P ( D ) cancel if we comparetwo formation scenarios with equal prior probability, so it isreally P ( D | F ) that is of interest. This quantity is given bythe marginalisation P ( D | F ) = (cid:80) g P ( D | g ) P ( g | F ) (2)where g represents galaxy properties. There will also be nui-sance parameters (call these ν ), such as the orientation of theellipticity of a galaxy, which are also to be marginalised over,thus P ( D | g ) = (cid:80) ν P ( D | g, ν ) P ( ν ) . (3)Conventional lens modelling consists of constructing g soas to optimise P ( D | g ). Here, there are two basic approaches.One is to assume some parametric form for the lensing massdistribution and fit to the data. The idea goes back to the veryfirst lens-modelling paper (Young et al. 1980). Recent para-metric lens models (such as Yıldırım et al. 2020) are moreelaborate, but still much simplified compared to a simulationfrom AGAMA or SEAGLE . Alternatively, one can let the lensingmass distribution be free-form, and sample the abstract spaceof mass distributions that fit the data. This approach is morecommon in cluster lensing (see e.g., Wagner et al. 2019), butalso used in galaxy lensing (e.g., K¨ung et al. 2018). Free-formmass models are more complex, but they are not necessar-ily dynamically plausible. Neither style of lens modelling hasmuch input from P ( g | F ). Some comparisons of lens mod-els against dynamical simulations of galaxies have been done(e.g. Saha et al. 2006; Barnab`e et al. 2009; Coles et al. 2014;Ding et al. 2020), as have some model-independent compar-isons of image statistics with substructure in ΛCDM (Gomer& Williams 2017), but all of these provide only qualitativeinformation with respect to P ( g | F ).In this work, we attempt for the first time a direct com-parison of lensing data and galaxy-formation without con-ventional lens models. We use SEAGLE lenses as samples of P ( g | F ) from two different galaxy-formation scenarios. Wethen formulate P ( D | g, ν ) so that a procedure for fittingsource brightness distributions (developed earlier for conven-tional lens modelling Denzel et al. 2020b; Denzel et al. 2020a)can be repurposed. This allows us to find EAGLE galaxiesthat can account for the observed images in a small test sam-ple of seven SLACS lenses (see Table 1). As this work isintended as proof of concept, we do not include data otherthan multiple images from extended sources.The following Section 2 introduces what we may callthe P ( D | F ) method. The subsequent Section 3 details the SEAGLE pipeline and summarizes how the catalogue of surface-density maps was compiled. The selected test-case lenses fromthe SLACS survey are presented in Section 4, and the resultsof these tests are reported in Section 5. Finally, a summaryand discussion, in particular about possible expansions andapplications of the lens-matching approach are given in Sec-tion 6.
To go beyond the simple abstractions above and discuss theactual method, let us rewrite Eqs. (2) and (3) as P ( D | F ) ≈ (cid:88) s,ξ P (cid:0) I obs (cid:12)(cid:12) α , s (cid:1) P (cid:0) α , (cid:12)(cid:12) ξ, F (cid:1) P ( s, ξ ) . (4)Rather than galaxy properties g in general, we are concernedwith a lensing deflection field α . The ν parameters consistof (a) location and rotation parameters (say ξ ) to produce α from a simulated EAGLE galaxy, and (b) the unlensedbrightness distribution s at the source redshift. The priors P ( s, ξ ) we take as flat. Hence it is on the factors P ( D | α , s )and P ( α | ξ, F ) that we must concentrate. We now describe the ingredients for the factor P (cid:0) α , (cid:12)(cid:12) ξ, F (cid:1) in Eq. (4).The convergence map (that is, the lensing mass distributionin dimensionless form) is given by the usual projection of the3D mass density as κ ( θ , ξ ) = 4 πGcH d LS d L d S (cid:90) ρ ( θ , ξ, z ) d z . (5)Here, θ is the angle on the observer’s sky, d LS is the di-mensionless angular-diameter distance from the lens to thesource, d L and d S are analogous, and ξ represents the loca-tion and orientation of the 3D density ρ ( θ , z ). A conventionalΛCDM cosmology is assumed.In this work, we have limited the analysis to two galaxy-formation scenarios from the EAGLE simulations (details arein Section 3 below). From the two simulations, SEAGLE pro-jected each of 554 simulated galaxies along three orthogonalaxes to produce maps of κ ( θ ) for the fiducial redshift values z L = 0 .
23 and z S = 0 .
8. The κ maps have 161 ×
161 squarepixels with a pixel size of 0 . (cid:48)(cid:48) , yielding an angular size ofabout 8 (cid:48)(cid:48) × (cid:48)(cid:48) for an entire map. The κ distributions differin size and in shape. Figure 1 shows the distribution of meanenclosed κ as a function of radial distance from lens centrefor the entire catalogue. The notional Einstein radius is thevalue of θ E for which (cid:104) κ (cid:105) θ E = 1. About 20 mass maps in thecatalogue are always below κ = 1 and hence are not stronglylensing for z L = 0 . , z S = 0 .
8. But most of the galaxiescan produce multiple images, and the Einstein radii go up to3 . (cid:48)(cid:48) .The convergence maps are then rescaled from z L =0 . , z S = 0 . κ maps with Einstein radius in the expected rangefor each system was then selected for further processing.The next step was the computation of the lens potential ψ ( θ ) = 2 ∇ − Q ( θ ). To reduce the computational time re-quired, the κ maps were discretised to 23 ×
23 tiles. Thepotential is then expressed as ψ ( θ ) = 2 (cid:88) n κ n ∇ − Q ( θ − θ n ) (6)where κ n is the density of the n -th tile and Q ( θ − θ n ) isthe contribution of a square tile with constant κ = 1 located MNRAS , 1–10 () atching observed and simulated lensing galaxies θ [arcsec] ›fi θ Figure 1.
The distribution of radial profiles of the mean enclosed (cid:104) κ (cid:105) within a given projected radius from the centre of the galax-ies, for the entire catalogue, assuming z L = 0 . , z S = 0 .
8. Thehorizontal line at κ = 1 indicates the notional Einstein radii. at θ n . The functional form of Q ( θ ) is given in AbdelSalamet al. (1998). Note that only the mass distribution is reducedin resolution in this way, but θ and ψ ( θ ) can still be evaluatedat any desired resolution. The effect of the approximation (6)is expected to be very small.Once the lens potential is known we have the deflectionangle as α ( θ ) = ∇ ψ ( θ ) . (7) We now consider the factor P (cid:0) I obs (cid:12)(cid:12) α , s (cid:1) in Eq. (4).As a result of the deflection (7) a light ray originating at asource at β on the sky will be observed at θ which is relatedto β by the usual lens equation β = θ − α ( θ ) . (8)The lens equation amounts to a mapping L ( θ , β ) between thesource and image planes, which can be discretised as a ma-trix. Any given θ corresponds to a unique β , whereas a given β may correspond to more than one θ . A source-brightnessdistribution s ( β ) produces an image-brightness distribution I ( θ ) = (cid:90) L ( θ , β ) s ( β ) d β . (9)The observed image brightness will involve a further convo-lution with the point-spread function (PSF) P ( θ − θ (cid:48) ) of thetelescope and camera. The result¯ I ( θ ) = (cid:90) P ( θ − θ (cid:48) ) I ( θ (cid:48) ) d θ (cid:48) (10)we will call the synthetic image, and it is what will get com-pared with the data.For the lens sample investigated here, appropriate PSFshave been employed which were modelled using tinytim (Krist et al. 2011).Assuming now that the detector noise is Gaussian withknown σ θ ∝ (cid:112) I obs ( θ ) we take P (cid:0) I obs (cid:12)(cid:12) α , s (cid:1) ∝ exp (cid:0) − χ (cid:1) (11) (cid:135) https://github.com/spacetelescope/tinytim where χ = (cid:88) θ σ − θ (cid:104) I obs ( θ ) − ¯ I ( θ , α , s ) (cid:105) (12)From Eqs. (9) and (10) it is clear that the synthetic im-age ¯ I ( θ ) is linear in the source-brightness distribution, eventhough it is completely non-linear in the mass distribution.Hence s ( β ) can be solved to optimise P (cid:0) I obs (cid:12)(cid:12) α , s (cid:1) by lin-ear least-squares. It is important, however, to mask the lightfrom the lensing galaxy, since it is not part of I obs ( θ ).As the lensed images are typically highly magnified, thesource or β plane needs much smaller pixels than the imageor θ plane. For this reason, the lens mapping L ( θ , β ) mapseach θ pixel to a cluster of β pixels. To simplify the com-putation, we replace each θ pixel by its central point for thepurposes of the lens mapping. Then each image pixel mapsto a single source pixel. This procedure leaves many sourcepixels ‘blank’, because they send light to edges and cornersof the image pixels. These blank pixels could be filled in byinterpolation, but in this paper we have not done so. As a re-sult, the reconstructed sources have a fragmented appearanceon small scales, as we will see later in Figs. 4–10.While the fitting of synthetic images is essentially the sameas in conventional lensing modelling (our implementation isthe same as in Denzel et al. 2020b; Denzel et al. 2020a),plausible-matching requires a further issue to be solved,namely the alignment and orientation of the lens system. Thenuisance parameters ξ = ( p rel , φ rel ), where p rel is the position, φ rel is the orientation of the mass map relative to the obser-vation, needs to be marginalised out. The marginalisation isdone using short Markov-Chain Monte-Carlo (MCMC) simu-lations. The result is an ensemble of plausible-matches, remi-niscent of model ensembles in free-form lens modelling (Saha& Williams 2004; Coles et al. 2014) but having a differentmeaning, because they arise from galaxy-formation simula-tions.The minimum of χ in Eq. (12) need not correspond to aunique α . In other words, very different galaxies can in prin-ciple produce identical synthetic images. This is the well-known problem of lensing degeneracies (for a review, seeWagner 2018). The plausible matching strategy automati-cally marginalises over simulated galaxies that are degener-ate in the observables, so lensing degeneracies as such are notan obstacle to the method. If, however, the differences be-tween galaxy-formation scenarios happen to be aligned alonglensing degeneracies, lensing observables would be ineffectiveas discriminators between galaxy-formation models. Such athing seems unlikely, but we cannot rule it out at present.In total, 11634 MCMC simulations had to be executed untilthe solutions for all simulated galaxies and lens systems con-verged. This was relatively easily achieved within about 4–8hours per lens through some optimisations and some compro-mises. The inclusion of a PSF increases the non-sparseness ofthe synthetic-image mapping considerably, makes the genera-tion of synthetics quite computationally intensive, and slowsdown the MCMC simulations by an average factor of ∼ p rel neverdeviated from the centre by more than 0 . (cid:48)(cid:48) , which leadus to discard that parameter in the final stage. The con- MNRAS000
8. Thehorizontal line at κ = 1 indicates the notional Einstein radii. at θ n . The functional form of Q ( θ ) is given in AbdelSalamet al. (1998). Note that only the mass distribution is reducedin resolution in this way, but θ and ψ ( θ ) can still be evaluatedat any desired resolution. The effect of the approximation (6)is expected to be very small.Once the lens potential is known we have the deflectionangle as α ( θ ) = ∇ ψ ( θ ) . (7) We now consider the factor P (cid:0) I obs (cid:12)(cid:12) α , s (cid:1) in Eq. (4).As a result of the deflection (7) a light ray originating at asource at β on the sky will be observed at θ which is relatedto β by the usual lens equation β = θ − α ( θ ) . (8)The lens equation amounts to a mapping L ( θ , β ) between thesource and image planes, which can be discretised as a ma-trix. Any given θ corresponds to a unique β , whereas a given β may correspond to more than one θ . A source-brightnessdistribution s ( β ) produces an image-brightness distribution I ( θ ) = (cid:90) L ( θ , β ) s ( β ) d β . (9)The observed image brightness will involve a further convo-lution with the point-spread function (PSF) P ( θ − θ (cid:48) ) of thetelescope and camera. The result¯ I ( θ ) = (cid:90) P ( θ − θ (cid:48) ) I ( θ (cid:48) ) d θ (cid:48) (10)we will call the synthetic image, and it is what will get com-pared with the data.For the lens sample investigated here, appropriate PSFshave been employed which were modelled using tinytim (Krist et al. 2011).Assuming now that the detector noise is Gaussian withknown σ θ ∝ (cid:112) I obs ( θ ) we take P (cid:0) I obs (cid:12)(cid:12) α , s (cid:1) ∝ exp (cid:0) − χ (cid:1) (11) (cid:135) https://github.com/spacetelescope/tinytim where χ = (cid:88) θ σ − θ (cid:104) I obs ( θ ) − ¯ I ( θ , α , s ) (cid:105) (12)From Eqs. (9) and (10) it is clear that the synthetic im-age ¯ I ( θ ) is linear in the source-brightness distribution, eventhough it is completely non-linear in the mass distribution.Hence s ( β ) can be solved to optimise P (cid:0) I obs (cid:12)(cid:12) α , s (cid:1) by lin-ear least-squares. It is important, however, to mask the lightfrom the lensing galaxy, since it is not part of I obs ( θ ).As the lensed images are typically highly magnified, thesource or β plane needs much smaller pixels than the imageor θ plane. For this reason, the lens mapping L ( θ , β ) mapseach θ pixel to a cluster of β pixels. To simplify the com-putation, we replace each θ pixel by its central point for thepurposes of the lens mapping. Then each image pixel mapsto a single source pixel. This procedure leaves many sourcepixels ‘blank’, because they send light to edges and cornersof the image pixels. These blank pixels could be filled in byinterpolation, but in this paper we have not done so. As a re-sult, the reconstructed sources have a fragmented appearanceon small scales, as we will see later in Figs. 4–10.While the fitting of synthetic images is essentially the sameas in conventional lensing modelling (our implementation isthe same as in Denzel et al. 2020b; Denzel et al. 2020a),plausible-matching requires a further issue to be solved,namely the alignment and orientation of the lens system. Thenuisance parameters ξ = ( p rel , φ rel ), where p rel is the position, φ rel is the orientation of the mass map relative to the obser-vation, needs to be marginalised out. The marginalisation isdone using short Markov-Chain Monte-Carlo (MCMC) simu-lations. The result is an ensemble of plausible-matches, remi-niscent of model ensembles in free-form lens modelling (Saha& Williams 2004; Coles et al. 2014) but having a differentmeaning, because they arise from galaxy-formation simula-tions.The minimum of χ in Eq. (12) need not correspond to aunique α . In other words, very different galaxies can in prin-ciple produce identical synthetic images. This is the well-known problem of lensing degeneracies (for a review, seeWagner 2018). The plausible matching strategy automati-cally marginalises over simulated galaxies that are degener-ate in the observables, so lensing degeneracies as such are notan obstacle to the method. If, however, the differences be-tween galaxy-formation scenarios happen to be aligned alonglensing degeneracies, lensing observables would be ineffectiveas discriminators between galaxy-formation models. Such athing seems unlikely, but we cannot rule it out at present.In total, 11634 MCMC simulations had to be executed untilthe solutions for all simulated galaxies and lens systems con-verged. This was relatively easily achieved within about 4–8hours per lens through some optimisations and some compro-mises. The inclusion of a PSF increases the non-sparseness ofthe synthetic-image mapping considerably, makes the genera-tion of synthetics quite computationally intensive, and slowsdown the MCMC simulations by an average factor of ∼ p rel neverdeviated from the centre by more than 0 . (cid:48)(cid:48) , which leadus to discard that parameter in the final stage. The con- MNRAS000 , 1–10 ()
Denzel et al. vergence to optimal alignment rotation angles on the otherhand was more relevant, especially for galaxies with high el-lipticity, whereas for round galaxies the rotation angles werearbitrary and usually settled around 0 ° .The subsequently described lens-matching method hasbeen implemented in the public software gleam (Gravita-tional Lens Extended Analysis Module) by PD. It is writtenin Python and thus comes with all of its flexibility and a largescientific library support. Computationally demanding taskssuch as the calculation of potential gradients are alternativelyalso implemented in a mixture of C and Cython (Behnel et al.2011). Similar to the lens modelling tool GLASS by Coleset al. (2014), the module encompasses more general features,some of which are still in development, but the lens-matchingtechnique lies at its core. In particular, the synthetic imagerdescribed in Subsection 2.2 is implemented in the sub-module gleam.reconsrc .The entire lens-matching method was intentionally keptrelatively simple and lightweight in order to keep it scalablefor a much bigger lens sample using larger catalogues andto minimize the input required from the outside. Figure 2shows a schematic graph which summarizes each key stepof the lens-matching method. The analysis presented hereaimed for a proof-of-concept only. With a working basis, fur-ther refinements and improvements can easily be explored inisolation and afterwards properly implemented. In Section 6,we give some suggestions of what aspects could be improvedfirst. Mukherjee et al. (2018) introduced the
SEAGLE pipeline tosystematically study galaxy formation via simulated stronglenses from the EAGLE simulations (Schaye et al. 2014; Crainet al. 2015; McAlpine et al. 2016).
SEAGLE used the
GLAMER ray-tracing package (Gravitational Lensing with AdaptiveMesh Refinement; Metcalf & Petkova 2014; Petkova et al.2014) to create realistic lensed images and calculate all otherlensing quantities used in their analysis.
SEAGLE aims to inves-tigate and possibly disentangle galaxy formation and evolu-tion mechanisms by creating, modelling, and analysing simu-lated strong lens-galaxies to compare them with observations.EAGLE is a suite of state-of-the-art hydrodynamical sim-ulations that explored several feedback scenarios and modelvariations giving us a set of galaxy evolution scenarios to as-sess their impact on the present-day universe. Crain et al.(2015) divided the simulations into two categories. The firstcomprises four simulations calibrated to yield the z = 0 . SEAGLE , the authors quan-tified that if the simulated lensed images are modelled similarto the observations, then the median total mass density slope (cid:135) https://github.com/phdenzel/gleam (cid:135) https://github.com/jpcoles/glass SEAGLEgleamhydrodynamical simulations(EAGLE)model variationsgalaxy mass projections in x, y, zobservational parameterscatalogue of convergence mapssource deprojectionssynthetic imagesBayesian MCMCplausible lens matcheslens observationsP(D | F)
Figure 2.
A schematic top-bottom graph summarizing the specificsteps of the lens-matching method in this work. The pipeline wasintentionally kept quite modular; in particular, the components inthe blue and yellow coloured subgraphs are specific choices of thiswork, and could in principle be replaced by equivalent operators.Red fields represent independent data inputs from observations orsimulations. of galaxies from an inefficient AGN feedback model (AG-NdT8: Reference variation) and a constant feedback model(FBconst: Calibrated simulation) that becomes inefficient atdenser environment gives slopes t =2.01 and t =2.07, respec-tively, in good agreement with the observations of SLACS,SL2S (Strong Lensing Legacy Survey), and BELLS (BaryonOscillation Spectroscopic Survey (BOSS) Emission-Line LensSurvey). Galaxies in the EAGLE Reference model (bench-mark model), however, tend to have a steeper median totalmass density slope ( t =2.24) than observed lens galaxies (i.e. t =2.08 for SLACS, t =2.11 for BELLS and t =2.18 for SL2S).The nomenclature of the SEAGLE -projected mass distribu-tions in the catalogue depends on their halo, subhalo, andprojection axis. A number following ’H’ refers to the halonumber, ’S’ gives the subhalo, and letters ’A/B/G’ refers tothe projection the galaxy has undergone in Cartesian coor-dinates i.e. α , β and γ respectively. The feedback model de-nominations are prepended in this nomenclature.For our analysis, we choose these two galaxy evolution sce-narios (AGNdT8 and FBconst) as they are most realistic tothe strong lensing observations. We briefly discuss the keyfeatures of these feedback models below.In the calibrated simulations, the models differ in terms oftheir adopted efficiency of feedback associated with star for-mation, and how this efficiency depends upon the local en-vironment. The general consensus shows that the propertiesof simulated galaxies are most sensitive to the efficiency of MNRAS , 1–10 () atching observed and simulated lensing galaxies baryonic feedback (see e.g., Schaye et al. 2010; Vogelsbergeret al. 2013). Below a certain resolution limit, the physicalprocesses cannot be simulated via the dynamics of the par-ticles. So they are incorporated via analytic prescriptions inall hydro-dynamic simulations including EAGLE. In EAGLEmodel variations, the efficiency of the stellar feedback andthe BH accretion were calibrated to broadly match the ob-served local ( z ≈
0) GSMF. Also, several studies establishedthat AGN feedback is a necessary ingredient for regulatingthe growth of massive galaxies (e.g. Crain et al. 2009; Schayeet al. 2010; Haas et al. 2013).Below we briefly describe the EAGLE galaxy formationmodels which were used in this work.
The simplest feedback model used in EAGLE is FBconst.In this calibrated model, independently from the local con-ditions, a fixed amount of energy per unit stellar mass isinjected into the ISM. This fixed energy corresponds to thetotal energy discharged by type-II SNe ( f th = 1). While thestellar feedback in this model was not calibrated, the modeldoes reproduce the observables used for the calibration. Crainet al. (2015) found that the thermal stellar feedback prescrip-tion employed in EAGLE becomes inefficient at high gas den-sities due to resolution effects (Vecchia & Schaye 2012). Thusin this model, there is a lack of compensation for more energyat higher gas density. Thus the stellar feedback will be lesseffective in high-mass galaxies (where the gas tends to havehigher densities) (Crain et al. 2015).Schaye et al. (2014) demonstrated that it is possible tocalibrate the Reference model to reproduce the GSMF andthe observed sizes (in different bands) of galaxies at z = 0.1.Crain et al. (2015) conducted a series of simulations (listedin the lower section of Table 1 therein) for which the valueof a single parameter was varied from that adopted in theReference model. One of the parameters varied was AGNtemperature. Schaye et al. (2014) have examined the role of the AGNheating temperature in EAGLE by adopting ∆ T AGN =10 . K and 10 K. They demonstrated that a higher heat-ing temperature produces less frequent but more energeticAGN feedback episodes. They concluded it is necessary toreproduce the gas fractions and X-ray luminosities of galaxygroups. Brun et al. (2014) also concluded that higher heatingtemperature yields more efficient AGN feedback. There aretwo Reference-model variation simulations with ∆ T AGN =10 K (AGNdT8) and ∆ T AGN = 10 K (AGNdT9), besides theReference model itself which adopted ∆ T AGN = 10 . K. Inmassive galaxies, the heating events (less frequent but moreenergetic) are more effective at regulating star formation dueto a higher heating temperature. AGNdT8 (AGNdT9) modelhas a higher (lower) peak star fraction compared to the Refer-ence model. The reduced efficiency of AGN feedback, when alower heating temperature is adopted, leads to the formationof more compact galaxies because gas can more easily accreteonto the centers of galaxies and form stars. Mukherjee et al. (2018) showed that for galaxy-galaxy strong lenses, AGNdT8produces closest analogs for SLACS. Thus, for this work, weuse galaxies from the AGNdT8 simulation, in addition to thegalaxies from the simpler FBconst model.
In order to test whether searching for plausible matches fromEAGLE simulations is at all feasible, we selected a small sam-ple of seven lens systems that have already been studied byother methods. The selection was based on three criteria.First, the system had to be clearly strongly lensed, with rel-atively easily identifiable images showing very clear evidenceof multiple imaging. Second, the observations needed to haveextended images and arcs with some imperfections (ratherthan point-like lensed quasars) so as to challenge the match-ing technique. Third, the sample had to be representative ofa larger sample of lenses. The third criterion made it naturalto choose from SLACS, and from the SLACS lenses of qual-ity category “A” we selected seven, having a wide range ofmean image radii, which is a rough proxy for Einstein radii.Figs. 4–10 in their top left panels show the lensed imagesin HST-image F814W bands. The most relevant informationabout the systems is listed in Table 1, including references tothe discovery papers. − SDSS
J0029 − appears to be a relatively small, doublylensing system observed on 12 September 2006. Initial re-ports by Bolton et al. (2008) classify it as a single, early-typegalaxy. The redshift of the foreground galaxy was spectro-scopically measured to z L = 0 . z S = 0 . (cid:48)(cid:48) . It also has a well measured stellar velocity dispersionof σ SDSS = 229 ±
18 kms − .The initial report presented a singular isothermal ellipsoidand light-traces-mass gravitational lens model which pro-vided best fits using two source-plane components. However,the present work indicates that a single-component source(see second image in the left column in Figure 4) is also pos-sible. The top image in Figure 4 in the left column shows thesystem from the HST/ACS-WFC1 observation (AdvancedCamera System Wide Field Channel 1) using the F814Wfilter. SDSS
J0737+3216 appeared in the first SLACS report byBolton et al. (2006). A successive report (Bolton et al. 2008)grades the quality of the single-multiplicity, early-type galaxyto be of type “A”. Its foreground and background redshiftswere measured to z L = 0 . z S = 0 . σ SDSS = 338 ±
17 kms − was provided.Parametric lens models from Bolton et al. (2008) used twosource-plane components to fit the astrometric data. The MNRAS000
17 kms − was provided.Parametric lens models from Bolton et al. (2008) used twosource-plane components to fit the astrometric data. The MNRAS000 , 1–10 ()
Denzel et al.
Name R.A. [hms] Decl. [dms] z l z s σ SDSS [km/s] R eff [arcsec] ReferenceSDSSJ0029 − − ±
18 2.16 (1)SDSSJ0737+3216 07:37:28.5 +32:16:18 0.32 0.58 310 ±
15 2.16 (2)SDSSJ0753+3416 07:53:46.2 +34:16:33 0.14 0.96 208 ±
12 1.89 (3)SDSSJ0956+5100 09:56:29.8 +51:00:06 0.24 0.47 299 ±
16 2.33 (2)SDSSJ1051+4439 10:51:09.4 +44:39:08 0.16 0.54 216 ±
16 1.66 (3)SDSSJ1430+6104 14:30:34.8 +61:04:04 0.17 0.65 180 ±
15 2.24 (3)SDSSJ1627 − − ±
12 2.08 (2)
Table 1.
Summary of the observed lensing systems. (1) Bolton et al. (2008); (2) Bolton et al. (2006); (3) Shu et al. (2017)
HST/ACS-WFC1 observation (on 21 September 2004) us-ing the F814W filter is displayed in the top image in the leftcolumn of Figure 5. It shows two extended images of whichthe brighter image is most likely the product of two mergedones, and a point-like image connected via a dim arc, whichwould most likely classify it as a short-axis quad. Besidesthe initial modelling, Ferreras et al. (2007) provided a free-form lens model for this lens system, along with a spatiallyresolved comparison to the stellar-mass surface distributionderived from population-synthesis models.
The top image of the left column in Figure 6 showsSDSS
J0753+3416 (HST/ACS WFC1 F814W on 8 Septem-ber 2010) as a clearly lensing system. It is a very interestingsystem with minimum eight (possibly even 12) lensed imagesof at least two sources. Shu et al. (2017) reports ellipsoid lensmodels using even four source-plane components. Either way,this lens promises a much lower degree of degeneracy due tothe high number of lensed images and sources.From the SDSS observations the lensing galaxy was classi-fied as an early-type, single-multiplicity foreground galaxywith a well measured velocity dispersion σ SDSS = 206 ±
11 kms − . The redshift estimates for the lens and source are z L = 0 . z S = 0 . In the left column of Figure 7, the top image showsSDSS
J0956+5100 from the HST/ACS-WFC1 F814W ob-servation from 1 November 2006. Bolton et al. (2006) des-ignates its lens an early-type, single foreground galaxy. Thespectroscopic survey yielded a velocity dispersion of σ SDSS =299 ±
16 kms − and redshifts of z L = 0 . z S = 0 . R M of M tot ( < R M ) = 66 . +25 . − . · M (cid:12) and M star ( < R M ) = 41 . +4 . − . · M (cid:12) , where the aperture radius is 2 R max − R min , the difference of projected radii of twice the outermost and innermost lensimages. Shu et al. (2017) reports SDSS
J1051+4439 as another early-type, single foreground-galaxy lens system. The SDSS datayields a velocity dispersion value of σ SDSS = 216 ±
16 kms − ;the redshifts of the lens and background source are reportedwith z L = 0 . z S = 0 . Figure 9 (first image in the left column) depictsSDSS
J1430+6104 (HST/ACS-WFC1 F813W) as a verynoisy lens system with faint lensed images, with consid-erable pollution by the host galaxy. Shu et al. (2017) re-ports the early-type galaxy with a velocity dispersion valueof σ SDSS = 180 ±
15 kms − . The SDSS redshifts for theforeground and background objects are z L = 0 . z S = 0 . . · M (cid:12) . − On 12 March 2006, the HST/ACS-WFC1 observedSDSS
J1627 − as a double with an almost completelyclosed ring. Bolton et al. (2006) reported it as an early-typeforeground galaxy in a single-multiplicity system with red-shifts z L = 0 . z S = 0 . σ SDSS = 290 ±
15 kms − . A picture of the lens system canbe found on the top panel of the left column in Figure 10. The results on plausible matches for the seven lens systemsconsidered are displayed in Figs. 4–10 and summarised inTable 2. Figs. 4–10 are devoted to one lens each, in the sameorder as in Table 1.Each of these figures has eight panels, arranged as follows.
MNRAS , 1–10 () atching observed and simulated lensing galaxies Observed image Synthetic imageSource brightness Residual ImageLensing mass Pixelized lensing massLensing Roche potential Mean enclosed convergenceWe now discuss properties of the most-plausible matchesas shown in Figs. 4–10.
The top row of Figs. 4–10 shows the observed lensed imagesand the synthetic image from the most-plausible match. Thelensing galaxy is masked out. The difference between these,scaled by the noise — in other words, the pixelwise χ fromEq. (12) — is shown in the right panel of the second row.For the MCMC over the orientation φ rel it is computa-tionally simpler to rotate the image rather than the lens. Asa result, there are some rotated-corner artifacts, less notice-able in the synthetic images, but at the edges of the sourceplane, especially for example in Figure 8. These are, however,harmless χ computation, for which only a circular region wasconsidered.In all of the second-row right panels, it is evident thatthe contributions to χ come mainly from an annular regionwhere the multiply-imaged features are. The black inner discis of course just the masked-out lensing galaxy. The outer partin the χ maps is dark (or at a lower level) because withoutmultiple images the source brightness has the trivial solution s ( θ − α ( θ )) = I obs ( θ ) (13)and then any contribution to χ comes only because there arefewer pixels in the source plane than in the image plane. Itwould be better to consider only the multiply-imaged regionwhen computing χ , but it is not clear how to do so efficiently. The left panel in the second row in each of Figs. 4–10 showsthe reconstructed s ( β ) for the most-plausible match. Thesources appear fragmented on small scales because of a dis-cretisation artefact explained in Subsection 2.2 which we havenot interpolated out.The source-fitting as implemented here does not guaran-tee that the reconstructed source will be blob-like and not arandom scatter plot. However, plausible matching lenses aregenerally associated with plausible looking source maps (dis-regarding the small-scale fragmentation). In cases where thedata are more noisy the source plane also tends to be noisy;this is especially noticeable in J0737+3216 , J1051+4439 ,and
J1430+6104 (Figures 5, 8, and 9), where the brightspecks from the data images (probably artifacts left by cosmicparticles) also appear in the source plane. While most sourcesseem to be rather symmetric,
J0753+3416 (Figure 6) and
J0956+5100 (Figure 7) appear to be amorphous with mul-tiple cores. This could be an indication of a merging systemor multi-component source, however further investigation isneeded to confirm this.A curious artifact appears in the cases of
J0956+5100 (Figure 7) and
J1051+4439 (Figure 8). There the source ap-pears to have bright edges in a curved diamond shape. Thecurved edges evidently correspond to the diamond caustic for four-image lenses, which correspond extreme magnification,and single pixels along these edges can map to large areas onthe image plane. We conjecture that the source-fitting pro-cedure is using this property of caustics to fit noise in theimages.In comparison with source reconstructions from previousworks, some differences are noticeable. In most cases, gen-eral shapes of the sources agree with previous works, whennoise is ignored, especially for
J0029 − . For J0753+3416 and
J1430+6104 the main cores exhibit similar shapes, butprevious works include more secondary sources compared tomost source reconstructions here. Contrarily,
J0956+5100 ,although being very noisy, seems to exhibit more componentsthan reconstructions from previous works.
The third row in each of Figs. 4–10 shows the κ maps from SEAGLE and the reduced-resolution κ n maps that we actu-ally used, for the most-plausible match. The dark contoursindicate κ = 1.Interestingly, while the catalogue did include manyprojected surface-densities with high ellipticity, the lens-matching approach seems to preferentially select rather roundmodels. However, this of course depends on the selection ofthe lens system and considering to the light profiles of thelenses in the data, mass distributions with low ellipticity wereto be expected. The mass models do, however, exhibit a mod-erate amount of substructure.The bottom-right panel in each of Figs. 4–10 shows themean enclosed density (cid:104) κ (cid:105) θ within a given angular radius forthe 10 most plausible matches in the sense of χ . As in Fig-ure 1 (cid:104) κ (cid:105) θ = 1 is understood as the Einstein radius. The valueis well-constrained, even if we consider the 50 most-plausiblematches as illustrated, or in a subset of best-matching modelswith χ ν < The bottom-left panels in Figs. 4–10 show another interestingquantity, a contour map of the lensing Roche potential P ( θ ) = θ − ψ ( θ ) (14)which we introduced in Denzel et al. (2020b). The lens equa-tion (8) is equivalent to β = ∇P ( θ ) (15)and consequently the points where ∇P = 0 are image lo-cations from a source at β = 0. These points are extrema(minima, maxima, and saddle-points) of ∇P and easy to dis-cern on a contour map. The actual image positions will besomewhat different, depending on the details of s ( β ), butnevertheless, the contours of the lensing Roche potential of-fer a simple confirmation that a plausible match is indeed astrongly lensing system, and that we have not simply stum-bled upon the trivial solution (13). Every pre-selected model was match-tested against the ob-servational data of each lens according to Eq. (12), which
MNRAS000
MNRAS000 , 1–10 ()
Denzel et al.
J0029-0055 J0737+3216
J0753+3416 J0956+5100
J1051+4439
J1430+6104 n m o d e l s J1627-0053 all modelsFBconstAGNdT8
Figure 3.
Cumulative χ ν histograms for all match-tested mod-els. The dotted lines show the fractions of models with feedbackscheme AGNdT8, whereas the dashed lines are for FBconst mod-els. The numbers of models n models were normalized by the totalnumbers of models, preselected for each lens individually. Mod-els with high χ ν are considered bad matches to the observations.For instance, for the lens systems J0737+3216 and
J0956+5100 only a few plausible matches have been found within the modelcatalogue. yielded distributions of reduced least squares χ ν . Figure 3shows these distributions as cumulative histograms, includ-ing the fractions of models from the two galaxy-formationscenarios, FBconst and AGNdT8.Subsets of most-plausible matches, that is, matches withminimal χ ν , are likely to contain models from both galaxy-formation scenarios, evident in Table 2 and Figure 3.Considering the most-plausible matches with e.g. χ ν < P (AGNdT8 | D ) P (FBconst | D )for each lens. Using the values from Table 2, this is between 0.6 and 0.7 for J0029 − , J0956+5100 , J1051+4439 ,and
J1430+6104 , meaning these systems show a slight ten-dency towards FBconst. For
J0737+3216 and
J0753+3416 the expression above evaluates to well above 1.0, indicatinga tendency towards AGNdT8, whereas for
J1627 − it isvery close to 1.0. Although, it should be noted that thesevalues are not significant yet (especially for J0737+3216 and
J0956+5100 ) and for better statistics more matchingtests should be performed. With more match-tests and bet-ter statistics, the criterion χ ν < χ ν above 5start to display various noticeable deficits in the source recon-structions and synthetic images and are therefore not suitableto estimate the relative posterior.Table 2 also lists the subsets’ median values of the Ein-stein radii which is a measurement of the total mass the lens.These values are consistent with previous studies (Boltonet al. 2008, 2006; Shu et al. 2017; Ferreras et al. 2007). Thecomparison of the median stellar masses of the model galax-ies with previous estimates also seem to agree well, if it isconsidered that previous estimates are within an Einstein ra-dius or half-light radius of the lensing galaxies, whereas forour models it is possible to estimate the entire mass in stars. Mass reconstructions in gravitational lensing are in gen-eral non-unique. Even for strong-lensing clusters with tensof multiply-imaged systems over a range of redshifts, thereis significant scatter among mass models even if they fitthe data equally well (see e.g., Meneghetti et al. 2017). Forgalaxy lenses the non-uniqueness of models is much more ev-ident, and indeed has been known since the earliest days oflens modelling (Young et al. 1981). This facts suggests thatthe large catalogues of simulated galaxies in recent galaxy-formation simulations may contain plausible matches to in-dividual observed lensing galaxies. In this work we search forand find plausible matches among EAGLE simulated galax-ies to seven observed lensing galaxies from SLACS. The maincomputational part is to fit for (a) an orientation of a givensimulated galaxy and (b) a source light distribution, such thatthe observed light distribution is reproduced. This is imple-mented in the new gleam code, but automated lens-modellingtools such as
AutoLens (Nightingale et al. 2018) and
Ensai (Hezaveh et al. 2017) could probably also be adapted for thepurpose, if required.The main conclusion of this work is that EAGLE — andpresumably other comparable galaxy-formation simulations— contain plausible matches for observed lensing galaxies.Hence it appears feasible to use observed lensing galaxies asconstraints on galaxy-formation scenarios, without conven-tional lens models. Obtaining statistically significant results,however, will need several issues to be addressed first, whichwe discuss briefly below.(i) In this work we have used single simulated galaxies, dis-regarding the environment and line-of-sight structures, andalso approximated the projected mass as consisting of 23 × MNRAS , 1–10 () atching observed and simulated lensing galaxies Lens N χ ν < N χ ν < χ ν χ ν δφ rel θ E M stel AGNdT8 FBconst AGNdT8 FBconst [ ° ] [ (cid:48)(cid:48) ] [10 M (cid:12) ]SDSSJ0029 − +3 . − . +0 . − . +0 . − . SDSSJ0737+3216 6 2 3.74 3.47 4.8 +0 . − . +0 . − . +1 . − . SDSSJ0753+3416 61 48 2.78 2.84 15.8 +3 . − . +0 . − . +0 . − . SDSSJ0956+5100 4 6 3.50 3.68 3.7 +1 . − . +0 . − . +1 . − . SDSSJ1051+4439 17 24 2.90 2.69 9.4 +2 . − . +0 . − . +0 . − . SDSSJ1430+6104 41 58 2.49 2.65 5.1 +1 . − . +0 . − . +0 . − . SDSSJ1627 − +5 . − . +0 . − . +0 . − . Table 2.
Results for a subset of the most plausible matches for each lens system with χ ν <
5. The reduced χ ν apply to the best syntheticimages of the matching tests, δφ rel are the average deviations in orientations about the line of sight of the 68% interval from the MCMCof all models in the subset. The θ E column contains medians of Einstein radii and the M stel column medians of the total mass in stars ofthe simulated galaxy models, with uncertainties covering the 68% interval of the model subset. about three orthogonal candidate lines of sight, rather thanarbitrary orientations in 3D. All these aspects of the imple-mentation need to be improved, while keeping it efficient forthe purpose of scaling up to larger lens samples.(ii) The source reconstruction is another area that can be im-proved. The advantage of the procedure used in this work isthat supervision at the level of individual lenses is not re-quired, though this will not be true if the observation data isdominated by noise and extraneous light, because additionalmasks would be needed. The disadvantage of the current pro-cedure is that the fitted source is just an arbitrary brightnessmap, and the principle of plausible matches is not being ap-plied.(iii) Since plausible-matching galaxies for any given lens alwayshave very similar Einstein radii, even though they may differin other ways, it is advantageous to pre-select the simulatedgalaxies to be within a suitable range of Einstein radii. Inthis work, we produced a conventional lens model first, buta more efficient method is desirable.(iv) Provided the lensing galaxy is clearly visible in the data,stellar mass estimated from multi-band images using popu-lation synthesis (cf. Leier et al. 2016) could be incorporatedinto the likelihood P ( D | g ). Stellar-maps from the simula-tions are, of course, known a priori. Ideally, the stellar lightdistribution would be subtracted from the entire observa-tional data using models of the galaxy light.(v) Stellar kinematics would be an important ingredient in P ( D | g ). Current simulations soften the gravitational dynam-ics on scales of order a kpc (see e.g., Table 2 in Schaye et al.2014), and it would be interesting to see if this strongly affects P ( D | g ). It would also be interesting to see if an equilibriumgalaxy-modelling framework like AGAMA , which resolves muchsmaller scales, yields higher P ( D | g ) than cosmological sim-ulation.(vi) Finally, although available for only a small fraction of lenses,time time-delays (for recent observations see ? ) would be in-teresting to incorporate in the plausible-match scheme. Lens-ing time delays are usually thought of as a way of measuringcosmological parameters, especially H . But the accuracy ofthe H inferences from lensing depends on how well P ( g | F )of the universe is constrained. Hence time delays could be use-ful (if they turn out to be not the best way to measure H )with cosmological parameter-values taken from other meth-ods, as a way of constraining P ( g | F ). ACKNOWLEDGMENTS
We would like to thank Liliya L. R. Williams for useful dis-cussions and comments on the paper.We also thank the anonymous referee for the constructivesuggestions to bring the paper to its final form.PD acknowledges support from the Swiss National ScienceFoundation. SM acknowledges the funding from the Euro-pean Research Council (ERC) under the EUs Horizon 2020research and innovation programme (COSMICLENS; grantagreement no. 787886).This research is based on observations made with theNASA/ESA Hubble Space Telescope obtained from the SpaceTelescope Science Institute, which is operated by the Associ-ation of Universities for Research in Astronomy, Inc., underNASA contract NAS 5–26555. These observations are asso-ciated with programs
DATA AVAILABILITY
The data underlying this article are available at the STScI(https://mast.stsci.edu/; the unique identifiers are cited inthe acknowledgements). The derived data generated in thisresearch will be shared on request to the corresponding au-thor, or can be replicated using the open-source softwareavailable at: (cid:135) https://github.com/phdenzel/gleam.
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Results for SDSS J0029-0055. See Section 5 for details. MNRAS000
Results for SDSS J0029-0055. See Section 5 for details. MNRAS000 , 1–10 () Denzel et al. SDSSJ0737+3216 χ ν : 3.4704 θ [arcsec] ›fi θ Figure 5.
Results for SDSS J0737+3216. See Section 5 for details.MNRAS , 1–10 () atching observed and simulated lensing galaxies SDSSJ0753+3416 χ ν : 2.7768 θ [arcsec] ›fi θ Figure 6.
Results for SDSS J0753+3416. See Section 5 for details. MNRAS000
Results for SDSS J0753+3416. See Section 5 for details. MNRAS000 , 1–10 () Denzel et al. SDSSJ0956+5100 χ ν : 3.4979 θ [arcsec] ›fi θ Figure 7.
Results for SDSS J0956+5100. See Section 5 for details.MNRAS , 1–10 () atching observed and simulated lensing galaxies SDSSJ1051+4439 χ ν : 2.6880 θ [arcsec] ›fi θ Figure 8.
Results for SDSS J1051+4439. See Section 5 for details. MNRAS000
Results for SDSS J1051+4439. See Section 5 for details. MNRAS000 , 1–10 () Denzel et al. SDSSJ1430+6104 χ ν : 2.4904 θ [arcsec] ›fi θ Figure 9.
Results for SDSS J1430+6104. See Section 5 for details.MNRAS , 1–10 () atching observed and simulated lensing galaxies SDSSJ1627-0053 χ ν : 2.3681 θ [arcsec] ›fi θ Figure 10.
Results for SDSS J1627-0053. See Section 5 for details. MNRAS000