Astrometric Perturbations in Substructure Lensing

0.46 MB·

by  Jacqueline Chen

Draftversion February5,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
ASTROMETRIC PERTURBATIONS IN SUBSTRUCTURE LENSING
Jacqueline Chen1, Eduardo Rozo2, Neal Dalal3, & James E. Taylor4
Draft versionFebruary 5, 2008
ABSTRACT
In recent years, gravitational lensing has been used as a means to detect substructure in galaxy-
sized halos, via anomalous flux ratios in quadruply-imaged lenses. In addition to causing anomalous
flux ratios, substructure may also perturb the positions of lensed images at observable levels. In this
paper,wenumericallyinvestigatethe scaleofsuchastrometricperturbationsusingrealisticmodels of
substructuredistributions. Substructure distributionsthatprojectclumps nearthe Einsteinradiusof
7 the lens result in perturbations that are the least degenerate with the best-fit smooth macromodel,
0 with residuals at the milliarcsecond scale. Degeneracies between the center of the lens potential
0
and astrometric perturbations suggest that milliarcsecond constraints on the center of the lensing
2
potentialboosttheobservedastrometricperturbationsbyanorderofmagnitudecomparedtoleaving

n the center of the lens as a free parameter. In addition, we discuss methods of substructure detection
a via astrometric perturbations that avoid full lens modeling in favor of local image observables and
J also discuss modeling of systems with luminous satellites to constrain the masses of those satellites.

1 Subject headings: cosmology: theory – dark matter – gravitationallensing

2
v 1. INTRODUCTION Another promising method of study is through the
9 gravitational lensing effects of substructure, referred to
The cold dark matter (CDM) paradigm predicts that
5 as substructure lensing. Much of the previous work
the dark matter (DM) halos hosting most galaxies con-
3 in this field has focused on the effects of substructure
tain a large number of low-mass, compact subhalos
6
on image magnifications and fluxes (Mao & Schneider
0 within their virialized regions. These subclumps, collec-
1998; Metcalf & Madau 2001; Dalal & Kochanek 2002;
6 tively referred to as substructure, may or may not con-
Kochanek & Dalal2004;Rozo et al.2006)orontimede-
0 tainluminousstellartracers. Sincethe observednumber
lays (Morgan et al. 2006). However, substructure can
/ ofdwarfgalaxysatellitesintheLocalGroupfallsshortof
h perturb the deflection angle,α, of lensed images as well.
the expected number of subhalos by more than an order
p Imagepositions in lensedsystems may not be subject to
of magnitude (Klypin et al. 1999b; Moore et al. 1999),
- the sameforegrounds– suchasdust absorption– asflux
o then either the CDM model is incorrect or dwarf galax-
anomalies. Further, since the astrometric perturbation,
r ies are biased tracers of DM in galaxy-scale halos (e.g.,
t δα, is a steeper function of subhalo mass than flux per-
s Spergel & Steinhardt 2000; Hannestad & Scherrer 2000;
a Hu et al. 2000; Bullock et al. 2000; Benson et al. 2002; turbations, it may provide a qualitatively distinct probe
: of substructure properties.
v Somerville 2002; Stoehr 2002; Nagai & Kravtsov 2005).
A study of astrometric perturbations is particularly
i If the CDM paradigmis correct, then the paucity of op-
X timely given the possibility of submilliarcsecond res-
ticalcounterpartsofsubstructuresleavesuswithfewav-
olution observations of strong lenses. For example,
r enues for detecting and investigating DM substructure.
a Biggs et al. (2004) have presented observations of a
One possibility for studying dark substructure
four image jet source system, CLASS B0128+437, in
around local galaxies is through its perturbative ef-
which milliarcseondastrometricperturbations may have
fect on kinematically cold systems like tidal streams
been detected. Previous studies of astrometric per-
(e.g., Mayer et al. 2002) or galactic disks (e.g.,
turbations by substructures have come to conflicting
Toth & Ostriker 1992; Benson et al. 2004). These tests
conclusions about their overall size and probability.
probe only the closest and most massive substructures,
Metcalf & Madau (2001) use lensing simulations of ran-
and are only applicable in very nearby galaxies. As a
dom realizations of substructure in regions near images
result,thismethodprovideslittleinformationtoaddress
to suggest that in order to change image positions by
fundamentalquestionssuchastheamplitudeofthemat-
a few tens of milliarcseconds (mas), there must be sub-
ter power spectrum on subgalactic scales or the detailed
properties of dark matter itself. clumps with masses & 108M⊙ in Milky Way sized ha-
los that are well aligned with the images they perturb.
1Argelander-Institut fu¨r Astronomie, Universita¨t Bonn, Auf ThisalignmentislikelytoberareinCDM,althoughthe
demHu¨gel71,D-53121Bonn;[email protected] probability would increase in systems where the source
2CCAPP Postdoctoral Fellow, Department of Physics, is elongated in a jet (Metcalf 2002). On the other hand,
The Ohio State University, 1040 Physics Research Build-
Chiba (2002) tests the size of astrometric perturbations
ing 191 West Woodruff Avenue Columbus, Ohio 43210-1117;
[email protected] in B1422+231 with a model of CDM subhalos as point
3CanadianInstituteforTheoreticalAstrophysics,Universityof masses and finds deflections of 10 to 20 mas, using sub-
Toronto, 60 St. George St., Toronto, Ontario, Canada M5S 3H8; clumps with masses greater than 2 108h−1M⊙. Ad-
[email protected] ∼ ×
4DepartmentofPhysicsandAstronomy,UniversityofWaterloo ditional studies of astrometric perturbations by single
200UniversityAvenueWest,Waterloo,Ontario,CanadaN2L3G1; perturbershavefocusedonthe detectionofsubstructure
[email protected] via distortion of a finite source (Inoue & Chiba 2003,
2
2005a,b). In addition, Pen & Mao (2005) studied the tems with substructure, finding the image positions in
effect of multiple lens planes on the rotation of lensed such systems, and fitting for the best-fit smooth model.
images.
Inthispaper,weestimatetheamplitudeofastrometric 2.1. Halo Model
perturbations produced by substructures using realistic We beginby specifying the macromodelparametersof
substructure models and test the feasibility of observ- thelenssystem–usingtypicallensparameters–andthe
ing such perturbations by comparison with the image background cosmology used to generate artificial lenses.
positions given by a best-fit lens macromodel. While In particular, we choose a lens redshift of z = 0.5, a
l
Metcalf & Madau(2001)suggestthatsingleclumpsneed source redshift of z =2.0, and a flat ΛCDM cosmology
s
to be well aligned with the images in order to produce with Ω = 0.3, Ω = 0.7, and h = 0.7. The halo of
m Λ
observableperturbations,measurableeffectsmaybepos- the lensing galaxy is modeled as a singular isothermal
sible in more general scenarios, given the collective ef- ellipsoid (SIE), the projected density profile of which is
fects of entire substructure distributions. In addition, given by
the kinds ofperturbations in these scenariosmay be dif- b
κ(ξ)= , (1)
ferent. For example, while the scenario of a nearby sub-
2ξ
halo that perturbs only a single image ensures that de-
generacies with best-fit smooth macromodels are small, where ξ is the projected radius, ξ2 = x2 + y2/q2 and
large numbers of more distant substructure may affect q = 0.9 is our fiducial value for the lens’s axis ratio.
multiple images and exhibit large degeneracies with the The particular values for the ellipticity and shear do
macromodel. In addition to using detailed lens model- not appear to be significant to the results of this pa-
ing, we discuss model-independent methods for identi- per. The length scale b corresponds to the Einstein ra-
fying substructures, proposing a method using systems dius of the lens for circularly symmetric profiles (q =1).
with multiply imaged jets. We take as our fiducial value b = 1.05′′, corresponding
Substructurelensingmayalsobeusedforcomparisons to the Einstein radius of a singular isothermal sphere
between luminous satellites and dark subhalos. For ex- of mass M = 1013M⊙ in our chosen cosmology and
ample, we canaddress one of the open questions regard- source and lens redshifts. Finally, we assume there is
ingCDMsubstructure: whyaresomesubhalosdarkand an external tidal shear of γ = 0.16 aligned with the
someluminous? Therehavebeenmanyproposedmecha- major axis of the halo (θγ = 0). Our particular choice
nismstoexplainwhichsubhaloshavestarsandwhichdo γ =0.16correspondstothebest-fitvaluefortheexternal
not. One possibility is that the efficiency of star forma- shear obtained by Bradaˇc et al. (2002) for the lens sys-
tion diminishes with decreasing halo mass, perhaps due tem B1422+231, a system known to exhibit anomalous
to photoionization squelching (Bullock et al. 2001). An- flux ratios.
other suggestion has been that luminous galaxies form
2.2. Substructure Models
only in the highest mass halos (M > 109M⊙), and over
time tidal stripping reduces subhalo masses to the low We employ four different subhalo catalogs to populate
values (M 108M⊙) inferred for the smallest local our fiducial halo with substructures. The catalogs de-
∼
dwarfs (Kravtsov et al. 2004b). Yet another possibility scribe the three-dimensional position in the parent halo
is that the low masses inferred for local dwarfs are in and the density profile of each subhalo. In order to
factmistaken,andinsteadaremuchlarger,M &109M⊙ use these catalogs in our halo model, the substructure
(Stoehr et al. 2002, 2003). Kazantzidis et al.(2004) find positions and density profiles must be projected along
that subhalos do not experience the significant mass re- an axis. The density profiles are projected by param-
distribution in their centers required to embed satellites eterizing the spherical density profiles in three dimen-
in massive subhalos, and studies stellar kinematics in sions, then using an approximate formula for the profile
thedwarfsalsodisputetheexistenceofmassivesubhalos in two dimensions. The following section describes the
(e.g., Wilkinson et al. 2004). Here, we propose directly substructure models in detail.
measuringthemassesassociatedwithluminoussatellites For two of these models, we use substructure catalogs
projectednear lensed images,the results of which would from a DM cluster simulation with Ω = 0.3, Ω =
m Λ
direcly test the Stoehr et al. (2002) hypothesis. 0.7, h=0.7,andσ =0.9. Theadaptive-refinement-tree
8
The layoutof the paper is as follows. We begin by de- (ART)codeisusedtorunthesimulation(Kravtsov et al.
scribing the host halo and substructure models we will 1997; Kravtsov 1999), using a 2563 uniform root grid
be employing in our analysis in 2. The statistical prop- covering the computational box of 80h−1 Mpc and res-
§
erties ofthe astrometricperturbationsin our models are imulating a cluster in the box with more particles and
presented in 3, and their degeneracy with macro-lens higher spatial resolution (Tasitsiomi et al. 2004). For
§
model parameters is discussed in 4. We discuss model- thishigherresolutionsimulation,theeffectivemassreso-
independent,astrometricsignature§sofsubstructurelens- lutionismp =3.95 107h−1M⊙andthesmallestcellsize
ing in 5. Finally, the effect of luminous substructures reached is 0.6h−1 c×omoving kiloparsecs. Details of the
§
on astrometric perturbations and lens modeling is dis- simulations can be found in Nagai & Kravtsov (2005).
cussed in 6, and we present a summary of our results A variant of the Bound Density Maxima halo finding
§
and conclusions in section 7. algorithm (Klypin et al. 1999a) is used to identify sub-
§
halos. Detailsofthe algorithmusedto findsubhaloscan
2. NUMERICALESTIMATES be found in Kravtsov et al. (2004a).
In this section we present the host halo and substruc- We draw two different subhalo catalogs from the clus-
turemodelsweusedtoanalyzeastrometricperturbations tersimulation,oneatz=0.5,astheclusterisundergoing
and describe our methods of creating artificial lens sys- a major merger, and another at z=0. The mass of the
3
parent halo itself is scaled down to 1013M⊙ and all sub-
clumps are scaled accordingly5. Finally, clumps within
the virial radius with mass M 108M⊙ (M/Mvir
10−5)arechosen. Theseclumps c≥orrespondtoN 8≥0.
P
≥
Nagai & Kravtsov (2005) show that the mass function
of the high resolution simulation converges with that of
a lower resolution simulation for N 80. We fit the
P
≥
clumps with a Moore profile (Moore et al. 1999),
ρ
s
ρ(r)= (2)
(r/r )1.5(1+(r/r )1.5)
s s
truncated at a radius r using a Levenberg-Marquardt
t
minimizationtechnique. Theresultingoverallmassfrac-
tion in substructure is 10% at z=0.5 and 13% at z=0.
The left-hand panels of Figures 1 and 2 show the num-
berdensityprofileandcumulativemassfunctionforboth
redshift samples.
Fig. 2.— Left: The cumulative mass function of subhalos from
numerical models enclosed within the virial radius, for subhalos
at z=0.5 (top) and z=0 (bottom). Right: The cumulative mass
functionofsubhalosfromsemi-analyticmodelsenclosedwithinthe
virial radius for subhalos for a young halo (top) and an old halo
(bottom). In all panels, the mass function from the three dimen-
sionalmassprofileisadottedline,themassfunctionderivedfrom
the parameterization of the subhalos using Eqn. 4 is a solid line
andthenormalizedn∼m−1.8 massfunctionisadashedline.
halothanagalaxy-scalesimulation. Inaddition,simula-
tionsmaysuffer fromovermerginginthe centerofhalos,
reducing the number of subclumps in the inner portions
ofthehalo(Moore et al.1996;Klypin et al.1999a). This
effect may be particularly important in lensing studies,
where substructure close to the projected center of the
system produces the strongest perturbations. To get an
independent estimate of how much substructure might
survive in the inner parts of galaxy halos, we consider
substructure distributions in two halos constructed us-
ingthesemi-analyticmodeldescribedinTaylor & Babul
Fig. 1.—Left: Thenumberdensityprofileofsubhalosfromnu- (2001, 2004); these models – like our simulation models
mericalmodelsnormalizedtothenumberdensitywithinthethree
– do not include a galaxy component, but contain much
dimensional virial radius, n(r)/hnviri, for z=0.5 (top) and z=0
(bottom). The dashed lineshows the darkmatter profile normal- more substructure in the central regions of the cluster
ized to the virial overdensity. Right: The number density profile (Taylor & Babul 2005), thus likely providing an upper
of subhalos from semi-analytic models normalized to the number limit to the amount of substructure in CDM halos.
densitywithinthevirialradius,n(r)/hnviri,forayounghalo(top) The semi-analytic models assume spherical symmetry
and an old halo (bottom), using all clumps with masses greater
than M = 108M⊙. The dashed line shows a Moore profile with in the input orbits and the potential, and all clumps are
concentration of6.4,normalizedtothescaleradius. parameterized by a Moore profile, truncated at rt and
decreasedin density by a fraction f as in Hayashi et al.
t
Compared to galactic halo, the scaled cluster sim- (2003) (equation 8):
ulation may have a greater number of substructures
f ρ
(Zentner et al.2005a)–althoughinoursubstructurecat- ρ(r)= t s . (3)
1+(r/r )3((r/r )1.5)(1+(r/r )1.5)
alogs, the substructure mass fraction does not seem sig- t s s
nificantly larger than expected at 10-20%. In addition, Kazantzidis et al.(2004)suggeststhattheHayashi et al.
we may expect that the concentration of the subclumps (2003)relationmayreflectloweredconcentrationsinthe
in the cluster simulation may be smaller than in a com- the subhalos simulation due to nonequilibrium initial
parablegalactichalo;theissueofsubhaloconcentrations conditions combined with numerical resolution rather
is further discussed in Section 4.4. than tidal shocks; the issue of subhalo concentrations
§
The radial distribution of subhalos may be weakly is further discussed in Section 4.4.
dependent upon the mass scale of the parent halo The two semi-analytical su§bstructure catalogs we
(Gao et al. 2004; Diemand et al. 2004), with a cluster- choose correspond to a dynamically old and a dynam-
scalesimulationwithfewersubclumpsinthecenterofthe ically young parent halo at z=0. The old halo has ac-
creted 50% of the parent halo’s mass by z=3.2 and 90%
5Althoughthetwocatalogsaredrawnfromdifferentepochs,we by z =0.67. The corresponding redshifts for the young
use both for ad hoc substructure distributions in the lens at the
halo are 0.54 and 0.04. Finally, the old halo has a sub-
lens redshift and, therefore, fix the masses and virial radii to the
samevalues. structuremassfractionof10%,whilethe younghalohas
4
a mass fraction of 21%. We scale the host halo mass – a We show the mass functions of the substructure distri-
galactic halo of 1.6 1012M⊙ – to 1013M⊙ and scale all butions using this approximation in Fig. 2 (solid lines).
×
the substructures accordingly, keeping only those sub- Note our approximation does not appreciably bias the
structureswithrescaledmasseslargerthan108M⊙. The substructure mass functions.
right-hand panels of Figures 1 and 2 show the resulting
2.3. Generating and Modeling Artificial Lenses
number densityprofile andcumulative massfunction for
both samples. Our host halo and substructure models are used to
In comparison to the simulation profiles, the semi- generate artificial lenses as follows. First, the image po-
analyticcatalogshavemanymoreclumpsinthecenterof sitions of a given point source are obtained by finding
thehalo,whilethenumberdensityprofilesofallthesub- all roots of the lens equation using a Newton-Raphson
structure models track the host halo dark matter profile root finder with a gridded set of initial guesses. Once
at large radii. In addition, the mass functions of all the the image positions of a source are obtained, we fit the
substructure models are similar, n m−1.8, as expected resulting lens system with a lens model that parameter-
∼
fromthe resultsofhighresolutionnumericalsimulations izes the smooth components of the lens only. We then
(Ghigna et al. 2000). In both the simulation substruc- compare the true image position with that obtained in
ture models and the semi-analytic models, one of the the absence of substructures and with that predicted by
distributions follows n m−1.8 to masses greater than the best-fit model for each image in the lens. We repeat
∼
1% of the halo mass, while the other distribution falls this experiment to statistically sample the source plane
off and has fewer large-mass clumps. We shall see that in order to obtain distributions for the astrometric per-
the presence or absence of large mass clumps can have turbations generated by the dark matter substructures.
important consequences on the distribution of position The results of such a statistical comparison obviously
perturbations. depend on the how the source plane is sampled. While
In addition to specifying the general properties of the naively one might expect uniform source plane sampling
hosthalo andits substructure population, lensing calcu- tobeadequate,observedlenssamplessufferfrommagni-
lations require we specify a line-of-sight projection axis. fication bias (brighter systems are more likely to be ob-
We havechosento projectallhalosalongthe majoraxis servedthan dimmer systems) and consequently magnifi-
of the host halo. This choice for line-of-sight projection cationweightedsamplingisthemostappropriatechoice.
is motivated by the fact that it results in the most com- Keeton & Zabludoff (2004) show that for sources with
pact, and therefore most effective, lenses. Consequently, a power law luminosity function, dN/dS S−ν with
one might expect most observed strong lens systems to ν = 2, magnification weighting in the sou∝rce plane is
be projected along this axis. Moreover, this projection equivalent to uniform sampling of the image plane. The
leads to the largest substructure densities in the cen- importance of this result rests on the fact that uniform
tral part of the halo, which should maximize the impact image plane sampling is easy to implement and the for-
substructures have on the lensed images (Zentner et al. tuitousfactthattheobservedquasarluminosityfunction
2005b). in the largest lens surveys is roughly a power law with
As expected, the projected substructure distributions an index, ν 2. Consequently, we have opted for dis-
obtainedfromthesimulationcatalogshavefewerclumps tributing sou≈rces along the source plane in accordance
inthecentralregionsofthehalothantheprojectedsemi- with uniform image plane weighting in order to provide
analytic models. For instance, within 2′′ of the center a closer match to observations.
of the lens, the z = 0.5 simulation projection has no Best-fit models to artificial lenses are obtained by χ2
clumps and the z = 0 simulation has 1 clump of mass minimization with a downhill simplex algorithm. If con-
3.8 108M⊙,or 0.5%ofthehalomassthat is enclosed vergence is not achieved within a prescribed number of
× ∼
within the Einstein radius. On the other hand, both steps,theoriginalinputparameters(themacromodelpa-
semi-analyticmodelshave7clumpswithinthisprojected rameters) are perturbed and the modeling is repeated
radius,withtotalmassesof6.5 109M⊙and7.1 109M⊙ untilconvergenceisachieved. The χ2 minimization puts
× ×
for the dynamically old and dynamically young catalogs priorsonparityagreementandexcessivelymagnifiedim-
respectively, or 5 - 10% of the halo mass that is en- ages during modeling. We eliminate extremely dim im-
∼
closed within the Einstein radius. For future reference, ages, µ < 0.01, and systems with excessively magnified
we note that the z =0 simulation model has the largest images, µ > 50. Finally, an appropriate best-fit model
subclump of all the models, 7% of the halo mass, and maynotbe found ifthe simplex travelstoanareaofpa-
∼
this subclump is projected far from the Einstein radius rameterspacethatproducesanumberofmodeledimages
of the host halo. that is different from the observed number of images.
Ratherthanmodelingthetwodimensionalmassprofile Thesmoothmodelsusedintheminimizationarecom-
of each individual substructure as a projected Moore or prisedofanSIEmassdistributionalongwithanexternal
Hayashi profile, we use the approximation, shear component. The model parameters are the Ein-
stein radius b, the projected axis ratio q, the external
3.5κ
s
κ(x)= , (4) shear γ, the orientation of the shear θ , the orientation
x1/2+x2 γ
oftheellipticityθ ,thecenterofthepotentialx and
q origin
where κ = 2ρ r for projected Moore profiles and y , and the coordinatesof the sourceposition x
s s s origin source
κ = 2f ρ r for the Hayashi profile, and x = 2.4R/r and y .
s t s s s source
where R is the projected radius. Our choice of profile is Notice that lens systems with a single source position
numerically motivated in that the deflection angle asso- have nine parameters to be fit whereas there are only
ciated with our chosen profile has a simple closed form eightobservablesinquadruplyimagedsystems. Tomake
expression,whereastheprojectedMooreprofiledoesnot. the system overconstrained, we artificially hold the ori-
5
gin (i.e., the position of the SIE) fixed at (x,y)=(0,0),
therebyreducingthenumberoffreeparameterstoseven.
Observationally, this can be accomplished by fixing the
center of the mass distribution to the observed position
of the lensing galaxy. Recent work by Yoo et al. (2005)
suggest that the position of the lensing galaxy is an ap-
propriate approximation of the center of the lensing po-
tential. For lens systems with more than a single source
position(i.e.,jetsources),quadruplyimagedsystemswill
always be overconstrainedwithout reducing the number
of parameters. Note that in all sections, we restrict our-
selves to four image lenses.
3. INTRINSICASTROMETRICPERTURBATIONS
In this section we investigate the intrinsic perturba-
tions generated by substructure: the position difference
between images generated by a parent halo alone and
images generated by a parent halo with substructure.
Fig. 3.— Histograms of the residuals between observed images
While observationally we are interested in the residuals
inmilliarcsecondsforsystemswithsubstructureandwithoutsub-
relativetothebest-fitmodel,comparisonstotheintrinsic structure for quadruply-imaged single point source systems. The
perturbations allow us to determine to what extent per- panelsshowthez=0.5simulationdistribution(top-left)dynami-
turbations by substructure are degenerate with changes callyyoungsemi-analyticsubstructuredistribution(top-right),the
dynamicallyoldsemi-analyticdistribution(bottom-right),andthe
in the macromodel parameters of the best-fit lens.
z=0simulationdistribution(bottom-left).
Our initial expectations for intrinsic perturbations are
abundant smaller clumps. However, even in the mod-
colored by our understanding of single perturbers. Con-
els where no such massivesubstructures are present, the
sider the case of adding a single perturber to a smooth
astrometric perturbations of the images is still consider-
macromodel; it is clear that the astrometric perturba-
able.
tion will scale with the the size and radial position of
the perturber, so that larger and more centrally located 4. MODELEDRESIDUALS
subclumps produce larger perturbations. In the case of
In general, the intrinsic perturbations just discussed
substructure distributions, however, it is unclear if the
are at least partly degenerate with macromodel param-
most massive/most centrally located substructures will
eters. For instance, substructure can change the macro-
dominate the astrometric perturbations. For instance,
modelbyaddingtothetotalprojectedmassdistribution
the steepness of the mass function means that there are
andincreasingtheEinsteinradius,b;bychangingtheel-
many more small perturbers than large, so if the for-
lipticityororientationofthemacromodel;andbyadding
mer act cooperatively they could in principle generate
externalsheartothe potential. Forexample,thepertur-
a large perturbation. Conversely, since oppositely posi-
bations from a single subclump placed far from the lens
tioned perturbers generateequaland opposite perturba-
are degenerate with an external shear. We expect then
tions, the net effect of a large number of substructures
that the modeled residuals, residuals between the ob-
maycanceloutensuringthatrare,massivesubstructures
servedimagepositionsandtheimagepositionspredicted
dominate the position perturbation of the images.
by the best fit smooth macro-model to the lens, will be
Figure 3 shows the intrinsic perturbation distribution
smaller than the intrinsic residuals discussed above.
for each of the fiducial substructure models considered;
Wetesttheextenttowhichtheintrinsicperturbations
each system is quadruply-imaged and the residual for
canbeaccommodatedwiththemacromodelandwhether
each image position is calculated. We note here that
the remaining perturbations (i.e., the perturbations rel-
in this figure – as in all of our subsequent figures in-
ative to the best-fit model) are large enough to be de-
volving histograms of residuals – shows the results in a
tectable. We consider only quadruply-imaged systems
logarithmic scale. Testing the entire subclump distribu-
and test two possible lens modeling scenarios: (1) a sin-
tions,wefindthattheresidualdistributionsallhavevery
large peak perturbations, & 10 mas. Interestingly, our gle point source system modeled with the 7 parameters
mentionedearlier(b,q,θ ,γ,θ ,~r ,withthecenterof
simulation-derived substructure models result in residu- q γ source
thepotentialfixed)and(2)ajetsource,approximatedby
als that are comparable to or larger than those of the
two source positions separated by 10 mas and modeled
semi-analytic models, demonstrating that the intrinsic
by 11 parameters, including the center of the potential
astrometric perturbations are not necessarily dominated
and two source positions. In jet source cases, we refer
bynearbyclumps. Inaddition,sincethesimulationmod-
to each pair of images produced by the source positions
els haveextremely few orno substructures nearthe Ein-
togetherasanimageofthe jet, andrefertothe imageof
steinradiusofthelensandthereforetheperturbersmust
each individual subsource as a subimage.
be located further away, we can infer that position per-
turbations of different images in any lens configuration
4.1. Automated Lens Fitting
maybestronglycorrelated. Afinalinterestingqualityof
our residual is that the two simulations with the largest Before we move on to results, a discussion of one of
peak residuals are also the two models that have very the difficulties in the automated lens fitting procedure
massive substructures. This suggests that rare, massive is required. The lens fitting is done through a simple
clumps may cause larger perturbations than the more downhill simplex algorithm. If our initial guess for the
6
Fig. 4.— Histograms of the residuals between modeled and Fig. 5.— Histograms of the residuals between modeled and
observedimagesinmilliarcsecondsforsinglepointsource systems observedimagesinmilliarcsecondsforsinglepointsourcesystems
using7fittedparametersandfixingthecenterofthelenspotential using7fittedparametersandfixingthecenterofthelenspotential
to (x,y)=(0,0). Only four image systems are plotted. In each to (x,y)=(0,0) (solid) and for jet source systems using 11 fitted
panel the solidline indicates images with Gaussian errors and no parameters (dotted). Only four image systems are plotted. The
substructure, while the dotted line is the same, where the initial panelsshowthez=0.5simulationdistribution(top-left)dynami-
guessfortheEinsteinradiusisunderestimatedby10%. Clockwise callyyoungsemi-analyticsubstructuredistribution(top-right),the
fromthetop-leftpanel,Gaussianerrorswithastandarddeviation dynamicallyoldsemi-analyticdistribution(bottom-right),andthe
of 0.1, 1, 10, and 100 mas are shown. We can see that a tail z=0simulationdistribution(bottom-left).
of large residuals is an artifact of our fitting procedure, though
thepeakperturbationscaleisrobustlydetermined. Consequently,
in all future plots of residual distributions we focus only on the ture distributions in Fig. 5. Here, the modeled residuals
scale at which the distribution peaks. The tail of high residuals are small and, for the most part, significantly smaller
comprisesabout10%ofthetotalnumberofsourcesconsideredfor thantheintrinsicresidualsdiscussedintheprevioussec-
eachmodel(≈10000).
tion, with the peak perturbations occurring between .1
best-fit model is not in the same χ2 valley as the true mas and 10 mas. The decrease in the size of the peak
best-fit model, our resulting formal best-fit model can perturbationswith respectto the intrinsic perturbations
be quite far from the true best-fit smooth macromodel. reflects the fact that there is significant degeneracy be-
We illustrate this point in Figure 4. For this figure, we tween the subclumps and the macromodel.
generatedlenses using the smooth hosthalo mass distri- Comparingthedifferentsubstructuredistributions,we
butiononly (no substructure), andthen addedGaussian seethatforthe singlesourcecase,thesemi-analyticsub-
perturbations of 0.1, 1, 10, and 100 mas to the image structure models produce larger perturbations than the
positions in successive panels. The resulting lenses were simulation substructure models. This difference is evi-
then fit startingfrom twodifferent initial guessesfor the dence for the importance of having substructures within
bestfitmodel. Forthe first,showninFigure 4asa solid the neighborhood of the Einstein radius of the lens in
line, we used the original mass distribution used to gen- order for the system to exhibit observable astrometric
eratethelensinthefirstplace(i.e.,the”correct”smooth perturbations. It is also worth noting that the z = 0
massmodel). Forthesecond,shownwiththedottedline simulation substructure model had the largest intrinsic
in Figure 4, we simply lowered our initial best guess for position residuals and has the largest fraction of poor
the Einsteinradiusby 10%. Inallcases,only fourimage modeled fits – demonstrating that a massive but distant
systemsareusedandtheresidualfromeachimageiscal- clump changes the macromodel enough to be more dif-
culated. AsisevidentfromFigure4,inthe firstcasethe ficult to find a best-fit model. The acceptable modeled
lens is alwaysfit correctly,andthe residualdistributions residuals,however,arecomparabletothatofthez =0.5
aresharplycutaroundthescaleofthedeviation,exactly simulations, showing that those perturbations induced
as we would expect. On the other hand, for the second by a massive but distant clump are degenerate with an
case we find that there is a large, unphysical tail of high external shear.
residuals. This tail was found in all of our substructure Inadditiontosubstructuremodeldifferences,thereare
runs, and accounts for about 10% of the systems fit. significant differences between single point source sys-
≈
This relation is true for Gaussian perturbations regard- tems,whereweholdthecenterofthelenspotentialfixed,
less of the size of the perturbations. and jet source systems, in which the center is allowedto
In the remaining sections of our work, we ignore this float. When jet sources are used, modeled residuals are
tail as an artifact of our lens fitting algorithm. More smaller than the corresponding residuals in the single
importantly, the peak of the residual distribution is cor- sourcecase. Thisresultiscontrarytoournaiveexpecta-
rectlyrecoveredbyouralgorithm,andthelocationofthe tions. One could imagine taking each set of four subim-
peak in our substructure runs has physical significance. ages, and fitting each individually with a mass model.
Since the best-fit models for each set of four subimages
4.2. Substructure Degeneracies with Macromodels
will in general differ, a single mass model for all eight
We present the residual distribution for the single images should result in larger residuals. The resolution
source and jet source scenarios using realistic substruc- to this problem becomes apparent when we realize that
7
ber ofa subimagepair is perturbed inthe samemanner,
however, the distribution of residuals changes dramati-
cally, and in particular the peak residual is an order of
magnitude lowerthan whatwe foundwhen the center of
the potential was held fixed. Thus, our observed resid-
ual distribution from the substructure lenses indicates
that the position perturbations generated by substruc-
tures are coherent on scales at least as large as the as-
sumed extent of our jet source.
Fig. 6.— Histograms of the residuals between modeled and
observedimagesinmilliarcsecondsforimageswithGaussianerrors
with a standard deviation of 1 mas and no substructure. For jet
sources, each subimage is perturbed separately in the dot-dashed
line but both subimages are perturbed by the same amount in
solid line. In both panels, the shaded, dotted histogram shows
thedistributionwhenemployingsinglepointsourcesystems,with
the center of the lensing potential fixed. Top: The center of the
lensingpotential isheld fixed forthe jet sourcesystems. Bottom:
Thecenterofthelensingpotentialisfitforthejetsourcesystems.
in fitting jet sources we have allowed the center of the
potential to float, so the added freedom could result in Fig. 7.— Histograms of the residuals between modeled and
a better fit. Now, in practice, we find that this added observedimagesinmilliarcsecondsforsinglepointsourcesystems
using7fittedparametersandGaussianpriorsonthecenterofthe
freedomdoes resultinbetter fits, butthis didn’t haveto
lenspotentialof5mas(dotted)and1mas(solid). Onlyfourimage
be the case, i.e., substructure position perturbations did systems are plotted. The panels show the z =0.5 simulationdis-
not have to be degenerate with a change in the position tribution (top-left) dynamically young semi-analytic substructure
of the lens’s center a priori. distribution(top-right),thedynamicallyoldsemi-analyticdistribu-
tion(bottom-right),andthez=0simulationdistribution(bottom-
We illustrate this point in Figure 6. The figure is pro- left).
duced by adding 1 mas Gaussian errors to images for
both single point source systems and jet source systems
Giventheimportanceoffixingthecenterofthepoten-
as lensed in the absence of substructures (i.e., we use
tial, an interesting question to ask is how well must the
only the smooth halo component to generate the artifi-
center of the lensing potential be known for astrometric
cial lenses). In addition, we test two methods of adding
perturbations to be sizeable. We address this question
the Gaussian errors to the jet source systems; in one,
in Figure 7, where we test the residual distributions of
each subimage is perturbed independently of the other,
lenses with substructure for single point source systems
while, in the other, each image is perturbed separately,
where the center of the potential is constrained with a
but the two subimages of the each lensed image are per-
Gaussian prior. As can be seen from the figure, a Gaus-
turbed in the same way. We then proceed to fit the
sianpriorof5 masdoesnotresultinmuchimprovement
artificial lenses. As before, single point source systems
relative to the case with no prior (the jet source case),
are modeled by holding the center ofthe potential fixed.
while a prior of 1 mas results in a residual distribution
Jetsourcesystems,onthe otherhand,aremodeledboth
similar to that obtained when we fixed the center of the
withthecenterofthepotentialfixedandwiththecenter
potential(ourfiducialsinglepointsourcecase). Thus,we
ofthepotentialallowedtofloat. Fortheformercase(top
expectthatthecenterofthepotentialmustbeknownto
panel), we find that, just as we would expect, the single
1 mas accuracy for substructure perturbations to be
source case (filled, dotted histogram) results in smaller ≈
non-degeneratewith anallowedshift ofthe centerof the
residualsthanthe jetsourcecase(dot-dashedline)when
lensing potential. As discussed previously, the results of
each subimage is perturbed independently. When each
Yoo et al. (2005) suggest that the assumption that the
subimage pair is perturbed in the same way, the result-
lensing galaxy represents the center of the lens poten-
ing histogram (solid line) is essentially identical to that
tial holds to within 5 to 10 mas. Detailed model fits
of the single point source case, again in agreement with
of Hubble Space Telescope (HST) data shows that lens
our expectations. Turning now to the bottom panel of
galaxy astrometry down to 2 mas are achievable (e.g.,
Figure 6 in which the center of the potential is allowed
Impey et al. 1998; Leh´ar et al. 2000).
tofloat,wefindthatthisaddedfreedomleavestheresid-
ual distribution of the jet source unchanged when each
4.3. Massive Substructures Near the Einstein Radius
subimage is perturbed independently. When each mem-
8
lens model, it is still interesting to ask what residuals
would a similar dark substructure produce.
To address this question, we have chosen an an alter-
nate projection of the z = 0 simulation model in which
the most massive halo substructure gets projected to
within 2′′ of the halo center. We choose the z = 0
∼
simulationmodelbecausethismodelcontainsthelargest
subclump among our four substructure realizationswith
amassofabout 7%ofthehost halomass. Adarksub-
∼
haloofthissizeisunlikely,butitisinterestingtotestthe
extremes necessary for large astrometric perturbations.
The resulting residual distributions for this substruc-
ture model are shownin Figure 8 for both a single point
source(toppanel,solidline)anda jetsource(toppanel,
dotted line). The same basic qualitative features as seen
in the fiducial cases are found in this projection. The
peak residual for the single source case is 10 mas and
≈
drops to 1 mas for the jet source case. The total
amount of≈substructure within 2′′ of the lens center in
Fig. 8.—Histogramsof theresidualsbetween modeledand ob-
this projection is significantly larger than those found
servedimagesinmilliarcsecondsforouralternateprojectionofthe
for the fiducial substructure models, but the peak resid-
z = 0 simulation substructure model that projects a large sub-
clumpnearthecenterofthesystem. Onlyfourimagesystemsare uals are only somewhatlarger. We also show in the bot-
plotted. Top: Singlepointsourcesystemsusing7fittedparameters tompaneltheresidualdistributionsobtainedusing1mas
and fixing the center of the lens potential are plotted in the solid
(solid) and 5 mas (dotted) Gaussian priors on the cen-
line and jet source systems using 11 fitted parameters are shown
in the dotted line. Bottom: Histograms of the residuals between ter of the lensing potential and using only a single point
modeled and observed images in milliarcseconds for single point source. We again find that the center of the lensing po-
source systems using 7 fitted parameters and Gaussian priors on tential must be known to within about 1 mas for the
thecenterofthelenspotentialof5mas(dotted)and1mas(solid).
position residuals to be sizeable.
4.4. Dependence on the Concentration of Subhalos
In the previous sections, we have preferentially chosen
substructure realizationsthat would maximize the num-
bersofsubhalosprojectednearthecenterofthelens–us-
ing semi-analytic substructure models and choosing the
line-of-sight along the major axis – thereby maximizing
theestimatedsignalfromsubstructure. Fordistributions
with significant numbers of subhalos projected near the
Einstein radius, the peak of the residual distribution is
a few milliarcseconds. As discussed previously, however,
the subhalos in both the simulation-based substructure
models and the semi-analytic substructure models could
bebiasedtolowerconcentrations. Higherconcentrations
may lead to larger deflections but may require better
alignment with images. In Figure 9, the residual dis-
tribution for the fiducial dynamically old semi-analytic
substructurerealizationiscomparedtothesamerealiza-
tion where the concentration of each subhalo is doubled
while the mass and tidal radius of each subhalo is held
fixed and to the realization where the concentration is
Fig. 9.— Histograms of the residuals between modeled and increasedbyanorderofmagnitude. Here,itcanbe seen
observed images in milliarcseconds for the dynamically old semi- thatthepeakoftheresidualdistributionisloweredwhen
analyticsubstructuremodel. Onlyfourimagesystemsareplotted.
the concentration of clumps is increased, so, in fact, our
Single point source systems using 7 fitted parameters and fixing
the center of the lens potential are plotted. The fiducial result is estimateslikelyrepresentthelargestvaluesforastromet-
shown by asolidline. Increasing the concentration by 2(at fixed ric perturbations.
mass) is shown by a dotted line and increasing the concentration
by10isshownbyadashedline. 4.5. Halo-to-Halo Variation
Oneadditionalpossibilityweconsideriswhathappens The fiducial substucture distributions show that the
when an extremely massive substructure projects near sizeofmodeledresidualsincreaseswithagreateramount
the Einstein radius of the lens. This question is rele- of subhalos projected near to image positions. Halo-to-
vant not only because one expect such cases to result in halo variation, however, could swamp this effect. A full
the largestresiduals,butalsobecause,observationally,a statisticalanalysisofhalo-to-halovariationwouldrequire
large fraction of the current lens sample is seen to have amoreextensiveunderstandingofbothobservationalbi-
luminous satellites projected near the Einstein radius. asesandsubstructuremodelparameters. Inthissection,
While luminous satellites canbe directly included in the wetestthepossiblerangeofthescatterusing48different
9
cult to detect on the basis of astrometric perturbations
alone. Yoo et al.(2005)havesuggestedthattheassump-
tion that the lensing galaxy represents the center of the
lenspotentialholdstowithin5to10mas. Astrometryof
lessthanafewmilliarcsecondsforlensedgalaxiesmaybe
achievedusing detailed model fits of Hubble Space Tele-
scope (HST) data (e.g., Impey et al. 1998; Leh´ar et al.
2000).
Interestingly,Biggs et al.(2004) havepresenteda four
image jet source system, CLASS B0128+437, in which
reproducingthe observationswith a smoothmass model
appears to be difficult, even while allowing the center of
the lensing potential float. More specifically, the largest
positionresidualsthey find areof ordera few mas. Such
large perturbations are comparable to the highest peak
residualswefound,andsuggestalargesubstructuremass
fraction at radii near the Einstein radius. The errors
in the originalDalal & Kochanek (2002) result are large
enoughtoincludebothsmallandlargesubstructuremass
fractions (0.6% to 7% within the 90% confidence inter-
Fig. 10.— The peak of the residual distributions for 4-image vals). In simulations, Mao et al. (2004) found that the
single point source lens systems for 48 semi-analytic substructure
substructure mass fraction at radii near the positions is
models using a random line-of-sight. The average residual peak
size is shown as square points, enclosed within the minimum and small, 0.5% – as small as the amount of substructure
∼
maximum peak residuals values. Top: The average peak residual foundinourprojectedsimulationhaloswhichhavesmall
comparedtothenumberofsubhaloswithin2′′ ofthehalocenter. astrometric residuals. In addition, studies of cusp rela-
Center: The average peak residual compared to the total mass
in subhalos within 2′′ of the halo center. Bottom: The average tion anomalies with ray-tracing of simulations of galac-
peakresidualcomparedtothemassofthelargestsubhaloineach tic halos with substructure have implied that there in
distribution. not enough substructure in simulations to account for
substructure distributions drawn from the semi-analytic theobservedlevelofcuspanomalies(Bradaˇc et al.2004;
models discussed previously and including the two fidu- Amara et al. 2006; Macci`o et al. 2006)
cial semi-analytic models. Testing single point source Yoo et al. (2005) measured the displacement between
systems with a random line-of-sight, these semi-analytic the observedposition of various lensing galaxiesand the
substucture models have between zero and fifteen sub- position of the best fit lensing model. Their finding of
halos within 2 arcsecs of the center of the halo. Most of displacements between the two of order 5 to 10 mas im-
the substructure realizations contain between three and ply that the center of the lens potential and the posi-
fiveclumpswithin2arcsecofthecenterofthehalo. The tionofthelensinggalaxyareatleastroughlycoincident,
number of models with a particular number of nearby but they also suggest that the displacements may in-
clumps cannot be ascribed any particular significance, dicate an alternative way to measure the substructure
however, a relationship between the number of nearby mass fraction. Displacements of this size are consistent
clumps and the mean of the peak of the residual distri- with the typical displacements we observed, modeling
bution may be seen in the top panel of Fig. 10. The our clumpy lenses with smooth macromodels. However,
range of residuals peaks in the bins is also shown in this whenweallowedthecenterofthepotentialtofloatusing
figure,where, for the mostpopulated bins, the scatter is jet sources,we found that that even when there were no
significant, but not as large as the total range of peak substructures or an extremely small substructure mass
distributions. For all the models, the residuals fall be- fraction within 2′′ of the center of the lens, the center of
tween0.1masand30mas. Asimilarcorrelationisfound the potential was still displaced by a few to tens of mil-
between the averagepeak residual and the total mass of liarcseconds. This suggests that such displacements are
subclumps within 2arcsec. Forcomparison,wealsoplot sensitive to substructure far from the Einstein radius of
the average peak residual compared to the mass of the the lens, and hence interpretation of such observations
largest subhalo. Here, no clear correlation can be ob- as limits on the substructure mass fraction in the cen-
served. tral regions of lenses might be subject to some caveats.
Further investigation of whether the substructure mass
4.6. Discussion
fraction can be robustly estimated from mis-alignment
Our results indicate that systems with massive sub- of the observed position of the lensing galaxy with the
structures within the area of the Einstein radius result best fit location for the center of the potential is clearly
in the largest residuals, of a few mas, which are an or- warranted.
der of magnitude larger than residuals for substructure
distributionwithnonearbyclumps. Justasimportantly, 5. DETECTINGSUBSTRUCTUREINJETSYSTEMS
however,wehavefoundthatwithoutmilliarcsecondcon- Given the difficulties in using lens modeling to probe
straints on the center of the lensing potential, astromet- substructure distributions, unambiguous detections of
ric perturbations from substructures are sufficiently de- substructureinindividuallensesmightbemorerobustly
generate with changes in the position of the lensing po- achieved through local measures of astrometric pertur-
tential so as to reduce modeled residuals by an order bations. By offering more observational constraints, jet
of magnitude and making the residuals extremely diffi- sourcesystemsseemparticularlypromisingavenueofre-
10
search. U has three free parameters or less. Here, we constrain
the form of the relative distortion matrix U using the
5.1. Astrometric Signatures of Substructure
well known fact that for smooth lenses fold images are
Givenaperfectlystraightjetsourcesmallenoughthat expected to have equal and opposite magnifications as
lensing can be locally linearized, its images will also be the image separation goes to zero. We thus set the con-
perfectly straight. If one were to observe a lensed jet dition det(U) = 1; violations of this condition may be
−
with two images, one straight and one with a kink, one indicative of substructure. It is worth noting that this
might be tempted to conclude that the kink must have constraint is coordinate independent, so fitting can be
beenintroducedbyalocalsmallscaleperturbationtothe done in any appropriate, observationally-defined coordi-
lensing potential. However,large kinks may also arise in nate system without any loss of generality. For com-
the absence of substructures since small deviations from pleteness,appendix Aexplicitly performsaTaylorseries
linearityinthejetcanbegreatlyamplifiedwhenthelens expansionofthelensingpotentialaroundaspecificpoint
mapping is nearly singular.6 In general, then, without a alongthelens’scriticalcurveandinaspecificcoordinate
priori knowledge of the linearity of a jet source, some system,thoughwefoundthismoredetailedanalysistobe
degree of modeling is required to detect substructure. less sensitive to substructure than the simpler approach
presented here.
5.2. Local Lens Modeling
The properties of the distribution of the residual ∆r
Inthis subsectionwe considerone possible method for also depend on the assumed properties of the jet source.
detectingastrometricperturbationsusingpurelylocalin- The magnitude of the residual scales with distance be-
formation from the images (i.e., eschewing global lens tween jet subcomponents, which we take to be 1 mas
modeling). We consider lensed images of sources where in our examples. More interestingly, we will find that
multiple source subcomponents are resolved and where the distribution of ∆r depends on the jet bending angle,
each of the subcomponents is multiply imaged. More- the angle between the two relative position vectors of
over, we assume the images are in a fold configuration, the source subcomponents. Thus, a zero degree bending
and focus our attention on the close pair of images. As- angle corresponds to a perfectly straight jet.
suming the size of the source is smaller than any length Figure 11 shows the ∆r distribution while enforcing
h i
scale associated with the lensing potential, the mapping det(U) = 1 for various jet bending angles. Solid lines
−
between the source and each of its images may be lin- are obtained in the no substructure case, and the cross-
earized. Letting A+ and A− denote the inverse magnifi- hatchedhistogramscorrespondtothecaseofthedynam-
cation tensors describing the mapping from the positive ically old semi-analytic substructure model. We can see
andnegativeparityimages,respectively,ontothesource, that for straightor nearly straightjets, the substructure
the two images must themselves be related via a linear and no substructure cases are indistinguishable. As the
transformation, δx+ = U δx−, where U = (A+)−1A− jetbecomesnon-linear(thebendingangleincreases),the
and δx± denotes the imag·e position vectors in any co- ∆r distributions for the substructure case starts to ex-
ordinate system chosen such that the origins in the lens tend to higher ∆r values, implying that nonlinear jets
plane maps to a single origin point in the source plane. may in principle detect substructure through this local
Moreover, the assumption that the images are in a fold test, but only if the substructure density is highenough.
configuration puts a constraint on the form of the ma- Indeed, the simulation substructure models are always
trix U, so deviations on the form of the matrix U from indistinguishable from the no substructure case, which
its expected structure may signal the presence of sub- is not unexpected given the lack of substructures in the
structures on scales comparable to or smaller than the central regions of the halo.
separationbetweenimages. The goalofthissectionisto Our result in Figure 11 might seem surprising: linear
investigate this possibility quantitatively. jets areineffective at detecting substructure without the
Operationally,we proceedas follows. Given a jet with full lens modeling. As suggested strongly by section 4,
§
three subimages, we define the best-fit linear transfor- the simple picture of substructure producing a bend in
mation U by minimizing the total residual ∆r defined a single image while leaving the other untouched seems
by h i overly naive. Moreover, jets that are bent to begin with
can detect substructure. We interpret the bending angle
1 N 2 dependence as follows: for a perfectly linear jet, obser-
h∆ri=vuuN Xi=1(cid:12)(x(+i)−x(+0))−U ·(x(−i)−x(−0))(cid:12) !, vthaetiosinnsgloefjtehteajxeitscoonnlys,trsaointhtehecolerrnescintgamdiosutonrttioofndaisltoonrg-
t (cid:12) (cid:12) (5) tion can be reproduced in the substructure realizations
(cid:12) (cid:12)
where the sum is over the position vectors, x(i) is the regardless of the det(U)= 1 constraint. For bent jets,
± −
imagepositionofvectori,x(0) istheimagepositioncho- however,thedistortionalongtwodifferentaxisisprobed,
± so both eigenvalues of the relative distortion matrix can
sen as the origin of the coordinate system, and N is the
be constrained by observations. Since the det(U) = 1
number of terms in the sum. For three jet subcompo- −
conditionspecifiesarelationbetweenthesetwoeigenval-
nents, N = 2, so fitting is overconstrained if the matrix
ues, it is not surprising that only when both eigenvalues
6 Whilebentradiojetsdonot,ingeneral,implythepresenceof areresolvedcantheeffectsofsubstructuresbediscerned.
substructure,somelensconfigurationscanbedescribedthatwould
unambiguouslysignalthepresenceoflocalizedperturbations. For 6. LUMINOUSSATELLITES
instance, if two jets in a fold configuration – where two of four In the previous sections, we have focused on cases
lensedimagesareclosetogether –arebentinthesamesense, one
where the substructure producing astrometric perturba-
wouldassociatethesameparitytobothimages,violatingageneric
predictionofsmoothlensingpotentials. tions is dark. However, luminous satellites can perturb