Flux ratios as a probe of dark substructures in quadruple-image gravitational lenses

6 Pages·0.12 MB·English

by  R. Benton Metcalf

DraftversionFebruary5,2008
PreprinttypesetusingLATEXstyleemulateapjv.21/08/00
FLUX RATIOS AS A PROBE OF DARK SUBSTRUCTURES IN QUADRUPLE-IMAGE
GRAVITATIONAL LENSES
R. Benton Metcalf1 & HongSheng Zhao1
InstituteofAstronomy,Cambridge,CB30HA,UK
Draft version February 5, 2008
ABSTRACT
Wedemonstratethatthefluxratiosof4-imagelensedquasarsprovideapowerfulmeansofprobingthe
small scale structure of Dark Matter (DM) halos. A family of smooth lens models can precisely predict
2 certain combinations of flux ratios using only the positions of the images and lens as inputs. Using 5
0 observed lens systems we show that real galaxies cannot be described by smooth singular isothermal
0 ellipsoids, nor by the more general elliptical power-law potentials. Large scale distortions from the
2 ellipticalmodelscannotyetberuledout. Neverthelesswefind,bycomparingwithsimulations,thatthe

n data can be accountedfor by a significant( 5 10%) amountofdark substructures within a projected
≥ −
a distanceofseveralkpcfromthecenteroflenses. ThisinterpretationfavorstheColdDarkMatter(CDM)
J model other the warm or self-interacting DM models.

1 Subject headings: cosmology: theory — dark matter — galaxies: formation — gravitationallensing
1

3 1. introduction detect this effect the model or family of models must ac-
v curately predict the magnification ratios at around the
The standard ΛCDM cosmological model has been
7 0.1 maglevel. Itmustbecertainthatthereisnosmooth
2 very successful in accounting for observations on scales ∼
lens model which can reproduce both the observed ratios
4 larger than around a Mpc. But mounting evidence
and image positions. This is the task that is taken up in
1 points towards difficulties for this theory on the scales
this paper.
1 ofgalaxiesanddwarfgalaxies(van den Bosch et al. 2000;
Mao & Schnieder (1998) first proposed that the mag-
1 Gnedin & Zhao 2001). Most interestingly for this paper,
0 CDM simulationsof the localgroupof galaxiespredictan nification ratios of B1422+231 are better fit by a model
/ with substructure in it than with a smooth model. They
h order of magnitude more dwarf galaxy halos with masses
fixed the smooth model and added point masses to repre-
p greater than 108 M⊙ than there are observed satel-
∼ sentglobularclustersandplanewaveperturbations. They
- lites ofthe Milky Way (MW) Galaxy and the M31 galaxy
o foundthattheycouldreproducetheratioswithsmallscale
(Moore et al 1999; Klypin et al 1999; Mateo 1998).
r fluctuations of relatively large amplitude. This does not
t This overprediction of dwarf halos could be a sign that
s howeverexplainthediscrepanciesbetweenradioandopti-
there is something fundamentally wrong with the CDM
a cal ratios in this system. Recently Chiba (2001) has done
: model – the variants include warm dark matter (e.g.,
v a similar analysis where the smooth model is fixed and
Bode, Ostriker & Turok 2001), self-interacting dark mat-
i masses are added. He concludes that globular clusters
X ter (Spergel & Steinhardt 2000) and unorthodox inflation
and dwarf galaxies (as represented by point masses) are
models(Kamionkowski& Liddle 2000). Alternatively,the
r notsufficientto reproducethe data. Boththese investiga-
a small DM clumps could exist, but not contain stars, so as
tions were restricted to the singular isothermal ellipsoidal
to escape detectionas observabledwarfgalaxies. This sit-
modelsforthelensgalaxy,anassumptionwhichwedonot
uationcaneasily,perhapsinevitably, come aboutthrough
make here. During the refereeing process of this paper a
theactionoffeedbackprocessesintheearlyuniverse(pho-
few otherpreprintsappearedonverysimilartopics(Dalal
toionizationandsupernovawinds)(e.g.,Somerville2001).
& Kochanek 2001, Keeton 2001 and Bradaˇcet al. 2001).
Several authors, e.g. Metcalf (2001), have argue that the
overabundance of DM clumps is likely to extend down to
2. method
smaller masses and larger fractions of the halo mass than
have thus far been accessible to numerical simulations. 2.1. The Lens model
Thesenearlypuredarkmatterstructureshavelargelybeen
The lensing equation, ~z = ~x α~(~x), relates a point on
considered undetectable. −
the source plane ~z to a point on the image plain ~x, where
Metcalf&Madau(2001)showedthatiftheCDMmodel
α~(~x) is the deflection angle. These are angular positions.
is correct and these substructures exist within the strong
The deflection angle can be expressedas the gradientof a
gravitationallensesresponsible formultiply imagedQSOs
lensingpotential: α~(~x)= ~ψ(~x). Thispotentialisrelated
they will have a dramatic effect on the image magnifica- ∇
to the surface density of the lens, Σ(~x), through the two
tions – compound gravitational lensing. It was proposed
dimensional Poissonequation
thattheimagepositionswhichareonlyweaklyaffectedby
substructure could be used to constrain a smooth model 2ψ(~x)=2κ(~x) , κ(~x)=Σ(~x)/Σ . (1)
c
∇
for the lens. The magnification ratios can then be com- whereΣ c2D (4πGD D )−1 isthecriticalsurfaceden-
pared with the predictions of this model. However, to c ≡ s l ls
sity. Here D , D , and D are the angular size distances
l s ls
1 Email: [email protected], [email protected]
1
2 Flux ratios as a probe of dark substructures
to the lens, source, and from the lens to the source, re- curvedinparameterspaceorhaslongthinregionsproject-
spectively. ingfromathickerregion. Themostdifficulttaskistofind
To model the lens galaxy we use an elliptical poten- the boundaries of the very high confidence regions. As a
tial with a power-law radial profile (EPL). The influence resultthe95%confidenceintervalsreportedherearemore
of other galaxies that might neighbor the primary lens is secure than the 99% ones although we do believe that the
included through a background shear (EPL+S). The po- calculation has convergedin all cases.
tential for these models is In this paper we are primarily interested in the magni-
fication ratios. For a 4-image system we can combine the
1 fluxes into three independent quantities of the form
ψ(x,y)=br2n+ γ (∆x2 ∆y2)cos2θ (2)
ǫ 2 (cid:2) − γ 4 4
+2∆x∆ysin2θγ] qj =Xajimi ; Xaji =0 (4)
i=1 i=1
where mi is the magnitude of the ith image. The con-
∆x (x x ) , ∆y (y y )
r2 ≡ ∆x−2(clos2θ +ǫ≡2sin2−θ )l straint on aji ensures that qj is independent of the source
ǫ ≡ +∆y2(sin2θǫ +ǫ2cos2ǫθ ) (3) luminosity. Inadditionwerequirethatthetransformation
ǫ ǫ from the observed magnitudes be orthogonal so that the
+∆x∆y(1−ǫ2)sin2θǫ qi’s arestillmeasuredinmagnitudes. We choosethree or-
where the center of the lens is located at ~x = (x ,y ). thogonalgeneralizedflux ratiossuchthatone ofthem has
l l l
The secondterminthe potential(2)is theexternalshear. the smallest confidence interval and one of them has the
With the addition of the source position the number of largest. Thisdisplaysthemaximumprecisiontowhichthe
freeparameterscomesto10. Asingularisothermalsphere magnification ratios can be predicted.
corresponds to n=0.5 and ǫ=1 and a point mass would
3. data
be indicated if n were close to zero (in this case ψ lnr).
∝
We model five 4-image systems (MG0414+0534,
2.2. Searching parameter space B1422+231, PG1115+080, Q2237+030 and H1413+117)
Typically the lens model and thus the likelihood are using publicly available data. For the positions of the
functionsofmanyparameters–inourcase10. Asaresult QSO images and the center of the lens galaxy we use
characterizing the properties of a family of models in any HST data available on the CASTLES Survey’s web site
more detail then simply finding the best fitting set of pa- (http://cfa-www.harvard.edu/castles/). We use the in-
rameters can be difficult. Most, perhaps all, authors have frared and visible extinction-corrected flux ratios from
taken a Monte Carlo approach where the image positions Falco et al. (1999). The errors in the IR/visible ratios
arechosenatrandomaccordingtotheobservationaluncer- are dominated by uncertainties in the extinction correc-
tainties. Then the best–fitmodel is found andthe param- tionratherthanphotometricerrors. Theradiodatacomes
eters of that model are recorded. This is not the correct fromKatz,Moore&Hewitt(1997)forMG0414,fromPat-
procedure. There is not generally a one-to-one correspon- naik el al. (1992) for B1422, and from Falco el al. (1996)
dencebetweenimagepositionsandmodelparameters;not for Q2237.
all sets of image positions correspond exactly to a set of
4. results
model parameters. The region over which one can legally
vary the positions is highly restricted (in 10 dimensions). In Table 1 some of the best–fit lens parameters for the
Inaddition,ifthemodelhasahighdegreeoffreedommul- five 4-image systems are given along with the coefficients
tiplesetsofparametersmaycorrespondtothesameimage forthe generalizedmagnificationratio. Itshouldbenoted
positions. that in practice the likelihood is often rather flat in some
To avoid these problems we generate random models directionsnearthemaximumasismostclearlythecasein
and then calculate the image positions. This is signifi- MG0414(seefigure1). OurMonteCarlotechniquetriesto
cantly more time consuming because one does not know accountforthesedifficulties. Inadditiontherecanalsobe
howwellaparametersetfitsthedatauntilallthecalcula- local maxima that are well separated in parameter space.
tions are finished. To achieve any degree of efficiency and For B1422 we find two well fitting regions of parameter
to ferret out the corners of confidence regions some adap- space. They do not appear to be connected at any rea-
tive sampling ofthe parametersmust be used. We do this sonable confidence level. In every case the best–fit model
by first finding the best–fit model and set a maximum χ2 fits the image and lens positions very well which is ex-
that we consider an acceptable fit. The models are cho- pected since we have an equal number of constraints and
senaccordingtotheprobabilityP(x)= p(x ,σ )where parameters. The image positions are all consistent with
Qi i i
each p(x ,σ ) is a normaldistribution. The parametersx the hypothesis that the EPL+S models correctly describe
i i i
are the distance from the best–fit model along eachof the the lens potentials.
principle directions of the distribution of models already Figure 1 shows the magnification ratio confidence re-
found in the acceptable–fit region. These directions are gions along with the data. The generalized magnification
periodically updated. The standard deviations σ are up- ratios have been shifted for convenience so that the con-
i
dated so that it is some fixed factor times the distance to fidence region is centered on zero magnitudes in all cases
mostextremeacceptable–fitmodelinthatdirection. This except B1422 where we have two acceptable–fit regions.
factor is increased until the final confidence region is no Here the ratios are shifted so that the region with the
longer dependent on it, typically √3. This algorithm globally best fitting solution is centered on 0. It is signif-
∼
workswell exceptwhen the acceptable–fitregionis highly icant that the width 99% confidence interval for the best
Metcalf & Zhao 3
generalizedratiois<0.06maginallcasesexceptMG0414 The known populations of small scale substructures in
where it is 1.1 mag. The data is also plotted in figure 1 theGalaxywouldbe unlikelytocausethe effectsreported
with the size of the symbols representing the errorstrans- here. The overdensitiesinspiralarms donot appearlarge
formed into the generalized ratios. One can immediately enough and, as pointed out by Mao & Schneider (1998),
see that many of the measurements are inconsistent with the fractionof the Galaxy halo’s mass in globularclusters
the smooth EPL+S model. Only MG0414 is fully consis- is only about 10−4. The mass in dwarfgalaxy satellites
∼
tent with the models. Of the two acceptable regions for is 1% of the halo, but 80 90% of this is in just two
∼ −
B1422 it is the one that fits the image positions less well objects. One would expect a chance alignment of these
that fits the flux ratios best. types of structures with the QSO images to be rare not
ubiquitous.
5. lensing simulations
Within the EPL+S models the magnification ratios are
For comparisonwe have done some simulations of what quite strongly constrained in some dimensions. In all but
is expected in the CDM model. We have not attempted MG0414 we were able to constrain one combination of
to account for the fact that the image positions do not magnification ratios to within 0.06 mag. Within a large
constrain the smooth model to a precise set of smooth family of galactic mass profiles the ratios can be an effec-
modelparameters. Becausethe effects ofsubstructureare tive tool for probing small scale structure. In simulations
stronglydependentonthesmoothmodelusedonlyaqual- with pure CDM, galaxies do have power-law radial pro-
itative comparison with the CDM model is now possible. files within the smallradial distances important for quad-
We will make a more systematic study of this problem in lenses. And observed lens galaxies are typically relaxed
future work. giant ellipticals. However, there is no compelling reason
The code used to do these simulations is described in to believe that CDM halos and their embedded galaxies
Metcalf&Madau(2001). Thesmoothmodels arefixedto should be precisely elliptical; bulges and inclined disks
the best fit models given in Table 1. The substructure is could make the lens mass distribution more boxy or more
modeledas singularisothermalspherescutoffatthe tidal disky in projection in the inner few kpc. Zhao & Pronk
radius. The massfunction is m−2 between104 M⊙ and (2001) have found that lens models with different degrees
106 M⊙ andthenormalization∝isfixedsothat5%and10% of boxiness can fit the image positions equally well, but
ofthelenssurfacedensityiscontainedinthesubstructure. predict different flux ratios and time delay ratios. This
Thephysicalradiusofthesourcesaresetto30pcwhichis leads to an ambiguity in our substructure interpretation
perhaps appropriate for the radio emission. We calculate with large scale distortion. Ultimately this degeneracy
1500randomrealizationoftheratiosforeachlens. Wedid can be lifted by either restricting the range of realistic
notsimulateH1413becausethe lens redshiftisnotknown halo profiles using simulations or by comparing observa-
Theresultsareshowninfigure2. Exceptforthesecond tions in different wavelengths – the magnifications should
ratio of Q2237, all of the anomalous ratios fall within 2 be wavelength dependent (Metcalf & Madau 2001).
σ of the regions containing 95% of the simulated ratios if The constraints on the predicted magnification ratios
10% of the mass is put in substructure. If anything the are good, but they are not negligible. In some cases the
data appears to favor even more substructure or higher uncertainty in the ratiosis muchlargerthan the expected
mass substructure relative to the source size. variations caused by substructure. The influence of sub-
structure on the ratios can be strongly dependent on the
6. discussion
precise values of smooth model parameters. This should
InallcasesexceptMG0414theEPL+Smodelsareruled be takeninto accountinfuture compoundlensingstudies.
out at greater than 95% confidence. The radio fluxes are This resultis inroughagreementwithwhatis expected
not affected by extinction or microlensing and they alone in the CDM model (Metcalf & Madau 2001) as shown by
rule out the models for B1422 and Q2237. The disagree- our simulations. In variants of the CDM model such as
ment between visible and radio ratios in Q2237 is confir- warm DM, hot DM or interacting DM small scale sub-
mation of the microlensingthat was alreadyknown to ex- structure is almost nonexistent. These models are not si-
ist through time variability studies (Wo´zniak el al. 2000, multaneouslyconsistentwith the EPL+Slens models and
andreferencestherein). Thestrongdisagreementbetween the observed flux ratios. Possible degeneracies with the
the radio and model predictions for Q2237 does indicate larger scale distortions of the lens make us unable to con-
thatthereissomeotherkindofsubstructurethereaswell. clusively discriminate between DM models at present.
Given the absence of microlensing-relatedstrong variabil-
ityinPG1115orH1413theobservedvisible/IRfluxratios
appearinconsistentwithEPL+Smodels. Thisisevidence Acknowledgements
that a significant amount of small scale structure must RBM would like to thank Piero Madau for very useful
exist either in the lenses or along the lines of sight. conversations.
REFERENCES
Bode,P.,Ostriker,J.P.,&Turok,N.2001,ApJ,556,93 Falco,E.E.,Impey,C.D.,Kochanek, C.S.,Leha´r,J.,McLeod,B.A.,
Bradaˇc,M.,Schneider,P.,Steinmetz,M.,Lombardi,M.,King,L.J., Rix, H., Keeton, C.R., Mun˜oz, J.A. & Peng, C.Y.,1999, ApJ,
&Porcas,R.,2001, preprint,astro-ph/0112038 523,617.
Chiba,M.2001,ApJ,inpress,(astro-ph/0109499) Falco,E.E.,Lehar,J.,Perley,R.A.,Wambsganss, J.&Gorenstein,
Dalal,N.&Kochanek,C.,2001,preprint,astro-ph/0111456 M.V.,1996,AJ,112,897
Gnedin, O. & Zhao, H., 2001, submitted to MNRAS, (astro-
ph/0108108)
4 Flux ratios as a probe of dark substructures
Keeton, C.R.,2001,preprint,astro-ph/0111595 Moore,B.,Ghigna, S.,Governato, F.,Lake,G.,Quinn,T.,Stadel,
Kamionkowski,M.&Liddle,A.,2000,Phys.Rev.Lett.,84,4525 J.,&Tozzi,P.1999,ApJ,524,L19
Katz,C.A.,Moore,C.B.&Hewitt,J.N.,1997,ApJ,475,512 Patnaik, A.R.,Browne, I.W.A.,Walsh, D.,Chaffee, F.H.& Foltz,
Klypin,A.,Kravtsov,A.V.,Valenzuela,O.,&Prada,F.1999,ApJ, C.B.,1992,MNRAS,259,1P
522,82 Somerville,R.S.,2001, submittedtoApJ(astro-ph/0107507)
Mao,S.,&Schneider,P.1998, MNRAS,295,587 SpergelD.N.,SteinhardtP.J.,2000,Phys.Rev.Lett.,84,3760
Mateo, M.1998, ARA&A,36,435 vandenBoschF.C.,RobertsonB.E.,DalcantonJ.J.,deBlokW.
Metcalf, R.B., 2001, “Where is the Matter?”, ed. Tresse, L. & J.G.,2000,AJ,119,1579
Treyer,M.,(astro-ph/0109347) Wo´zniak, P.R.,Alard,C.,Udalski,A.,Szyman´ski,M.,Kubiak,M.,
Metcalf,R.B.&Madau,P.,2001, ApJinpress,(astro-ph/0108224) Pietrzyn´ski,G.&Zebrun´,K.,2000,ApJ,529,88–92.
Zhao,H.&Pronk,D.,2001,MNRAS,320,402
Metcalf & Zhao 5
Best–Fit Lens Models Generalized Ratios
Lens P(χ2) b γ ǫ2 n a a a
1 2 3
B1422+231 0.00 0.91 0.25 1.58 0.32 -0.61 0.76 -0.20
0.61 0.62 0.49
0.00 2.00 0.36 2.38 0.11 -0.73 0.69 -0.03
0.61 0.62 0.49
PG1115+080 0.00 14.7 0.21 1.13 0.04 -0.76 0.65 0.08
0.65 0.76 0.05
Q2237+030 0.00 1.44 0.05 1.37 0.28 0.77 0.01 -0.63
-0.56 -0.45 -0.69
H1413+117 0.21 0.45 0.10 1.21 0.97 -0.63 0.57 0.52
0.77 0.48 0.41
MG0414+0534 0.00 0.10 0.85 0.52 0.65 -0.004 -0.01 -0.999
-0.68 -0.73 0.01
Table 1
Theai coefficientsareforthemostwell constrained generalizedmagnificationratioon top andtheleast well
constrained ratioon thebottom (a4=−P3i=1ai). The thirdgeneralizedratiois orthoginalto thesetwo vectors. The
numberingof imagesis inthe sameorder as thosein theobservationalliturature. P(χ2)is theexpected probability
(roundedoff)of χ2 beingsmallerthantheone measuredwith 10 degreesof freedom. Of thetwo models forB1422the
second one isa slightly betterfittothe imagepositions.
Fig. 1.— The confidence ranges for the EPL+S in the optimal three independent combinations of the magnification ratios for five lens
systems. The95% and99% confidence regionsareshownbythetwosets ofhorizontal barsoneacherrorbar. Thebow ties markthe radio
ratios at 5 GHz except for B1422 where both 5 GHz and 8 GHz ratios are shown. The size of the bow ties are the reported errors. The
diamondsmarktheextinction–corrected infraredandvisibleratioswiththelengthsgivingtheerrors(Falco et al. 1999).
6 Flux ratios as a probe of dark substructures
Fig. 2.— Same as figure 1 except that the smooth models are fixed and the range of magnification ratios expected in the presence of
substructureareshownasrectangles. Therectanglesshowtheregioninwhich95%ofthesimulatedrealizationsfall. Therearetworectangles
foreachratio,oneinsidetheother. Thesmalleronesareforthecaseof5%ofthesurfacedensityinsubstructureandthelargeronesarefor
10%. Thescalehasbeenchanged fromfigure1forMG0414forclarity.