The impact of Λ CDM substructure and baryon-dark matter transition on the image positions of quad galaxy lenses
MMNRAS , 1–17 (2017) Preprint 8 August 2018 Compiled using MNRAS L A TEX style file v3.0
The impact of Λ CDM substructure and baryon-darkmatter transition on the image positions of quad galaxylenses
Matthew R. Gomer and Liliya L.R. Williams
School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis MN, 55455, USA
ABSTRACT
The positions of multiple images in galaxy lenses are related to the galaxy mass dis-tribution. Smooth elliptical mass profiles were previously shown to be inadequate inreproducing the quad population. In this paper, we explore the deviations from suchsmooth elliptical mass distributions. Unlike most other work, we use a model-freeapproach based on the relative polar image angles of quads, and their position in 3Dspace with respect to the Fundamental Surface of Quads. The FSQ is defined by quadsproduced by elliptical lenses. We have generated thousands of quads from syntheticpopulations of lenses with substructure consistent with Λ CDM simulations, and foundthat such perturbations are not sufficient to match the observed distribution of quadsrelative to the FSQ. The result is unchanged even when subhalo masses are increasedby a factor of ten, and the most optimistic lensing selection bias is applied. We thenproduce quads from galaxies created using two components, representing baryons anddark matter. The transition from the mass being dominated by baryons in inner radiito being dominated by dark matter in outer radii can carry with it asymmetries, whichwould affect relative image angles. We run preliminary experiments using lenses withtwo elliptical mass components with nonidentical axis ratios and position angles, per-turbations from ellipticity in the form of nonzero Fourier coefficients a and a , andartificially offset ellipse centers as a proxy for asymmetry at image radii. We showthat combination of these effects is a promising way of accounting for quad populationproperties. We conclude that the quad population provides a unique and sensitive toolfor constraining detailed mass distribution in the centers of galaxies. Key words: gravitational lensing: strong – galaxies: structure
The internal structure of early-type galaxies is of great in-terest in the context of galaxy formation. Inner regions ofgalaxies are composed of a combination of baryonic and darkmatter, forming an elliptical smooth density profile whichdeclines with radius. There are different models which areused to describe the shape of such a profile, for exampleSersic, NFW, and SIE (many are cataloged and describedin Keeton (2001)). These simple profiles do not account forseveral effects which could complicate the picture, such as Λ CDM substructure (Klypin et al. 1999; Moore et al. 1999;Springel et al. 2008), baryons that are distributed differentlythan dark matter, or line-of-sight effects in the case of lensing(Jaroszynski & Kostrzewa-Rutkowska 2012; McCully et al.2017). How big a role these effects play and how well realgalaxies are described by simple mass distributions at 0.5-2 effective radii are the central questions this paper seeks toexplore.One of the tools that is capable of extracting informa-tion about galaxies’ mass distributions is gravitational lens-ing (Blandford & Narayan 1986). In the context of this paperwe will specifically discuss quads, which are produced in thestrong lensing regime. Five total images can be created, butthe central image is demagnified and difficult to detect. Asa result, only four images are observed. The positions of thefour images of quads are directly related to the distributionof mass in the main lens.In this paper we work with image polar coordinates withrespect to the lens center. We use angular coordinates be-cause they have much less dependence on the exact radialprofile of the galaxy while still carrying information aboutthe level of symmetry in the lens (Williams et al. 2008).Like Woldesenbet & Williams (2012, 2015), we define θ as © a r X i v : . [ a s t r o - ph . GA ] F e b Gomer and Williams
Figure 1.
An example quad to depict the relative image angles.The left panel shows the projected isodensity contours for anelliptical lens with the images as magenta solid dots. The middlepanel labels the images based on arrival order and denotes therelevant angles between the images. The right panel shows thelocation within the diamond caustic for the source from which theimages arise. This figure is meant for illustration and all scalesare arbitrary. the angle between the first-arriving image and the second-arriving image, and likewise for θ and θ , for each quad.Figure 1 shows an example of a quad with the angles labeled.The first panel shows the projected isodensity contours of asynthetic elliptical lens. The second panel indicates the or-der of arrival for each image and the definitions of relativeimage angles. The last panel shows the location inside thecaustic for which a source gives rise to this quad.Woldesenbet & Williams (2012) describe an empiricalrelationship between the relative image angles of quad im-ages that holds true for any lens model where the mass dis-tribution is symmetric about two orthogonal axes (double-mirror symmetric) and is convex at all radii with no wavyfeatures (i.e. simple). Most parametric lens models to date fitthese very general criteria, including psuedo-isothermal el-liptical mass distributions (Kassiola & Kovner 1993), NFWprofiles (Navarro et al. 1996, 1997), de Vaucouleurs profiles(de Vaucouleurs 1948), Hernquist profiles (Hernquist 1990),and others (Keeton 2001).For such lenses, plotting the relative image angles ofquads in 3D results in every point lying on a surface calledthe Fundamental Surface of Quads (FSQ). We emphasizethat all double-mirror symmetric lenses with elliptical iso-density contours, independent of ellipticity or density profile,generate quads that lie very closely to the FSQ. Figure 2shows the angles of several thousand quads generated froman elliptical lens, plotted in 3D. The quads all lie on thiswell-defined surface.Aside from being an interesting feature which revealssome nuances in the solutions to the lens equation, the FSQalso provides a point of comparison with observations. Wold-esenbet & Williams (2012) catalog the known population of40 quads from galaxy lenses taken from a variety of surveys. There is a small caveat to the statement that all double-mirror-symmetric, convex mass distributions generate quads which lieon the FSQ, independent of mass profile. Different distributionswithin these criteria will generate surfaces which mathematicallydiffer from one another, albeit only slightly. In the most drasti-cally different models tested in Woldesenbet & Williams (2012),deviation from the FSQ is ∼ . ◦ in this 3D space, which translatesto roughly 0.01” on the sky. While some observational methodscan measure to this precision or greater, the effect is negligiblefor our purposes. From now on we will use a 2D projection of the FSQ (as inFigure 3). The horizontal axis of this projection is θ . Thevertical axis, ∆ θ , is the difference between the position ofthe quad and the FSQ, given θ and θ of the quad. Thisstyle of plotting will be used because it allows us to visual-ize deviation from the FSQ, with the FSQ itself representedas the ∆ θ = horizontal line. The observed quad popula-tion with error bars from Woldesenbet & Williams (2012)is also plotted for comparison. When one plots the observedquads in comparison to the FSQ, the points do not lie ex-actly on the surface. This distribution of observed quadsrelative to the FSQ is what we will refer to as “observeddeviations” or “observed discrepancy” from the FSQ. Thesedeviations from the FSQ can only mean that at least somefraction of the lenses which created the observed quads arenot perfect simple double-mirror symmetric lenses and musthave some perturbations from these lens models (Woldesen-bet & Williams 2012). Such perturbations could be anythingwhich causes the mass profile to be either not double-mirrorsymmetric or not universally convex. In addition to externalshear, candidates for such effects include Λ CDM substruc-ture, baryonic distributions which are different from that ofdark matter halos, or line-of-sight effects.External shear in and of itself is unlikely to account forthe deviations of the population of observed quads. Wolde-senbet & Williams (2015) showed that the presence of ex-ternal shear in an otherwise pure elliptical lens causes thepopulation of quads to split into two surfaces– one aboveand one below the FSQ– instead of lying on the FSQ it-self. An example is shown in Figure 3, which depicts quadsfrom an elliptical lens with varying levels of shear relative tothe FSQ. To get deviations from the FSQ of similar magni-tude as observations, rather exceptional values of shear arerequired.There is an additional problem with trying to explainthe FSQ deviations using shear alone. The natural temp-tation is to argue that any individual observed quad whichdoes not lie on the FSQ can be explained by a certain ex-ternal shear tuned to describe that particular system. Thiswould imply that the true population of quads is best repre-sented by a series of split surfaces of varying heights aboveand below the FSQ, similar to the gray populations in Fig-ure 3. This would mean that quads exist which have largedeviations from the FSQ for all values of θ . However, theobserved quad population seems to have larger deviationsfrom the FSQ for smaller θ and smaller deviations forlarger θ . This effect cannot be explained by external shearalone, although external shear may be a piece of the puzzle.This thought experiment shows the advantage of usinga population of quads rather than a single quad: some ex-planations which may work for a single system cannot solvethe problem for the population. To gain insight as to whatkinds of lenses are likely to be common, one must create apopulation of quads which is consistent with the observedpopulation. This is the goal of the present paper.Williams et al. (2008) studied the distribution of quadimage angles and found statistical evidence for substructure.They found that simulated populations of quads from sim-ple double-mirror symmetric elliptical lens models are un-able to match the observed quad population (Woldesenbet& Williams 2012, 2015).Though, in the context of image position angles, the MNRAS000
An example quad to depict the relative image angles.The left panel shows the projected isodensity contours for anelliptical lens with the images as magenta solid dots. The middlepanel labels the images based on arrival order and denotes therelevant angles between the images. The right panel shows thelocation within the diamond caustic for the source from which theimages arise. This figure is meant for illustration and all scalesare arbitrary. the angle between the first-arriving image and the second-arriving image, and likewise for θ and θ , for each quad.Figure 1 shows an example of a quad with the angles labeled.The first panel shows the projected isodensity contours of asynthetic elliptical lens. The second panel indicates the or-der of arrival for each image and the definitions of relativeimage angles. The last panel shows the location inside thecaustic for which a source gives rise to this quad.Woldesenbet & Williams (2012) describe an empiricalrelationship between the relative image angles of quad im-ages that holds true for any lens model where the mass dis-tribution is symmetric about two orthogonal axes (double-mirror symmetric) and is convex at all radii with no wavyfeatures (i.e. simple). Most parametric lens models to date fitthese very general criteria, including psuedo-isothermal el-liptical mass distributions (Kassiola & Kovner 1993), NFWprofiles (Navarro et al. 1996, 1997), de Vaucouleurs profiles(de Vaucouleurs 1948), Hernquist profiles (Hernquist 1990),and others (Keeton 2001).For such lenses, plotting the relative image angles ofquads in 3D results in every point lying on a surface calledthe Fundamental Surface of Quads (FSQ). We emphasizethat all double-mirror symmetric lenses with elliptical iso-density contours, independent of ellipticity or density profile,generate quads that lie very closely to the FSQ. Figure 2shows the angles of several thousand quads generated froman elliptical lens, plotted in 3D. The quads all lie on thiswell-defined surface.Aside from being an interesting feature which revealssome nuances in the solutions to the lens equation, the FSQalso provides a point of comparison with observations. Wold-esenbet & Williams (2012) catalog the known population of40 quads from galaxy lenses taken from a variety of surveys. There is a small caveat to the statement that all double-mirror-symmetric, convex mass distributions generate quads which lieon the FSQ, independent of mass profile. Different distributionswithin these criteria will generate surfaces which mathematicallydiffer from one another, albeit only slightly. In the most drasti-cally different models tested in Woldesenbet & Williams (2012),deviation from the FSQ is ∼ . ◦ in this 3D space, which translatesto roughly 0.01” on the sky. While some observational methodscan measure to this precision or greater, the effect is negligiblefor our purposes. From now on we will use a 2D projection of the FSQ (as inFigure 3). The horizontal axis of this projection is θ . Thevertical axis, ∆ θ , is the difference between the position ofthe quad and the FSQ, given θ and θ of the quad. Thisstyle of plotting will be used because it allows us to visual-ize deviation from the FSQ, with the FSQ itself representedas the ∆ θ = horizontal line. The observed quad popula-tion with error bars from Woldesenbet & Williams (2012)is also plotted for comparison. When one plots the observedquads in comparison to the FSQ, the points do not lie ex-actly on the surface. This distribution of observed quadsrelative to the FSQ is what we will refer to as “observeddeviations” or “observed discrepancy” from the FSQ. Thesedeviations from the FSQ can only mean that at least somefraction of the lenses which created the observed quads arenot perfect simple double-mirror symmetric lenses and musthave some perturbations from these lens models (Woldesen-bet & Williams 2012). Such perturbations could be anythingwhich causes the mass profile to be either not double-mirrorsymmetric or not universally convex. In addition to externalshear, candidates for such effects include Λ CDM substruc-ture, baryonic distributions which are different from that ofdark matter halos, or line-of-sight effects.External shear in and of itself is unlikely to account forthe deviations of the population of observed quads. Wolde-senbet & Williams (2015) showed that the presence of ex-ternal shear in an otherwise pure elliptical lens causes thepopulation of quads to split into two surfaces– one aboveand one below the FSQ– instead of lying on the FSQ it-self. An example is shown in Figure 3, which depicts quadsfrom an elliptical lens with varying levels of shear relative tothe FSQ. To get deviations from the FSQ of similar magni-tude as observations, rather exceptional values of shear arerequired.There is an additional problem with trying to explainthe FSQ deviations using shear alone. The natural temp-tation is to argue that any individual observed quad whichdoes not lie on the FSQ can be explained by a certain ex-ternal shear tuned to describe that particular system. Thiswould imply that the true population of quads is best repre-sented by a series of split surfaces of varying heights aboveand below the FSQ, similar to the gray populations in Fig-ure 3. This would mean that quads exist which have largedeviations from the FSQ for all values of θ . However, theobserved quad population seems to have larger deviationsfrom the FSQ for smaller θ and smaller deviations forlarger θ . This effect cannot be explained by external shearalone, although external shear may be a piece of the puzzle.This thought experiment shows the advantage of usinga population of quads rather than a single quad: some ex-planations which may work for a single system cannot solvethe problem for the population. To gain insight as to whatkinds of lenses are likely to be common, one must create apopulation of quads which is consistent with the observedpopulation. This is the goal of the present paper.Williams et al. (2008) studied the distribution of quadimage angles and found statistical evidence for substructure.They found that simulated populations of quads from sim-ple double-mirror symmetric elliptical lens models are un-able to match the observed quad population (Woldesenbet& Williams 2012, 2015).Though, in the context of image position angles, the MNRAS000 , 1–17 (2017) mpact of substructure/baryon-DM transition on quads Figure 2.
The surface shown is the Fundamental Surface of Quads (FSQ), a curved surface described via a 4th-order polynomial. Twodifferent viewing angles of the FSQ are plotted in 3D, with the right panel looking “down the barrel” from above. Additionally, 10,000quads are generated from an double-mirror symmetric elliptical lens, specifically an Einasto profile with α = . and an axis ratio of0.82. The angles between images are plotted in 3D as blue points. All points lie very close to the FSQ. exact radial density profile does not have a strong effect(Woldesenbet & Williams 2015), lenses in this paper will beconstructed using Einasto profiles, which fit halos well ac-cording to simulations (Navarro et al. 2010; Springel et al.2008). Such profiles feature a changing logarithmic densityslope, γ = − d ln ρ / d ln r , which is shallower in the innermostregions and steepens at farther radii based on the shape pa-rameter α . Einasto profiles can be parameterized such that ρ ( r ) = ρ − exp (cid:26) − α (cid:20)(cid:18) rr − (cid:19) α − (cid:21)(cid:27) (1)where r − is the scale radius at which γ =2, correspondingto an isothermal sphere analog. ρ − is the density at r − (Einasto 1965; Springel et al. 2008).Our goal is to construct a population of quads which isconsistent with the observed discrepancy from the FSQ viaa population of galaxies which are physically motivated. It isclear that a simple ellipsoidal smooth profile will be unableto provide enough asymmetry at the radius where imagesare located to create significant deviations from the FSQ. Assuch, we must consider effects which cause the galaxy massdistribution to be more complicated. We consider two maintypes of asymmetries: those arising from Λ CDM substruc-ture and those arising from the transition region where themass distribution changes from being dominated by baryonsto being dominated by dark matter. In Section 2 we will dis-cuss the synthesis of galaxies with Λ CDM substructure andthe effects of substructure on the quad distribution relativeto the FSQ. It will turn out that substructure at the Λ CDMlevel will be insufficient to explain the FSQ deviations, so inthis section we also conduct an experiment by increasing themass of substructure by a factor of ten. In Section 3 we willdiscuss the potential effects of selection biases on the quadpopulation and the role these biases play with regard to theFSQ. Section 4 discusses the effects of adding a baryon com-ponent to the mass distribution which is not identical to thedark matter. Finally Section 5 summarizes these results anddiscusses their implications.Throughout the paper, we use H = km s − Mpc − , Ω m = . , and Ω Λ = . . Of all phenomena potentially capable of causing deviationsfrom the FSQ, the first candidate we explore is that ofdark matter subhalos as predicted by Λ CDM simulations(Springel et al. 2008; Navarro et al. 2010) and detected bymultiple observations (Dalal & Kochanek 2002; Vegetti et al.2010, 2012; Hezaveh et al. 2016). First we will discuss theprocess by which we generate lenses with such substructure,then we will examine the effect this substructure has on thedistribution of quads relative to the FSQ.
We synthesized lenses in 3D with properties similar to thosefrom the Aquarius Project simulations (Springel et al. 2008;Navarro et al. 2010) in that they consist of many assortedsubhalos within a single main halo. We need to be able toconstruct these subhalos in terms of Einasto profiles, pa-rameterized only by the halo mass and shape parameter. Tothis end, we utilize the information gleaned about subhalomasses and distributions from Springel et al. (2008), whopresent several relations which can be used to connect scaleradius, central density, and total mass of a halo. Comingfrom fits to simulated subhalos, the first of such relation-ships is between the maximum circular velocity of particleswithin a subhalo, V max , and the total mass of that subhalo, M sub : M sub (cid:39) . × (cid:18) V max km s − (cid:19) . [ M (cid:12) ] (2)The maximum circular velocity is also presented as V max = . Gr − ρ − , where ρ − is the density at the scaleradius. Eliminating V max relates M sub , r − , and ρ − , but toeliminate ρ − an additional equation is needed: the Einastoprofile itself, specifically the enclosed mass within a givenradius of such a profile, again listed in Springel et al. (2008). M ( r , α ) = π r − ρ − α exp (cid:18) α + − ln 8 α (cid:19) γ (cid:20) α, α (cid:18) rr − (cid:19) α (cid:21) (3)where γ ( a , x ) is the lower incomplete gamma function. We MNRAS , 1–17 (2017)
Gomer and Williams
Figure 3.
Observed population of galaxy-scale quads from Wold-esenbet & Williams (2012), plotted relative to FSQ in projection(red) with error bars determined from astrometric errors in imageposition. Rather than displaying the quad angles in 3D, only the θ -axis is shown (horizontal axis). The ∆ θ -axis (vertical) de-picts the difference between a quad’s value for θ and the valuewhich it would need in order to lie on the FSQ. The FSQ itself isrepresented by a horizontal line at ∆ θ =0. It is clear that the dis-tribution of observed quads deviates considerably from the FSQ.This deviation indicates, perhaps unsurprisingly, that a purely el-liptical model is too simplistic to describe the observed quad lenspopulation. Additionally, quads are generated from the same lensas Figure 2 but with varying levels of external shear and plottedin grayscale. The light gray corresponds to a shear of 0.1, thedarker gray a shear of 0.2, and the black a shear of 0.5. As shearincreases, the quads lie not on the FSQ itself but on two surfacesincreasingly split above and below the FSQ. would like to be able to set the Einasto profile enclosed massequal to the subhalo mass M sub , but to do this we need tomake an approximation. M sub is defined in Springel et al.(2008) as the mass enclosed within the radius such that thedensity of the particles bound to the subhalo drops below thelocal density at the subhalo’s location. This is not strictlyequivalent to the mass determined by integrating a profileto infinity, but it should be close enough for our purposes,which are not so much to exactly model the Aquarius Projectas much as to see if Λ CDM subhalos are a reasonable sourcefor the deviations from the FSQ. With this in mind, we canapproximate M sub (cid:39) M ∞ , where M ∞ = lim r →∞ M ( r , α ) . Thisallows us to eliminate ρ − and find a relationship between r − and M ∞ . r − kpc = . × − A ( α ) (cid:18) M ∞ M (cid:12) (cid:19) . (4)where A ( α ) = α exp (cid:18) α + − ln 8 α (cid:19) Γ (cid:20) α (cid:21) with Γ ( x ) being the complete gamma function. A ( α ) ∼ forrealistic values of α . The next step is to project this Einasto profile into 2D for lensing. Dhar & Williams (2010) showedthat the 2D projected Einasto profile can be analyticallyapproximated as Σ ( X , α ) = Σ Γ [ n + ] (cid:26) n Γ (cid:104) n , b ( ζ X ) n (cid:105) (5) + b n X (cid:16) − n (cid:17) γ (cid:20) , X n (cid:21) e − bX n − δ b n Xe − b (cid:16) √ + (cid:15) X (cid:17) n (cid:27) where Γ ( x , a ) is the upper incomplete gamma function and X = Rr − , with R now being the 2D projected radius from thecenter and r − being the same 3D scale radius as before. Thecentral surface mass density at R = is Σ , while n = α and b = n ; δ , (cid:15) , and ζ are functions of n and X . To calculate themass within R one needs to integrate: M ( R , α ) = ∬ Σ ( R , α ) R dR d θ = π Σ r − ∫ Rr − Xs ( X , α ) dX (6)where s ( X , α ) = Σ ( X , α )/ Σ . Taking R to infinity and numeri-cally integrating, we can combine Equation 6 with Equation4, resulting in a relationship between total mass and centraldensity Σ M (cid:12) kpc − = . × A ( α ) ∫ ∞ Xs ( X , α ) dX (cid:18) M ∞ M (cid:12) (cid:19) . = . × B ( α ) (cid:18) M ∞ M (cid:12) (cid:19) . (7)where all of the α − dependence is absorbed into a function B ( α ) ∼ − . With this, we can readily convert a desiredmass for a halo into a corresponding 2D central density andEinasto scale radius.Before continuing, it is prudent to check that the ap-proximations above yield results that are reasonable. Specif-ically, we should confirm that these formulas, which are de-rived in part from information about the subhalos, can rea-sonably accurately describe the main halo as well. We cantest this by inserting values quoted in the Aq-A-1 simula-tion, the highest resolution simulation discussed in Springelet al. (2008). M , the mass enclosed within the radius atwhich the enclosed density is 50 × the average density of theuniverse, is quoted for Aq-A-1 as . × M (cid:12) . The shapeparameter is quoted in Navarro et al. (2010) as α = . .Similar to above, one can approximate M (cid:39) M ∞ and plugthis into Equation 4 to yield a scale radius of 12.3 kpc, whileSpringel et al. (2008) report a scale radius of 13.0 kpc, whichgives an idea of the level of error in the above approxima-tions.Now that the relationships between α , Σ , and M havebeen established, a small deviation is made from the Aquar-ius Project, in that the shape parameter of the main lens ischanged from α = . to α = . , reducing the scale radiusto 8.5 kpc. The reasoning for this is as follows. Since the goalis to compare simulated lenses which have only a single masscomponent with observations which have both baryons anddark matter, it is necessary to alter the shape of the single-component profile to be less like dark-matter-only simula-tions and more like observations. Koopmans et al. (2006)have measured the average 3D logarithmic density slopewithin the Einstein radius, (cid:104) γ (cid:48) D (cid:105) = (cid:104)− d log ρ / d log r (cid:105) = . + . − . . The slope of 2.0 indicates that the effects of baryonsand dark matter have “conspired” to make the density slope MNRAS000
Observed population of galaxy-scale quads from Wold-esenbet & Williams (2012), plotted relative to FSQ in projection(red) with error bars determined from astrometric errors in imageposition. Rather than displaying the quad angles in 3D, only the θ -axis is shown (horizontal axis). The ∆ θ -axis (vertical) de-picts the difference between a quad’s value for θ and the valuewhich it would need in order to lie on the FSQ. The FSQ itself isrepresented by a horizontal line at ∆ θ =0. It is clear that the dis-tribution of observed quads deviates considerably from the FSQ.This deviation indicates, perhaps unsurprisingly, that a purely el-liptical model is too simplistic to describe the observed quad lenspopulation. Additionally, quads are generated from the same lensas Figure 2 but with varying levels of external shear and plottedin grayscale. The light gray corresponds to a shear of 0.1, thedarker gray a shear of 0.2, and the black a shear of 0.5. As shearincreases, the quads lie not on the FSQ itself but on two surfacesincreasingly split above and below the FSQ. would like to be able to set the Einasto profile enclosed massequal to the subhalo mass M sub , but to do this we need tomake an approximation. M sub is defined in Springel et al.(2008) as the mass enclosed within the radius such that thedensity of the particles bound to the subhalo drops below thelocal density at the subhalo’s location. This is not strictlyequivalent to the mass determined by integrating a profileto infinity, but it should be close enough for our purposes,which are not so much to exactly model the Aquarius Projectas much as to see if Λ CDM subhalos are a reasonable sourcefor the deviations from the FSQ. With this in mind, we canapproximate M sub (cid:39) M ∞ , where M ∞ = lim r →∞ M ( r , α ) . Thisallows us to eliminate ρ − and find a relationship between r − and M ∞ . r − kpc = . × − A ( α ) (cid:18) M ∞ M (cid:12) (cid:19) . (4)where A ( α ) = α exp (cid:18) α + − ln 8 α (cid:19) Γ (cid:20) α (cid:21) with Γ ( x ) being the complete gamma function. A ( α ) ∼ forrealistic values of α . The next step is to project this Einasto profile into 2D for lensing. Dhar & Williams (2010) showedthat the 2D projected Einasto profile can be analyticallyapproximated as Σ ( X , α ) = Σ Γ [ n + ] (cid:26) n Γ (cid:104) n , b ( ζ X ) n (cid:105) (5) + b n X (cid:16) − n (cid:17) γ (cid:20) , X n (cid:21) e − bX n − δ b n Xe − b (cid:16) √ + (cid:15) X (cid:17) n (cid:27) where Γ ( x , a ) is the upper incomplete gamma function and X = Rr − , with R now being the 2D projected radius from thecenter and r − being the same 3D scale radius as before. Thecentral surface mass density at R = is Σ , while n = α and b = n ; δ , (cid:15) , and ζ are functions of n and X . To calculate themass within R one needs to integrate: M ( R , α ) = ∬ Σ ( R , α ) R dR d θ = π Σ r − ∫ Rr − Xs ( X , α ) dX (6)where s ( X , α ) = Σ ( X , α )/ Σ . Taking R to infinity and numeri-cally integrating, we can combine Equation 6 with Equation4, resulting in a relationship between total mass and centraldensity Σ M (cid:12) kpc − = . × A ( α ) ∫ ∞ Xs ( X , α ) dX (cid:18) M ∞ M (cid:12) (cid:19) . = . × B ( α ) (cid:18) M ∞ M (cid:12) (cid:19) . (7)where all of the α − dependence is absorbed into a function B ( α ) ∼ − . With this, we can readily convert a desiredmass for a halo into a corresponding 2D central density andEinasto scale radius.Before continuing, it is prudent to check that the ap-proximations above yield results that are reasonable. Specif-ically, we should confirm that these formulas, which are de-rived in part from information about the subhalos, can rea-sonably accurately describe the main halo as well. We cantest this by inserting values quoted in the Aq-A-1 simula-tion, the highest resolution simulation discussed in Springelet al. (2008). M , the mass enclosed within the radius atwhich the enclosed density is 50 × the average density of theuniverse, is quoted for Aq-A-1 as . × M (cid:12) . The shapeparameter is quoted in Navarro et al. (2010) as α = . .Similar to above, one can approximate M (cid:39) M ∞ and plugthis into Equation 4 to yield a scale radius of 12.3 kpc, whileSpringel et al. (2008) report a scale radius of 13.0 kpc, whichgives an idea of the level of error in the above approxima-tions.Now that the relationships between α , Σ , and M havebeen established, a small deviation is made from the Aquar-ius Project, in that the shape parameter of the main lens ischanged from α = . to α = . , reducing the scale radiusto 8.5 kpc. The reasoning for this is as follows. Since the goalis to compare simulated lenses which have only a single masscomponent with observations which have both baryons anddark matter, it is necessary to alter the shape of the single-component profile to be less like dark-matter-only simula-tions and more like observations. Koopmans et al. (2006)have measured the average 3D logarithmic density slopewithin the Einstein radius, (cid:104) γ (cid:48) D (cid:105) = (cid:104)− d log ρ / d log r (cid:105) = . + . − . . The slope of 2.0 indicates that the effects of baryonsand dark matter have “conspired” to make the density slope MNRAS000 , 1–17 (2017) mpact of substructure/baryon-DM transition on quads approximately “isothermal.” The authors model the pro-file as a power law, meaning the average logarithmic densityslope (cid:104) γ (cid:48) D (cid:105) is the same as the local slope at the Einsteinradius, γ (cid:48) D | R E . It is reasonable to assume that the 2D log-arithmic density slope γ (cid:48) D = − d log Σ / d log R ≈ γ (cid:48) D − ≈ ,although due to projection effects and a changing slope, thevalue at the Einstein radius would not be exactly γ (cid:48) D − .Without changing the total mass of the halo, altering α to 0.14 makes the mass profile more centrally concentratedthan the Aquarius simulations, which increases the size ofthe Einstein radius, and steepens the local slope at that ra-dius from γ (cid:48) D = . to 0.91. Since this is closer to the targetof ≈ , it is considered more realistic. In the Aquarius simulations, subhalos are fit by several dif-ferent shape parameters with slightly higher values thanthe main halo shape parameters, roughly ranging from α sub = α sub is set to 0.18 for allsubhalos, similar to what is done in Springel et al. (2008),which the authors argue is roughly equivalent to having α sub in the range of . − . .For substructure to be analogous to that in the Aquariussimulations, subhalos must follow a mass function: dNdM = a (cid:18) Mm (cid:19) n (8)with n = − . and a = . × / M where m = − M . M refers to the main halo and will again be approximatedas M ∞ . Subhalo populations analogous to the Aq-A-1 sim-ulation run by Springel et al. (2008) are constructed using M = . × M (cid:12) The fractional mass in subhalos f sub is oforder 0.1, so a population of subhalo masses is synthesizedto have a cumulative mass of . × M (cid:12) . Optimally, apopulation would include subhalo masses all the way downto the free-streaming limit of dark matter (Springel et al.2008), but this is not feasible since the inclusion of lowermass subhalos requires many more subhalos themselves, andtherefore is computationally expensive. Additionally, smallerhalos are less important because they are less likely to pro-duce the densities or shears necessary to affect the position ofimages. With this in mind, each synthesized main halo con-tains a population of 133,400 subhalos with masses rangingfrom M (cid:12) to M (cid:12) , distributed in 3D.In our lensing simulations themselves, however, very fewof these numerous subhalos are actually included. This isbecause we are interested only in the regions which are likelyto produce quads, which limits us to the central region of thegalaxy. Only the subhalos which are positioned along the lineof sight and near the center in projection will be relevant.The 3D distribution of subhalos in space is produced by anEinasto profile with α = . and r − = kpc, consistentagain with Springel et al. (2008). Subhalos which are farther “Isothermal” refers to the fact that the slope is equivalent tothat of an isothermal sphere, but the quotation marks are usedbecause the term in this context makes no claim as to the dynam-ics of the system, such as whether or not the system is actuallyat the same temperature throughout. than 17 kpc away from the line-of-sight axis– a somewhatarbitrary value chosen to be slightly larger than the windowof the 2D simulations– are considered too far from the centerand are omitted. This typically leaves − subhalosremaining to be included in the lens itself.To check the effect of halos outside the simulation win-dow, one can estimate the amount by which the image posi-tions, and therefore angles, would change. We estimate thisby treating the subhalo as a point mass, and placing it at twowindow radii. The largest possible mass for such a subhalowould be M (cid:12) . We then calculate the deflection anglefor a single image at the Einstein radius. For our lens andsource redshifts, this corresponds to a deflection of 0.05 kpc= 0.007”. To have the most extreme effect on releative im-age angles, we imagine the deflector is on the x-axis and theimage is on the y-axis, so that the deflection is nearly com-pletely in the azimuthal direction with respect to the centerof the lens. It turns out that the polar image angle is alteredby ∼ ◦ . Additionally, since image angles are defined withrespect to one another, there could be up to a factor of 2additional effect if another image lies opposite the first (at y (cid:39) − R E ). This means that if a subhalo were placed out-side the window in such a way to optimally alter our imageangles, it could only change angles by ∼ ◦ , which is sig-nificantly less than the level of FSQ deviation necessary tomatch observations ( ∼ ◦ ). We therefore choose to simplyignore the effects of subhalos outside of the window radius. Each main lens is identical, with α = . and M ∞ = . × M (cid:12) , corresponding to r − = . kpc. The lenses areelliptical, with an axis ratio of q = . . The critical lensingdensity is set to Σ crit = . × M (cid:12) kpc − , correspondingto the source being at redshift z = . and the lens being at z = . , making the Einstein radius R E = . kpc. For nowlenses have zero external shear. Stacked on top of the mainhalos are ∼ subhalos that happen to lie along the line ofsight within the virial radius. Different lenses are created viadifferent random seeds for the subhalos. An example lens isdepicted in Figure 4.Near the Einstein radius, our simulated lenses have pro-jected mass fractions (cid:104) f (cid:105) ranging from 0.1% to 1.0% in sub-halos. This is consistent with the Aquarius halos, which have (cid:104) f (cid:105) = . + . − . % (Springel et al. 2008; Xu et al. 2009; Vegettiet al. 2012), shown in Figure 5. The number of halos in eachmass bin is consistent with the Aquarius simulations (Xuet al. 2015).It is worth noting that recent observations have detectedthis type of substructure in real lenses and have inferredmass fractions which are higher than Λ CDM simulations.In a study of seven lens systems, Dalal & Kochanek (2002)deduced a local mass fraction of subhalos at the image ra-dius between 0.6% and 7% with a 90% confidence level.Vegetti et al. (2010) found a dark substructure in the lensSDSSJ0946+1006, which is one of the 40 quads in Woldesen-bet & Williams (2012). For this galaxy, Vegetti et al. (2010)infer (cid:104) f (cid:105) = . + . − . % at the Einstein radius when assuming n = − . ± . . When comparing their value with simulations,they find a likelihood of 0.51, which is consistent given theirsole detection. Vegetti et al. (2012) found a dark satellite inthe JVAS B1938+666 system implying an average subhalo MNRAS , 1–17 (2017)
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Figure 4.
Density contours and caustic for a lens with Λ CDMsubstructure. This lens was chosen as an example because it haslarger perturbations due to substructure relative to some of theother realizations. Scale is in arcseconds. At z=0.6, 1 arcsecondcorresponds to 6.7 kpc. Typical quad images are at (cid:39) . kpcfrom the center. Source positions within the diamond caustic areshaded according to resulting θ values, with darker gray in-dicating an angle less than 40 degrees, lighter gray indicatinggreater than 60 degrees, and gray indicating the intermediate an-gles. Quads more similar to “Einstein crosses” come from sourcesin the lighter gray central region. 10,000 quads are created fromsources within the diamond caustic. mass fraction (cid:104) f (cid:105) = . + . − . % within the Einstein radius anda different mass function slope n = − . + . − . . Had they as-sumed n = − . ± . they would be closer to the Aquariussimulated mass fraction with their result of (cid:104) f (cid:105) = . + . − . % ,arguing that the remaining discrepancy is due to the factthat their galaxy is at a different redshift than those in theAquarius Project, which are at z=0. These findings are com-pared visually with our simulated lens population in Figure5. Finally, Hezaveh et al. (2016) report the detection of adark subhalo in the SDP.81 system. They claim their mass Figure 5.
The distribution of subhalo mass fractions localizednear the Einstein radius for the population of 20 lenses is depictedas a solid blue histogram. The same 20 lenses, but with their sub-structure mass amplified by a factor of 10, have the mass fractiondistribution shown as the white open histogram. Our simulated Λ CDM lenses are consistent with the values for the Aquariussimulations from Xu et al. (2009) (black circle), but are on theedge of the 68% C.L. for the observations of Vegetti et al. (2010)(green square) and Vegetti et al. (2012) (red triangle) and the90% C.L. for the observations of Dalal & Kochanek (2002) (bluediamond). Vertical positions for these values are arbitrary. Thisis not necessarily inconsistent, but it does provide some inspira-tion for our × Λ CDM test in Section 2.4, where substructureproduces mass fractions closer to these observations, similar tothe white histogram. function is consistent with simulations, based on their onedetected subhalo at M (cid:12) and upper limits established atlower masses. It remains to be seen whether the minor ten-sion between some observations and simulations indicates aproblem with theory or is simply a result of having a smallnumber of observations.More recent simulations which include baryon effectshave been done by Fiacconi et al. (2016), which found thatlocal projected mass fractions at the Einstein radius for ∼ M (cid:12) halos can exceed 2% if the lens is at a redshiftof 0.7. Since the halos are both more massive and at ahigher redshift than the Aquarius halo, they are dynami-cally younger, as their subhalos are accreted more recently,and therefore clumpier, with higher mass fractions.Our lenses are meant to represent the population of ob-served galaxies, so it is useful to compare them with typi-cal quad-producing galaxies. Unlike the real galaxy popula-tion, these synthetic lenses represent a population of galax-ies which would all have the same mass, redshift, and profileshape. This may seem like an oversimplification, but notethat these parameters would not cause any asymmetries inthe lenses. Similar galaxies at different redshifts, for exam-ple, would change the critical densities necessary for lensingand therefore the radial positions of the images, but, since MNRAS000
The distribution of subhalo mass fractions localizednear the Einstein radius for the population of 20 lenses is depictedas a solid blue histogram. The same 20 lenses, but with their sub-structure mass amplified by a factor of 10, have the mass fractiondistribution shown as the white open histogram. Our simulated Λ CDM lenses are consistent with the values for the Aquariussimulations from Xu et al. (2009) (black circle), but are on theedge of the 68% C.L. for the observations of Vegetti et al. (2010)(green square) and Vegetti et al. (2012) (red triangle) and the90% C.L. for the observations of Dalal & Kochanek (2002) (bluediamond). Vertical positions for these values are arbitrary. Thisis not necessarily inconsistent, but it does provide some inspira-tion for our × Λ CDM test in Section 2.4, where substructureproduces mass fractions closer to these observations, similar tothe white histogram. function is consistent with simulations, based on their onedetected subhalo at M (cid:12) and upper limits established atlower masses. It remains to be seen whether the minor ten-sion between some observations and simulations indicates aproblem with theory or is simply a result of having a smallnumber of observations.More recent simulations which include baryon effectshave been done by Fiacconi et al. (2016), which found thatlocal projected mass fractions at the Einstein radius for ∼ M (cid:12) halos can exceed 2% if the lens is at a redshiftof 0.7. Since the halos are both more massive and at ahigher redshift than the Aquarius halo, they are dynami-cally younger, as their subhalos are accreted more recently,and therefore clumpier, with higher mass fractions.Our lenses are meant to represent the population of ob-served galaxies, so it is useful to compare them with typi-cal quad-producing galaxies. Unlike the real galaxy popula-tion, these synthetic lenses represent a population of galax-ies which would all have the same mass, redshift, and profileshape. This may seem like an oversimplification, but notethat these parameters would not cause any asymmetries inthe lenses. Similar galaxies at different redshifts, for exam-ple, would change the critical densities necessary for lensingand therefore the radial positions of the images, but, since MNRAS000 , 1–17 (2017) mpact of substructure/baryon-DM transition on quads this affects all parts of the lens equally, the double-mirrorsymmetry of the lens remains unaltered. Since the double-mirror symmetry is unchanged, the deviations from the FSQwill be no different (Woldesenbet & Williams 2015). Thisfurther justifies the assumptions about the mass profile ofthe lenses made in Section 2.1.The ellipticity and external shear for each galaxy arealso held constant for now, with an axis ratio of 0.82 and noexternal shear. Unlike the parameters above, these proper-ties can cause deviations in the FSQ, provided the shear isat an oblique angle with respect to the ellipse major axis.In the terminology of Woldesenbet & Williams (2015), theseparameters create Type II lenses, in that they break thedouble-mirror symmetry only once, as opposed to substruc-tured lenses (Type III), which have no remaining symme-tries. The reason these parameters are held constant for nowis not because they have no effect on image angles, but thatwe seek to isolate the effect of substructure. In Section 2.5we will relax the restriction that ellipticity and shear be soconstrained.From the example lens we created 10,000 quads anddetermined their deviations from the FSQ. The FSQ is de-scribed by a polynomial fit explicitly shown in Woldesenbet& Williams (2012) which expresses θ as a function of θ and θ . For each quad, the difference between the FSQ-predicted θ and the actual value for each quad is used asthe measure of the deviation. In Figure 6, the resulting de-viations from the FSQ for the example lens are shown. Thepopulation of 40 observed quads cataloged in Woldesenbet& Williams (2012) is plotted alongside our simulated quadsin the bottom panel of Figure 6. It is evident by eye thatthe scale of deviations provided by Λ CDM substructure issimply too small to account for observations.
Because substructure at the scale of Λ CDM simulations istoo weak of a perturbation to create the observed deviationsfrom the FSQ, we now increase the mass of the subhalos tosee what the effects would be. This idea is motivated inpart by the potential tension mentioned above between thesubhalo mass fractions of simulations and observations ofVegetti et al. (2010, 2012); Hezaveh et al. (2016), where ob-servations may hint at more mass in subhalos than what Λ CDM simulations predict. The logic is essentially that ifsubhalos are increased in mass, they may be able to repro-duce the deviations from the FSQ. That would hint thatperhaps the predictions from Λ CDM simulations do not cre-ate sufficiently large subhalos to match reality. On the otherhand, if even larger subhalos are still unable to recreate theFSQ deviations, that would indicate that Λ CDM subhalosare undeniably not responsible for the observed deviationsfrom the FSQ. With this in mind, we dial up the normal-ization for subhalos by a factor of ten above the Λ CDMprediction and re-run the experiment. The same examplelens with now × the subhalo mass as Λ CDM is shown inFigure 7, in addition to a second lens with a different seed.A complication that arises for some subhalo configura-tions is that since subhalos are a factor of ten larger thanthose produced with Λ CDM, they are more likely to havecentral densities high enough to be their own strong lenses.We are not interested in these types of scenarios because the
Figure 6.
The top panel shows the deviations of the quad relativeimage angles from the FSQ in a fashion similar to Figure 3, but forthe single lens with Λ CDM subhalo perturbers shown in Figure4. Each point represents a quad. For a purely elliptical lens, thedeviations would be very nearly zero. The bottom panel is thesame except the vertical scale is increased and observed quadsare included as red circles. It is immediately apparent that thesimulated deviations, while nonzero, are insufficient to explain theobserved deviations. While not depicted here, all other attemptedrandom realizations of substructure positions have similar results. images are so close together that they are unlikely to be re-solved in observations. To prevent situations like these twosteps are taken. First, if the last-arriving image (central, 5thimage) is more than 0.5 R E from the lens center the quad isnot included. Second, when comparing with observations, wewill use quads only within a selection window which omits MNRAS , 1–17 (2017)
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Figure 7.
The left lens is the same as in Figure 4 except with the mass normalization for subhalos multiplied by a factor of 10. Notethe disturbed caustic. The × CDM-population consists of 200 lenses similar to this one, with different subhalos generated drawn fromthe full 3D population. The lens on the right is also × CDM, with this particular realization generating a more tame caustic. Again, 1arcsecond corresponds to 6.7 kpc. the quads with θ < ◦ , where these anomalies are mostlikely to reside. This selection also omits one observed quad.The deviations from the FSQ for the simulated lens areplotted in Figure 8. The deviations are much larger now andare of the same order as observations, so now more care isnecessary to confirm or rule out consistency. The metric weuse to test consistency is the two-dimensional Kolmogorov-Smirnoff (KS) test originally presented by Peacock (1983),expanded on by Fasano & Franceschini (1987), and madereadily available by Press et al. (1997). Unlike the one-dimensional KS test, the 2D KS test is not strictly indepen-dent of the shape of the distribution because the CumulativeDistribution Function (CDF) is not uniquely defined in morethan one dimension (Peacock 1983). Fortunately, this effectis minuscule for cases where the x and y values for the dataset are not strongly correlated, and is not measurable in thecase of our data.The test works by taking a point from one of the distri-butions and counting the fractional number of points fromeach distribution in each of the quadrants around the start-ing point. It then repeats this process using each point as itsstarting point and finding the one which creates the quad-rant with the maximum discrepancy between the two distri-butions. This discrepancy can then be evaluated for statisti-cal significance (Peacock 1983; Fasano & Franceschini 1987).Like the traditional KS test, the 2D KS test returns a p-valuewhich is the probability of obtaining a more extreme discrep-ancy than measured, assuming the null hypothesis is true. Inthis case, the null hypothesis is the claim that the observedpopulation is simply drawn randomly from the synthesizedpopulation. For our purposes, a p-value less than 5% will indicate that the simulated population and observed popu-lation are indeed different.The uncertainites in observations are accounted for bytaking each observed quad and replacing it with 100 pointsdistributed in a 2D Gaussian with σ x and σ y correspondingto the astrometric uncertainties in observations and centeredon the intersection of the error bars. This spreads out thedensity of points in accordance with the error bars. Sincethis process artificially gives the population 100 times thenumber of points, for the purpose of testing statistical sig-nificance it is still considered to consist of only the 39 quads.When applied to the simulated quads from the lens inFigure 8, the 2D KS test returns a p-value of 4.1%, meaningthat the population of quads produced by this single syn-thetic lens is almost consistent with the observed populationof quads.We should not necessarily expect that any single lenswould be able to reproduce the entire population of ob-served quads, since they themselves come from many dif-ferent galaxies. Instead it makes more sense to compare apopulation of simulated lenses with the observed quad pop-ulation. Specifically we created 200 lenses with different ran-dom seeds in the manner described in the previous sectionand plotted their deviations from the FSQ in Figure 9. It isimmediately apparent that the deviations from the FSQ areeven less than that for the single lens. This is because thereare more lenses similar to the lens on the right in Figure 7than that on the left in the same figure, which has largerperturbations. This makes the number of quads that devi-ate from the FSQ more diluted and the population an evenweaker match to observations. The 2D KS test confirms this, MNRAS000
The left lens is the same as in Figure 4 except with the mass normalization for subhalos multiplied by a factor of 10. Notethe disturbed caustic. The × CDM-population consists of 200 lenses similar to this one, with different subhalos generated drawn fromthe full 3D population. The lens on the right is also × CDM, with this particular realization generating a more tame caustic. Again, 1arcsecond corresponds to 6.7 kpc. the quads with θ < ◦ , where these anomalies are mostlikely to reside. This selection also omits one observed quad.The deviations from the FSQ for the simulated lens areplotted in Figure 8. The deviations are much larger now andare of the same order as observations, so now more care isnecessary to confirm or rule out consistency. The metric weuse to test consistency is the two-dimensional Kolmogorov-Smirnoff (KS) test originally presented by Peacock (1983),expanded on by Fasano & Franceschini (1987), and madereadily available by Press et al. (1997). Unlike the one-dimensional KS test, the 2D KS test is not strictly indepen-dent of the shape of the distribution because the CumulativeDistribution Function (CDF) is not uniquely defined in morethan one dimension (Peacock 1983). Fortunately, this effectis minuscule for cases where the x and y values for the dataset are not strongly correlated, and is not measurable in thecase of our data.The test works by taking a point from one of the distri-butions and counting the fractional number of points fromeach distribution in each of the quadrants around the start-ing point. It then repeats this process using each point as itsstarting point and finding the one which creates the quad-rant with the maximum discrepancy between the two distri-butions. This discrepancy can then be evaluated for statisti-cal significance (Peacock 1983; Fasano & Franceschini 1987).Like the traditional KS test, the 2D KS test returns a p-valuewhich is the probability of obtaining a more extreme discrep-ancy than measured, assuming the null hypothesis is true. Inthis case, the null hypothesis is the claim that the observedpopulation is simply drawn randomly from the synthesizedpopulation. For our purposes, a p-value less than 5% will indicate that the simulated population and observed popu-lation are indeed different.The uncertainites in observations are accounted for bytaking each observed quad and replacing it with 100 pointsdistributed in a 2D Gaussian with σ x and σ y correspondingto the astrometric uncertainties in observations and centeredon the intersection of the error bars. This spreads out thedensity of points in accordance with the error bars. Sincethis process artificially gives the population 100 times thenumber of points, for the purpose of testing statistical sig-nificance it is still considered to consist of only the 39 quads.When applied to the simulated quads from the lens inFigure 8, the 2D KS test returns a p-value of 4.1%, meaningthat the population of quads produced by this single syn-thetic lens is almost consistent with the observed populationof quads.We should not necessarily expect that any single lenswould be able to reproduce the entire population of ob-served quads, since they themselves come from many dif-ferent galaxies. Instead it makes more sense to compare apopulation of simulated lenses with the observed quad pop-ulation. Specifically we created 200 lenses with different ran-dom seeds in the manner described in the previous sectionand plotted their deviations from the FSQ in Figure 9. It isimmediately apparent that the deviations from the FSQ areeven less than that for the single lens. This is because thereare more lenses similar to the lens on the right in Figure 7than that on the left in the same figure, which has largerperturbations. This makes the number of quads that devi-ate from the FSQ more diluted and the population an evenweaker match to observations. The 2D KS test confirms this, MNRAS000 , 1–17 (2017) mpact of substructure/baryon-DM transition on quads Figure 8.
Deviations from the FSQ, projected along the θ axis, are plotted for a single lens shown in the left panels of Figure 7. Here,the left panel shows a scatter plot of quads, while the middle and right panels show simulated quads by density. Grayscale corresponds tothe simulated quad population while red circles with error bars corresponds to observations. This shaded plot style will be used from nowon, since it is easier to display densities of points, particularly when the number of points is large. Quads with θ < ◦ are omitted withthe cutoff shown as a vertical dashed line. Interestingly, for this particular lens, there exist some quads with θ > ◦ , which normallydoes not happen. This can occur if one of the images lies near a particularly large subhalo, which delays that image and changes thearrival time order. This only happens for a handful of quads in only the most extreme lenses and only for × Λ CDM. It is not a concernin our analysis. The population of quads in the right panel is the same population, after the most effective selection bias from Table 1is applied (See Section 3.1). As before, the vertical dashed line represents the cutoff removing quads with θ < ◦ . This particular biasuses the 0th percentile (minimum of the data) as ξ and the 75th percentile as ξ . The biased population is consistent with the observedpopulation (p=32%) while the unbiased is not (p=4.1%). This is mostly because the bias has made quads with lower θ more likely,moving the denser part of the gray population to the left making it more consistent with the denser part of the red population. returning a p-value of . . This indicates that even with × Λ CDM substructure, the observed deviations from theFSQ cannot be explained.We consider this experiment to be the most critical ofthe experiments done in this paper, and therefore have com-mitted the computational resources necessary to synthesize200 lenses. Other populations within this paper are synthe-sized using less than 200 lenses. Curious about whether apopulation of 20 galaxies would return the same p-value asone of 200, we divided the 200 lenses into 10 sets of 20 andcalculated the p-value 10 times. Values returned typicallyranged from 0.03% to 0.23%. We interpret this to mean thatwhen only 20 galaxies are used in other tests, the p-value canvary by ∼ . (for the unbiased case- when selection biasesare applied in Section 3 this value will be different). External shear is a common feature in lens models because itis easy to express analytically and it seems to fit many lenses,although it may not necessarily correspond to a readily iden-tifiable physical entity. Wong et al. (2011) found disagree-ment between fitted values for external shear in lens modelsand measured values from the environments of those lenses,indicating that the external shear inferred from model lensesin reality corresponds to a handful of environmental factorsincluding not only external shear, but also line-of-sight ef-fects, as well as compensates for simplifying assumptionsabout the main lens. Rather than thinking of external shearas a physical observable quantity, perhaps it makes mostsense to instead think of it as a simple first order fitting pa-rameter that represents information from several unknown effects. Whatever the case, Woldesenbet & Williams (2015)document the effects of external shear on the distribution ofquads relative to the FSQ. Since this shear can provide devi-ations from the FSQ, we can experiment with nonzero valuesof shear and see if this is able to better match observations.Bolton et al. (2008) modeled 63 lenses discovered in theSloan Lens ACS Survey (SLACS) and fit values for the ex-ternal shears of each lens, ranging from 0 to 0.27 with amedian of 0.05. Since the authors argue that their popula-tion of lenses is statistically consistent with being drawn atrandom from the survey, we can assume that the shears andaxis ratios they found are representative of typical exter-nal shear values for these types of lenses. The distributionof shear values from SLACS is consistent with the valuesdetermined from lens environments (Wong et al. 2011), andthe distribution of axis ratios from SLACS is consistent withthat of nearby ellipticals (Ryden 1992). Both of these con-sistency checks come from methods which are independentfrom lensing models. It is therefore justified to use these val-ues when synthesizing a population of lenses. For the same63 lenses, the authors also list the axis ratios for each lensfrom their model, ranging from 0.51 to 0.97 with a medianof 0.79, providing a natural way to make the synthesizedpopulation of main halos more representative of a true pop-ulation.We created a population of 20 lenses. The subhalos aremade the same way as above, with × Λ CDM substructure,while the main halos have axis ratios and external shearsrandomly drawn from the 63 values in Bolton et al. (2008)and given a random shear orientation angle. Since the axisratios are not all the same, the caustic size differs on a lens-by-lens basis. This is because in the limiting case of axis ratio
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Figure 9.
Deviations from the FSQ for the population of over100,000 quads from 200 synthetic galaxies with × Λ CDM sub-structure and no external shear (grayscale), compared to observa-tions (red). Since the majority of the 200 galaxies have few largeperturbations from elliptical contours, only small deviation fromthe FSQ results when considering a population. = 1 the inner caustic becomes a point, so nearly circularlenses have smaller diamond caustics than elliptical ones.This means the lensing cross section for quads is differentfor each galaxy, so it no longer makes sense to construct apopulation with simply 10,000 quads for each lens. Instead,the number of quads for each lens is proportional to thecaustic area. Once again the deviations from the FSQ aretoo small to match observations. The 2D KS test confirmsthis, returning a p-value of . . Even the combinationof × Λ CDM substructure, realistic external shears, andrealistic axis ratios are unable to produce the deviations fromthe FSQ necessary to explain observations.
Observational selection biases affect all surveys. The quadsample we are using in this paper is very heterogeneous:some quads were discovered as part of a well-defined sur-vey while others were discovered individually. This meansthat correctly accounting for biases is impossible. In lieuof a known selection function, we have devised a makeshiftmodel which biases quad selection in a probabalistic sense.In the future, quads will be discovered by the Large SynopticSurvey Telescope (LSST), with well-defined selection crite-ria. In the meantime, our treatment is sufficient to mimicselection effects and gain intuition as to their general impli-cations.In Section 4 populations of quads will be generatedwhich are closer to the observed distribution and so selectionbiases will be important. As such, we consider these biases now, and apply them to the population of quads generatedin Sections 2.4 and 2.5.Three main biases are considered. First, quads which arebrighter are more likely to be detected. The source luminos-ity is uncorrelated with the lens properties and is thereforenot relevant to this analysis. What matters in this contextis the total magnification of all the images. If the images arehighly magnified, the quad will likely be detected. Anotherpotential source for bias is the separation between images(Oguri et al. 2006; Pindor et al. 2003). If 2, 3, or even all 4 im-ages are close together, they may not be resolved as distinctimages and the quad may instead look like a triple, doubleor a point source. Such a case would not be included in theobserved galaxy quad population. Finally, quads which havea large contrast between the magnification of the brightestimage and the dimmest will also be less likely to be resolvedas having distinct images (Oguri et al. 2006; Pindor et al.2003). These cases are unlikely to register as more than apoint source in a survey and may not be followed up withdeeper observations. These inherent biases in the way quadsare observed could select quads whose properties are differ-ent from those of an unbiased population. It is not hard toimagine this affecting the distribution of quads around theFSQ, so we examined the consequences of these biases.Optimally one would simply know the limiting resolu-tion and magnitude of one’s survey and omit synthetic quadsthat are outside of that range. However, the population ofknown quads comes from an amalgam of many different sur-veys, making it difficult to systematically identify the degreeof lensing biases (Oguri et al. 2006). Although quads discov-ered may have a biased population due to inability to resolveclosely spaced images or images of drastically different mag-nifications, this paper will largely ignore these biases in favorof the total magnification bias. The main reason for this isthat the central question of this exercise asks if it is at allpossible that biases could explain the observed populationof quads via selection effects. When addressing this ques-tion, it makes sense to look at the populations in the bestpossible light, and our tests seem to indicate that the totalmagnification bias results in the highest p-values comparedto the flux ratio and image separation biases. From now on,discussion of biases will be limited to the total magnificationbias.To properly account for the magnification bias, onewould need knowledge of the quasar luminosity function andlens mass distributions for all redshifts (Han & Park 2015).Even then, it is infeasible to identify any particular mag-nitude cutoff that applies to all surveys. Conscious of ourignorance, we use the following logic to approximate thegeneral effects of the bias. First, imagine that above (below)a certain threshold for the summed image magnifications aquad is guaranteed to be (not be) detected. Between thesetwo thresholds, suppose the probability of detection scaleslinearly with the summed magnification. Since those thresh-olds are difficult to pinpoint, we set them to a percentileof the data, ξ and ξ , e.g. the 25th percentile and 75thpercentile. This process is shown visually in Figure 10. Theexact value of these percentiles is unknown in actual surveysand the effects of changing them will be an important part ofanalyzing the effects of the bias. The application of this biasoccurs before the θ < ◦ cutoff selection. Our bias will beapplied to all populations presented in following sections. MNRAS000
Observational selection biases affect all surveys. The quadsample we are using in this paper is very heterogeneous:some quads were discovered as part of a well-defined sur-vey while others were discovered individually. This meansthat correctly accounting for biases is impossible. In lieuof a known selection function, we have devised a makeshiftmodel which biases quad selection in a probabalistic sense.In the future, quads will be discovered by the Large SynopticSurvey Telescope (LSST), with well-defined selection crite-ria. In the meantime, our treatment is sufficient to mimicselection effects and gain intuition as to their general impli-cations.In Section 4 populations of quads will be generatedwhich are closer to the observed distribution and so selectionbiases will be important. As such, we consider these biases now, and apply them to the population of quads generatedin Sections 2.4 and 2.5.Three main biases are considered. First, quads which arebrighter are more likely to be detected. The source luminos-ity is uncorrelated with the lens properties and is thereforenot relevant to this analysis. What matters in this contextis the total magnification of all the images. If the images arehighly magnified, the quad will likely be detected. Anotherpotential source for bias is the separation between images(Oguri et al. 2006; Pindor et al. 2003). If 2, 3, or even all 4 im-ages are close together, they may not be resolved as distinctimages and the quad may instead look like a triple, doubleor a point source. Such a case would not be included in theobserved galaxy quad population. Finally, quads which havea large contrast between the magnification of the brightestimage and the dimmest will also be less likely to be resolvedas having distinct images (Oguri et al. 2006; Pindor et al.2003). These cases are unlikely to register as more than apoint source in a survey and may not be followed up withdeeper observations. These inherent biases in the way quadsare observed could select quads whose properties are differ-ent from those of an unbiased population. It is not hard toimagine this affecting the distribution of quads around theFSQ, so we examined the consequences of these biases.Optimally one would simply know the limiting resolu-tion and magnitude of one’s survey and omit synthetic quadsthat are outside of that range. However, the population ofknown quads comes from an amalgam of many different sur-veys, making it difficult to systematically identify the degreeof lensing biases (Oguri et al. 2006). Although quads discov-ered may have a biased population due to inability to resolveclosely spaced images or images of drastically different mag-nifications, this paper will largely ignore these biases in favorof the total magnification bias. The main reason for this isthat the central question of this exercise asks if it is at allpossible that biases could explain the observed populationof quads via selection effects. When addressing this ques-tion, it makes sense to look at the populations in the bestpossible light, and our tests seem to indicate that the totalmagnification bias results in the highest p-values comparedto the flux ratio and image separation biases. From now on,discussion of biases will be limited to the total magnificationbias.To properly account for the magnification bias, onewould need knowledge of the quasar luminosity function andlens mass distributions for all redshifts (Han & Park 2015).Even then, it is infeasible to identify any particular mag-nitude cutoff that applies to all surveys. Conscious of ourignorance, we use the following logic to approximate thegeneral effects of the bias. First, imagine that above (below)a certain threshold for the summed image magnifications aquad is guaranteed to be (not be) detected. Between thesetwo thresholds, suppose the probability of detection scaleslinearly with the summed magnification. Since those thresh-olds are difficult to pinpoint, we set them to a percentileof the data, ξ and ξ , e.g. the 25th percentile and 75thpercentile. This process is shown visually in Figure 10. Theexact value of these percentiles is unknown in actual surveysand the effects of changing them will be an important part ofanalyzing the effects of the bias. The application of this biasoccurs before the θ < ◦ cutoff selection. Our bias will beapplied to all populations presented in following sections. MNRAS000 , 1–17 (2017) mpact of substructure/baryon-DM transition on quads Figure 10.
A schematic presentation of how the bias is ap-plied. The blue solid curve shows the CDF of the magnification(summed from all four images) for the unbiased quad population.The red dashed line represents the probability that a quad willbe detected and kept in the population after the bias is applied.The minimum threshold value below which detection is impossi-ble, µ , is ξ , the 25th percentile in this example. Likewise themaximum threshold value above which detection is certain, µ , is ξ , the 75th percentile here. Once the thresholds are set, the prob-ability of detection scales linearly with the value of the summedmagnification between the thresholds, shown by the red dashedline. Before applying the bias to all synthesized quad populations,we will first apply it to a single population to convey its gen-eral effects. The population that we believe depicts this bestis that of the first single lens with × Λ CDM subhalos,(Figure 8). This is because this population has appreciabledeviations from the FSQ leading to visible distributions inboth the θ and ∆ θ dimensions. It will turn out that thebias notably affects only the distribution in the θ dimen-sion, but this is most readily seen when the spread of pointsin both dimensions is large. The trends we see here apply toall populations of quads.The bias makes the quads with the larger total mag-nification more likely to be detected, while throwing outthe fainter quads which are unlikely to be detected. Table1 shows the resulting p-values from the 2D KS test afterthis bias has been applied with various values for minimumpercentile, ξ , below which detection is impossible and themaximum percentile, ξ , above which detection is certain.Since the strength of the bias is unknown, we will imagine itis as effective as it possibly could be in making the observedand synthetic populations consistent with one another. Thehighest p-value corresponds to the best-case scenario. Withthis being said, we did not feel it was necessary to run anexhaustive search for the maximum because the exact speci-ficity in threshold percentiles chosen is not particularly use- Single Lens Example:p-values for Biases (%) ξ
99 – – – – –75 – – – – .
50 – – – . .
25 – –
29 13 0 . . . ξ Table 1.
Optimizing the effect of the selection bias. P-value re-sults are presented for various minimum and maximum cutoffsfor the total magnification bias. p-values greater than 5% indi-cate that the populations are consistent with the null hypothesis,which claims the observed population comes from selecting quadsfrom the synthesized population. For this example lens there areseveral cases with such p-values, with the highest value ocurringin the case where the minimum cutoff for the bias is the mini-mum of the data and the threshold for certain detection is the75th percentile of the data. This case puts the population in themost positive light, with 5177 out of 8907 quads detected, and isplotted in Figure 8 (right panel). The number of quads remainingin the population after the bias is applied is not depicted, butis lowest in the upper right corner (312 remain out of 8907) andhighest in the bottom left. (8233 remain) ful. Instead, we simply use the highest p-value in Table 1, which is 32%. This value is greater than the 5% signifi-cance threshold, meaning that, when constructed from thissingle lens with an optimistic bias applied, this particularpopulation of quads is consistent with observations.The effect of this bias is shown in Figure 8, but is per-haps easier to see in the marginalized distributions, shownin Figure 11. The bias has little effect on the distributionin ∆ θ but a significant effect on the distribution in θ .Both figures show that the bias selectively removes quads ina way that shifts the remaining population to smaller θ .This makes sense because smaller θ means that the 2ndand 3rd arriving images are close together, which comes fromthe quads where the source is near the caustic line, resultingin high magnification. These quads are more magnified andmore likely to remain after the bias is applied.Though we have only depicted the case returning thehighest p-value, in reality there are many different realiza-tions possible in which the distributions, biased at differentlevels, would take on intermediate forms between the twocases depicted in the middle and rightmost panels of Figure8. It is also possible to have a stronger bias with more pro-nounced effects than those shown in the right panel, but thiswould result in a lower p-value than what is shown here.Independent of the strength of the bias, it is importantto note that the bias strongly affects the θ distributionand only weakly affects the ∆ θ distribution. The result isthat the ∆ θ distribution is the more important one whenattempting to decipher whether or not a population is con-sistent with observations: if there is a mismatch between the The 99th percentile is chosen as the highest ξ rather thanthe 100th percentile because if the latter is used, a single large-magnification outlier, which would likely be code-resolution arti-fact, could drastically decrease the slope of the linear detectabilityfunction (Figure 10) and artificially cause nearly all quads to havea low probability of detection.MNRAS , 1–17 (2017) Gomer and Williams
Figure 11.
Marginalized distributions are shown for both the θ (x-axis of Figure 8) and ∆ θ (y-axis of Figure 8) for the popula-tion of quads created from the single × Λ CDM lens (blue), com-pared to the observed population (open white histogram). The toptwo panels show the unbiased population while the panels on thebottom show the biased population which yields the best match toobservations. The cutoff removing quads with θ < ◦ is shownas the dashed line. The bias strongly skews the θ distributionto lower values, matching up more closely with the larger peak inthe observed data, which is the largest factor in why the p-valueimproves. Meanwhile the general shape of the ∆ θ distribution isonly slightly affected by the bias. synthesized population’s θ distribution and that of the ob-servations, there may exist a bias that could bring the pop-ulation into the realm of plausible consistency, however ifthere is a similar mismatch between the synthesized popu-lation’s ∆ θ distribution and that of the observations, nobias will fix the problem. Now that the effects of such a selection bias have beendemonstrated for a single lens, it is time to apply the samebias to the population of lenses which is meant to representthe galaxy lens population. The table analogous to Table 1 isnot presented, but the same analysis is run, this time for thepopulation of 200 lenses with × Λ CDM substructure withno shear. The most optimistic bias leaves the simulated quadpopulation with a p-value of . , which is still inconsistentwith the null hypothesis. A similar exercise as in Section 2.4–where we recalculate p-values with 10 subsamples using 20galaxies each– yields p-values that typically vary from 0.8%to 2.3%. This means for populations using only 20 lenses,the p-value can be expected to vary by ∼ in the biasedcase. Using the population of 20 lenses from the case withnonzero shear (Section 2.5), the best match occurs underthe same cutoff values, again with a p-value of 1.3%. Evenwhen the most optimistic bias is applied, these synthesizedpopulations from the × Λ CDM substructure scenario are completely inconsistent with observations. It therefore seemsunlikely for Λ CDM substructure to account for the observeddeviations from the FSQ.
Aside from Λ CDM substructure, there are other effectswhich are capable of producing asymmetry in the lens tocreate significant deviations from the FSQ. If the mass ofthe galaxy is not relaxed into a single smooth profile, forexample, then there could be inherent asymmetries. Chaeet al. (2014) found that in order to fit the density pro-files for early-type galaxies a two-component mass modelwas required. Young et al. (2016) found that simulations ofboth dark matter only and dark matter + hydrodynamicsresulted in mass distributions which are not fully relaxed.If the baryons and dark matter do not coalesce into identi-cal distributions then it would be possible to have a galaxywhich has two related but not identical distributions. Withtwo non-identical distributions, there are several possible re-alizations which could break the double-mirror symmetry orgive rise to “wavy” features in the lens projected isodensitycontours.The quad image circle in halos of (cid:39) M (cid:12) , slightlylarger than that in our halos, has a radius of around 6 kpc.It is a remarkable coincidence that this radius happens tocorrespond to a transition region from baryons to dark mat-ter, illustrated in Figure 12. At smaller radii, the baryonsare the dominant mass component while the dark matterdominates at outer radii. It just so happens that the radiuswhere they have comparable mass lies at a similar radius asthe position of images, which is fortunate because the im-age circle is the region lensing can most precisely probe. Italso complicates matters, because if there are any inherentasymmetries in the baryon and dark matter distributions,this transition area is where they will have the most drasticeffects on image positions. It stands to reason that variousperturbations arising from ellipticity transitioning betweenthe baryon and dark matter distributions could result in de-viations from the FSQ. This motivates an additional seriesof experiments.This time, we construct lenses with no Λ CDM substruc-ture, but using two superimposed elliptical Einasto profilesinstead of just one. The first profile represents the dark mat-ter, in which shape parameter α is changed to 0.18 but oth-erwise the same as before. This slightly larger shape param-eter makes the profile less concentrated than before, makingthe scale radius 13.6 kpc. The central density for the profilerepresenting the baryons is set to × the dark matter Σ ,motivated by the Illustris simulations (Vogelsberger et al.2014; Young et al. 2017), but the profile drops off muchmore steeply than that of the dark matter. The baryon pro-file is given a scale radius of 1 kpc and a shape parameterof 0.6, which have been chosen to make the slope near theimage radius more realistic, as in Section 2.1. This meansthe baryons will be the dominant mass component in thevery central regions, but at radii near the images the darkmatter has become dominant.The (3D) transition radius in this setup actually liesnear 3 kpc instead of 6 kpc, which is mostly due to thesmaller halo size. The transition radius for a . M (cid:12) MNRAS000
Aside from Λ CDM substructure, there are other effectswhich are capable of producing asymmetry in the lens tocreate significant deviations from the FSQ. If the mass ofthe galaxy is not relaxed into a single smooth profile, forexample, then there could be inherent asymmetries. Chaeet al. (2014) found that in order to fit the density pro-files for early-type galaxies a two-component mass modelwas required. Young et al. (2016) found that simulations ofboth dark matter only and dark matter + hydrodynamicsresulted in mass distributions which are not fully relaxed.If the baryons and dark matter do not coalesce into identi-cal distributions then it would be possible to have a galaxywhich has two related but not identical distributions. Withtwo non-identical distributions, there are several possible re-alizations which could break the double-mirror symmetry orgive rise to “wavy” features in the lens projected isodensitycontours.The quad image circle in halos of (cid:39) M (cid:12) , slightlylarger than that in our halos, has a radius of around 6 kpc.It is a remarkable coincidence that this radius happens tocorrespond to a transition region from baryons to dark mat-ter, illustrated in Figure 12. At smaller radii, the baryonsare the dominant mass component while the dark matterdominates at outer radii. It just so happens that the radiuswhere they have comparable mass lies at a similar radius asthe position of images, which is fortunate because the im-age circle is the region lensing can most precisely probe. Italso complicates matters, because if there are any inherentasymmetries in the baryon and dark matter distributions,this transition area is where they will have the most drasticeffects on image positions. It stands to reason that variousperturbations arising from ellipticity transitioning betweenthe baryon and dark matter distributions could result in de-viations from the FSQ. This motivates an additional seriesof experiments.This time, we construct lenses with no Λ CDM substruc-ture, but using two superimposed elliptical Einasto profilesinstead of just one. The first profile represents the dark mat-ter, in which shape parameter α is changed to 0.18 but oth-erwise the same as before. This slightly larger shape param-eter makes the profile less concentrated than before, makingthe scale radius 13.6 kpc. The central density for the profilerepresenting the baryons is set to × the dark matter Σ ,motivated by the Illustris simulations (Vogelsberger et al.2014; Young et al. 2017), but the profile drops off muchmore steeply than that of the dark matter. The baryon pro-file is given a scale radius of 1 kpc and a shape parameterof 0.6, which have been chosen to make the slope near theimage radius more realistic, as in Section 2.1. This meansthe baryons will be the dominant mass component in thevery central regions, but at radii near the images the darkmatter has become dominant.The (3D) transition radius in this setup actually liesnear 3 kpc instead of 6 kpc, which is mostly due to thesmaller halo size. The transition radius for a . M (cid:12) MNRAS000 , 1–17 (2017) mpact of substructure/baryon-DM transition on quads Figure 12.
Comparison of the baryon-dark matter transition ra-dius with the Einstein radius. The histogram displays the Einsteinradius for the 33 observed quads which have known lens redshift.The average and median of this distribution are presented in thefigure. The radius of where the dark matter component becomesdominant over the baryon component depicted here is calculatedin Chae et al. (2014) for a M (cid:12) halo, a typical mass for galaxieswhich host quads. . It is fortuitous that these radii values shouldcoincide. galaxy, approximately the mass of our halos, is closer to4 kpc (Chae et al. 2014). We consider this slight mismatchbetween 3 and 4 kpc acceptable, recognizing that if the radiimatched better, any asymmetries created would have a morepronounced effect on quad image angles. This transition ra-dius is identifiable in Figure 13, which shows the density asa function of radius for the two components.Within the simulation window, (cid:39) of the mass is inbaryons. At infinity, only about 4% of the mass is in baryons.Depending slightly on the exact values for the ellipticity, theEinstein radius increases to (cid:39) . kpc due to this additionalbaryonic mass. Axis ratios for the elliptical profiles are sepa-rately drawn from Bolton et al. (2008). As before, the num-ber of quads per galaxy is again proportional to the causticsize, and quads with θ < ◦ will again be omitted. Ex-ternal shear is left at zero. Perturbations will be applied tothe elliptical structure of the two profiles– dark matter andbaryons– as described in the next sections. The first form of mass perturbations from pure ellipses weexamined are motivated by observations. Bender & Moellen-hoff (1987) measured deviations from ellipticity in isophotesin terms of Fourier expansion in the polar angle. Radialisophote deviations from a perfect ellipse are parameterizedby coefficients of sines and cosines, ∆ R = (cid:213) k = a k cos ( k φ ) + b k sin ( k φ ) , Figure 13.
Density profile for the two components from whichlenses will be constructed. The blue solid line represents darkmatter and orange represents baryons. The dashed line depictsan isothermal slope. The arrow indicates the transition radiuswhere the densities are equal, at 3.3 kpc. The axes are scaledwith respect to the critical density of the universe at lens redshift(z=0.6) and r so that the graph can be readily compared withFigure 12 of Chae et al. (2014). It should be noted that this isthe analytical form from Equation 1 and is spherically symmetric.Once ellipticities and perturbations from ellipticities are applied(Section 4), the true profile will differ as a function of positionangle, but should be reasonably close to this when sphericallyaveraged. where φ is the angle with respect to the ellipse axis. Theindex starts at 3 because a , a , b , and b are already con-strained by ellipse parameters such as axis ratios, semimajoraxis, and center position. Since then, numerous studies havefollowed their notation. Though in principle there could ex-ist higher order deviations, the index is typically cut off at k = . The values of the coefficients change for each isophoteand are therefore a function of radius, but this dependenceis complicated and for our purposes we will just use aver-age values. The most commonly discussed deviation is the a term. Positive values deform the ellipse into a diamond-shaped “disky” isophote while negative values deform the el-lipse into a “boxy” shape. Mitsuda et al. (2016) analyzed the a values for early-type galaxies at z ∼ and 0, finding valuesroughly distributed from -0.02 to +0.04. Corsini et al. (2016)measured the Fourier coefficients for three nearby galaxiesand found a to range from -0.03 to 0 in one case, between0 and 0.06 in another case, and 0 and 0.03 in the third case.They found a and b to be largely consistent with 0, but inone case had b and b range from 0 to about 0.03 and twocases where a ranges from 0 to approximately 0.03. Theydid not include information as to a or b . Kormendy et al.(2009) found values of a typically between -0.02 to 0.02 forgalaxies in the Virgo cluster, where a was as large as 0.09in one case. Nonzero a values were also found, but were notas extreme as the a values measured. These general results MNRAS , 1–17 (2017) Gomer and Williams give an impression for the order of deviation from ellipticityfor realistic galaxies and provide a framework from which toconstruct a galaxy population.We construct this population by having a uniformly se-lected between -0.04 and 0.04 and a uniformly selected be-tween − . and . . Unlike real galaxies, the selected valuefor the Fourier coefficients is kept constant as a function ofradius, but should still provide insight into the effects thesedeviations from ellipticity have on quad deviations from theFSQ. Though the observations above only apply to the light,it is assumed in this context that the dark matter profileslikewise have deviations from ellipticity of similar order. Assuch, both profiles get different values chosen for a and a .The other coefficients are left at zero. For now, the majoraxes of the two elliptical distributions are colinear and thecenters coincide.One hundred galaxy lenses are constructed and the re-sulting population of quads is analyzed using the 2D KS test. The result is a p-value of 0.00044% in the unbiased caseand 0.013% when the same bias as before is applied. Thisis less than the 5% threshold, indicating that the Fourierperturbations alone are insufficient to match observations.
Another case to explore is that in which the baryons andthe dark matter have elliptical projected mass distributionsbut their ellipse major axes do not necessarily line up per-fectly i.e. the two distributions have different position angles(PAs). Perhaps a population of lenses with features like thiscould be responsible for the observed deviations from theFSQ. In this test, the dark matter profile has an ellipse PAthat is tilted with respect to the x-axis by an angle ran-domly selected between 0 and 45 degrees. The PA of thebaryon profile remains aligned with the x-axis. Fourier co-efficients introduced in Section 4.1 are set to zero. A popu-lation of quads is synthesized from 100 lenses like this andsubjected to the 2D KS test. Before biasing, the p-value re-turned is 0.0017% and after the bias is applied the p-valuereturned is 0.020%. Thus, the effect of misaligned ellipses isalso nowhere near sufficient to create the necessary devia-tions from the FSQ.Another variant of this idea of having two elliptical pro-files give rise to a non-elliptical mass distribution is to offsetthe centers of the ellipses themselves. Image positions aremost sensitive to mass perturbations near the image radius,so it is the perturbations near the image radii that we trulywish to emulate. If the centers of the profiles were not coin-cident it would cause a non-elliptical perturbation even atradii farther out than the centers. In this way, artificially off-setting the centers can serve as an easy-to-generate asym-metry. In reality, the centers of baryonic and dark matterdistributions are thought to be nearly coincident. In lens-ing models it is usually assumed that the centers coincide(Gavazzi et al. 2008; Bolton et al. 2008). However, since the One hundred lenses are now used to create a population as op-posed to twenty used for most of the cases before because as moretypes of perturbations are added, parameter space gets largerthan before, so having more lenses in a population is necessary. positions of quads are more sensitive to structure at the im-age radius than structure in the lens center, we will acceptthat this model is inaccurate in most central region of thelens in return for the structure at image radius it gener-ates. That is to say, we will offset the centers of the twodistributions in the simulations, but this is not necessarilya claim that the centers are in reality offset so drastically.Instead, the offset centers are an artificial way to introducenon-ellipticity near the image radius, which could exist inreal lenses. Having said that, we note that it is possible forthe centers of dark matter and baryonic distributions to benon-coincident, within the framework of self-interacting darkmatter (Kahlhoefer et al. 2014, 2015). Offsetting these cen-ters could be thought of as additional Fourier perturbationsof lower order than 3, since these coefficients are constrainedby the ellipse center position among other parameters.We create a population consisting of 100 simulatedgalaxies, each with a baryon and dark matter componentdescribed above. The ellipse PAs of the two distributionsare again tilted by an angle between 0 and 45 degrees whilethe centers are offset by a radius randomly selected between0 and 15 pixels (1 kpc) in a random direction in the 2D lensplane. The center of the lens is considered to be the center ofthe baryon distribution since this is the distribution whichan observer would see and assume to be the center. The 2DKS test for the population compared to observations resultsin a p-value of 0.053%. After the bias, the KS test returns ap-value of 3.4%, still indicating inconsistency with the nullhypothesis.
Each of the types of perturbations described in Sections 4.1& 4.2 carries potential FSQ deviations with them. It wouldbe remiss not to test the combination of effects. This time1000 lenses are created with axis ratios from Bolton et al.(2008) for both the dark matter and baryon distributions.The elliptical distributions are tilted with respect to one an-other by an angle between 0 and 45 degrees. The center forthe dark matter distribution is offset by between 0 and 1kpc, and each distribution shape is altered via a (between-0.04 and 0.04) and a (between -0.02 and 0.02). Four ex-ample galaxies are shown in Figure 14. We invite the readerto compare these synthetic distributions to observed galax-ies in Figure 12 of Mitsuda et al. (2016), which bear visualresemblance to one another.The deviations from the FSQ generated by this popu-lation of galaxies are shown in Figure 15. The 2D KS testresults in a p-value of 0.16% for the unbiased distributionand 6.2% once the bias is applied. This result implies that acombination of the above perturbations from pure ellipticitycould be consistent with observations if observational biasesare favorable. We have sought to construct a population of quadruple im-age systems, generated by synthetic lensing galaxies, whichis consistent with the observed distribution of quads relativeto the Fundamental Surface of Quads (FSQ) (Figure 3). Weattempted to do so using physically motivated perturbations
MNRAS000
MNRAS000 , 1–17 (2017) mpact of substructure/baryon-DM transition on quads Figure 14.
Four example lenses and caustics from the population which includes offset centers, tilted ellipse axes, and nonzero a and a for both the dark matter and baryon distributions. Significant deviations from ellipticity are clear, which produce noticable changesin the caustic and thereby the image positions and deviations from the FSQ. Note the resemblance of these mass density contours to theshape of the observed isophotes of Mitsuda et al. (2016). The thick curve is the boundary outside of which the mass is set to zero. Itsshape arises from the combination of a cut along a particular isodensity contour and a circular cut at the edge of the simulation window(16.7kpc). 1 arcsecond corresponds to 6.7 kpc.MNRAS , 1–17 (2017) Gomer and Williams
Figure 15.
Deviations from the FSQ generated from the popula-tion of galaxies like those in Figure 14, with the most optimisticbias applied. The deviations from the FSQ are consistent withobservations with a p-value of 6.2%. from a simple ellipsoidal projected mass distribution, firstwith Λ CDM substructure (Figure 6), then with 10 × Λ CDMsubstructure (Figure 9), and later using a superposition ofdark matter and baryon profiles with Fourier perturbations,misaligned PAs, and/or offset centers to alter the shape ofthe mass isodensity contours (Figure 15). We devised a se-lection bias that mimics observational bias, based on thesummed magnifications of the images, and applied it to oursynthetic quads before comparing with observations. Table2 catalogs each experiment discussed herein and the corre-sponding p-values and relevant figures for each set of pertur-bations from purely elliptical lenses. One caveat to note isthat the p-values quoted do not take into account the num-ber of parameters for each model, so they cannot be properlycompared to each other in any attempt to select a “correct”model, but only as a tool to judge which quad populationsare consistent with the null hypothesis.The first finding of this study is that substructure aspredicted from Λ CDM is unable to generate quads with suf-ficient spread in ∆ θ from the FSQ to match observations.Even if subhalos are ten times as massive as Λ CDM sim-ulations predict, the resulting population of lenses is stillinsufficient to generate quads consistent with observations.Factoring in external shear using realistic shear values fromBolton et al. (2008) does not alleviate this mismatch.The second finding is that there exist some perturba-tions from pure ellipsoidal mass distributions which are ca-pable of generating a population of quads consistent withobservations, if an optimistic selection bias is applied. Themost effective perturbation appears to be the one that mod-els galaxies as two superimposed elliptical distributions,one representing dark matter and the other representingbaryons, with the two centers offset by up to 1 kpc. Sucha perturbation does not necessarily imply that the centers of baryon and dark matter distributions are not coincident,but rather introduces an asymmetry which causes a devi-ation from ellipticity near the image circle. When this per-turbation is applied in conjunction with misaligning the PAsand applying realistic Fourier perturbations, the populationconsists of lenses like those in Figure 14. These galaxies bearvisual resemblance to observed galaxies depicted in Figure12 of Mitsuda et al. (2016). After the bias is applied, the gen-erated population of quads is consistent with the observedpopulation with a p-value of 6.2%.It is interesting that these non-elliptical lens profiles cre-ate a better match with observations than elliptical ones.This is not the first finding to indicate that this may be thecase. Biggs et al. (2004) studied a radio jet lensed by a galaxyin which spectral features made it possible to associate im-ages with various features in the source. They found thata Singular Isothermal Ellipsoid with external shear was un-able to account for the positions of all images simultaneously.The model they found which fit the image positions requireddrastic azimuthal dependence using a sum of Fourier coef-ficients, implying that the mass distribution for the galaxyhad considerable “wavy” features.Why galaxies should be structured this way is an inter-esting question. For one reason or another, the dark mattermass distributions of these galaxies, though in equilibrium,are not relaxed (Young et al. 2016), and additional compli-cations from baryon-dark matter gravitational interactions,or dark matter self-interactions, are likely to make these sys-tems even less relaxed. Perhaps galaxy mergers are frequentand elliptical profiles are disrupted by such events. The an-swers to these questions will be vital to an understanding ofgalaxy formation.There remain other potential explanations which couldcause the population of quads to not lie on the FSQ. Weexplored one in particular, albeit not at the level of detailwith which we explored substructure or superimposed butnonidentical baryon and dark matter profiles. This possibil-ity is that of supermassive black holes (SMBHs) which aredisplaced from the center of the galaxy due to recoil fromgravitational wave emission. These SMBHs are thought tobe formed via merged black holes, which results in an asym-metry in the gravitational wave emission, imparting a recoilvelocity on the SMBH which can be on the order of the es-cape velocity, kicking the SMBH far from the center of thegalaxy (Blecha et al. 2016). It has been predicted that ifsuch systems exist, SMBHs would be present ∼ − kpcaway from the galactic center. To simulate this, we added a M (cid:12) point mass to the synthetic lenses at a distance of 3kpc, right at the image radius where it would have the mosteffect on image positions. Any alteration to the mass distri-bution was not visible on the density contour plot, and nosubstantial deviations from the FSQ were produced. Whileinteresting, recoiling SMBHs are unlikely to have an effecton the quad population.Another possible way for quads which lie off of the FSQto be formed is for mass to be present between the sourceand the observer along the line of sight (LoS) (McCully et al.2017). Such structure would make the thin-lens approxima-tion used in many lens models inaccurate. Quads lensed withline of sight structure will be explored more thoroughly ina coming paper, although the contribution of LoS and en-vironmental structures near the main lens is unlikely to be MNRAS000
Deviations from the FSQ generated from the popula-tion of galaxies like those in Figure 14, with the most optimisticbias applied. The deviations from the FSQ are consistent withobservations with a p-value of 6.2%. from a simple ellipsoidal projected mass distribution, firstwith Λ CDM substructure (Figure 6), then with 10 × Λ CDMsubstructure (Figure 9), and later using a superposition ofdark matter and baryon profiles with Fourier perturbations,misaligned PAs, and/or offset centers to alter the shape ofthe mass isodensity contours (Figure 15). We devised a se-lection bias that mimics observational bias, based on thesummed magnifications of the images, and applied it to oursynthetic quads before comparing with observations. Table2 catalogs each experiment discussed herein and the corre-sponding p-values and relevant figures for each set of pertur-bations from purely elliptical lenses. One caveat to note isthat the p-values quoted do not take into account the num-ber of parameters for each model, so they cannot be properlycompared to each other in any attempt to select a “correct”model, but only as a tool to judge which quad populationsare consistent with the null hypothesis.The first finding of this study is that substructure aspredicted from Λ CDM is unable to generate quads with suf-ficient spread in ∆ θ from the FSQ to match observations.Even if subhalos are ten times as massive as Λ CDM sim-ulations predict, the resulting population of lenses is stillinsufficient to generate quads consistent with observations.Factoring in external shear using realistic shear values fromBolton et al. (2008) does not alleviate this mismatch.The second finding is that there exist some perturba-tions from pure ellipsoidal mass distributions which are ca-pable of generating a population of quads consistent withobservations, if an optimistic selection bias is applied. Themost effective perturbation appears to be the one that mod-els galaxies as two superimposed elliptical distributions,one representing dark matter and the other representingbaryons, with the two centers offset by up to 1 kpc. Sucha perturbation does not necessarily imply that the centers of baryon and dark matter distributions are not coincident,but rather introduces an asymmetry which causes a devi-ation from ellipticity near the image circle. When this per-turbation is applied in conjunction with misaligning the PAsand applying realistic Fourier perturbations, the populationconsists of lenses like those in Figure 14. These galaxies bearvisual resemblance to observed galaxies depicted in Figure12 of Mitsuda et al. (2016). After the bias is applied, the gen-erated population of quads is consistent with the observedpopulation with a p-value of 6.2%.It is interesting that these non-elliptical lens profiles cre-ate a better match with observations than elliptical ones.This is not the first finding to indicate that this may be thecase. Biggs et al. (2004) studied a radio jet lensed by a galaxyin which spectral features made it possible to associate im-ages with various features in the source. They found thata Singular Isothermal Ellipsoid with external shear was un-able to account for the positions of all images simultaneously.The model they found which fit the image positions requireddrastic azimuthal dependence using a sum of Fourier coef-ficients, implying that the mass distribution for the galaxyhad considerable “wavy” features.Why galaxies should be structured this way is an inter-esting question. For one reason or another, the dark mattermass distributions of these galaxies, though in equilibrium,are not relaxed (Young et al. 2016), and additional compli-cations from baryon-dark matter gravitational interactions,or dark matter self-interactions, are likely to make these sys-tems even less relaxed. Perhaps galaxy mergers are frequentand elliptical profiles are disrupted by such events. The an-swers to these questions will be vital to an understanding ofgalaxy formation.There remain other potential explanations which couldcause the population of quads to not lie on the FSQ. Weexplored one in particular, albeit not at the level of detailwith which we explored substructure or superimposed butnonidentical baryon and dark matter profiles. This possibil-ity is that of supermassive black holes (SMBHs) which aredisplaced from the center of the galaxy due to recoil fromgravitational wave emission. These SMBHs are thought tobe formed via merged black holes, which results in an asym-metry in the gravitational wave emission, imparting a recoilvelocity on the SMBH which can be on the order of the es-cape velocity, kicking the SMBH far from the center of thegalaxy (Blecha et al. 2016). It has been predicted that ifsuch systems exist, SMBHs would be present ∼ − kpcaway from the galactic center. To simulate this, we added a M (cid:12) point mass to the synthetic lenses at a distance of 3kpc, right at the image radius where it would have the mosteffect on image positions. Any alteration to the mass distri-bution was not visible on the density contour plot, and nosubstantial deviations from the FSQ were produced. Whileinteresting, recoiling SMBHs are unlikely to have an effecton the quad population.Another possible way for quads which lie off of the FSQto be formed is for mass to be present between the sourceand the observer along the line of sight (LoS) (McCully et al.2017). Such structure would make the thin-lens approxima-tion used in many lens models inaccurate. Quads lensed withline of sight structure will be explored more thoroughly ina coming paper, although the contribution of LoS and en-vironmental structures near the main lens is unlikely to be MNRAS000 , 1–17 (2017) mpact of substructure/baryon-DM transition on quads important. A number of papers have shown that LoS sub-structure contributes less to the lensing optical depth thanthe substructure around the main lens (Metcalf 2002; Chenet al. 2003; Wambsganss et al. 2005) , and if 10 × Λ CDM doesnot come close to observations in the space of quad relativeimage angles, LoS is very unlikely to do so.As this has been a preliminary study, there are sev-eral opportunities for future work. It is somewhat discon-certing that our synthesized population which most closelymatches observations has a relatively low p-value of only6.2%. We suspect that this is because the variables used tocreate non-ellipticity in the present paper span a large pa-rameter space which has not been fully explored. This maybe a task suitable for machine learning, which could poten-tially identify new ways to create alterations from ellipticityin a way that results in a closer match between synthesizedand observed quad populations. Additionally, as larger sur-veys come online, we can expect to find many more quadlenses. Oguri & Marshall (2010) predict that the Large Syn-optic Survey Telescope will find ∼ lensed quasars. As-suming a quad fraction similar to the present observed frac-tion, 0.154 (Oguri 2007), one should anticipate over 1000new quads to be discovered. Additionally, since all of thesequads will be found in the same survey, the observationalselection biases will be much easier to quantify and can bemore reliably applied. An analysis similar to ours using thelarger sample size with well-determined biases will be moreconclusive as to what type of galaxy lenses are consistentwith observations. REFERENCES
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R., Hjorth J., 2016, J. CosmologyAstropart. Phys., 5, 010Young A. M., Williams L. L. R., Hjorth J., 2017, (in preparation)de Vaucouleurs G., 1948, Annales d’Astrophysique, 11, 247This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS , 1–17 (2017) Gomer and Williams D a r k m a tt e r o n l y B a r y o n s a ndd a r k m a tt e r P o pu l a t i o n ? Λ C D M × Λ C D M Sh e a r A x i s R a t i o s F o u r i e r M i s a li g n e d A x e s O ff s e t C e n t e r s B i a s ? p - v a l u e F S Q F i g . M a ss D i s t . F i g . x . × − % F i g . F i g . x . % F i g . ( m i dd l e ) F i g . xx . % F i g . F i g . xxxx . % –– xx % F i g . ( r i g h t) F i g . xxx . % – F i g . xxxxx . % –– xxx . % –– xxxx . % –– xxx . % –– xxxx . % –– xxxx . % –– xxxxx . % –– xxxxx . % – F i g . xxxxxx . % F i g . F i g . T a b l e : A t a b l e s u mm a r i z i n g t h e c o m b i n a t i o n s o f e ff e c t s e x p l o r e d i n t h i s p a p e r . R e a d i n g h o r i z o n t a ll y a c r o ss d e s c r i b e s e a c h l e n s o r p o pu l a t i o n o f l e n s e s e x p l o r e d , w h e r e a n “ x “ d e n o t e s w h i c h e ff e c t s w e r e i n c l ud e d o n t h a t e x p e r i m e n t . E x p e r i m e n t s a r e li s t e d i n o r d e r o f a pp e a r a n c e w i t h i n t h i s p a p e r , w i t h t h e fi r s t b e i n g t h e s i n g l e l e n s w i t h o n l y Λ C D M s ub s t r u c t u r e a nd t h e l a s t b e i n g t h e p o pu l a t i o n o f l e n s e s w i t h t h e c o m b i n a t i o n o f e ff e c t s du e t o b a r y o n s li s t e d i n . . T h ee ff e c t s li s t e d a r e ,i n o r d e r , w h e t h e r a p o pu l a t i o n o f l e n s e s w a s u s e d a s o pp o s e d t oa s i n g l e l e n s , w h e t h e r o r n o t Λ C D M s ub s t r u c t u r e w a s i n c l ud e d , w h e t h e r o r n o t × Λ C D M s ub s t r u c t u r e w a s i n c l ud e d , w h e t h e r e x t e r n a l s h e a r s w e r e d r a w n f r o m B o l t o n e t a l. ( ) o r l e f t a s z e r o , w h e t h e r t h e a x i s r a t i o s w e r e d r a w n f r o m B o l t o n e t a l. ( ) o r a ll s e tt o0 . , w h e t h e r o r n o t a b a r y o np o pu l a t i o n w i t hn o n z e r o a a nd a F o u r i e r p e r t u r b a t i o n s w e r e i n c l ud e d i n a dd i t i o n t oa d a r k m a tt e r c o m p o n e n t w i t h s i m il a r F o u r i e r p e r t u r b a t i o n s , w h e t h e r o r n o tt h e b a r y o np o pu l a t i o nh a d a m i s a li g n e d m a j o r a x i s w i t h r e s p e c tt o t h e d a r k m a tt e r , w h e t h e r o r n o tt h e b a r y o np o pu l a t i o nh a d a n o ff s e t c e n t e r f r o m t h e d a r k m a tt e r , a nd w h e t h e r o r n o tt h e m o s t o p t i m i s t i c b i a s w a s a pp li e d . T h e r e m a i n i n g c o l u m n s d e p i c tt h e p - v a l u e s f o r e a c h e x p e r i m e n t a s w e ll a s t h e fi g u r e s w h e r e o n e c a nfind t h e d i s t r i bu t i o n o f q u a d s r e l a t i v e t o t h e F S Q a nd /o r t h e m a ss d i s t r i bu t i o n c o n t o u r s , w h e r e a pp li c a b l e . C a u t i o n i s n e c e ss a r y w h e n c o m p a r i n g p - v a l u e s a s t h e r e h a s b ee nn oa cc o un t i n g f o r t h e a dd i t i o n o f p a r a m e t e r s , s o t h e p - v a l u e s a r e n o t d i r e c t l y c o m p a r a b l e . MNRAS000