Is There a Microlensing Puzzle

a r X i v :a s t r o -p h /0505167v 2 6 O c t 2006

Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 2February 2008

(MN L A T E X style ?le v2.2)

The Microlensing Optical Depth Towards the Large

Magellanic Cloud:Is There A Puzzle?

N.Wyn Evans 1,Vasily Belokurov 1

1

Institute of Astronomy,Madingley Rd,Cambridge,CB30HA,UK

2February 2008

ABSTRACT

Using neural networks,Belokurov,Evans &Le Du (2003,2004)showed that 7out

of the 29microlensing candidates towards the Large Magellanic Cloud (LMC)of the MACHO collaboration are consistent with blended microlensing and added Gaussian noise.We then estimated the microlensing optical depth to the LMC to be 0.3×10?7～

<τ～<0.5×10?7,lower than the value τ=1.2+0.4?0.3×10?7claimed by the MACHO

collaboration (Alcock et al.2000).There have been independent claims of a low optical depth to the LMC by the EROS collaboration,who have most recently reported τ<0.36×10?6(Tisserand et al.2006).

Griest &Thomas (2005)have contested our calculations.Unfortunately,their paper contains a number of scienti?c misrepresentations of our work,which we clarify here.We stand by our application of the neural networks to microlensing searches,and believe it to be a technique of great promise.Rather,the main cause of the disparity between Griest &Thomas (2005)and Belokurov et al.(2004)lies in the very di?erent datasets through which these investigators look for microlensing events.Whilst not everything is understood about the microlensing datasets towards the LMC,the latest downward revisions of the optical depth to (1.0±0.3)×10?7(Bennett 2005)is within ～<2σof the theoretical prediction from stellar populations alone.

E?ciency calculations can correct for the e?ects of false negatives,but they cannot correct for the e?ects of false positives (variable stars that are mistaken for microlens-ing).In our opinion,the best strategy in a microlensing experiment is to eschew a decision boundary altogether and so sidestep the vagaries of candidate selection and e?ciency calculations.Rather,each lightcurve should be assigned a probability that it is a bona ?de microlensing event and the microlensing rate calculated by summing over the probabilities of all such lightcurves.

Key words:gravitational lensing –stars:variables:others –dark matter

1INTRODUCTION

The microlensing puzzle is:what is the origin of the microlensing events towards the Large Magellanic Cloud (LMC)?Speci?cally,what fraction of the microlensing events are caused by known stellar components in the Milky Way and by self-lensing of the LMC,and what fraction by a compact dark matter component in the Milky Way halo.Griest &Thomas (2005)argue that there is evidence for an excess of events above and beyond the contribution of the known stellar components in the Milky Way and LMC and hence there is evidence for compact dark objects in the halo.There are two microlensing collaborations who have heroically monitored the Magellanic Clouds over many years.They have reported rather di?erent numbers of events.After 8years of monitoring,the EROS collabora-tion announced just 3microlensing candidates towards the

LMC (Lasserre et al.2000).By contrast,the MACHO col-laboration (Alcock et al.1997)?rst published an analysis of their 2-year dataset.They found a high microlensing opti-cal depth (τ=2.9+1.4?0.9×10

?7

)based on an 8event sample.They suggested that this was consistent with about 50%of the halo within 50kpc being made of objects with mass ～0.5M ⊙.This optical depth value was superseded by the analysis of 5.7years of data,which indicated a somewhat

lower optical depth of τ=1.2+0.4?0.3×10

?7

based on either 13or 17events (Alcock et al.2000).

Belokurov,Evans &Le Du (2004)re-analyzed 22000publicly available MACHO lightcurves with neural net-works,and provided alternative sets of microlensing events.The subset reanalyzed contained all the microlensing can-didates of Alcock et al.(2000),but is only a small frac-tion of the entire public archive of 9million MACHO lightcurves We argued that at least some of the events iden-

2N.W.Evans&V.Belokurov

ti?ed as microlensing by Alcock et al.(2000)may in fact be contaminants.We roughly estimated the optical depth as0.3×10?7～<τ～<0.5×10?7(Evans&Belokurov2004) Subsequently,the EROS collaboration(Tisserand& Milsztajn2005)reported an optical depth to the Large Mag-ellanic Cloud ofτ=(0.15±0.12)×10?7based on3candi-dates found in6.7years of data–a remarkably low result. Here,the error is calculated using Han&Gould’s(1995)for-mula withη=2.0.Note that the EROS estimate provides an upper bound to the contribution of compact dark halo ob-jects to the total optical depth,as an obvious disk lens event was removed.Very recently,EROS have reported further in-teresting results based on the clever use of a bright subsam-ple of source stars to minimise contamination(Tisserand et al.2006).They?nd only one microlensing candidate in this subsample and suggest that the optical depth due to such lenses isτ<0.36×10?7at the95%con?dence level.If these low values for the optical depth are accepted,then the stellar populations in the outer Galaxy and the LMC must provide most of the lenses for the known events–as in fact is true in all instances where the location of the lens can be identi?ed.There are two exotic events towards the LMC, for which the location of the event can be more or less in-ferred.They are the binary caustic crossing event studied by Bennett et al.(1996)and the xallarap event studied by Alcock et al.(2001a).In both cases,the lens preferentially lies in the Magellanic Clouds.In addition,there has been the direct imaging of another of the microlenses by Alcock et al.(2001b),revealing it to be a nearby low-mass star in the disk of the Milky Way.

The claims of Belokurov et al.(2003,2004)and espe-cially Evans&Belokurov(2004)were challenged by Griest &Thomas(2005).Of course,there is no need to re-enact the epic battle between the mice and the frogs(Homer,8th century BC)in the pages of this Journal.Nonetheless,Gri-est&Thomas(2005)did make a number of scienti?cally incorrect statements regarding our neural network compu-tations.The main aim of this paper is simply to set the record straight with regard to the event selection(§2)and the e?ciency calculation(§3).In our discussion(§4),we de-lineate the remaining causes of scienti?c disagreement and discuss ongoing experiments that may provide a resolution. 2REMARKS ON EVENT SELECTION

Any treatment of this subject should begin with some hum-bling remarks.Neither Griest&Thomas(2005)with pow-erful statistical methods nor Belokurov et al.(2004)with neural networks can really claim to have devised methods for microlensing detection that are completely successful.A striking indication of this is provided by the EROS collabo-ration’s discovery that the event MACHO LMC-23is a vari-able star(Tisserand&Milsztajn2005).The lightcurve for this event is a good?t to a blended microlensing curve.Al-cock et al.(2000)report aχ2of1.452per degree of freedom. The event was included in their set of con?dent microlensing events(“set A”).Likewise,Belokurov et al.(2004)assessed the probability of microlensing as P=0.99.Therefore,both methods failed.

The implications of this for microlensing surveys are worrisome.There exist classes of variable stars whose lightcurves are good?ts to blended microlensing.They can-not be distinguished from microlensing,except by more ac-curate photometric measurements or by long-baseline mon-itoring for repeat variations.

Belokurov et al.(2003,2004)pioneered the use of neu-ral networks to identify microlensing by single lenses.Our calculations showed that–using the publicly available data –only7of the events of Alcock et al.(2000)are consis-tent with blended microlensing and added Gaussian noise. These calculations are correct,but the noise in the actual experiment is more complicated than Gaussian.

Notice that the selection of events by Alcock et al. (2000)makes the same assumption that the noise is close to Gaussian,in order to proceed with lightcurve?tting and the use of theχ2statistic.However,the data through which Alcock et al.(2000)search for events is not the publicly available data,but is derived from the publicly available data by a cleaning process(see Alcock et al.1997).We refer to this as the cleaned dataset;it is not publicly available.

Let us compare the noise properties of the public data with the cleaned data.1To estimate the amount of vari-ability in the lightcurve,we calculate theχ2value of the constant baseline model.The empirical cumulative proba-bility distributions P(χ<χ0)are then constructed and shown in Figure1as full and dotted curves.The vertical axis is the fraction of the dataset(and so runs from0to 1).In a typical cumulative probability distribution,the hor-izontal axis would be the chi-squared value,χ0.However, here we have converted this to a probability using the fact that,if the errors are normally distributed,the theoreti-cal cumulative probability distribution is known and is the incomplete gamma functionΓ(0.5,χ0/2)(see Press et al. 1992).In Figure1,the cumulative probability distributions in the full black and dashed black lines refer to the pub-lic data in the MACHO instrumental R and B?lters.The full and dashed grey lines refer to the cleaned data in John-son V and Kron-Cousins R(see Alcock et al.1997).We see immediately that the noise properties of the two datasets are very di?erent.In the ideal case of Gaussian noise and non-varying lightcurves,the probability distributions should have a similar shape but pass through the point(0.5,0.5). For the public data,～90%of the lightcurves correspond to varying objects.By contrast,the noise properties of the cleaned data are much closer to Gaussian.

Let us illustrate the points at issue with an example. Figure2shows the R and B band data of MACHO LMC-4,using the public data.The upper panels show the un-binned data,the lower panels show the binned data.This ?gure should be compared with the published version of the lightcurve of the same event in Alcock et al.(2000).In par-ticular,the lightcurve of Figure2shows secondary activity outside the microlensing bump,for example,at t≈1400 days in the B band.These datapoints are not present in Alcock et al.(2000)and so they must have been rejected at

1One tile only(roughly3000lightcurves)of the cleaned data was made available to us for comparison pur-poses.We thank Andrew Drake for making this pos-sible The public but uncleaned data are available at http://wwwmacho.mcmaster.ca/Data/MachoData.html.

The Microlensing Optical Depth Towards the Large Magellanic Cloud:Is There A Puzzle?

3

Figure 1.The χ2value of the constant baseline model is computed for each lightcurve using (i)the publicly available data from the MACHO website (black curves)and (ii)the cleaned data (grey curves)used in Alcock et al’s (2000)analysis.The full and dashed curves are the empirical cumulative probability distributions P (χ<χ0)in the red and blue passbands respectively.The vertical axis is the fraction of the dataset.The horizontal axis would normally be the chi-squared value χ0in a cumulative probability distribution.However,if the errors are normally distibuted,the theoretical cumulative probablity distribution is known and is the incomplete gamma function P =Γ(0.5,χ0/2).This is plotted as the horizontal axis.If the noise is Gaussian,then the probability distribution passes through the point (0.5,0.5),whose location is marked by the dotted lines.As the cleaned data pass close to this point,we conclude that their noise properties are nearly Gaussian.By contrast,the noise properties of the publicly available data are not close to Gaussian at all.

an early stage in the photometric reduction.There may of course have been good reasons to remove these datapoints.

Griest &Thomas (2005)argue that the greater num-ber of lightcurves identi?ed as microlensing by Alcock et al.(2000)is due to the power of lightcurve ?tting.This is not the case.Lightcurve ?tting will not identify the data in Figure 2as a microlensing event.To show this,we perform exactly the lightcurve ?tting that Griest &Thomas (2005)advocate.The χ2per degree of freedom is 1.73for the pub-licly available lightcurve of MACHO LMC-4.This is higher than the value of 1.38reported by Alcock et al.(2000).In fact,there are no lightcurves in Alcock et al.’s (2000)set A with as high a reduced χ2as 1.73.

In other words,the main di?erences between the re-sults of Belokurov et al.(2004)and those of Griest &Thomas (2005)are caused by the fact that these investi-gators search through di?erent datasets.More succinctly,the publicly available MACHO data are badly polluted with photometric outliers.

Of course,the MACHO collaboration has much more information on photometric problems than is publicly avail-able.For example,they can remove datapoints based on photometric problems that recur for hundreds of stars in the same ?eld for a troubling exposure (Griest 2005,private communication).At least as judged from Figure 1,they seem to have done a good job.Hence,we believe that the cleaned data are more useful than the public data –it is just a pity that the cleaned data are not generally available.

Belokurov et al.(2004)realised that there are problems with the public data and discarded any data-point that de-viates more than 3σfrom its neighbours.Griest &Thomas (2005)suggest that this procedure removes fast,short dura-tion events and so is partly responsible for the discrepancy.

For each lightcurve,Belokurov et al.(2004)analyse both the public data and the data from which 3σoutliers have been removed and quote the maximum output from the neural networks (i.e.,the one for which the probability of microlens-ing is the greatest).Hence,if the public data has a very high probability of microlensing,the event is recognized.Griest &Thomas (2005)assert that Belokurov et al.’s (2004)cross and auto-correlations assume that the photo-metric data are evenly spaced in time.Neural networks are pattern recognition machines.So,they can be trained to recognize the pattern of a sparsely sampled microlensing lightcurve,without explicit accounting for missing data.Of course,it is a di?erent pattern to a microlensing lightcurve with full time-sampling –but a pattern none the less.All that is needed is to include examples of such patterns with sparse sampling in the training set,as Belokurov et al.(2004)do.Besides,one of the features input to the neural net-work,namely the Lomb-Scargle periodogram,does not as-sume equally spaced datapoints (see for example,Press et al.1992).

Griest &Thomas (2005)have also miscalculated the false positive rate of the algorithm in Belokurov et al.(2004).The false positive rates stated in Belokurov et al.(2004)refer to the rates at which the common classes of vari-able stars in the training set are mistakenly identi?ed as microlensing.They do not refer to the rate at which any lightcurve (whether variable or constant baseline)is mistak-enly identi?ed as microlensing.To compute the number of false positives for the whole of the 9million lightcurves in the MACHO database,we must ?rst calculate how many vari-able stars are expected?We can estimate this using results

from the OGLE–II survey of the LMC (˙Zebru′n et al.2001).

OGLE-II found that 0.8%of all stars towards the LMC are

4N.W.Evans &V.

Belokurov

Figure 2.The publicly available R and B band data for the event MACHO LMC-4.The upper two panels show the unbinned data,the lower two panels the binned data (with bin-size 1day).The horizontal axis is time in JD -2448623.5(2Jan 1992),whilst the vertical axis is the linear magni?cation A R,B .The peak of the event,which is discernible from the microlensing ?t superposed on the lightcurves,is at 1023.0days and the Einstein crossing time t E is 43days.The corresponding lightcurves using the cleaned data are displayed in Alcock et al.(2000).

variables with amplitude variations exceeding the measure-ment errors and with periods less than a few years.The MA-CHO collaboration monitored ～11.9million lightcurves,of which 20%occur in ?eld overlaps (Alcock et al.2000).So,this implies that the number of true variable stars is ～85000.From the experiment reported in Figure 3of Be-lokurov et al.(2004),the false positive rate is 2in 22000or 0.9×10?4.Given that the total number of variable stars is ～85000,this means that the false positives amount to less than 8for the whole dataset of 9million lightcurves.However,the experiment reported in Figure 3of Be-lokurov et al.(2004)used the public data,not the cleaned data.As we have shown in Figure 1here,this public dataset

is very noisy and contains over an order of magnitude too many variable objects (mainly caused by artefacts).Hence,when applied to the dataset that the MACHO collabora-tion actually use,our false positive will certainly diminish,probably by at least an order of magnitude.

Griest &Thomas (2005)point out that the MACHO selection procedure uses around 20statistical methods,whereas Evans &Belokurov (2004)use only 5as inputs to the neural networks.They suggest that their suite of statis-tical methods may be more powerful.In fact,it is di?cult to tell unless both methods are run on the same data.However,as Figure 2of Belokurov et al.(2004)shows,the 5statistical methods already identify 95%of microlensing events in the

The Microlensing Optical Depth Towards the Large Magellanic Cloud:Is There A Puzzle?5 test set.This is an excellent result by any standards–and

it is easy to incorporate any further statistics as inputs to

the neural networks,if desired.

Finally,Griest&Thomas(2005)mistakenly argue that

Evans&Belokurov(2004)only looked for microlensing

events among the22MACHO candidates and so the e?-

ciency is necessarily reduced.Let us clarify the calculation

that was actually done.Evans&Belokurov(2004)examined

22000lightcurves,including the22MACHO candidates.

This number includes the“set A”or stronger microlens-

ing candidates and the“set B”or weaker candidates from

Alcock et al.(2000).We made the additional,strong as-

sumption that there are no further microlensing events in

the rest of the data.This assumption may be valid or in-

valid–it is impossible to say unless all the data are made

available for testing with neural networks2.However,there

is some supporting evidence for Evans&Belokurov’s(2004)

point of view,as none of the“set B”MACHO candidates

passed our neural network selection criterion.Having made

this assumption,Evans&Belokurov(2004)proceed–as

usual in any e?ciency calculation–by calculating the frac-

tion of simulated events that are identi?ed by the algorithm.

Such an e?ciency calculation does not depend on whether

all or part of the original dataset was analyzed.

3REMARKS ON F ALSE POSITIVES

Any treatment of this subject should begin with some cau-

tionary remarks on the limitations of e?ciency calculations.

The optical depth is usually calculated from the data as a

sum over events using

τ=π

NT?(t0,i)

(1)

where N is the number of stars monitored,T is the du-ration of the experiment and?is the e?ciency as a func-tion of timescale.There is an important assumption in this formula.The assumption is that the false positive rate is completely negligible.An e?ciency calculation can correct for false negatives(that is,missed microlensing events)but it cannot correct for false positives(that is,variable objects wrongly classi?ed as microlensing).

By altering the threshold in any selection procedure,we simply generate more false positives at the expense of less false negatives–or vice versa.Even after correcting with the e?ciency,di?erent selection procedures will not give the same optical depth,unless the false positive rate is com-pletely negligible for all the thresholds.Alcock et al.(1997) and Alcock et al.(2000)used di?erent cuts and obtained quite di?erent values for the optical depth.This is a broad hint that the false positive rate is not negligible in the Alcock et al.(1997)sample.

Everyone now accepts that MACHO LMC-23is a vari-able star(Tisserand&Milsztajn2005).This false positive was included in the more con?dent“set A”of Alcock et al.

2At present,lightcurves can be downloaded on an individual basis(or in small groups)from the MACHO website,thus making downloads of12million lightcurves practically impossible.In any case,what is really needed is the capability to make downloads of entire?elds of the cleaned data.(2000).“Set B”has a lower threshold and logically must con-tain still more false positives.In any case,let us emphasise that there is nothing exotic about MACHO LMC-23.It is a variable star that is able to masquerade as a microlensing event because of photometric noise.There is already some indication of this in the comparatively high value of itsχ2 of1.452per degree of freedom.

Given the fact that there are false positives in the Al-cock et al.(2000)samples,what is the best methodology for correcting the optical depth?There have been three at-tempts to do this so far,namely by Griest&Thomas(2005), Bennett(2005)and Evans&Belokurov(2004).All three computations have inadequacies.

Griest&Thomas(2005)attempted to correct for the ef-fects of contamination by removing the contribution of MA-CHO LMC-23.To see why this inappropriate,let us consider a model problem in which the cut is only based on aχ2per degree of freedom.Then three further events(candidates5,8 and21)would be discarded,as theirχ2per degree of free-dom is worse than MACHO LMC-23.This is because the de-cision boundary between microlensing and non-microlensing cannot have arti?cially created holes,or any abrupt or sharp features.Of course,the actual algorithm that the MACHO collaboration use is more sophisticated than a cut onχ2.Our point is merely that a misclassi?ed event inside the decision boundary a?ects its entire neighbourhood.It is not enough simply to remove by hand the contribution of the misclassi-?ed event,as Griest&Thomas(2005)do.Their calculation is not a proper accounting of the e?ects of contamination, even in the optimistic case in which MACHO LMC-23is the only false positive.

Bennett(2005)tried to correct the optical depth calcu-lations of Alcock et al.(2000)for the e?ects of contamination by introducing a likelihood estimator.His likelihood estima-tor is tantamount to assuming that the contamination rate is 1event out of every5.Although Bennett(2005)does recom-pute the e?ciencies,this is to take into account a systematic error in the e?ciencies used by Alcock et al.(2000).How-ever,Bennett does not take into account the change in the e?ciencies caused by the di?erent event selection required to eliminate the contaminants.

Evans&Belokurov(2004)published an estimate of the optical depth allowing for contamination.We argued for more contaminants than either Bennett(2005)or Griest &Thomas(2005)and concluded that the optical depthτsatis?ed0.3×10?7～<τ～<0.5×10?7,where the range cor-responds to the±1σinterval.Evans&Belokurov’s(2004) analysis has two problems.First,as stated earlier,we did not have access to the whole dataset,but to～<1%of it. Second,in our e?ciency calculation,we used a simple mi-crolensing lightcurve model with added Gaussian noise and MACHO sampling to test whether events would be found by neural networks.This procedure would be better applied to the cleaned data,which are not available,and so Evans &Belokurov(2004)perforce used the polluted public data.

If a decision boundary between microlensing and non-microlensing is introduced,then it is crucial to know the false positive rate.We have not been able to?nd such a calculation in the literature for the MACHO experiment.To compute the false positive rate,it is important to use the full gamut of possible variable star lightcurves.In Belokurov et al.(2004),the false positive rate was calculated using

6N.W.Evans&V.Belokurov

standard libraries of variable stars.Another possibility is to apply Feeney et al.’s(2005)adaption of the technique of K fold cross-validation,which uses the entire dataset itself to provide the range of variable lightcurves.

The alternative to introducing a decision boundary is to assign probabilities to each lightcurve using,for example, the outputs of neural networks.The microlensing rate can be calculated directly from the outputs,without introduc-ing an explicit decision boundary.Every lightcurve makes a weighted contribution to the microlensing rate.

One way of carrying this out is described in Belokurov et al.(2004,see Appendix A).Brie?y,the formula

?P(microlensing)≈1

T

?P(microlensing)(3) The advantage of this algorithm is that the rate can be computed directly from the dataset,without the intervening steps of candidate selection and e?ciency estimation.

This is very di?erent to the approach of all microlensing experiments so far,which have categorized events as either microlensing or non-microlensing.The probabilities assigned are therefore either1or0.Not merely are marginal events incorporated into the optical depth with the same weight as unambiguous events,but–worse still–their contribution is ampli?ed by the e?ciency factor as well.The e?ciency nat-urally tends to be low for the marginal events.This may well be part of the reason for the continuing mismatch between theoretical estimates and observational results in microlens-ing.

4DISCUSSION

Evans&Belokurov’s(2004)suggestion that the optical depth to the LMC may have been over-estimated because of contamination by false positives deserves serious consid-eration.Indeed,the suggestion receives support from the subsequent results of the EROS collaboration,which was also monitoring the Large Magellanic Cloud.Tisserand& Milsztajn(2005)?nd the low optical depthτ=(0.15±0.12)×10?7based on6.7years of data.They also report that MACHO LMC-23–included in Alcock et al.’s(2000)“set A”of con?dent events–is actually a variable star. Very recently,Tisserand et al.(2006)exploited the idea of a bright subsample to minimise the e?ects of contamination and obtainedτ<0.38×10?7

The main reason for the di?erence between the results of Belokurov et al.(2004)and those of Griest&Thomas (2005)lies in the treatment of the photometric outliers.Be-lokurov et al.’s(2004)calculations use the public data and are valid in the case that the noise is Gaussian.In fact,the noise properties of the public data are non-Gaussian.Gri-est&Thomas(2005)use the cleaned data–a version of the data in which many photometric outliers have been re-moved–so that the noise is closer to Gaussian.The cleaned data are not publicly available.The events that Belokurov et al.(2004)claimed as microlensing are reasonably trust-worthy.If an event is identi?ed in noisy data,then use of the cleaned data will only improve matters.More problematic are the events for which Belokurov et al.(2004)failed to identify as microlensing,at variance with the original judge-ment of Alcock et al.(2000).It is impossible to say anything further about these events until either the cleaned data or the algorithm for cleaning the public data are made public. Belokurov et al.(2004)therefore give a?nal sample of events whose microlensing nature is almost beyond question.This is valuable,as false positives are destructive and cannot be corrected by the e?ciency.

Let us also remark that the events Belokurov et al. (2004)claimed as non-microlensing may be incorrectly des-ignated(c.f.Bennett,Becker&Tomaney2005).If so,this is not a fault of the neural network methods,but a conse-quence of the use of the polluted public data.

Are Griest&Thomas(2005)correct to claim a mi-crolensing puzzle?It is true that the experimental deter-minations of the optical depth to the LMC are presently uncertain to almost an order of magnitude(fromτ= (0.15±0.12)×10?7based on EROS data by Tisserand& Milsztajn(2005)to(1.0±0.3)×10?7based on MACHO data by Bennett(2005)).However,the EROS experiment mon-itors a wider solid angle of less crowded?elds in the LMC than the MACHO experiment.So,blending and contamina-tion by LMC self-lensing are less important for the EROS experiment than for MACHO.The EROS result is there-fore an average value of the optical depth over a wide area of the LMC disk,whilst the MACHO value is the optical depth in the central parts.Nonetheless,this cannot be the whole story.The contribution to the optical depth of lens-ing objects lying in the Milky Way halo varies only weakly across the face of the LMC.So,if the claims that20%of the dark halo is in the form of compact objects are correct (e.g.,Alcock et al.2000;Griest&Thomas2005),then this optical depth contribution of this lensing population(ap-proximatelyτ～0.6×10?7)should be largely independent of position.

The theoretical estimates of the optical depth of the known Galactic components in the direction of the LMC have been computed anew and are listed in https://www.sodocs.net/doc/29481927.html,-ing the latest models of the thin and thick disk(e.g., Binney&Evans2001),we?nd that their contribution is τ=0.10×10?7.This is a middle-of-the-range value,and both larger(e.g.,Alcock et al.1997;Evans et al1998)and smaller numbers(Alcock et al.2000)can be found in the literature.The optical depth of the spheroid is uncontrover-sial and isτ=0.02×10?7.There is much more dispute about the LMC self-lensing optical depth.Accordingly,we list a number of recent estimates in the Table–our preferred value is0.55×10?7,corresponding to the zero o?set model of Zhao&Evans(2001),which is again a reasonable middle-of-the-range value.Notice from Figure2of Zhao&Evans

The Microlensing Optical Depth Towards the Large Magellanic Cloud:Is There A Puzzle?7 Table1.The theoretical value of the microlensing optical depth towards the LMC for various known stellar populations in the Milky Way and the Large Magellanic Clouds.

Thin and Thick disk0.10×10?7Eqn.(2)of Binney&Evans(2001)using a local column density of

27M⊙pc?2and a radial scalelength of3.0kpc Spheroid0.02×10?7Standardρ=1.18×10?4(r/R0)?3.5M⊙pc?3spheroid of Giudice et al.(1994) LMC disk/bar0.55×10?7Zero O?set Model of Zhao&Evans(2001)

LMC disk/bar1.0×10?7Non-Zero O?set Model of Zhao&Evans(2001)

LMC disk/bar0.05?0.80×10?7Models of Gyuk,Dalal&Griest(2000)

8N.W.Evans&V.Belokurov

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