Quasar Microlensing at High Magnification and the Role of DarkMatter: Enhanced Fluctuations and Suppressed Saddlepoints

Paul L. SchechterDepartment of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA

02139Institute for Advanced Study, Olden Lane, Princeton, NJ 08540

and

Joachim WambsganssUniversitat Potsdam, Institut fur Physik, Am Neuen Palais 10, 14467 Potsdam, Germany

ABSTRACT

Contrary to naive expectation, diluting the stellar component of the lensing galaxy in a highlymagnified system with smoothly distributed “dark” matter increases rather than decreases themicrolensing fluctuations caused by the remaining stars. For parameters typical of a quadruplyimaged QSO, saddlepoints (as opposed to minima) of the arrival time surface are much morestrongly affected. For a mass ratio of smoothly distributed (dark) matter to clumpy (microlensing)matter of 4:1, a saddlepoint with a macro-magnification of µ = 9.5 will spend half of its time morethan a magnitude fainter than predicted. The anomalous flux ratio observed for the close pair ofimages in MG0414+0534 is a factor of five more likely than computed by Witt, Mao and Schechter(1995) if the dark matter fraction is as high as 93%. The magnification probability histogramsfor each macroimage exhibit structure that varies with the dark matter content, providing ahandle on the dark matter fraction. Enhanced fluctuations can manifest themselves either inthe temporal variations of a lightcurve or as flux ratio anomalies in a single epoch snapshot of amultiply imaged system. While the milli-lensing simulations of Metcalf and Madau (2001) alsogive larger anomalies for saddlepoints than for minima, the effect appears to be less dramaticfor extended subhalos than for point masses. Furthermore, micro-lensing is distinguishable frommilli-lensing because it will produce noticeable changes in the magnification on a time scale of adecade or less.

Subject headings: cosmology: gravitational lensing, dark matter; quasars: MG0414+0534

1. Introduction

Simple gravitational lens models have provenremarkably successful at reproducing the observedpositions of multiply imaged objects. But theagreement between the flux ratios predicted bythese models and the flux ratios actually observedis mediocre at best. At the level of several tenthsof a magnitude, far worse than the observationalaccuracy, the models begin to fail.

These shortcomings have variously been at-

tributed to the effects of intervening dust (Lawrenceet al. 1995), microlensing by the stars which com-prise the lens (Witt, Mao and Schechter 1995,hereafter WMS) or milli-lensing by galactic sub-structure (Mao and Schneider 1998; Dalal andKochanek 2002; Metcalf and Madau 2001).

The discrepancies can be particularly obviousfor the case of a quadruple lens with a close pair ortriplet of images. The best known example is thatof MG0414+0534 (Hewitt et al. 1992; Schechterand Moore 1993) for which the observed A2/A1

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flux ratio is a factor of two smaller than the 1:1that is observed at radio wavelengths (Trotter etal. 2000) and is typical of most models. Assum-ing that the lensing galaxy was comprised entirelyof stars, it was argued in WMS that microlensingwould, on rare occasions, produce such a dramaticflux ratio. Witt et al. (1995) also emphasizedthat the expected fluctuations are larger for im-ages that are saddlepoints (as opposed to minima)of the arrival time surface.

Our interest in anomalous flux ratios wasreawakened by four recent discoveries (Reimerset al. 2002; Inada et al., in preparation; Burleset al., in preparation; Schechter et al., in prepara-tion) of quadruple systems with similar anomalouspairs of images. In three cases (and possibly inthe fourth), the fainter image is a saddlepoint andthe brighter image is again a minimum, as wasthe case in MG0414+0534. While there has been,as yet, no systematic survey, the present paperproceeds on the working hypothesis that this phe-nomenon may be fairly general.

We find that the expected brightness fluctua-tions for highly magnified macro-images of quasarsare enhanced as one relaxes the assumption thatthe lensing galaxy is comprised entirely of starsand admits a substantial, smoothly distributed“dark” component. While the fluctuations forminima are only slightly larger, those for saddle-points are very much larger. In both cases themagnification distributions become asymmetric,with a substantial probability that saddlepointswill be very much fainter than predicted by macro-models.

Our argument builds on the finding by Deguchiand Watson (1987, 1988) and Seitz and Schneider(1994) that as one increases the surface densityof microlenses – at fixed shear – the microlensingfluctuations at first increase and then decrease asone approaches infinite magnification. Here we fo-cus on the fluctuations resulting from a differentslice through the plane spanned by surface densityand shear. This locus traces the values of the “ef-fective” surface density κeff and “effective” shearγeff (Paczynski 1986) obtained as one graduallysubstitutes a smoothly distributed (dark) matterdensity κc for the clumpy stellar mass density κ∗.The effect that we find can be reconstructed (atleast with the benefit of hindsight) from the pre-viously published microlensing lensing simulations

(Wambsganss 1992; Lewis and Irwin 1995, 1996).In §2 we review a few basics of macro- and

micro-lensing. In §3 we consider a toy model forthe two effects we seek to explain. In §4 we ex-amine microlensing simulations for parameters ap-propriate to quadruple systems. In §5 we com-pare results of simulations to the specific case ofMG0414+0534. In §6 we extend our discussion toother systems. In §7 we discuss how ensembles ofsystems might be used to estimate the dark matterfraction in galaxies. In §8 we consider the conse-quences of our findings for milli-lensing by sub-halos, fitting lens models and the measurement ofH0.

2. Macro- and Micro-lensing Basics

2.1. macro-models

Following Witt et al. (1995), we restrictourselves to modelling macro-lenses as singularisothermal spheres (SIS) with an external tidalfield. Images form at minima and saddlepointsof the Fermat travel-time surface (Blandford andNarayan 1986). Formally there is also an im-age at or near the central maximum, but for theSIS model this maximum is infinitely demagni-fied. An approximate relation is derived in WMSbetween the dimensionless surface density (theconvergence), κtot, and the combined effect of alltides – the shear, γ – at the positions at whichimages form in such a model,

γ ≈ 3κtot − 1 . (1)

The magnification µmacro of a macro-image isgiven by the inverse of the product of the eigen-vectors of the curvature matrix,

µmacro =1

[(1− κtot) + γ][(1− κtot)− γ]. (2)

An image is a minimum of the travel-time surfaceif 1− κtot − γ > 0, a saddlepoint if 1− κtot − γ <0, and a maximum if 1 − κtot + γ < 0. Whilemaxima and saddlepoints may have |µmacro| < 1,indicating demagnification, minima must alwaysbe magnified.

For parameters typical of quadruple systems,the images are magnified by factors µmacro ≈5−20. The typical close pair of macro-images in aquad might have (κtot, γ) = (0.475, 0.425) for the

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minimum and (0.525, 0.575) for the saddlepoint,giving them magnifications of µminimum = 10.5and µsaddle = −9.5, respectively, with the negativemagnification indicating the parity flip associatedwith a saddlepoint. We shall use these values forsimulations presented in §4.

2.2. micro-images

Introducing small scale perturbations in thelens potential produces new hills, valleys andridges that are stationary points of the Fermat sur-face thereby producing micro-images. For a pointsource, what we think of as a coherent macro-image will then be comprised of many such mi-croimages. Paczynski’s (1986) Figure 1 illustratesthis especially well. For point mass perturbers themicroimages are either minima or saddlepoints.

For both macro-saddles and macro-minima,there is a nearly infinite number of faint micro-saddlepoints, one for every star. In general thereare extra negative/positive parity pairs of micro-images, with the mean number increasing with in-creasing macro-magnification. Wambsganss, Wittand Schneider (1992) give an expression for thedependence of the mean number of extra imagepairs on the density of microlenses in the absenceof external shear. Depending upon the accidentaldistribution of microlenses, a macro-saddlepointmay or may not have micro-minima. By contrast,a macro-minimum must have at least one. Aswith macro-images, micro-saddlepoints may behighly demagnified but micro-minima must havemagnifications greater than unity.

3. Suppression of Saddlepoints: Explana-tions

Witt et al. (1995) noticed that the fluctuationsin their simulations of microlensing were largerfor saddlepoints (standard deviation σsaddle ∼ 0.9mag) than for minima (σminimum ∼ 0.6 mag).They offered the following explanation.

The macroimages of positive parity havesmaller fluctuation because their magnifi-cations must be larger than unity whereasthe macroimages of negative parity have nolower limit in magnification.This sounds plausible but for the simulations

presented in the next section we find that the fluc-tuations are much larger for the saddlepoints even

though the magnifications rarely, if ever, dip belowunity. Here we offer a toy model which we believecaptures the elements of the effect. Readers whoprefer not to be toyed with may wish to skip to §4and then return to it.

3.1. a toy model

We consider the extreme case of a lens in whichall but an infinitesimal fraction of the mass is ina smooth dark component. We then introduce asingle point mass perturber and examine its effectson the total magnification of macro-saddlepointsand macro-minima. Our treatment is essentiallythat of Chang and Refsdal (1979; 1984), thoughwith different emphasis and notation.

A macro-image is characterized by a local valueof the convergence κtot and the shear, γ. In the ab-sence of the perturber, the magnification is givenby equation (2) above. For the highly magnifiedimages typical of quadruple systems, the curva-ture matrix for a saddlepoint (or minimum) hasa steep minimum directed approximately radiallyoutward from the lens center and a broad max-imum (or minimum) in the tangential direction.In the absence of microlenses each macro-imageis comprised of exactly one micro-image with theexact properties expected for the macro-image.

For both cases, macro-minimum and macro-saddlepoint, we introduce a point mass perturber,which for simplicity, we place directly along theline of sight to the macro-image. The macro-minimum is split into 4 microimages: two micro-minima along the tangential direction with mag-nifications

µminimum =1

4γ[(1− κtot)− γ], (3)

and two micro-saddlepoints along the radial direc-tion with magnifications

µsaddle =1

4γ[(1− κtot) + γ]. (4)

The same perturber along the line of sight to themacro-saddlepoint produces only the two micro-saddlepoints along the radial direction, with thesame magnification as in the case of the macro-minimum.

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Summing over the microimages we have

µtot,minimum =(1− κtot)

γ[(1− κtot) + γ][(1− κtot)− γ](5)

for the macro-minima and

µtot,saddle =12 [(1− κtot)− γ]

γ[(1− κtot) + γ][(1− κtot)− γ](6)

for the macro-saddlepoints, where we have deliber-ately expanded the denominator to show how themacro-magnification, given by equation (2) factorsout of these. Taking γ ≈ κtot, typical of the SIS,and letting κtot → 1

2 , we find

µtot,minimum → µmacro (7)

for the macro-minimum and

µtot,saddle → 1 (8)

for the macro-saddlepoint. The perturbed macro-minimum gets brighter by a small factor, whilethe perturbed macro-saddlepoint gets fainter bya large factor, exactly cancelling out the macro-magnification. This is the essence of our ex-planation for the different behavior of macro-saddlepoints and macro-minima.

Of course the probability of a such direct hitis vanishingly small. Moving the perturber awayfrom the macro-minimum, the total magnifica-tion increases, slowly at first but then rapidly.A pair of micro-images merge and the magnifi-cation drops to something close to the predictedmacro-magnification. As the perturber moves offto infinity, a single micro-minimum is left whichasympotically approaches the predicted macro-magnification.

Moving the perturber away from the macro-saddlepoint along the radial direction, one sad-dlepoint decreases in brightness while the otherincreases, asymptotically approaching the macro-magnification. Moving it away along the tangen-tial direction, the total magnification increasesat first slowly but then dramatically with thecreation of a new pair of micro-images. Itthen falls dramatically after the newly createdmicro-minimum merges with one of the originalmicro-saddlepoints. One of the two remainingmicro-saddlepoints asymptotically approaches themacro-magnification.

In both cases, moving the perturber off theline of sight introduces some additional fluctua-tions in the sum of the micro-images, but it doesnot change the fundamental behavior. Most ofthe time the perturbed macro-minimum is slightlybrighter than it otherwise would have been, whilefor much of the time the macro-saddlepoint is verymuch fainter than it would have been.

What happens as we increase the mass fractionin microlenses? As long as the microlenses aresparsely distributed, they don’t interfere with eachother. The average number of extra micro-minimais less than unity and their magnification distribu-tion looks much as it did for the single perturber.But as the microlens density increases, higher or-der caustics begin to form, the average number ofextra micro-minima grows, and their magnifica-tions are influenced by local density fluctuationsrather than the global parameters. The total mag-nification is proportional to the number of micro-minima (Wambsganss et al. 1992), with fractionalfluctuations decreasing as 1/

√N . The fractional

fluctuations therefore have a maximum somewherebetween 100% smoothly distributed (dark) matterand 100% clumpy (microlensing) matter.

Though macro-maxima are only rarely seen inlensed systems (and not in the systems consideredhere) our toy model helps in their interpretationas well. A perturber directly along the line ofsight to a macro-maximum makes the curvatureof the maximum infinite, reducing the magnifica-tion to zero. Moving the perturber away from themacro-maxium at first produces no change – noflux. At some point a new micro-maximum/micro-saddlepoint pair is created. The micro-maximummoves toward the macro-maximum and the micro-saddlepoint moves closer to the perturber.

4. Microlensing Simulations

The handwaving of the preceding section can betested by numerical simulations of microlensing, aspioneered by Paczynski (1986), Schneider & Weiss(1987), Kayser, Refsdal, Stabell (1987), and elab-orated by Wambsganss (1990), Witt(1993), Lewiset al. (1993).

4.1. choice of parameters

At first it might seem that the space of sim-ulations should be three dimensional, as parame-

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terized by the shear, γ, the convergence providedby smoothly distributed dark matter, κc, and theconvergence contributed by a clumpy stellar com-ponent, κ∗, with κc + κ∗ = κtot. Paczynski (1986)has shown that there is a scaling such that for anychoice of these there is an equivalent model withan effective shear

γeff =γ

(1 − κc)(9)

and an effective convergence,

κeff∗ =

κ∗(1− κc)

. (10)

but no smooth component. One must also scaleeach dimension of the source plane by a factor(1 − κc)−1 and the resulting magnifications by afactor (1− κc)−2. To investigate the effect of sub-stituting continuous dark matter for stellar mat-ter, one starts with a microlensing simulation inwhich κ∗ = κtot and γ are given by the macro-model, then gradually increases κc and decreasesκ∗ keeping κtot constant.

In Figure 1 we show the (κeff∗ , γeff) plane.

Macro-minima lie in the shaded region in the lowerleft; macro-maxima lie in the cross-hatched regionto the lower right; macro-saddlepoints lie in the re-maining triangular region. The dotted line showsthe γeff = κeff locus, which is appropriate to theunperturbed SIS model. The hyperbolae marklines of constant µeff , defined by

µeff =1

[(1 − κeff∗ ) + γeff ][(1− κeff∗ )− γeff ].

(11)Note that µeff = (1 − κc)2µtot and is equal to thetotal magnification only in the absence of darkmatter. The symbols show the effect of increas-ing the dark matter fraction in a pair of models.They terminate on the κeff

∗ = 0 axis for 100% darkmatter.

We have carried out two pairs of microlensingsimulations. The first of these follows a minimumand a saddlepoint with total magnifications ∼ 10as we add increasing proportions of dark matter,with κeff and γeff given by the symbols in Figure1. The second was chosen to permit direct com-parison with the WMS models for MG0414+0534,which have total magnifications ∼ 25, and are alsocarried out with increasing proportions of darkmatter. The relevant parameters are given in Ta-ble 1.

4.2. image magnification |µtot| ∼ 10

The character of the brightness fluctuationsis revealed by mapping the magnification of thesource as a function of its position in the sourceplane (Wambsganss, Paczynski & Schneider 1990).In Figure 2 we show the magnification maps forthe “typical” macro-minimum (on the left) andmacro-saddlepoint (on the right) with dark mat-ter fractions of 0%, ∼ 85% and ∼ 98%, from topto bottom.

Figure 3 presents magnification histogramscomputed for each of these. These are the top,middle and bottom panels. In the second andfourth panels we include intermediate histogramsfor dark matter fractions ∼ 75% and ∼ 95%. Forthe sake of comparison with our toy model, we dis-cuss the 98% dark matter model first and proceedto higher densities of microlenses.

Both the magnification maps and the his-tograms for the 98% dark matter cases confirmthe qualitative results of our toy model. For themacro-minimum, the magnification is mostly justthat given by the smooth macromodel. Thereare isolated diamond shaped caustics that sur-round “plateaus” of slightly higher magnificationwhich correspond to the 4 microimage regions de-scribed in the toy model. They have one extramicro-minimum. The sharp transitions betweenthe typical magnification and the plateaus arethe caustics, marking the points at which a newmicro-saddlepoint/micro-minimum pair is createdor annihilated. Students of stellar and planetarymicrolensing will recognize these configurations asthe magnification map of a planet lying just out-side the Einstein ring of its parent star (Changand Refsdal 1979, 1984; Mao & Paczynski 1991;Wambsganss 1997).

For the macro-saddlepoint, the magnification isagain mostly just that given by the smooth macro-model. But there are occasional “lagoons” of verylow magnification, bracketted by small triangu-lar “calderas” of high magnification. The lagoonsappear whenever a microlens lies directly alongthe line of sight. The calderas mark the 4 mi-croimage regions described in the toy model, andhave one extra micro-minimum. The lagoons haveno micro-minima. This lagoon/twin-triangle con-figuration corresponds to the magnification mapof a planet inside the Einstein ring of its parent

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Table 1

Simulation Parameters

ID κtot γ µtot

M10 0.475 0.425 10.5S10 0.525 0.475 -9.5

M25 0.472 0.488 24.2S25 0.485 0.550 -26.8

star (Chang and Refsdal 1979, 1984; Wambsganss1997).

An important difference between the magnifi-cation maps (and histograms) for the 98% (and95%) cases is that while the macro-saddlepoint isskewed heavily toward demagnification, it also un-dergoes strong magnifications. By contrast themacro-minimum undergoes only modest demagi-fications and somewhat stronger magnifications.It will therefore not be surprising if, with moremicrolenses, the magnification histogram for themacro-saddlepoint is broader than that for themacro-minimum.

The magnification maps for the 85% casesbear some resemblance to the 98% dark mat-ter maps, but are far more complex. For themacro-minimum the diamond caustics now beginto overlap, sometimes multiple times, producingsuccessively higher plateaus. The area completelyoutside the caustics, regions that produce onlyone micro-minimum, now have clearly lower thanaverage magnification. The corresponding magni-fication histogram shows two distinct peaks asso-ciated with regions of one and two micro-minima(cf. Rauch et al. 1992). In the 75% dark mat-ter histogram this bifurcation is even more pro-nounced.

For the macro-saddlepoint the lagoons, regionswith no micro-minima, have now grown to thepoint at which they dominate the magnificationmap. The triangular calderas have also grown,to the point at which their caustics sometimescross, producing regions with two, three and fourmicro-minimum images. The magnification his-togram shows two distinct peaks, corresponding toregions with zero and one micro-minima, respec-tively, and perhaps a plateau corresponding to re-

gions with two micro-minima. The histogram hasa long tail toward high magnifications and dropsabruptly at low magnifications. This low mag-nification dropoff occurs very near the minimummagnification allowed by our toy model, µtot = 1.Again, the 75% dark matter histogram shows ayet more pronounced bifurcation.

In 0% dark matter simulations the macro-minimum and macro-saddlepoint have begun toresemble each other. For the macro-saddlepoint,the low magnification lagoons, still with no micro-minima, have been crowded out by the expandingweb of caustics. Most of the magnification map iscovered by regions with large numbers of micro-minima. The web of caustics has also expandedfor the macro-minimum. The region outside allcaustics looks much the same as for the macro-saddlepoint, though these regions still produceone micro-minimum.

We get the general impression that the dia-mond and lagoon/twin-triangle configurations arethe building blocks from which the higher densityconfigurations are constructed. We start out withrelatively isolated features. As the surface densityof perturbers increases, the fluctuations increaseas the building blocks begin to cover the sourceplane. Increasing the density of perturbers yetfurther, the building blocks overlap and begin toaverage out, decreasing the fractional amplitudeof the fluctuations.

4.3. image magnification |µtot| ∼ 25

Witt et al. (WMS) considered the specific caseof MG0414+0534, a quadruple system for whichthe flux ratio of the close bright pair of images,A2/A1, was 0.9 in the radio (Katz and Hewitt1993) and 0.45 ± 0.06 in the optical (Schechter

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and Moore 1993). By contrast their predicted fluxratio was 1.1.

They investigated whether the difference in fluxratios might be due to microlensing, assuming nosmoothly distributed dark matter. Their a poste-riori assessment was that the flux difference be-tween the observed optical flux ratio and the pre-dicted ratio is “unlikely, but not ridiculously so.”

We carried out a second suite of simulations,M25 and S25 in Table 1, using the WMS macro-parameters, but adding increasing proportions ofdark matter, up to 97.5%.

The magnification maps and histograms aresimilar in broad outline but different in detail fromthe M10 and S10 models. For the 0% dark mattercase, the caustic networks are very much denser.The histograms for the 0% case are narrower thanfor models M10 and S10, and are more nearly sym-metric. The M25 and S25 are more similar to eachother.

As the dark matter percentage is increased bothhistograms widen. The low magnification tail forthe macro-saddlepoint drops sharply at µtot ∼ 1,consistent with our toy model. The saddlepointhistogram bifurcates, with the bifurcation mostprominent and the distribution broadest for the93% dark matter model.

There is a simple reason why this is the value forwhich the probability distribution has the largestdispersion: It corresponds to the case where κc =1 − γ. Going back to the definition of the mag-nification (equation 2), this behaviour can be eas-ily explained. If κtot is expressed as the sum ofκc + κ∗, and κc = 1− γ, then the terms (1− κc)2

and −γ2 cancel, and the (local) magnification isdetermined by the small and highly variable (lo-cal) value of κ∗, which leads to very large fluctu-ations of the (local) magnification and hence to alarge dispersion in the magnification probabilitydistribution.

5. Microlensing and MG0414+0534

5.1. the A2/A1 flux ratio

Witt, Mao and Schechter (1995) present his-tograms of magnitude differences,

∆mA1/A2 ≡ −2.5 logA1

A2(12)

for three values of the ratio of the Gaussian sourceradius, rs to the Einstein ring radius ξE . They es-timate that the effect of letting rs/ξE → 0 wouldbroaden the histogram for rs/ξE = 0.04, theirsmallest value, by an additional 10%.

Figure 4 shows ∆mA1/A2 histograms computedfor four of our simulations (cases M25 and S25)for which we have taken rs/ξE = 0.04. The firstof these, with no dark matter, satisfactorily repro-duces the WMS histogram, with small differences,consistent with differences in the simulation tech-nique. The second, with 87.5% dark matter, isconsiderably broader. Moreover, it is asymmetric,with A2 more likely to be fainter than A1.

The third, for 92.5% dark matter, shows thebroadest distribution. With still increasing frac-tion of dark matter, the distributions quickly getnarrower again, as the fourth column for 97.5%dark matter illustrates, though it does show asmall plateau.

For the case of 0% dark matter, the probabil-ity that A2 will be more than 0.87 magnitudesfainter than A1 is 0.068 (cf. WMS). For the 87.5%dark matter case, that probability has grown to0.28, and to 0.35 for the 93% dark matter case,an increase of more than a factor of five. We con-clude that if the dark matter surface mass den-sity along the line of sight were 87.5% or 93% ofthe total, flux ratios as extreme as that seen inMG0414+0534, would not be uncommon as longas the saddlepoint image is the fainter of the twoimages.

5.2. dark matter and source size

Witt Mao and Schechter (1995) obtained a95% confidence upper limit on the source size for

MG0414+0534 of 1016 cm ×(< M > /0.1M�)12

for the case of no dark matter. As increasing thedark matter content increases the fluctuation am-plitude, that upper limit on the source size is re-laxed.

Most simulations of microlensing use a finitesource size, parameterized by rs/ξE . This adds athird dimension to one’s model space. A roughidea of the effect of source size can be had by tak-ing the source to be composed of two components– one very compact, so that point source simula-tions suffice, and one very extended, so that themacro-magnification holds. The resulting magni-

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fication map is then just a luminosity weightedaverage of a point source map and a uniform map.

The effect of increasing the extended fractionof a source is to compress the magnification his-tograms of Figure 3 horizontally, preserving theirshapes. Thus while increasing the extended frac-tion of the source would to first order mimicdecreasing the dark fraction of the lens, detailsof the magnification histograms will be differentand residuals from an ensemble of identical sys-tems would in principal permit further tighter con-straints on both the dark matter fraction and theextended source fraction. The source and lens inMG0414+0534 are surely moving with respect toeach other. If we wait long enough, we will get anew, statistically independent set of flux ratios.

6. Extension to Other Systems

We have studied the case of MG0414+0534 indetail because its flux ratio discrepancy is wellknown and has been discussed at length in the lit-erature. But we are aware of three recently discov-ered systems (Inada et al., in preparation; Burleset al., in preparation; Schechter and Wisotzki, inpreparation) which exhibit a similar anomaly. Ourdecision to revisit the case of MG0414+0534 wasinspired by these discoveries.

6.1. HS0810+2554

Reimers et al. (2002) recently reported thediscovery of HS0180+2554, a quadruple systemvery much like MG0414+0534, but a factor of 2.5smaller in angular size. It has a bright close pairof images, A and B whose magnitudes differ by 0.7magnitudes – a factor of two. The brief letter byReimers et al. (2002) does not include positionsfor the four images or the lensing galaxy, but thesecan be measured by counting pixels on the pub-lished image and adopting a scale of 0.′′05 per pixel.Our measurements are reported in Table 2.

We have produced a number of models for thissystem, none of which is entirely satisfactory. Thesystem is so nearly symmetric that it is difficultto ascertain the parity of the images – it dependsupon the errors assigned to the galaxy positionand whether or not one uses the flux ratios as con-straints. Results for one such model are given inTable 2. While the parity is uncertain, the mod-els have many features in common. In particular

the close pair of images is highly magnified, theshear is quite small, the (projected) position ofthe source is very close to that of the lens, and thetwo fainter images are comparable in brightness.The 0.7 magnitude difference between A and Bmight be due to absorption by dust, as was arguedfor MG0414+0534 (Lawrence et al. 1995), but itmight also be the product of microlensing. Such astrong microlensing effect for such a high magnifi-cation would be very unlikely – unless there weresubstantial amounts of dark matter to broaden themagnification histogram.

The magnifications in our model are unusuallylarge, and the shear unusually small. The pre-dicted magnification would be reduced a factor of2 if we adopted a model in which the quadrupoleterm in the potential was due to the lensing galaxyrather than external tides.

7. The Dark Matter Fraction as Deter-mined from Ensembles

As noted in our discussion of MG0414+0534,one could in principal measure the dark mattercontent of a single lens by sampling the magnifica-tion histograms many times, waiting for the sourceand lens to move with respect to each other, givena macro-model with accurate values of shear andconvergence. As this could be inconvenient, analternative would be to assemble an ensemble oflenses and to look at the distribution of intensityratio residuals from best fitting models. Severalsurveys for lenses are underway that might pro-duce such samples. In particular the Hamburg-ESO survey (Wisotzki et al. 1996) and the SloanDigital Sky Survey (Schneider et al. 2002) willultimately yield large number of lenses.

7.1. point sources

The treatment of an ensemble of lenses is moststraightforward in the limit where rs/ξE is small.For every image in every quadruple in the sam-ple, one would obtain a model (ideally not usingthe flux ratios as constraints) and compute mag-nification histograms for each of the four imagesfor a range of values of κc/κtot. One would thenconstruct a likelihood function from the productof the probabilities obtained from the histogram.There would be an additional free parameter foreach system – a number which converts observed

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Table 2

Positions for HS0812+2554

Image x(”) y(”) µpred

A -0.450 -0.100 114B -0.365 -0.250 -117C 0.300 -0.375 20D 0.165 0.450 -15

flux into magnification. Maximizing the likelihoodwould give a best value for the dark matter frac-tion, κc/κtot.

A variant of this approach would be to assign arange of mass-to-light ratios to the observed sur-face brightness profile for each lensing galaxy, at-tributing the shortfall in surface density to darkmatter.

7.2. finite sources

There are both theoretical and observationalreasons to think that rs/ξE is not vanishinglysmall, at least for some systems. The one-size-fits-all approach would be to assign a single valueof rs/ξE to every system (or a fixed physical sizers, scaled properly by the lens-dependent ξE), andthen maximize the likelihood function with thisadditional variable. One might use a Gaussiansurface brightness profile for the sources or, forthe sake of computational speed, treat the quasaras a point source embedded in a very extendedsource. More realistically, but at the cost of con-siderable additional complexity, one could alloweach system to have a different (e.g., luminosity-dependent) source size and profile.

7.3. double image systems

We have emphasized quadruple systems ratherthan doubles because the effects of dark mattermight be expected more important at high mag-nifications. The results of the previous sectionswould argue that even doubles that are not highlymagnified might still suffer substantial microlens-ing.

However, doubles suffer from another difficulty– a dearth of model constraints. One needs 5 con-

straints to obtain the simplest SIS with externalshear model, and doubles have only four positionalconstraints – the two image positions relative tothe lens. The flux ratio is often taken as the fifthconstraint, but using it as a constraint will give aperfect fit to the model. Doubles are not beyondredemption however; one might use radio flux ra-tios, emission line ratios (Wisotzki et al. 1993), ormid-IR flux ratios (Agol et al. 2000) to constrainthe model on the hypothesis that the emission re-gions are large compared to the Einstein rings ofthe microlenses1.

8. Further Consequences

8.1. milli-lensing and mini-halos

The present work is fundamentally similar to(though on the surface quite different from) thatof Dalal and Kochanek (2002) and Metcalf andMadau (2001), who argue that the brightness ratioanomalies observed in quadruple systems are dueto the presence of dark matter mini-halos.

The differences are obvious. They substituteincreasing amounts of clumpy dark matter forsmoothly distributed luminous matter. We sub-stitute increasing amounts of smoothly distributeddark matter for clumpy luminous matter. But thephenomenon we seek to explain is the same.

We find, as they do, that a small micro- (ormilli-) lensing fraction produces surprisingly largeflux ratio anomalies. In our case we find this tobe especially true for saddlepoints.

But the situation for extended micro- or milli-lenses is somewhat different than for point mass

1Such observations might also be used as additional con-straints in models for quadruple systems.

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perturbers. If we follow the same line of reason-ing as in the toy model, but taking the perturbersto be singular isothermal spheres directly alongthe line of sight to the macro-images, we find thatboth the micro-minima and micro-saddlepointshave twice the magnification that the correspond-ing point mass perturber would have had. Inthe limit as γ ≈ κtot → 1

2 , the SIS perturberdoubles the magnification of a macro-minimumand reduces the maco-saddlepoint but still leavesit a factor of two brighter than the unmagnifiedsource. We therefore expect extended sourcesto produce larger fluctuations for macro-minimaand smaller fluctuations for macro-saddlepointsthan would point mass perturbers. The anomaliesought to be more uniformly distributed among themultiple images, and not so heavily weighted to-ward the saddlepoints. Perhaps this is why Dalaland Kochanek (2002) made no distinction amongmacro-minima and macro-saddlepoints at theirdiscussion (though they appear to have treatedthese correctly in their simulations). Metcalf andMadau (2001) do note an asymmetry between thetwo, and their figures show it to be in the samesense, if not as exaggerated, as those we find forpoint perturbers.

Dalal and Kochanek (2002) argue that theanomalous flux ratios observed at radio wave-lengths are due to milli-lensing by subhalos com-prised entirely of dark matter. Metcalf and Zhao(2001), building on the work of Metcalf & Madau(2001), argue for a similar effect including opticalas well as radio fluctuations in their discussion.

If either the radio or optical continuum emis-sion regions in quasars were small compared tothe Einstein rings of the stars in the lensing galax-ies, the results of the previous sections would un-dermine the conclusions regarding subhalos, sincesome fraction of the observed fluctuations wouldbe due to micro- rather than milli-lensing.

8.2. milli-lensing vs. micro-lensing

In fact, there is a way to distinguish microlens-ing effects that we propose here as an explanationfor the discrepant intensity ratios in close doubleimages, from millilensing, as suggested by Mao &Schneider (1998), Metcalfe & Madau (2001), Met-cald & Zhao (2001), or Dalal & Kochanek (2002).Since quasar, lensing galaxy and observer moverelative to each other, the brightness of one partic-

ular quasar image will change with time due to thefact that the focussing matter in front of it changesposition. The time scale of such fluctuations is oforder a few years up to a decade for microlens-ing by solar-mass stars. Since this timescale isproportional to the Einstein radius of the lenses,it increases with the square root of the mass ofthe lensing objects. Typical masses of substruc-ture clumps range from about 102M� to 106−9M�(Metcalf & Madau 2001, Dalal & Kochanek 2002).This means milli-lensing effects of such substruc-ture clumps should not modify the intensity ratiosof affected multiple-quasars over the professionallifetime of an astronomer, whereas microlensingshould produce variable intensity ratios.

8.3. modelling lenses

The magnification histograms presented in Fig-ures 3 and 4 show that flux ratios are reliable con-straints only in the absence of micro- and milli-lensing. If either is suspected, one will wantto take the expected magnifications into account.Since the histograms can be quite broad and heav-ily skewed, proper treatment demands a full-blownmaximum likelihood analysis rather than the as-sumption of Gaussian residuals. Fitting for fluxesshould work better than fitting for magnitudes(i.e., log flux), since the mean magnitude resid-uals can be quite different from zero.

8.4. the Hubble constant

It has been long known (and has recently be-come well known) that the largest source of sys-tematic uncertainty in time delay estimates ofH0 is the degree of central concentration of lens-ing galaxies (Refsdal and Surdej 1994; Kochanek2002). The available constraints usually give largeuncertainties in the concentrations of lensing po-tentials. While flux ratios add little to constrain-ing potentials, absolute magnifications do, withhigher concentrations giving less magnification.The sharp edges on the magnification histogramsof Figure 3 occur at absolute rather than relativemagnifications. Cases of extreme demagnificationmight therefore place upper limits on the degreeof central concentration. Less extreme flux ratiosmight still help narrow the allowable range.

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9. Summary and Conclusions

We have shown that the substitution of asmoothly distributed dark matter component forthe clumpy stellar surface density increases ratherthan decreases the amplitude of microlensingbrightness fluctuations for the close pair imagesassociated with quadruple gravitational lenses.We have also shown that the magnification prob-abilites for saddlepoints and minima of the Fer-mat arrival time surface are skewed in the op-posite sense, making it likely that saddlepointswill undergo substantial demagnification. Inclu-sion of a dark matter component in models forMG0414+0534 makes the large optical flux ratioof the bright pair of images considerably morelikely under the microlensing hypothesis. Whileno single case is likely to establish the presenceand relative contribution of smoothly distributeddark matter, an ensemble of lensed systems, or along time series for a single system, might do so.

We thank our colleagues who have graciouslyshared news of their discoveries in advance of pub-lication. We gratefully acknowledge generous sup-port from the Institute for Advanced Study andPrinceton University Observatory. PLS is grate-ful to the John Simon Guggenheim Foundation forthe award of a Fellowship.

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Fig. 1.— κeff-γeff-plane. Images corresponding to minima in the arrival time surface form in the hatchedregion; maxima form in the cross-hatched region; saddlepoint images form in the remaining triangular region.Lines of constant magnification are hyperbolae. The symbols indicate the models we investigated (M10 –squares; S10 – stars). The symbols approach the vertical axis as increasing amounts of dark matter aresubstituted for microlensing matter, keeping κtot constant.

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Fig. 2.— Two-dimensional microlensing magnification distribution in the quasar plane, for aminimum (left, models M10) and a saddlepoint image (right, models S10). The color scale rangesfrom dark blue (large demagnification) - light blue - green - red - yellow (large magnification).The total convergence κtot remains constant for each column, whereas the fraction of smoothlydistributed matter increases downward from 0% at the top through 85% in the middle to 98% atthe bottom (corresponding to rows 1, 3 and 5 of Figure 3).14

Fig. 3.— Magnification probability distribution for a minimum (left, models M10, γ = 0.425) and a saddle-point image (right, models S10, γ = 0.475). The total convergence κtot remains constant for each column.The dark matter increases from top to bottom, with fractional contributions of 0%, 75%, 85%, 95% and98%, respectively. The three vertical lines indicate the following: short-dashed: ∆m = 0mag (theoreti-cally expected macro-magnification); dotted: < ∆m > (average magnification in magnitudes); long-dashed:µabs = 1.0 (absolute magnification unity, i.e. unlensed case).

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Fig. 4.— Probability distributions for the magnitude difference between images A1 and A2 of MG0414+0534,with parameters as chosen by WMS: κA1,tot = 0.472, γA1 = 0.488 (our model M25), and κA2,tot = 0.485,γA2 = 0.550 (our model S25). Column 1 illustrates the case of all matter in clumpy form (same as treatedin Figure 4 of WMS); the panel at the top shows the differential probability, the bottom panel presents thecorresponding integrated probability. Columns 2, 3 and 4 show the corresponding probability distributionsfor 88.5% dark matter, 93% dark matter, and 97.5% dark matter, respectively.

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