A nom alous F lux R atios in G ravitational Lenses: For or A gainst
CDM ?
Shude M ao1,Y ipeng Jing2,Jerem iah P.O striker3,Jochen W eller3
arXiv:astro-ph/0402149v1 6 Feb 2004
A B ST R A C T
W e review the evidence for substructures from the anom alous ux ratios in
gravitationallenses. U sing high-resolution num ericalsim ulations,we show that
attypicalim agepositions,the fraction ofsurface m assdensity in substructuresis
. 0:5% w ith m assabove 10 4 virialm assesin the \concordance" C D M cosm ology. Substructures outside the virialradius (but projected at typicallens im age
positions) only increase the fraction m oderately. Several e ects, in particular
baryonic settling and the requirem ent ofcom pactness,m ay further decrease the
predictions by a factor of few . T he predicted fraction w ith appropriate properties thus appears to be lower than that required by lensing, although both
are stilluncertain. M ore speculative substructures such as m assive black holes
(M
105 106M ) in the halo m ay o er viable alternatives.
Subject headings: gravitational lensing - cosm ology: theory - dark m atter galaxies: structure,evolution
1. IN T R O D U C T IO N
G ravitational lenses on arcsecond scales provide a unique sam ple to probe the m ass
distribution in the lensing galaxies at interm ediate redshift (z 0:5 1). Im age positions
in m ost lenses can be tted adequately using sim ple sm ooth galaxy m ass m odels. But
observed ux ratios are m ore di cult to m atch (e.g., K ochanek 1991). T he discrepancy
between the predicted and observed ux ratios is com m only referred to as the \anom alous
1
U niversity of M anchester, Jodrell B ank O bservatory, M accles eld, C heshire SK 11 9D L, U K ;
sm ao@ jb.m an.ac.uk
2
ShanghaiA stronom icalO bservatory;the Partner G roup ofM PA ,N andan R oad 80,Shanghai200030,
C hina;ypjing@ center.shao.ac.cn
3
Institute of A stronom y, C am bridge U niversity, M adingley R oad, C am bridge, U K ; (jpo,
jw 249)@ ast.cam .ac.uk
{2{
ux ratio problem ." T he m ost apparent cases are found in quadrupole lenses w here we
observe a close pairora close triple ofim ages. H ere we know thatthe lensed source is close
to either a fold or a cusp caustic. T he asym ptotic m agni cation behavior in such cases is
wellunderstood { a close pair m ust have equalbrightness,w hile for a close triple,the ux
ofthe m iddle im age should be equalto the total uxes ofthe two outer im ages. V irtually
allthe observed pairs and triples disagree w ith these relations (x2). T his has been argued
as evidence for substructures on the scale ofthe separations ofthe im ages (a few tenths of
arcseconds,e.g.,M ao & Schneider 1998;M etcalf& Zhao 2002). A nother piece ofevidence
forsubstructuresisthefactthatsaddleim agesarepreferentially dim m ed com pared to m odel
predictions (K ochanek & D alal2003). T his is expected from m illi-lensing by substructures
(K eeton 2003)orm icrolensing by stars(Schechter& W am bsganss2002)1;such a preferential
de-m agni cation ofsaddle im ages is not expected from other propagationale ects.
T he C old D ark M atter(C D M )structure form ation m odelpredicts the existence ofjust
such substructuresfrom both sem i-analyticalstudiesand num ericalsim ulations(e.g.,K au m ann etal.1993;K lypin etal.1999;M ooreetal.1999;G higna etal.2000).A bout5-10% of
the m assispredicted to be in substructures w ith a m assspectrum ofn(M )dM
M 1:8dM .
Intriguingly, the predicted num ber of subhaloes in C D M exceeds the observed num ber of
lum inous satellite galaxies in a M ilky-W ay type galaxy (e.g.,K lypin et al. 1999;M oore et
al. 1999;see also Stoehr et al. 2002). O ne solution for this disagreem ent m ay be that som e
substructures, especially those oflowest m ass,are dark. Ifthis is true,then gravitational
lensing m ay be the bestway to detectthem . In thispaper,we exam ine the required am ount
ofsubstructures in gravitationallenses (x2) and com pare it w ith the predictions from C D M
(x3).Finally,in x4,we discussseverale ectsthata ectthe predictionsofnum ericalsim ulations.T hroughoutthispaper,we adoptthe \concordance" C D M cosm ology (e.g.,O striker
& Steinhardt 1995; Spergelet al. 2003 and references therein), w ith a density param eter
2
m ;0 = 0:3,a cosm ologically constant
;0 = 0:7,a baryon density param eter b = 0:024h ,
and we take the power-spectrum norm alization 8 = 0:9. W e w rite the H ubble constant as
h = H 0=(100km s 1 M pc 1) w ith h = 0:7.
2. R equired A m ount of Substructures From G ravitationalLenses
A num berofpapershave discussed evidence forsubstructuresfrom gravitationallenses;
m ost concentrated on the anom alous ux ratios (e.g., M ao & Schneider 1998; M etcalf &
1
W e refer to lensing by substructures and lensing by stars as m illi-lensing and m icrolensing respectively
because the angular scales involved are m as and as in these two cases.
{3{
M adau 2001;C hiba 2002;Bradac et al. 2002;D alal& K ochanek 2003;K eeton 2003),w hile
several papers discussed astrom etric signatures, such as bent jets in B1152+ 199 (M etcalf
2002), and unusual V LBI structures for M G 2016+ 112 (K ochanek & D alal2002; see also
K oopm ans et al. 2002) and B0128+ 437 (Phillips et al. 2000; Biggs et al. 2003). It is
illustrative to see the issues using the largest hom ogeneous lens survey { the C osm ic Lens
A ll-Sky Survey (C LA SS,M yers et al. 2003 and Brow ne et al. 2003). T his radio survey has
wellde ned selection criteria and does not su er from the e ect ofdust extinction. R adio
lenses are also not substantially a ected by stellar m icrolensing (see K oopm ans et al. 2003
and references therein).
In total, there are 22 new lenses discovered in C LA SS. A m ong these, 7 are sim ple
quadrupole lenses,including 5 close pairs (B0128+ 437,Phillips et al. 2000;M G 0414+ 0534,
H ew itt et al. 1992; B0712+ 472, Jackson et al. 1998; B1608+ 656, M yers et al. 1995;
B1555+ 375,M arlow et al. 1999),and two close triples (B2045+ 265,Fassnacht et al. 1999;
B1422+ 231,Patnaik etal.1992).Forthe ve pairslisted above,the observed ux ratiosare
0:56 (5G H z),0:88 (15G H z),0:75 (15G H z),0:51 (8.4G H z),0:56(15G H z);for each lens,the
highestfrequency w herethepairhasbeen observed isshow n in thebracket.Forthetwo close
triples,the ratios ofthe ux ofthe m iddle im age to the total ux ofthe two outer im ages
are,0:32 (14.9G H z)and 0:70 (15G H z,Patnaik & N arasim ha 2001)respectively. T he typical
errors on the ux ratios are 1 2% . Except the case of M G 0414+ 0534,the observed
values are alldi erent from the asym ptotic value (unity).
H owever,not allthese ux ratios are \anom alous" because som e ofthese system s m ay
not yet reach the asym ptotic regim e. A lso som e radio lenses m ay have been a ected by
scattering by free electrons along the line ofsight. In addition,m ost lens m odelers adopt
isotherm al ellipsoidal m odels or variants as an approxim ation for the lensing potentials.
M odels w ith m ore com plex radialand angular structures can usually better reproduce the
observed ux ratios(e.g.,Evans& W itt2003;however,see K ochanek & D alal2003).O bservationally,itis clearthatsom e lens system s are com plex. Forexam ple,B1608+ 656 hastwo
lensing galaxies and a m odelw hich accounts for this fact can reproduce the observed ux
ratio (K oopm ans & Fassnacht 1999). For B0712+ 472,a singular isotherm alellipsoid (SIE)
m odelcannotreproduce the ux ratio,but a foreground group ofgalaxieswas subsequently
found along the line of sight of the prim ary lensing galaxy (Fassnacht & Lubin 2002),so
a m ore realistic m odelm ay m atch the ux ratio. T he close pair ux ratio in B1555+ 575
cannot be reproduced by an SIE m odel(M eyers et al. 1995),but a m ore com plex m odel
(e.g.,w ith an additionalshear) m ay be able to explain it. For B0128+ 437,an SIE m odel
cannot t the ux ratio,but the observed ux ratio can be reproduced w ith an additional
shear. H owever,in this case,the relative orientations ofjets in this system appear di cult
to m atch w ith a sm ooth m odel(Biggs et al. 2003),so substructures m ay yet be called for.
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Forthe two close triple lenses,the ux ratiosappear di cult to m atch w ith sm ooth m odels
(M ao & Schneider 1998;Fassnacht et al. 1999).
M ao & Schneider (1998) showed that substructures preferentially a ect the ux ratios
ofhighly m agni ed im ages. In order to reproduce the ux ratios in B1422+ 231,they found
that the required perturbation in the dim ensionless surface m ass density, = ,is ofa few
percent and roughly corresponds to a physicalsurface density ofa few tens M pc 2. D alal
& K ochanek (2002) perform ed a statisticalstudy ofseven radio lenses using M onte C arlo
sim ulations,six ofw hich show anom alous ux ratios. T hey nd that the fraction ofm ass
required in substructures should be in the range offsub = 0:6% 7% (90% con dence lim it)
w ith a best t offsub
2% . M ore recently, M etcalf et al. (2003) applied the m ethod of
M oustakas & M etcalf(2003) to the quadrupole system 2237+ 0305. T hey conclude that,in
orderto m atch the observed ux ratiosin the radio,infrared and narrow and broad em ission
lines,4% -7% ofthe surface m ass density (95% con dence lim it) m ust be in substructures
w ith m ass between 104M
108M .
3. P redicted Substructures in N um erical Sim ulations
W e use the high resolution halo sim ulationsofJing & Suto (2002;2000)to constrain the
fraction ofm assin substructures;sim ilarresults(butw ith largererrorbars)are found using
the sim ulation data obtained w ith the tree-particle-m esh code ofBode & O striker (2003).
T welve halos were selected from a cosm ologicalsim ulation ofbox of100h 1 M pc,w ith four
each atgalactic,group,and clusterm asses,respectively. T hese sim ulationsare evolved w ith
a nested-grid PPPM code w hich was designed to sim ulate high-resolution halos. T he force
resolution istypically 0:004rvir,w here rvir isthe virialradius. A tthe end ofeach sim ulation,
about(0:5 1) 106 particlesare w ithin rvir ofeach halo (see Jing & Suto 2000 fordetails).
W e adopt the SUBFIND routine of Springel et al. (2001) to nd disjoint self-bound
subhalos w ithin a parent halo. A llsubhalos w ith m ore than 10 particles are included in our
analysis. O ur num ericalresolution im plies that we can only identify subhaloes w ith m ass
largerthan about10 4 ofthe parenthalo m ass;the m ostm assive subhalo hasabout10% of
the parenthalo m ass. Foreach halo,we m ake 30 random projectionsand calculate the total
m assw ithin di erentannuli;theannuliareequally spaced in logR =rvir from 2:2 to 0 w ith a
step size of0.2,w here R isthe projected radiusand the sphericalradiusisr.T he totalm ass
in substructures w ithin an annulus is calculated by sum m ing up allthe m ass of subhalos
w hose centers fallin it. D ividing the totalsubstructure m ass by the totalm ass w ithin the
annulusyieldsthe fraction ofthe projected surface density in substructures,fsub .T he m ean
value and variance offsub are found from the 12 haloes and the 30 random projections. In
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the upper left panelofFig. 1,we show fsub as a function ofR =rvir. T he m ean fraction can
be approxim ated asfsub 0:25(R =rvir).T he scatteraround the m ean am ong haloesisquite
large. A t R
rvir,the scatter is about 40% ;at sm aller R the scatter is larger due to fewer
subhaloes along the line of sight. A s lensing concerns only the projected surface density,
we also checked w hether substructures outside the sphericalvirialradius can contribute to
the surface density at a given R (Fig. 1,top right). T he increase is about a factor oftwo
at R
rvir (cf. K lypin et al. 2001),but m ore m odest (. 1:3) at typicalim age positions
(R 0:01rvir 0:03rvir),consistent w ith the analyticalestim ate by C hen et al. (2003).
A lm ost allsubhaloes in the sim ulation have a sphericalradius r > 0:1rvir. M ost substructures at typicalim age positions (R
0:01rvir 0:03rvir) only appear along the line of
sight due to projection. T hese subhaloes, especially those at r rvir,are extended, thus
notallthe bound m ass w ithin the subhaloes can e ciently cause the ux anom aly (see x4).
To illustrate this e ect,fsub is recalculated by including only the m ass w ithin a spherical
radius of0:025rvir for each subhalo. D ue to this com pactness requirem ent,at R = 0:03rvir,
the average fsub value (forsubstructuresw ith r < rvir)islowered from 0.8% to 0.6% (see the
lower panels in Fig. 1). W e return to this com pactness issue in the discussion.
To com pare the predicted fraction w ith lensing requirem ent we need an estim ate ofthe
virialradius for the lensing galaxies. Ifthe lens and source redshifts are know n,then the
separation between im ages (/ 2) can be used to determ ine the velocity dispersion ( ) and
p
the approxim ate halo circularvelocity (Vc
2 ),w hich in turn allow susto determ ine the
halo virialradius (e.g.,eqs. 2-3 in M o,M ao & W hite 1998). T he lens and source redshifts
are know n for M G 0414+ 0534,B0712+ 472,B1608+ 656,B2045+ 265 and B1555+ 375. For
these system s,the im age positionsin unitsofthe virialradiusare,0.027,0.012,0.018,0.017
and 0.012,respectively. From Fig. 1 we can then inferthe fraction ofm ass in substructures
is only around 0.2-0.8% ,w ith a large scatter am ong di erent haloes.
So far we have only considered the fraction ofdark m atter surface density in substructures. H owever,the innerpartsofgalaxiesare likely dom inated by baryons. To addressthis
issue,we consider the cooling and settling ofbaryons in an N FW halo (N avarro,Frenk &
W hite 1997),sim ilarto the procedure used by K eeton (2001).Initially the baryonsand dark
m atter follow the sam e pro le. T he baryons then cooland condense into the center to form
a stellar com ponent w ith a de Vaucouleurs pro le w hile the dark m atter responds to the
baryonic settling adiabatically. T he m odelis described by the concentration param eter c in
the N FW pro le,the e ective radius (re )in units ofthe halo virialradius,and the ratio of
the stellar m ass to the totalm ass,fcool (or equivalently,a m ass-to-light ratio).
Forthe ve sources w ith know n lens and source redshifts (M G 0414+ 0534,B0712+ 472,
B1608+ 656, B2045+ 265 and B1555+ 375), the e ective radii have been determ ined using
{6{
H ST photom etry (K ochanek etal. 2000).In unitsofthe virialradiusthey are,respectively,
0.018,0.006,0.01,0.006 and 0.005. W e take c = 10 (appropriate for a galactic-sized halo)
and then carry out the procedure described above. For the ve system s,we nd that the
stellar com ponent contributes about20% -50% ofthe projected surface density atthe im age
positions (R 1:5re
3re ) for fcool ranging from 0.05 to 0.16 (
b = 0). T he fraction of
surface m ass density in dark m atter is consistent w ith that required by stellar m icrolensing
( 70 90% at 1.5re ,Schechter & W am bsganss 2003),but at odds w ith the recent claim
ofR om anow sky et al. (2003). T he e ect ofbaryons hence reduces the the m ass fraction in
substructures in typicallensing system s to . 0:5% .
4. D iscussion
W e have reviewed the evidence forsubstructures from close pairsand triplesin quadruple lenses. A s em phasized by K ochanek & D alal(2003), the fact that saddle im ages are
frequently dim m er than expected is di cult to accom m odate by other m eans. Q uantitatively,the anom alous ux ratiosin lensesappearto require a few percentofthe surface m ass
density in substructures at typicalim age positions (D alal& K ochanek 2002;M etcalfet al.
2003).T he required fraction ishigherthan thatprovided by globularclusters and lum inous
satellite galaxies(M ao & Schneider 1998;C hiba 2002)and italso appearsto be higherthan
the predicted values (fsub . 0:5% ) from the C D M cosm ology at typicalim age positions.
H owever,at present it is unclear how serious the discrepancy is because ofuncertainties in
both observations and theoreticalpredictions.
T here are a num ber ofissues that need to be understood better in current num erical
sim ulations. Even the basic question ofthe identi cation ofsubstructures needs to be explored further.T heSUBFIND algorithm weadopted only identi esbound subhaloes,however,
tidalstream s from disrupted system s (for exam ples in the M ilky W ay,see,e.g.,Ibata et al.
2001;Yanny etal.2003)m ay also contributeto thebudgetofsubstructures. A notherim portant issue is w hether our results have achieved convergence as a function ofthe spatialand
m ass resolutions. N ew sim ulations are underway to address this issue. Presum ably w hen
the num ericalresolution becom es higher, the inner parts of subhaloes are resolved better
into higher-density regions that can survive tidaldisruptions longer. H owever,the survival
ofsubstructures m ay be linked to another sm all-scale problem ofC D M :the centraldensity
pro les oflow -surface brightness galaxies (usually w ith circular velocities of. 100km s 1)
seem to be too concentrated com pared w ith observed galaxies (e.g., Bolatto et al. 2002;
W eldrake et al. 2003;see,however,Swaters et al. 2003). T herefore,ifwe put in observed
m asspro les,substructuresm ay actually bem oreeasily destroyed by tidalforcesdueto their
{7{
lowercentralconcentrations. T here isanothere ectthatm akesthesurvivalofsubstructures
in the centralpart m ore di cult. In collisionless num ericalsim ulations,the density pro le
can be approxim ated by an N FW pro le;the density scales as / r 1 out to 0:1rvir ( 25
kpc) for typicalgalactic-sized haloes. H owever, the observed velocity dispersion is nearly
constant in the inner part im plying / r 2, i.e., the density in real galaxies rises m ore
quickly as the radius decreases. H ence substructures w illbe disrupted m ore easily ifthey
com e close to the center and dynam icalfrictions dragging them into the center would also
be larger in realistically sim ulated galaxies.
O ursim ulationsresolvesubstructurem assesfrom 108M to 1011M fora 1012M parent
halo. H owever,according to M etcalfet al. (2003),the required substructure m ass is in the
range of104M
108M in the case of2237+ 0305. Ifthe m ass spectrum ofsubstructures
n(M )dM / M 1:8dM extendsalltheway dow n to 104M ,onecan estim atethatthefraction
ofm assin substructures w ith 104M < M < 108M isabouta factorof3 sm allerthan that
in substructures w ith 108M < M < M 11M ,i.e.,the m ass fraction in substructures in the
range required by M etcalfet al. (2003) w illbe even lower than our predicted value. T his
conclusion depends on the m ass spectrum . H owever, for any power-law spectrum w ith a
slope shallower than 2,the m ass fraction in substructures w ith 104M < M < 108M is
likely sm aller than that in higher-m ass substructures identi ed in num ericalsim ulations.
T here is another e ect that m ay reduce the utilizable m ass in substructures even further: the need for com pactness, an issue we already touched upon in x2 (see Fig. 1). In
order to a ect the ux ratios signi cantly,the physicalsize ofsubstructures m ust be su ciently com pact (cf. M etcalfet al. 2003). T he m ost e cient substructures should be ofthe
sam e order ofthe im age separation ofthe pairs or triples. For the ve C LA SS quadrupole
lenses that have know n lens and source redshifts (see x4),the closest pairs have projected
separations from 0:6h 1kpc (forB1422+ 231)to 2:3h 1kpc (forM G 0414+ 0534).A tredshift
of0.5,dark m atter haloes w ith M . 4 106M w illhave rvir 1h 1kpc,and so they w ill
be e cient in causing ux anom alies ifthey are located between close pairs or triples. For
larger m asses,the e ect is reduced. W e estim ate the reduction by assum ing that only the
m assenclosed w ithin 1h 1kpc w illa ectthe ux ratios.W e adopta m assspectrum fordark
m atter haloes of n(M )dM / M 1:8 for 104M < M < 1011M . Each halo is described
by an N FW pro le w ith a concentration param eter given by c 11(M =1012h 1)0:15 (e.g.,
Zhao et al. 2003). N ote that the substructures in the centralparts are tidally truncated
and so cannot be tted wellby N FW pro les,but as m ost substructures have r > 0:1rvir
in sphericalradius,the e ect oftidaltruncation m ay be m odest. W e nd that the m ass in
substructures that can e ciently cause ux anom alies is reduced by an additionalfactor of
ve com pared w ith the totalm assin allsubstructures { m ostsubstructures w hich are in the
outer parts are too extended to cause ux anom alies e ciently. O ur rough estim ate show s
{8{
that the com pactness requirem ent is an issue that needs to be addressed m ore carefully.
T he two additionale ects noted can reduce the likely m assfraction in substructures having
the required m asses and sizes to as low as 0:03% ,uncom fortably low com pared w ith the
observationalrequirem ent.
Progress can be m ade from both the observationaland theoreticalfronts to reduce the
uncertainties. O bservationally, in the radio, the e ect of scattering can be studied w ith
observations at high frequency w here it is expected to be unim portant. In the infrared,
it would be interesting to have m ore observations w ith integral eld spectroscopy sim ilar
to that reported by M etcalf et al. (2003). T his m ethod o ers a way to separate stellar
m icrolensing from substructure m illi-lensing. M ore astrom etric signatures ofsubstructures
w illbe im portant as well(see x2). T heoretically,higher-resolution sim ulations are needed
and are already under-way. Iffuture observations and num ericalsim ulations stillindicate
a discrepancy between lensing requirem ents and C D M predictions,then alternatives m ust
be sought. O ne possibility is that these substructures are m assive black holes w ith M
105 106M (Lacey & O striker 1985; X u & O striker 1994), w hich satisfy the m ass and
com pactness requirem ents. W e require only a few percent of the surface density in the
substructures,so thedensity param eterin theseblack holesisonly fsub 0 0:012(fsub =0:04),
w hich is about 30% (fsub =0:04) ofthe baryon density in the universe. T hese m assive black
holes w illhave other observable signatures (e.g.,W am bsganss & Paczynski1992;Trem aine
& O striker 1999) and can be further tested.
W e thank I.Brow ne,H .J.M o,T .York and S.W hite for helpfuldiscussions,X .K ang
for identifying subhalos and V .Springelfor providing us w ith his SUBFIND code. Y PJ is
supported in part by N K BR SF (G 19990754)and by N SFC .
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T his preprint was prepared w ith the A A S LATEX m acros v5.2.
{ 11 {
Fig. 1.| Predicted fractions of dark m atter surface m ass density in substructures, fsub ,
as a function of the projected radius, R , in units of the virial radius, rvir. T he lensed
im ages typically have R =rvir = 1%
3% . T he left panels show fsub as a function of R
for all substructures w ithin the spherical radius r < rvir w hile the right panels include
all substructures w ith r < 2rvir. In the lower panels we only include the m ass w ithin
0:025rvir foreach subhalo. T he solid line in each panelindicates an unweighted least-square
t(logfsub = logA + logR =rvir,i.e.,fsub = A R =rvir). T he A values are,0.25,0.31,0.20 and
0.17,clockw ise from the top left to the bottom right panels,respectively.