Model-independent and model-based local lensing properties of B0128+437 from resolved quasar images
AAstronomy & Astrophysics manuscript no. aa_unblinded_v6 c (cid:13)
ESO 2019September 5, 2019
Model-independent and model-based local lensing properties ofB0128+437 from resolved quasar images
Jenny Wagner and Liliya L. R. Williams Universität Heidelberg, Zentrum für Astronomie, Astron. Rechen-Institut, Mönchhofstr. 12–14, 69120 Heidelberg, Germanye-mail: [email protected] School of Physics and Astronomy, University of Minnesota, 116 Church Street, Minneapolis, MN 55455, USAe-mail: [email protected]
Received XX; accepted XX
ABSTRACT
The galaxy-scale gravitational lens B0128 +
437 generates a quadrupole-image configuration of a background quasar that shows milli-arcsecond-scale subcomponents in the multiple images observed with VLBI. As this multiple-image configuration including thesubcomponents has eluded a parametric lens-model characterisation so far, we determine local lens properties at the positions ofthe multiple images with our model-independent approach. Using
PixeLens , we also succeed in setting up a global free-form massdensity reconstruction including all subcomponents as constraints. We compare the model-independent local lens properties withthose obtained by
PixeLens and those obtained by the parametric modelling algorithm
Lensmodel . A comparison of all three ap-proaches and a model-free analysis based on the relative polar angles of the multiple images corroborate the hypothesis that ellipticallysymmetric models are too simplistic to characterise the asymmetric mass density distribution of this lenticular or late-type galaxy.Determining the local lens properties model-independently, the sparsity and the strong alignment of the subcomponents yield broad1- σ confidence intervals ranging from 8% to over 1000% of the local lens property values. The lens model approaches yield compara-bly broad confidence intervals. Within these intervals, there is a high degree of agreement between the model-independent local lensproperties of our approach based on the subcomponent positions and the local lens properties obtained by PixeLens . In addition, themodel-independent approach e ffi ciently determines local lens properties on the scale of the quasar subcomponents, which are com-putationally intensive to obtain by free-form model-based approaches. Relying on the quadrupole moment of each subcomponent,these small-scale local lens properties show tighter 1- σ confidence bounds by at least one order of magnitude on the average witha range of 9% to 535% of the of the local lens property values. As only 40% of the small-scale subcomponent local lens propertiesoverlap within the confidence bounds, mass density gradients on milli-arcsecond scales cannot be excluded. Hence, aiming at a globalreconstruction of the deflecting mass density distribution, increasingly detailed observations require flexible free-form models thatallow for density fluctuations on milli-arcsecond scale to replace parametric ones, especially for asymmetric lenses or lenses withlocalised inhomogeneities like B0128. Key words. cosmology: dark matter – gravitational lensing: strong – methods: analytical – galaxies: individual: B0128 +
437 –galaxies: luminosity function, mass function – galaxies: quasars: general
1. Introduction
B0128 +
437 is a galaxy-scale gravitational lens that has beencontroversially discussed in the literature. It was discovered byPhillips et al. (2000) as a gravitational lensing configuration offour multiple images of a quasar. Three of these images showthree subcomponents each in the radio band, Norbury (2002),Biggs et al. (2004). So far, no lens model with a simple, smoothmass distribution has been found that can explain the positions ofall four quasar images and their subcomponents. Using the datafrom observations described in Biggs et al. (2004), we further in-vestigate this problem with the model-independent approach asdeveloped in Wagner & Tessore (2018) to determine the ratiosof scaled mass densities (convergences) and the reduced shearcomponents at the angular positions of the multiple images.Subsequently, we compare these values to the ratios of scaledmass densities and reduced shear components that the paramet-ric lens model approach
Lensmodel (Keeton (2001), Keeton(2004)) and the free-form lens model approach
PixeLens (Saha& Williams (2004)) predict. Compared to previous lens models, we take into account the current best-fit redshifts of the lens andthe source.This comparison is analogous to the one carried out in Wag-ner et al. (2018) on the galaxy-cluster scale lens CL0024 andshows that the model-independent approach can be applied togravitational lensing configurations of any size in the same way.For both lens scales, the ratios of convergences and the reducedshear components of the model-independent approach show ahigh degree of agreement to the same local lens properties ob-tained by lens modelling approaches. In this work, we investigatein addition if the model-independent approach is able to furtherconstrain the local lens properties on the level of the subcompo-nents in the multiple images of the quasar behind B0128. Lensreconstructions like
PixeLens describe non-smooth irregularlyshaped mass density distributions on a pixelised grid. Therefore,such methods usually require computationally intensive pixelisa-tions to determine these small-scale properties. For lens recon-structions based on analytical models like
Lensmodel , solvingthe lens equation as an optimisation problem may also requirea highly resolved sampling grid to determine small-scale detailsof the lens. Thus, a more e ffi cient way to obtain small-scale lo- Article number, page 1 of 15 a r X i v : . [ a s t r o - ph . GA ] S e p & A proofs: manuscript no. aa_unblinded_v6 cal lens properties is highly desired for the increasing amount ofdata in upcoming surveys.The paper is organised as follows: Section 2 introduces theinformation about B0128 that has become available so far. It ismainly based on the works of Phillips et al. (2000), Norbury(2002), Biggs et al. (2004), and Lagattuta et al. (2010). Sub-sequently, we list all observational data that we employ to cal-culate the model-independent, local lens properties and that areused to constrain our lens models in Section 3. In Section 4, webriefly outline the model-independent algorithm, which is fur-ther detailed in Wagner & Tessore (2018) before we show themodel-independent, local lens characterisation of B0128. Analo-gously, Sections 5 and 6 describe the lens modelling based on theparametric code
Lensmodel , and the free-form code
PixeLens ,respectively, before applying it to B0128. To avoid any poten-tial confirmation bias due to exchanging the values to be com-pared at an early stage, the evaluation is blinded in the sameway as the evaluation performed in Wagner et al. (2018). Thismeans that the values of the local lens properties are deter-mined independently for the model-independent and the model-based approaches and only revealed for the comparison at thevery end, after all modelling is finished. In Section 7, we ap-ply the model-free comparison as established in Woldesenbet &Williams (2012, 2015) and Gomer & Williams (2018) to B0128to investigate its degree of asymmetry with respect to lenses withdouble mirror symmetry based on the relative polar angles be-tween the multiple image positions. In Section 8, we compare allresults obtained in Sections 4, 5, 6, and 7. Finally, we concludeby assembling a consistent picture of the lensing configurationin B0128 and summarising the methodological results that canbe deduced from the comparison of the model-independent andthe model-based lens reconstructions.
2. Related work on B0128
In the discovery paper, Phillips et al. (2000), high-resolutionMERLIN observations at 5 GHz from B0128 were obtained inthe course of the Cosmic Lens All-Sky Survey (CLASS). Fourunresolved multiple images, i.e. without visible substructuresdown to the scale of 30 mas, with a maximum image separationof 0.54 arcsec, arranged in a classic quad-lens formation weredetected (see Figure 1 (right)). Long-term observations showedno hint of a time-variability of the quasar.B0128 was modelled by a singular isothermal ellipse (SIE),using the angular positions and flux ratios of the four (unre-solved) multiple images to constrain the lens model. Redshiftsof z l = . z s = . Ω m = A , C , and D in Figure 1 (left))could be further decomposed into three subcomponents. Subsequently, Biggs et al. (2004) summarised the resultsfound in Norbury (2002) to conclude that image B in Figure 1(right) is most likely scatter-broadened and hence, it is di ffi cult todecompose it into subcomponents. They also performed VLBI-observations at 2.3, 5, and 8.4 GHz to obtain further details aboutthe subcomponents and found that the observations could rathershow a core and jet instead of a compact symmetric object, asoriginally assumed in Phillips et al. (2000). The detailed struc-ture of all four images with the labelling of subcomponents ac-cording to Norbury (2002) and Biggs et al. (2004) is shown inFigure 2. Like the lens models by Norbury (2002), the lens mod-els by Biggs et al. (2004), based on the relative image positionsas constraints, were not able to fully describe the lensing con-figuration to sub-milliarcsecond precision, even when di ff erentalgorithms and modelling principles were employed.McKean et al. (2004) finally determined the redshift of thesource to be z s = . ± . z l = . , . , or 0 . z l = .
645 or 1 . Ω m = . Ω Λ = . Ω Λ (cid:44)
0. They assumed the lens at a redshift of z l = .
6, which is less likely than z l = . A and B are caused by yet unobserved dark matter substructuresin the lens and concluded that the oversimplified or improperlens model in addition to the scatter-broadening is more likely tocause the flux anomaly than additional small-scale dark-matterhalos.Summarising the previous results, observations indicate that – B0128 is a potentially lenticular or late-type galaxy mostlikely located at redshift z l = .
145 that generates four mul-tiple images of a quasar at redshift z s = . – three of these multiple images, A , C , and D in Figure 1(right), can be resolved into three clearly identifiable sub-components, the fourth, image B in Figure 1 (right), is mostlikely scatter-broadened, – no smooth lens model (even including the shear galaxy lo-cated 7.8 arcseconds away, which is shown in Figure 1 (left))can fully explain the positions of all sub-components to sub-mas-precision simultaneously.
3. Observed data from the multiple images
For our analysis, we use the data, as already stated in Phillipset al. (2000), Biggs et al. (2004), and Lagattuta et al. (2010). Ta-bles 1 and 2 summarise the image positions and flux densitiesfor the unresolved and resolved observations. Since it is verydi ffi cult to determine the relative angular positions for the hardlyvisible subcomponents in image B , we assume an uncertainty inthese o ff sets of 1 mas and also investigate the impact, if we in-crease the uncertainty to 3 mas. A similar approach was pursuedin Biggs et al. (2004). Article number, page 2 of 15enny Wagner and Liliya L. R. Williams: Local lens properties of B0128 + Fig. 1: Left: HST I-band observation from Norbury (2002); Right: MERLIN 5GHz observation by Phillips et al. (2000).Fig. 2: VLBA 8.4 GHz details of all four multiple images from Biggs et al. (2004).In order to align the coordinates of the KECK observationsin the optical from Lagattuta et al. (2010) with the MERLIN observations from Phillips et al. (2000), we have to identify areference point in both observations. To do so, we choose the
Article number, page 3 of 15 & A proofs: manuscript no. aa_unblinded_v6
Table 1: MERLIN 5 GHz measurements as in Table 2 of Phillips et al. (2000) (cols. 2–4), VLA 8.4 GHz measurements as in Table 1of Phillips et al. (2000) (cols. 5–7), and as in Table 2 of Lagattuta et al. (2010) using the
K p -filter of the Near Infrared Camera 2 onthe Keck II telescope along with the LGS AO system (cols. 8–10), both showing unresolved multiple images.Image F (cid:2) mJy (cid:3) ∆ α [arcsec] ∆ δ [ (cid:48)(cid:48) ] F (cid:2) mJy (cid:3) ∆ α [ (cid:48)(cid:48) ] ∆ δ [ (cid:48)(cid:48) ] m K (cid:2) mag (cid:3) ∆ α K [ (cid:48)(cid:48) ] ∆ δ K [ (cid:48)(cid:48) ] A . ≡ . ≡ .
00 14 . ≡ . ≡ .
00 21 . ± . ≡ . ≡ . B . .
098 0 .
094 – – – 23 . ± .
22 0 .
099 0 . C . . − .
172 3 . . − .
188 22 . ± .
04 0 . − . D . . − .
250 5 . . − .
266 22 . ± .
12 0 .
109 -0.260
Notes.
Col. 1: Name; Col. 2: Flux densities F with their uncertainties of a few percent measured in the MERLIN 5 GHz band; Cols. 3 and 4:Relative angular image positions w.r.t. image A at α =
01 : 31 : 13 . δ = + ◦ (cid:48) (cid:48)(cid:48) .
14 (J2000.0); Col. 5: Flux densities F with theiruncertainties of a few percent measured in the VLA 8.4 GHz band, images A and B not separable; Cols. 6 and 7: Relative angular image positionsw.r.t. image A at α =
01 : 31 : 13 . δ = + ◦ (cid:48) (cid:48)(cid:48) .
938 (J2000.0); Col. 8: Apparent magnitudes m and their uncertainties as measured in the K p -filter of NIRC2; Cols. 9 and 10: Relative angular image positions w.r.t. image A , reference position not given, uncertainties on images B , C ,and D are 0.001. angular position of image C , since the resolved components lieclosest together and, as the counter image opposite to the three-image configuration B , A , D , it is farther from the critical curveas those images and is therefore subject to the least amount ofmagnification.The spectroscopic observation of McKean et al. (2004) didnot find any hints of a second source and the similarity of thesubcomponents in the images A , C , and D is high, so that wetreat the images A , B , C , and D as multiple images from thesame source at z s = . B , A , and D form a cusp configuration and that image C is of the same par-ity as image A , see Wagner & Tessore (2018) for further detailsabout fold and cusp configurations and their parity.Mapping the subcomponents of the images onto each othershould then work analogously to the mapping of reference pointsin CL0024, as done in Wagner et al. (2018). The subcomponentsin the images in B0128 are much more aligned, which will causehigher uncertainties. In addition, three subcomponents are theminimum number of reference points required in each image toapply the method outlined in Wagner & Tessore (2018) and Wag-ner et al. (2018). Considering images B and A as a fold configu-ration and images A and D as another fold configuration of twoimages mirror-inverted at the critical curve between them, wenotice that the matching of subcomponents between A and D asshown in Figure 2 does not yield a fold configuration on the scaleof the subcomponents. Matching the subcomponents by their in-tensity led to this labelling. Yet, the multi-band observations car-ried out in Norbury (2002) and Biggs et al. (2004) strongly hintat dust in the lens, so that image B might not be the only im-age that is subject to attentuation due to scatter-broadening. Asa consequence, the matching of the subcomponents of image A and D could be di ff erent than proposed in Biggs et al. (2004).In Section 4, we systematically investigate the impact of di ff er-ent matchings on the local model-independent properties and thelens model.Table 3 shows the flux density ratios of images B , C , and D with respect to image A for all available bands, F i , i = B , C , D .Biggs et al. (2004) assume less than 5% measurement uncertain-ties on their flux ratios. Lagattuta et al. (2010) obtain ∆ F B = ∆ F C = ∆ F D = B , C , and D mustbe better visible, i.e. have a higher (flux) density, when the nat-ural weighting scheme is applied. Furthermore, the fact that F B is much smaller for the uniform than for the natural weightingscheme, strongly hints at scatter-broadening due to the dust inthe lens, as already noted by Biggs et al. (2004).
4. Model-independent reconstruction E ff ectively describing the gravitational lens as a two-dimensional mass distribution in a single lens plane at redshift z l , the standard gravitational lensing formalism (see e.g. Petterset al. (2001) or Schneider et al. (1992) for an introduction) treatsthis projected deflecting mass distribution in terms of the con-vergence κ ( x ). This is the two-dimensional mass density at theposition x ∈ R in the lens plane scaled by the critical density Σ cr which is the su ffi cient mass density to generate multiple images,given a lens at z l and a source at z s . While κ ( x ) only enlarges ordiminishes the source, the shear γ ( x ) = ( γ ( x ) , γ ( x )) addition-ally distorts the images. Both κ ( x ) and γ ( x ) can be determinedas second-order derivatives from the two-dimensional gravita-tional deflection potential ψ ( x ) as detailed in Wagner & Tessore(2018).Usually, global lens reconstructions, as detailed in Sections 5and 6 are set up, using the observables from the multiple imagesas constraints to reconstruct the deflecting mass density distri-bution as a whole. As observables, relative image positions, thequadrupole moment of the images around their centre of light,the flux ratios, and the time delays, if available, are employed.These lens models are subject to a lot of degeneracies, see Wag-ner (2018) for a mathematical derivation and Wagner (2019) forthe physical explanation of all degeneracies arising in the gen-eral lensing formalism. To avoid the model-based degeneracies,Tessore (2017) derived the most general information which canbe obtained from multiple images of a background source with-out assuming a specific gravitational lens model. He found thattransforming the multiple images onto each other yields ratios of Article number, page 4 of 15enny Wagner and Liliya L. R. Williams: Local lens properties of B0128 + Table 2: VLBI observations as in Table 1 from Biggs et al. (2004).Image ν [GHz] F (cid:2) mJy (cid:3) ∆ α [mas] ∆ δ [mas] a [mas] r θ (cid:2) deg (cid:3) A . ± . − . . . ± . . ± .
04 27 . ± . A . ± . − . . . ± . . ± .
09 28 . ± . A . ± . − . . . ± . . ± .
03 30 . ± . B − . . B − . . B − . . C . ± . . − . . ± . . ± . − . ± . C . ± . . − . . ± . − . ± . C . ± . . − . . ± . . ± . − . ± . D . ± . − . − . . ± . . ± . − . ± . D . ± . − . − . . ± . . ± . − . ± . D . ± . − . − . . ± . . ± . − . ± . A . ± . − . . . ± . . ± .
11 29 . ± . C . ± . . − . . ± . − . ± . D . ± . − . − . . ± . . ± . − . ± . Notes.
Col. 1: Name of subcomponent; Col. 2: Observing frequency; Col. 3: Flux density and uncertainty; Cols. 4 and 5: Relative image positionsto α =
01 : 31 : 13 . δ = + ◦ (cid:48) (cid:48)(cid:48) .
805 (J2000), extended by the components for image B as read o ff Figure 6 in Biggs et al. (2004). Weassume an uncertainty of 0.1 mas for all subcomponent positions of images A , C , and D and estimate an uncertainty of 1 mas for the subcomponentpositions of image B ; Col. 6: Extension of semi-major axis of elliptical Gaussian fitted by O mfit ; Col. 7: Axis ratio of semi-minor to semi-majoraxis; Col. 8: Position angle measured from north to east. Table 3: Observed flux density ratios F i with respect to image A for images i = B , C , D in di ff erent bands.Band F B F C F D VLBA 5 GHz (uni) 0.26 0.45 (1) (2)
VLBA 5 GHz (nat) 0.65 0.56 0.59VLBA 2.3 GHz 0.49 0.34 0.47MERLIN 5 GHz 0.56 0.49 0.47
K p
Notes.
Rows 1 and 2: In the 5-GHz VLBA maps with uniform (uni) andnatural (nat) weighting as in Table 3 of Biggs et al. (2004); Row 3: In2.3 GHz VLBA maps; Row 4: In 5 GHz MERLIN maps as taken fromTable 4 of Biggs et al. (2004); Row 5: In the
K p -band (fifth row) astaken from Table 3 of Lagattuta et al. (2010); (1) flux density ratios for the subcomponents are F C , = . F C , = . F C , = . (2) flux density ratios for the subcomponents are F D , = . F D , = . F D , = . convergences f i j ≡ − κ ( x i )1 − κ ( x j ) ≡ − κ i − κ j (1)between all multiple images i , j and reduced shears g ( x i ) ≡ g i ≡ ( g i , , g i , ) ≡ γ i − κ i (2)at the positions of the multiple images. These local lens proper-ties in Equations (1) and (2) are invariant under the mass-sheettransformation. Thus, as further elaborated in Wagner (2019), they represent the information about the lens that all lens modelsshould have in common at leading order. Derived from the f i j sand g i s, the magnification ratios J i j ≡ µ j µ i = det( A i )det( A j ) (3)for image pairs i , j can be calculated because the magnification µ i of an image i is given as the inverse of the determinant of thedistortion matrix A i = (1 − κ i ) (cid:32) − g i , − g i , − g i , + g i , (cid:33) . (4)As minimum requirements, the brightness profiles of the multi-ple images must contain clearly identifiable substructures, e.g. atleast three linearly independent reference points like star form-ing regions, that can be matched across all multiple images. Fur-thermore, at least three multiple images with these identifiablesubstructures are needed which are not aligned like in an ax-isymmetric deflection potential.Our C-implementation employs the centroid of all referencepoints as the position of a multiple image, also called “anchorpoint". Apart from this choice, any point in the convex hull ofthe reference points can be used as anchor point, since the ap-proach assumes that the convergence and the shear are approxi-mately constant over the extension of a multiple image. The po-sition of the anchor point of one image, the so-called “referenceimage" remains fixed. Then, a system of equations is set up bylinearly mapping the reference image onto all other multiple im-ages. Solving the system of equations by a χ -parameter estima-tion as detailed in Wagner et al. (2018) yields all f i j and g i for allmultiple images, and the most likely anchor point positions in all available at https://github.com/ntessore/imagemap Article number, page 5 of 15 & A proofs: manuscript no. aa_unblinded_v6 remaining images. Wagner & Tessore (2018) and Wagner et al.(2018) further detail the algorithmic implementation and the pro-cedure to obtain confidence bounds on the local lens propertiesby sampling their covariance matrix close to the most probablevalues of the f i j , g i , and the anchor points. Applying the method as outlined in Section 4.1 to the case ofB0128, we first find that the four non-aligned multiple imageswith their three subcomponents fulfil the requirements. The sub-components are almost linearly aligned, yet, not in a mathemat-ically rigorous way to cause the optimisation problem which de-termines the local lens properties to be under-constrained andthus degenerate. But, from the systematic analyses performed inWagner et al. (2018), we expect broad confidence bounds for thelocal lens properties determined by the subcomponents due totheir alignment and the small area that they span.We choose image A as the reference image and set up trans-formations between all remaining images to image A in orderto determine the local lens properties. To investigate the impactof this choice, we also determine the local lens properties in thethree-image configuration of images A , C , and D mapping thethree identifiable subcomponents onto each other and using im-age C or image D as reference image. In all three cases, we ob-tain the same local lens properties, yet with di ff erent confidencebounds. The analysis shows that using image C as reference im-age yields smaller confidence bounds for the three-image con-figuration ACD . Yet, it yields comparable to slightly worse con-fidence bounds when matching the individual Gaussian fits ofthe subcomponents in image A , C , and D onto each other and weobtain larger confidence bounds for the four-image configuration ABCD . Thus, image C is only slightly less suitable as referenceimage than image A . Contrary to that, using image D as referenceimage yields smaller confidence bounds for the three-image con-figuration, yet with a very low e ff ective number of samples in theimportance sampling process.To simplify the notation, we will omit A in the subscripts ofthe local lens properties and simply write f j ≡ f A j = − κ A − κ j , J j ≡ J A j = µ j µ A , j = B , C , D . (5)Instead of using the positions of the three subcomponentsin each multiple image as reference points, we also reconstructlocal lens properties on the scale of the subcomponents them-selves. For this, we choose image A again as reference imageand use the centre of light of the Gaussian fitted to the subcom-ponent i as first reference point. Then, we use the end pointsof the semi-major and semi-minor axis of the fitted Gaussian asfurther reference points in each subcomponent. The calculationshow to obtain these positions from the values of the semi-majoraxis, the axis ratio, and the position angle can be found in Ap-pendix A. In this way, we can match the subcomponents 1 and3 across images A , C , and D to infer the individual local lensproperties at their positions. These values can be compared tothe ones obtained for the convex hull of all subcomponents. Thematching for the image- and subcomponent-scale is sketched inFig. 3.In the following we start with matching the subcomponentsfor images A , C , and D (see Section 4.2.1). To investigate theimpact of a relabelling, we systematically interchange the 1 and3 labels for images C and D . Then, in Section 4.2.2, we includeimage B and systematically interchange its subcomponents 1 and Fig. 3: Visualisation of image-scale matching: using the posi-tions of the three subcomponents (red dots) in each image, themultiple images A , B , C , and D can be matched onto each other(red arrows) to determine the local lens properties on the imagescale (indicated by the larger red areas for visualisation purposes,as the convex hull spanned by the reference points in each imageis too small to be drawn). Visualisation of subcomponent-scalematching (highlighted in blue): using the Gaussians fitted to asubcomponent (here: 1, highlighted in dark grey), the subcom-ponents can be matched onto each other to determine the locallens properties within the area of the Gaussians. The first refer-ence point in each Gaussian is its centre of light (red dots). Thetwo reference points that are further required are the end pointsof the axes of the elliptical isocontour (black dots). North is tothe top and east to the left.3 to determine the most likely local lens properties for all fourimages. In Section 4.2.3, we match the Gaussians fitted to sub-components 1 and 3 across the images A , C , and D to investigatethe local lens properties on the subcomponent-scale and discusspotential biases due to dust in the lens plane, as assumed in Nor-bury (2002), Biggs et al. (2004), and Lagattuta et al. (2010). A , C , and D Employing the positions of the three subcomponents 1, 2, and 3in images A , C , and D listed in Table 2 as reference points, wedetermine the local lens properties. The most likely lens proper-ties, the mean values and the 1- σ confidence bounds are listed inthe second, third, and fourth column of Table 5, respectively.As mentioned in Section 3, the labelling of the subcompo-nents according to Table 2 is not compatible with image A and D being a fold configuration on the level of subcomponents. Tosystematically investigate which subcomponents across the im-ages should be matched, we interchange the labelling of the sub-components 1 and 3 systematically as indicated in Table 4. Theresulting lens properties can be found in Appendix B. None ofthe configurations obtains the correct relative parities betweenimages A , C , and D in the signs of the relative magnifications J i , i = C , D .Thus, the labelling of subcomponents according to Biggset al. (2004) is the only one yielding results which are consis-tent with leading order lensing theory. Considering the ratios ofconvergences f i , we note that the highly negative value for f D could indicate that images A and D lie on opposite sides of theisocontour κ ( x ) =
1. The positive sign for f C could indicate, thatimages A and C are located at the same side of κ ( x ) =
1. Since f C and f D are both subject to broad confidence bounds several times Article number, page 6 of 15enny Wagner and Liliya L. R. Williams: Local lens properties of B0128 + Table 4: Configurations of di ff erently matched subcomponentsacross images A , C , and D . Resulting local lens properties canbe found in Table B.1. Configuration 0 is the matching accordingto Table 2. In all configurations, image A is the reference image.Conf. Subcomp. 1 Subcomp. 2 Subcomp. 3 A A A C C C D D D A A A C C C D D D A A A C C C D D D A A A C C C D D D their absolute value, these statements are further investigated inSections 4.2.2 and 4.2.3. In Section 4.2.4, they are also checkedfor consistency in a comparison with our lens models as set upin Sections 5 and 6. Building upon the configuration of three multiple images A , C ,and D with the three subcomponent positions as shown in Ta-ble 2, we include the three subcomponent positions in image B and determine the local lens properties from it. The results can beread o ff columns 5–7 in Table 5 and are mostly subject to confi-dence bounds that exceed their absolute values. These large con-fidence bounds can partly be caused by the scatter-broadening ofimage B which is not accounted for in our local lens reconstruc-tion.Swapping the labelling of subcomponent B with B , we findthat both results agree within their confidence bounds except for g A . We favour the labelling as proposed by Biggs et al. (2004)due to the slightly lower confidence bounds of the resulting locallens properties and because the most likely value for J B has thecorrect parity, which is not the case when we interchange thelabels.In the four-image configuration, images A , C , and D mostlikely lie on the same side of the isocontour κ ( x ) = B seems to be located on the opposite side. Yet, analogously tothe previous results, f B has a large confidence bound, which alsoincludes the possibility to lie on the same side as image A .Increasing the uncertainty in the positions of the subcompo-nents of image B from 1 mas to 3 mas leads to an increase in theconfidence bounds. The local lens properties still agree withintheir confidence bounds except for g A . Assuming an uncertaintyof 3 mas in the position of all subcomponents, the confidencebounds do not increase, yet, the e ff ective number of samples inthe importance sampling is reduced below 10. In this case, thelocal lens properties except for g A , coincide with the ones us-ing 1 mas as uncertainty in the positions of the B i , i = , ,
3. Weshow all results for the interchange of B and B and the increaseof the measurement uncertainties in Appendix C. and ff erent configurations of refer-ence points taken from the end points of the semi-major andsemi-minor axes yield the same local lens properties. As detailedin Appendix A, this is the case for both subcomponents 1 and 3.Subsequently, we determine the local lens properties at the po-sitions of subcomponent 1 in images A , C , and D as listed incolumns 8–10 of Table 5 and the ones at the positions of sub-component 3 as listed in columns 11–13 of Table 5. Contrary tothe image-scale local lens properties, all f - and g -values haveconfidence bounds that are smaller than their absolute value ex-cept for g C , and g D , of subcomponent 3. We find that half of thelens properties at the position of subcomponent 1 agree with theones at the position of subcomponent 3 within their confidencebounds.Concerning their relative positions with respect to the iso-contour κ ( x ) =
1, the most likely f i , i = C , D inTable 5 implythat all images are supposed to lie on the same side to a muchhigher degree of confidence because the confidence bounds donot include negative values anymore. From Section 4.2.1, we can draw the conclusion that it is possibleto determine local f i and g i , i = A , C , D , such that their valuesare constant over the area spanned by the three subcomponentsin each image. The matching of the subcomponents accordingto Table 2 is the only one yielding the correct relative paritiesin the relative magnifications J C and J D . These findings are inagreement with previous results as summarised in Section 2.Including the subcomponent positions of image B , we findthat the local lens properties for images C and D agree withintheir confidence bounds and all relative confidence bounds donot significantly increase. These results indicate that it shouldbe feasible to construct a smooth lens model that explains thefour-image configuration with its subcomponents. So far, onlythe specific approaches detailed in Biggs et al. (2004) have beenunsuccessful.In addition, we note that we obtained similarly large relativeconfidence bounds for the case of the galaxy-cluster-scale lensCL0024, see Wagner et al. (2018), when we reduced the numberof reference points to four that were almost aligned and spannedonly a small area. Thus, apart from the scatter-broadening due tothe comparably large amount of dust in the lens, the sparsity ofthe data and their alignment also contributes significantly to thelarge confidence bounds. The smaller confidence bounds deter-mined for the local lens properties of the subcomponents due tothe orthogonally oriented vectors between the reference points,support this hypothesis. Without further, improved multi-bandobservations, it is hard to disentangle the impact of the individ-ual e ff ects.Comparing the image-scale local lens properties as obtainedin Section 4.2.1 to the local lens properties at the positions ofthe subcomponents 1 and 3, as derived in Section 4.2.3, we findthat the local lens properties at the positions of the two subcom-ponents agree with the ones determined over their entire con-vex hull with the exceptions of g A at both positions and J C and J D at the position of subcomponent 1. Comparing the local lensproperties at the positions of the two subcomponents with eachother, we observe that half of them agree within their confidencebounds. This puts a very weak upper limit to the scale of poten- Article number, page 7 of 15 & A proofs: manuscript no. aa_unblinded_v6
Table 5: Synopsis of local lens properties obtained by the various configurations detailed in Sections 4.2.1, 4.2.2, and 4.2.3. f i , g i are listed with their most likely value, their mean and 1 − σ confidence bound. ConfigACD is the image-scale three-imageconfiguration of Section 4.2.1, ConfigABCD is the image-scale four-image configuration of Section 4.2.2, ConfigA C D is thesubcomponent-scale three-image configuration at the position of subcomponent 1, ConfigA C D is the subcomponent-scale three-image configuration at the position of subcomponent 3. The total number of subcomponents (SCs) involved to determine the f i and g i is listed below the name of the configuration. Config. ACD ABCD A C D A C D Lens prop. 9 SCs 12 SCs 3 SCs 3 SCs J A f A g A , -0.56 -0.56 0.04 -0.44 -0.44 0.05 3.53 3.62 0.73 1.32 1.40 0.39 g A , J B -1.47 -1.44 1.53 f B -16.75 -0.64 122.09 g B , -8.77 0.31 66.85 g B , -6.09 -0.90 27.62 J C f C g C , -0.54 -0.58 3.03 -0.16 -0.16 0.24 1.57 1.63 0.35 0.11 0.17 0.30 g C , J D -0.17 -0.17 0.10 -0.09 -0.10 0.09 -0.90 -0.90 0.11 -0.31 -0.31 0.06 f D -6.92 -1.05 102.20 0.16 0.31 2.23 0.25 0.25 0.05 0.26 0.25 0.06 g D , -1.54 -0.08 29.03 0.15 0.21 0.55 0.21 0.20 0.09 -0.01 0.03 0.16 g D , A , C , and D .Local lens properties obtained at the positions of the subcom-ponents 1 and 3 indicate that all images lie on the same side ofthe isocontour κ ( x ) =
1. Most probably, they lie outside κ ( x ) = κ ( x ) = /
5. Model-based reconstruction: parametric
The most common way to model galaxy lenses is to fit imagepositions using a simple parametric form for the galaxy massdistribution. We use publicly available software,
Lensmodel , de-scribed in Keeton (2001) and Keeton (2004).
Lensmodel o ff ersa range of lens models; we choose to work with analytic poten-tials, instead of mass distributions, because the former have ana-lytic expressions for all relevant lensing quantities. We note thatthe reconstructions presented in this section use a very di ff erentansatz from the ones in Section 6, in that the latter recover a massdistribution, instead of a potential, and do not assume any fixedparameteric form. The alphapot lensing potential is a softenedpower law potential given by φ a ( x ) = b ( s + ξ ( x ) ) α/ , (6)where b is the normalisation constant, s is the core radius, whichis set to a small non-zero value, and ξ ( x ) = x + x / q , with q being the axis ratio of the potential. The boxy power law poten-tial, called boxypot , expressed in polar coordinates x = ( r , θ )has the form φ b ( x ) = br α [1 − (cid:15) cos 2( θ − θ (cid:15) )] β , (7) where (cid:15) is the ellipticity and θ (cid:15) is its position angle . In the re-constructions, both potentials are augmented with external shear. We run several lens reconstructions for each of the two imageconfigurations (abbreviated as configABCD and configACD; weuse the naming convention as introduced in Table 5), and for thecases with and without fixing the galaxy lens centre. The recon-structions use di ff erent initial parameter guesses, and alphapot and boxypot potentials.If only subcomponent 1 is used as input, configA C D andconfigA B C D can be fit perfectly with either alphapot or boxypot , if the lens centre coordinates are left as free modelparameters. This is summarised in the second column of thefirst two rows of Table 6, which shows typical root-mean-squaredeviations (rms) between the observed image positions and themodel-predicted ones in the lens plane. If the lens centre is heldfixed, configA C D can still reproduce the images perfectly(third column of the first row). If configA B C D is used, typ-ical rms becomes 0 . (cid:48)(cid:48) . When the other two subcomponents,2 and 3, are included, for a total of 9 and 12 subcomponentsfor each of configACD and configABCD (last two rows), neither alphapot nor boxypot provide good fits, for floating or fixedlens centre. These lens plane rm are larger than our assumed un-certainty of 0.1 mas for images A, C, and D.The resulting Lensmodel local lens properties forconfigA B C D and configA C D as defined in Equations (1)and (2) are extracted at the positions of subcomponent 1, andare listed in the columns 4–7 of Table 7. The tightest confidencebounds are found for configA B C D with fixed lens centre. We note that
Lensmodel actually uses φ b = br α [1 − (cid:15) cos 2( θ − θ (cid:15) )] αβ to do the calculations, which is not exactly the same form as presentedin the Keeton (2004) manual.Article number, page 8 of 15enny Wagner and Liliya L. R. Williams: Local lens properties of B0128 + Fig. 4:
PixeLens reconstructed maps of lensing convergence. Left: using all 12 subcomponents of B0128; Right: using only the 9subcomponents of images A , C , and D . The subcomponents used as constraints in each case are indicated with magenta circles. Thegrey iso-convergence contours are spaced logarithmically. The blue contour is the iso-convergence contour at κ ( x ) = . (cid:48)(cid:48) , i.e., substantially smaller than HST pixelsize. If the source in B0128 were extended, with lensed imagescovering one or more HST-sized pixels in the lens plane, therewould be no indication that simple models do not fit.Table 6: Lensmodel results: typical lens plane image rms. Nam-ing convention according to the constraining observables as inTable 5.Configuration lens centre lens centrefloating fixedconfigA C D (3 SCs) 0 . (cid:48)(cid:48) . (cid:48)(cid:48) configA B C D (4 SCs) 0 . (cid:48)(cid:48) . (cid:48)(cid:48) configACD (9 SCs) 0 . (cid:48)(cid:48) . (cid:48)(cid:48) configABCD (12 SCs) 0 . (cid:48)(cid:48) . (cid:48)(cid:48)
6. Model-based reconstruction: free-form
Because simple parametric models, like elliptical mass distribu-tions with external shear presented in Section 5, cannot repro-duce all 12 subcomponents in B0128, here we use a free-formmethod, called
PixeLens (Saha & Williams 2004), to recon- struct the mass density distribution in B0128.
PixeLens is pub-licly available, and has an easy-to-use GUI interface .For any given basis set, the lens equation can be written as aset of linear equations in the unknowns, which are the weights ofthe basis functions, and the source positions. PixeLens breaksup the lens plane into equal size square mass pixels (its basisset), and imposes a few constraints. The positions of the imagesare specified with respect to the centre of the lens, which servesas the centre of the reconstruction. The parities of the imagesare also specified as input and are strictly enforced. The massgradient should point not more than ± ◦ away from purely ra-dial. Except for the central pixel, the mass of no other pixel canexceed twice the average mass of all its neighbours. These con-straints act to regularise the mass distribution. Because many setsof pixel weights, i.e., many mass distributions, can reproduce theimages exactly, there is a plethora of solutions. In this paper werun PixeLens for 250 models, and discard the first 50 “burn-in” models, as is sometimes done for MCMC runs. The final
PixeLens lens reconstruction is then taken as an average over200 individual solutions.The confidence bounds for the local lens properties are calcu-lated as the rms dispersion between 20 sets of 10 individual mod-els. We use 10, instead of 1, because, averaging over a handfulof individual mass density models suppresses one-o ff astrophysi-cally unrealistic features and enhances common features. We ex-periment with averaging 10, 20 and 40 individual models to ob-tain one final PixeLens model (with the corresponding numberof 20, 10, and 5 sets of
PixeLens models to determine the rms).As expected, we found that, the rms between the sets decreases.In the limit, if averaging is done over a very large number ofindividual reconstructions, the di ff erence between sets will goto zero, and so will the rms. Our choice of averaging over 10 https: // / psaha / lens / pixelens.phpArticle number, page 9 of 15 & A proofs: manuscript no. aa_unblinded_v6
Fig. 5: The result of subtracting mass density distribution of con-figABCD from that of configACD. Light blue (green) contoursrepresent negative (positive) mass di ff erences. Contour levels areat ∆ κ values of ± . , ± . , ± . , ± . , ± . ff erences are statistically significant.individual models to obtain one final model is somewhat conser-vative, because the corresponding rms is on the high side. First, we carry out
PixeLens reconstructions using all 12 sub-components of all four images in B0128. Relative images fluxesare not used. A region of radius 0.675 arcseconds around thecentral reference point in Table 2 is divided into 41 by 41 pix-els, such that each
PixeLens pixel covers an area with an edgelength of 33 milli-arcseconds. The average projected lensingconvergence map is shown in the left panel of Figure 4. It isthe average over 200
PixeLens solutions. The recovered massdistribution is not very circularly symmetric, and would be hardto represent with a simple parametric model. This is not surpris-ing, and is consistent with parametric models not being able toreproduce all 12 subcomponents. Put di ff erently, to reproduceall 12 subcomponents, one requires significant deviations from apurely elliptical projected mass distribution.Mindful of the fact that mass reconstructions depend criti-cally on the quality of the image data, we also carry out a recon-struction that does not include any subcomponents of the most-likely scatter-broadened image B . The reconstruction based onjust the 3 subcomponents of each of images A , C , and D is shownin the right panel of Figure 4. The local lens properties of con-figABCD and configACD as set up in Equations (1) and (2) atthe positions of subcomponent 1 are displayed in the second andthird column of Table 7.Figure 5 shows the di ff erences between the convergencemaps of the two models configACD and configABCD shownin Figure 4. The light blue (green) contours represent negative(positive) di ff erences between the convergences. To investigatewhether these di ff erences are signficant, we calculate the statis- Fig. 6: Distribution of quads in the space of relative image an-gles. Here, a 2D projection of that 3D space is used. The Fun-damental Surface of Quads is the horizontal line at ∆ θ = ∝ r − . , ellipticity (cid:15) = .
25, and external shear γ = . ◦ .tical significance at each location in the lens plane as S = κ A − κ B (cid:113) σ A + σ B , (8)where the σ ’s are the rms obtained using 20 sets of 10 individ-ual PixeLens reconstructions. In fact, none of the di ff erencesshown in Figure 5 are statistically significant; the value of S is always less than 1. Consistently, the f - and g -values for thetwo models (see the first two models in Table 7) all agree withintheir 1- σ confidence bounds. Using alternative labelling of sub-components, for example Conf. 1 in Table 4, yields arrival timesurfaces with very contorted contours, which is consistent withthe results of Section 4.2.1.
7. Model-free analysis
Finally, we perform another type of analysis on the 12 subcom-ponents of B0128, which is not a mass reconstruction, and sodoes not yield values of surface mass density or shear. The anal-ysis is described in Woldesenbet & Williams (2012, 2015) andGomer & Williams (2018). It is based solely on the relativeimage polar angles of quadrupoly-imaged quasars (quads), asviewed from the centre of the galaxy lens. The three angles θ i j are measured between the i th and j th arriving images: θ , θ ,and θ . Woldesenbet & Williams (2012) show that all quadsgenerated by lenses with double mirror symmetry, regardless of Article number, page 10 of 15enny Wagner and Liliya L. R. Williams: Local lens properties of B0128 + Table 7: Local lens properties as defined by Equations (1) and (2) of
PixeLens and
Lensmodel reconstructions for the di ff erentmultiple-image configurations with subcomponents (details about the configurations, see Sections 5 and 6; naming conventionsaccording to Table 5). Method
PixeLens PixeLens Lensmodel Lensmodel Lensmodel Lensmodel
Config. ABCD ACD A B C D A B C D A C D A C D
12 SCs 9 SCs 3 SCs 3 SCs 3 SCs 3 SCscentre fixed centre fixed centre fixed centre floating centre fixed centre floatingLens prop. J A ± ± ± ± ± ± f A ± ± ± ± ± ± g A , -0.46 ± ± ± ± ± ± g A , ± ± ± ± ± ± J B -0.26 ± ± ± ± ± ± f B ± ± ± ± ± ± g B , ± ± ± ± ± ± g B , ± ± ± ± ± ± J C ± ± ± ± ± ± f C ± ± ± ± ± ± g C , -0.63 ± ± ± ± ± ± g C , -0.17 ± ± ± ± ± ± J D -0.16 ± ± ± ± ± ± f D ± ± ± ± ± ± g D , ± ± ± ± ± ± g D , -2.08 ± ± ± ± ± ± θ versus the deviation of quads from the FSQ, ∆ θ . Note that θ is singled out because second and third arriving images arethe ones that approach each other in the lens plane and vanishwhen the source moves further away from the lens centre and aquad becomes a double. Therefore these two images distinguisha quad from a double. Figure 6 shows the three quads of the subcomponents in B0128as green squares, together with 40 galaxy-scale quads presentedin Woldesenbet & Williams (2012). In general, the more a givenlens deviates from being purely double mirror symmetric, themore its quads will deviate from the FSQ. Deviations from dou-ble mirror symmetry can be of two general types: one can addexternal shear to an elliptical lens, or one can add non-ellipticalmass density perturbations to an otherwise elliptical lens.As a reference, we plot a few thousand quads from a syn-thetic lens with projected mass density profile ∝ r − . , elliptic-ity (cid:15) = .
25, and external shear γ = .
25, which is misalignedwith the ellipticity position angle by 80 ◦ . By comparing the lo-cation of the B0128 quads to the quads of the synthetic lens,we estimate that if B0128 is fitted with a simple lens model, itsellipticity and / or shear can be approximately 0 .
25. This is con-sistent with the findings of Biggs et al. (2004), whose two mod-els have γ = .
26 and 0 .
22, as well as our own findings using
Lensmodel (Section 5.2), where ellipticity and external shearhave similar magnitudes.Since this is a rather large shear, an alternative interpreta-tion of the location of the B0128 quads in Figure 6 is that thegalaxy lens has non-elliptical density perturbations. The role ofsuch perturbations on the relative image angles of galaxy-scaleobserved quads was explored by Gomer & Williams (2018). The authors concluded that observed deviations from FSQ by ∆ θ ∼ ◦ − ◦ are possible if realistic perturbations of the den-sity profile from a purely elliptical model are included in themass model.The possible presence of non-negligible perturbations fromellipticity in the case of B0128, indicated by Figure 6 isconsistent with the mass distribution produced by free-form PixeLens , in Section 6, and shown in Figure 4.
8. Conclusion
We present four di ff erent types of analyses and reconstruc-tions of the galaxy-scale lenticular or late-type gravitational lensB0128 constrained by the quadrupole-image configuration of abackground quasar. Previous multi-band observations revealedthat each of the four quasar images shows three bright sub-component features. Radio observations resolve these subcom-ponents, which are separated by less than 10 milli-arcseconds,giving constraints on the lensing mass distribution on verysmall scales. All approaches to find a global mass density re-construction of B0128 based on lens models to reproduce thesubcomponent-structure within the multiple images, mainly pur-sued by Biggs et al. (2004), have been unsuccessful. In contrastto that, the four multiple images observed at a lower resolutionat which no subcomponents are resolved can be reproduced by asingular isothermal ellipse lens plus external shear.Section 8.1 summarises the methodological progress thatcould be made to explain the multiple-image configuration atsubcomponent-scale by analysing B0128 with di ff erent lenscharacterisation approaches and comparing their results witheach other. Subsequently, we conclude in Section 8.2 by sum-marising the consistent lens description that can be set up withall methods discussed and compared in Section 8.1. Lastly, weput our findings in the context of similar cases. Article number, page 11 of 15 & A proofs: manuscript no. aa_unblinded_v6
Using the model-independent approach, as detailed in Wagner& Tessore (2018) and Wagner et al. (2018), we found lens-model-independent leading-order ratios of convergences and re-duced shear values for all multiple images in B0128 based onthe positions of the three subcomponents in the images (seeSection 4.2.4 and Table 5). Subsequently, we succeeded in set-ting up a global free-form
PixeLens lens model using the po-sitions of the subcomponents as constraints (see Section 6.2).Due to scatter-broadening and a strong alignment of the subcom-ponents, the local lens properties of all approaches are subjectto broad confidence bounds. Within the 1- σ confidence bounds,the model-independent ratios of convergences and reduced shearvalues agree to the values obtained by the PixeLens reconstruc-tion in all but one. So there is a similarly high degree of agree-ment between the model-independent local lens properties andthe model-based values as was found in Wagner et al. (2018) forthe galaxy-cluster-scale lens CL0024. Similar to Wagner et al.(2018), we conclude that the overall width of confidence inter-vals of the local lens properties is decreased for the
PixeLens re-construction with its additional global regularisation constraintscompared to the model-independent ones. The tendency of tight-ening confidence intervals for an increasing amount of additionalmodel assumptions and of regularisation constraints is supportedby the findings of Williams & Liesenborgs (2019) on galaxy-cluster scale as well.Determining the ratios of convergences and reduced shearvalues at the individual positions of the subcomponents, i.e.on milli-arcsecond scale, we find larger mean reduced shearvalues than on image-scale. In addition, the local lens proper-ties between the subcomponents within one image only over-lap in 50% of the cases within their 1- σ confidence bounds.The same degree of agreement is found when comparing themodel-independent local lens properties at subcomponent 1to the ones at the same positions obtained by the parametric Lensmodel reconstruction using only subcomponent 1 in im-ages A , C , and D as multiple image constraints. The high amountof required milli-arcsecond-sized pixels in PixeLens preventsus from setting up a
PixeLens model to determine the locallens properties at the individual subcomponents with their confi-dence bounds. Hence,
PixeLens is more robust than the model-independent approach in returning tighter confidence bounds dueto additional regularisation constraints. Vice versa, the model-independent approach has the advantage over
PixeLens that itis highly e ffi cient in returning local lens properties and their con-fidence bounds at any scale with a minimum amount of compu-tational e ff ort.On the whole, we conclude that the suitability of di ff erentglobal lens reconstruction approaches decisively depends on theresolution and the quality of the observations: on the scale ofunresolved multiple images (e.g. for the data summarised in Ta-ble 1), the mass density distribution in B0128 still has ellipticalsymmetry, so that parametric lens models like Lensmodel areable to reproduce the multiple-image configuration, if the lenscentre is not fixed. The resolved subcomponent structures inthe multiple images reveal asymmetries in the deflecting massdensity distribution which require more sophisticated free-formmodelling approaches like
PixeLens to explain the multiple-image configuration. Constraining local lens properties at themilli-arcsecond scale of the subcomponents to probe small-scaledark matter properties is computationally more e ffi cient to pur-sue with the model-independent approach. It only yields locallens properties, i.e. does not pursue a global reconstruction, but the local lens properties give the maximum information at lead-ing order, which all lens model agree upon. Statistical screen-ing methods that probe the symmetries of the observables ofthe multiple-image configuration like the one put forward inWoldesenbet & Williams (2012) can serve as consistency checksor provide initialisations for the global lens reconstruction ap-proaches to increase the modelling e ffi ciency. So far, only a few multiple-image configurations have been ob-served at milli-arcsecond level and have been found to showsubstructures on this scale. The quad-configuration in B0128is one of these rare cases. B0128 is also special in a secondway because its deflecting mass density distribution is a high-redshift lenticular or late-type galaxy and not an early-type one.As many recent works have consistently shown, see e.g. Hsuehet al. (2018), Gomer & Williams (2018), Nightingale et al.(2019), and Gilman et al. (2019), smooth symmetric parametriclens models may not be a su ffi cient means to describe observedhighly resolved multiple-image configurations on galaxy-scalemuch longer. The findings summarised in Section 8.1 consis-tently show that B0128 is such an example.Based on the PixeLens model of Section 6.2, we confirmthe hypothesis stated in Xu et al. (2015) that the lens models asset up in Norbury (2002) or Biggs et al. (2004) are too simplisticto resolve the asymmetric deflecting mass density distributionthat causes the multiple-images including their substructures onmilli-arcsecond scale. Given the type of the deflecting galaxy, itis not surprising that mass density isocontours change their mor-phology for increasing distance from the galactic centre. Takinginto account the estimates of Xu et al. (2015) about potentiallyexisting small-scale dark matter inhomogeneities in B0128 andthe e ff ects of the baryonic part of the mass density, the higherreduced shear values at the subcomponent-level and their poten-tial variations between the subcomponents look plausible, butremain to be corroborated by further examples.Comparing the magnification ratios obtained by the model-independent approach (see J i , i = A , B , C , D , in Table 5), themagnification ratios as determined by PixeLens (see J i in Ta-ble 7), and the observed flux ratios (see Table 3), we find ahigh degree of agreement between the observed flux ratios andthe PixeLens values within the broad confidence bounds ofthe
PixeLens reconstruction. The values based on the model-independent approach have tighter confidence bounds and onlyagree for the subcomponent 3 in image D with the observedones. While the comparison between the flux ratios and the PixeLens magnification ratios is considered over areas that havethe same order of magnitude, the model-independent approachdetermines the magnification ratios over the triangle spanned bythe three subcomponent positions, which is less than 1% of thearea of a
PixeLens pixel. For the subcomponents, the di ff er-ences between observed flux ratios and model-independent mag-nification ratio values shrink, which corroborates the hypothesisthat the di ff erent sizes over which the quantities are determinedcauses discrepancies between the results. Yet, further investiga-tions on the way that the flux ratios are calculated are necessaryto confirm this. In addition, the observed flux ratios can be influ-enced by microlensing, scatter-broadening and absorption, seeBiggs et al. (2004) and Lagattuta et al. (2010) for further details.On the whole, we can conclude that, at the current obser-vational accuracy and precision, we have arrived at a consis-tent reconstruction of the deflecting mass density distributionand model-independent local lens properties of B0128 which Article number, page 12 of 15enny Wagner and Liliya L. R. Williams: Local lens properties of B0128 + are able to explain the observed multiple-image configurationincluding the subcomponent structure on milli-arcsecond scale.The findings are also in accordance with observations and mod-elling results of previous works.The high-precision astrometry and existence of subcompo-nents in the radio bands give an unprecedented view of thegalaxy-scale lens. Evidence for deviations from a simple lensis found at arcsecond and milli-arcsecond scales. On milli-arcsecond scale, there are high shear values and shear gradi-ents, which imply gradients in mass density. On arcsecond scale,simple parametric mass distributions, like Lensmodel modelscannot reproduce images within the astrometric precision (doesthat sound better? for me, it’s strange to produce somethingwithin some error) if the lens centre is fixed (see Section 5.2).Furthermore, the external shear required (as determined by
Lensmodel and the model-free approach in Figure 6) is 0.22-0.26. Such large shears are unlikely to arise from nearby galax-ies. For comparison, Bolton et al. (2008) modelled 63 SLACSlenses and found that shears range from 0 to 0.27, but the me-dian is only 0.05. The large shear value in B0128 could suggestthat shear subsumes in it other complexities of the mass distribu-tion, such as those suggested by Gomer & Williams (2018), andillustrateed in their Figure 14.A similar case like B0128 is B1933 + + Acknowledgements.
The authors would like to thank Jori Liesenborgs for kindlyagreeing to be the mediator of the unblinding process. JW gratefully acknowl-edges the support by the Deutsche Forschungsgemeinschaft (DFG) WA3547 / References
Biggs, A. D., Browne, I. W. A., Jackson, N. J., et al. 2004, MNRAS, 350, 949Bolton, A. S., Burles, S., Koopmans, L. V. E., et al. 2008, ApJ, 682, 964Cohn, J. D., Kochanek, C. S., McLeod, B. A., & Keeton, C. R. 2001, ApJ, 554,1216Gilman, D., Birrer, S., Treu, T., Nierenberg, A., & Benson, A. 2019, MNRAS,1618Gomer, M. R. & Williams, L. L. R. 2018, MNRAS, 475, 1987Hsueh, J.-W., Despali, G., Vegetti, S., et al. 2018, Monthly Notices of the RoyalAstronomical Society, 475, 2438Keeton, C. 2004, gravlens 1.06 Software for Gravitational Lensing, 9th edn.,uRL: http: // / keeton / gravlens / manual.pdfKeeton, C. R. 2001, ArXiv Astrophysics e-prints [ astro-ph/0102341 ]Lagattuta, D. J., Auger, M. W., & Fassnacht, C. D. 2010, ApJ, 716, L185McKean, J. P., Koopmans, L. V. E., Browne, I. W. A., et al. 2004, MNRAS, 350,167Nightingale, J. W., Massey, R. J., Harvey, D. R., et al. 2019, arXiv e-prints[ arXiv:1901.07801 ]Norbury, M. A. 2002, PhD thesis, The University of Manchester (United King-domPetters, A. O., Levine, H., & Wambsganss, J. 2001, Singularity Theory and Grav-itational Lensing, Progress in Mathematical Physics, Volume 21 (Birkhäuser)Phillips, P. M., Norbury, M. A., Koopmans, L. V. E., et al. 2000, MNRAS, 319,L7Rusin, D., Kochanek, C. S., Norbury, M., et al. 2001, ApJ, 557, 594 Saha, P. & Williams, L. L. R. 2004, AJ, 127, 2604Schneider, P., Ehlers, J., & Falco, E. E. 1992, Gravitational Lenses, Astronomyand Astrophysics Library (New York: Springer)Sluse, D., Chantry, V., Magain, P., Courbin, F., & Meylan, G. 2012, A&A, 538,A99Suyu, S. H., Hensel, S. W., McKean, J. P., et al. 2012, ApJ, 750, 10Tessore, N. 2017, A&A, 597, L1Wagner, J. 2018, A&A, 620, A86Wagner, J. 2019, MNRAS, 487, 4492Wagner, J. 2019, Universe, 5Wagner, J., Liesenborgs, J., & Tessore, N. 2018, A&A, 612, A17Wagner, J. & Tessore, N. 2018, A&A, 613, A6Williams, L. L. R. & Liesenborgs, J. 2019, MNRAS, 482, 5666Woldesenbet, A. G. & Williams, L. L. R. 2012, MNRAS, 420, 2944Woldesenbet, A. G. & Williams, L. L. R. 2015, MNRAS, 454, 862Xu, D., Sluse, D., Gao, L., et al. 2015, MNRAS, 447, 3189 Appendix A: Derivation of the reference pointpositions on the subcomponent-scale
Appendix A.1: Derivation of coordinate positions
Fig. A.1: Determining the reference points from an ellipticalGaussian fitted to the subcomponents: the centre of light is givenby the relative coordinates ∆ α and ∆ δ in Table 2, here denotedby ( α , δ ). For images A and C with the same parity, we use thepositions of the semi-major and semi-minor axes as denoted by( α , δ ) and ( α , δ ). Since image D has opposite parity, we em-ploy ( α , δ ) and ( α , δ ) as reference point positions in additionto ( α , δ ).Fig. A.1 depicts the quantities measured in the elliptical Gaus-sians fitted to the subcomponents: the length of the semi-majoraxis a , the axis ratio between the semi-minor and the semi-majoraxis r , and the position angle θ measured with respect to thenorth axis. Denoting the centre of light of the Gaussian at therelative coordinates ( ∆ α, ∆ δ ) from a global reference point (seeTable 2) as ( α , δ ), we obtain the four end points at the axes ofthe Gaussian by the following trigonometric relations: α = α + a sin( θ ) , δ = δ + a cos( θ ) , (A.1) α = α + ra cos( θ ) , δ = δ − ra sin( θ ) , (A.2) α = α − a sin( θ ) , δ = δ − a cos( θ ) , (A.3) α = α − ra cos( θ ) , δ = δ + ra sin( θ ) . (A.4)To estimate the uncertainty of the ( α i , δ i ), i = , ...,
4, we as-sume that the uncertainties in α , δ , r , and a were uncorrelated.Due to the fitting procedure, this is not the case. Yet, it yields theorder of magnitude to which the reference points can be deter-mined. Given the uncertainties in Table 2, we find uncertaintieson the order of 0.1 to 0.2 mas for the subcomponents in Table 2. Article number, page 13 of 15 & A proofs: manuscript no. aa_unblinded_v6
Table A.1: Configurations of reference points in the subcompo-nents i = , A , C , and D that should all yield thesame local lens properties.Conf. A i C i D i ( α , δ ) ( α , δ ) ( α , δ )0 ( α , δ ) ( α , δ ) ( α , δ )( α , δ ) ( α , δ ) ( α , δ )( α , δ ) ( α , δ ) ( α , δ )1 ( α , δ ) ( α , δ ) ( α , δ )( α , δ ) ( α , δ ) ( α , δ )( α , δ ) ( α , δ ) ( α , δ )2 ( α , δ ) ( α , δ ) ( α , δ )( α , δ ) ( α , δ ) ( α , δ )( α , δ ) ( α , δ ) ( α , δ )3 ( α , δ ) ( α , δ ) ( α , δ )( α , δ ) ( α , δ ) ( α , δ ) Appendix A.2: Impact of the uncertainties in the positions onthe local lens properties
When mapping one elliptically Gaussian subcomponent to theone of another multiple image, the mapping should be indepen-dent of the reference points used. Only the relative parity be-tween the multiple images must be obeyed. Thus, for the caseof B0128, the configurations of reference points as listed in Ta-ble A.1 should all yield the same local lens properties. Determin-ing the reference points from the semi-major and semi-minoraxes according to Appendix A.1 and calculating the local lensproperties for all configurations listed in Table A.1, we corro-bate this assumption.Without loss of generality, we use configuration 0 of Ta-ble A.1 for all analyses described in Section 4.2.3.
Appendix B: Local lens properties forsystematically interchanged subcomponentlabelsAppendix C: Local lens properties for thefour-image configuration
ABC D
Article number, page 14 of 15enny Wagner and Liliya L. R. Williams: Local lens properties of B0128 + Table B.1: Local lens properties as obtained using the three subcomponents in images A , C , and D with the matching across theimages as listed in Table 4.Lens prop. Conf. 1 Conf. 2 Conf. 3 J A f A g A , -0.63 -0.63 0.05 -0.61 -0.62 0.05 -0.51 -0.51 0.02 g A , J C -0.20 -0.19 0.06 0.20 0.19 0.04 -0.20 -0.19 0.06 f C -0.77 -1.51 67.44 -0.61 -0.60 109.97 -2.29 -3.62 127.75 g C , g C , J D -0.17 -0.17 0.11 0.17 0.17 0.08 0.17 0.16 0.11 f D -0.60 -1.28 52.54 -0.20 -0.22 1.72 2.29 -13.26 1235.95 g D , g D , -0.60 -0.25 28.05 -1.11 -1.12 0.88 0.14 -7.68 622.89Table C.1: Local lens properties, their most likely value, mean, and the 1 − σ uncertainty bound (in columns 1, 2, and 3 of eachconfiguration, respectively), as obtained using the three subcomponents in images A , B , C , and D with the matching across theimages as listed in Table 4. Configuration 1 uses interchanged subcomponents 1 and 3 in image B , Configuration 2 uses a 3 masuncertainty in the positions of the subcomponents in image B , Configuration 3 increases the uncertainty in the subcomponentposition to 3 mas for all subcomponent positions of all images.Lens prop. Conf. 1 Conf. 2 Conf. 3 J A f A g A , -0.63 -0.63 0.04 -0.55 -0.56 0.04 0.03 -0.09 0.22 g A , J B f B -0.41 -0.23 2.85 -2.50 -3.92 93.65 -0.72 -13.23 48.15 g B , g B , J C f C -2.86 1.50 395.80 1.30 1.53 33.65 0.05 -0.17 2.62 g C , g C , -4.70 0.49 465.50 0.38 0.65 39.23 -1.00 -0.71 1.55 J D -0.15 -0.14 0.11 -0.17 -0.16 0.11 -0.05 -0.07 1.13 f D -0.41 -0.47 0.89 -330.07 -0.60 377.50 0.02 0.31 6.31 g D , g D , -0.73 -0.69 0.53 191.67 -0.52 227.50 -1.00 -0.92 2.14-0.73 -0.69 0.53 191.67 -0.52 227.50 -1.00 -0.92 2.14