Gravitational Lens Recovery with GLASS: Measuring the mass profile and shape of a lens

1.5 MB·

by  Jonathan P. Coles

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed1September2014 (MNLATEXstylefilev2.2)
Gravitational Lens Recovery with Glass: Measuring the
mass profile and shape of a lens
Jonathan P. Coles,1,3(cid:63)
Justin I. Read2,
4
1 and Prasenjit Saha3
0
1Exascale Research Computing Lab, Campus Teratec, 2 Rue de la Piquetterie, 91680 Bruyeres-le-Chatel, France
2
2Department of Physics, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom

g 3Physik-Institut, Universita¨t Zu¨rich, 190 Winterthurerstrasse, 8057, Zu¨rich, Switzerland
u
A
1September2014
9
2

] ABSTRACT
O We use a new non-parametric gravitational modelling tool – Glass – to determine
C what quality of data (strong lensing, stellar kinematics, and/or stellar masses) are
requiredtomeasurethecircularlyaveragedmassprofileofalensanditsshape.Glass
.
h uses an under-constrained adaptive grid of mass pixels to model the lens, searching
p throughthousandsofmodelstomarginaliseovermodeluncertainties.Ourkeyfindings
-
o are as follows: (i) for pure lens data, multiple sources with wide redshift separation
r give the strongest constraints as this breaks the well-known mass-sheet or steepness
t degeneracy; (ii) a single quad with time delays also performs well, giving a good
s
a recovery of both the mass profile and its shape; (iii) stellar masses – for lenses where
[ the stars dominate the central potential – can also break the steepness degeneracy,

givingarecoveryfordoublesalmostasgoodashavingaquadwithtimedelaydata,or
2
v multiplesourceredshifts;(iv)stellarkinematicsprovidearobustmeasureofthemass
0 at the half light radius of the stars r1/2 that can also break the steepness degeneracy
9 if the Einstein radius r =r ; and (v) if r r , then stellar kinematic data can
E 1/2 E 1/2
9 be used to probe the stell(cid:54)ar velocity anisotropy∼β – an interesting quantity in its own
7 right. Where information on the mass distribution from lensing and/or other probes
.
1 becomes redundant, this opens up the possibility of using strong lensing to constrain
0 cosmological models.
4
1 Key words: gravitational lensing: strong, methods: numerical, methods: statistical
:
v
i
X
r 1 INTRODUCTION mann2010;Amendolaetal.2013).However,extractingdark
a
matter properties or cosmological constraints from these
Strong gravitational lenses are rare. Since the discovery of
lensingdatawillrequiresophisticatedmodelling.Inparticu-
thefirstlensQ0957+561(Walshetal.1979),just∼400have
lar,withanunprecedenteddatasetimminent,itisprudent
beendiscoveredtodate1.However,thisnumberisexpected
to look again at systematic errors in the lens models to de-
to increase to several thousand over the next ten years as
termine what quality of data (in particular complementary
new surveys, both ground-based2,3 and space-based4 – to-
data from stellar/gas kinematics, lens time delays and/or
getherwithacommunityofcitizen-sciencevolunteersexam-
stellar mass constraints) are required to address problems
ining the image data for candidates5 – come online.
of interest. It is towards that goal that this present work is
Sincelensingdependsonlyongravity,stronglensesoffer
directed.
a unique window onto dark matter and cosmology (Bartel-
To see why lens modelling details are of crucial impor-
tance, let us recall the essential quantities that appear in
(cid:63) [email protected] lensing (see also §2 for a more detailed exposition). First
1 See,e.g.,http://masterlens.astro.utah.eduforacatalogue. we have the distances. Let DL, DS, DLS be the angular-
2 http://pan-starrs.ifa.hawaii.edu diameter distances to the lens, source, and from lens to
3 http://www.darkenergysurvey.org source; these are all proportional to c/H but have factors
4 http://www.euclid-ec.org 0
5 http://spacewarps.org
(cid:13)c 0000RAS
that depend on the particular choice of cosmology6. Typi- Being under-constrained, it is then necessary to explore
cally: model degeneracies rather than finding a single ‘best-fit’
solution. Free-form models are more commonly used with
c D
DL ≈zLH and DS ∼1. (1) cluster lenses (Saha et al. 2006; Saha & Read 2009; Merten
0 LS etal.2009;Sendraetal.2014),butcanbeusedwithgalaxy
where zL is the redshift of the lens. For multiple images, lensesaswell,wheretheirlessrestrictiveassumptionscanbe
the sky-projected density must exceed the critical lensing important. For example, in time-delay galaxy lenses, para-
density in some region: metric model measures of the Hubble parameter H have
0
c2 1kgm−2 historicallybeenattensionwithindependentmeasures(e.g.,
Σcrit = 4πGD ∼ z (2) Kochanek 2002a,b); these are resolved once the less restric-
L L
tive assumptions of free-form models are permitted (Read
where G is Newton’s gravitational constant. The angular
et al. 2007). Hybrid methods, using a mass grid on top
separationbetweenthelensedimagesisofordertheEinstein
a parametric model, have also been explored (e.g., Vegetti
radius θ , which is related to the mass by:
E et al. 2010).
θ ∼ RGDLS (3) (ii) Modelensembles:Modelensembles,exploringadi-
E D D verse range of possible mass distributions that nonetheless
L S
all fit the data, are a way of combating the non-uniqueness
where R = GM/c2 (with M the projected mass enclosed
G ofmodels.Suchensemblesarepossibleinparametricmodels
within θ ) is the gravitational radius. If the source is a
E (e.g., Bernstein & Fischer 1999; Jullo et al. 2010; Richard
quasarorotherwiserapidlyvariable,atimedelay∆tinthe
et al. 2014; Johnson et al. 2014; Coe et al. 2014), but are
variability will be present where:
morecommoninfree-formmodels,where–sincesuchmod-
∆t∼R /c (4) els are deliberately under-constrained – they become vital
G
(Williams&Saha2000;Saha&Read2009;Lubini&Coles
So in principle, one can not only measure the mass of
2012).
the lens, one can use the dependence on the cosmology-
(iii) Stellar kinematic constraints:Thiswasfirstsug-
dependentD factorsto extractthe cosmologicalmodeland
gested by Treu & Koopmans (2002) as a means to break
all its parameters. Zwicky (1937) drew attention to the for-
lensingdegeneracies.Theideaisthatstellarkinematicscan
mer, and Refsdal (1964, 1966) pointed out the latter, all
provide an independent estimate of the Einstein radius, via
long before lenses were discovered. The difficulty with ac-
the virial theorem:
tually doing this, however, became apparent soon after the
(cid:104)v2 (cid:105) θ D
discoveryofthefirstlensbyWalshetal.(1979).Inthefirst los ≈ E S (5)
everpaperonlensmodelling,Youngetal.(1981)foundthat c2 6πDLS
manyplausiblemassdistributionscouldreproducethedata. where (cid:104)v2 (cid:105) is the line of sight stellar velocity dispersion,
los
Young et al. (1981) were remarkably prescient about and the above relation becomes exact for isothermal lenses.
the subsequent development of lens modelling. First, they This can then be used to probe cosmological parameters if
introduced the technique of choosing a parametric form for lensesareknowntobeisothermal(e.g.,Collettetal.2012);
the lensing mass and then fitting for the parameters, which or to break the steepness degeneracy in the more general
is still the most common strategy (see for example Keeton situation (see §2.3). The technique has since been applied
2010; Kneib & Natarajan 2011). Second, they pointed out to many lenses (e.g., Koopmans et al. 2006; Bolton et al.
the non-uniqueness of lens models – lensing degeneracies. 2008).Goingfurther,theuseoftwo-dimensionalkinematics
Third, they suggested combining lensing data with stellar (Barnab`e et al. 2011) is especially interesting.
kinematicsandX-rays,toreducetheeffectofthedegenera- (iv) Stellar mass constraints: The stellar mass in a
cies. Later work, as well as following up these suggestions, lens can be inferred from photometry and compared with
has introduced some further new ideas. Five of these are the total mass (e.g., Keeton et al. 1998; Kochanek et al.
important for the present work: 2000; Rusin et al. 2003; Ferreras et al. 2005, 2008; Leier
et al. 2011). Since the inferred stellar mass depends on the
(i) Free-form modelling: In ‘free-form’ or non-
assumedIMF,lensesinwhichstellarmassdominatescanbe
parametric modelling, thereis no specified parametric form
used to derive upper bounds on the stellar M/L (Ferreras
forthemassdistribution.Therearestillassumptions(orpri-
etal.2010).LowerboundsonstellarM/Lhavealsorecently
ors) on the mass distribution, such as smoothness or being
been claimed by fitting ΛCDM semi-analytic models to the
centrally concentrated (Saha & Williams 1997; Diego et al.
tilt of the fundamental plane (Dutton et al. 2013).
2005;Mertenetal.2009;Coeetal.2010)butthesearemuch
(v) Testing modelling strategies:Usingmockdatato
lessrestrictivethanparametricforms.Aparticularlyelegant
see how well a given model can recover simulated lenses
priorisimplementedbyLiesenborgsetal.(2006),requiring
is increasingly being recognised as essential. Simple blind
that the mass distribution to be non-negative and no extra
testshaveappearedinearlierwork(forexample,Figure2in
images allowed. To be concrete, we define from here on:
Williams&Saha2000),butmorerecently,testsagainstdy-
Non-parametric, or ‘free-form’ ≡ more parameters than data namicallysimulatedgalaxiesorclustersarefavoured(Read
constraints (i.e. deliberately under-constrained) et al. 2007; Liesenborgs et al. 2007; Merten et al. 2009;
Barnab`e et al. 2009; Coe et al. 2010).
6 Here, c is the speed of light in vacuo and H0 is the Hubble There are three further key modelling ideas in the lit-
parameter. erature that we will not touch upon in this present work:
to use X-ray intensity and temperature profiles as a mass
constraint (e.g., Newman et al. 2013); and to model mul- The lens equation:
tiple lenses simultaneously, with one or more cosmological
D
parametersvariablebutsharedbetweenthelenses.Thislat- β=θ− LSα(θ)
D
ter strategy has been used to constrain H0 from time delay 4G (cid:90)S (θ−θ(cid:48)) (6)
lenses(Sahaetal.2006;Coles2008;Paraficz&Hjorth2010) α(θ)= Σ(θ(cid:48)) d2θ(cid:48)
c2D |θ−θ(cid:48)|2
and recently the cosmological parameters Ω as well (Sereno L
& Paraficz 2014). Third, it is in principle possible to esti- maps an observed image position θ to a source position β.
mate the Ω parameters even from a single lens, if there are Using the thin lens approximation, the lens can be thought
lensedsourcesatmultipleredshifts(Lubinietal.2014)orby of as a projected surface density Σ which diverts the path
using additional priors (Jullo et al. 2010; Suyu et al. 2014). of a photon instantaneously through the bending angle α.
In this paper, we introduce a new non-parametric lens TheD factors,asintheprevioussection,areangulardiam-
modellingframework–Glass(GravitationalLensingAnal- eter distances, which depend on the cosmological density-
ysiS Software). This shares some aspects with an earlier parametersΩ,theredshiftsz ,z ofthelensandthesource,
L S
code PixeLens (Saha & Williams 2004; Coles 2008). How- and the Hubble parameter H , thus
0
egevrraoelru,nkGedyluawpsasy–s–s:iwgnhiifichcacnotnlytaiimnsparollvnesewupcoondePwixrietLteennsfroinmstehve- DLS = Hc011++zzLS (cid:90)zLzS (cid:112)Ωm(1+dzz)3+ΩΛ (7)
andD ≡D ,D ≡D .Onewaytounderstandthelens
(i) At the heart of Glass is a new uniform sampling al- L 0,L S 0,S
equation is via Fermat’s principle. We can think of light as
gorithmforhighdimensionalspaces(Lubini&Coles2012).
travelling only along extremum paths where lensed images
This allows for large ensembles of > 10,000 models to be
occur.Suchpathsoccurattheextremaofthephotonarrival
efficiently generated.
time t(θ) that depends on the geometric path the photon
(ii) Glassprovidesamodularframeworkthatallowsnew
takesandthegeneralrelativisticgravitationaltimedilation
priors to be added and modified easily.
due to a thin lens at redshift z :
(iii) The basis functions approximating a model can be L
easily changed (in this paper, we assume pixels as in Pixe- ct(θ) = 1|θ−β|2· DS
Lens). (1+zL)DL 2 DLS (8)
(iv) With so many models in the final ensemble, we can 4GD (cid:90)
− L Σ(θ(cid:48))ln|θ−θ(cid:48)|d2θ(cid:48)
afford to apply non-linear constraints (for example stellar c2
kinematicdata;ortheremovalofmodelswithspuriousextra
Wecansimplifytheaboveequationbyintroducingadimen-
images) to accept/reject models in a post-processing step.
sionless time τ and density κ:
(v) Thecentralregionofthemassmapcanhaveahigher
resolution to more efficiently capture steep models. ct(θ) Σ(θ)
τ(θ)= ; κ(θ)≡ (9)
(vi) Stellar density can be used as an additional con- (1+z )D Σ
L L crit
straint on the models.
and hence rewrite Eq. (8) as:
(vii) Pointorextendedmassobjectscanbeplacedinthe
field. τ(θ)= 1|θ−β|2· DS − 1 (cid:90) κ(θ(cid:48))ln|θ−θ(cid:48)|d2θ(cid:48) (10)
2 D π
As a first application, we use Glass on mock data to de- LS
termine which combination of lensing, stellar mass and/or Thescaledarrivaltimeτ islikeasolidangle.Itisoforderthe
stellar kinematic constraints best constrain the projected area(insteradians)ofthefulllensingsystem.Theexpression
massprofileandshapeofagravitationallens.Wewillapply |θ−β|2 is of order the image-separation squared, and the
Glass to real lens data in a series of forthcoming papers. othertermsareofsimilarsize.Forthisreason,isconvenient
This paper is organised as follows. In §3, we describe to measure τ in arcsec2.
theGlasscode.In§2,wereviewthekeyelementsoflensing Lensing observations provide information only at θ
theory,stellarpopulationsynthesis,andstellardynamicswe wherethereareimages.Hence,thearrival-timesurfaceτ(θ)
willneed.In§4,wedescribeourmockdata.In§5,wepresent isnotitselfobservable.Itsusefulnessliesinthatobservables
ourresultsfromapplyingGlasstothesemockdata.Finally, canbederivedfromit.Animageobservedatθ1impliesthat
in §6 we present our conclusions. ∇τ(θ1)=0.Ameasurementoftimedelaysbetweenimages
at θ and θ implies that t(θ )−t(θ ) is known. Interest-
1 2 1 2
ingly,boththesetypesofobservationsgiveconstraintsthat
are linear in κ and β.
2 THEORY
The rather complicated dependence of lensing observ-
2.1 Lensing essentials ablesonthemassdistributionκ(θ)hasanimportantconse-
quence:verydifferentmassdistributionscanresultinsimilar
In the following summary, we follow Blandford & Narayan
observables.Thisisthephenomenonoflensingdegeneracies.
(1986) with some differences in notation, in particular
While the non-uniqueness of lens models noted by Young
putting back the speed of light c and the gravitational con-
et al. (1981) already hinted at degeneracies, their existence
stant G.
was first derived by Falco et al. (1985). The most impor-
tant is the so-called mass-sheet degeneracy, which is that
image positions remain invariant if τ(θ) is multiplied by an
arbitraryconstant.Thiscorrespondstorescalingthesurface
densityattheimagesκ(θ).Infactthereareinfinitelymany
degeneracies(Saha2000)becauseanytransformationofthe of the stars; ν(r) is the three dimensional stellar density;
arrival-time surface away from the images has no effect on σ (r) are the radial and tangential velocity dispersions,
r,t
the lensing observables. In particular, there are degenera- respectively; β(r) = 1−σ2/2σ2 = const. is the velocity
t r
ciesthatinvolvetheshapeofthemassdistribution(Saha& anisotropy(hereassumedtobeconstant,andnottobecon-
Williams 2006; Schneider & Sluse 2014). Degeneracies tend fused with β(θ) from lensing); G is Newton’s gravitational
to be suppressed if there are sources at very different red- constant;andM(r)isthemassprofilethatwewouldliketo
shifts or ‘redshift contrast’ (AbdelSalam et al. 1998; Saha measure. By convention, we always write R for a projected
& Read 2009), because the presence of different factors of radius, and r for a 3D radius.
D /D intheimageplanemakesitmoredifficulttochange ItisimmediatelyclearfromEq.(12)that,evenassum-
S LS
the mass distribution and the arrival-time surface without ing spherical symmetry, we have a degeneracy between the
affecting the lensing observables. But degeneracies are still enclosedmassprofileM(r)andthevelocityanisotropyβ(r).
present with multiple source redshifts (Liesenborgs et al. This can be understood intuitively since β(r) measures the
2008; Schneider 2014). relativeimportanceofradialversuscircularorbitsandisin-
trinsically difficult to constrain given only one component
of the velocity vector for each star. Nonetheless, β(r) can
2.2 Stellar populations
be constrained given sufficiently many stars, since radial
For many galaxy lenses, the gravitational potential in the Doppler velocities sample eccentric orbits as r → 0 and
inner region is dominated by the stellar mass. Stellar mass tangential orbits as r → ∞ (e.g., Wilkinson et al. 2002).
canbeestimatedbycombiningphotometryandcolourswith ItcanalsobeestimatedifanindependentmeasureofM(r)
modelsofthestellarpopulations.Suchestimatesarereason- is available – for example coming from strong lensing.
ablyrobust,evenifthestar-formationhistoryisveryuncer- While M(r) is difficult to measure from stellar kine-
tain: given a stellar-population model (such as Bruzual & matics alone, the mass within the half light radius is ro-
Charlot2003)andaninitialmassfunction(IMF),thestellar bustly recovered (e.g., Walker et al. 2009; Wolf et al. 2010;
mass can be inferred to 0.1 to 0.2 dex using just two pho- Agnello & Evans 2012) since stellar systems in dynamic
tometric bands (see, e.g., Figure 1 in Ferreras et al. 2008). quasi-equilibriumobeythevirialtheorem(equation5).This
By comparing the lensing-mass and stellar-mass profiles in meansthatstellarkinematicscanbreakthesteepnessdegen-
elliptical galaxies, it is possible to extract the radial depen- eracyifr1/2 (cid:54)=rE,whererE =DLθEisthephysicalEinstein
denceofthebaryonicvsdark-matterfraction(Ferrerasetal. radius. We test this expectation in §5.
2005, 2008; Leier et al. 2011). We describe our numerical solution of Eq. (12) in §3.7
The major uncertainty at present in the stellar mass and present tests applied to mock data in §5.
is probably the IMF. In the lensing galaxy of the Einstein
Cross, the IMF cannot be much more bottom-heavy than
Chabrier (2003), because otherwise the stellar mass would 3 NUMERICAL METHODS
exceedthelensingmassFerrerasetal.(2010).Moremassive
3.1 A new lens modelling framework: Glass
galaxies, however, do appear to have more of their stellar
massinlow-massstars.Thisisindicatedbymolecularspec- Glass is the Gravitational Lensing AnalysiS Software. It
tralfeaturescharacteristicoflowmassstars(Cenarroetal. extends and develops some of the concepts from the free
2004;Conroy&vanDokkum2012;Ferrerasetal.2013).The formmodellingtoolPixeLens(Saha&Williams2004;Coles
Chabrier (2003) IMF would, however, still provide a robust 2008), but with all new code. The most compute intensive
lowerlimitonthestellarmassandhence,alsoalimitonthe portion was written in C but Python was chosen because
total mass. Accordingly, Glass allows a constraint of the of its flexibility as a language and for its large scientific li-
form brary support. The flexibility allows Glass to have quite
sophisticated behavior while at the same time simplifying
M(θ)≥M (θ) (11)
stel the user experience and reducing the overall development
on the total mass. time. One of the striking features is that the input file to
GlassisitselfaPythonprogram.UnderstandingPythonis
not necessary for the most basic use, but this allows a user
2.3 Stellar kinematics
tobuildcomplexanalysisofamodeldirectlyintotheinput
Another useful constraint follows from the velocity of stars file. Glass may furthermore be used as an external library
within the lensing galaxy. Assuming spherical symmetry, to other Python programs. The software is freely available
stars obey the projected Jeans equations (e.g., Binney & for download or from the first author.7
Tremaine 2008): The key scientific and technical improvements are:
(i) A new uniform sampling algorithm for high dimen-
2 (cid:90) ∞ (cid:18) R2(cid:19) νσ2r sional spaces.
σ2(R)= dr 1−β √ r ; (12)
p I(R) r2 r2−R2
R AttheheartofGlassliesanewalgorithmforsamplingthe
highdimensionallinearspacethatrepresentsthemodelling
r−2β (cid:90) ∞ GM(r(cid:48))
σ2(r)= r(cid:48)2βν dr(cid:48) (13) solution space. This algorithm was described and tested in
r ν r(cid:48)2
r
where σ is the projected velocity dispersion of the stars as
p 7 http://www.jpcoles.com
afunctionofprojectedradiusR;I(R)isthesurfacedensity
Lubini & Coles (2012); it is multi-threaded allowing it to close to the main lens. The substructure may have only a
run efficiently on many-cored machines. small effect if the lens is a single galaxy, but if the lens is a
group or cluster then a potential can be added for each of
(ii) A modular framework that allows new priors to be
the known member galaxies. A few standard functions are
added and modified easily.
alreadyincludedinGlassincludingthoseforapointmass,
Each prior is a simple function that adds linear constraints apowerlawdistribution,oranisothermal(aparticularcase
that operate on either a single lens object or the entire en- of the power law).
sembleofobjects.Glasscomeswithanumberofusefulpri-
ors (the default ones will be described in §3.3), but a user
3.2 Analysis Tools
canwritetheirowndirectlyintheinputfile,orbymodifying
the source code. Glass is not only a modeling tool but also an analysis en-
gine. Glass provides many functions for viewing and ma-
(iii) The basis functions approximating a model can be
nipulatingthecomputedmodels.Thesefunctionscaneither
changed.
becalledfromaprogramwrittenbytheuserorbyusingthe
Glass currently describes the lens mass as a collection of program viewstate.py included with Glass. There is also
pixels,butthecodehasbeendesignedtosupportalternative a tool, lenspick.py for creating a lens, either analytically
methods. In particular, there are future plans to develop a or from an N-body simulation file. To load the simulation
module using Bessel functions. This will require a new set data, Glass relies on the Pynbody library (Pontzen et al.
of priors that operate on these functions. 2013)andcanthusloadanyfilesupportedbythatpackage.
(iv) Non-linear constraints can be imposed in an auto-
mated post-processing step. 3.3 Pixelated models
OnceGlasshasgeneratedanensembleofmodelsgiventhe Forthispaper,wewillrestrictourselvestousingapixelated
linear constraints, any number of post processing functions basissetasusedbyPixeLens(Saha&Williams2004;Coles
can be applied. Not only can these functions be used to 2008),butnotethatitisstraightforwardtoaddotherbasis
derive new quantities from the mass models, they can also functionexpansionstoGlass.Thealgorithmforgenerating
beusedasafiltertoacceptorrejectamodelbasedonsome models in Glass samples a convex polytope in a high di-
non-linear constraint. For example, we can reject models mensional space whose interior points satisfy both the lens
that have spurious extra images (§3.6), or models that do equation and other physically motivated linear priors (Lu-
notmatchstellarkinematicconstraints(§3.7).Theplotting bini & Coles 2012). A limitation of our sampling strategy
functions within Glass will correctly display models that is that only linear constraints may be applied when build-
have been accepted or rejected. ingthemodelensemble;however,non-linearconstraintscan
beappliedinpost-processing(see§3.6and§3.7).Wethere-
(v) The central region can have a higher resolution to
foreformulateallofourequationsasequationslinearinthe
capture steep models.
unknowns. We describe the density distribution κ as a set
With the default basis set of pixels, the mass distribution of discrete grid cells or pixels κ and rewrite the potential
i
of the lens is described by a uniform grid. However, in the (Eq. A4) as:
centralregionofalensinggalaxywherethemassprofilemay
(cid:88)
rise steeply, the center pixel uses a higher resolution. This ψ(θ)= κnQn(θ) (14)
allows the density to increase smoothly but still allow for n
a large degree of freedom within the inner region without where the sum runs over all the pixels and Q is the inte-
n
allowing the density to be arbitrarily high. gral of the logarithm over pixel n. The exact form for Q is
described in Appendix B. We can find the discretized lens
(vi) Stellar density can be used as an additional con-
equation by simply taking the gradient of the above equa-
straint.
tions.
Themassininnerregionsofgalaxiesisoftendominatedby The pixels only cover a finite circular area with physi-
the stellar component which one can estimate using stan- cal radius R and pixel radius R with the central cell
map pix
dard mass-to-light models. This data can be added to the centered on the lensing galaxy. To account for any global
potentialasdescribedlaterin§5.3.Byusingthestellarmass shearingoutsidethisregionfrom,e.g.,aneighboringgalaxy,
onecanplacealowerboundonthemassandhelpconstrain we also add to Eq. (14) two shearing terms:
the inner most mass profile.
γ (θ2−θ2)+2γ θ θ . (15)
1 x y 2 x y
(vii) Pointorextendedmassobjectscanbeplacedinthe
We can continue adding terms to account for other poten-
field.
tials.Forinstance,wemaywanttoimposeabasepotential
A shear term can be added to the potential, as shown later over the field, or add potentials from the presence of other
in Eq. (15), to account for mass external to the modelled galaxies in the field. Glass already includes potentials for
region.Thisisusefultocapturethegrosseffectsofadistant apointmassoranexponentialform,butcustompotentials
neighbour,sincethereisadegeneracybetweentheellipticity are straightforward to add and can be included directly in
of a lens and its shear field (the greater the allowed shear, theinputfile.Ifthestellardensityκ hasbeenestimatedwe
s
the more circular the lens may be). Glass also allows fur- canusethisasalowerboundwherethestellarpotentialisa
ther analytic potential components to be included. These knownconstantoftheformEq.(14),e.g.,κ =κ +κ
n dm,n s,n
can be used to model substructure or multiple neighbours for a two-component model.
3.3.1 Priors 3.4 Building the model ensemble
The lens equation and the arrival times alone are typically In the simplest form, a single model for a lens is a tuple
notenoughtoformaclosedvolumeinthesolutionspace.We M =(κ,β,γ ,γ ). Asingle model represents asingle point
1 2
thereforerequireadditionallinearconstraints–priors.Some in the solution space polytope. Using the MCMC sampling
oftheseare‘physical’inthesensethattheyareunarguable– strategy described in Lubini & Coles (2012) we uniformly
forexampledemandingthatthemassdensityiseverywhere sample this space. Collectively, the sampled models are re-
positive;othersaremoresubjective,forexampledemanding ferred to as an ensemble E ={M }, where we usually gen-
i
that the mass map is smooth over some region. Such ‘regu- erate |E|∼1000 models. One can choose to further process
larisation’ priors may be switched off for all or some of the thesemodelstoimposepriorsthatmaybedifficulttoenforce
mass map if the data are sufficiently constraining. during the modeling process. For instance, non-linear con-
ThepriorsbuiltintoGlassaresimilartothoseusedin straints,orsimplyfilteringofmodelsthatdonotmeetsome
PixeLens(Coles2008).Thephysicalpriorsarealwaysused criteria can be excluded, or weighted against as discussed
bydefault;theregularisationpriorsareusedsparingly–i.e. previously.Inthispaper,wedonotexcludeanymodelsand
only if the data are not sufficiently constraining to obtain treat all models as equally likely.
sensible solutions without them: The time to generate the model ensemble is mostly a
function of the size of the parameter space. The MCMC
Physical priors
algorithmhasa“warm-up”phasewhereitestimatesthesize
(i) The density must be non-negative everywhere. andshapeofeachdimensioninthesolutionspace.Oncethis
(ii) Image parity is enforced. has been completed, the models are sampled very quickly.
Infact,thereislittledifferencebetweengenerating1,000or
Regularisation priors 10,000 models, although we find little statistical difference
after1,000models.Forthemocklenses,thetypical“warm-
(i) The local gradient everywhere must point within 45◦
up” time was about 4s, and the modelling time was 20s
of the center.
usingaparallelshared-memorymachinewith40cores.The
(ii) Theazimuthallyaverageddensityprofilemusthavea
abilitytorapidlygeneratesomanymodelsiswhatallowsus
slope everywhere ≤0.
tothenaccept/rejectmodelstoapplynon-linearconstraints
(iii) The density is inversion symmetric.
(see §3.6 and §3.7). This is a key advantage over our earlier
For typical lens data, the regularisation priors are very im- pixelated strong lens tool PixeLens.
portant for creating physically sensible solutions. Prior (i)
demands that the peak in the mass density is at the centre
3.5 Raytracing
ofthemassmap.Secondary‘plateaus’inthemassmapare
possible, but not secondary peaks. Note that this prior still Glass can also determine the position of images and time
successfully allows merging galaxy systems to be correctly delaysfromparticle-basedsimulationoutputgivenasource
captured, provided that the two galaxies are not equally position β. This is used to generate the lens configurations
dense in projection (see, for example the PixeLens model usedintheparameterstudy.Theparticlesarefirstprojected
of the merger system B1608 in Read et al. 2007); and for ontoaveryhighresolutiongridrepresentingthelensplane.
the successful detection of ‘meso-structure’ in strong lens- Thecentersθ ofeachofthegridcellsaremappedbackonto
i
ing galaxy clusters (Saha et al. 2007). Prior (ii) is arguably thesourceplaneusingEq.(6).Ifthelocationonthesource
a physical prior since a positive slope in the azimuthally plane β is within a user specified ε of β then θ is
i accept i
averaged density profile would be unstable (e.g., Binney & accepted and further refined using a root finding algorithm
Tremaine2008).Notethatthispriordoesnotprecludesuc- untilthedistancetoβisnearlyzero.Ifmultiplepointscon-
cessfulmodellingofmergersorsubstructureunlessthetotal vergetoanε ofeachotherthenonlyonepointistaken.
root
projected mass in substructure is comparable to the pro- Care must be taken that the grid resolution is high enough
jectedmassofthehostinanazimuthalannulus(Readetal. that the resulting image position error is below the equiva-
2007; Saha et al. 2007). Prior (iii) is only used for doubles lent observational error. Time delays are then calculated in
that ought to be inversion symmetric and quads where in- order of the arrival time at each image (Eq. A2).
version symmetry is clear from the image configuration.
Finally,weremindthereaderthatalloftheregularisa-
tionpriorscanbeswitchedofforchanged/improveddepend- 3.6 Removing models with extra images
ing on the data quality available. For clusters, substructure
While linear constraints are applied in Glass by the na-
can be explicitly modelled by adding analytic potentials at
tureofthesamplingalgorithm,non-linearconstraintsmust
the known locations of galaxies; furthermore the above pri-
be applied in post-processing. Models that are inconsistent
orscanberelaxedinregionsofthemassmapwherethedata
with such constraints must then be statistically discarded
areparticularlyconstraining(forexampleneartheimages).
via a likelihood analysis. An example of such a non-linear
We will apply Glass to a host of strong lensing clusters in
constraint is the spurious presence of unobserved images.
forthcomingwork,wherewewillexplicitlytesttheprioron
This ‘null-space’ prior was first proposed and explored by
mock data that has significant substructure.
Liesenborgs et al. (2006) and found to be extremely pow-
erful. We find that our gradient prior in Glass (see §3.3),
performs much of the same function as Liesenborgs et al.’s
null-space prior, but some models can still rarely turn up
spurious images. We reject these in a post-processing step,
where we sweep through the model ensemble applying the Galaxy γ(cid:63) M(cid:63) γDM MDM Rmap
ray tracing algorithm described in 3.5.
star1.0-dmCore 1 4 0.05 112.95 50kpc
star1.0-dmCusp 1 4 1 112 50kpc
star1.5-dmCore 1.5 21.5 0.16 112.84 50kpc
3.7 A post-processing module for stellar
star1.5-dmCusp 1.5 21.5 1 112 10kpc
kinematics
Table1.Profileparametersforthefourmockgalaxies.Thename
Similarly to the null-space constraint (§3.6), stellar kine-
indicates whether the galaxy is centrally dark matter or stellar
matic constraints constitute a non-linear prior on the mass dominated with a shallow or cuspy dark matter density profile.
map and must be applied in post-processing. We sweep Masses are in units of 1.8×1010M(cid:12). The scale lengths for all
through the model ensemble performing an Abel deprojec- lenses are (a(cid:63),aDM) = (2,20)kpc. Rmap is the 2D projected
tion to determine M(r) from the projected surface den- radius used to generate the lens configurations. In the case of
sity Σ(R) assuming spherical symmetry (e.g., Binney & star1.5-dmCusp, the profile is sufficiently steep that the profile
Tremaine 2008; Broadhurst & Barkana 2008): couldbetruncatedatRmap=10kpc.
(cid:90) π/2 (cid:20) 1
M(r) = Mp(<r)−4r2 Σ(x) cos2θ dark matter contribute equally to the total mass at a∗.
0 The dark matter scale length is fixed for all models at
(cid:18) (cid:19)(cid:21)
sinθ cosθ
− arctan dθ (16) aDM = 20kpc. These values were chosen to closely resem-
cos3θ sinθ
blethelensinggalaxyPG1115+080(Weymannetal.1980).
where We place the galaxy at a redshift of zL = 0.31 for lens-
(cid:90) r ing.Throughout,weassumeacosmologywhereH0−1 =13.7
Mp(<r)=2π RΣ(R)dR (17) Gyr,ΩM =0.28,andΩΛ =0.72.Thecriticallensingdensity
0 is κ ∼1.8×109M /kpc2.
crit (cid:12)
istheprojectedenclosedmassevaluatedat3Dradiusr;and Thegalaxiesweregeneratedasthreedimensionalparti-
x=r/cosθ. cle distributions as in Dehnen (2009). Each component fol-
This de-projection algorithmwas testedon triaxial fig- lows the profile:
ures in Saha et al. (2006). They found that for triaxiali-
M
tiestypicalofourcurrentcosmology,themethodworksex- ρ(r˜)= (3−γ)(r˜/a)−γ(1+r˜/a)γ−4 (18)
4πa3
tremely well unless the triaxial figure is projected directly
whereaisthecomponentscaleradiusmentionedinTable1;
alongthelineofsightsuchthatweseethegalaxyorgalaxy
r˜2 = (x/λ )2+(y/λ )2+(z/λ )2 is the ellipsoidal radius;
cluster ‘down the barrel’. Such a situation is unlikely, but 1 2 3
and the axis ratios are λ : λ : λ = 6 : 4 : 3. In the case
in any case avoidable since the resultant figure appears 1 2 3
wherethecentraldensityprofileindexγ isunity(andinthe
sphericalinprojection.Thisleadstotheseeminglycounter-
limit of spherical symmetry), this is the Hernquist profile
intuitiveresultthatthekinematicconstraints–thatrelyon
(Hernquist 1990). The four combinations of profile indices
the above de-projection – are most secure for systems that
are shown in Table 1.
do not appear spherical in projection (unless independent
InFigure1,weshowthe3Dradialdensity,the2Dpro-
datacanconfirmthethreedimensionalshapeisindeedvery
jected density, and the 2D enclosed mass for each galaxy.
round).
We use the deprojected mass to numerically solve
Eq. (12) for constant β(r), assuming either β(r) = 1 or
4.2 Lens configurations
β(r)=0atallradiitobracketthetwoextremumsituations.
Where the data are good enough, these two may be distin- For each of the four galaxies, we used the raytracing fea-
guished giving dynamical information about β(r). In more ture of Glass described in §3.5 to construct 6 basic lensing
typical situations, however, we seek to simply marginalise morphologies:
over the effect of β(r), using the stellar kinematics as a ro-
(i) one double and one extended double;
bust measure of M(r ) (see §2.3).
1/2 (ii) one quad and one extended quad;
(iii) two 2-source quads with varying redshift contrast.
The ‘extended’ configurations use multiple point sources at
4 THE MOCK DATA
the same redshift to simulate an extended source that will
We now present a study of four mock galaxies with known produces an arc-like image. Figure 2 shows the lens con-
analytic forms. These are used to verify that Glass is able figurations for the star1.5-dmCusp galaxy. The configura-
tocorrectlyrecoverthemassprofile,and–moreimportantly tions for the other galaxies are similar. The labels Z1, Z2,
–todeterminewhattypeandqualityofdatabestconstrain Z3withinthenamesrefertotheredshiftofthesources.We
the mass profile and shape of a lens. have chosen Z1=1.72, Z2=0.72, and Z3=0.51 so that the
radial distribution of the images is roughly equally spaced.
For all mocks, we do not apply any external shear field.
4.1 The triaxial N-body mock galaxies
Only the central image of the Z1 source is used to avoid
We generate four two-component mock galaxies, where the over-constraining the models, otherwise all the central im-
darkmatterandstellarprofilesareallowedtobebothsteep ageswouldfallwithinthecentralpixelandnosolutionexists
and shallow. The enclosed mass of the stars and dark mat- thatsatisfiesalllocationssimultaneouslyforonepixelvalue.
ter are both fixed to be M = 1.8 × 1010M at the Each of these configurations were modelled with and
∗,DM (cid:12)
stellar scale radius a = 2kpc, such that the stars and withouttimedelays;withandwithoutacentralimage;and
∗
109 1010 1012
108
3pc)107 109 R)1011
/k(cid:12)106 Σ <
(M105 M(
ρ104 star1.5-dmCore 108 1010
star1.5-dmCusp
103 star1.0-dmCusp
star1.0-dmCore
100 101 100 101 100 101
r(kpc) R(kpc) R(kpc)
Figure 1. Profilesofthefourmockgalaxiesshowingthestellar(dotted)anddarkmatter(dashed)componentsandthetotal(solid).
Left: The spherically averaged density. The stars in models star1.5-dmCore and star1.5-dmCusp contribute significantly to the
central potential. Middle: The radially averaged two-dimensional projected density. The critical lensing density at zL =0.31, κcrit ∼
1.8×109M(cid:12)/kpc2,ismarkedbythehorizontalline.Right:Theenclosedprojectedmass.
DoubleZ1a ExtendedDoubleZ1 ExtendedQuadZ1
1.0
1.0 1.0
0.5
0.5 0.5
0.0 0.0 0.0
0.5 0.5
− − 0.5
−
1.0 1.0
− − 1.0
−
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
− − − − − −
QuadZ1a ZContrastZ1Z2 ZContrastZ1Z3
1.0 1.0 1.0
0.5 0.5 0.5
0.0 0.0 0.0
0.5 0.5 0.5
− − −
1.0 1.0 1.0
− − −
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
− − − − − −
Figure2.Thelensconfigurationsforthesixtestcasesusingthestar1.5-dmCuspmockgalaxy.Theothermockgalaxiesproducesimilar
results.Here,thecentralimageisshown,althoughnotalltestsincludeit.Thenamingconventionindicatestheredshiftofthesources
withZ1=1.72,Z2=0.72,andZ3=0.51.ThecentralimageonlybelongstotheZ1sourcetoavoidover-constrainingthemodels(see§4.2
for further details). Small diamonds identify the location of the source(s) and images of the same shape share a common source. The
extended source examples have been constructed so that the images will form arclets. The maximum separation of the sources in the
sourceplaneis2.23kpcintheextendeddoubleand0.92kpcintheextendedquad.Greycirclesareavisualaidtohelpdetermineradial
separationbetweenimages.Theaxesareinarcseconds.
with and without the stellar mass as a lower bound, for a masswasradiallysymmetric(Priorvi).Forourmockdata,
totalof 48testcases. (The central image istypically highly this is known to be true; it is most often the case with real
demagnified.Forgalaxylensesitisverydifficulttofindsince galaxies,unlessthereisanobviousobservedasymmetry.(We
it lies along the sight line to the bright lensing galaxy; in explore the effect of switching off the symmetry prior in
clusters, however, such images have been seen – e.g., Inada Appendix C. For the quads, the difference is small; for the
et al. 2005). We assumed for all our tests that the lensing doubles–asexpected–theresultsaresignificantlydegraded
without this prior.) We use, by default, 8 pixels from the ofmockdatateststodeterminetheroleofdataversusprior
centre to the edge of the mass map; the central pixel was in strong lensing.
further refined into 5×5 pixels to capture any steep rise Figure6andFigure7showtheresultsforourfullmock
in the profile (two of the four mock galaxies have a steeply dataensemble.Eachsubplotcorrespondstoadifferentmock
rising inner profile). We demonstrate that our results are galaxy,asmarked.Weshowthefractionalerrorofthemass
robust to changing the grid resolution in Appendix D. distribution for each of the test configurations with (red)
In all cases – despite applying no external shear to the and without (black) stellar mass. In Figure 6 we define the
mocklenses–weallowabroadrangeofexternalshearinour error:
loernzsemroodshelearercionnsatlrluccatsieosn.s.ItGilsapsosssciobrlreectthlaytremtuorrenscoamsmplaelxl (cid:80)i(cid:12)(cid:12)(cid:12)M(i)−M(cid:99)(i)(cid:12)(cid:12)(cid:12)
f = (19)
shear fields present in real lensing galaxies could introduce R (cid:80)M(cid:99)(i)
further degeneracies beyond those discussed here. However,
anysuchshearfieldcan,atleastinprinciple,beconstrained basedonthemassM(i)ofeachpixelringiandthemassM(cid:99)
bydata(e.g.,combiningweaklensingconstraints,orassum- from the mock galaxy. In Figure 7 the error is defined over
ingthattheshearfieldcorrelateswithvisiblegalaxies–e.g., all the pixels θ:
Merten et al. 2009; Wong et al. 2011). (cid:12) (cid:12)
(cid:80)θ(cid:12)(cid:12)M(θ)−M(cid:99)(θ)(cid:12)(cid:12)
f = (20)
θ (cid:80)
M(cid:99)(θ)
θ
5 RESULTS
Since both error measurements consider the mass of each
5.1 Radial profile recovery pixel, we are implicitly weighting the recovered density by
the varying size of the pixels. The value f emphasises the
Figure 3 shows some example reconstructions of the radial R
error one would see from radial profiles, while f is useful
profile of our mock lenses. The left column shows the en- θ
as a measure of how well each individual pixel is recovered.
semble average arrival time surface with images marked as
For both f and f , we only consider mass up to one pixel
circles and the inferred source positions as diamonds. The R θ
length passed the outermost image, since there is no longer
centre column shows the radial density profile. The error
any lensing information beyond that point. This means we
bars cover a 1σ range around the median; the grey bands
typicallyuse8bins,linearlyspaced,ignoringtheoutermost
show the full ensemble range. The true density profile from
3bins.Thespacingchanges,however,attheborderbetween
the mock data is also plotted for comparison. The vertical
the high resolution region in the middle.
lines mark the radial position of the images. The right col-
The abundance of strong lensing data increases from
umnshowstheenclosedmass.Fromtoptobottom,therows
left to right within each plot. As a result, there is a gen-
correspond to an extended double for star1.5-dmCusp; an
eral trend for the reconstruction quality to increase (and
extended double with stellar mass constraints for star1.5-
therefore for f to decrease). When both time delays and a
dmCusp;aquadwithtimedelaydataforstar1.5-dmCusp;
centralimagearepresent(TD+central),thequalityishigh-
andaquadwithtimedelaysforstar1.0-dmCore.Figure5
est. A double is known to provide very little constraint on
shows an example 2D reconstruction for star1.5-dmCusp
the mass distribution. This is particularly evident in galax-
for a quad; we discuss shape recovery further in §5.2.
ies star1.0-dmCusp and star1.5-dmCusp where the mass
As expected, the accuracies and precisions are best in
profile is steepest and the reconstruction of the double is
the range of radii with lensed images where the most infor-
poorest.However,the addition ofan arcfrom theextended
mation about the lens is present. Even in the weakly con-
source is sufficient to correct this. Notice that, as in Fig-
strainedcaseoftheextendeddoublewheretheradialprofile
ure 3, the recovery for star1.5-dmCusp quickly saturates;
ispoor,thetrueenclosedmassM(<R)iswellrecoveredat
there is little improvement as the data improves beyond a
theimageradiiandourensemblealwaysencompassesit.We
single quad. This occurs because the Glass sample prior
haveverifiedthisisthecaseinallofourtests,althoughfor
in the absence of data favours steep models like star1.5-
brevity we have not included the plots here. In all cases,
dmCusp over shallower models like star1.0-dmCore (see
there is a dip in the profile at large R due to the cut off in
also Figure 3).
massinthelensingmap.Thisisoflittleimportance,though,
as there is no lensing information there.
Notice that the extended double (top row) gives the 5.2 Shape recovery
poorest constraints, as expected. Adding stellar mass (sec-
Figure 7already givesus important information abouthow
ond row) significantly improves the constraints, for this ex-
wellwecanrecovertheshapeofalens.Thetrendsarevery
amplewherethestarscontributesignificantlytothepoten-
similarto theradialprofilerecovery inFigure6,suggesting
tial. Moving to a quad with time delays gives constraints
thatiftheradialprofileiswell-recoveredthen,typically,the
almost as strong as the double with stellar mass, but note
shapeistoo.Anotableexceptionisforthestar1.5-dmCusp
that focussing only on the goodness of the fit can be mis-
modelswhereaddingstellarmassconstraintsaidstheshape
leading. In the third row of Figure 3, we obtain a better
recovery,butlittle-improvestheradialmassprofile.Avisual
recoverythaninthebottomrowforprecisely the same data
example of the shape recovery is given in Figure 5.
quality.ThisoccursbecausetheGlasspriorfavourssteeper
We can also more directly probe the recovery of the
models consistent with star1.5-dmCusp, but not star1.0-
shape of the mass distribution by considering the ratio of
dmCore.ItistheGlassprior,ratherthanthedatathatis
the major and minor axes λ ,λ of the inertia ellipse. If
drivingthegoodrecoveryforstar1.5-dmCuspinthisexam- 1 2
they are equal, the mass is distributed uniformly on the
ple. This emphasises the importance of using a wide range
2 star1.5-dmCusp ExtendedDoubleZ1 1011 DM
1013 N-bodyDM+Stars
1 1010 1012
c R)
arcse 0 Σ109 M(<1011
1 1010
− 108
2 109
− 107
2 1 0 1 2 100 101 100 101
− − arcsec R(kpc) R(kpc)
2 star1.5-dmCusp ExtendedDoubleZ1 1011 Total(DM+Stars)
1013 DM
Stars
N-bodyDM+Stars
1 1010 1012
c R)
arcse 0 Σ109 M(<1011
1 1010
− 108
2 109
− 107
2 1 0 1 2 100 101 100 101
− − arcsec R(kpc) R(kpc)
Figure 3. Two reconstructions of the mock galaxy star1.5-dmCusp for an extended double without stellar mass (Top) and with
stellarmass(Bottom).Notimedelayswereassumed.Theimprovedconstraintsonthemassdistributionwhenalowerboundisgiven
by the stellar mass is evident in the reduced range of allowable models. Left: The ensemble average arrival time surface with just the
iso-contoursforthesaddlepointsdrawn.Thecentraldiamondsshowthereconstructedsourcepositions.Middle:Thesurfacedensityof
thedarkmatter(DM;magenta);thestars(yellow);andthetotal(black).TheoriginalN-bodymassmodel(withstars)usedtocreate
the lens is shown in green. The vertical lines mark the radial positions of the images. The higher resolution feature of Glass has been
usedonthecentralpixelallowingthesteepprofiletobecaptured.Right:Thecumulativemass.Theerrorbarsonallplotsare1σ;the
greybandsshowthefullrangeofmodels.
projecteddisc.Themoredissimilartheyare,themoreellip- 5.3 Stellar mass
ticalthemassdistribution.Wedefinetheglobalmeasureof
Thestellarmassdistributiongivesalowerboundontheto-
lens shape as:
talmass.Wherethestarsdominatethecentralpotential,it
s≡λ /λ (21) canprovideapowerfulconstraintextratothestronglensing
1 2
data. We took the stellar mass directly from the generated
whereλ andλ aretheeigenvaluesofthe2Dinertiatensor:
1 2 galaxies and projected the particles onto the pixels. Glass
(cid:18) (cid:80) M(θ)θ2 −(cid:80) M(θ)θ θ (cid:19) also offers an option to interpolate any map of stellar mass
−(cid:80)θM(θ)θyθ (cid:80)θM(θ)θx2 y (22) (e.g., from an observation) onto the pixels. The linear con-
θ x y θ x
We always take λ to be the largest value. As with f and straint is added to Glass by writing κn = κdm,n+κs,n as
1 R the sum of the dark matter and stellar mass components
f , we only consider mass up to one radial position passed
θ in the potential (Eq. 14). Since each κ is just a constant
the outermost image and compute the fractional error as: s,n
we do not add new, separate equations for each pixel. Al-
f =|s−s|/s (23) though we assume a perfect recovery of the stellar mass
shape (cid:98) (cid:98)
with no error on the lower mass bound, it is straightfor-
where s is the shape of the mock galaxy. The distribution
(cid:98) ward to add errors as the stellar mass constraint remains
of f for each mock galaxy and each test case is shown
shape linear: κ = κ +(cid:15)κ , where (cid:15) ∼ 1 is an additional
n dm,n s,n
in Figure 8. Interestingly, for this global shape parameter
error parameter.
recoveryitappearsmoreimportanttohavetimedelaydata
Withthestellarmasslowerbound,thereisasignificant
and/or a central image (TD,TD+central) than to have a
improvementofthereconstructionqualityshowninFigure6
quad or multiple sources with wide redshift separation. In
and Figure 7 for the doubles in the steepest mock galaxies
all cases, the stellar mass little-aids the recovery, reflecting
(star1.0-dmCusp and star1.5-dmCusp). This is because
thefactthatsisheavilyweightedtowardstheshapeatthe
thesemodelsaredominatedbystarsintheinnerregion.By
edge of the mass map, rather than at the centre where the
contrast,theothertwogalaxies–wherethestarscontribute
stars may dominate the potential (see Eq. (22)).
negligibly to the potential – are largely unaffected.