A generalisation of the mass-sheet degeneracy producing ring-like artefacts in the lens mass distribution

1.1 MB·English

by  J. Liesenborgs

Mon.Not.R.Astron.Soc.000,1–6(2008) Printed4February2008 (MNLATEXstylefilev2.2)
A generalisation of the mass-sheet degeneracy producing
ring-like artefacts in the lens mass distribution
J. Liesenborgs1⋆, S. De Rijcke2†, H. Dejonghe2 and P. Bekaert1
8 1 Expertisecentrum voor Digitale Media, UniversiteitHasselt, Wetenschapspark 2, B-3590, Diepenbeek, Belgium
0 2 Sterrenkundig Observatorium, UniversiteitGent, Krijgslaan 281, S9, B-9000, Gent, Belgium
0
2

n
a
J
ABSTRACT

8
The inversion of a gravitational lens system is, as is well known, plagued by the
2
so-called mass-sheet degeneracy: one can always rescale the density distribution of

the lens and add a constant-density mass-sheet such that the, also properly rescaled,
]
h source plane is projected onto the same observed images. For strong lensing systems,
p it is often claimed that this degeneracy is broken as soon as two or more sources at
- differentredshiftsareavailable.Thisisdefinitelytrueinthestrictsensethatitisthen
o
impossible to add a constant-density mass-sheet to the rescaled density of the lens
r
t withoutaffectingtheresultingimages.However,oftenonecaneasilyconstructamore
s
a general mass distribution – instead of a constant-density sheet of mass – which gives
[ rise to the same effect: a uniform scaling of the sources involvedwithout affecting the

observedimages.We show that this canbe achievedby adding one or more circularly
1
symmetric mass distributions, each with its own center of symmetry, to the rescaled
v
massdistributionoftheoriginallens.Asitusescircularlysymmetricdistributions,this
5
5 procedurecanleadtotheintroductionofringshapedfeaturesinthemassdistribution
2 of the lens. In this paper, we show explicitly how degenerate inversions for a given
4 stronglensingsystemcanbeconstructed.Itthenbecomesclearthatmanyconstraints
. are needed to effectively break this degeneracy.
1
0 Key words: gravitationallensing – dark matter
8
0
:
v
i
X 1 INTRODUCTION suchacase,weshowinthispaperhowamoregeneralmass
r distribution, to be added to the original lens mass distri-
Being an ill-posed problem, it is not surprising that
a bution, can be constructed which gives rise to a similar
gravitational lens inversion is plagued by degeneracies.
type of degenerate solutions. In particular, the alternative
Several classes of degeneracies were first identified by
name of steepness degeneracy is still applicable since the
Gorenstein et al. (1988) and were later reinterpreted by
construction of degenerate solutions still requires the origi-
Saha (2000) in terms of changes in the arrival-time sur-
nal mass distribution to be rescaled. Although the method
face. Of these, the most widely known degeneracy is usu-
described below is quite straightforward, the authors, hav-
ally called the mass-sheet degeneracy, although recently
ingperformed a thorough literature study,are not aware of
the more correct name of steepness degeneracy has been
thisresult being published before.
suggested (Saha & Williams 2006). This degeneracy was
In section 2 the necessary equations from the gravita-
first mentioned in the context of the strong lensing system
tional lensing formalism will be reviewed. These equations
Q0957+561 (Falco et al. 1985), but has also been studied
willbeusedtoderivethemass-sheetdegeneracyinsection3
inweaklensingsystems(e.g.Schneider& Seitz(1995))and
andtoexplaintheextensiontomultipleredshiftsinsection
even in thecontext of microlensing (Paczynski1986).
4.Finally,insection5,theresultsandtheirimplicationsare
In strong lensing systems, it is often claimed that the
discussed.
presenceoftwosourcesatdifferentredshiftssufficestobreak
the mass-sheet degeneracy (e.g. Abdelsalam et al. (1998)).
Whileitisdefinitelytruethataconstant-densitymass-sheet
can no longer be used to construct degenerate solutions in 2 LENSING FORMALISM
We shall briefly review the necessary equations related to
⋆ Correspondingauthor:[email protected] thegravitationallensingformalism.Theinterestedreaderis
† PostdoctoralFellowoftheFundforScientificResearch-Flan- referred to Schneideret al. (1992) for a thorough review of
ders(Belgium)(F.W.O) gravitational lensing theory.
2 J. Liesenborgs, S. De Rijcke, H. Dejonghe and P. Bekaert
In the thin lens approximation, the gravitational lens 4 EXTENSION TO MULTIPLE REDSHIFTS
effect isessentially a mappingofthesourceplane(β-space)
Aninfinitesheetofmass,however,isnottheonlymassdis-
ontotheimage plane(θ-space), described by thelens equa-
tributionwhichcanbeusedtoproducedegeneratesolutions.
tion:
D Ifthemassdensityiscircularly symmetricandequaltoΣcr
β(θ)=θ− dsαˆ(θ). (1) in an area large enough to encompass all the images, the
D
s samesourcescalingeffectwill arise, thankstoequation(2).
Here, αˆ(θ) describes the instantaneous deflection of a light The center of symmetry of such a distribution determines
ray that punctures the lens plane at the position θ, and the center of the scaling (which is the origin of the coor-
depends linearly on the two-dimensional projected surface dinate system in case of an infinite sheet). This way, the
mass densityΣ(θ) of the lens. mass-sheet degeneracy is easily transformed into a mass-
For a circularly symmetric projected mass density cen- disk degeneracy. In fact, the added mass density need not
tered on theorigin, the defection angle reducesto: beconstantinsidesuchadisktoproducethesameeffect.As
4GM(θ) long as the total mass inside each image point is the same
αˆ(θ)= θ, (2) asforthemass-disk,equation(2)ensuresthatthedistribu-
c2D θ2
d
tion can beused to construct adegenerate solution as well.
in which M(θ) is the total mass within a radius θ. The ge-
ThisconstraintautomaticallyimpliesadensityequaltoΣcr
ometry of the situation is described by the angular diame-
inside the annuli in which the images reside, but otherwise
ter distances Dd, Dds and Ds. As a result, for a circularly allows a lot of freedom.
symmetric projected mass distribution, only the total mass
Thisfreedomallowsustoconstructamassdistribution
inside a specific radius contributes to thedeflection of light
which, when added to a scaled version of an existing solu-
rays.
tion for the lens mass density, is equally compatible with
theobservedimages,butwhichwillrescalethesources.The
effect is therefore verysimilar to that of themass-sheet de-
3 THE MASS-SHEET DEGENERACY
generacy, but this degeneracy is not necessarily broken by
thepresenceofadditionalimagesofsourcesatdifferentred-
Let us first consider a strong lensing system with images
shifts.
coming from a single source. A uniform sheet of mass with
density Σs produces a deflection described by To illustrate the procedure, consider the two sources
and their respective images in Fig. 1. The two sources are
αˆs(θ)= DDdssΣΣcsrθ, (3) aprlaeccerdeaattedrebdyshaiftnsonz1-si=ng1u.l2arainsodthze2rm=a1l.e8llaipnsde tahtezim=a0g.e5s.
in which the critical mass density for the current geometry This non-singular isothermal ellipse then provides us with
is defined as follows: the initial mass density Σ0(θ). A flat cosmological model
Σcr = 4πGc2D DDs . (4) wusiethdΩtomca=lc0u.l2a7t,eΩthΛe=ne0c.7es3saarnydaHng0u=la7r0dkiammse−t1erMdpisct−a1ncweass.
d ds
aNnogtuelathradtiaΣmcretdeerpdeinstdasnocenstDhesarenddshDidfts.oLfetthΣe0s(oθu)rcbeevaiamtahses respoTnhdeincgircMulgaernl(yθ)sytmhamtewtreicshmaallsscodnenstsriutyctΣ,gmenu(sθt)haanvdectohre-
sameeffectasamass-sheetforbothsources.Thismassden-
distribution that is compatible with the observed images.
sity will serve as the generator of thetransformation which
This means that thecorresponding lens equation
creates a degenerate solution Σ1(θ) from an existing solu-
D
β0(θ)=θ− Ddssαˆ0(θ) (5) tciaosne:Σ0(θ).The procedureisverysimilar tothemass-sheet
projectstheimagesontothesourceplaneinsuchawaythat
they overlap exactly. Without further constraints, this im- Σ1(θ)=λΣ0(θ)+(1−λ)Σgen(|θ−θc|), (8)
mediately yieldsaninfinitenumberofalternativesolutions. inwhichθc isthecenterofsymmetryofthegenerator.The
Indeed,if the mass distribution is replaced by
mass distribution of the generator must satisfy constraints
Σ1(θ)=λΣ0(θ)+(1−λ)Σcr, (6) provided by the images: the mass enclosed by each image
point must equal the mass of the corresponding constant-
thenew lens equation becomes
densitymass-sheet.Therefore,ifaspecificimageofasource
β1(θ)=θ−λDDdssαˆ0(θ)−(1−λ)DDdssαˆs(θ)=λβ0(θ). (7) oauttreerdrsahdifituszθloiuets,itnheancoannstnrualiunstwpritohvidinenderbyrasdaiiudsimθinagaenids
thefollowing:
The transformation (6) describes the so-called mass-sheet
degeneracy andsimply rescales thesourceplanebythefac- ∀θ∈[θin,θout]:Mgen(θ)=πDd2θ2Σcr(z), (9)
tor λ, producing an equally acceptable source reconstruc-
tion.Notethatmerelyaddingamass-sheetisnotsufficient; in which the radii are measured with respect to the chosen
one also needs to rescale the original mass distribution by centerof symmetryθc. Consequently,within suchan annu-
the same factor λ, which justifies the alternative name of lus the mass density must equal the critical density for an
steepness degeneracy. The density of the mass-sheet has to image at redshift z and in the region enclosed by the an-
be precisely the critical mass density for this to work. For nulus, the mean density must equal the critical density. In
this reason, a mass-sheet cannot be used when there are theleft panel of Fig. 2, we plot theannuliof theimages, as
sources at different redshifts, since these would require dif- seen from the center of the non-singular isothermal ellipse.
ferent critical densities. Lookingatthefurthestimageofeachsource,itisclearthat
A generalised mass-sheet degeneracy 3
20 40
15 30
10 20
5 10
ec) ec)
Arcs 0 Arcs 0
Y ( Y (
-5 -10
-10 -20
-15 -30
-20 -40
-20 -15 -10 -5 0 5 10 15 20 -40 -30 -20 -10 0 10 20 30 40
X (Arcsec) X (Arcsec)
Figure 1. Left panel: original sources used to illustrate the construction of degenerate solutions. The source surrounded by a box is
placedatredshiftz1=1.2,thesecondsourceisatz2=1.8.Thecausticscreatedbythenon-singularisothermalellipseplacedatz=0.5
are also visible. The solid line corresponds to z1 = 1.2, the dotted line to z2 = 1.8. Right panel: images of the two sources used to
illustratetheconstructionofdegeneratesolutions.Thecriticallinesarealsoshown.
Figure2.Leftpanel:theannuliinwhichtheimagesreside,asseenfromtheoriginofthecoordinatesystem,aredisplayedasgreyrings.
The darker ringindicates the region inwhichthe annuli of the outer images overlap. Because of this overlap, no suitable mass density
canbeconstructed(seetext).Rightpanel:similartotheleftpanel,butnowtheannuliarecenteredon(0,−5).Thiscentercanbeused
intheconstructionofadegenerate solutionsincetherearenolongeroverlappingannuli.
no Σgen can be constructed. The mass density would have of the distribution. The rest of the mass map can easily be
to be equal to Σcr(z1) inside the annulus of one image and interpolated, after which the full density profile of Σgen(θ)
Σcr(z2) inside the annulus of the other image. Since these can be derived. In the left panel of Fig. 3, a third degree
regions overlap, as is indicated by the darker ring, this is polynomialwasusedtointerpolatebetweentheconstrained
impossible. However, if we take (0,−5) as the center, there regions. The resulting density profile is plotted in in the
are no overlapping annuli as can be seen in the right panel right panel of the same figure and the critical densities for
of Fig. 2. the two sources are indicated with dotted lines. Note that
although this particular example does not require negative
Onceanappropriatecenterhasbeenidentified,thepo- densities, in general it is possible that this is indeed neces-
sitionsoftheimagesofeachsourcecanbeusedtocalculate sary. This need not be a problem, since the resulting mass
partsof thetotalmass map Mgen(θ),as specified by(9).In distribution will becombined with theexistingdistribution
ourexample,theseconstraintsareillustratedbythickblack Σ0(θ) (the non-singular isothermal ellipse in this example)
lines in Fig. 3 (left panel), when using (0,−5) as the center
4 J. Liesenborgs, S. De Rijcke, H. Dejonghe and P. Bekaert
5e+14 12
10
4e+14
8
M (M)genO 23ee++1144 -2Σ (kg m)gen 6
4
1e+14
2
0 0
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Radius (Arcsec) Radius (Arcsec)
Figure 3.Left panel: the positions of the images of each source place constraints on the enclosed mass Mgen (thick lines on the mass
profile).Theregions inbetween caneasilybeinterpolated. Rightpanel:the total massprofileintheleftpanel givesrisetothedensity
profileΣgen shownhere.Thedotted linesindicatethecriticalmassdensitiesforthetwosources.
40 6
35
4
30
2
25
-2m) ec)
Σ (kg 1 20 Y (Arcs 0
15
-2
10
-4
5
0 -6
0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12
Radius (Arcsec) X (Arcsec)
Figure 4. The profile of the degenerate solution (thick line) is Figure 5. Sources recreated by the degenerate solution (thick
comparedtotheprofileoftheoriginalmassdistribution,anon- solid lines); the original sources are indicated by dotted lines.
singularisothermalellipse.Inthisexample,λ=0.75wasused. Thedirectionofthescalingisclearlytowards(0,−5).
and may still yield an overall positive density profile. Still, a specific source overlap perfectly when projected onto the
placing a positivity constraint on theoverall density profile sourceplaneprovesthattheconstructedmassdistributionis
may help to alleviate thisdegeneracy. stillcompatiblewiththeobservedimagesandcantherefore
correctly be identified as a degenerate solution. Since each
By construction, the procedure (8) has the same effect
dimension is scaled by λ = 0.75, the reconstructed sources
asthemass-sheetdegeneracy:theobservedimagesareiden-
aresmaller thantheoriginal ones(dottedlines).Theimage
ticalbutthereconstructedsourcesarescaledversionsofthe
alsoclearlyshowsthatthedirectionofthescalingistowards
original ones while the density profile of the lens has be-
(0,−5), the center of the circularly symmetric Σgen which
come less steep. The resulting density profile for λ = 0.75
was constructed.
can be seen in Fig. 4, in which the original profile is shown
as well. Clearly, the central peak has become weaker while
at larger radii a ring of excess density has been introduced. Of course, since the positions of the images are not af-
This figureillustrates nicely that theterm steepness degen- fected, one is free to repeat the entire procedure using the
eracystillappliestothiskindofdegeneratesolution.When newlyacquiredΣ1 asthe“original”solution.Ingeneral,ifit
the images of Fig. 1 are projected back onto their source ispossibletocreateN differentcircularlysymmetricdensity
planes using the new mass distribution Σ1, the sources in distributions Σgen,i, each with another center of symmetry
Fig.5(solidlines)areretrieved.Thefactthattheimagesof θc,i, it is easily derived that for any λ, the following mass
A generalised mass-sheet degeneracy 5
20 40
15 30
10 20
5 10
ec) ec)
Arcs 0 Arcs 0
Y ( Y (
-5 -10
-10 -20
-15 -30
-20 -40
-20 -15 -10 -5 0 5 10 15 20 -40 -30 -20 -10 0 10 20 30 40
X (Arcsec) X (Arcsec)
Figure 6. Left panel: sources and caustics predicted by a degenerate solution. Comparing with the left panel of Fig. 1 one sees that
both sources and caustics are scaled versions of their original counterparts. Right panel: the reconstructed sources and caustics shown
inthe left panel predict the images and critical lines shown here. The same images as inthe rightpanel of Fig. 1 can be seen and the
criticallinesstillresembletheoriginalones.
ΣN (kg m-2)
7
30
25 6
20
15
5
10
5
0 60 -2m) 4
40 Σ (kg N 3
20
-60
0
-40 2
-20 -20 Y (Arcsec)
0
20 -40 1
X (Arcsec) 40
6-060
0
0 10 20 30 40 50 60
Radius (Arcsec)
Figure7.Leftpanel:thedegeneratesolutionwhichgivesrisetothesourceandimageplanesshowninFig.6.Severalring-likefeatures
can be seen, the most prominent one being centered on (−8,−8). Right panel: density profile as seen from (−8,−8). Apart from the
peakofthenon-singularisothermalellipse,onecanclearlyseearing-likefeature.
distributionwillstillprojecttheimagesbackontoconsistent tainedbyusingN =100differentΣgen,i canbeseeninFig.
sources: 6. Each source is scaled by a factor λN = 0.75 in each di-
N mension; the caustics are scaled as well. The critical lines
ΣN(θ)=λNΣ0(θ)+(1−λ)XλN−iΣgen,i(|θ−θc,i|).(10) stillshowthesamegeneralstructure.Themassdistribution
of the degenerate solution can be seen in Fig. 7 (left panel)
i=1
andcontainsseveralring-likestructures,themostprominent
This way, the mass distribution which is added to Σ0 need onebeingcenteredon(−8,−8).Thiscanalsobeclearlyseen
notpossesscircularsymmetryanymoreanditsdensitypro-
when thedensity profile is calculated using (−8,−8) as the
file can become much more complex. Equation (10) is im-
center (right panel of Fig. 7). The first peak in this plot is
portant from a practical point of view: the target scale λN
due to the non-singular isothermal ellipse, the second one
caneasilybereachedbyusingN generators,eachproducing
iscaused bythering-likesubstructureofthedegenerateso-
onlyaverysmalleffect.Ifalargenumberofsuitablecenter
lution. Note that the ring is not caused by one particular
positionscanbefound,thiscanseverelyreducetheamount
generator, but is a combined effect.
of substructureintroduced bytheprocedure.
An exampleof degenerate source and image planes ob- As an aside, even when it is impossible to rescale the
6 J. Liesenborgs, S. De Rijcke, H. Dejonghe and P. Bekaert
sources, it may still be possible to introduce ring-shaped andthecreatedmassdistributionwillnolongerbecompat-
features. Oneonly needsa ring-shaped region without data ible with the observed situation. However, by using several
points.Acircularlysymmetricdistributionwhichiszeroev- versions of Σgen, each with a different center of symmetry,
erywhere but which fluctuates in the ring-shaped region in the amount of substructure that needs to be introduced to
such a way that the total mass inside the region is zero as obtain a specific scaling can belimited.
well, can simply be added to the original, again thanks to It is straightforward to extend the procedure to more
equation (2). Doing so can obviously introduce a ring-like than two sources, but at some point it will no longer be
feature in themass distribution of thelens. possible to construct such degenerate solutions. How many
sourcesareneededwill dependon anumberof factors. The
numberoftheimagesandtheirsizesplayanimportantrole,
as this determines how easily their annuli will overlap and
5 DISCUSSION AND CONCLUSION
therefore how difficult it will be to find appropriate scaling
Inthisarticlewehavepresentedastraightforwardextension centers. The precise redshifts of the sources will determine
ofthemass-sheetdegeneracytomultipleredshifts.Although how much substructure needs to be introduced, since the
thereisnoactualsheetofmassinvolved,theprocedureand redshifts will determine if the corresponding critical densi-
effect of the degeneracy are the same. In both cases, the ties lie close to each other or differ a lot. If the latter is the
existing mass distribution is rescaled and a component is case,automatically moresubstructurewillneedtobeintro-
added, leading to a rescaling of the sources. This justifies duced,allowingonlyalimitedrangeofλvaluestoavoidthe
placing the degeneracy described above in the same cate- prediction of extra images.
gory as the mass-sheet or steepness degeneracy. Although Although at a first glance the mass-sheet or steepness
weillustratedtheconstructionofdegeneratesolutionsusing degeneracy is easily broken, this article shows that it is in
astronglensing example,it isclear that weak lensingstud- factalotmoredifficulttodosoandarelativelylargenumber
iescanbeaffectedaswell.Inparticular,ifweaklensingdata ofconstraintsmaybeneeded.Theexamplesabovealsosug-
aloneleadstosolutionspronetothemass-sheetdegeneracy, gestthatcircularlysymmetricfeaturesshouldbedistrusted,
itwillnotsufficetoaddstronglensingdataofasinglesource as they are easily introduced in degenerate solutions.
to break the degeneracy if strong and weak lensing regions
donotoverlap.Otherwise,asimilarconstructionasaboveis
possible,usingonecriticaldensityintheweaklensingregion ACKNOWLEDGMENT
and another in thestrong lensing region.
We thank our referee, Prasenjit Saha, for his comments on
However, as was shown by Bradaˇc et al. (2004), using
thisarticle.
weak lensing data it is in principle possible to break the
mass-sheet degeneracy – including the extension described
here – if individual source redshifts are available and if
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