SEAGLE--II: Constraints on feedback models in galaxy formation from massive early type strong lens galaxies
Sampath Mukherjee, Leon V. E. Koopmans, R. Benton Metcalf, Crescenzo Tortora, Matthieu Schaller, Joop Schaye, Giorgos Vernardos, Fabio Bellagamba
MMNRAS , 1–23 (2018) Preprint April 15, 2020 Compiled using MNRAS L A TEX style file v3.0
SEAGLE–II: Constraints on feedback models in galaxyformation from massive early-type strong lens galaxies
Sampath Mukherjee , (cid:63) , L´eon V. E. Koopmans , R. Benton Metcalf , ,Cresenzo Tortora , , Matthieu Schaller , Joop Schaye , Georgios Vernardos ,Fabio Bellagamba , Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700AV Groningen, The Netherlands STAR Institute, Quartier Agora - All´ee du six Ao ˆ u t, 19c B-4000 Li`ege, Belgium Dipartimento di Fisica e Astronomia, Universit`a di Bologna, via Gobetti 93/2, I-40129 Bologna, Italy INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129 Bologna, Italy INAF – Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125, Firenze, Italy Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We use nine different galaxy formation scenarios in ten cosmological simulation boxesfrom the EAGLE suite of Λ CDM hydrodynamical simulations to assess the impact offeedback mechanisms in galaxy formation and compare these to observed strong grav-itational lenses. To compare observations with simulations, we create strong lenseswith M ∗ > M (cid:12) with the appropriate resolution and noise level, and model themwith an elliptical power-law mass model to constrain their total mass density slope.We also obtain the mass-size relation of the simulated lens-galaxy sample. We findsignificant variation in the total mass density slope at the Einstein radius and in theprojected stellar mass-size relation, mainly due to different implementations of stel-lar and AGN feedback. We find that for lens selected galaxies, models with eithertoo weak or too strong stellar and/or AGN feedback fail to explain the distributionof observed mass-density slopes, with the counter-intuitive trend that increasing thefeedback steepens the mass density slope around the Einstein radius ( ≈ − d log ( ρ )/ d log ( r ) ≈ . ) and slope distributions statis-tically agreeing with observed strong lens galaxies in SLACS and BELLS. Agreementis only slightly worse with the more heterogeneous SL2S lens galaxy sample. Observa-tions of strong-lens selected galaxies thus appear to favor models with relatively weakfeedback in massive galaxies. Key words: gravitational lensing: strong – methods: numerical – galaxies: evolution– galaxy formation – galaxies: elliptical and lenticular, cD – galaxies: structure
Large-scale numerical simulations have established theCold Dark Matter (CDM) paradigm as a viable frameworkfor galaxy formation (e.g. Davis et al. 1985; Frenk et al.1988). The CDM model predicts that galaxies form indark matter halos having a Navarro-Frenk-White (NFW)density profile (Navarro et al. 1996, 1997) and predict the (cid:63) [email protected] abundance and distribution of substructures within thesehalos (e.g. Gao et al. 2004; Springel 2010). The physicsof galaxy formation, however, complicates the descriptionof the matter distribution on small (several kpc) scales.Moreover, the central regions of CDM halos can also bestrongly modified by baryonic matter and their associatedphysical processes. Baryons settle into the centers of densityconcentrations due to dissipation, thereby modifying theinner DM slopes (e.g. Duffy et al. 2010; Sonnenfeld et al.2012; Grillo 2012; Remus et al. 2013; Cappellari et al. 2013; © a r X i v : . [ a s t r o - ph . GA ] A p r Mukherjee et al.
Tortora et al. 2009, 2010, 2014a; Pontzen & Governato2014). Because a complete analytic theory of baryonicphysics is lacking, hydrodynamic simulations that includemany physical processes have emerged as the dominanttool to study the complex non-linear interactions takingplace during galaxy formation (e.g. Schaye et al. 2010;Vogelsberger et al. 2014; Schaye et al. 2015; Dubois et al.2016; Hopkins et al. 2016). State-of-the art hydrodynamicalsimulations with improved stellar and AGN feedback, forexample, can reproduce the cosmic star formation historyof the Universe and the galaxy stellar mass function.Hydrodynamic simulations are currently working onlyabove certain mass and spatial resolutions, however, andphysical processes on smaller scales are implemented viaanalytic prescriptions known as ‘sub-grid physics’. Theimpact of varying sub-grid physics prescriptions on largerepresentative populations of stellar systems was firstsystematically explored in the ‘OverWhelmingly LargeSimulations’ project (OWLS; Schaye et al. 2010), a suiteof over fifty large cosmological hydrodynamical simulationswith varying sub-grid physics. Calibration of sub-gridprescriptions to reproduce a limited number of observableshas been explored extensively (Vogelsberger et al. 2014;Schaye et al. 2015; Crain et al. 2015; McCarthy et al.2017), showing that their exact parameterizations are veryimportant.Strong gravitational lensing is one of the most robustand powerful techniques to measure the total mass and itsdistribution in galaxies on kpc scales (Kochanek 1991; Koop-mans et al. 2006), allowing their inner structure and evolu-tion over cosmic time to be studied in detail (Treu et al.2006, 2009; Koopmans et al. 2006, 2009; Dutton & Treu2014), independently of the nature of the matter or its dy-namical state. In particular, the mass density profile of mas-sive lensing galaxies at z > . can trace their formationand evolution mechanisms (e.g. Barnab`e et al. 2009, 2011).The last two decades have seen major progress in observa-tional studies of strong lensing thanks to surveys such as theLenses Structure and Dynamics survey (LSD; Treu & Koop-mans 2004), the Sloan Lens ACS Survey (SLACS; Boltonet al. 2006; Koopmans et al. 2006; Bolton et al. 2008a,c;Koopmans et al. 2009; Auger et al. 2010a,b; Shu et al. 2015,2017), the Strong Lensing Legacy Survey (SL2S; Cabanacet al. 2007; Ruff et al. 2011; Gavazzi et al. 2012; Sonnenfeldet al. 2013a,b, 2015) and the BOSS Emission-Line Lens Sur-vey (BELLS; Brownstein et al. 2012). Future surveys suchas the Euclid (Laureijs et al. 2011) and the Large SynopticSurvey Telescope (LSST; Ivezi´c et al. 2008), as well as theongoing Kilo Degree Survey (KiDS; de Jong et al. 2015) andthe Dark Energy Survey (DES; The Dark Energy SurveyCollaboration 2005), are expected to increase the number ofknown strong lenses by several orders of magnitude (Petrilloet al. 2017; Metcalf et al. 2018; Treu et al. 2018) and revo-lutionize strong lensing studies.The paper is structured as follows. In Section 2, wesummarize the EAGLE galaxy formation simulations andthe relevant codes that are used in this paper. Section 3describes the simulation and analysis pipeline. The massmodels used are described in Section 4. We give a brief de-scription of the strong lensing observations in Section 5. InSection 6, we compare mock lens samples with observations, in terms of their mass-size relations and the total matterdensity slopes. The implications of our results are discussedand summarized in Section 7. Throughout the paper, we useEAGLE simulations that assume a Chabrier stellar InitialMass Function (IMF, Chabrier 2003) and compare these toobservables derived under the same IMF assumption. Thevalues of the cosmological parameters are Ω Λ = 0.693, Ω b =0.0482519, Ω m = 0.307, h = H /(
100 km s − Mpc − ) = 0.6777and σ = 0.8288. These are taken from the Planck satellitedata release (Planck Collaboration et al. 2014). Although there have been simulation studies of strong lens-ing focusing on the mass-size relations, the total densityslope and other observables (e.g. Remus et al. 2017; Peiraniet al. 2017; Xu et al. 2017), the impacts of varying sub-gridphysics (in particular baryonic feedback) on lensing statis-tics, their mass density slopes and stellar masses and sizeshave not been studied comprehensively yet (Peirani et al.2018). Duffy et al. (2010) analyzed the impact of baryonphysics on dark matter structure but only had low-resolutionmodels at low redshift.Mukherjee et al. (2018) (hereafter M18), introduced the
SEAGLE pipeline to systematically study galaxy formationvia simulated strong lenses.
SEAGLE aims to investigateand possibly disentangle galaxy formation and evolutionmechanisms by comparing strong lens early-type galaxies(ETGs) from hydrodynamic simulations with those ob-served, analyzing them in a similar manner (although thisis not always exactly possible).As in M18, we make use of the Evolution and Assem-bly of GaLaxies and their Environments (EAGLE) simula-tions (Schaye et al. 2015; Crain et al. 2015; McAlpine et al.2016) – a suite of state-of-the-art hydrodynamical simula-tions – to create, model and analyze simulated strong lens-galaxies and compare them with observations. Throughoutthis study, we use ten selected galaxy formation scenarios(i.e. having different sub-grid physics prescriptions; Schayeet al. 2015; Crain et al. 2015), the
GLAMER ray-tracing pack-age (Metcalf & Petkova 2014; Petkova et al. 2014), and the
LENSED lens-modeling code (Tessore et al. 2016). We pres-elect potential strong lenses based on their stellar massesand create projected mass maps for three different orienta-tions. We calculate the half-mass radius from the simulatedmass maps. We create mock lenses by ray tracing throughthe mass maps, placing an analytic Sersic (1968) source, ata higher redshift, having observationally motivated parame-ters. We ignore line-of-sight effects, which for massive ETGsis expected to be a good approximation (see e.g., Koop-mans et al. 2006). We use a single-orbit HST-ACS F814Wnoise level and PSF to mimic strong lenses found in SLACSand BELLS observations (Auger et al. 2010a; Bolton et al.2012a).Throughout this work, we also discuss possible observa-tional systematics (e.g. differences in model-fitting method-ologies, differences in filters/bands of the observational sur-veys, possible lens selection biases, etc.) as well as resolutioneffects in the simulations, that might affect their compari-son. The main aim of this study, however, is to illustrate the
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EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios Table 1.
Main sub-grid parameters of the EAGLE simulations used in this work. Columns (left to right) are the name ofthe simulation, L : the comoving side length of the volume, N : the number of particles for individual type i.e. gas and DM, γ eos : the power-law slope of the polytropic equation of state, n (cid:63) H : the star formation density threshold, f th : the star formationfeedback efficiency, f th , max : the asymptotic maximum and f th , min : minimum values of f th , n H , : the density-term denominatorfor the Reference model , n n : the Reference model density-term exponent (from equation 4), C visc : the sub-grid accretion discviscosity parameter (from equation 7 in Crain et al. 2015), and ∆ T AGN : the temperature increment of stochastic AGN heating.The calibrated models reproduce the GSMF at z = . . The reference variation models adopt a single-parameter variationof the Reference simulation (varied parameters are highlighted in bold). Except for FB σ (which uses the parameter n T ), allother models have n Z = / ln 10 with the same numerical value (see equation 2). For FBconst, this parameter is not applicable.This Table is partially reproduced from Crain et al. (2015). Identifier Side length N γ eos n (cid:63) H f th -scaling f th , max f th , min n H , n n C visc / π ∆ T AGN L [cMpc] [ cm − ] [ cm − ] log [K] Calibrated models
FBconst 50 752 / Eq. 1 − . . − − . FB σ
50 752 / Eq. 1 σ . . − − . FBZ 50 752 / Eq. 1 Z . . − − . Ref (FBZ ρ ) 50 752 / Eq. 1 Z , ρ . . .
67 2 / ln 10 10 . Ref-100 (FBZ ρ ) 100 1504 / Eq. 1 Z , ρ . . .
67 2 / ln 10 10 . Reference-variations
ViscLo 50 752 / Eq. 1 Z , ρ . . .
67 2 / ln 10 . ViscHi 50 752 / Eq. 1 Z , ρ . . .
67 2 / ln 10 − . AGNdT8 50 752 / Eq. 1 Z , ρ . . .
67 2 / ln 10 10 . AGNdT9 50 752 / Eq. 1 Z , ρ . . .
67 2 / ln 10 10 . NOAGN 50 752 / Eq. 1 Z , ρ . . .
67 2 / ln 10 10 − effects of the sub-grid physics parametrization adopted bythe EAGLE models, and the strong sensitivity of a numberof strong lens observables e.g. total mass density slope, mass-size relation, and Einstein radius to the variation of the keysub-grid physics. In future work, we will analyze other prop-erties such as the dark matter fractions and the stellar Ini-tial Mass Function (IMF). Although we assume a ChabrierIMF in this work, the impact of assuming a different IMF(e.g. stellar mass and feedback) is partially removed duringthe process of calibration (see Section 2.3). The impact of achanging IMF should therefore be very carefully examinedand will be done in a future publication for the Referencemodel for which these models are available (see e.g. Barberet al. 2018). A full analysis is currently not possible for theother models and well beyond the scope of this work, wherewe focus on the impact of galaxy-formation models.In this section we describe the EAGLE simulations usedin this study. In Section 2.1, we broadly describe the types ofmodel-variations that have been chosen and in Section 2.2,we describe the simulation setup and the sub-grid physicsrecipes that are used in those model variations. Section 2.3describes the calibrated simulations and reference modelsvariation are summarized in Section 2.4. The details pre-sented here are kept concise, yet informative, to make thispaper self-contained. The simulations explored in this paper are taken from Crainet al. (2015) plus the 100cMpc-Reference run from Schayeet al. (2015). Crain et al. (2015) divided the simulationsinto two categories. The first comprises four simulations cal-ibrated to yield the z = . galaxy stellar mass function(GSMF) and central black hole (BH) masses as a function of galaxy stellar mass. The second category comprises sim-ulations that each vary a single sub-grid physics parameterwith respect to the Reference model but without consider-ing whether they match the GSMF (i.e. they are not cali-brated). In the calibrated simulations, the models differ interms of their adopted efficiency of feedback associated withstar formation, and how this efficiency depends upon the lo-cal environment. In the Reference variation simulations, thesensitivity of the resulting galaxies to these variations are as-sessed. We note that similar variations have previously beendone in the OWLS project (Schaye et al. 2010). The generalconclusion from previous work has been that the propertiesof simulated galaxies are most sensitive to the efficiency ofbaryonic feedback (see e.g., Schaye et al. 2010; Scannapiecoet al. 2012; Haas et al. 2013a,b; Vogelsberger et al. 2013).This has motivated us to largely focus in this study on theeffect of baryonic feedback on lensing observables, in partic-ular on the total mass density profile in the inner regions ofmassive ETGs ( ∼ Any simulation has a certain resolution limit below whichthe physical processes cannot be simulated via the dynamicsof the particles. Similarly, the physical processes on scalessmaller than the resolution of the EAGLE simulations areincorporated via analytic prescriptions. In EAGLE elevenchemical elements have been considered in the simulations.The calculations of radiative cooling and heating rates us-ing the
CLOUDY (version 07.02) code of Ferland et al. (1998),account for variations in metallicity and for variations inthe relative abundances of individual elements. The cool-
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Mukherjee et al. ing rates are specified as a function of density, tempera-ture and redshift. While implementing the cooling in EA-GLE simulations, it is assumed that the optically thin gasis in a state of ionization equilibrium and is exposed to theCMB and an instantaneous, spatially uniform, temporally-evolving (Haardt & Madau 2001) UV/X-ray background(Wiersma et al. 2009a). Stochastic star formation, as for-mulated by Schaye & Dalla Vecchia (2008), has been imple-mented, but with the metallicity-dependent density thresh-old of Schaye (2004). A density threshold for star formation, n (cid:63) H , was imposed because star formation occurs only in cold( T (cid:28) K ), dense gas. Because the transition from a warm,neutral phase to a cold, molecular phase only occurs at lowerdensities and pressures in more metal-rich (and hence dust-rich) gas, the metallicity-dependent star formation thresholdput forward by Schaye 2004 (see his equations 19 and 24)was adopted: n (cid:63) H ( Z ) = min (cid:34) . (cid:18) Z0 . (cid:19) − . ,
10 cm (cid:35) , (1)where Z is the gas metallicity. Every star particle consti-tutes a stellar population with a fixed Chabrier (2003) IMF.The mass-to-light (M/L) ratio includes all the stellar rem-nants. The stellar evolution and mass loss implemented inEAGLE, is based on the prescription proposed in Wiersmaet al. (2009b). The simulations adopt the stochastic thermalstellar feedback scheme of Dalla Vecchia & Schaye (2012),in which the temperature increment, (cid:52) T SF , of heated reso-lution elements is specified. The fraction of the supernovaenergy budget that is available for feedback determines theprobability that a resolution element neighboring a youngstar particle is heated. This fraction is referred to as f th (Dalla Vecchia & Schaye 2012). According to the conven-tion, f th = equates to . × erg M − (cid:12) , being the levelof injected energy per stellar mass formed. Lastly, AGN feed-back has been implemented via a single mode, where energyis injected thermally and stochastically, analogous to energyfeedback from star formation. In EAGLE model variations, the efficiency of the stellar feed-back and the BH accretion were calibrated to broadly matchthe observed local ( z ≈
0) GSMF, subject to the constraintthat galaxy sizes must be in agreement with observations.We explain why calibration was needed and then we brieflydescribe the calibrated simulations of Crain et al. (2015),that are also used in this paper. Table 1 provides a conciseoverview of all the important parameters and a brief de-scription of the four calibrated EAGLE simulations, adaptedfrom the above-mentioned work.
The choice of sub-grid routines and the adjustment of theirparameters can result in substantial alterations of the simu-lation outcomes. Schaye et al. (2015) argued that the appro-priate methodology for cosmological simulations is to cali-brate the parameters of the uncertain sub-grid routines forfeedback with a small number of key observations, in orderthat simulations reproduce those representative observables, and then compare properties (between simulations and ob-servations) whose quantities that are not considered duringthe calibration. The total mass density slope, examined inthis paper, is one of those which was not used in calibration.The results thus obtained can reasonably be considered be-ing a consequence of the implemented astrophysics. On theother hand, the impact of changing the IMF (e.g. Barberet al. 2018) is partly calibrated out, and will be more care-fully considered in a separate paper for the Reference model.
This is the simplest feedback model where, independentlyfrom the local conditions, a fixed amount of energy perunit stellar mass is injected into the ISM. This fixed energycorresponds to the total energy discharged by type-II SNe( f th = ). The thermal stellar feedback prescription employedin EAGLE becomes inefficient at high gas densities due toresolution effects (Dalla Vecchia & Schaye 2012). Model Ref-erence (see Section 2.3.5) compensates for this known arti-fact by injecting more energy at higher gas density. Becausethis is not done in FBconst, the stellar feedback will be lesseffective in high-mass galaxies (where the gas tends to havehigher densities) (Crain et al. 2015). σ ) This model prescribes stellar feedback based on the localconditions, inferred from neighboring DM particles. The ef-ficiency, f th , is calibrated as a function of the square of the3-dimensional velocity dispersion of the DM particles withina stellar particle’s smoothing kernel at the time of its birth( σ ).The prescription of f th in its functional form, is a logistic(sigmoid) given by, f th = f th , min + f th , max − f th , min + (cid:16) T DM K (cid:17) n T . (2) T DM is the temparature of the characteristic virial scale ofenvironment of the star particle. The parameter n T > con-trols how rapidly f th transitions as the dark matter ‘temper-ature’ scale deviates from K . This model makes the radiative losses, f th , a function ofthe metallicity of the ISM. Energy dissipation associatedwith star formation feedback are likely to be more signifi-cant when the metallicity is sufficient for cooling from metallines to dominate over the cooling contribution from H andHe. The transition of outflowing gas in the simulations isexpected to occur at Z ∼ . Z (cid:12) for a temperature range K < T < K (Wiersma et al. 2009a). This phenomenoncan be numerically depicted by equation 2, but only after re-placing ( T DM , n T , K ) with ( Z , n Z , . Z (cid:12) ) to obtain, f th = f th , min + f th , max − f th , min + (cid:16) Z . Z (cid:12) (cid:17) n Z , (3)where Z (cid:12) = . is the solar metallicity and n Z = n T = / ln 10 . MNRAS000
This is the simplest feedback model where, independentlyfrom the local conditions, a fixed amount of energy perunit stellar mass is injected into the ISM. This fixed energycorresponds to the total energy discharged by type-II SNe( f th = ). The thermal stellar feedback prescription employedin EAGLE becomes inefficient at high gas densities due toresolution effects (Dalla Vecchia & Schaye 2012). Model Ref-erence (see Section 2.3.5) compensates for this known arti-fact by injecting more energy at higher gas density. Becausethis is not done in FBconst, the stellar feedback will be lesseffective in high-mass galaxies (where the gas tends to havehigher densities) (Crain et al. 2015). σ ) This model prescribes stellar feedback based on the localconditions, inferred from neighboring DM particles. The ef-ficiency, f th , is calibrated as a function of the square of the3-dimensional velocity dispersion of the DM particles withina stellar particle’s smoothing kernel at the time of its birth( σ ).The prescription of f th in its functional form, is a logistic(sigmoid) given by, f th = f th , min + f th , max − f th , min + (cid:16) T DM K (cid:17) n T . (2) T DM is the temparature of the characteristic virial scale ofenvironment of the star particle. The parameter n T > con-trols how rapidly f th transitions as the dark matter ‘temper-ature’ scale deviates from K . This model makes the radiative losses, f th , a function ofthe metallicity of the ISM. Energy dissipation associatedwith star formation feedback are likely to be more signifi-cant when the metallicity is sufficient for cooling from metallines to dominate over the cooling contribution from H andHe. The transition of outflowing gas in the simulations isexpected to occur at Z ∼ . Z (cid:12) for a temperature range K < T < K (Wiersma et al. 2009a). This phenomenoncan be numerically depicted by equation 2, but only after re-placing ( T DM , n T , K ) with ( Z , n Z , . Z (cid:12) ) to obtain, f th = f th , min + f th , max − f th , min + (cid:16) Z . Z (cid:12) (cid:17) n Z , (3)where Z (cid:12) = . is the solar metallicity and n Z = n T = / ln 10 . MNRAS000 , 1–23 (2018)
EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios Z ρ ) The feedback associated with FB σ and FBZ becomes nu-merically inefficient in the centers of high-mass galaxies be-cause a significant fraction of the star particles form at den-sities more than the resolution-dependent critical density( n H , t c ) above which radiation loss of the feedback energy isquick (Dalla Vecchia & Schaye 2012). These spurious energylosses can be partly compensated when a density dependenceis introduced in the expression for f th : f th = f th , min + f th , max − f th , min + (cid:16) Z . Z (cid:12) (cid:17) n Z (cid:16) n H , birth n H , (cid:17) − n n , (4)where n H , birth is the gas particle’s density at the time when itgets converted into a star particle. Hence, at a fixed metal-licity f th increases with density. Such a density dependencemay have a physical basis, because the star formation lawand hence the feedback energy injection rate per unit vol-ume, has a supra-linear dependence on surface density, whichmay result in smaller radiative losses at higher densities. Inthis work we use both the 50 and 100 cMpc boxes of the Ref-erence model. The 100 cMpc box has a much larger numberof massive galaxies for comparison to strong lens observa-tions, whereas we use the Reference-50 boxes to comparewith other model variations. Schaye et al. (2015) demonstrated that it is possible to cal-ibrate the Reference model to reproduce the Galaxy StellarMass Function (GSMF) and the observed sizes (in differentbands) of galaxies at z = 0.1. However, a systematic study ofthe model’s key sub-grid parameters and sensitivity of thismodel to the variations of sub-grid parameters are critical.In order to quantify these effects, Crain et al. (2015) con-ducted a series of simulations (listed in the lower section ofTable 1) for which the value of a single parameter was variedfrom that adopted in the Reference model. Here, we brieflysummarize the five Reference model variations that are usedin this work. There are five more Reference-model variationsavailable, but those have a smaller box size (25 cMpc) thatprovide insufficient numbers of high-mass galaxies for com-parisons to observed strong lens galaxies. The viscosity parameter C visc governs two important physi-cal processes: (a) the angular momentum scale at which gasaccretion onto BHs switches from the relatively inefficientviscosity-limited regime to the Bondi-limited regime, and (b)the rate (only during the viscosity-limited regime) at whichgas transits through the accretion disc (Rosas-Guevara et al.2015). It is important to note that in both cases (viscosity-limited and the Bondi-limited regime) are subjected to theEddington limit. A lower value of the viscosity parameter C visc , corresponding to a higher sub-grid viscosity. When thesub-grid viscosity is high, an earlier onset of the dominanceof AGN feedback is triggered at a larger energy injectionrate during the viscosity-limited regime. The viscosity pa-rameter could thus affect the efficiency of galaxy formationand the scale of the halo mass at the peak of the stellar fraction. Lower (higher) values for the viscosity increase (de-crease) both of them. However, we note that Bower et al.(2017) showed that the transition from slow to fast black-hole growth, which leads to the quenching of star formation,occurs when the halo is sufficiently massive to make stellarfeedback inefficient and depends only very weakly on C visc . Schaye et al. (2015) have examined the role of the AGNheating temperature in EAGLE by adopting ∆ T AGN = . K and 10 K . They demonstrated that a higher heat-ing temperature produces less frequent but more energeticAGN feedback episodes. They concluded it is necessaryto reproduce the gas fractions and X-ray luminosities ofgalaxy groups. Le Brun et al. (2014) also concluded that ahigher heating temperature yields more efficient AGN feed-back. We analyze two Reference-model variation simulationswith ∆ T AGN = K (AGNdT8) and ∆ T AGN = K (AG-NdT9), besides the Reference model itself which adopted ∆ T AGN = . K . In massive galaxies, the heating events (lessfrequent but more energetic) are more effective at regulatingstar formation due to a higher heating temperature. AG-NdT8 (AGNdT9) model has higher (lower) peak star frac-tion compared to the Reference model. The reduced effi-ciency of AGN feedback, when a lower heating temperatureis adopted, leads to the formation of more compact galax-ies, because gas can more easily accrete onto the centers ofgalaxies and form stars. The final model that we consider has no AGN feedback andis the most extreme EAGLE model variation for massivegalaxies. It appears unrealistic because the lack of AGN feed-back is expected to dramatically increase the baryon con-centration in the inner regions of galaxies, producing overlymassive and concentrated galaxies. The reason that this vari-ation is included, is to clearly demonstrate the effect of theabsence of AGN activity. All other parameters are kept thesame as in the Reference run.
Here, we explain the
SEAGLE (Simulating EAGLE LEnses)pipeline in more detail. We briefly summarize the selectioncriteria of the (lens) galaxies, the extraction of the galax-ies from the simulations, the impact of projection on thelens galaxy convergence map (Section 3.1), ray-tracing with
GLAMER to create mock lensed images (Section 3.2), and fi-nally the automatic process to create masks around thelensed images used in the lens modeling (Section 3.3). Theflow diagram shown in Figure 1 of M18 describes the
SEA-GLE pipeline and the resulting data products. The reader isreferred to M18 for more details on the pipeline.
The initial down-selection of (lens) galaxies is based on thelens redshift ( z l ) and stellar mass ( M (cid:63) ) range from SLACS. MNRAS , 1–23 (2018)
Mukherjee et al.
Auger et al. (2010a) find a broad lens redshift range of . < z l < . and a lower limit on the total stellarmass of M (cid:63) ≥ . × M (cid:12) . The luminosities and effectiveradii of SLACS lens galaxies are based on a de Vaucouleursprofile fit to the galaxy brightness distribution as observedwith Hubble Space Telescope (HST). We choose their I-bandfilter value, assuming it is closest to the bulk of the stellarmass. These are turned into stellar masses assuming eithera Chabrier or Salpeter stellar IMF (Salpeter 1955). We usethe former in this paper to remain consistent with EAGLE.We also use a lower limit on both the line-of-sight stellar ve-locity dispersion ( σ >
120 km s − ) and the half stellar massradius ( R > ) from the EAGLE snapshot catalogs toavoid blatant outliers e.g., due to mergers. Table 2 summa-rizes these initial selection criteria.We select all sub-haloes that match these selection cri-teria and extract all their particles from the snapshot. We dothis for a single redshift roughly in the middle of the SLACSredshift range, i.e. z l = . . We reiterate, as in M18, thatthe lens redshift is fixed at z =0.271 for all mock lenses, de-spite having a range of observed lens redshifts. This redshiftis intermediate between that of SLACS at somewhat lowerredshifts, and SL2S plus BELLS at somewhat higher red-shifts. Choosing simulation boxes at different redshifts for alllenses, to account for the minor effect of evolution, is com-putationally not feasible. We expect the effect of evolutionto be small around this redshift (Furlong et al. 2015, 2017)and to be smaller than the observed scatter in the inferredquantities for all galaxies. Although this neglects the effectof evolution in the simulated sample, this redshift is roughlyin the middle of the bulk of the redshifts of the combinedset of SLACS, BELLS and SL2S lenses. For more details onthe galaxy extraction we refer to Section 3.2 of M18. Wefinally rotate the particle position vectors in several direc-tions around the center of the lens galaxy. In the currentpaper, each galaxy is projected along the three simulationbox axes. The particles using the same SPH kernel as usedin the simulation are exported into projected surface den-sity maps (for more specifications see Trayford et al. 2017).For each galaxy, we separately calculate the surface densitymaps for the individual particle types (DM, stars and gas),as well as their total surface density map. Stellar remnantsare included in the star particles. The surface density maps are created in units of solar massesper pixel on a square-pixel grid of 512 ×
512 (Table 2). Theyform the input to the ray-tracing lensing code
GLAMER (Met-calf & Petkova 2014; Petkova et al. 2014). The size in properkilo persec (pkpc) (100 pkpc) and pixel scale ( ≈ z =0.271, corresponding to ≈ GLAMER to convert thesemass maps into convergence maps, by dividing the surfacedensity maps by the critical surface density which is set bythe lens and source redshifts (Meylan et al. 2006). We choosea fixed redshift of z s = . , typical for SLACS lenses. Similar to the lens redshift, we choose a fixed source redshift to re-duce computational overhead, although this restriction canbe let go in the future. The dependence of the Einstein ra-dius on source redshift is weak, however, increasing by < from z s = . to 1.0. Since all quantities in this work are de-termined inside fractions of the effective radius, the impactof the choice of the source redshift is very small. To describethe source, we use an elliptical S´ersic brightness profile withan index n = , apparent magnitude = 23 in the HST-ACSF814W filter (AB system), an effective radius of 0.2 arc-sec, a position angle φ s = , and a constant axis ratio q s =0.6. We set the parameters as such to keep close resem-blance to sources found in SLACS (see Figure 4 in Newtonet al. 2011). As shown in M18, the choice of the source sizehas negligible influence on the quantities of interest in thisanalysis. Furthermore, in Section 4 of Tessore et al. (2016),it is shown that there is only a negligible impact on the re-covered parameters when using a realistic source as opposedto using a pixellated or parametric source model. They alsoshow that LENSED recovers the source parameters well forboth an exact model (i.e. the truth is part of the model fam-ily) and an inexact model. Thus our constant-size analyticsource model is expected to have a negligible impact on ourconclusions related to the mass density slopes as is furthermotivated in Appendix B.For each convergence map, the critical curves and caus-tics are calculated, using
GLAMER . We then randomly putthe S´ersic source inside the diamond caustics of the lens tocreate multiple lensed images. This helps to maximize thenumber of arc and ring-like systems in the simulations (thisroughly mimics the large magnification bias in the observa-tions). The pixel scale of the grid – representing the lensedimages – is set to 0.05 arcsec with the PSF and noise cor-responding to an HST-ACS F814W exposure of typically2400 s. The final resulting images have sizes of 161 × To mask large areas of noisy pixels in the image and includeonly regions around the lensed images in the lens model-ing (see Figure 4 in M18), we automatically create a maskfor each lens system. The noisy lensed images are convolvedwith a Gaussian having FWHM of 0.25 arcsec to decrease thenoise by about a factor of 5 and obtain a slightly larger foot-print of the lensed images. A surface brightness thresholdis set at typically 2.5–5 times below the original noise level.This threshold defines the edge of the mask, faithfully tracesthe lensed images below the noise, and sufficiently extendsoutside the lensed images to include some noise-dominatedpixels in the original image (see e.g. middle panel of Figure 4in M18). The central 7 × MNRAS , 1–23 (2018)
EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios Table 2.
Summary of the simulation settings and output products.
Galaxy Selection
Observable Value Name Comments M (cid:63) ≥ . × M (cid:12) Stellar mass threshold Taken from Auger et al. (2010a) σ >
120 km/sec Stellar velocity dispersion Kept lower than SLACS R > Lens Candidates M (cid:63) threshold M (cid:63) thresholdfor follow-up work for this work- - - - - - - - - - - - - - - - - - - - - - - After 3Simulation ≥ . × M (cid:12) > M (cid:12) projections CommentsReference-100cMpc - 67 201 100 cMpc box.Reference-50 (FBZ ρ ) 252 25 75 50 cMpcboxFBconst 279 22 66 (cid:48)(cid:48) FB σ
259 22 66 (cid:48)(cid:48)
FBZ 312 19 57 (cid:48)(cid:48)
ViscLo 289 29 87 (cid:48)(cid:48)
ViscHi 188 14 42 (cid:48)(cid:48)
AGNdT8 276 27 81 (cid:48)(cid:48)
AGNdT9 194 8 24 (cid:48)(cid:48)
NOAGN 312 37 111 (cid:48)(cid:48)
Object-properties Value Type CommentsOrientation 3 x, y, z Projected surface density mapsRedshift z l = . - Consistent with SLACS’mean lens-redshift of 0.3 Source Properties
Parameters Value Unit CommentsSource Type S´ersic - Consistent with SLACS lenses(Newton et al. 2011)Brightness 23 apparent mag. (cid:48)(cid:48)
Size ( R eff ) 0.2 arcsec (cid:48)(cid:48) Axis ratio ( q s ) 0.6 - (cid:48)(cid:48) S´ersic Index 1 - (cid:48)(cid:48)
Redshift z s =0.6 - (cid:48)(cid:48) Position Random Within caustics Producing rings and arcs lens systemsconsistent with SLACS
Instrumental Settings
Parameters Type Value CommentsPSF Gaussian FWHM=0.1 arcsec -Noise HST ACS-F814W 2400 sec -
Image Properties
Map used Properties ValueSurface density (a) Size 512 ×
512 pixels(b) Units pkpc κ , Inv. mag. map and Lens (a) Size 161 ×
161 pixels(b) Units degrees the potential), thereby de-magnifying the central lensed im-age, in the mock lenses it leads to a too bright central image.To avoid a bias in the lens model, we mask this central re-gion. This artificial core has however little impact on theouter images near the Einstein radius. The resulting mask is used in all subsequent modeling and only image data insidethe mask are used for the lens modeling.
MNRAS , 1–23 (2018)
Mukherjee et al. R e f e r e n c e F B c o n s t F B Z F B σ V i s c - L o V i s c - H i A G N d T A G N d T N O A G N Figure 1.
Mosaic of a randomly selected sub-sample of six strong lenses from each of the nine EAGLE model variations ( z l = 0.271, z s = . ). Their morphologies (for a source randomly placed inside the diamond caustic) covers that of quads, rings and arcs, and visuallyresemble SLACS lenses remarkably well. In this section we describe the selection of the final mocklens sample (Section 4.1), and the subsequent gravitationallens modeling and convergence-map fitting, i.e. the modelingof the surface mass density as directly obtained from thesimulations (Section 4.2).
Implementing an automated recipe for the lens modeling ofgalaxies with stellar masses M (cid:63) < M (cid:12) has proven diffi-cult due to the finite resolution effect of the particles duringprojection causing an artificial ‘core’ in the inner densityprofile, which in turn creates prominent but artificial im-ages in the central regions of the lenses during ray tracing.These artificial images are not observed in real lens systemsand are particularly pronounced in lower-mass galaxies that MNRAS000
Implementing an automated recipe for the lens modeling ofgalaxies with stellar masses M (cid:63) < M (cid:12) has proven diffi-cult due to the finite resolution effect of the particles duringprojection causing an artificial ‘core’ in the inner densityprofile, which in turn creates prominent but artificial im-ages in the central regions of the lenses during ray tracing.These artificial images are not observed in real lens systemsand are particularly pronounced in lower-mass galaxies that MNRAS000 , 1–23 (2018)
EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios are more affected by the finite resolution of the simulations.As in M18, we therefore restrict ourselves to galaxies withtotal stellar masses M (cid:63) > M (cid:12) . These galaxies are farless affected by any resolution effects and still significantlyoverlap with the massive lensing galaxies of SLACS andSL2S. Moreover, the disc to total ratio (D/T) distributionsalso matches well between SLACS and EAGLE (Reference-100) and thus we should statistically select comparable ETGcandidates. Of these massive galaxies, about 80% are cen-tral galaxies (the most massive subhalo of a given halo) andabout 20% are satellites (subhalos other than the main sub-halo) in the 100 cMpc box. For the 50 cMpc boxes they aremostly ( > Having created the mock lens systems, we model each lenssystem with the open source, publicly available lens mod-eling code:
LENSED (Tessore et al. 2016; Bellagamba et al.2017). We use either an Elliptic Power Law (EPL; Tessore& Metcalf 2015) or a Singular Isothermal Ellipsoid (SIE;Kormann et al. 1994) mass model, including external shear.We use the corresponding mask, noise level and PSF for eachsystem. A total of 14 or 15 parameters are sampled using aNested Sampling MCMC method for the SIE or EPL mod-els, respectively. The EPL mass model (which includes theSIE) has been utilised in several previous studies and hasproven to describe very well the underlying mass model ofstrong gravitational lenses in various observational studies(Treu & Koopmans 2004; Koopmans et al. 2006, 2009; Barn-ab`e et al. 2009; Newton et al. 2011; Barnab`e et al. 2011).When modeling with an SIE plus external shear, we usethe prior settings tabulated in Table 3 of M18. The SIEmodel’s (dimensionless) surface mass density can be numer-ically stated as: κ ( R ) = b R , (5)where b equates to the measured radius of the Einstein ring(formally only for q =1) and R is the elliptical radius definedby: R = (cid:112) qx + y / q , where q is the axis ratio (minor overmajor axis) and x , y are Cartesian coordinates on the imageplane. The lens is allowed to vary in position angle and masscentroid as well. We perform the lens modelling on the lenseswith an EPL mass model. From Tessore & Metcalf (2015)we write the convergence as κ ( R ) = ( − t L ) (cid:18) bR (cid:19) t L , (6)where 0 < t L < ρ ( r ) ∝ r − t , where t = t L + . Both models also include external shear param-eters. Statistically we aim to compare the SLACS, BELLS and SL2S lenses with those from the simulated lenses viathe ensemble of density slopes obtained from the EPL im-plemented lens-modelling technique.However, many of the SLACS density slopes were ob-tained from a joint lensing and dynamics analysis, ratherthan only from lensing (Koopmans et al. 2009; Barnab`e et al.2009; Auger et al. 2010b; Barnab`e et al. 2011). We assumehere that there is no significant bias between the lensingand lensing plus dynamics analyses (Tortora et al. 2014a;Xu et al. 2017). A direct comparison of the model param-eters with the convergence map fitting can be performedwith the same model, which we do not discuss further inthis work but was extensively studied in M18. As in the cre-ation of the mock lenses, we use a S´ersic profile to model thesource. Even though some of the SLACS, BELLS and SL2Ssources show irregular morphologies, our main objective isto calculate the global properties of the galaxies acting aslenses, and the exact choice of the source model does not biasthe lens parameters for different (and inexact) source mod-els (see section 4.4 of Tessore et al. 2016). We also comparethe recovered source size between SIE and EPL and foundnegligible difference that does not bias our results (see Ap-pendix B). In Figure B3 we also demonstrate that there isno such correlation between source-size density slope whichmight bias our analyses. Additional tests were carried out inM18, where we found no change in the distribution or thevalue of the model parameters when changing the sourcemodel parameters between lens systems (see Appendix A ofM18). The priors used in the lens and source modeling arelisted in Appendix A (see also Table 3 of M18). The priorswere chosen such that the convergence of lens modeling pa-rameters occurs faster in the Nested Sampling optimizationand leads to minimal biases. We note that the priors aregenerally much wider than the inferred errors, hence theymostly guide the convergence rather than impact the pa-rameter errors. The overall modelled parameters give con-siderably good fit to the lens and optimised residuals (fordetails see SEAGLE-I). Here we summarize the strong lensing observational surveysthat we use to compare with our results. In Section 5.1 webriefly describe the observations. Section 5.2 describes theweighting scheme that is used to compare simulated lensensemble properties with observation. We note that in ourcomparison between simulated and observed lenses, we showall of the SLACS lens galaxies for visual purposes, but onlyquantitatively compare these galaxies with simulated galax-ies for the restricted range M (cid:63) > M (cid:12) . In the SLACS survey, Bolton et al. (2006) selected po-tential lens candidates spectroscopically from SDSS. Sincethen SLACS has successfully identified more than a hun-dred confirmed strong lens systems, with HST follow-up.The SLACS galaxies are massive ETGs, specifically Lumi-nous Red Galaxies (LRGs) with star-forming backgroundsources emitting strong emission lines. The advantage of theSLACS survey is that for all lenses spectroscopic lens and
MNRAS , 1–23 (2018) Mukherjee et al. source galaxy redshifts are available. The mean Einstein ra-dius of SLACS lenses is 1.2 arcsec (Koopmans et al. 2006;Auger et al. 2010a) with sources having a typical size ofabout 0.2 arcsec (Koopmans et al. 2006; Newton et al. 2011)and typically being at z s ≈ . . Although it is the largestcomplete strong lens sample, SLACS has a relatively lim-ited lens redshift range with the bulk of the lenses in therange of z l ≈ . − . .The SL2S survey was initiated to increase the numberof known lenses by a different methodology than SLACS. InSL2S, Cabanac et al. (2007) performed a dedicated searchin the CFHTLS to find strong gravitational lenses. They fo-cused on mostly galaxy-scale and group-scale lenses. SL2Saimed at providing a larger sample of strong lenses at higherredshift. RingFinder (Gavazzi et al. 2014), an automatedsoftware was used in SL2S to find lenses by searching 170square degrees of the sky.
RingFinder performed a search forblue arcs that are elongated tangentially and ring-like struc-tures around red galaxies to select lens candidates. The mostpromising systems were followed up with HST and spec-troscopy (Gavazzi et al. 2012). Even though SL2S lensescombined with SLACS provided evidence of structural evo-lution (Ruff et al. 2011), the SL2S sample is limited by alack of source-galaxy redshifts for a considerable number ofsystems.In BELLS, Brownstein et al. (2012) utilized the samespectroscopic methodology implemented in SLACS, to selectthe strong lenses, but they used Baryon Oscillation Spectro-scopic Survey (BOSS; Eisenstein et al. 2011) spectra. BELLSdiscovered a sample of strong galaxy-galaxy lenses, at some-what higher redshifts, that is of comparable size and ho-mogeneity to that of SLACS at lower redshift. BELLS isalso comparable in stellar mass to the SLACS lens galaxies.Both the BELLS and SLACS samples are complete in theirspectroscopic lens and source galaxy redshifts. The lens red-shifts of the three lens samples are within a similar range of0.1–0.65, but the source redshifts cover a wide range from0.3 to 3.5. Bolton et al. (2012b) have reported evidence formild evolution in the mass density slope between BELLSand SLACS. We ignore this in the sample of mock lenses andcompare observations with simulations only at z = . , inbetween the two samples, as discussed earlier. Differences in lens-galaxy selection and follow-up can lead todifferences in the population of lenses in the SLACS, BELLSand SL2S samples. For example, due to the relatively smallfiber opening used in SDSS spectroscopic observations ( . (cid:48)(cid:48) radius), the SLACS spectroscopic survey typically limitsthe search to lenses with an equivalent or smaller Einsteinradius (although larger lenses could be found if one ofthe lensed images is inside the fiber and bright enough),and finite source effects play a role as well. SL2S on theother hand can select lenses directly from images and overa larger Einstein radius range, i.e. mass scale, typicallyyielding Einstein radii greater than (cid:48)(cid:48) , because they areless well resolved in ground-based images. These selectioneffects are hard to quantify though (see e.g., Dobler et al.2008, for SLACS).Observational selection biases often hinder a proper comparison between simulations and lens surveys, stronglensing being no exception. In this work, we assume that lensselection biases are dominated by the lens-galaxy mass andcorrelate sub-dominantly with the lens and source redshifts,and with the lens-galaxy mass density profile and ellipticity.This is a reasonable assumption if the lens mass models areclose to isothermal (i.e., the caustics are shape invariant as afunction of redshift and only scale in cross-section) and thesource size is small compared to the Einstein radius (Dobleret al. 2008). Massive ellipticals also do not vary stronglyin their ellipticity. The observed lens sample properties arethen mainly affected by the lensing cross-section (Marshallet al. 2007), which is mass dependent, and by the magnifica-tion bias, which can be different between surveys. A preciseanalysis is difficult to implement and beyond the scope ofthis paper. We therefore take an empirical approach andonly correct for the lens cross-section and we assume thatthe magnification bias does not correlate with galaxy mass .The square of the Einstein radius varies proportionally withthe cross-section of lensing for the EPL model for a fixedellipticity (generally also close to the SIE model). Assum-ing the Faber-Jackson relation (Faber & Jackson 1976) anda constant mass-to-light ratio, the Einstein radius is againproportional to the stellar mass of the respective galaxy.Hence we arrive at a direct observable (i.e. the stellar mass)in both the simulations and observations.Motivated by the above arguments, we propose the fol-lowing weighting scheme per lens: W ( M (cid:63) ) ≡ (cid:18) M (cid:63) (cid:104) M (cid:63) (cid:105) (cid:19) α , (7)with (cid:104) M (cid:63) (cid:105) being the average stellar mass of the galaxies inthe sample and α being the exponent of the weight function.We re-weight each simulated strong lens (which we assumeto be volume limited) when comparing distributions (i.e. his-tograms) of the mass-density slopes between observed lensesfrom SLACS, BELLS, SL2S and simulated lenses. Hence aweight W i for simulated lens i implies it counts as W i galaxies(note that the weights are non-integers). Most of the lensesare massive systems, and in general drawn from the expo-nential tail of the mass function. Hence re-weighting shouldhave a limited impact on the massive end of the distributionfunctions, but it does strongly affect the low-mass end. Wetest values of α = . , . , and 1.5 to show that the weight-ing scheme does not affect the conclusions and are only tomimic the observation selection bias of the lenses dependingon their stellar mass. Other options for re-weighting the lensgalaxies, to account for their lens cross-section, are using ei-ther their Einstein radii or their stellar velocity dispersions,which we have not done in this work and leave for futureimprovements in the analysis when we study the redshiftevolution of these lenses. In this section we compare the simulated EAGLE lenses withthose from SLACS, BELLS and SL2S, in terms of their sur- This holds exactly for SIE models if the source is a point sourceand the galaxy mass model (i.e. ellipticity for the SIE) does notvary with galaxy mass. MNRAS000
RingFinder performed a search forblue arcs that are elongated tangentially and ring-like struc-tures around red galaxies to select lens candidates. The mostpromising systems were followed up with HST and spec-troscopy (Gavazzi et al. 2012). Even though SL2S lensescombined with SLACS provided evidence of structural evo-lution (Ruff et al. 2011), the SL2S sample is limited by alack of source-galaxy redshifts for a considerable number ofsystems.In BELLS, Brownstein et al. (2012) utilized the samespectroscopic methodology implemented in SLACS, to selectthe strong lenses, but they used Baryon Oscillation Spectro-scopic Survey (BOSS; Eisenstein et al. 2011) spectra. BELLSdiscovered a sample of strong galaxy-galaxy lenses, at some-what higher redshifts, that is of comparable size and ho-mogeneity to that of SLACS at lower redshift. BELLS isalso comparable in stellar mass to the SLACS lens galaxies.Both the BELLS and SLACS samples are complete in theirspectroscopic lens and source galaxy redshifts. The lens red-shifts of the three lens samples are within a similar range of0.1–0.65, but the source redshifts cover a wide range from0.3 to 3.5. Bolton et al. (2012b) have reported evidence formild evolution in the mass density slope between BELLSand SLACS. We ignore this in the sample of mock lenses andcompare observations with simulations only at z = . , inbetween the two samples, as discussed earlier. Differences in lens-galaxy selection and follow-up can lead todifferences in the population of lenses in the SLACS, BELLSand SL2S samples. For example, due to the relatively smallfiber opening used in SDSS spectroscopic observations ( . (cid:48)(cid:48) radius), the SLACS spectroscopic survey typically limitsthe search to lenses with an equivalent or smaller Einsteinradius (although larger lenses could be found if one ofthe lensed images is inside the fiber and bright enough),and finite source effects play a role as well. SL2S on theother hand can select lenses directly from images and overa larger Einstein radius range, i.e. mass scale, typicallyyielding Einstein radii greater than (cid:48)(cid:48) , because they areless well resolved in ground-based images. These selectioneffects are hard to quantify though (see e.g., Dobler et al.2008, for SLACS).Observational selection biases often hinder a proper comparison between simulations and lens surveys, stronglensing being no exception. In this work, we assume that lensselection biases are dominated by the lens-galaxy mass andcorrelate sub-dominantly with the lens and source redshifts,and with the lens-galaxy mass density profile and ellipticity.This is a reasonable assumption if the lens mass models areclose to isothermal (i.e., the caustics are shape invariant as afunction of redshift and only scale in cross-section) and thesource size is small compared to the Einstein radius (Dobleret al. 2008). Massive ellipticals also do not vary stronglyin their ellipticity. The observed lens sample properties arethen mainly affected by the lensing cross-section (Marshallet al. 2007), which is mass dependent, and by the magnifica-tion bias, which can be different between surveys. A preciseanalysis is difficult to implement and beyond the scope ofthis paper. We therefore take an empirical approach andonly correct for the lens cross-section and we assume thatthe magnification bias does not correlate with galaxy mass .The square of the Einstein radius varies proportionally withthe cross-section of lensing for the EPL model for a fixedellipticity (generally also close to the SIE model). Assum-ing the Faber-Jackson relation (Faber & Jackson 1976) anda constant mass-to-light ratio, the Einstein radius is againproportional to the stellar mass of the respective galaxy.Hence we arrive at a direct observable (i.e. the stellar mass)in both the simulations and observations.Motivated by the above arguments, we propose the fol-lowing weighting scheme per lens: W ( M (cid:63) ) ≡ (cid:18) M (cid:63) (cid:104) M (cid:63) (cid:105) (cid:19) α , (7)with (cid:104) M (cid:63) (cid:105) being the average stellar mass of the galaxies inthe sample and α being the exponent of the weight function.We re-weight each simulated strong lens (which we assumeto be volume limited) when comparing distributions (i.e. his-tograms) of the mass-density slopes between observed lensesfrom SLACS, BELLS, SL2S and simulated lenses. Hence aweight W i for simulated lens i implies it counts as W i galaxies(note that the weights are non-integers). Most of the lensesare massive systems, and in general drawn from the expo-nential tail of the mass function. Hence re-weighting shouldhave a limited impact on the massive end of the distributionfunctions, but it does strongly affect the low-mass end. Wetest values of α = . , . , and 1.5 to show that the weight-ing scheme does not affect the conclusions and are only tomimic the observation selection bias of the lenses dependingon their stellar mass. Other options for re-weighting the lensgalaxies, to account for their lens cross-section, are using ei-ther their Einstein radii or their stellar velocity dispersions,which we have not done in this work and leave for futureimprovements in the analysis when we study the redshiftevolution of these lenses. In this section we compare the simulated EAGLE lenses withthose from SLACS, BELLS and SL2S, in terms of their sur- This holds exactly for SIE models if the source is a point sourceand the galaxy mass model (i.e. ellipticity for the SIE) does notvary with galaxy mass. MNRAS000 , 1–23 (2018)
EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios face mass density profiles. Even though SL2S and BELLSlenses are typically at somewhat higher redshifts, we com-pare the simulated lenses at z l =0.271 assuming limited ETGevolution within the redshift range 0 < z <
1, as discussed ear-lier. This assumption is reasonable as it was pointed out byboth Sonnenfeld et al. (2013b) and Koopmans et al. (2006),that the total mass density slopes (which are close to isother-mal) do not strongly evolve with time in observed ETGlenses (although see Bolton et al. 2012a). We compare themass-size relation in Section 6.1, the total density slopes inSection 6.2, and the Einstein radius in Section 6.3. We com-pare our results with OWLS simulation in Section 6.4. Table2 summarizes the number of galaxies, lenses and projectedmass maps. Tables 3, 4 and 5 give the effect of magnificationbias (mimicked by a weighting scheme) on the total massdensity slope ( t ) values, the average Einstein radii, the ratioof Einstein radius over effective radius and several other rel-evant quantities of the simulated strong lenses from differentmodel variations of EAGLE. Observationally the stellar mass (or luminosity to beprecise) of an ETG correlates with its size (e.g. Baldry et al.2012). Similarly, in our simulations the stellar masses ofgalaxies correlate with their sizes (Furlong et al. 2017). Toassess whether a similar relation holds for the mock lensesat z l = . , we define the effective radius ( R eff ) as thestellar projected half-mass radius in the simulations, henceassuming a constant mass-to-light ratio. As demonstratedby Remus et al. (2013, 2017), this might lead to a slightoverestimation of the actual size of the galaxy compared toobservations (e.g. in the case of SLACS the effective radiusis derived from a de Vaucouleur fit to the galaxy brightnessdistribution), but we ignore this minor ( < σ and FBZ, which(except for FBconst) were calibrated on the GSMF but noton galaxy sizes, all have similar effective radii as SLACS, ex-cept for two outliers around the lowest stellar mass end andabove the relation that have unusually large effective radii .Due to the relatively low efficiency of stellar feedback in theFBconst, FB σ , FBZ models and the absence of AGN feed-back in the NOAGN model, stars tend to form somewhatcloser to the center of the galaxy (see Crain et al. 2015).The NOAGN model, however, leads to much more compactgalaxies, with some systems even straddling the resolutionlimit of the simulations. The galaxies from the AGN model We note that each mock lens is shown three times (once foreach principle-axis orientation), as discussed earlier, and hencethe number of independent outliers is very small. variations (AGNdT8 and AGNdT9) both have larger effec-tive radii than the NOAGN model. When ∆ T = K (AG-NdT8) about half of the galaxies are more compact in sizeand in good agreement with SLACS, whereas for ∆ T = K (AGNdT9) hardly any galaxy matches the observations. Thehigher temperature in the AGNdT9 model leads to more ef-fective AGN feedback, keeping gas away from the center andincreasing the size of the galaxy. For comparison, the Ref-erence model assumes ∆ T = . K , explaining its positionhalfway between AGNdT8 and AGNdT9 in mass-size rela-tion (Figure 2). A low black hole accretion disc viscosity(ViscLo), i.e a high viscosity parameter ( C visc ), delays theonset of AGN feedback, allowing gas to settle closer to thegalaxy center before star formation. The ViscHi model hasthe opposite effect, increasing the size of the galaxy.Overall, we conclude that simulated galaxies from EA-GLE better match the mass-size relation of SLACS lensgalaxies when there is moderately low AGN activity or stel-lar feedback driving the galaxy formation, with only a mildimpact from variations in the type of stellar feedback model.This trend is consistent with previous studies (Remus et al.2017; Figure 1 in Peirani et al. 2018). Finally, we find thatchanges in the viscosity have a stronger impact by indirectlyaffecting AGN feedback. We now compare the inferred mass-size relation with theresults by Schaye et al. (2015), Crain et al. (2015) and Fur-long et al. (2017). This comparison is necessary to assessany selection bias within the simulations. If we are select-ing an ETG population that is significantly different thanthe total galaxy sample analyzed in other aforementionedEAGLE works, this might invoke a bias in our lensing ETGsample and their properties. Moreover, our calculations areperformed on mass maps and not directly on the catalogedparticles. Schaye et al. (2015) compared the mass-size rela-tion of the Reference model by Shen et al. (2003) and Baldryet al. (2012), and found excellent agreement. Similarly, Crainet al. (2015) compared the mass-size relation from the cal-ibrated models (Figure 3 therein) and found ≈ ≈ M (cid:63) > M (cid:12) , due to small number statistics for Ref-50 (see also Crain et al. 2015). However, some systematicdifferences are still present with strong-lens galaxies tend- MNRAS , 1–23 (2018) Mukherjee et al.
Figure 2.
The galaxy mass-size relation for nine EAGLE model variations from simulations with a box-size of 50 cMpcat z l =0.271, as compared to the observed mass-size relation of SLACS lens galaxies. The stellar masses and effective radiifor the observed and simulated lenses are derived using slightly different methods (fitting profiles versus inference from thesimulations), but using the same stellar IMF (i.e. Chabrier). The simulated galaxies are only shown for stellar masses > M (cid:12) ,whereas for visual comparison, we show all of the SLACS lenses, although only a few of the lenses have lower stellar masses.ing to be more compact than non-lensing galaxies. SLACSgalaxies, therefore, appear about 0.2 dex smaller in size thannon-lensing galaxies of similar mass (see right panel of Figure4). In paper III of SEAGLE series we will explore the sys-tematics and compare with non-lensing ETGs from SPIDERsurvey (La Barbera et al. 2010; Tortora et al. 2014a), whichwe will show, have sizes that agree much better with EA-GLE, and we point out the methodological differences (e.g.,measurements with different bands of observations, differ-ent fitting algorithm, etc.) that could potentially bias theanalysis. Keeping the mass-size results discussed in the the previ-ous section in mind, in this section we assess whether thesame galaxy formation models that (visually) reproduce themass-size relation of SLACS lens galaxies also reproducetheir mass density slopes, which is not an observable used inthe calibration of the EAGLE simulations. We use the EPLsurface mass density profile to model the simulated stronglenses with
LENSED , closely mimicking real lens observations(see Section 4 for details). This allows for a more unbiasedand systematic comparison with strong lens observations.
MNRAS000
MNRAS000 , 1–23 (2018)
EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios Figure 3.
Comparison of the mass-size relation obtained inthis work, Furlong et al. (2017) for Reference 50 and 100 Mpcsimulation box for galaxies with M (cid:63) > M (cid:12) , Shen et al.(2003) and Baldry et al. (2012) are shown. The shaded regionindicates the standard deviation of the spread in values. As a first check, we confirm that the lens galaxies fromthe Reference-100 cMpc model show a similar distributionof density slopes as presented in M18 where the smaller50 cMpc box was used. The latter has a much smaller num-ber of massive galaxies. We confirm that EAGLE galaxiesfrom the Reference model tend to have slightly steeper den-sity slopes than SLACS, BELLS and SL2S (see left panel inFigure 4 and also Figure 12 in M18). However, the ratio of R Ein / R eff can play a crucial role in this respect because thelens modeling is mainly dependent on information obtainednear the Einstein radius. Since the total mass density canbe sensitive to the radial scale at which it is measured (Xuet al. 2017), we will explore this aspect in Section 6.3.In Figure 5 the density slopes for all EAGLE modelvariations are shown for the smaller 50 cMpc boxes. TheFBconst model appears to yield galaxies most similar toSLACS with the total mass density profile being very closeto isothermal. This can be attributed to its less efficient stel-lar feedback, which yields a mass profile, different than theReference model. The FBZ and FB σ models have more darkmatter in the center of the galaxy compared to the FBconstand Reference models, leading to a shallower total densityslope in their central regions. Hence, whereas the FBZ andFB σ models visually reproduce the mass-size relation ofSLACS rather well, they fail to reproduce their mass densityslopes. We find the rather counterintuitive trend that whenfeedback efficiency increases from the FBZ, FB σ , FBconstto Reference models, the average total mass density slopesteepens. We will see that variations in AGN feedback showthe same trend and we will discuss the cause in the nextsection. In Section 6.3 we will also study the correlation of the ratio of R Ein / R eff with the total mass density slope fordifferent model variations. There is a clear dependence of the total mass density slopeon AGN feedback. As the stochastic temperature incrementin AGN models increases from ∆ T = K (AGNdT8) to ∆ T = . K (Reference) and ∆ T = K (AGNdT9) the to-tal density slope steepens. Generally, we would expect theopposite, since stronger AGN activity (i.e. temperature in-crements) should move or keep gas particles away from thegalaxy center, preventing star formation. As mentioned inLe Brun et al. (2014), more energetic heating events asso-ciated with a higher heating temperature, even-though lessfrequent, are more effective at regulating star formation inmassive galaxies. Crain et al. (2015) also pointed out thatthe peak galaxy formation efficiency decreases with increas-ing AGN temperature. The reduced efficiency of AGN feed-back thus, counter-intuitively, manifests itself in a steepertotal mass density slope. A similar trend is found when theviscosity parameter is increased, which impacts AGN feed-back at fixed mass as discussed earlier. This trend is con-sistent with previous simulation studies (e.g. Remus et al.2017; Xu et al. 2017). In short, the AGNdT8 model with itsweaker AGN feedback (compared to the Reference model)produces lensing galaxies that are closer to isothermal and inbetter agreement with the results from SLACS, BELLS andSL2S lens galaxies. Table 5 summarizes the mean, medianand standard deviation of the density slopes for all EAGLEmodel variations used in this work. The evolutionary trendswill be studied in details in a forthcoming paper.
We correlate the total mass density slope and the stellarmass of the three prominent simulation models compared inour analysis, namely, Ref-50, FBconst and AGNdT8. Figure6 shows the distribution of the density slopes calculated fromlens modeling from both simulations and SLACS (Koop-mans et al. 2009). We find at most a very mild trend inthe total mass density slope with the stellar mass, consis-tent with strong lensing observations of SLACS (Koopmanset al. 2009). More massive galaxies tend to have a slightlylower total density slope than less massive galaxies in allthree model variations (see also Tortora et al. 2014a, wherethis trend, with shallower (isothermal) profile at high massand steeper profiles at lower masses are found). However,the intrinsic scatter in the distribution in each of the modelvariations is too large to draw any significant conclusion, es-pecially since the high-mass end of the distribution containsvery few galaxies in the simulations. This very mild trend isalso consistent with theoretical work by Remus et al. (2017)and Xu et al. (2017).
We test different values of the α parameter in our weightingscheme to demonstrate the robustness of our results againstthe selection effects in the observations. In Figure 5, we showthe variations in the median total mass density slope for MNRAS , 1–23 (2018) Mukherjee et al.
Figure 4.
Left panel: Histograms of the total mass density slopes (i.e. t = − log ( Σ )/ log ( R ) ; Σ ( R ) being the surface mass densityof the lens galaxies) of galaxies from the EAGLE model variation Reference-100 cMpc at z l =0.271 (having M (cid:63) > M (cid:12) ),compared to those from SLACS, BELLS and SL2S. The mean density slope from the simulations is 2.10 and the medianvalue is 2.31. The EAGLE distributions have been obtained from lens modeling with the code LENSED , similar in setup to theobservations (see text) and have been re-weighted by a proxy of their lens cross-section to correct for the larger lens selectionbias. The total mass density slopes of observations are taken from Auger et al. (2010b) for SLACS, Sonnenfeld et al. (2013b)for SL2S and Bolton et al. (2012a) for BELLS. For SLACS and BELLS, the density slopes are derived from a combination oflensing and stellar-kinematic constraints. Right panel: The mass-size relation from the same simulation compared with SLACS.A comparative study of all the total mass density slopes (from the 50 cMpc boxes) for all other simulations is presented inFigure 5.three different values of α =0.5,1.0, and 1.5. Although, themedian density slope is sensitive to the weighting scheme,relative changes are well within the spread calculated foreach of the model variations. This result implies that ourconclusions do not strongly depend on the observational se-lection bias. We note that we do not separately compensatefor the magnification bias, as a function of galaxy mass, butassume this effect is folded into the weighting scheme. Theat-most mild trend of the density slope with galaxy mass,however, suggest that any re-weighting based on galaxy masswill make little difference in the conclusions. Tables 3 and5 list the median values of the total mass density slope fordifferent values of α parameter, and their relative changecompared to the nominal model with α = . We note thatwe have not considered the errors on the measured slopein Figure 5. The errors on the measured slopes will slightlybroaden the distributions. However, the rms error on theslopes is typically well below 0.2 (see Auger et al. 2010b), i.e.inside our chosen bin-size, and considerably smaller than thespread in the distribution. In addition, the slope measure-ments from the simulations have a similar spread, mimickingpartly this broadening effect, thus reducing its impact. Thechanges in galaxy-formation processes is by far the mostprominent source differences in the distributions. The Einstein radius ( R Ein ) is a fundamental observable instrong gravitational lensing. However, to compare between
Table 3.
The median values of mass density slopes, t , of thesimulated lenses in different galaxy formation models sub-jected to weighting scheme with α =0.5 and 1.5 and their re-spective fractional change. Table 5 have the value for α =1.0. Simulation α =0.5 α =1.5 | ∆ t | / t Ref-50 2.16 2.20 0.02FBconst 1.98 2.08 0.05FB σ strong lenses simulated from different model variations ofEAGLE having a range of effective radii and strong lensingsurveys having different observing strategies (e.g. SLACS,SL2S and BELLS), we need to compare the ratio of R Ein / R eff (see Li et al. 2018). For SLACS, the values of R Ein / R eff ra-tios populate ≈ R Ein with similar sized lensing ETGs as SLACS, due to the largespread in redshift-range of the lensing galaxies ( z l = . − . )and the lensed sources ( z s = − . ) (Sonnenfeld et al. 2013a).In BELLS, the R Ein / R eff values mainly range from 0.5 to 1.5 MNRAS000
The median values of mass density slopes, t , of thesimulated lenses in different galaxy formation models sub-jected to weighting scheme with α =0.5 and 1.5 and their re-spective fractional change. Table 5 have the value for α =1.0. Simulation α =0.5 α =1.5 | ∆ t | / t Ref-50 2.16 2.20 0.02FBconst 1.98 2.08 0.05FB σ strong lenses simulated from different model variations ofEAGLE having a range of effective radii and strong lensingsurveys having different observing strategies (e.g. SLACS,SL2S and BELLS), we need to compare the ratio of R Ein / R eff (see Li et al. 2018). For SLACS, the values of R Ein / R eff ra-tios populate ≈ R Ein with similar sized lensing ETGs as SLACS, due to the largespread in redshift-range of the lensing galaxies ( z l = . − . )and the lensed sources ( z s = − . ) (Sonnenfeld et al. 2013a).In BELLS, the R Ein / R eff values mainly range from 0.5 to 1.5 MNRAS000 , 1–23 (2018)
EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios Figure 5.
Histograms of the total mass density slopes (i.e. t = − d log ( Σ )/ d log ( R ) ; Σ ( R ) being the surface mass density ofthe lens galaxies) of galaxies from EAGLE model variations (having M (cid:63) > M (cid:12) ) compared to those from SLACS, BELLSand SL2S. The EAGLE distributions have been obtained from lens modeling with the code LENSED , similar in setup to theobservations (see text) and have been re-weighted by a proxy for their lens cross-section to correct for the larger lens selectionbias. The median values for different values of α , see equation 7, are shown in colored vertical dashed lines: α =0.5 (green), α =1.0 (cyan) and α =1.5 (magenta). The shaded region shows the respective ± rms range centered on the median value (for α =1.0) for each scenario. Table 3 contains the most extreme values of α and their fractional difference. The total mass densityslopes of observations are taken from Auger et al. (2010b) for SLACS, Sonnenfeld et al. (2013b) for SL2S and Bolton et al.(2012a) for BELLS. For SLACS and BELLS, the density slopes are derived from a combination of lensing and stellar-kinematicconstraints.with a sharp drop below 0.5, primarily due to a wide rangeof the source redshift from z s =0.8 to 3.5 with mean lens red-shift of z l =0.52 (Li et al. 2018). We find that our best models,FBconst and AGNdT8, are closest in their R Ein / R eff to themean value of SLACS. Table 4 gives a complete overview ofthe mean of R eff , R Ein , the ratio R Ein / R eff and their respec-tive rms values for different model variations of EAGLE andobservations (e.g. SLACS, SL2S and BELLS).Figure 7 shows the correlation between the average to-tal mass-density slope ( t ) and R Ein / R eff ratios from differ-ent model variations of EAGLE. We find that as the feed- back becomes stronger, the effective radius increases (con-sistent with Sales et al. 2010). This in-turn decreases theratio R Ein / R eff and steepens the total density slope since t is calculated at the R Ein . The larger sizes of Einstein radiusfor strong lenses in SL2S are primarily due to the differencein observing strategy from SLACS. SLACS and BELLS se-lected lens candidates from spectroscopic signatures comingfrom two objects at different redshifts on the same line ofsight in the SDSS spectra. The relatively small fiber usedin SDSS spectroscopic observations, 1.5 (cid:48)(cid:48) for SLACS and 1 (cid:48)(cid:48) for BELLS in radius, they tend to select strong lenses with
MNRAS , 1–23 (2018) Mukherjee et al.
Figure 6.
The total mass density slopes correlation with stellar mass from Reference, FBconst and AGNdT8 model variationof EAGLE and SLACS lenses. The mass density slope and stellar mass of SLACS lenses are obtained from Auger et al.(2010b). The dashed green line is given at SLACS mean slope at t = . with the gray area being ±
10% intrinsic scatter asobtained from Koopmans et al. (2009).small Einstein radii. SL2S finds considerably more stronglenses with Einstein radii above 1 (cid:48)(cid:48) , since they can be moreclearly resolved in ground-based images. For similar com-parison of R Ein / R eff in SLACS, BELLS and SL2S, readersare refereed to Figure 1 in Sonnenfeld et al. (2013a) and Liet al. (2018). Koopmans et al. (2006) tested the assumption of the shapeof the density-profile itself, i.e. the power-law model. If thedensity profiles of lens galaxies are different from a power-law, but have the same shape for each galaxy (scaled to acommon scale), or, if they are different from a power-lawand different between lens galaxies, the power-law assump-tion might give biased results. In either case, it is expectedthat the inferred (average) logarithmic total mass densityslope inside R Ein will change with the ratio ( R Ein / R eff ) for aparticular model variation. In the case of the total mass den-sity slope is a broken power-law with a change in slope inside R Ein , one expects t to change depending on where the changein slope occurs with respect to R eff . Thus one is expected tofind some “average” slope weighed by the luminosity andkinematic profile, varying as function of ( R Ein / R eff ). This isdue to the dependence of R Ein mostly on the relative dis-tances of the lens and the source and is not a physical scaleof the lens galaxy itself. Koopmans et al. (2006) found noevidence of any clear systematic correlation between t and R Ein / R eff ratio (see Figure 5 therein). Figure 8 shows thetrend in the total mass density slope and the ratio R Ein / R eff for individual lenses. We also find no evidence of any corre-lation between t and R Ein / R eff ratio for both FBconst andAGNdT8 models, thus showing that our results are not bi-ased by the power-law assumption. The small deviations of t from 2.0 further support this. We conclude that our as-sumption of a single power-law shape for the total density profile is valid and reliable, consistent with the finding ofKoopmans et al. (2006). In a previous study using five model variations from OWLS(Schaye et al. 2010) and also the DM-only simulation, Duffyet al. (2010) probed the mass density slope at z =2 andcompared the results with SLACS lenses (Figure 3 therein).They found that implementation of AGN feedback, orextremely efficient feedback from massive stars, is necessaryto match the observed stellar-mass fractions in groups andclusters. However, that made the inner density profilesshallower than isothermal. They concluded that a weak orno feedback produces galaxies with an isothermal profile.This is consistent with the results in this work, where wealso found that weaker feedback leads to better agreementof the total mass density slope with SLACS, BELLS andSL2S observations. However, they also conclude that otherobservables, such as the stellar fractions, rule out thoseweak feedback models (e.g. see Crain et al. 2015). Oneway to explain this conundrum is that all the models misssomething critical, which may well be the case. Anotherexplanation could be that the strong lenses are a biasedsample of the total ETG population in a volume limitedsample. Previously, Sales et al. (2010) explored differentfeedback models in OWLS (Schaye et al. 2010) and foundlarge variations in the abundance and structural propertiesof bright galaxies at z = 2. They showed that models withinefficient or no feedback lead to the formation of overlymassive and compact galaxies with a large fraction (up-wards of 50 percent) of all available baryons (gas, stars, andstellar remnants) being retained in each halo. Increasing theefficiency of stellar or AGN feedback reduces the baryonicmass fraction fraction and increases the size of the simu- MNRAS000
10% intrinsic scatter asobtained from Koopmans et al. (2009).small Einstein radii. SL2S finds considerably more stronglenses with Einstein radii above 1 (cid:48)(cid:48) , since they can be moreclearly resolved in ground-based images. For similar com-parison of R Ein / R eff in SLACS, BELLS and SL2S, readersare refereed to Figure 1 in Sonnenfeld et al. (2013a) and Liet al. (2018). Koopmans et al. (2006) tested the assumption of the shapeof the density-profile itself, i.e. the power-law model. If thedensity profiles of lens galaxies are different from a power-law, but have the same shape for each galaxy (scaled to acommon scale), or, if they are different from a power-lawand different between lens galaxies, the power-law assump-tion might give biased results. In either case, it is expectedthat the inferred (average) logarithmic total mass densityslope inside R Ein will change with the ratio ( R Ein / R eff ) for aparticular model variation. In the case of the total mass den-sity slope is a broken power-law with a change in slope inside R Ein , one expects t to change depending on where the changein slope occurs with respect to R eff . Thus one is expected tofind some “average” slope weighed by the luminosity andkinematic profile, varying as function of ( R Ein / R eff ). This isdue to the dependence of R Ein mostly on the relative dis-tances of the lens and the source and is not a physical scaleof the lens galaxy itself. Koopmans et al. (2006) found noevidence of any clear systematic correlation between t and R Ein / R eff ratio (see Figure 5 therein). Figure 8 shows thetrend in the total mass density slope and the ratio R Ein / R eff for individual lenses. We also find no evidence of any corre-lation between t and R Ein / R eff ratio for both FBconst andAGNdT8 models, thus showing that our results are not bi-ased by the power-law assumption. The small deviations of t from 2.0 further support this. We conclude that our as-sumption of a single power-law shape for the total density profile is valid and reliable, consistent with the finding ofKoopmans et al. (2006). In a previous study using five model variations from OWLS(Schaye et al. 2010) and also the DM-only simulation, Duffyet al. (2010) probed the mass density slope at z =2 andcompared the results with SLACS lenses (Figure 3 therein).They found that implementation of AGN feedback, orextremely efficient feedback from massive stars, is necessaryto match the observed stellar-mass fractions in groups andclusters. However, that made the inner density profilesshallower than isothermal. They concluded that a weak orno feedback produces galaxies with an isothermal profile.This is consistent with the results in this work, where wealso found that weaker feedback leads to better agreementof the total mass density slope with SLACS, BELLS andSL2S observations. However, they also conclude that otherobservables, such as the stellar fractions, rule out thoseweak feedback models (e.g. see Crain et al. 2015). Oneway to explain this conundrum is that all the models misssomething critical, which may well be the case. Anotherexplanation could be that the strong lenses are a biasedsample of the total ETG population in a volume limitedsample. Previously, Sales et al. (2010) explored differentfeedback models in OWLS (Schaye et al. 2010) and foundlarge variations in the abundance and structural propertiesof bright galaxies at z = 2. They showed that models withinefficient or no feedback lead to the formation of overlymassive and compact galaxies with a large fraction (up-wards of 50 percent) of all available baryons (gas, stars, andstellar remnants) being retained in each halo. Increasing theefficiency of stellar or AGN feedback reduces the baryonicmass fraction fraction and increases the size of the simu- MNRAS000 , 1–23 (2018)
EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios Figure 7.
Correlation of the total mass density slope ( t ) with R Ein / R eff for nine different model variations of EAGLE andcomparison with SLACS, SL2S and BELLS. The symbols used here are: FBconst (blue down-filled-triangle), FBZ (cyanleft-filled-triangle), FB σ (green up-filled-triangle), Ref (red filled-circle), AGNdT8 (blue filled-hexagon), AGNdT9 (greenfilled-star), ViscLo (magenta right-filled-triangle), ViscHi (orange filled-octagon), NOAGN (brown filled-hexagon), SLACS(black open-square), SL2S (black open-diamond) and BELLS (black open-pentagon). Table 4.
The mean values of effective radius, R eff , of the lensing galaxies in different galaxy formation models and theirrespective mean Einstein radius, R Ein . The ratio R Ein / R eff gives a good estimate of the type of strong lenses simulated fromEAGLE and observations. Simulation < log ( R eff ) > rms < log ( R Ein ) > rms R Ein / R eff rmsRef-50 0.91 0.21 0.65 0.34 0.71 0.23FBconst 0.84 0.26 0.68 0.35 0.83 0.22FB σ , 1–23 (2018) Mukherjee et al.
Figure 8.
Correlation of the total mass density slope ( t )with R Ein / R eff for individual lensing galaxies in FBconst andAGNdT8 model variations of EAGLE. The red circles arethe lenses from FBconst and blue squares are from AGNdT8.The green dashed line is the mean total mass density slopeof SLACS (Koopmans et al. 2009) with ±
10% rms (shadedregion).lated galaxies. This trend is also consistent with our findings.The conclusion in Duffy et al. (2010) that NOAGN feed-back produces an isothermal profile is in contradiction withour analysis. One reason could be that our analysis is carriedout at a redshift of z = . , however, closer in redshift towhere these lens galaxies are observed and is consistent withthe results of several other simulation studies (Xu et al. 2017;Remus et al. 2017). Analysis of Duffy et al. (2010) is doneat a significantly higher redshift of z =2. In the next section,we will discuss the possible reasons for these differences inlight of potential systematics. There could be several effects that play a role in the com-parison between observations and simulations. We describethree of these below.
Evolution of the density profile:
The inclusion ofbaryons results in differences in the total density profilesthat depend on the efficiency of the radiative cooling andfeedback. As pointed out in Remus et al. (2017) and Xuet al. (2017), there could be a significant steepening of thetotal mass density slope in the simulations at lower red-shifts which might affect the density-slope analysis. Eventhough Koopmans et al. (2006) have shown that there isno strong evidence for evolution in the total mass densityslope in SLACS with redshift, this only holds for the red-shift range of . (cid:46) z (cid:46) . where the bulk SLACS lensesare found. Evolution might exist as we move to higher red-shifts (Bolton et al. 2012a; Sonnenfeld et al. 2013b). Thispotentially could explain the differences between this work( z =0.271) and the analysis in Duffy et al. (2010) which wascarried out at a higher redshift ( z =2). Moreover, the galaxiesanalyzed in Duffy et al. (2010) are less massive than thoseused in our analysis, mostly due to the significant differencein the redshifts of both the analysis. Also, for a randomlens system, we measured the density profile with the lens-ing galaxy at three lens redshifts of z l =0.101, z l =0.271 andz=0.474, with the source redshift remaining at z s =0.6. We found the difference in the slope parameter to be 0.02 and0.03 respectively i.e. much below the rms error. So we as-sume the effects are currently not significant in our case. Asimilar result is also reported recently by Wang et al. (2019)where they find the density slope to remain nearly invariantafter z =1 with a mild increase towards z = . However, inour case, a full-scale redshift evolution study is beyond thescope of this work. Simulation resolution bias:
Duffy et al. (2010) foundthat the resolution of the simulations can strongly affect theregion where the mass density slope is measured. Their den-sity slope measurement, however, was typically done aroundan Einstein radius of ∼ kpc, only just above the resolu-tion limit in the highest-resolution OWLS run at z = .Similarly, Schaller et al. (2015) showed that below a radiusof roughly ∼ ∼
10% (i.e. the conver-gence radius) and the enclosed mass to within ∼ M (cid:63) > M (cid:12) . Observational biases:
Dobler et al. (2008) found that themost significant instrumental selection effect is the finite sizeof the spectroscopic fiber, which selects against large sepa-ration lenses and results in a non-monotonic dependence ofthe rogue line probability (defined as the probability that agiven luminous red galaxy (LRG) has a rogue [O II ] line inits spectrum) on velocity dispersion. The situation is furthercomplicated by the effects of atmospheric seeing, which canadd flux from images outside or remove flux from imagesinside the fiber. Dobler et al. (2008) also reported that thelensing probability has a fairly weak dependence on the sizeof the source (see also the appendix of M18). Hence, whereasit is clear that lens galaxies are mass-selected and biased tohigher-mass galaxies, some of the most massive lenses mightnot have been found in SLACS due to the above-mentionedeffects. These massive systems are already rare to begin withand their absence would not bias the bulk of the lens pop-ulation which peaks around M (cid:63) = . M (cid:12) (Auger et al.2010b). As was shown by Bolton et al. (2008b), SLACS lensgalaxies also appear in all observational aspects to be sim-ilar to their LRG parent population, suggesting that theyare not a biased LRG sub-sample. Also, BELLS is very sim-ilar to SLACS in the type of lens galaxy, given the moreheterogeneous nature of the lenses and their environments MNRAS000
Dobler et al. (2008) found that themost significant instrumental selection effect is the finite sizeof the spectroscopic fiber, which selects against large sepa-ration lenses and results in a non-monotonic dependence ofthe rogue line probability (defined as the probability that agiven luminous red galaxy (LRG) has a rogue [O II ] line inits spectrum) on velocity dispersion. The situation is furthercomplicated by the effects of atmospheric seeing, which canadd flux from images outside or remove flux from imagesinside the fiber. Dobler et al. (2008) also reported that thelensing probability has a fairly weak dependence on the sizeof the source (see also the appendix of M18). Hence, whereasit is clear that lens galaxies are mass-selected and biased tohigher-mass galaxies, some of the most massive lenses mightnot have been found in SLACS due to the above-mentionedeffects. These massive systems are already rare to begin withand their absence would not bias the bulk of the lens pop-ulation which peaks around M (cid:63) = . M (cid:12) (Auger et al.2010b). As was shown by Bolton et al. (2008b), SLACS lensgalaxies also appear in all observational aspects to be sim-ilar to their LRG parent population, suggesting that theyare not a biased LRG sub-sample. Also, BELLS is very sim-ilar to SLACS in the type of lens galaxy, given the moreheterogeneous nature of the lenses and their environments MNRAS000 , 1–23 (2018)
EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios in the SL2S survey (which were morphologically and notspectroscopically selected), a lesser agreement with SL2S ismaybe not entirely unexpected. Nevertheless, previous ob-servational (e.g. Auger et al. 2010b; Sonnenfeld et al. 2013b;Li et al. 2018) and simulation analyses (Xu et al. 2016; Re-mus et al. 2017) these surveys have been compared amongeach other with the assumption that different observationalselections does not hinder a fair comparison.Moreover, as pointed out in Tortora et al. (2014b) (Ta-ble 1), strong lensing galaxies tend to be more compact thannon-lensing galaxies (e.g. SPIDER sample). However, SPI-DER uses K-band data and S´ersic fitted values of R eff , whileSLACS uses V-band and de Vaucouleurs fit. This can givedifferent results. But Auger et al. (2010b) showed that us-ing different fitting profiles gives negligible difference in R eff values. Even though this is consistent with the argumentthat strong lensing prefers weaker feedback which in turnforms galaxies with relatively smaller sizes at fixed stellarmass compared with more efficient feedback models, it mightbias correlations between galaxy properties. It could be thatLRGs are a biased sub-sample of galaxies with respect to vol-ume limited samples. We will explore this trend of galaxysizes in light of dark matter fraction and explore possiblesystematics that might be causing the differences in a forth-coming work. Even though we find qualitatively and visually quite simi-lar distributions between some of the model variations (i.e.,FBconst, AGNdT8) and observations, we have not quan-tified this (dis)agreement. The Kolmogorov–Smirnov (KS)test (Kolmogorov 1933) is a nonparametric test of the equal-ity of continuous, one-dimensional probability distributionsthat can be used to compare a sample with a reference prob-ability distribution, or be used to compare two samples. TheKS statistics (D-value) quantifies the maximum probabilitydifference between the cumulative probability distributionfunctions of two samples. A KS test also yields a p-value, be-ing the probability that two distributions are in fact drawnfrom the same underlying distribution and are dissimilar atthe current level (D) or larger, by random chance. In thiswork, we use the standard 1D KS test to compare the massdensity slopes and we use the 2D KS test of Peacock (1983)to compare the mass-size relations. Table 5 summarizes theKS D- and p-values by comparing the results from the EA-GLE model variations with those of SLACS, BELLS andSL2S, respectively.We indeed find that the FBconst, AGNdT8 and Vis-cHi models which visually appeared most consistent withthe observations, also have consistently high p-values (weassume a lower limit of acceptance of p > . ). When wecombine our analysis with the p-values from the 2D KS testfor the mass-size relation, we find that only the FBconstand AGNdT8 model variations remain viable. The Referencemodel, even though displaying similarity to observations ofthe mass-size relation from SLACS, performs poorly in themass density slope KS test. In addition, we can clearly ruleout the NOAGN, ViscLo, FBZ and FB σ model variationsbased on their failure to reproduce the observed strong lensdistributions in slope, mass and size. This confirms our ear-lier visual inspection. In this paper, we have systematically explored the impactof different galaxy formation processes used in the EAGLEhydrodynamical simulations – in particular stellar andAGN feedback – on strong lens observables in massiveearly-type galaxies with M (cid:63) > M (cid:12) . Simulations ofvarious mock-lens ensembles with the SEAGLE pipeline(M18) allow us to quantify in particular the (dis)agreementbetween the total mass density slopes around the Einsteinradius and the stellar mass-size relation between these mocklens ensembles and observations from the SLACS, BELLSand SL2S lens surveys. We compared these observableswith the outcome of a range of EAGLE model variations,varying stellar & AGN feedback and black hole accretiondisc viscosity parameters (Schaye et al. 2015; Crain et al.2015).We select potential strong lenses based on the stellar mass( M (cid:63) > M (cid:12) ) at a redshift of z l = . and createprojected mass maps for three different orientations. Wecreate mock lenses by ray tracing through the mass maps,placing an analytic Sersic (1968) source with observation-ally motivated parameters at a higher redshift ( z s = . ).We add realistic HST noise and PSF to mimic stronglenses found in observations. We calculate the projectedhalf-mass radius for each individual mass map. We alsomodel these lenses with an elliptic power-law model (EPL)and obtain their mass density slopes around their respectiveEinstein radii. Their strikingly similar visual appearance(see Figure 1) and similar stellar mass function to SLACS,SL2S and BELLS, motivates us to compare these observedlens samples to the simulated lens systems. This allows usto compare our findings with observations and draw thefollowing main conclusions:(1) The stellar mass-size relation and total mass densityslope of strong lens galaxies from SLACS, BELLS andSL2S agree best with EAGLE galaxy formation modelsthat have weak or mild AGN activity or in which stellarfeedback becomes inefficient at high gas densities (FB-const). In particular, the AGN model with a moderatetemperature increment during active periods, ∆ T = K (AGNdT8), shows excellent agreement with the observa-tions. Models with no or high-temperature increments agreeconsiderably less well in statistical KS tests. Similarly, thestellar-feedback model with a constant supernova energyinjection per unit stellar mass into the surrounding medium(i.e. FBconst) also shows excellent agreement with theobservations. Our finding that more efficient feedback yieldslarger galaxy sizes for a fixed galaxy mass is consistentwith previous work by Sales et al. (2010), based on OWLS(Schaye et al. 2010).(2) Models in which the energy injection per unit stellarmass formed depends either on metalicity or local environ-ment perform less well. Models with a high viscosity alsoreproduce the total mass density slopes of observed lensgalaxies, but perform poorly in reproducing the mass-sizerelation. The EAGLE Reference model (the benchmarkmodel) also does not perform well, most likely due to a tooefficient AGN feedback model. We note that agreement with MNRAS , 1–23 (2018) Mukherjee et al.
Table 5.
Mean, standard deviation and median values of mass density slopes inferred from lens modelling, t , of the simulatedlenses in different galaxy formation models. The KS test results for the mass density slopes (1D) and mass-size relation (2D)compared to SLACS, BELLS and SL2S, are also listed. The p-values that exceed 0.05, and hence indicate an acceptableagreement between the simulations and observations, are shown in bold. log M (cid:63) / M (cid:12) = . − . Mass density slope ( t ) Mass density slope KS test Mass-size KS test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Simulation Mean Std. Median SLACS SL2S BELLS SLACSD-value p-value D-value p-value D-value p-value D-value p-valueRef-100 2.09 0.26 2.24 0.26 0.53e-2 0.43 0.46e-3 0.42 0.17e-2 0.44 0.57e-2Ref-50 2.19 0.25 2.20 0.35 0.15e-5 0.51 0.27e-5 0.48 0.59e-5 0.41 FBconst 2.00 0.22 2.06 0.15 FB σ FBZ 1.60 0.21 1.65 0.82 5.08e-27 0.84 2.23e-14 0.63 1.24e-7 0.53 0.02ViscLo 1.64 0.25 1.61 0.68 1.2e-22 0.65 0.9e-10 0.46 0.001 0.52 0.002ViscHi 2.09 0.23 2.24 0.17
AGNdT9 2.18 0.24 2.25 0.23 0.01 0.24 0.10 0.22
SL2S is in general worse for all models, which we expectis due to its more heterogeneous selection (as opposed toSLACS and BELLS, they were not selected to be lenses).(3) Quantitatively, we find that if the simulated lensedimages are modeled using an elliptical power law (EPL)profile plus external shear, then the median total massdensity slope of galaxies from the AGNdT8 and FBconstmodels, which have the highest p -values in the KS tests, are t =2.01 and t =2.07, respectively, in good agreement withthe observations of SLACS, SL2S and BELLS. Galaxiesin the EAGLE Reference model, however, tend to havea steeper median total mass density slope ( t =2.24) thanobserved lens galaxies (i.e. t =2.08 for SLACS, t =2.11 forBELLS and t =2.18 for SL2S). This trend in mass densityslope agrees well with the results from other independentanalyses (e.g. Remus et al. 2017; Peirani et al. 2018).(4) We also assess whether in the best model variations thatemerged in our analysis (FBconst and AGNdT8) and thebenchmark model (Reference), t correlates with stellar massand found only a mild trend of slopes being shallower thanisothermal at higher stellar mass. This is consistent withobservations (Auger et al. 2010b; Tortora et al. 2014a) andsimulations (Remus et al. 2017; Xu et al. 2017). However,we find no evidence of correlation at any significant levelbetween R Ein / R eff ratios and t . This is consistent withKoopmans et al. (2006), Auger et al. (2009), Koopmanset al. (2009) and Treu et al. (2009). Thus any selection biasbased on mass should therefore not affect the conclusions.(5) We also find that the mean R Ein / R eff ratios in Reference,FBconst and AGNdT8 models are the closest to SLACS.We see a trend in the total mass density slope and R Ein / R eff ratio where increasing the feedback efficiency, increases the R eff thereby decreasing the value of R Ein / R eff and steepeningthe total mass density slope ( t ) as in the lens modeling t iscalculated around R Ein . Overall we conclude that more efficient feedback in massivegalaxies yields steeper total mass density slopes at a radiusof ≈ z =2) and forlower-mass galaxies. Also, they did not create simulatedlenses and model them with an EPL model, as done inthis work, which might lead to some additional biases. Wenote that LRGs could have other observational selectionbiases and might not represent volume limited samples. Ourconclusions are not biased by this trend as the evolutionof R eff is considerably small (Furlong et al. 2017) in EAGLE.Our results prefer galaxy-formation models that have beenruled out in Crain et al. (2015) after comparison withnon-lensing observations. Furlong et al. (2017) found thatthe Reference model agrees well with the observed mass-sizerelation when compared to non-lensing galaxies. Thisfinding is also seen in Duffy et al. (2010), who found thatweak feedback is required to match the lensing observations(consistent with our work) but also pointed out that otherobservables, such as the stellar fractions, rule out those weak MNRAS000
SL2S is in general worse for all models, which we expectis due to its more heterogeneous selection (as opposed toSLACS and BELLS, they were not selected to be lenses).(3) Quantitatively, we find that if the simulated lensedimages are modeled using an elliptical power law (EPL)profile plus external shear, then the median total massdensity slope of galaxies from the AGNdT8 and FBconstmodels, which have the highest p -values in the KS tests, are t =2.01 and t =2.07, respectively, in good agreement withthe observations of SLACS, SL2S and BELLS. Galaxiesin the EAGLE Reference model, however, tend to havea steeper median total mass density slope ( t =2.24) thanobserved lens galaxies (i.e. t =2.08 for SLACS, t =2.11 forBELLS and t =2.18 for SL2S). This trend in mass densityslope agrees well with the results from other independentanalyses (e.g. Remus et al. 2017; Peirani et al. 2018).(4) We also assess whether in the best model variations thatemerged in our analysis (FBconst and AGNdT8) and thebenchmark model (Reference), t correlates with stellar massand found only a mild trend of slopes being shallower thanisothermal at higher stellar mass. This is consistent withobservations (Auger et al. 2010b; Tortora et al. 2014a) andsimulations (Remus et al. 2017; Xu et al. 2017). However,we find no evidence of correlation at any significant levelbetween R Ein / R eff ratios and t . This is consistent withKoopmans et al. (2006), Auger et al. (2009), Koopmanset al. (2009) and Treu et al. (2009). Thus any selection biasbased on mass should therefore not affect the conclusions.(5) We also find that the mean R Ein / R eff ratios in Reference,FBconst and AGNdT8 models are the closest to SLACS.We see a trend in the total mass density slope and R Ein / R eff ratio where increasing the feedback efficiency, increases the R eff thereby decreasing the value of R Ein / R eff and steepeningthe total mass density slope ( t ) as in the lens modeling t iscalculated around R Ein . Overall we conclude that more efficient feedback in massivegalaxies yields steeper total mass density slopes at a radiusof ≈ z =2) and forlower-mass galaxies. Also, they did not create simulatedlenses and model them with an EPL model, as done inthis work, which might lead to some additional biases. Wenote that LRGs could have other observational selectionbiases and might not represent volume limited samples. Ourconclusions are not biased by this trend as the evolutionof R eff is considerably small (Furlong et al. 2017) in EAGLE.Our results prefer galaxy-formation models that have beenruled out in Crain et al. (2015) after comparison withnon-lensing observations. Furlong et al. (2017) found thatthe Reference model agrees well with the observed mass-sizerelation when compared to non-lensing galaxies. Thisfinding is also seen in Duffy et al. (2010), who found thatweak feedback is required to match the lensing observations(consistent with our work) but also pointed out that otherobservables, such as the stellar fractions, rule out those weak MNRAS000 , 1–23 (2018)
EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios feedback models. These seemingly opposing conclusionscould be due to either differences in the precise method-ologies adopted in the strong-lensing (Duffy et al. 2010,this work) and their non-lensing studies (Crain et al. 2015;Furlong et al. 2017), or additional observational selectionbiases in the galaxy samples, or even from missing crucialphysics. This also might indicate that LRGs that acts aslensing galaxy, might have different formation history thanthe rest. A complete redshift evolution study of the totalmass density slope will be addressed in a forthcoming work.In this work, we have demonstrated that observables ofstrong lens galaxies, in particular their total mass densityprofiles in the inner 3-10 kpc radial range, are very sensi-tive to variations in the feedback in galaxy formation mod-els. Although strong lensing analysis could have systemat-ical difference from non-lensing analysis in the methods ofthe modeling. We stress again that SLACS lens galaxies arenot different from the parent population of non-lens galaxiesfrom which they were drawn (Treu et al. 2006; Bolton et al.2008b). In paper III of the SEAGLE series we will explorethe systematic errors and compare simulated lenses to non-lensing ETGs from SPIDER survey (La Barbera et al. 2010;Tortora et al. 2012, 2014a) and, we will show that mass-sizerelation of EAGLE matches very well with it. Whereas inthis paper we have concentrated on the mass-size and massdensity slopes, in forthcoming papers we will investigate theinner mass regions in more detail, focusing in particular onthe effects of the dark matter distribution and the stellarIMF. ACKNOWLEDGEMENTS
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APPENDIX A: PRIOR USED FOR DENSITYSLOPE
Here in Table A1, we give the prior values used for mod-elling the simulated strong lenses with
LENSED . We use acombination of uniform and Gaussian priors. In Mukherjeeet al. (2018), we have explained for the motivation of pri-ors used and also demonstrated the tests that we performedwith different prior combinations for the EPL model.
APPENDIX B: SOURCE-SIZE RELATED TESTS
Here we present some results to demonstrate that the recov-ered source-sizes do not bias our conclusions. We compare
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EAGLE-II: Strong lensing in EAGLE galaxy formation scenarios Table A1.
The priors used in the modeling with an EPL plus shear mass model, using
LENSED . Parameter Prior type (cid:63)(cid:63)
Prior range Description µ σ min max x L norm 80.0 5.0 - - Lens position: x coordinate y L norm 80.0 5.0 - - Lens position: y coordinate r L unif - - 5.0 70.0 Einstein radius in pixel units t L norm 1.1 0.1 - - Surface mass density slope q L unif - - 0.2 0.99 Lens axis ratio φ L unif - - 0.0 180.0 Lens position angle in degrees, wrapped around γ norm 0.0 0.01 - - Shear vector γ norm 0.0 0.01 - - Shear vector x S norm 80.0 30.0 - - Source position: x coordinate y S norm 80.0 30.0 - - Source position: y coordinate r S unif - - 0.1 10.0 Source size in pixel units mag S unif - - -5.0 0.0 Source magnitude, adjusted with the background magnitude n S norm 1.0 0.1 - - S´ersic index q S norm 0.5 0.1 - - Source axis ratio φ S unif - - 0.0 180.0 Source position angle in degrees, wrapped around (cid:63) All values are in pixels except q , γ , t L , mag S , n S , and φ . (cid:63)(cid:63) norm = Gaussian (with mean µ and standard dev. σ ), unif = Uniform Source’s real magnitude = Background magnitude - mag s , where background magnitude is flux due to background in mag/arcsec the source-sizes between SIE and EPL and assess the source-size versus slope correlation in those models. These tests arein addition to those carried out in Mukherjee et al. (2018).Readers can consult the Appendix in the latter paper.In Figure B1, we present the histograms of source-sizecomparison between SIE and EPL for Reference-100 simu-lation. We show that the recovered source-sizes agree withthe input ones within the error limits for both the models.Also the SIE and EPL modelling provide consistent results.The difference between source-sizes from these two differentmodels is on average 0.008 arcsec, i.e. only 0.4% of the sourcesize. In Figure B3, furthermore, we compare the source-sizeof SLACS and the EAGLE Reference-100 model against thedensity slope. No clear source-size versus density slope cor-relation is seem, either for the SIE or the EPL models (whichalso agree with each other). Thus no obvious bias is foundin our analyses and hence we believe the conclusions to berobust. In Figure B2 We also compared the Source-sizes be-tween Reference, FBconst and AGNdT8 with mean of 0.218,0.217 and 0.213 arcsec respectively. Thus there is an overallgood agreement.Finally, even if there were a small bias, such biaseswould occurs in real lenses as well (see Newton et al. 2011),and hence would broaden both the observed and simulatedslope distributions and not impact the inference on the for-mation scenarios. APPENDIX C: COMPARISON WITH DIRECTFITTING
Previously we performed this test between density slopes in-ferred via convergence fitting, t N M and LENSED, t LE NSED in SEAGLE-I and reported (Figure. 8 therein) that therecould be a difference of 10% in Einstein radius (see alsoK¨ung et al. 2015) and demonstrated that we find a meanratio of 0.91 for t N M / t LE NSED , with a standard deviationof 0.17 (Figure 9 therein). Eventhough, the lens modeling
Figure B1.
Source size comparison between SIE and EPL inReference-100. fits the density profile (more precisely that of the potential)near the lensed images, whereas the direct fit is mostly fit-ting the higher density regions inside the mask, we do notfind any biased results from these two different methods. InFigure C1 we have shown the mean density slope comparisonbetween Reference, FBconst and AGNdT8 models.
MNRAS , 1–23 (2018) Mukherjee et al.
Figure B2.
Source size comparison between Reference-100, FB-const and AGNdT8 sub-grid models.
Figure B3.
Source size vs Mass density Slope for Reference-100simulation and SLACS.This paper has been typeset from a TEX/L A TEX file prepared bythe author.
Figure C1.
Comparison of mean density slopes for Refer-ence, FBconst and AGNdT8 simulation from direct fitting andLENSED, where the error bars are the 1 σ scatter of the sampledistributions. The black dashed line is the one-to-one mappingline. The dark and grey regions show the 1 σ and 2 σ , respec-tively, where in this case σ is the lens modeling uncertainty, i.e.0.05. MNRAS000