Gravitational .. Lensing in .. Astronomy \ centerlineJoachim Wambsganss{ Gravitational \quad Lensing in \quad Astronomy } Astrophysikalisches Institut Potsdam \ centerlineAn der Sternwarte{Joachim 1 6 Wambsganss } 14482 Potsdam Gravitational Lensing in Astronomy \ centerlineGermany { Astrophysikalisches InstitutJoachim Potsdam Wambsganss} jwambsganss at aip period de Astrophysikalisches Institut Potsdam \ centerlinePublished 2{ NovemberAn der Sternwarte 1 998 1 6 }An der Sternwarte 1 6 open parenthesis Last Amended 31 August 200114482 closing Potsdam parenthesis \ centerlineLiving Reviews{14482 in Relativity Potsdam } Germany www period livingreviews period org slash Articlesjwambsganss slash Volume@ aip 1 . slash de 1 998 hyphen 12 wamb \ centerlinePublished by{Germany the Max hyphen} Planck hyphenPublished Institute 2 November for Gravitational 1 998 Physics A lbe r-t Einstein Institute comma Potsdam( Last comma Amended Germany 31 August 2001 ) \ centerlineAbstract {jwambsganss $ @ $ aipLiving . de Reviews} in Relativity Deflection of lightwww by gravity . livingreviews was predicted . org by General / Articles Relativity / Volume and 1 / 1 998 - 12 wamb \ centerlineobservationally{ Published confirmedPublished in 2 1 November 9 1 9 by period the Max1 .. 998 In - Planck the} following - Institute decades for Gravitational comma various Physics as hyphen pects of the gravitational lens effectA lbe werer − exploredt Einstein theoretically Institute , Potsdam period .. , GermanyAmong \ centerlinethem were :{ ..( the Last possibility Amended of multiple 31 August or ring 2001 hyphenAbstract ) like} images of background sources comma the use ofDeflection lensing as of a light gravitational by gravity telescope was predicted on very by General faint and Relativity and \ centerlinedistant obj ects{ Living commaobservationally Reviews and the possibility in confirmed Relativity of in determining 1 9 1} 9 . 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It has in theoretically con - . \quad Among } the recent past we predicttributed an even significant more luminous new results future in areas for gravitational as different as the cosmological \ centerlinelensing period{them weredistance : \quad scale ,the the large possibility scale matter distribution of multiple in the or universe ring , mass− like images of background } circlecopyrt-c 1 998 Maxand hyphen mass distribution Planck hyphen of galaxy Gesellschaft clusters , the and physics the authors of quasars period , dark Further mat - information on \ centerlinecopyright is{ givensources at httpter , in : theslash galaxy use slash halos of www , and lensing period galaxy l structure as ivingreviews a gravitational . Lookingperiod org at these slash telescope successes Inf o slash in on Copyright very faint slash period and } For permission to reproducethe the recent article past please we predict contact an livrev even more at aei luminous hyphen future potsdam for gravitational period mpg period de period \ centerline { distant obj ects , and the possibilitylensing . of determining Hubble ’ s constant with } circlecopyrt − c1998 Max - Planck - Gesellschaft and the authors . Further information on \ centerlinecopyright{ l e n s i n is g given . \quad at httpIt : is / only / www relatively . l ivingreviews recently . org , \ /quad Inf o( / after Copyright the discovery/ of the first } . For permission to reproduce the article please contact livrev @ aei - potsdam . mpg . de . \ centerline { doubly imaged quasar in 1 9 79 ) , that gravitational lensing has became an }

\ centerline { observational science . Today lensing is a booming part of astrophysics . }

\ centerline { In addition to multiply − imaged quasars , a number of other aspects of }

\ centerline { lensing have been discovered : \quad For example , giant luminous arcs , quasar }

\ centerline { microlensing , \quad Einstein rings , \quad galactic microlensing events , \quad a r c l e t s , \quad and }

\ centerline {weak gravitational lensing . \quad At present , literally hundreds of individual }

\ centerline { gravitational lens phenomena are known . }

\ centerline {Although still in its childhood , lensing has established itself as a very }

\ centerline { useful astrophysical tool with some remarkable successes . \quad I t has con − }

\ centerline { tributed significant new results in areas as different as the cosmological }

\ centerline { distance scale , the large scale matter distribution in the universe , mass }

\ centerline {and mass distribution of galaxy clusters , the physics of quasars , dark mat − }

\ centerline { ter in galaxy halos , and galaxy structure . \quad Looking at these successes in }

\ centerline { the recent past we predict an even more luminous future for gravitational }

\ centerline { l e n s i n g . }

\noindent $ circlecopyrt −c 1 998 $ Max − Planck − Gesellschaft and the authors . Further information on copyright is given at http : / / www . l ivingreviews . org / Inf o / Copyright / . For permission to reproduce the article please contact livrev $ @ $ aei − potsdam . mpg . de . Article .. Amendments \ centerline31 August 2001{ A r .. t i Errata c l e \quad requestedAmendments by author and} minor editorial revision of reference list open parenthesis ref hyphen \ beginerence{ c numbers e n t e r } have changed closing parenthesis comma .. URLs 31updated August period 2001 \quad Errata requestedArticle by author Amendments and minor editorialCase 1 On author revision request of aLiving reference Reviews list article ( can ref be− amended to include errata Case 2 and small additions to ensure 31 August 2001 Errata requested by author and minor editorial revision of reference list ( that\end the{ c e most n t e r accurate} and up hyphen to hyphen date infor hyphen Case 3 mation possible is provided period For detailed ref - documentation of amendments comma Case 4 please go to the article s online version at Case 5 http : slash slash www period l \noindent erence numbers have changed ) , \quad URLs ivingreviews perioderence org numbers slash Arti have c changed le s slash ) Vol , e URLs 1 slash updated 1 998 hyphen . 1 2 w m-a b slash period Case 6 Owing to the fact that aupdated Living Reviews . article can evolve over t ime comma we Case 7 recommend to cite the article as follows : Case 8 Wambsganss comma J period comma Case 9 Gravitational Lensing in Astronomy comma Case 10 Living Rev period Relativity comma 1 comma\ [ \ l e f open t .quotedblleft parenthesis 1 998−quoteright closing parenthesis−quoteright comma− 1u 2− periodm−quotedblright open square bracket−hyphen OnlineOnauthorrequesta\ begin Article{ a l closing i g n eLivingReviews d square} & bracket On articlecanbeamendedtoincludeerrata author request a Living Reviews article can be amended to include errata \\ : cited on less date greater comma Case 11 http : slash slash www period livingreviews periodandsmalladditionstoensurethatthemostaccurateandup org slash Articles slash Volume − to − dateinfor− & and small additions to ensure that the most accurate and up − 1 slash 1 998 1 2 wamb slash period Case 12 The date in quoteright cited on less date greatermationpossibleisprovided then uniquely identifies. Fordetaileddocumentationofamendments the , versionto − of thedate Case 13 i article n f o r you− are \\ referring to period & mation possible is provided . For detailed documentationpleasegotothearticlesonlineversion@ of amendments , \\ http : //www.livingreviews.org/Articles/Vol e1/1998 − 12wm − ab/. & please go to the article s online version @ \\ OwingtothefactthataLivingReviewsarticlecanevolveovertime, we & http : / / www . l ivingreviews . org / Arti c le s / Vol equotedblleft 1 / 1− quoteright 998 −− quoteright1 2− wu − m m−−aquotedblright b /− . hyphen\\ recommendtocitethearticleas : & Owing to the fact that a Living Reviews articleWambsganss can, J., evolve over t ime , we \\ GravitationalLensinginAstronomy , & recommend to cite the article as \succ : \\ LivingRev. Relativity, 1, (1998), 12. [OnlineArticle] : citedon < date >, & Wambsganss , J . , \\ & Gravitational Lensing in Astronomy , \\ http : //www.livingreviews.org/Articles/Volume1/199812wamb/. & Living Rev . Relativity , 1 , ( 1 998 )Thedatein , 10citedon 2< .date [> Onlinethenuniquelyidentifiestheversionofthe Article ] : cited on < date > , \\ articleyouarerefer˚ to. & http : / / www . livingreviews . org / Articles / Volume 1 / 1 998 1 2 wamb / . \\ & The date in ’ cited on < date > then uniquely identi fi es the version of the \\ & article you are \ mathring { r e f e r } to . \end{ a l i g n e d }\ right . \ ] 3 .... Gravitational Lensing in Astronomy \noindentContents 3 \ h f i l l Gravitational Lensing in Astronomy 1 .... Introduction .... 4 \noindent2 .... HistoryContents of Gravitational Lensing .... 7 3 .... Basics3 of Gravitational Lensing .... 10 Gravitational Lensing in Astronomy \noindent3 period 1Contents1 ..\ Lensh f i equation l l Introduction period .. period\ h f .. i l period l 4 .. period .. period .. period .. period .. period .. period .. period .. period .. period1 .. period .. period .. period .. period Introduction .. period .. period .. period .. period .. period .. period 4 .. period .. period\noindent .. period22 ..\ h period f i l l ..History period .. of period Gravitational History .. period of .. Gravitational 1 0 Lensing Lensing\ h f i l l 7 7 3 period 23 .. Einstein radius .. period .. Basics period of .. period Gravitational .. period Lensing.. period .. period .. period .. period 10 .. period .. period ..\noindent period .. period3 \3h. .. f 1i period l l LensBasics .. equation period of .. . Gravitational period . .. . period . . .. Lensing period . . .. . period\ h . f i l .. . l period10 . . .. period . . .. . period . .. . period . .. period .. period .. period. .. . period . ... period . . .. period . . .. . 1 3 . . 1 0 3 . 2 Einstein radius ...... 3 .3 period1 \quad 3...... 13 ..Lens Critical equation surface mass . \ densityquad ... period\quad ... period\quad .. period. \quad .. period. \quad .. period. \quad .. period. \ ..quad period. ..\quad period. .. \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 0 period3 . 2.. period\quad ..3Einstein period. 3 Critical .. period radius surface .. period\ massquad .. density period. \quad .. period .. .\quad .. . period .. \ ...quad period . . ..\ periodquad ...... period\quad . .... period .\quad . .. 1. 3 \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 3 3 period 4. .. Image . . positions . . and . magnifications 1 3 3 . 4 Image .. period positions .. period and .. magnifications period .. period ... period . . .. period . . .. . period .. period ..3 period . 3 \ ..quad period...... 14Critical .. period .. surface period .. periodmass density.. period ..\ periodquad ... period\quad ... period\quad .. 1. 4 \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 3 33 . period 4 \quad 5 .. openImage3 . 5parenthesis positions ( Non - Non) Singularhyphen and isothermalmagnifications closing parenthesis sphere\ .quad Singular . .. isothermal\ .quad .. . sphere\quad . . ... period.\quad . .. . period. .\quad .. period. \quad .. . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 4 period .. period. .. . period . . .. period. . .. 1 period5 .. period .. period .. period .. period .. period .. period .. period .. period .. period\ hspace .. period∗{\ f i .. l3 l 1.}3 56 . Lens 5 \quad mapping( Non. .− .) Singular . . . isothermal . . . . sphere . .\ .quad .. .\quad . .. .\quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 5 3 period 6. .. Lens . . mapping . . period . . .. period. . .. . period . 1 .. 6 period 3 . 7 .. Time period delay .. period and .. “ period Fermat .. ’ period s ” theorem .. period .. period .. period3 . 6 ..\quad period...... 17Lens .. period mapping .. period . ..\quad period. ..\ periodquad ... period\quad ... period\quad .. period. \quad .. period. \quad .. period. \quad .. period. \ ..quad period. ..\quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 6 period3 . 7.. period\quad4 ..Time period delay .. period and .. period\quad .. period‘‘ Lensing Fermat .. 1 6 Phenomena ’ s ’’ theorem \quad . \quad . \quad 20 . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 7 3 period 7 ..4 Time . 1 delay Multiply and .. - quotedblleft imaged quasars Fermat . quoteright . . . s quotedblright . . . theorem . . .. . period . .. . period . .. period .. period\noindent .. period.4 ..\ .h period f i l . l ..Lensing . period . .. . Phenomena period . .. . period .\ h .. f . i period l l 2020 4 .. . period 1 . 1 .. period The first .. period lens : .. Double period quasar .. period Q .. period .. 1 7 4 .... Lensing957 Phenomena+ 561 ...... 20 22 4 . 2 Quasar microlensing ...... 4 .4 period 1 \quad 1. ..Multiply . . . hyphen− .imaged . imaged . quasars. quasars . . period . .\quad ... period .. .\ ..quad .period .. ..\ .quad period . 25. ..\ 4periodquad . 3 .. Einstein. period\quad ..rings. period\quad .. period. \quad .. . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 20 period4 . 1 .. period. 1...... \quad .. periodThefirst .. period .. lens period :DoublequasarQ .. period .. period .. period $957 .. period + .. 56 period 1 .. period . ... period . .. . period . .. period. . .. period ...... 20 . 22 . $ 32 4 . 4 Giant luminous arcs and arclets ...... 44 . period 2 \quad 1. period .Quasar . 1 .. . The microlensing . first . lens . :. Double .\quad 35 quasar. \quad Q 957. plus\quad 56 1 period. \quad period. \ periodquad period. \quad period. \ periodquad period. \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 25 period4 . 3period\quad 22 4Einstein . 5 Weak rings / st atistical\quad lensing. \quad .. \quad . .. .\quad . .. .\quad . .. \ .quad . .. \ .quad . . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 32 44 . period 4 \quad 2. .. Quasar .Giant . microlensing . luminous . . . .. arcs period . 39 and .. 4 . period arclets 5 . 1 .. Cluster period . \quad.. mass period reconstruction. \ ..quad period. ..\quad period . .. ..\ . periodquad . .. .. period\ .quad ... period\quad .. . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 35 period .. period. .. . period . . .. period. . .. . period . . .. period . . .. 40 period .. period .. period .. period .. period .. period .. period .. period4 . 5 ..\ periodquad Weak ..4 period. 6 / Cosmological .. st 25 atistical aspects lensing of ( strong\quad ) lensing. \quad . .. \ .quad .. \quad . .. .\quad . .. .\quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 39 44 . period 5 . 13. ..\quad Einstein . .Clusterrings 4 1 4 .. . 7period mass Galactic .. reconstruction period microlensing .. period .. .\ periodquad . . ... period .\quad . .. . period. \ .quad .. . period. .\quad . .. period .. .\quad .. .period. \ ..quad period. \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 40 .. period .. period. . .. .period . .. . period . . .. period . . .. . period . . .. period 44 .. period .. period .. period .. period .. period .. period .. period4 . 6 ..\ periodquad5 ..CosmologicalF period− uture ..Gravitational period aspects .. period Lensing .. of period ( strong .. 32 ) lensing \quad . \quad . \quad . 49\quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 4 1 44 . period 7 \quad 46 .. GiantGalactic luminous microlensing arcs and arclets . Acknowledgementsperiod\quad ... period\quad .. period. \quad .. period. \quad .. period. \ ..quad period. ..\ 53quad period. ..\ periodquad ... \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 44 period .. period .. period .. period .. period .. period .. period .. period .. period .. periodReferences .. period .. 35 54 \noindent4 period 5 ..$ Weak 5 slashF−u st{ atisticalture } lensing$ Gravitational .. period .. period LensingLiving .. period Reviews\ ..h period f i inl l Relativity49 .. period( ..1998 period - 12.. period) .. period .. period .. period .. period .. period .. period .. period .. periodhttp .. : period / / www.. period . l .. ivingreviews period .. period . .. org period .. period .. period\noindent .. period6 ..\ h 39 f i l l Acknowledgements \ h f i l l 53 4 period 5 period 1 .. Cluster mass reconstruction .. period .. period .. period .. period .. period .. period .. period .. period\ hspace .. period∗{\ f i .. l l period} References .. period ..\quad period54 .. period .. period .. period .. period .. period .. 40 4 period 6 .. Cosmological aspects of open parenthesis strong closing parenthesis lensing .. period .. period .. period .. period\ hspace .. period∗{\ f i .. l l period} Living .. period Reviews .. period in .. Relativity period .. period ( 1998 .. period− 12 .. period ) .. period .. period .. 4 1 4 period 7 .. Galactic microlensing period .. period .. period .. period .. period .. period .. period .. period .. period .. period\ hspace .. period∗{\ f i l.. l } periodhttp .. : period / / www .. period . l .. ivingreviews period .. period . .. org period .. period .. period .. period .. period .. period .. period .. period .. period .. 44 5 F-u sub ture Gravitational Lensing .... 49 6 .... Acknowledgements .... 53 References .. 54 Living Reviews in Relativity open parenthesis 1998 hyphen 12 closing parenthesis http : slash slash www period l ivingreviews period org J period Wambsganss .... 4 \noindent1 .. IntroductionJ . Wambsganss \ h f i l l 4 Within the last 20 years comma gravitational lensing has changed from being consid hyphen \noindentered a geometric1 \quad curiosityIntroduction to a helpful and in some ways unique tool of modern astrophysicsJ . period Wambsganss Although the deflection of light at the solar limb was very success hyphen 4 \noindentfully hailed1Within as the Introduction first the experiment last 20 to years confirm , a gravitational prediction of Einstein lensing quoteright has s changedtheory from being consid − eredof General a geometricWithin Relativity the curiosityinlast 1 20 9 1 years 9 comma , to gravitational a .. helpfulit took morelensing and than has in half changed some a century ways from tobeing unique establish consid tool - ered of a moderngeometric this phenomenoncuriosity observationally to a helpful and in insome some other ways environment unique tool period of modern By now almost a \noindentdozen differentastrophysicsastrophysics realiz ations . 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In Figure for two the 1 the gravdecades number - itational of of publications telescopeexistence . in This as scientific an trend observational jis our reflected hyphen in the branch growing of astro − nals that dealnumber with of gravitational people working lensing in the is plotted field . over In t Figure ime period 1 the It number is obvious of publications that in scientific j \noindentlensing is boomingourphysics - nals as that , an the areadeal field withof investigation gravitational of gravitational period lensing is plotted lensing over t has ime been. It is obvious continuously that lensing growing is . Every fewAlthough years therebooming a new had as realisationbeen an area a slight of investigation sense of of the disappointment phenomenon . in the is astronom discovered hyphen . Multiple quasars , gianti cal community luminousAlthough a fewarcs years there , quasar ago had because been microlensing a slightlensing sense had ofnot disappointment, yet Einstein solved all rings the in the big astronom , galactic - i cal microlensing community , weakproblems lensing ofa few astrophysics years, galaxy ago open because− parenthesisgalaxy lensing lensing had e period not yet openg period solved up determination all very the big different of the Hubble regimes constant for semicolon the grav nature− of itationaldark matterproblems telescope semicolon of astrophysics physics . \quad slash (This sizee . g of . trend quasarsdetermination is closing reflected of parenthesis the Hubble in comma constant the growing this ; naturefeeling number ofhas apparently of people reversed period Withworking indark the matter field ; physics . \quad / sizeIn of quasars Figure ) , 1 this the feeling number has apparently of publications reversed . With in scientific j our − nalsits many that applicationsits deal many with applications and gravitational quantitative and quantitative results lensing comma results lensing is , plotted lensing has started has over started to fulfill t ime to its fulfill . It its is astrophysical obvious that lensingastrophysical ispromises booming promises . period as an area of investigation . We shall start with a brief lookWe back shall in start time andwith mention a brief look some back historic in time as hyphen and mention some historic as - Althoughpects of light therepects deflection of had light been deflection and lensing a slight and in lensing Section sense in 2 Section period of disappointment 2.. . We We then then attempt attempt in to explainto the explain astronom the basic− features ithe cal basic community featuresof gravitational of a gravitational few lensing years quantitatively lensing ago quantitativelybecause , deriving lensing comma some had ofderiving the not some yet of solved the all the big relevant equationsrelevant equations open parenthesis ( Section Section 3 ) . A 3 whole closing variety parenthesis of lensing period observations A whole and variety phe of - nomena lensing observationswhich and phe hyphen\noindent curvedproblems space of - time astrophysics provides for us ( is e presented . g . determination in Section 4 , for example of the , multipleHubble versions constant of ; nature of nomena whichquasars curved , gigantically space hyphen distorted time provides images of for galaxies us is presented , and highly in Section magnified 4 comma stars for . Additionally , \noindentexample commawedark explain multiple matter and versions discuss ; physicsthe of quasars astrophys / size comma - ofi cal gigantically quasars applications distorted) , of this lensing images feeling which of galaxies show has the comma apparently use of this reversed . With and highlytool magnified . This sectionstars period will be Additionally the most detailed comma one we explain. Finally and , in discuss the concluding the astrophys Section hyphen 5 we try to \noindenti cal applicationsextrapolateits many of lensing and applications speculate which show about the and the use quantitative future of this development tool period results This of the section field , lensing . will be has started to fulfill its astrophysicalthe most detailedBy design promises one period , this Finally .review cancomma only in touch the concluding upon the issues Section relevant 5 we try in the to extrapolate astro - physical field of and speculategravitational about the lensing future . development This article of should the field serve period as a guide and general introduction and provide \ hspaceBy design∗{\a commaf numberi l l }We this of shall usefulreview links start can only and with referencestouch a upon brief . the It issues look is entirely relevant back impossible in in the time astro to andbe hyphen complete mention in any some sense historic . as − physical fieldSo the of gravitational selection of topics lensing and period .. This article should serve as a guide and \noindentgeneral introductionLivingpects Reviews of and light provide in Relativity deflection a number( 1998 of and useful - 12 lensing links) and inreferences Section period 2 ... It\quad is We then attempt to explain theentirely basic impossiblehttp features : to / /be ofwww complete gravitational . l in ivingreviews any sense lensingperiod . org .. So quantitatively the selection of topics , deriving and some of the Living Reviews in Relativity open parenthesis 1998 hyphen 12 closing parenthesis \noindenthttp : slashrelevant slash www equations period l ivingreviews ( Section period 3 org ) . A whole variety of lensing observations and phe − nomena which curved space − time provides for us is presented in Section 4 , for example , multiple versions of quasars , gigantically distorted images of galaxies , and highly magnified stars . Additionally , we explain and discuss the astrophys − i cal applications of lensing which show the use of this tool . This section will be the most detailed one . Finally , in the concluding Section 5 we try to extrapolate and speculate about the future development of the field .

By design , this review can only touch upon the issues relevant in the astro − physical field of gravitational lensing . \quad This article should serve as a guide and general introduction and provide a number of useful links and references . \quad I t i s entirely impossible to be complete in any sense . \quad So the selection of topics and

\noindent Living Reviews in Relativity ( 1998 − 12 )

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\noindent http : / / www . l ivingreviews . org 7 .... Gravitational Lensing in Astronomy \noindent2 .. History7 ..\ ofh f Gravitational i l l Gravitational .. Lensing Lensing in Astronomy The first written account of the deflection of light by gravity appeared in the \noindentquotedblleft2 Berliner\quad AstronomischesHistory \quad Jahrbuchof Gravitational auf das Jahr 1 804\quad quotedblrightLensing .. in an article entitled : quotedblleft7 Ueber die Ablenkung eines Lichtstrals von seiner geradlinigen Gravitational Bewegung Lensing comma in durch Astronomy \noindentdie Attraktion2The eines first History Weltk written dieresis-o of account rpers Gravitational comma of the an welchem deflection er Lensingnahe of vorbeigeht light quotedblright by gravity open appeared parenthesis in quoted- the blleft‘‘ BerlinerOn theThe De hyphen Astronomisches first written account Jahrbuch of the deflection auf das of light Jahr by 1 gravity 804 ’’ appeared\quad inin the an “Berliner article As- entitled : flection oftronomisches a Light Ray from Jahrbuch its Straight auf das Motion Jahr 1 due 804 to” the in Attraction an article entitled of a World : ‘‘Body Ueber which die it“ Ablenkung Passes Ueber die Closely Ablenkung eines quotedblright Lichtstrals eines Lichtstrals closing parenthesis von von seiner seiner .. geradlinigen open geradlinigen square Bewegung bracket Bewegung 1 , durch 42 closing die , Attrak- square durch bracket period ..die Johann Attraktion Soldnertion eines endash eines Weltk a German Weltko ¨ rpers geodesist , an $ \ welchemddot comma{o er} nahe$ rpers vorbeigeht , an ” ( welchem “ On the De er - nahe vorbeigeht ’’ ( ‘‘ On the De − mathematicianflection and of astronomer a Light Ray then from working its Straight at the Motion Berlin Observatory due to the Attraction endash ex ofhyphen a World Body which \noindentplored thisitflection effect Passes and Closely inferred of a ” Light) that [ a1light 42 Ray ] . ray from Johannclose its to Soldner the Straight solar – limb a German Motion would geodesistbe due to , mathematician the Attraction and of a World Bodydeflected which byastronomer an it angle Passes alpha-tilde then Closely working = 0 at period ’’ the ) Berlin\ 84quad arcsec Observatory[ period 1 42 .. ] – It ex . is\ - veryquad plored interestingJohann this effect to Soldner and read inferred how−− care thata hyphen German a geodesist , mathematicianfully and cautiouslylight ray and close he astronomer investigated to the solar this thenlimb idea would working and its be consequences deflected at the by Berlin an on angle practical Observatoryα ˜ = 0.84 arcsec−− .ex It− is very ploredastronomy thisinteresting period effect to andread how inferred care - fully that and a cautiously light ray he investigated close to this the idea solar and its limb consequences would be deflectedIn the yearon 1 by practical9 1 an 1 endash angle more $ than\ tilde a century{\alpha later} endash= Albert 0 . Einstein 84$ open arcsec. square bracket\quad 4 1 closingIt is square very bracket interesting to read how care − directlyfully andastronomy cautiously . he investigated this idea and its consequences on practical In the year 1 9 1 1 – more than a century later – Albert Einstein [ 4 1 ] directly addressed the influence of gravity on light open parenthesis¨ quotedblleft U-dieresis ber den Einflu germandbls der Schw- erkraft\noindent addressedastronomy the .influence of gravity on light ( “ U ber den Einflu ßder Schwerkraft auf die Ausbreitungauf die Ausbreitung des Lichtes des quotedblright Lichtes ” .... open parenthesis ( “ On the quotedblleft Influence On of Gravity the Influence on the of Propa Gravity - on the Propa hyphen\ hspace ∗{\gationf i l l } ofIn Light the ” year ) . At 1 this 9 1 time 1 −− , themore General than Theory a century of Relativity later was−− notAlbert fully developed Einstein . [ 4 1 ] directly gation of LightThis isquotedblright the reason why closing Einstein parenthesis obtained period – unaware At this of time the comma earlier theresult General – the sameTheory value of Relativity for the was not fully \noindent deflectionaddressed angle the as Soldner influence had calculated of gravity with on light ( ‘‘ $ \ddot{U} $ ber den Einflu \ ss der Schwerkraft developed period .. This is the reason why Einstein obtained endash unaware of the earlier 2 result endashNewtonian the same physics value .for the deflection angle In this as Soldner paper , Einsteinhad calculated foundα with˜ = 2GM /c R = 0.83 \noindentNewtonianarcsecauf physics die for period the Ausbreitung deflection .... In this angle paper des of aLichtes comma ray grazing Einstein ’’ the\ h found sun f i l l ( alpha-tildehere( ‘‘M Onand = the 2R GM Influenceare sub the odot mass slash of and Gravity c the to the power on the of 2 Propa − R sub odot =radius 0 period of the 83 sun , c and G are the velocity of light and the gravitational constant , respectively \noindentarcsec for) thegation . deflection Einstein of Lightangle emphasized of ’’a ray ) his grazing . wish At that this the as sun time - open , parenthesis the General here M Theory sub odot of and Relativity R sub odot are was not fully developedthe mass andtronomers . the\quad radius investigateThis of the is sun this the comma question reason c (and “ Es why G w area ¨ Einsteinre the dringend velocity obtainedzu of w lightu ¨ nschen and−− the , daunaware ßsich Astronomen of the der earlier ¨ u¨ r egravitational s u l t −−hierthe constant aufgerollten same comma value Frage .. for respectively anna¨ thehmen deflection closing , auch wenn parenthesis dieangle im period vorigen as Soldner .. gegebenen Einstein had emphasizedU berlegungen calculated his wish ungen with that as hyphen tronomersgend investigate fundiert this oder question gar abenteuerlich open parenthesis erscheinen quotedblleft sollten Es . ” w dieresis-a ( “ It would re dringend be very zu desirable w u-dieresis that nschen comma da\noindent germandblsastronomersNewtonian sich address physics the question. \ h f i l un l In - rolled this here paper , even , if theEinstein considerations found should $ \ tilde seem to{\ bealpha} = 2Astronomen GM {\insufficientlyodot der hier} aufgerollten/ founded c ˆ or{ Frage entirely2 } annR speculative a-dieresis{\odot .hmen ”} ) comma= Recently 0 auch it . waswenn 83 discovered die $ im vorigen that Einstein had gegebenenderived U-dieresis the berlegungen lens equation ungen , the topossibility the power of aof double dieresis-u image gend and fundiert the magnifications oder gar abenteuerlich of the images erscheinen \noindentsollten periodinarcsec a notebook quotedblright for in the the .. year deflection open 1 9 parenthesis 1 2 [ 1 20angle ] . quotedblleft Inof 1 a 9 1 ray 3 It Einstein would grazing evenbe very the contacted desirable sun the( here that director astronomers $M of the{\ addressodot }$ the questionand $ un R hyphenMt{\ .odot Wilson}$ Observatory are , George Ellery Hale , and asked him whether it would be possible to therolled mass heremeasure and comma the positionseven radius if the of considerations ofstars the near sun the should sun $ during , seem c the to $ be day and insufficiently in order $G$ to foundedestablish are the the velocity deflection effect of light and the gravitationalor entirelyof speculative the sun constant . period quotedblright, \quad respectively closing parenthesis ) . ..\quad RecentlyEinstein it was discovered emphasized that Einstein his wish had derived that as − the lens equationSee [ comma 9 ] to view the a possibility facsimile of a a letter double Einstein image and wrote the to magnifications G . E . Hale on of October the 1 4 , 1 9 1 3 . \noindentimages in aIntronomers notebook the letter in ,investigateEinstein the year 1asked 9 1 2 Hale openthis whether square question bracket it would ( 1 be ‘‘ 20 possible closing Es w square to $ determine\ddot bracket{a} the period$ light re .. deflection dringendIn 1 9 1 3 Einstein zu w even $ \ddot{u} $ contactednschen , daat the\ ss solars i c hlimb during the day . However , there was a “ wrong ” value of the deflection angle Astronomenthe directorin of a der sketch the hierMt Einstein period aufgerollten Wilson included Observatory in the Frage letter comma ann . $George\ddot Ellery{a} Hale$ hmen comma , and auch asked wenn him die im vorigen gegebenenwhether it would $There\ddot be actually possible{U} $ were to measure berlegungen plans to positions test Einstein $ of ungen stars ’ s near wrong ˆ{\ the predictionddot sun during{u}} of$ the gend deflection fundiert angle during oder gar abenteuerlich erscheinen s othe l l t day e n in .a order solar’ ’ \ toquad eclipse establish( in ‘‘ 1 the 9 It 1 deflection 4 would on the Russian effectbe very of Crimea the desirable sun peninsula period that . However astronomers , when the address observers were the question un − rolledSee open here squarealready , bracket even in Russia if 9 closing , the World considerations square War I bracket broke out to viewand should they a facsimile were seem captured of to a letter be by insufficiently EinsteinRussian soldiers wrote to [ 27G founded period ] . So E period Hale onor October entirely, fortunately speculative for Einstein . ’’ , the) \ measurementquad Recently of the it deflection was discovered angle at the solarthat limb Einstein had to be had derived the1 4 lenscommapostponed equation .. 1 9 1 3 for period , a the few .. years In possibility the . letter comma of aEinstein double asked image Hale andwhether the it wouldmagnifications be possible to of the imagesdetermine in the aWith lightnotebook the deflection completion in at the the of year solar the General limb 1 9 during 1 Theory 2 [the 1 of day 20 Relativity period ] . \ Howeverquad , EinsteinIn comma was 1 9 the 1there first 3 Einstein to derive the even contacted thewas director a quotedblleftcorrect of deflection wrong the Mt quotedblright angle . Wilsonα ˜ of a value light Observatory ofray the passing deflection at , a George distanceangle in Ellery a sketch EinsteinHale , included and asked in the him whetherletter period it would be possible to measure positionsLiving of Reviews stars in near Relativity the sun( 1998 during - 12 ) theThere day actually in order were plans to establish to test Einstein the quoteright deflection s wronghttp effect prediction : / /of www of the the . sun deflection l ivingreviews . . org angle during a solar eclipse in 1 9 1 4 on the Russian Crimea peninsula period However comma Seewhen [ 9 the ] observers to view were a facsimile already in Russia of a comma letter World Einstein War I broke wrote out and to G they . Ewere . Hale on October 1captured 4 , \quad by Russian1 9 1 soldiers 3 . \quad open squareIn the bracket letter 27 closing , Einstein squarebracket asked periodHale whether So comma it fortunately would be for possible Einstein to commadetermine the measurement the light deflection at the solar limb during the day . However , there wasof the a deflection‘‘ wrong angle ’’ atvalue the solar of limbthe had deflection to be postponed angle for in a few a sketchyears period Einstein included in the l eWith t t e r the . completion of the General Theory of Relativity comma Einstein was the first to derive the correct deflection angle alpha-tilde of a light ray passing at a distance ThereLiving actually Reviews in were Relativity plans open to parenthesis test Einstein 1998 hyphen ’ s 12 wrong closing prediction parenthesis of the deflection anglehttp : during slash slash a www solar period eclipse l ivingreviews in 1 9 period 1 4 org on the Russian Crimea peninsula . However , when the observers were already in Russia , World War I broke out and they were captured by Russian soldiers [ 27 ] . So , fortunately for Einstein , the measurement of the deflection angle at the solar limb had to be postponed for a few years .

With the completion of the General Theory of Relativity , Einstein was the first to derive the correct deflection angle $ \ tilde {\alpha} $ of a light ray passing at a distance

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .. 8 \noindentr from an objectJ . Wambsganss of mass M as \quad 8 $Equation: r $ from open an parenthesis object 1 of closing mass parenthesis $M$ .. asalpha-tilde = 4 GM c to the power of 2 1 r comma where G is the constant of gravity and c is the velo city of light period The additional \ beginfactor{ ofa l itwo gJ n .∗} openWambsganss parenthesis 8 comparedr from an to object the quotedblleft of mass M as Newtonian quotedblright value closing parenthesis reflects the spatial\ tilde curvature{\alpha} = 4 GM { c ˆ{ 2 }} 1 { r } , \ tag ∗{$ ( 1 ) $} \endopen{ a l parenthesis i g n ∗} which is missed if photons are just treated as particles closing parenthesis period With the solar values α˜ = 4GM 2 1 , (1) for radius and mass Einstein obtained open square bracket 42c commar 43 closing square bracket : \noindentEquation:where open parenthesis $ G $ 2 isclosing the parenthesis constant .. ofalpha-tilde gravity sub and odot = $ 4 c GM $ c isto the the power velo of 2city to the of power light of odot . 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Today by Einstein such a lens comma con - Fritzfiguration Zwicky is called open square bracket 00 1 74 comma 1“ 75 Einstein closing− square˚ although bracketmore pointed correctly out in it should be called “ Chwolson - ring ” . Influenced by In1 the 937 thatfollowing galaxies decadesopen, parenthesis , light quotedblleft deflection extragalactic or gravitational nebulae quotedblright lensing closing was parenthesis only are much more likelyvery to rarely be graviEinstein hyphen the , topic Fritz Zwicky of a [ 1research 74 , 1 75 ] paper pointed :out In in 1 924 , Chwolson [ 3 1 ] mentioned the ideat ationally o f a1 lensed\ 937quad that than‘‘ galaxies stars fictitous and ( “ that extragalactic double one can use starnebulae the gravitational ’’ ” )\quad are muchand lens more the effect likely mirror as to be− gravireversed - t ationally nature of the secondary imagea quotedblleft . \quad naturalHe also Case 1mentioned quotedblright the Case symmetric 2 period case of st ar exactly behind star00 , re − sulting inlensed a circular than stars and image that . one Einstein can use the also gravitational reported lens in effect 1 as936 a “ about natural thetelescope appearance In the 1 960 s comma a few partly independent theoretical studies showed the use hyphen . of a ‘‘ luminous circle ’’ for perfect alignment between source and lens [ 44 ] , and of fulness of lensingIn the for 1 astronomy960 s , a few open partly square independent bracket theoretical 73 comma studies 84 comma showed 85 comma the use 93- fulness comma of 1lensing 1 3 comma .. 1 1 4 closing squarefor bracket astronomy period [ ..73 In , 84 particular , 85 , 93 comma , 1 1 3 Sjur , 1 1 4 ] . In particular , Sjur Refsdal derived the \noindent two magnified images for slightly displaced $ positions ˆ{ 1 } . $ \quad Today such a lens con − Refsdal derivedbasic equationsthe basic equations of gravitational of gravitational lens theory lens and theory subsequently and subsequently showed how the gravitational lens figuration is called \quad ‘‘ Einstein $ − \ l e f t . \ mathring {}\ begin { array }{ c} ’’ \\ , \end{ array } although \ right . $ showed howeffect the can gravitational be used to lens determine effect can Hubble be used ’ s con to determine - stant by Hubble measuring quoteright the t ime s con delay hyphen between two more correctly it should be called stant by measuringlensed images the t . ime He delay followed between up this two work lensed with images interesting period applications .. He followed of up lensing [ 1 1 6 , 1 1 ‘ ‘ Chwolson − ring ’’ . Influenced by Einstein , Fritz Zwicky [ 1 74 , 1 75 ] pointed out in this work with5 , interesting 1 1 7 ] . Theapplications math - ematical of lensing foundation open square of how bracket a light 1 1 bundle 6 comma is distorted .. 1 1 5 comma on its passage .. 1 1 7 closing square bracket periodthrough .. The themath universe hyphen had been derived in the context of gravitational radiation even before [ 1 24 \noindent 1 937 that galaxies ( ‘‘ extragalactic nebulae ’’ ) are much more likely to be gravi − ematical foundation]. of how a light bundle is distorted on its passage through t ationally lensed than stars and that one can use the gravitational lens effect as the universe hadOriginally been derived , gravitational in the context lensing of was gravitational discussed forradiation stars or even for galaxies . When quasars were a ‘‘ natural $\ l e f t . t e l e s c o p e \ begin { a l i g n e d } &’’ \\ before opendiscovered square bracket in the 1 24960 closing s , Barnothy square [ bracket 1 5 ] was period the first to connect them &. \end{ a l i g n e d }\ right . $ Originallywith comma the gravitational gravitational lensing lens effect was . discussed In the for late stars 60 s or / earlyfor galaxies 70 s , a period few groups When and individuals quasars wereexplored discovered various in aspects the 1 960 of lensing s comma further Barnothy , for example open square , statistical bracket effects 1 5 of closing lo cal inhomogeneities square bracket was the first to In the 1 960 s , a few partly independent theoretical studies showed the use − connect themon the propagation of light [57, 58, ?]; lensing fulness of lensing for astronomy [ 73 , 84 , 85 , 93 , 1 1 3 , \quad 1 1 4 ] . \quad In particular , Sjur with the gravitational lens effect1 stated period above .. In . Onlythe late very 60 recently s slash it was early shown 70 s that comma Einstein a few had groups derived theseand equa - Refsdal derived the basicAs equations of gravitational lens theory and subsequently individualst ionsexplored as early various as 1 9 1 aspects 2 , but did of lensing not bother further to publish comma them for [ 1 example 20 ] . comma statistical showed how the gravitational lens effect can be used to determine Hubble ’ s con − effects of loLiving cal inhomogeneities Reviews in Relativity on the( propagation1998 - 12 of) light open square bracket 57 comma 58 comma ? closing square stant by measuring the t ime delay between two lensed images . \quad He followed up bracket semicolonhttp lensing : / / www . l ivingreviews . org this1 sub work As stated with above interesting period Only applications very recently it was of shown lensing that Einstein [ 1 1 6 had , derived\quad these1 1 equa 5 , hyphen\quad 1 1 7 ] . \quad The math − ematicalt ions as early foundation as 1 9 1 2 comma of how but a did light not bother bundle to publish is distorted them open square on its bracket passage 1 20 closing through square bracket period theLiving universe Reviews hadin Relativity been derived open parenthesis in the 1998 context hyphen of12 closing gravitational parenthesis radiation even beforehttp : slash [ 1 slash 24 ] www . period l ivingreviews period org Originally , gravitational lensing was discussed for stars or for galaxies . When quasars were discovered in the 1 960 s , Barnothy [ 1 5 ] was the first to connect them

\noindent with the gravitational lens effect . \quad In the late 60 s / early 70 s , a few groups and individuals explored various aspects of lensing further , for example , statistical effects of lo cal inhomogeneities on the propagation of light $ [ 57 , 58 , ? ] ; $ l e n s i n g

\ hspace ∗{\ f i l l } $ 1 { As }$ stated above . Only very recently it was shown that Einstein had derived these equa −

\noindent t ions as early as 1 9 1 2 , but did not bother to publish them [ 1 20 ] .

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 9 .... Gravitational Lensing in Astronomy \noindentapplied to9 quasars\ h f i and l l Gravitational clusters of galaxies Lensing.... open square in Astronomy bracket 34 comma .... 1 0 comma .... 1 25 closing square bracket semicolon .... development of a \noindentformalismapplied for transparent to quasars lenses .... and openclusters square bracket of ? galaxies comma 32\ closingh f i l l square[ 34 bracket , \ h f semicolon i l l 1 0 .... , \ andh fi the l l effect1 25 ] ; \ h f i l l development of a of an inhomogeneous9 Gravitational Lensing in Astronomy \noindentuniverse onappliedformalism the distance to quasars for hyphen and transparent redshift clusters relations of galaxies lenses open\ squareh [ 34 f i l , l bracket$ 1 [0 37 , closing ? 1 25 , square ] ; 32 bracket development ] period ; $ of\ ah f i l l and the effect of an inhomogeneous But only informalism 1 979 did for the transparent whole field lenses receive a real boost [?, 32]; when the first and double the effect of an inhomogeneous \noindentquasar wasuniverseuniverse discovered on the andon distance theconfirmed distance - redshift to be a− relations realredshift gravitational [ 37 ] .relations lens by Walsh [ 37 comma ] . Carswell ampersandBut only Weymann in 1 979 open did the square whole bracket field 1receive 60 closing a real square boost bracket when the period first .. double This discovery quasar was comma .. and the developmentBut only in ofdiscovered lensing 1 979 and did confirmed the whole to be field a real gravitational receive a lens real by Walsh boost , Carswellwhen the & Weymann first double [ 1 60 ] quasarsince then was. comma discovered This will discovery be described and , and confirmed in the Section development 4 to period be of a lensing real since gravitational then , will be described lens by in Walsh Section ,4 . Carswellbullet .. Going $ •\& furtherGoing $ Weymann period further .. There [ . 1 are60There a ] few are . \ historic aquad few historicThis accounts accounts discovery of lensing of lensing which , \quad which are and are more the detailed development of lensing sincemore detailed thenthan , than the will one the be presented one described presented herehere . in Some period Section examples .. Some 4 can examples. be found can in be found [ 1 22 ] , in [ 1 35 ] , and in .. openin square [ 1 bracket 6 1 ] . 1 22 The closing complete square history bracket of gravitational comma .. in .. lensing open square has yet bracket to be written 1 35 closing . square bracket comma ..$ and\ bullet in .. open$ square\quad bracketGoing 1 6 further 1 closing square . \quad bracketThere periodLiving are .. The a Reviews few complete historic in Relativity history accounts of gravitational( 1998 - of 12 lensing) which are morelensing detailed has yet to than be written the periodone presented here . http\quad :Some / / www examples . l ivingreviews can be found . org inLiving\quad Reviews[ 1 22in Relativity ] , \quad openin parenthesis\quad [ 1998 1 35 hyphen ] , 12\quad closingand parenthesis in \quad [ 1 6 1 ] . \quad The complete history of gravitational lensinghttp : slash has slash yet www to period be written l ivingreviews . period org \ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 1 0 \noindent3 .. BasicsJ .. .of WambsganssGravitational ..\ Lensingh f i l l 1 0 The path comma the size and the cross section of a light bundle propagating through \noindentspacetime3 in\ principlequad Basics are affected\quad by allof the Gravitational matter between the\quad light sourceLensing and the observerJ . Wambsganss period For most practical purposes comma we can assume that the lensing 1 0 \noindentaction is dominated3The path Basics by , a singlethe size matter of Gravitational and inhomogeneity the cross at sectionsome lo Lensing cation of a between light bundle propagating through spacetimesource andThe observer in path principle period, the size .. areThis and is affected the usually cross called section by theall of quotedblleft a the light matter bundle thin propagatingbetween lens approximation the through light quotedblright spacetime source in : .. All andthe the action observerprinciple of deflection are . isaffected For thought most by to all practical take the place matter at purposes abetween single dist the , ance light we period can source assume .. and This the ap that observer hyphen the . lensing For most actionproach is is validpractical dominated only purposes if the by relative a, we single velo can cities assume matter of lens that inhomogeneity comma the lensing source action and at observer is some dominated are lo cation by a single between matter sourcesmall compared andinhomogeneity observer to the velocity at . some\quad of lo light cationThis v ll between cis and usually if source the Newtonian and called observer potential the . ‘‘ This is thin is usually lens called approximation the “ thin ’’ : \quad All thesmall action bar Capitallens of approximation deflection Phi bar ll c ” to : is the All thought power the action of 2 to period of take deflection These place two is thought assumptions at a to single take are place j dist ustified at a ance single in all . dist astronomical\quad ance This cases ap of− proachinterest is period. valid This The ap sizeonly - proach of a if galaxythe is valid comma relative only eif period the velo relative g period cities velo comma cities of oflens is lens of order , , source source 50 kp and c observerand comma observer even are small a cluster are of galaxies small comparedcompared to to the the velocity velocity of light ofv light c and $if v the\ Newtonianl l c $ potential and if is the Newtonian potential is is not much larger than 1 Mpc2 period This quotedblleft lens thickness quotedblright is small compared to the typical distancessmall | ofΦ | orderc few. These Gpc between two assumptions observer are and j lens ustified or lens in all and astronomical back hyphen cases of interest . The \noindentground quasarsizesmall of slash a galaxy $ galaxy\mid , e comma . g\ .Phi , isrespectively of order\mid 50 period kp\ cl l, .... even Wec a ˆ clusterassume{ 2 } of that galaxies. the $ underlying These is not much two spacetime larger assumptions than 1 are j ustified in all astronomical cases of interestis well describedMpc . The . This by size a“ perturbed lens of thickness a galaxy Friedmann ” is small , hyphen e compared. g Robertson . , to is the of hyphen typical order Walker distances 50 kpmetric of c order to, even the few power Gpc a clusterbetween of 2 : of galaxies isEquation: not muchobserver open larger parenthesis and lens than 3 or closing 1lens Mpc and parenthesis . back This - .. ‘‘ ds lens to the thickness power of 2 = parenleftbigg ’’ is small 1 plus compared 2 c to the to power the of 2 Capital typical distancesground quasar of / ordergalaxy , few respectively Gpc between . observer We assume and lens that the or underlying lens and spacetime back − Phi parenrightbigg c to the power of 2 dt to the power of 2 minus a to the power of 22 open parenthesis t closing parenthesis parenleftbiggis 1minus well described 2 c to the by power a perturbed of 2 Capital Friedmann Phi parenrightbigg - Robertson - d Walker sigma metric to the power: of 2 period \noindent3 period 1ground .. Lens equation quasar / galaxy , respectively . \ h f i l l We assume that the underlying spacetime The basic setup for such a simplified gravitational2 lens2 scenario2 2 involving a point2 ds = (1 + 2 2 Φ)c dt − a (t)(1 − 2 2 Φ)dσ . (3) \noindentsource andis a point well lens described is displayed by in Figure a perturbed 2 periodc .. Friedmann The three ingredients− cRobertson in such− Walker $ metric ˆ{ 2 } : $ a lensing situation3 . 1 are Lens the source equation S comma the lens L comma and the observer O period Light rays emitted from the source are deflected by the lens period .. For a point hyphen like lens comma there \ begin { a l i gThe n ∗} basic setup for such a simplified gravitational lens scenario involving a point source and a will alwayspoint be open lens isparenthesis displayed at in leastFigure closing 2 . parenthesis The three ingredients two images in S such sub 1a lensingand S sub situation 2 of the are source the period With externalds ˆ{ shear2 } = ( 1 + 2 { c ˆ{ 2 }}\Phi ) c ˆ{ 2 } dt ˆ{ 2 } − a ˆ{ 2 } ( t ) (source 1 S− , the2 lens{ Lc , and ˆ{ the2 }}\ observerPhi O . Light) rays d emitted\sigma fromˆ{ the2 } source. \ aretag deflected∗{$ ( by 3 ) $} endash duethe to lens the .tidal For field a point of objects - like outside lens , there but near the light bundles endash there can \endbe{ morea l i g nimages∗} period .. The observer sees the images in directions corresponding to will always be ( at least ) two images S1 and S2 of the source . With external shear – due to the tangents to the real incoming light paths period \noindent the3 . tidal 1 \ fieldquad ofLens objects equation outside but near the light bundles – there can be more images . The In Figure 3observer the corresponding sees the images angles in and directions angular corresponding diameter distances to the D tangents sub L comma to the D real sub incoming S comma light D sub LS arepaths indicated . to the power of 3 period .. In the thin hyphen lens approximation comma the hyperbolic paths are \noindentapproximatedThe by basic their asymptotes setup for period such In a the simplified circular hyphen gravitational symmetric case the lens deflection scenario involving a point source and a point lensIn is Figure displayed 3 the corresponding in Figure angles 2 . and\quad angularThe diameter three distances ingredientsDL,DS, in such angle is given as 3 a lensingDLS situationare indicated are. theIn the source thin - Slens , approximation the lens L , , the and hyperbolic the observer paths are O approximated . Light rays Equation:by open their parenthesis asymptotes 4 closing. In the parenthesis circular - symmetric .. alpha-tilde case open the deflection parenthesis angle xi closing is given parenthesis as = 4 GM c to the poweremitted of 2 open from parenthesis the source xi closing are parenthesis deflected 1 xi by comma the lens . \quad For a point − like lens , there where M open parenthesis xi closing parenthesis is the mass inside a radius xi period In this depiction the origin is chosen \noindent will always be ( at least ) two images $ S { 1 }$ and $ S { 2 }$ of the source . With external shear at the observer period .. From the diagram it canα˜( beξ) = seen 4GM thatc2 ( theξ)1ξ, following relation (4) −−Linedue 1 holds to the : Line tidal 2 theta field D sub of S = objects beta D sub outside S plus alpha-tilde but near D sub the LS light open parenthesis bundles −− 5 closingthe re parenthesis can be2 sub more A detailed imageswhere M description .(ξ)\quad is the of massThe optics inside observer in curved a radius sees spacetimesξ. In the this andimages depiction a derivation in thedirections origin of the is chosen lens equation corresponding at the observer . to thefrom tangents EinsteinFrom quoteright tothe diagramthe real s field it can incoming equations be seen canthat light be the found following paths in Chapter . relation 3 and 4 of open square bracket 1 35 closing square bracket period \ hspace3 sub In∗{\ cosmologyf i l l } In comma Figure the 3 various the corresponding methods to define anglesdistanceholds diverge and : angular open parenthesis diameter see comma distances e period $ g D period{ L } ,D { S } , $ Chapters 14 period 4 θDS = βDS +αD ˜ LS (5) of open square bracket 1 66 closing square bracket or 3 period 5 of open square bracket 1 35 closing square bracket closing 2 detailed description of optics in curved spacetimes and a derivation of the lens equation from Einstein ’ s parenthesis\noindent period$ DA ....{ TheLS relevant}$ are distances $ indicated for gravitational ˆ{ 3 lensing} . are $ the\quad angularIn the thin − lens approximation , the hyperbolic paths are field equations can be found in Chapter 3 and 4 of [ 1 35 ] . approximateddiameter distances by comma their see asymptotes open square .bracket In the 96 closing circular square− bracketsymmetric period case the deflection 3 cosmology , the various methods to define distance diverge ( see , e . g . Chapters 14 . 4 angleLiving is Reviews given in Relativityas In open parenthesis 1998 hyphen 12 closing parenthesis of [ 1 66 ] or 3 . 5 of [ 1 35 ] ) . The relevant distances for gravitational lensing are the angular http : slash slash www period l ivingreviews period org \ begin { a l i gdiameter n ∗} distances , see [ 96 ] . \ tilde {\alphaLiving} Reviews( in\ xi Relativity)( =1998 4 - 12 GM) { c ˆ{ 2 }} ( \ xi ) 1{\ xi } , \ tag ∗{$ ( 4 ) $} http : / / www . l ivingreviews . org \end{ a l i g n ∗}

\noindent where $ M ( \ xi ) $ is the mass inside a radius $ \ xi . $ In this depiction the origin is chosen at the observer . \quad From the diagram it can be seen that the following relation

\ [ \ begin { a l i g n e d } holds : \\ \theta D { S } = \beta D { S } + \ tilde {\alpha} D { LS } ( 5 ) \end{ a l i g n e d }\ ]

$ 2 { A }$ detailed description of optics in curved spacetimes and a derivation of the lens equation from Einstein ’ s field equations can be found in Chapter 3 and 4 of [ 1 35 ] .

\ hspace ∗{\ f i l l } $ 3 { In }$ cosmology , the various methods to define distance diverge ( see , e . g . Chapters 14 . 4

\noindent of [166] or3.5of [135] ) . \ h f i l l The relevant distances for gravitational lensing are the angular

\noindent diameter distances , see [ 96 ] .

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 1 1 .... Gravitational Lensing in Astronomy \noindentFigure 2 :1 .... 1 Setup\ h f iof l l a gravitationalGravitational lens s Lensingituation : .... in The Astronomy lens L located between s ource S .. and o bs erver O produces two .. images S sub 1 .. and S sub 2 .. of the .. ba ckground \noindents ource periodFigure 2 : \ h f i l l Setup of a gravitational lens s ituation : \ h f i l l The l e n s $ L $ locatedLiving Reviews between1 1 in Relativity open parenthesis 1998 hyphen 12 closing Gravitationalparenthesis Lensing in Astronomy http : slashFigure slash 2 www : periodSetup l ivingreviews of a gravitational period org lens s ituation : The lens L located between \noindent ss ource ourceS $and S o$ bs\ erverquad Oandproduces o bs two erver images $ O $S1 producesand S2 twoof the\quad ba ckgroundimages $ S { 1 }$ \quad ands $ource S .{ 2 }$ \quad o f the \quad ba ckground s ource . Living Reviews in Relativity ( 1998 - 12 ) http : / / www . l ivingreviews . org \ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 1 2 \noindentopen parenthesisJ . Wambsganss for theta comma\ h f i beta l l 1 comma 2 alpha-tilde ll 1 semicolon .. this condition is fulfilled in practically all astrophysically \noindentrelevant situations( f o r closing $ \theta parenthesis, period\beta .. With, the\ definitiontilde {\ ofalpha the reduced}\ l deflection l 1 angle ; $ as\quad this condition is fulfilled in practically all astrophysically relevantalpha openJ situations parenthesis . Wambsganss theta ) . closing\quad parenthesisWith the = open definition parenthesis of D sub the LS reduced slash D sub deflection S closing parenthesis angle1 2 as alpha-tilde open$ parenthesis\alpha( for theta( θ, closing\ β,thetaα˜ parenthesis )1; = this comma condition( this D can{ isLS be fulfilled expressed} /D in practically as : { S } all astrophysically) \ tilde {\ relevantalpha} ( \theta )Equation: , $ thissituations open can parenthesis ) be . expressed With 6 closing the definition parenthesis as : of the .. reduced beta = deflectiontheta minus angle alpha as α open(θ) = parenthesis (DLS/DS)˜ thetaα(θ), this closing parenthesis period can be expressed as : \ beginThis{ relationa l i g n ∗} between the positions of images and source can easily be derived \betafor a non= hyphen\theta symmetric− mass \alpha distribution( as well\theta period ..). In that\ casetag ∗{ comma$ ( .. all 6 angles ) $ are} β = θ − α(θ). (6) \endvector{ a l i hyphen g n ∗} valued period The two hyphen dimensional lens equation then reads : Equation:This open relation parenthesis between 7 closing the positions parenthesis of images .. beta-vector and source = vector-theta can easily beminus derived alpha-vector for a non open - sym- parenthesis to the power\noindent of vector-thetaThis relation closing parenthesis between period the positions of images and source can easily be derived f o r a nonmetric− symmetric mass distribution mass distribution as well . In that as case well , . all\ anglesquad areIn vector that - case valued , .\ Thequad twoall - angles are Figure 3 :dimensional .. The re lationlens between equation thethen various reads angles : and distances involved in the vectorlensing− s evalued tup .. can . .. The be derived two − fordimensional the .. cas e alpha-tilde lens equation ll 1 .. and then formulated reads in the : .. lens equation open parenthesis 6 closing parenthesis period \ begin { a l i g n ∗} ~ Living Reviews in Relativity open parenthesis 1998β~ hyphen= θ~ − 12~α(θ closing). parenthesis (7) \vechttp{\ :beta slash} slash= www\ periodvec{\ ltheta ivingreviews} − period \vec org{\alpha} ( ˆ{\vec{\theta}} ). \ tag ∗{$ ( 7 ) $} Figure 3 : The re lation between the various angles and distances involved in the lensing s e \end{ a l i g ntup∗} can be derived for the cas e α˜  1 and formulated in the lens equation ( 6 ) . \noindent LivingFigure Reviews 3 : \ inquad RelativityThe re( 1998 lation - 12 between) the various angles and distances involved in the lensing shttp e tup : /\quad / wwwcan . l\quad ivingreviewsbe derived . org for the \quad cas e $ \ tilde {\alpha}\ l l 1 $ \quad and formulated in the \quad l e n s equation ( 6 ) .

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 1 3 .... Gravitational Lensing in Astronomy \noindent3 period 21 .. 3Einstein\ h f i l radius l Gravitational Lensing in Astronomy For a point lens of mass M comma .... the deflection angle is given by Equation .... open parenthesis 4 closing parenthesis period\noindent 3 . 2 \quad Einstein radius Plugging into1 3 Equation open parenthesis 6 closing parenthesis and using Gravitational the relation Lensing xi = D insub Astronomy L theta open parenthesis cf\noindent period Figure3For . 3 2 closing a point Einstein parenthesis lens radius of comma mass one $M , $ \ h f i l l the deflection angle is given by Equation \ h f i l l ( 4 ) . Line 1 obtainsFor a : point Line 2lens beta of openmass parenthesisM, thetathe closing deflection parenthesis angle is given = theta by Equationminus D sub L to the ( 4 power ) . of D sub LS D\noindent sub S 4 subPluggingPlugging c sub 2 sub into into theta Equation toEquation the ( 6 power ) and ( of using 6 GM ) the periodand relation using openξ parenthesis= theDLθ relation( cf 8. Figure closing $ 3 parenthesis )\ ,xi one = D { L }\theta ( $ cf . Figure 3 ) , one For the special case in which the source lies exactly behind the lensobtains open parenthesis: beta = 0 closing parenthesis comma due to the symmetry a ring hyphen like image o ccurs whose angular radius is called \ [ \ begin { a l i g n e d } obtains : \\ DLS GM Einstein radius theta sub E : β(θ) = θ − DL DS4c 2θ . (8) \Equation:beta ( open\ parenthesistheta 9) closing = parenthesis\theta ..− thetaD sub ˆ{ ED = radicalbigg{ LS }} of{ 4L GM} cD to the{ S power} 4 of 2{ Dc sub}ˆ L{ toGM the} { 2 {\theta }} . ( 8For ) the\end special{ a l i case g n e ind }\ which] the source lies exactly behind the lens (β = 0), power of D subdue LS to D the sub symmetry S sub period a ring - like image o ccurs whose angular radius is called Einstein radius The Einstein radius defines the angular scale for a lens situation period For a massive θE : galaxy with a mass of M = 1 0 to the power of 1 2 M sub odot at a redshift of z sub L = 0 period 5 and a source at \noindentredshift z subFor S the = 2 period special 0 comma case open in which parenthesis the we source used here lies H = exactly 50 km sec behind to the power the of lens minus $ 1 (Mpc to\beta the = 0 ) , $ q power of minus 1 as the value of the DLS θE = 4GMc2 D DS (9) Hubble constant and an Einstein hyphen deSitter universe closingL parenthesis. comma the Einstein radius is \noindentEquation:due open to parenthesis the symmetry 1 0 closing a parenthesis ring − like .. theta image sub E o thickapprox ccurs whose 1 period angular 8 radicalBigg radius of 1is 0to called the power EinsteinThe radius Einstein $ radius\theta defines{ E the} angular: $ scale for a lens situation . For a massive galaxy with a of M 1 2 M sub odot arcsec 12 mass of M = 10 M at a redshift of zL = 0.5 and a source at open parenthesis note that for cosmological distances in general−1 D− sub1 LS negationslash-equal D sub S minus D sub L ! \ begin { a l i gredshift n ∗} zS = 2.0, ( we used here H = 50 km sec Mpc as the value of the Hubble constant closing parenthesisand an period Einstein For - a deSitter galactic universe ) , the Einstein radius is \thetamicrolensing{ E scenario} = in\ sqrt which{ stars4 in GM the{ diskc of ˆ{ the2 Milky}} D Way ˆ{ actD as{ lensesLS }} { L } D { S }} { . }\ tag ∗{$ ( 9for ) bulge $} stars close to the center of the Milky Way comma the scale defined by the \end{ a l i g n ∗} p M Einstein radius is θE ≈ 1.8 10 12 M arcsec (10) Equation: open parenthesis 1 1 closing parenthesis .. theta sub E thickapprox 0 period 5 radicalBigg of M sub odot to the \noindent The Einstein radius defines the angular scale for a lens situation . For a massive power of M milliarcsec( note that period for cosmological distances in general DLS 6= DS − DL!). For a galactic microlensing galaxyAn application withamassscenario and in some which of i llustrations stars $M in the = of disk the 1 point of 0ˆ the lens{ Milky1 case Way can2 }be actM found as{\ lenses inodot for bulge}$ at stars a close redshift to the of $ z { L } =Section 0 .4 periodcenter 5$ of7 on the andasourceat galactic Milky Way microlensing , the scale period defined by the Einstein radius is 3 period 3 .. Critical surface mass density \noindent r e d s h i f t $ z { S } = 2 . 0 , ($ weusedhere $H = 50$ km $ \sec ˆ{ − In the more general case of a three hyphen dimensionalq mass distribution of an extended 1 } Mpc ˆ{ − 1 }$ as the value of the M lens comma the density rho open parenthesisθ vector-rE ≈ 0.5 closingM milliarcsec parenthesis. can be projected along the line(11) of sight onto the lensHubble plane constant and an Einstein − deSitter universe ) , the Einstein radius is to obtain theAn applicationtwo hyphen and dimensional some i llustrations surface mass of density the point distribution lens casecan Capital be found Sigma in open Section parenthesis 4 . 7 on xi-vector closing parenthesis\ begin { a l comma i ggalactic n ∗} as microlensing . \thetaEquation:{3 openE .}\ 3 parenthesisthickapprox Critical 1 2 surface closing 1 parenthesis mass . 8 density ..\ sqrt Capital{ 1 Sigma 0 ˆ open{ M parenthesis{ 1 2 xi-vector}} M closing{\odot parenthesis}} a r =c s e c \ tag ∗{$ ( integral1 0 sub ) 0In $ to} the the more power general of D sub case S of rho a openthree parenthesis - dimensional r-vector mass distribution closing parenthesis of an extended dz period \endHere{ a lvector-r i g nlens∗} is , athe three density hyphenρ(~r) dimensional can be projected vector along in space the comma line of sight and vector-xi onto the is lens a two plane hyphen to obtain dimensional the vector in the lenstwo plane - dimensional period .. The surface two hyphen mass density dimensional distribution deflection Σ(ξ~) angle, as vector-alpha-tilde is then given as the \noindentsum over all( mass note elements that for in the cosmological lens plane : distances in general $ D { LS }\not= D { S } − Equation:D { L open} ! parenthesis ) .$ 1 3 closing Foragalactic parenthesis .. vector-alpha-tilde open parenthesis vector-xi closing parenthesis = 4 Z DS cmicrolensing to the power of 2 Gscenario integral open in parenthesiswhich stars xi-vector in~ the bar to disk the power of the of minus Milky sub Way vector-xi act xi-vector as lenses minus to the power for bulge stars close to the center ofΣ(ξ the) = Milkyρ(~r) Waydz. , the scale defined by the(12) of prime closing parenthesis sub vector-xi to the power of prime0 bar to the power of 2 to the power of Capital Sigma open parenthesisEinstein xi-vector radius to is the power of prime closing parenthesis d to the power of 2 xi to the power of prime period ~ Living ReviewsHere ~r inis Relativity a three -open dimensional parenthesis vector 1998 in spacehyphen , and 12 closingξ is a twoparenthesis - dimensional vector in the lens ~ \ beginhttp{ :a slash l i gplane n slash∗} . www The period two - dimensional l ivingreviews deflection period org angle α˜ is then given as the sum over all mass elements \theta {inE the}\ lensthickapprox plane : 0 . 5 \ sqrt { M ˆ{ M } {\odot }} milliarcsec . \ tag ∗{$ ( 1 1 ) $} \end{ a l i g n ∗} Z − 0)Σ(ξ~0) 2 0 ~ ~ 2 ~ ~ α˜(ξ) = 4c G (ξ |~ ξ−~0 2 d ξ . (13) \noindent An application and some i llustrationsξ ofξ | the point lens case can be found in Section 4 . 7 on galactic microlensing . Living Reviews in Relativity ( 1998 - 12 ) http : / / www . l ivingreviews . org \noindent 3 . 3 \quad Critical surface mass density

\noindent In the more general case of a three − dimensional mass distribution of an extended

\noindent lens , the density $ \rho ( \vec{ r } ) $ can be projected along the line of sight onto the lens plane to obtain the two − dimensional surface mass density distribution $ \Sigma ( \vec{\ xi } ) , $ as

\ begin { a l i g n ∗} \Sigma ( \vec{\ xi } ) = \ int ˆ{ D { S }} { 0 }\rho ( \vec{ r } ) dz . \ tag ∗{$ ( 1 2 ) $} \end{ a l i g n ∗}

\noindent Here $ \vec{ r } $ i s a thr ee − dimensional vector in space , and $ \vec{\ xi } $ i s a two − dimensional vector in the lens plane . \quad The two − dimensional deflection angle $ \vec{\ tilde {\alpha}} $ is then given as the sum over all mass elements in the lens plane :

\ begin { a l i g n ∗} \vec{\ tilde {\alpha}} ( \vec{\ xi } ) = 4 { c ˆ{ 2 }} G \ int ( \vec{\ xi }\mid ˆ{ − } {\vec{\ xi }} \vec{\ xi } { − }ˆ{\prime ) }ˆ{\Sigma ( \vec{\ xi } ˆ{\prime } ) } {\vec{\ xi } ˆ{\prime } \mid ˆ{ 2 }} d ˆ{ 2 }\ xi ˆ{\prime } . \ tag ∗{$ ( 1 3 ) $} \end{ a l i g n ∗}

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .. 1 4 \noindentFor a finiteJ circle . Wambsganss with constant\quad surface1 mass 4 density Capital Sigma the deflection angle can Forbe written a finite : circle with constant surface mass density $ \Sigma $ the deflection angle can Equation: open parenthesis 1 4 closing parenthesis .. alpha open parenthesis xi closing parenthesis = D sub LS D sub S 4 c\noindent to the powerJbe of. Wambsganss 2 written G Capital : Sigma 1 4 For pi xia finite xi to circle the power with constantof 2 period surface mass density Σ the deflection angle With xi =can D sub L theta this simplifies to \ beginEquation:{ a l i gbe open n ∗} written parenthesis : 1 5 closing parenthesis .. alpha open parenthesis theta closing parenthesis = 4 pi sub c sub 2 to\ thealpha power of( G Capital\ xi Sigma) = D sub D L D{LS sub{ SD to} the{ S power}} of4 D sub{ c LS ˆ{ theta2 }} periodG \Sigma{\ pi }\ xi {\ xi }ˆ{ 2 } . \ tag ∗{$ ( 1 4 ) $} With the definition of the critical surface mass density Capital Sigma2 sub crit as α(ξ) = D 4 2 GΣπξξ . (14) \endEquation:{ a l i g n ∗} open parenthesis 1 6 closing parenthesis .. CapitalLSDS c Sigma sub crit = 4 pi to the power of c to the power of 2 G D sub L to the power of D sub D sub LS to the power of S comma With ξ = DLθ this simplifies to \noindentthe deflectionWith angle $ for\ xi a such= a mass D distribution{ L }\theta can be expressed$ this as simplifies to Equation: open parenthesis 1 7 closing parenthesis .. alpha-tilde open parenthesis theta closing parenthesis = Capital \ begin { a l i g n ∗} Sigma sub crit to the power of Capital Sigma theta period GΣ DLS α(θ) = 4πc 2DLD Sθ. (15) \alphaThe critical( surface\theta mass density) =is given 4 by\ thepi lens{ massc }ˆ M{ quotedblleftG \Sigma smeared} { 2 out} quotedblrightD {L{ D }}ˆ{ D { LS }} { S } \theta . \ tag ∗{$ ( 1 5 ) $} over the areaWith of the definitionEinstein ring of the : Capitalcritical Sigma surface sub crit mass = Mdensity slash openΣcrit parenthesisas R sub E to the power of 2 pi closing parenthesis\end{ a l i g comman ∗} where R sub E = theta sub E D sub L period .... The

value of the critical surface mass density is roughly Capital2 Sigma sub crit thickapprox 0 period 8 g cm to the power of \noindent With the definition of theΣ critical= 4πc GD surfaceDS , mass density $ \Sigma (16){ c r i t }$ as minus 2 for lens crit L DLS and source redshifts of z sub L = 0 period 5 and z sub S = 2 period 0 comma respectively period .. For an arbitrary \ beginmass{ distributiona l i gthe n ∗} deflection comma angle the condition for a such Capital a mass Sigma distribution greater can Capital be expressed Sigma sub as crit at any point is sufficient to produce \Sigmamultiple{ imagesc r i t period} = 4 \ pi ˆ{ c ˆ{ 2 }} G D ˆ{ D } { L }ˆ{ S } { D { LS }} , \ tag ∗{$ ( 13 period 6 ) 4 $ ..} Image positions and magnifications α˜(θ) = ΣΣ θ. (17) \endThe{ a lens l i g n equation∗} open parenthesis 6 closing parenthesis cancrit be re hyphen formulated in the case of a single point lens : Equation: open parenthesis 1 8 closing parenthesis .. beta = theta minus E from theta to theta to the power of 2 period \noindent Thethe critical deflection surface mass angle density for is a given such by a the mass lens mass distributionM “ smeared can out be” expressed as Solving this for the image positions theta one finds that an isolated2 point source over the area of the Einstein ring : Σcrit = M/(REπ), where RE = θEDL. The always produces two images of a background source period The positions of the images−2 value of the critical surface mass density is roughly Σcrit ≈ 0.8 g cm for lens and source \ beginare given{ a l i byg n the∗} two solutions : redshifts of zL = 0.5 and zS = 2.0, respectively . For an arbitrary mass distribution , the \ tildeEquation:{\alpha open} parenthesis( \theta 1 9 closing) parenthesis = \Sigma .. thetaˆ{\ subSigma 1 comma} { 2c = r i 2 t to}\ thetheta power of 1. parenleftbigg\ tag ∗{$ ( beta 1 condition Σ > Σcrit at any point is sufficient to produce multiple images . plusminux7 ) $} radicalBig of beta to the power of 2 plus 4 theta sub E to the power of 2 parenrightbigg period \end{ a l i g n3∗} . 4 Image positions and magnifications The magnificationThe lens of equation an image ( 6 is ) defined can be re by - the formulated ratio between in the the case solid of a angles single point lens : of the image and the source comma since the surface brightness is conserved period .. Hence \noindentthe magnificationThe critical mu is given surface as mass density is given by the lens mass $M$ ‘‘ smeared out ’’ θ Equation: open parenthesis 20 closing parenthesis ..β mu= θ =− thetaEθ2 . beta d to the power of d sub beta to the(18) power of period to\noindent the power ofover theta the area of the Einstein ring $ : \Sigma { c r i t } = M / ( R ˆ{ 2 } { E } \ piIn the) symmetricSolving , $ where casethis for above the $ commaR image{ E positions the} image= θ magnificationone\theta finds that{ cananE isolated} beD written{ pointL as} source open. parenthesis $ always\ h fproduces i l l byThe two using the lensimages equation of a background closing parenthesis source . The: positions of the images are given by the two solutions : \noindentEquation:value open parenthesis of the 2 critical 1 closing parenthesis surface .. mass mu 1 density comma 2 = is open roughly parenthesis $ \ 1Sigma minus bracketleftbigg{ c r i t }\ subthickapprox theta 0 . 8 $ g $ cm ˆ{ − 2 }$ f o r l e n s sub 1 comma 2 to the power of theta sub E bracketrightbigg 4q closing parenthesis minus 1 = 2 u to the power of u surd of u and source redshifts of $ z { L } =1 0 .2 5 $2 and $ z { S } = 2 . 0 , $ to the power of 2 plus 4 to the power of 2 plus 2θ plusminux1,2 = 2 (β ± 2 toβ the+ power 4θE). of 1 period (19) respectivelyLiving Reviews . in\quad RelativityFor open an parenthesis arbitrary 1998 hyphen 12 closing parenthesis masshttp :distribution slashThe slash magnification www , period the of l condition an ivingreviews image is defined period $ \Sigma by org the ratio> between\Sigma the solid{ c r angles i t }$ of the at image any point and is sufficient to produce multiplethe images source . , since the surface brightness is conserved . Hence the magnification µ is given as

\noindent 3 . 4 \quad Image positions and magnifications dθ µ = θβd β. (20) \noindent The lens equation ( 6 ) can be re − formulated in the case of a single point lens : In the symmetric case above , the image magnification can be written as ( by using the lens \ begin { a l i gequation n ∗} ) : \beta = \theta − E ˆ{\theta } {\theta ˆ{ 2 }} . \ tag ∗{$ ( 1 8 ) $} \end{ a l i g n ∗} √ θE u 2 2+2 1 µ1, 2 = (1 − [θ 1,2]4) − 1 = 2u u + 4 ± 2 . (21) \noindent Solving this for the image positions $ \theta $ one finds that an isolated point source always producesLiving Reviews two images in Relativity of a( 1998 background - 12 ) source . The positions of the images are givenhttp by the: / two / www solutions . l ivingreviews : . org

\ begin { a l i g n ∗} \theta { 1 , 2 } = 2 ˆ{ 1 } ( \beta \pm \ sqrt {\beta ˆ{ 2 } + 4 \theta ˆ{ 2 } { E }} ). \ tag ∗{$ ( 1 9 ) $} \end{ a l i g n ∗}

\noindent The magnification of an image is defined by the ratio between the solid angles of the image and the source , since the surface brightness is conserved . \quad Hence the magnification $ \mu $ i s given as

\ begin { a l i g n ∗} \mu = \theta{\beta } d ˆ{ d }ˆ{\theta } {\beta ˆ{ . }}\ tag ∗{$ ( 20 ) $} \end{ a l i g n ∗}

\noindent In the symmetric case above , the image magnification can be written as ( by using the lens equation ) :

\ begin { a l i g n ∗} \mu 1 , 2 = ( 1 − [ {\theta }ˆ{\theta { E }} { 1 , 2 } ] 4 ) − 1 = 2 u ˆ{ u }\surd{ u ˆ{ 2 } + 4 }ˆ{ 2 + 2 }\pm 2 ˆ{ 1 } . \ tag ∗{$ ( 2 1 ) $} \end{ a l i g n ∗}

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 1 5 .... Gravitational Lensing in Astronomy \noindentHere we defined1 5 \ uh as f i the l l quotedblleftGravitational impact Lensing parameter in quotedblright Astronomy comma the angular separation between lens and source in units of the Einstein radius : u = beta slash theta sub E period .. The magnification \noindentof one imageHere open we parenthesis defined the $ one u inside $ as the the Einstein ‘‘ impact radius closing parameter parenthesis ’’ is , negative the angular period This separation means it has between lensnegative and parity1 source 5 : It is in mirror units hyphen of the inverted Einstein period For radius beta right $arrow : Gravitational u 0 the = magnification\ Lensingbeta in diverges/ Astronomy\theta period In{ E } . $ \quadthe limitThe ofHere magnification geometrical we defined opticsu as comma the “ impact the Einstein parameter ring ” of , a the point angular source separation has infinite between lens and source magnificationin units to the ofpower the Einstein of 4 !The radius sum : ofu the= β/θ absoluteE. The values magnification of the two image magnifications \noindentis the measurableofof one one image total image ( magnification the ( one the inside one mu the : inside Einstein the radius Einstein ) is negative radius . This means ) is it negative has negative . parity This means it has negativeEquation:: parityopen It is parenthesismirror : It - inverted is 22 mirrorclosing . For parenthesisβ−→inverted0 the magnification .. mu . = For bar mu diverges $ 1\ barbeta . plus In the bar\rightarrow limit mu 2 of bar geometrical = u to0 the optics$ power the of magnification 2 plus 2 u diverges . In the limit, the of Einstein geometrical ring of a optics point source , the has Einstein infinite ring of a point source has infinite surd of u to the power of 24 plus 4 sub period Note thatmagnification this value is open! The parenthesis sum of the absolute always closing values parenthesisof the two image larger magnifications than 1 to the is power the measurable of 5 ! .. The difference between\noindent the total$ magni magnification fi cationµ : ˆ{ 4 } ! $ The sum of the absolute values of the two image magnifications istwo the image measurable magnifications total is unity magnification : $ \mu : $ Equation: open parenthesis 23 closing parenthesis .. mu 1 plus mu√ 2 = 1 period µ =| µ1 | + | µ2 |= u2 + 2u u2 + 4 (22) \ begin3 period{ a l i5 g .. n ∗} open parenthesis Non hyphen closing parenthesis Singular isothermal. sphere \mu = \mid \mu 1 \mid + \mid \mu 2 \mid = u ˆ{ 2 } + 2 { u \surd{ u ˆ{ 2 } A handy and popular model for galaxy lenses is the singular5 isothermal sphere + 4 }} {Note. }\ thattag this∗{$ value ( is 22 ( always ) $)} larger than 1 ! The difference between the two image magni- with a threefications hyphen is dimensional unity : density distribution of \endEquation:{ a l i g n ∗} open parenthesis 24 closing parenthesis .. rho open parenthesis r closing parenthesis = v sigma to the power of 2 2 pi G 1 r to the power of 2 comma \noindent Note that this value is ( always ) larger than $ 1 ˆ{ 5 } ! $ \quad The difference between the where sigma sub v is the one hyphen dimensional velocityµ1 + µ2 dispersion = 1. period .. Projecting the matter on(23) twoa plane image comma magnifications one obtains the circularly is unity hyphen : symmetric surface mass distribution Equation:3 open . 5 parenthesis ( Non 25 - closing ) Singular parenthesis isothermal .. Capital Sigmasphere open parenthesis xi closing parenthesis = v sigma to \ begin { a l i g n ∗} the power of 2A 2 handy G 1 xi and period popular model for galaxy lenses is the singular isothermal sphere \mu 1 + \mu 2 = 1 . \ tag ∗{$ ( 23 ) $} With M openwith parenthesis a three - dimensional xi closing parenthesis density distribution = integral of sub 0 to the power of xi Capital Sigma open parenthesis xi to the\end power{ a l i gof n prime∗} closing parenthesis 2 pi xi to the power of prime d xi to the power of prime plugged into Equation open parenthesis 4 closing parenthesis comma one obtains the de hyphen \noindent 3 . 5 \quad ( Non − ) Singular isothermal2 sphere flection angle for an isothermal sphere comma whichρ(r) is = a v constantσ2πG1r2 , open parenthesis i period e period independent(24) of the impact parameter xi closing parenthesis : \noindentEquation:Awhere open handy parenthesisσv is and the one popular 26 - closing dimensional model parenthesis velocity for .. galaxy alpha-tilde dispersion lenses open. Projecting parenthesis is the the singular xi matter closing on parenthesis isothermal a plane , = one 4 v sphere from c to the power of 2 toobtains pi to the the power circularly of sigma - symmetric to the power surface of 2 mass period distribution \noindentIn quotedblleftwith practical a three units− quotedblrightdimensional for density the velocity distribution dispersion this can of be expressed as : Equation: open parenthesis 27 closing parenthesis .. alpha-tilde open parenthesis xi closing parenthesis = 1 period 1 5 Row Σ(ξ) = v2 2G1ξ. (25) 1\ begin sigma{ suba l iv g Row n ∗} 2 200 kms to the power of minus 1 . 2 arcsecσ period \rhoTwo generalizations( r ) of this = isothermal v {\sigma model are}ˆ{ commonly2 }{ 2 used}\ :pi ModelsG with 1 { r ˆ{ 2 }} , \ tag ∗{$ ( 24 ) $} With M(ξ) = R ξ Σ(ξ0)2πξ0dξ0 plugged into Equation ( 4 ) , one obtains the de - finite cores are more realistic0 for open parenthesis spiral closing parenthesis galaxies period .. In this case the deflection \endangle{ a l is i g modified nflection∗} to angle open for parenthesis an isothermal core sphere radius ,xi which sub c is closing a constant parenthesis ( i . e : . independent of the impact Equation:parameter open parenthesisξ): 28 closing parenthesis .. alpha-tilde open parenthesis xi closing parenthesis = 4 v from c to the power\noindent of 2 to piwhere to the power $ \sigma of sigma{ tov the}$ power i s ofthe 2 open one parenthesis− dimensional xi sub c tovelocity the power dispersion of 2 xi plus xi .to\ thequad powerProjecting of the matter on a plane , one obtains the circularly − symmetric surface mass distribution 2 closing parenthesis to the power of 1 slash 2 period c2 α˜(ξ) = 4v σ 2 . (26) 4 sub Due to the fact that physical objects have a finite size commaπ and also because at some limit \ begin { a l i g n ∗} wave opticsIn has “ practical to be applied units ”comma for the in velocity reality dispersionthe magnification this can stays be expressed finite period as : \Sigma5 sub This( does\ notxi violate) energy = vconservation{\sigma comma}ˆ{ since2 }{ this2 is} theG magnification 1{\ xi } relative. \ tag to∗{ an$ ( 25 ) $} \endquotedblleft{ a l i g n ∗} empty quotedblright universe and not relative to a quotedblleft smoothed out quotedblright universe period This issue is treated in detail  σ  α˜(ξ) = 1.15 v 2arcsec. (27) \noindentin comma eWith period $ g periodM ( comma\ xi open) square = bracket\ int200kms 1 30ˆ{\ closing−xi1 square} { 0 bracket}\Sigma or in Chapter( 4\ periodxi ˆ{\ 5 ofprime open square} ) bracket2 \ pi 1 35 closing\ xi ˆ square{\prime bracket} periodd \ xi ˆ{\prime }$ plugged into Equation ( 4 ) , one obtains the de − Living ReviewsTwo generalizations in Relativity open of this parenthesis isothermal 1998 model hyphen are 12 commonly closing parenthesis used : Models with finite cores are \noindenthttp : slashmoreflection slash realistic www angle period for ( spiral l for ivingreviews an ) galaxies isothermal period . Inorg this sphere case the , which deflection is angle a constant is modified ( to i ( . core e . independent of the impactradius parameterξc): $ \ xi ) : $

\ begin { a l i g n ∗} c2 2 1/2 \ tilde {\alpha} ( \ xi ) =α 4˜(ξ) = v 4v ˆ{πσ 2c(ξc ˆξ{+ξ2) }}. {\ pi ˆ{\sigma }ˆ{ 2 }}(28) . \ tag ∗{$ ( 26 ) $} \end{ a l i g n ∗} 4Due to the fact that physical objects have a finite size , and also because at some limit wave optics has to be applied , in reality the magnification stays finite . \noindent In5This ‘‘ practicaldoes not violate units energy conservation ’’ for the , since velocity this is the magnification dispersion relative this to an can “ empty be ” universeexpressed as : and not relative to a “ smoothed out ” universe . This issue is treated in detail in , e . g . , [ 1 30 ] or in Chapter \ begin { a l i g4 n . 5∗} of [ 1 35 ] . \ tilde {\alpha} ( \ xi ) = 1 . 1Living 5 \ l Reviewse f t (\ begin in Relativity{ array }{( 1998c}\sigma - 12 ) { v }\\ 200 kms ˆ{ − 1 }\end{ array }\ right ) 2 a r c s e chttp . :\ tag / /∗{ www$ ( . l27 ivingreviews ) $} . org \end{ a l i g n ∗}

\noindent Two generalizations of this isothermal model are commonly used : Models with finite cores are more realistic for ( spiral ) galaxies . \quad In this case the deflection angle is modified to ( core radius $ \ xi { c } ) : $

\ begin { a l i g n ∗} \ tilde {\alpha} ( \ xi ) = 4 v ˆ{ c ˆ{ 2 }} {\ pi ˆ{\sigma }ˆ{ 2 }} ( \ xi ˆ{ 2 } { c } \ xi { + \ xi ˆ{ 2 }} ) ˆ{ 1 / 2 } . \ tag ∗{$ ( 28 ) $} \end{ a l i g n ∗}

$ 4 { Due }$ to the fact that physical objects have a finite size , and also because at some limit wave optics has to be applied , in reality the magnification stays finite .

$ 5 { This }$ does not violate energy conservation , since this is the magnification relative to an ‘‘ empty ’’ universe and not relative to a ‘‘ smoothed out ’’ universe . This issue is treated in detail in , e . g . , [ 130 ] or in Chapter4 . 5 of [ 135 ] .

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .. 1 6 \noindentFurthermoreJ comma. Wambsganss a realistic\ galaxyquad lens1 6 usually is not perfectly symmetric but is slightly elliptical period Depending on whether one wants an elliptical mass distribu hyphen \noindentt ion or anFurthermore elliptical potential , a comma realistic various formalismsgalaxy lens have usuallybeen suggested is not period perfectly Detailed symmetric but is slightlytreatmentsJ elliptical of . Wambsganss elliptical lenses . Depending 1 can 6 be found on in whether open square one bracket wants 1 4 an comma elliptical 23 comma mass? comma distribu 7 1 comma− 78 comma 1t 38 ion closing or squareFurthermore an elliptical bracket , period a realistic potential galaxy lens , various usually is formalisms not perfectlysymmetric have been but suggested is slightly elliptical . Detailed 3 period 6. .. Depending Lens mapping on whether one wants an elliptical mass distribu - t ion or an elliptical potential , \noindentIn the vicinityvarioustreatments of an formalisms arbitrary of have elliptical point been comma suggested the lenses lens . Detailed mapping canbefound as shown in in Equation $ [ open 1 parenthesis 4 , 7 23 closing , parenthesis ? , 71can be described ,treatments 78 by, its of Jacobian elliptical 1 38 matrixlenses ] can A : .$ be found in [14, 23, ?, 71, 78, 138]. A = partialdiff3 . 6 sub partialdiff Lens mapping sub vector-theta to the power of beta-vector = open parenthesis delta sub ij minus partialdiff alpha\noindent sub i partialdiffIn3 the . 6 vicinity\ subquad theta ofLens an sub arbitrary j mapping to the point power , of the open lens parenthesis mapping as to shown the power in Equation of theta-vector ( 7 ) closing parenthesis closing parenthesis =can open be parenthesis described by delta its Jacobian sub ij minus matrix partialdiffA : sub partialdiff theta to the power of 2 sub i to the power of \noindent In the vicinity of an arbitrary point , the lens mapping as shown in Equation ( 7 ) partialdiff theta j to the power of psi open~ parenthesis to theθ~ power of vector-thetaθ~ closing parenthesis closing parenthesis A = ∂β = (δ − ∂α ∂( )) = (δ − ∂2 ψ( )). (29) period open parenthesis 29 closing parenthesis∂ θ~ ij i θj ij ∂θi∂θ j \noindentHere we madecan use be of described the fact open by parenthesis its Jacobian see open matrix square bracket $ A ? : comma $ 1 3 1 closing square bracket closing Here we made use of the fact ( see [?, 131]), that the deflection angle can be expressed~ asthewhere parenthesis comma that the deflection angle can be ∇θ ψ=~α, gradient of an effective two - dimensional scalar potential ψ : \ [expressed A = nabla-vector\ partial sub{\ thetapartial psi = vector-alpha}ˆ{\vec sub{\beta comma}} as{\ the wherevec{\ gradienttheta}} of an= effective ( two\ delta hyphen dimensional{ i j } − \ partial \alpha { i }{\ partial }ˆ{ ( ˆ{\vec{\theta}} ) } {\theta { j }} ) = scalar potential psi : ~ Z ( \ delta { i j } − \ partial ˆθ{ 2 }DLS{\ partial \theta }ˆ{\ psi ( ˆ{\vec{\theta}} psi open parenthesis to the power of theta-vectorψ( ) = D closingL DS2 parenthesisc2 Φ(~r)dz, = D(30) sub L to the power of D sub LS D sub S 2 c to the power) } { ofi 2 integral ˆ{\ partial Capital Phi\ opentheta parenthesis} j } r-vector) .closing ( parenthesis 29 ) dz\ ] comma open parenthesis 30 closing parenthesis and Capitaland Phi Φ( open~r) is theparenthesis Newtonian vector-r potential closing of the parenthesis lens . is the Newtonian potential of the lens period The determinant of theThe Jacobian determinant A is the of theinverse Jacobian of theA magnificationis the inverse : of the magnification : \noindent Herewemadeuse of the fact ( see $ [ ? , 1 3 1 ] ) ,$ that the deflection angle canbe mu = determinant to the power of 1 A to the power of1 period open parenthesis 3 1 closing parenthesis $Let expressed us define {\vec{\nabla} {\thetaµ}\= detpsiA. (31)= \vec{\alpha} { , }} as the { where }$ gradientpsi sub ij of = partialdiff an effective partialdiff two theta− dimensional sub i sub partialdiff scalar theta potential sub j to the power $ \ psi of 2 sub: psi $ period open parenthesis Let us define 32 closing parenthesis \ [ \ psi ( ˆ{\vec{\theta}} ) = D ˆ{ D { LS }} { L } D { S } 2 { c ˆ{ 2 }}\ int The Laplacian of the effective potential psi is twice the convergence2ψ : \Phi ( \vec{ r } ) dz , (ψ 30ij = ∂∂ ) \∂θ] j . (32) psi 1 1 plus psi 2 2 = 2 kappa = tr psi sub ij period openθi parenthesis 33 closing parenthesis With the definitionsThe Laplacian of the of components the effective of potential the externalψ is twice shear the gamma convergence comma : gamma 1 open parenthesis to the power of theta-vector closing parenthesis = 2 to the power of 1 open parenthesis psi 1 \noindent and $ \Phi ( \vec{ r } ) $ is the Newtonian potential of the lens . 1 minus psi 2 2 closing parenthesis = gammaψ open11 + parenthesisψ22 = 2κ = to trψ theij. power(33) of vector-theta closing parenthesis cosine open square bracket 2 phi open parenthesis to the power of theta-vector closing parenthesis closing square bracket comma open With the definitions of the components of the external shear γ, parenthesis\ centerline 34 closing{The determinant parenthesis of the Jacobian $ A $ is the inverse of the magnification : } and ~ ~ ~ γ1(θ) = 21(ψ11 − ψ22) = γ(θ) cos[2φ(θ)], (34) \ [ gamma\mu 2= open\ parenthesisdet ˆ{ 1 to} theA power ˆ{ . of} theta-vector( 3 closing 1 ) parenthesis\ ] = psi 1 2 = psi 2 1 = gamma open parenthesis to the power ofand vector-theta closing parenthesis sine open square bracket 2 phi open parenthesis to the power of theta-vector closing parenthesis closing square bracket open parenthesis 35 closing parenthesis ~ ~ ~ \noindentopen parenthesisLet us where define the angle phiγ reflects2(θ) = ψ the12 direction = ψ21 = ofγ(θ the) sin[2 shearφ(θ hyphen)] (35) inducing tidal force relative to the coordinate system closing parenthesis comma the Jacobian matrix can be written ( where the angle φ reflects the direction of the shear - inducing tidal force relative to the coor- \ [ Line\ psi 1 A{ =i Row j } 1 1= minus\ partial kappa minus\ gammapartial 1 minus{\theta gamma 2{ Rowi }} 2 minusˆ{ 2 gamma{\ psi 2 1}} minus{\ kappapartial plus gamma\theta 1 { j }} dinate system ) , the Jacobian matrix can be written .. = open ( parenthesis 32 ) \ 1] minus kappa closing parenthesis Row 1 1 0 Row 2 0 1 . minus gamma Row 1 cosine 2 phi sine 2 phi Row 2 sine 2 phi minus cosine 2 phi . period Line 2 open parenthesis 36 closing parenthesis The magnification can now1 − beκ expressed− γ1 − asγ a2 function of the lo cal1 convergence 0   cos 2φ sin 2φ  \noindent The LaplacianA = of the effective= potential (1 − κ) $ \ psi− γ $ is twice the. convergence : kappa and the lo cal shear gamma−γ2 : 1 − κ + γ1 0 1 sin 2φ − cos 2φ mu = open parenthesis determinant A closing parenthesis to the power of minus 1 = open parenthesis(36) 1 minus 1 closing parenthesis\ [ \ psi kappa1 sub 1 2 +minus\ gammapsi to2 the 2 power = of 2 2 period\kappa open parenthesis= t r 37 closing\ psi parenthesis{ i j } . ( 33 ) \ ] Living ReviewsThe magnification in Relativity canopen now parenthesis be expressed 1998 as hyphen a function 12 closing of the parenthesis lo cal convergence κ and the lo cal http : slashshear slashγ www: period l ivingreviews period org \noindent With the definitions of the components of the external shear $ \gamma , $ µ = (det A)−1 = (1 − 1) − γ2. (37) \ [ \gamma 1 ( ˆ{\vec{\theta}} ) = 2 ˆ{κ21 } ( \ psi 1 1 − \ psi 2 2 ) = \gammaLiving Reviews( ˆ{\ invec Relativity{\theta( 1998}} -) 12 )\cos [ 2 \phi ( ˆ{\vec{\theta}} )] , ( 34http ) :\ ] / / www . l ivingreviews . org

\noindent and

\ [ \gamma 2 ( ˆ{\vec{\theta}} ) = \ psi 1 2 = \ psi 2 1 = \gamma ( ˆ{\vec{\theta}} ) \ sin [ 2 \phi ( ˆ{\vec{\theta}} ) ] ( 35 ) \ ]

\noindent ( where the angle $ \phi $ reflects the direction of the shear − inducing tidal force relative to the coordinate system ) , the Jacobian matrix can be written

\ [ \ begin { a l i g n e d } A = \ l e f t (\ begin { array }{ cc } 1 − \kappa − \gamma 1 & − \gamma 2 \\ − \gamma 2 & 1 − \kappa + \gamma 1 \end{ array }\ right ) = ( 1 − \kappa ) \ l e f t (\ begin { array }{ cc } 1 & 0 \\ 0 & 1 \end{ array }\ right ) − \gamma \ l e f t (\ begin { array }{ cc }\cos 2 \phi & \ sin 2 \phi \\\ sin 2 \phi & − \cos 2 \phi \end{ array }\ right ) . \\ ( 36 ) \end{ a l i g n e d }\ ]

\noindent The magnification can now be expressed as a function of the lo cal convergence $ \kappa $ and the lo cal shear $ \gamma : $

\ [ \mu = ( \det A ) ˆ{ − 1 } = ( 1 − 1{ ) {\kappa }} { 2 } − \gamma ˆ{ 2 } . ( 37 ) \ ]

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 1 7 .... Gravitational Lensing in Astronomy \noindentLocations at1 7which\ h f det i l l AGravitational = 0 have formally Lensinginfinite magnification in Astronomy period .. They are called critical curves in the lens plane period .. The corresponding lo cations in the \noindentsource planeLocations are the caustics at which period ..det For spherically $ A = symmetric 0 $ have mass formallydistributions infinite comma magnification . \quad They are calledthe critical critical1 curves 7 are curves circles period in the .. For lens a point plane lens . comma\quad theThe caustic Gravitational corresponding degenerates Lensing into lo a in cations Astronomy in the sourcepoint period planeLocations .. For are elliptical at the which caustics lenses det A or= spherically 0. have\quad formally symmetricFor infinitespherically lenses magnification plus external symmetric . shear They mass comma are called distributionscritical , thethe criticalcausticscurves can curves consistin the of arelens cusps plane circles and . folds The . period\ correspondingquad In FigureFor a lo4 point the cations caustics lens in the and, source critical the plane caustic are the degeneratescaustics into a pointcurves . for\.quad an elliptical ForFor spherically lenselliptical with symmetric a finite lenses core mass are distributions or displayed spherically period , the critical symmetric curves are lenses circles . plus For external a point shear , the3 period caustics 7lens .. Time , canthe delay caustic consist and degenerates .. quotedblleft of cusps into a Fermatand point folds . quoteright For . elliptical In squotedblright Figure lenses 4 or the spherically .. theorem caustics symmetric and lenses critical curvesThe deflection forplus an angleexternal elliptical is the shear gradient , the lens caustics of an with effective can a consist finite lensing of potentialcore cusps areand psi folds displayed open . parenthesis In Figure . 4 as the was caustics and first showncritical by open curves square for bracketan elliptical 1 3 1 lens closing with square a finite bracket core are semicolon displayed see . also open square bracket ? closing square bracket\noindent closing33 parenthesis . . 7 7 \quad Time periodTime delay Hence delay and the lensand equation “\quad Fermat can‘‘ Fermatbe ’ s rewritten ” ’ theorem s as ’’ \quad theorem Equation:The open deflection parenthesis angle 38 closing is the gradient parenthesis of an .. openeffective parenthesis lensing potential to the powerψ( as of wastheta-vector minus vector-beta closing parenthesis\noindent minusfirstThe shown nabla-vector deflection by [ 1 sub3 1 angle ] theta ; see psialso is = [?]) the 0. commaHence gradient the lens of equation an effective can be rewritten lensing as potential $ \ psi ( $ asor was Equation: open parenthesis 39 closing parenthesis~ .. nabla-vector sub theta parenleftbigg sub 2 to the power of 1 open (θ−β~) − ∇~ ψ = 0, (38) parenthesis\noindent tofirst the power shown of vector-theta by [ 1 3 minus 1 ] beta-vector ; see also closingθ $ parenthesis [ ? to] the ) power .of $ 2 Henceminus psi the parenrightbigg lens equation = 0 can be rewritten as period or \ beginThe term{ a l i gin n brackets∗} appears as well in the physical time delay function for (gravitationally ˆ{\vec{\theta lensed} images } − : \vec{\beta} ) − \vec{\nabla} {\theta }\ psi = 0 , \ tag ∗{$ ( 38 ) $} ~ 1 θ~ ~ 2 Equation: open parenthesis 40 closing parenthesis∇θ( ..2( tau−β open) − ψ parenthesis) = 0. to the power of theta-vector(39) comma vector-beta closing\end{ a parenthesis l i g n ∗} = tau sub geom plus tau sub grav = 1 plus c z sub L D sub L D sub LS D sub S parenleftbigg sub 2 to the power of 1The open term parenthesis in brackets to theappears power as of well theta-vector in the physical minus time vector-beta delay function closing for parenthesis gravitationally to the lensed power of 2 minus psi open\noindent parenthesisimagesor theta : closing parenthesis parenrightbigg period This time delay surface is a function of the image geometry open parenthesis to the power of theta-vector comma vector-beta closing\ begin parenthesis{ a l i g n ∗} comma the gravi hyphen \vec{\nabla} {\theta θ~} ~ ( { 2 }ˆ{ 1 } ( ˆ{\vec{\theta1 θ~ ~}2 } − \vec{\beta} ) ˆ{ 2 } tational potential psi commaτ( and, β) = theτgeom distances+ τgrav D= sub 1 + Lc z commaLDLDLS DDS sub(2 S( comma−β) − andψ(θ)) D. sub LS period(40) .. The first part −endash \ psi the ) = 0 . \ tag ∗{$ ( 39 ) $} ~ \endgeometrical{ a l i g nThis∗} time time delay delay tau surface sub geom is a endashfunction reflects of the imagethe extra geometry path length (θ, β~), comparedthe gravi - to tational the potential ψ, direct lineand between the distances observerD andL,D sourceS, and periodDLS. ..The The first second part part – the endash geometrical the gravitational time delay τgeom – reflects \noindenttime delaytheThe tau extra sub term grav path in endash length brackets is compared the retardation appears to the directasduewell to line gravitational betweenin the observer potentialphysical and of thetime source lens delay . hyphen The function second for gravitationally lensed images : ing mass openpart parenthesis– the gravitational known andtime confirmed delay τgrav as– Shapiro is the retardation delay in the due solar to gravitational system closing potential parenthesis of the period .... From Equationslens open - parenthesis 39 comma 40 closing parenthesis comma it follows that the gravitationally lensed images appear at\ begin { a l i ging n ∗} mass ( known and confirmed as Shapiro delay in the solar system ) . From \taulo cations(Equations thatˆ{\ correspondvec{\ (theta 39 , to 40}} extrema ) , it, follows in the\vec light that{\beta travel the gravitationally} time) comma = .. lensed\ whichtau images reflects{ geom appear} at+ lo\ cationstau { grav } = 1Fermat + { quoterightcthat} correspondz s{ principleL } toD in extrema gravitational{L{ D in} the{ hyphen lightLS }} travel lensingD time{ opticsS ,} whichperiod( { reflects2 }ˆ{ Fermat1 } ’( s principle ˆ{\vec in{\theta} } − \vecThe{\ openbetagravitational parenthesis} ) ˆ{ angular2 - lensing} − hyphen optics \ psi diameter . ( closing\theta parenthesis)). distances\ thattag ∗{ appear$ ( in Equation 40 ) open$} parenthesis 40 closing\end{ a parenthesis l i g n ∗} depend on The ( angular - diameter ) distances that appear in Equation ( 40 ) depend on the value ofthe the value Hubble of the constant Hubble open constant square [ bracket 1 66 ] 1 ; 66 therefore closing square, it is bracketpossible semicolon to determine therefore the latter comma it is possible to\noindent determine thebyThis measuring time delaythe time surface delay between is a different function images of and the using image a good geometry model for $ the ( effective ˆ{\vec{\theta}} , latter\vec by{\ measuringgravitationalbeta} the) potential time ,$ delayψ the betweenof the gravi lens different (− see [ images79 , 1 14 and , 1 using 69 ] a good tationalmodel forand potentialthe Section effective 4 gravitational . $ 1 )\ .psi potential, $ and psi the of the distances lens open parenthesis $D { L see} open,D square{ bracketS } 79, $ comma and 1 14 $ D { LS } comma. $ \ 1quad 69 closingThe• square firstGoing bracket furtherpart −− . theThis section followed heavily the elegant presentation of the basics of geometricaland Sectionlensing 4 period time in Narayan 1delay closing and parenthesis$ Bartelmann\tau { periodgeom [ 96 ] .} Many$ −− morereflects details can the be found extra there path . More length complete compared to the directbullet .. line Goingderivations between further of period the observer lensing This section properties and followed source are also heavily . provided\quad the elegantThe in all second the presentation introductory part of−− textsthe mentioned gravitational in timethe basics delaySection of lensing $ \ Chaptertau in Narayan{ 1grav , in and particular}$ Bartelmann−− inis [ 1 the 35open ] . retardation square bracket 96 due closing to gravitational square bracket period potential Many more of details the lens − can be foundMore there onperiod the Moreformulation complete of gravitational derivations of lens the theory lensing in properties terms of aretime - delay and Fermat ’ s \noindentalso providedprincipleing in mass all can the ( beintroductory known found in and Blandford texts confirmed mentioned and Narayan as in Section Shapiro [?] Chapter delay 1 comma in the solar system ) . \ h f i l l From in particular in open square bracket 1 35 closing square bracketLiving period Reviews in Relativity ( 1998 - 12 ) \noindentMore on theEquations formulation ( of 39 gravitational , 40 ) , lens it theory follows in termshttp that of : time the / / hyphen gravitationally www . l ivingreviews lensed . org images appear at lodelay cations and Fermat that quoteright correspond s principle to extrema can be found in the in Blandford light travel and Narayan time open , \quad squarewhich bracket reflects ? closing square bracketFermat ’ s principle in gravitational − lensing optics . Living Reviews in Relativity open parenthesis 1998 hyphen 12 closing parenthesis \ hspacehttp :∗{\ slashf islash l l }The www ( period angular l ivingreviews− diameter period ) org distances that appear in Equation ( 40 ) depend on \noindent the value of the Hubble constant [ 1 66 ] ; therefore , it is possible to determine the latter by measuring the time delay between different images and using a good model for the effective gravitational potential $ \ psi $ of the lens ( see [ 79 , 114 , 169 ]

\noindent and Section 4 . 1 ) .

$ \ bullet $ \quad Going further . This section followed heavily the elegant presentation of the basics of lensing in Narayan and Bartelmann [ 96 ] . Many more details can be found there . More complete derivations of the lensing properties are also provided in all the introductory texts mentioned in Section Chapter 1 , in particular in [ 1 35 ] .

More on the formulation of gravitational lens theory in terms of time − delay and Fermat ’ s principle can be found in Blandford and Narayan $ [ ? ] $

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 1 8 \noindentFigure 4 :J .. The . Wambsganss critical curves\ openh f i l parenthesis l 1 8 upper panel closing parenthesis and caustics open parenthesis lower panel closing parenthesis for an \noindente l lip ticalFigure lens period 4 : ..\ Thequad numbersThe incritical the right panelscurves identify ( upper regions panel in the ) s ourceand caustics ( lower panel ) for an ep l lan lip e that ticalJ . co Wambsganss rrespond lens . to\ 1quad commaThe 3 or numbers 5 images commain the respectively right panels period ..identify The smooth regions lin es in in the1 8 the s ource pright lan hand e thatFigure panel co are4 : rrespond calledThe fold critical caustics to curves 1 , semicolon 3 ( upperor 5 thepanel images tips ) and at , which caustics respectively in the ( lower inner panel curve . \quad ) two for anThe e l smooth lip tical lin es in the rightfold caustics handlens connect panel . 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Discussions of the concept of quotedblleft distance quotedblright in relation andto Schneider cosmology slash [ 1 curved 3 1 space] . \ canquad be foundDiscussions in Section of 3 period the 5concept of open square of ‘‘ bracket distance 1 35 closing ’’ in square relation bracket or Sectionto cosmology1 9 / curved space can be found in Section 3 Gravitational . 5 of [ 1Lensing 35 ] in or Astronomy Section 14.4of[166]1 4 period 4 ofand open Schneider square . bracket[ 1 3 1 ] 1 . 66 Discussions closing square of the bracket concept period of “ distance ” in relation to cosmology Living Reviews/ curved in Relativity space can open be found parenthesis in Section 1998 3 .hyphen 5 of [ 1 12 35 closing ] or Section parenthesis 1 4 . 4 of [ 1 66 ] . \ hspacehttp :∗{\ slashf islash l l } Living www period Reviews l ivingreviews in Relativity period org ( 1998Living− Reviews12 ) in Relativity ( 1998 - 12 ) http : / / www . l ivingreviews . org \ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 20 \noindent4 .. LensingJ .. . Phenomena Wambsganss \ h f i l l 20 In this section comma we describe different groups of gravitational lens observations period \noindentThe subdivision4 \quad is pragmaticLensing rather\quad thanPhenomena entirely logical period .. It is done partly by lensed objectJ . Wambsganss comma or by lensing object comma or by lensing strength period The ordering roughly 20 \noindentreflects the4In chronological this Lensing section appearance , wePhenomena of describe different sub different hyphen disciplines groups to oflensing gravitational period The lens observations . Thefollowing subdivision sectionsIn this section are is on pragmatic ,: we describe rather different than groups entirely of gravitational logical lens observations . \quad It . The is subdivision done partly by lensedbullet .. object Multiplyis pragmatic , hyphen or rather by imaged lensing than quasars entirely object period logical , or . by It is lensing done partly strength by lensed . object The ordering , or by lensing roughly reflectsbullet .. Quasarobject the chronological microlensing , or by lensing period strength appearance . The ordering of different roughly reflects sub − thedisciplines chronological appearanceto lensing of . The followingbullet Einsteindifferent sections rings sub period - are disciplines on : to lensing . The following sections are on : bullet .. Giant luminous arcs and arclets period• Multiply - imaged quasars . \ centerlinebullet Weak{ lensing$ \ bullet slash Statistical$ \quad lensingMultiply period• Quasar− imaged microlensing quasars . . } bullet .. Cosmological aspects of open parenthesis• strongEinstein closing rings parenthesis . lensing period \ centerlinebullet .. Galactic{ $ \ microlensingbullet $ period\quad •QuasarGiant luminous microlensing arcs and . arclets} . Comprehensive reviews could be written• onWeak each lensing separate / Statistical subject period lensing .. Hence. \ centerlinethe treatment{ $ here\ bullet can be only$ very Einstein• cursoryCosmological ringsperiod aspects . } of ( strong ) lensing . 4 period 1 .. Multiply hyphen imaged quasars• Galactic microlensing . \ centerlineIn 1 979 comma{ Comprehensive$ gravitational\ bullet $ reviews lensing\quad couldbecameGiant be an written luminous observational on each arcs science separate and when subject arclets the dou . . hyphen Hence} the treatment ble quasarhere Q 957 can plus be only56 1 verywas discovered cursory . period .. This was the first example of a lensed \ centerlineobject open4{ square .$ 1\ bullet bracket Multiply 1$ 60 closingWeak - imaged squarelensing bracket quasars / Statistical period .. The discovery lensing itself . } happened rather by accident semicolon the discovererIn 1 979 , gravitational lensing became an observational science when the dou - ble quasar Q \ centerlineDennis Walsh957{ + describes$ 561\ bullet was in discovered a$ nice\ accountquad . ThisCosmological how was this the branch first example aspectsof astrophysics of ofa lensed ( came strong ) lensing . } into beingobject open square [ 1 60 ] bracket . The 1 discovery 59 closing itself square happened bracket rather period by accident ; the discoverer Dennis Walsh \ centerlineIt was notdescribes entirely{ $ \ bullet clear in anice at the$ account beginning\quad howGalactic comma this branch though microlensing of astrophysics comma whether came . } the two quasar images reallyinto were being an [ illusion1 59 ] . provided by curved space hyphen time endash or rather physical Comprehensivetwins period ..It Butreviews was intensive not entirely could observations clear be atwritten the soon beginning confirmed on each , the though separate almost , whether identical subject the spectra two . quasar period\quad imagesHence really theThe treatment interveningwere an quotedblleft here illusion can provided lensing be only quotedblrightby curved very cursoryspace galaxy - time was . – found or rather comma physical and the twins quotedblleft . But supporting intensive quotedblright cluster was observations soon confirmed the almost identical spectra . The intervening “ lensing ” galaxy was \noindentidentified asfound4 well . 1 , period and\quad the .... “Multiply Latersupporting very− similar ” clusterimaged lightcurves was quasars of the two images open parenthesis modulo offsets in timeidentified and magnitude as well . closing parenthesis Later confirmed very similar this system lightcurves beyond of any the doubt two images as a ( modulo \noindentbona fide gravitationaloffsetsIn 1in 979 time , lens and gravitational period magnitude ) confirmed lensing this became system beyondan observational any doubt as a sciencebona fide gravita- when the dou − bleBy nowquasarQ abouttional two lens $ dozen 957 . multiply + 56 hyphen 1 imaged $ was quasar discovered systems have . \ beenquad foundThis comma was the first example of a lensed plus another ten good candidatesBy now open about parenthesis two dozen updated multiply tables - imaged of multiply quasar hyphen systems imaged have been quasars found , \noindentand gravitationalplusobject another lens [ candidates 1 ten 60 good ] . candidates are\quad providedThe ( updated comma discovery e tables period of g itself multiply period comma happened - imaged by the quasars rather CASTLE by group accident open square ; the bracket discoverer 47Dennis closing square Walshand bracket gravitational describes closing lens inparenthesis candidates a nice period account are provided how , e this . g . , branch by the CASTLE of astrophysics group [ 47 ] ) came. This is This is notnot really really an an exceedingly exceedingly large large number number comma , considering considering a 20 a year 20 year effort effort to find lensed quasars . The \noindent reasonsinto being for this [ “ modest 1 59 ] ” success . rate are : to find lensed quasars period The reasons for this quotedblleft modest quotedblright4 success rate are : 1 period .. Quasars1 are. rareQuasars and arenot rare easy and to find not open easy parenthesisto find ( by by now now roughly roughly 10 1are 0 to known the power ) . of 4 are known closing parenthesisIt was not period entirely2 . The clear fraction at the of quasars beginning that are , lensed though is small , whether ( less than the one twopercent quasar ) . images2 period really ..Living The fractionReviews wereof an in quasars illusionRelativity that are( provided1998 lensed - 12 is small by) curved open parenthesis space − lesstime than−− one percentor rather closing physical parenthesis period twinsLiving . Reviews\httpquad in:But Relativity / / intensive www open. l parenthesis ivingreviews observations 1998 . hyphen soonorg 12 confirmed closing parenthesis the almost identical spectra . Thehttp intervening : slash slash www ‘‘ period lensing l ivingreviews ’’ galaxy period was org found , and the ‘‘ supporting ’’ cluster was \noindent identified as well . \ h f i l l Later very similar lightcurves of the two images ( modulo

\noindent offsets in time and magnitude ) confirmed this system beyond any doubt as a bona fide gravitational lens .

\ hspace ∗{\ f i l l }By now about two dozen multiply − imaged quasar systems have been found ,

\noindent plus another ten good candidates ( updated tables of multiply − imaged quasars

\noindent and gravitational lens candidates are provided , e . g . , by the CASTLE group [ 47 ] ) . This is not really an exceedingly large number , considering a 20 year effort to find lensed quasars . The reasons for this ‘‘ modest ’’ success rate are :

\ centerline {1 . \quad Quasars are rare and not easy to find ( by now roughly $ 1 0 ˆ{ 4 }$ are known ) . }

\ centerline {2 . \quad The fraction of quasars that are lensed is small ( less than one percent ) . }

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 2 1 .... Gravitational Lensing in Astronomy \noindent3 period ..2 It 1 is\ noth f trivial i l l Gravitational at all to identify the Lensing lensed open in Astronomy parenthesis i period e period multiply hyphen imaged closing parenthesis quasars 3 .among\quad theIt known isones not period trivial at all to identify the lensed ( i . e . multiply − imaged ) quasars amongGravitationally the2 known 1 lensed ones quasars . come in a variety of classes : .. double commaGravitational triple Lensing in Astronomy and quadruple3 systems . It is semicolon not trivial .. at symmetric all to identify and asymmetricthe lensed ( image i . e . configurations multiply - imaged are ) quasars among Gravitationallyknown periodthe known lensed ones . quasars come in a variety of classes : \quad double , triple andFor quadruple an overviewGravitationally of systems the geometry ; lensed\quad of multiply quasarssymmetric comehyphen in andimaged a variety asymmetric quasar of classes systems : image comma double configurations ,see triple the and quadruple are knowncollection . systems of images ; found symmetric at open and square asymmetric bracket image1 0 1 closing configurations square bracket are known period . A recurring problemFor an overviewconnected of with the double geometry quasars of multiply is the question - imaged whether quasar systems , see the collection of Forthey an are overview twoimages images found of of the a at single [ 1geometry 0 source 1 ] . or of rather multiply a physical− associationimaged quasar of two systems , see the collectionobjects open ofparenthesisA recurring images with problem found three at connected or more [ 1 0images with 1 ] double it . is more quasars and more is the likely question that it whether is lensed they are two system closingimages parenthesis of a single period source Few or systems rather a are physical as well association established of as two thedouble objects quasar ( with Q three 957plus or more 56 1 semicolon A recurringbut manyimages are problem considered it is more connected quotedblleft and morewithsafe likely quotedblright that double it is lensed quasars lenses as is well the period question .. Criteria whether for quotedblleft fair quotedblright commathey .. are quotedblleftsystem two images ) good. Few quotedblright of systems a single are as comma well source established or or rather as the double a physical quasar Q association 957 + 561; but of many two are objectsquotedblleft (considered with excellent three “ quotedblright safe or ” lenses more lensed as images well quasar . it Criteria candidates is more for “ comprise and fair ” more , the “ good likelyfollowing ” , or that : “ excellent it is ” lensed lensed bullet .. Therequasar are candidates two or more comprise point hyphen the following like images : of very similar optical color period \noindentbullet Redshiftssystem open )• parenthesis .There Few systemsare or two distances or aremore closing as point well parenthesis - like established images of of both very quasar similar as the images optical double are color identical quasar . or veryQ $ similar 957 period + 56bullet 1 .. 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The differently answer period de - pends .. Hence on the comma amount it is of not certainty easy one requires $ to\ bullet say how. many$ In a If recent gravitationally the compilation quasar lensed is , Keeton intrinsically quasar and systems Kochanek exist variable [ period 70 ] put .. together The , the answer fluxes29 quasars de hyphen measured as lenses or from lens the two ( or more ) images follow a very similar00 light curve , except for certain lags pends on thecandidates amount in of three certainty probability one requires “ classes period In a recent compilation comma Keeton and Kochanek open square bracket 70 closing square bracket. put together 29 quasars as lenses or lens candidates in three \ centerlineprobability{−− quotedblleftGravitationallythe time Caselensed delays1 quotedblright quasar−− and systems Case an 2 are overallperiod studied offsetindividually in in brightness great detail to ( get cf a . better Figure 8 ) . } Gravitationallyunderstanding lensed quasar of both systems lens and are source studied ( individually so that , e . in g . great , a measure detail - Forto most get a better ofment the understanding of theknown time multiple delay of both can be lens quasar used and to source determine systems open parenthesisthe , only Hubble some soconstant that of comma )the . Asabove e period an ensemble cri g period− , the comma a measure hyphenteria arelens fully systems confirmed are also analysed . And statistically there are in orderalso to good get in reasons - formation not about to the require population perfect of agreementment of thelenses with time ( delay and this quasars can list beused ) in. \ the toquad determine universeFor , example thetheir Hubble , constant the lensing closing parenthesis galaxy could period be .. As superposed toan one ensemble quasardistribution comma image the in lens distance and systems make ( i . arethe e . also cosmic quasar analysed time appear ) statistically and mass extended , inand order hence ; to color about get in the hyphen/ spectra cos - could beformation affectedmological about by the dust model population absorption ( more of about lenses in that open the in parenthesis Section lensing 4 and. 6 galaxy ) quasars . Here and closing we appear will parenthesis have not a close in identical the look universe on one ; comma their thedistribution lensparticularly could in distance be well too open investigated faint parenthesis to system be i period detectable . e period ( cosmic or even time aclosing real parenthesis dark lens and mass $ ? comma ) and ; $ hence\quad the aboutquasar the cos could hyphen be variable on time scales shorterLiving than Reviews the time in Relativity delay ;( 1998 microlens - 12 ) − ingmological can affect model open the parenthesis lightcurves more about of the that images in Sectionhttp differently 4 : period / / 6 www closing . \ lquad parenthesis ivingreviewsHence period , . it .. org Hereis not we will easy have a closeto say how many gravitationally lensed quasar systems exist . \quad The answer de − pendslook on on one the particularly amount well of investigated certainty system one requires period . In a recent compilation , Keeton andLiving Kochanek Reviews in[ Relativity70 ] put open together parenthesis 29 1998 quasars hyphen as 12 lenses closing parenthesis or lens candidates in three probabilityhttp : slash slash ‘‘ www $\ l period e f t . c l livingreviews a s s e s \ begin period{ a l i orgg n e d } &’’ \\ &. \end{ a l i g n e d }\ right . $

Gravitationally lensed quasar systems are studied individually in great detail to get a better understanding of both lens and source ( so that , e . g . , a measure −

\noindent ment of the time delay can be used to determine the Hubble constant ) . \quad As an ensemble , the lens systems are also analysed statistically in order to get in − formation about the population of lenses ( and quasars ) in the universe , their

\noindent distribution in distance ( i . e . cosmic time ) and mass , and hence about the cos −

\noindent mological model ( more about that in Section 4 . 6 ) . \quad Here we will have a close look on one particularly well investigated system .

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 22 \noindentFigure 5 :J A recent. Wambsganss example for\ theh f i identification l l 22 of the lensing galaxy in a double quasar system open square bracket 35 closing square bracket : .. The left panel shows on infrared open parenthesis J hyphen\noindent band closingFigure parenthesis 5 : A recent o bs ervation example of the for the identification of the lensing galaxy in a double quasartwo images systemJ of . Wambsganss double [ 35 quasar ] : HE\quad 1 1 4The hyphen left 1 825 panel open parenthesis shows on z infrared sub Q = 2 period ( J − 3 1band 6 comma ) o Capital bs22 ervation Delta theta of the = 3 period 2 arcsFigure ec 5 closing : A recent parenthesis example period for the .. identification The of the lensing galaxy in a double quasar system \noindentright panel[two o35 b ] tain images : ed The with of left s double panelome new shows quasar deconvolution on infrared HE t$( echnique J 1 - band 1 ) nicely o 4 bs ervation reveals− 1 the of the 825 ( z { Q } = 2 .lensing 3 1galaxytwo 6 images open , parenthesis of\ doubleDelta quasar at z\ subtheta HE G =114 1= period− 1825( 3 66zQ closing .= 2.316 2$ parenthesis, ∆θ arcsec).= 3.2 betweenarcs ec) the\quad . quasar TheThe images right period .. open parenthesisright panel Creditspanel o o : b b.. tain European ed ed with with s ome s new ome deconvolution new deconvolution t echnique nicely t echnique reveals the nicely lensing revealsgalaxy ( at the lensingSouthern galaxy ObservatoryzG = 1.66) (between at open $ square zthe{ quasar bracketG } images= 4 6 closing . 1 ( . square Credits 66 bracket : ) European $ period between closing theparenthesis quasar images . \quad ( Credits : \quad European 4 period 1Southern period 1 Observatory.. The first lens [ 4 6: .. ] . Double ) quasar Q 957 plus 561 \noindentThe quasar4Southern Q . 1 957 . 1 plus Observatory The561 was first originally lens [ : 4 found 6Double ] in . a )radio quasar survey Q 957 comma + 561 subsequently an optical counterpartThe quasar was Q 957+561 identified was as well originally period found .. After in a the radio confirmation survey , subsequently of its lens an optical counterpart \noindentnature openwas4 square . identified 1 . bracket 1 \ asquad well 1 60The . comma After first 1 the 59 lens confirmation closing : square\quad of bracket itsDouble lens comma nature quasar [this 1 60 Q quasar , 1$ 59 957 attracted ] , this + quasar quite 561 some $ attention period .. Q 957attracted plus 56 quite 1 has some attention . Q 957 + 561 has been looked at in all available wavebands , \noindentbeen lookedfromThe at inX quasar - all rays available to Q radio $ wavebands frequencies957 + comma . More 561 .. than $ from was 1 X 0 scientifichyphen originally rays papers to radio havefound appearedfrequencies in a on radio period Q 957+561( survey , subsequently an opticalMore than counterpartcf 1 .0 [scientific 1 8 ] ) , papers many was more have identified appearedthan on any on as Q other well 957 gravitationalplus . \ 56quad 1 openAfter lens parenthesis system the . confirmationcf Here period we open will summarize square of bracketits lens 1 8 closing squarenature bracket [what 1 closing 60 is , known parenthesis 1 59 about ] , comma this this system manyquasar from attracted optical and radio quite observations some attention . . \quad Q $ 957 + 56more 1 than $ has onIn any the other optical gravitational light , Q 957 lens + system 561 appears period as Here two we point will images summarize of roughly what 1 7 mag ( R band beenis known looked about) separated at this in system by all 6 . from available 1 arcseconds optical and wavebands ( see radio Figure observations 6 , )\ .quad The periodfrom spectra X − of therays two to quasars radio reveal frequencies both . MoreIn the than opticalredshifts 1 light0 scientific to comma be zQ Q= 957 1. papers41 plus. Between 56 have1 appears the appeared two as images two point onQ, not images $ 957 of roughly + 1 756 1 ( $ cf . [ 1 8 ] ) , many moremag openthanquite parenthesis on anyon the other R connecting band gravitational closing line parenthesis , the lensing lens separated galaxy system by ( with 6 period . redshift Here 1 arcseconds wezG will= 0.36) open summarize appears parenthesis as awhat fuzzy see Figure 6 closing parenthesisis known periodpatch about .. close The this to spectra the system component of the from B optical . This galaxy and is radio part of observations a cluster of galaxies . at about the same redshift . This is the reason for the relatively large two quasars reveal both redshifts to be z sub Q = 1 period 41 period Between the11− two12 images comma not Inquite the on optical theseparation connecting light for a line , galaxy Q comma $ - 957 type the lens lensing + ( typical 56 galaxy galaxies 1 open $ appearswith parenthesis masses as of with two 10 redshift pointM z images sub G = of 0 period roughly 36 closing 1 7 parenthesismag ( R appears bandproduce ) splittingseparated angles by of 6only . about 1 arcseconds one arcsecond ( , see see Equation Figure ( 6 1 0 ) ) ) . .\quad In thisThe lens spectrasystem of the twoas a quasars fuzzy, patch the reveal mass close in to the both the galaxy component redshifts cluster B helps period to to be This increase $ galaxy z the{ is deflectionQ part} of= a angles cluster 1 to of. this 41 large separation . $ Between . the two images , not galaxies at aboutA recent the same image redshift of Q 957 period + 561 .. taken This iswith the the reason MERLIN for the radio relatively telescope large is shown in Figure 7 . \noindentseparationThequite for apositions galaxy on the hyphen of the connecting two type point lens - open like lineobjects parenthesis , the in this lensing typical radio galaxies ob galaxy - servation with ( masses withcoincide of redshift 1 with 0 to the the optical power $ z of{ 1G 1 minus} = 1 20 M sub . odot 36sources )$ . There appears is no radio emission detected at the position of the galaxy center , i . e . the asproduce a fuzzy splittinglensing patch anglesgalaxy close of isonly radio to about - the quiet one component . arcsecond But B comma . This see Equation galaxy open is part parenthesis of a 1 cluster 0 closing parenthesis of closing parenthesisgalaxies periodLiving at about .. Reviews In the in same Relativity redshift( 1998 -. 12\quad) This is the reason for the relatively large this lens systemhttp : comma / / thewww mass . l in ivingreviews the galaxy cluster . org helps to increase the deflection \noindentangles to thisseparation large separation forperiod a galaxy − type lens ( typical galaxies with masses of $ 1 0 ˆ{ 1 1 A− recent1 image 2 of} QM 957{\ plusodot 56 1 taken}$ with the MERLIN radio telescope is shown in Figure 7 period The positions of the two point hyphen like objects in this radio ob hyphen \noindentservation coincideproduce with splitting the optical sourcesangles period of only There about is no radio one emission arcsecond detected , see Equation ( 1 0 ) ) . \quad In thisat the lens position system of the , galaxy the center mass comma in the i period galaxy e period cluster the lensing helps galaxy to increase is radio hyphen the quiet deflection period .. But anglesLiving Reviews to this in Relativity large separation open parenthesis . 1998 hyphen 12 closing parenthesis http : slash slash www period l ivingreviews period org A recent image of Q $ 957 + 56 1 $ taken with the MERLIN radio telescope is shown in Figure 7 . The positions of the two point − like objects in this radio ob − servation coincide with the optical sources . There is no radio emission detected at the position of the galaxy center , i . e . the lensing galaxy is radio − q u i e t . \quad But

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 23 .... Gravitational Lensing in Astronomy \noindentFigure 6 :23 .. In\ thish f i lfals l Gravitational e color Hubble Space Lensing Telescope in image Astronomy of the double quasar Q 957 plus 561 A comma B period .. The two images A open parenthesis bottom closing parenthesis and B open parenthesis top\noindent closing parenthesisFigure 6are : s\ eparatedquad In by this6 period fals 1 e color Hubble Space Telescope image of the double quasar Qarcs $957 econds23 period + 561$ .. Image A,B. B is a bout\ 1quad .. arcsThe econd two away images from the A Gravitational core( bottom of the galaxy ) Lensing and comma B in ( Astronomy top ) are s eparated by 6 . 1 arcsand hence econdsFigure s een . ..\6quad : quotedblleftInImage this fals through B e color is a quotedblrightHubble bout Space 1 \quad Telescope the haloarcs of image econdthe ofgalaxy the away double period from quasar .. open the Q parenthesiscore957 + of 561 the Credits galaxy : .. E , periodand henceE periodA s , .. Been Falco .\quad e The t al two period‘‘ images through A ( ’’bottom the ) halo and B of ( top the ) are galaxy s eparated . \quad by 6 .( 1 Credits arcs econds : \quad E.E. \quad Falco e t a l . (open CASTLE parenthesis. colla Image CASTLE boration B is a colla bout [ boration 41 7 arcs ] ) open econd and square NASA awaybracket from . ) the 4 core 7 closing of the square galaxy bracket , and hence closing s een parenthesis “ and NASA period closingthrough parenthesis ” the halo of the galaxy . ( Credits : E . E . Falco e t al . ( CASTLE colla \ hspaceLiving∗{\ Reviewsborationf i l l } inLiving Relativity [ 4 7 ] ) Reviews and open NASA parenthesis in . ) Relativity 1998 hyphen ( 199812 closing− 12 parenthesis ) http : slash slash www period l ivingreviews period org Living Reviews in Relativity ( 1998 - 12 ) \ hspace ∗{\ f i l l } http : / / www . l ivingreviewshttp . : org / / www . l ivingreviews . org J period Wambsganss .... 24 \noindentthis also tellsJ . us Wambsganss that a possible\ thirdh f i l limage24 of the quasar must be very faint comma below the detection limit of all the radio observations to the power of 6 period In Figure 7 comma a quotedblleft j et quotedblright\noindent this can also tells us that a possible third image of the quasar must be very faint , be seen emergingJ . Wambsganss from image A open parenthesis at the top closing parenthesis period It is not unusual24 for radio quasars \noindent thisbelow also the tells us detection that a possible limit third of image all of the the quasarradio must $ observations be very faint , ˆ{ 6 } . $ In Figure 7 , a ‘‘ j et ’’ can to have such a quotedblleft jet quotedblright feature period This6 is most likely matter that is ej ected from the central enginebelow of the the detection quasar with limit very of all high the speed radio along observations the polar. axisIn Figure of the 7 , a “ j et ” can \noindentcentral blackbebe seen hole seen emerging period emerging .. Thefrom reason image from thatA image ( at this the A jet top ( is at) seen . It the only is not top around unusual ) .one forIt image radio is notis quasars unusual to have for such radio a quasars tothat have it lies such“ outside jet ” a feature the‘‘ jetcaustic . This ’’ region is feature most inlikely the . source matterThis plane is that mostcomma is ej ected likely which from marks matter the the central part that engine is of ej the ected quasar from the centralthat is multiply enginewith very imaged of high the period speed quasar ..along Only the with the polar compact very axis high of core the of centralspeed the quasar black along lies hole theinside . polarThe the reason axis that of this the jet is centralcaustic and blackseen is doubly only hole around imaged . \ onequad period imageThe is reason that it lies that outside this the jet caustic is region seen in only the source around plane one , which image is thatFigure it 7 : liesmarks Radio outside the image part of that Q the 957 is caustic plusmultiply 561 from imaged region MERLIN . in Only the te lesthe source cope compact period plane core .. Itof , clearly the which quasar shows marks lies inside the the part thatthe two is po multiplycaustic int like and images imaged is doubly of the . imaged quasar\quad . coreOnly and the the jet compact emanating core only of from the the quasar lies inside the causticNorthern and partFigure isperiod 7 doubly : Radio .. open image imaged parenthesis of Q . 957 Credits + 561 :from N period MERLIN Jackson te les comma cope .Jodrell It clearly Bank period shows closing the two parenthesis As statedpo above int likecomma images a virtual of the .. quasar quotedblleft core and proof the quotedblright jet emanating .. only of a from gravitational the lens system is a mea hyphen \noindentsurement ofNorthernFigure the .. quotedblleft part7 : . Radio ( Creditstime image delay : N of quotedblright . Jackson Q $ 957 , Jodrell .. Capital + Bank 561 Delta . ) $ t comma from MERLIN the relative te shift les of cope the light . \ curvesquad ofIt clearly shows thethe two po intAs stated like above images , a virtual of the “ proofquasar ” core of a gravitational and the jetlens system emanating is a mea only - surement from of the two or morethe images “ time comma delay I sub ” A ∆ opent, the parenthesis relative shift t closing of the parenthesis light curves and of the I sub B open parenthesis t closing parenthesis comma\noindent so thattwoNorthern I sub or more B open partimages parenthesis .,I\Aquad(t) and t closing(IB Credits(t), parenthesisso that :I NB(t =) . = const Jacksonconst times× IA , I(t sub Jodrell+ ∆ At) open. Bank parenthesis . ) t plusAny Capital Delta t closing parenthesisintrinsic period fluctuation .... Any of the quasar shows up in both images , in general with an Asintrinsic stated fluctuation above , of a the virtual quasar6There shows\existsquad a up theorem‘ in ‘ both proof that images gravitational ’ ’ comma\quad lenses inof general should a gravitational produce with an an odd number lens of images system is a mea − surement6 sub There( of e exists . g the . , [ a 90\ theoremquad ] ) . ‘‘ that time gravitational delay ’’ lenses\quad should$ produce\Delta an oddt number , $ of the images relative shift of the light curves of the open parenthesisLiving Reviews e period in g period Relativity comma( 1998 open - square 12 ) bracket 90 closing square bracket closing parenthesis period \noindentLiving Reviewshttptwo inor: Relativity / more / www images open. l parenthesis ivingreviews $ , I 1998{ . hyphenA org} ( 12 closing t parenthesis ) $ and $ I { B } ( t ) , $ sohttp that : slash $ I slash{ wwwB } period( l t ivingreviews ) = period const org \times I { A } ( t + \Delta t ) . $ \ h f i l l Any

\noindent intrinsic fluctuation of the quasar shows up in both images , in general with an

\ hspace ∗{\ f i l l } $ 6 { There }$ exists a theorem that gravitational lenses should produce an odd number of images

\noindent (e.g.,[90]).

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 25 .... Gravitational Lensing in Astronomy \noindentoverall offset25 in\ apparenth f i l l Gravitational magnitude and an Lensing offset in time in Astronomyperiod Q 957 plus 56 1 is the first lens system in which the t ime delay was firmly est ab hyphen \noindentlished periodoverall .. Afteroffset a decadein long apparent attempt and magnitude various groups and claiming an offset either of in two time . favorable values25 open square bracket 1 6 comma ? comma 1 28 comma Gravitational 1 58 closingLensing square bracket in Astronomy comma Kundi acute-c Qet al $ period 957 openoverall + square 56offset bracket in 1 apparent $ 79 is closing the magnitude square first and bracket lens an offset confirmed system in time in the . which shorter theof t ime delay was firmly est ab − l ithe s h e two d . open\quadQ parenthesis 957+561After is cf a the period decade first lensFigure long system 8 semicolon attempt in which see the and also t ime various Oscoz delay et was al groups firmly period est openclaiming ab - square lishedeither. bracket After 1 of a 3 closing two square bracketfavorablevalues and Schilddecade ampersand long attempt $[ Thomson 1and various 6 open , squaregroups ? bracketclaiming , 1 either 29 closing 28 of two square , favorable 1 bracket 58 values closing ] [16, parenthesis ,$?, 128, Kundi158] :, $ \acute{c} $ etEquation: al . [Kundi 79 open ] parenthesisc ´ confirmedet al . [ 79 4 ] 1 the confirmed closing shorter parenthesis the shorter of .. of Capital Delta t sub Q 0 9 5 7 plus 5 6 1 = open parenthesis 4 1 7 plusminux 3 closingthe two parenthesis ( cf . Figure days 8 ; see also Oscoz et al . [ 1 3 ] and Schild & Thomson [ 1 29 ] ) : \noindentWith a modelthe of two the lens ( cf system . Figure comma 8 the ; time see delay also can Oscoz be used et to al determine . [ 1 3 ] and Schild $ \&$ Thomson [ 1 29 ] ) : the Hubble constant to the power of 7 period .. In Q 957 plus 56 1 comma the lensing action is done by an indi hyphen ∆t = (417 ± 3)days (41) \ beginvidual{ a galaxy l i g n ∗} plus an associated galaxy clusterQ open0957+561 parenthesis to which the galaxy belongs closing parenthesis period \DeltaThis providest some{ Q additional 0 9 problems 5 7 comma + a 5degeneracy 6 1 in} the= determination ( 4 of 1 7 \pm 3 ) days \ tag ∗{$ ( 4 1 ) $} With a model of the lens system , the time delay can be used to determine the Hubble constant open square7 bracket ? closing square bracket period The appearance of the double quasar system \end{ a l i g nthe∗} Hubble constant . In Q 957 + 561, the lensing action is done by an indi - vidual galaxy including plus an associated galaxy cluster ( to which the galaxy belongs ) . This provides some additional the time delay could be identical for different partitions of the matter between \ hspace ∗{\problemsf i l l }With , a degeneracy a model inof the the determination lens system of the , Hubble the time constant delay [?]. The can appearance be used of to the determine galaxy anddouble cluster quasar comma system but the including derived thevalue time of the delay Hubble could constant be identical could for be different quite partitions of the different period However comma this degeneracy can be .. quotedblleft broken quotedblright comma once the fo cussing \noindent matterthe Hubble between $ constant ˆ{ 7 } . $ \quad InQ $ 957 + 56 1 , $ the lensing action is done byan indi − contri hyphengalaxy and cluster , but the derived value of the Hubble constant could be quite different . vidualbution of galaxy the galaxy plus cluster an canassociated be determined galaxy independently cluster period ( to And which the latter the galaxyhas belongs ) . This providesHowever some , this additional degeneracy can problems be “ broken , a ” degeneracy , once the fo cussing in the contri determination - bution of the galaxy of been attemptedcluster recently can be open determined square bracket independently 49 closing . And square the bracket latter hasperiod been .. The attempted resulting recently value for [ 49 the ] Hubble. constant openthe square Hubble bracket constant 79 closing $ square [ bracket? ] . $ The appearance of the double quasar system including the timeThe delay resulting could value be for identical the Hubble for constant different [ 79 ] obtained partitions by employing of the a detailed matter lens between model obtained byby employing Grogin and a Narayandetailed lens [?] and model the by measured Grogin veloand Narayancity dispersion open square of the bracketlensing galaxy ? closing [ 48 square ] is bracket and the measured velo city dispersion of the lensing galaxy open square bracket 48 closing square bracket is \noindentEquation:galaxy open parenthesis and cluster 42 closing , parenthesis but the derived .. H sub 0 value = open ofparenthesis the Hubble 67 plusminux constant 1 3 closing could parenthesis be quite km different . However , this degeneracy can be \quad−1 ‘‘− broken1 ’’ , once the fo cussing contri − sec to the power of minus 1 Mpc to the powerH of0 = minus (67 ± 113)km comma sec Mpc , (42) butionwhere the of uncertainty the galaxy comprises cluster the 95 can percent be determined confidence level independently period . And the latter has beenbullet attempted .. Goingwhere further the recently uncertainty period .. [ Incomprises 49 addition ] . the\ toquad 95%the aboveThe confidence resulting mentioned level . CASTLES value for list of the Hubble constant [ 79 ] obtainedmultiple quasars by employing with detailed• a informationdetailedGoing further onlens observations . modelIn addition by comma Grogin to models the aboveand and Narayan mentioned CASTLES $ [ ? list ] of $ and thereferences measured openmultiple squarevelo quasars city bracket dispersion with 47 closing detailed square informationof thebracket lensing commaon observations a galaxy similar , compilation models[ 48 ] and is of references lens candidates [ 47 ] , subdivideda into quotedblleftsimilar presently compilation accepted of quotedblright lens candidates and subdivided quotedblleft into additional “ presently proposed accepted quotedblright ” and “ additional multiply imaged objects \ begincan be{ a found l i gproposed n ∗} at open ” multiply square bracket imaged 1 objects 0 1 closing can be square found bracket at [ 1 0 period 1 ] . .. More More time time delays delays are are becoming becoming available for otherH { 0 } available= ( for 67 other lens\pm systems1 ( 3 e . g) . , km [ 22\ ,sec 1 27ˆ{ ] − ) .1 Blandford} Mpc &ˆ{ Kundi − 1c ´ }[24], \ tag ∗{$ ( 42lens ) systems $} provide open a parenthesis nice review e in period which g they period explore comma the .. potential open square to get bracket a good 22 determination comma .. 1 27of closing square bracket closing\end{ a parenthesis l i g n ∗} the period extragalactic .. Blandford distance ampersand scale by combiningKundi acute-c measured open square t ime delays bracket with 24 good closing models square ; bracket see .. provide a nice also [ 1 26 ] and [ 1 69 ] for very recent summaries of the current situation on time delays \noindentreview in whichandwhere determination they the explore uncertainty ofthe the potential Hubble comprises to constant get a good the determination $ 95 \% of $ confidence level . the extragalactic distance scale by combining measuredfrom t lensing ime delays . with \ hspacegood models∗{\4f i . semicolon l l 2} $ \ Quasarbullet .. see also microlensing$ .. open\quad squareGoing bracket further 1 26 closing . \quad squareIn bracket addition and .. open to squarethe above bracket mentioned 1 69 closing CASTLES list of square bracketLight for very bundles recent from summaries “ lensed of the ” quasars are split by intervening galaxies . With multiplecurrent situation quasarstypical onseparations with time delays detailed of order and determination one information arcsecond of between the on Hubble observations center constant of galaxy , and models quasar and references [ 47 ] , a similar compilation of lens candidates subdivided into from lensing period7This can be seen very simply : Imagine a lens situation like the one displayed in Figure 2 . If now all length ‘‘4 period presently 2scales .. Quasar are accepted reduced microlensing by a’’ factor and of two ‘‘ and additional at the same time proposed all masses are ’’ reduced multiply by a factor imaged of two objects , then for an canLight be bundles foundobserver from at the .... [ angular 1 quotedblleft 0 1configuration ] . lensed\quad in the quotedblrightMore sky would time appear .... delays quasars exactly are are identical split becoming . by But intervening the available total length galaxies of the for period light other .... With lenstypical systems separationspath is ( reduced e of . order by g a . factorone , arcsecond\ ofquad two .[ Now between 22 , since , \quad center the time of1 delay galaxy 27 between ] )and . the quasar\quad two pathsBlandford is the same fraction $ \& of $ the Kundi $ \acute{c} [7 sub 24 This ]total can $ lengths be\quad seen in veryprovide either simply scenario a : ,Imagine nice a measurement a lens of situation this fractional like thelength onedisplayed allows to determine in Figure the 2 total period length , reviewIf now all inand length which hence scales the they Hubble are explore reduced constant by , the the a factor constant potential of of two proportionality and to at get the between same a good time distancedetermination all masses and redshift are . of reduced by a factor of two comma then for an observer the angularLiving configuration Reviews in Relativity in the sky( would1998 - 12 ) theappear extragalactic exactly identical distance period But scale the total by length combining of thehttp light measured : path / / is www reducedt ime . l delaysby ivingreviews a factor with of two . period org goodNow modelscomma since ; \quad the timesee delay a l sbetween o \quad the[ two 1 paths 26 ] is and the same\quad fraction[ 1 of69 the ] total for lengths very recent in summaries of the currenteither scenario situation comma on a measurement time delays of this and fractional determination length allows of to the determine Hubble the constanttotal length comma and hence the Hubble constant comma the constant of proportionality between distance and redshift period \ centerlineLiving Reviews{from in Relativity lensing open . } parenthesis 1998 hyphen 12 closing parenthesis http : slash slash www period l ivingreviews period org \noindent 4 . 2 \quad Quasar microlensing

\noindent Light bundles from \ h f i l l ‘ ‘ l e n s e d ’ ’ \ h f i l l quasars are split by intervening galaxies . \ h f i l l With

\noindent typical separations of order one arcsecond between center of galaxy and quasar

$ 7 { This }$ can be seen very simply : Imagine a lens situation like the one displayed in Figure 2 . If now all length scales are reduced by a factor of two and at the same time all masses are reduced by a factor of two , then for an observer the angular configuration in the sky would appear exactly identical . But the total length of the light path is reduced by a factor of two . Now , since the time delay between the two paths is the same fraction of the total lengths in either scenario , a measurement of this fractional length allows to determine the total length , and hence the Hubble constant , the constant of proportionality between distance and redshift .

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 26 \noindentFigure 8 :J .. Optical . Wambsganss Lightcurves\ h of f i images l l 26 Q 957 plus 561 A .. and B open parenthesis top panel : .. g hyphen band semicolon .. bottom panel : .. r hyphen band closing parenthesis period .. The b lue curve is the one of leading image A\noindent comma Figure 8 : \quad Optical Lightcurves of images Q $ 957 + 561 $ A \quad and B ( top panel : \quad g − bandthe red ; \ onequadJ .the Wambsganssbottom trailing image panel B period : \quad Noter the− sband t eep drop) . that\quad occuredThe in b December lue curve is the one26 of leading image A , the1 994 red in image oneFigure the A 8 and : trailing wasOptical s een Lightcurves imagein February B of. 1 Noteimages 996 in the image Q 957 s B t+ period561 eepA drop .. 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A\quad and ( Credits : \quad Tomislav KundiLiving Reviews $ was\acute s in een Relativity{c in} February;$ open 1 seealso[79]) parenthesis996 in image 1998 B . hyphen The 12light closing curves parenthesis are shi f − t ed in time by about http : slash4 1slash 7 days www re period lative to l ivingreviews each other . period ( Credits org : Tomislav Kundi c´; s ee als o [ 79 ] ) \noindent LivingLiving Reviews Reviews in Relativity in Relativity( 1998 - 12 ( 1998) − 12 ) http : / / www . l ivingreviews . org \noindent http : / / www . l ivingreviews . org 27 .... Gravitational Lensing in Astronomy \noindentimage comma27 this\ h f means i l l Gravitational that the quasar light Lensing bundle in passes Astronomy through the galaxy and slash or the galaxy halo period .. Galaxies consist at least partly of st ars comma .. and galaxy haloes \noindentconsist possiblyimage of compact, thisobjects means as that well periodthe quasar light bundle passes through the galaxy and / or theEach galaxy of these27 halo stars open . \quad parenthesisGalaxies or other consist compact at objects least comma partly Gravitational like black of st holes Lensingars comma , \ inquad brown Astronomyand dwarfs galaxy comma haloes consistor planets possiblyimage closing , parenthesis this of means compact acts that as the objectsa quotedblleft quasar light as compactwell bundle . passes lens quotedblright through the or galaxy quotedblleft and / or microlens the galaxy quotedblright and produces at leasthalo one . Galaxies consist at least partly of st ars , and galaxy haloes consist possibly of compact \ hspacenew image∗{\objectsf of i l the l }Each assource well of period . these .. 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But splitting open because parenthesis is proportional the image 4 closing to the splitting parenthesis lens closing is proportional parenthesis comma to the these lens microimages are only of ordermass a (microarcsecond see Equation ( 4 ) ) , these microimages are only of order a microarcsecond apart and can \noindentapart andnotmass can be not resolved( be see resolved Equation . Various period ..( aspects Various 4 ) ) of aspects microlensing, these of microlensing microimages have been have ad -are been dressed only ad afterhyphen of the order first double a microarcsecond apartdressed and afterquasar can the firstnothad beendouble be resolveddiscovered quasar had [29 . been,\30quad, discovered?, 68Various, ?, 137 open, aspects square bracket of microlensing 29 comma 30 comma have ? been comma ad 68− comma ? commadressed 1 37 aftercomma1 6 1 ] .the first double quasar had been discovered $ [ 29 , 30 , ? , 681 6 1, closing ?Figure square , 9 : 1 bracket“ Micro 37 period - Images, $ ” : The top left panel shows an assumed “ unlensed ” s ource Figure 9 :profile .. quotedblleft of a quasar Micro . hyphen The Images o th er quotedblright three panels i : l lustrate.. The top the leftmicro panel - image shows configuration an assumed .. quotedblleft unlensed\noindent quotedblrightas1 it6 would 1 ] . be produced by s t e l lar o bjects in th e foreground . The surface mass density s ource profileof the .. lens of a es quasar is 20%( periodtop .. right The o), th50%( er threebottom panels left ) i andl lustrate80% the micro hyphen image \noindentconfiguration(Figure bottom as it would right 9 : )\ be ofquad produced the critical‘ ‘ Micro by density s t e− llar (Images cf o . bjects Equation ’ ’ in : th (\ e1quad foreground 6 ) )The . topperiod left .. The panel shows an assumed \quad ‘‘ unlensed ’’ ssurface ource mass profile density\quad of theo lensThe f a es surface quasar is 20 percentmass . \ densityquad open inThe parenthesis front o of th a toper multiply rightthree imaged closing panels quasar parenthesis i is l of lustrateorder comma the 50 the percent micro open− image parenthesisconfiguration bottom left as closing it would parenthesis be produced and 80 percent by s t eLiving l lar Reviews o bjects in Relativity in th e( 1998 foreground - 12 ) . \quad The surfaceopen parenthesis mass density bottom right of closing the lens parenthesis es is of the $ critical 20http density\ :%( / / open www$ parenthesistop . l r ivingreviews i g h cf t period$ ) Equation . , org 50 open parenthesis\%( $ 1bottom 6 closing left parenthesis ) and closing$ 80 parenthesis\% $ period The surface mass density in front of a multiply imaged quasar is of order the \noindentLiving Reviews( bottom in Relativity right open ) of parenthesis the critical 1998 hyphen density 12 closing ( cf parenthesis . Equation ( 1 6 ) ) . http : slash slash www period l ivingreviews period org \ hspace ∗{\ f i l l }The surface mass density in front of a multiply imaged quasar is of order the

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 28 \noindentquotedblleftJ critical . Wambsganss surface mass\ h density f i l l 28 quotedblright comma see Equation open parenthesis 1 6 closing parenthesis period .... Hence microlensing should \noindentbe o ccuring‘‘ basically critical all the surface t ime period mass .. Thisdensity can be ’’ visualized , see in Equation the following ( 1 way 6 )period . \ h f i l l Hence microlensing should If one assignsJ . Wambsganss each microlens a little disk with radius equal to the Einstein 28 \noindentring comma“be critical then o ccuring the surface fraction mass basically of skydensity which ” all , is see covered the Equation t by ime these ( 1 6. disks )\ .quad correspondsThis can Hence to the be microlensing visualized should in the following way . Ifsurface one mass assignsbeo density ccuring each in basically units microlens of allthe the criticalt a ime little density . This semicolondisk can be with visualized this radius fraction in the isequal sometimes following to waythe . Einstein If one assigns ringalso called , theneach the thequotedblleft microlens fraction a little optical ofdisk Case sky with 1 which radiusquotedblright equal is covered to Case the Einstein 2 period by these ring , then disks the fractioncorresponds of sky which to the surfaceThe microlenses massis covered density produce by these a in complicated disks units corresponds of two the hyphen to critical the dimensional surface density mass magnification density ; this in units dis fraction hyphen of the critical is densitysometimes ; also called the ‘‘ optical $\ l e f t . depth \ begin { a l i g n e00 d } &’’ \\ tribution inthis the fraction source is plane sometimes period alsoIt consists called ofthe many “ optical causticsdepth comma locations that corre hyphen &.spond to\end formally{ a l i g infinitely n e d }\ right high magnification. $ period . An example forThe such microlenses a magnification produce pattern a complicated is shown two in Figure - dimensional 1 0 period magnification It is de hyphen dis - tribution in the Thetermined microlenses withsource the plane produceparameters . It consists a of complicated image of many A of the caustics twoquadruple− , locationsdimensional quasar that Q 2237 corre magnification plus - spond 305 to formally dis infinitely− tributionopen parenthesishigh in magnification the surface source mass . density plane kappa . It = 0consists period 36 semicolon of many external caustics shear ,gamma locations = 0 period that 44 closing corre parenthesis− periodspond .... to Color formally indicatesAn example theinfinitely for such a magnification high magnification pattern is shown . in Figure 1 0 . It is de - termined with magnificationthe parameters: .... blue is of relatively image A low of the magnification quadruple quasaropen parenthesis Q 2237 + 305 slightly demagnified com hyphen Anpared example to mean( surface for closing such mass parenthesis a density magnificationκ comma= 0.36; green external pattern is slightly shear isγ magnified= shown 0.44). andin Figurered and yellow 1 0Color . is highlyIt indicates is magni de − the hyphen terminedfied periodmagnification with the parameters : of image blue is A relatively of the low quadruple magnification quasar ( slightly Q demagnified $ 2237 com + - 305 $ Due to thepared relative to mean motion ) , between green is observer slightly magnified comma lens and and red source and yellow comma is the highly quasar magni - fied . \noindentchanges its( position surfaceDue to relative the mass relative to densitythis motion arrangement between $ \kappa ofobserver caustics ,= lens comma and0 isource period. 36, the e period quasar ;$ the changes externalshear apparent its position $ \gamma =brightness 0 .relative of 44 the quasar to ) this changes arrangement . $ \ withh f i ltime of l causticsColor period , indicates A i . one e . hyphen the apparent the dimensional cut through such a magnificationbrightness pattern of the comma quasar convolved changes with time a source . A one profile - dimensional of the quasar cut comma through results such a magnification \noindentin a microlensedpatternmagnification lightcurve , convolved period : with\ h .. f a Examplesi l source l blue profile for is microlensed relativelyof the quasar lightcurves ,low results magnification taken in a along microlensed ( lightcurve slightly . demagnified com − the yellowExamples tracks in Figure for microlensed 1 0 can be lightcurves seen in Figure taken 1 along 1 for thetwo yellow different tracks quasar in Figure 1 0 can be seen in \noindentsizes periodFigurepared 1 to 1 for mean two different ) , green quasar is slightly magnified and red and yellow is highly magni − f iIn e d particular . sizes when . the quasar track crosses a caustic open parenthesis the sharp lines in Fig hyphen ure 1 0 for whichIn particular the magnification when the formally quasaris track infinite crosses comma a caustic because ( the the sharp determinant lines in Fig - ure 1 0 for Dueof to the theJacobianwhich relative disappears the magnification motion comma between formally cf period is observerEquation infinite , open because , lens parenthesis the and determinant 3source 1 closing , parenthesis the quasar closing parenthesis comma achanges pair of highly itsof magnifiedthe position Jacobian mi disappearshyphen relative , cf to . Equation this arrangement ( 3 1 ) ) , a pair of of caustics highly magnified , i . mi e - . the apparent croimagescroimages appears newly appears or newly merges or and merges disappears and disappears open parenthesis ( see [?]). Such see open a microlensing square bracket event ? can closing easily square bracket closing\noindent parenthesisbebrightness detected period as Such a of strong a the microlensing peak quasar in the changes lightcurve with of the time quasar . A one − dimensional cut through such aevent magnification canimage easilybe . detected pattern as , a strong convolved peak in with the lightcurve a source of the profile quasar of the quasar , results inimage a microlensed period In most lightcurve simulations it . is\quad assumedExamples that the for relative microlensed positions of the lightcurves mi - crolenses taken is fixed along theIn most yellow ..and simulations tracks the lightcurves itin is Figure .. assumed are produced 1 that 0 can theonly be relative by seen the positions bulk in motion Figure of the between mi1 1 hyphen forquasar two , galaxy different and observer quasar crolenses is. fixedA visualiz and the ation lightcurves of a situation are produced with chang only - ingby the microlens bulk motion positions can be found in Figure 1 2 \noindentbetween quasarfors i z three e s comma . different galaxy values and of observer the surface period mass A visualiz density ation : of a situation with chang hyphen ing microlens positions can be found in Figure 1• 2 forκ = three 0.2 in different 1 2 a . values of Inthe particular surface mass when density the : quasar track crosses• κ = 0. a5 in caustic 1 2 b . ( the sharp lines in Fig − urebullet 1 0kappa for = which 0 period the 2 in magnification 1 2 a period formally• κ = 0.8 in is 1 2infinite c . , because the determinant bullet kappa =The 0 period change 5 of in caustics 1 2 b period shapes due to the motion of individual stars which can be looked at \noindentbullet kappawhenof = the 0 clicking period Jacobian on 8 in one 1 2 of disappears c the period three panels , cf of Figure . Equation 1 2 produces ( 3 1 ) ) , a pair of highly magnified mi − The changeadditional of caustics fluctuations shapes due in to the the lightcurve motion of [? individual, ?]. stars which \noindentcan be lookedcroimages at when clicking appearsThis on change one newly of theof or caustics three merges panels shapes and of due Figure disappears to the 1 2 motion produces ( of see individual $ [ stars ? produces ] ) . $ Such a microlensing eventadditional canadditional fluctuations easily fluctuations be in detected the lightcurve in the as lightcurve open a strong square [?, bracket peak?]. ?in comma the lightcurve ? closing square of bracket the quasar period This change of caustics shapesMicrolens due to the - induced motion fluctuations of individual in stars the observed produces brightness of quasars contain \noindentadditionalinformationimage fluctuations . both in the about lightcurve the light open - emitting square bracket source ?( size comma of continuum ? closing square region bracketor period MicrolensLiving hyphen Reviews induced in fluctuations Relativity in( 1998 the observed - 12 ) brightness of quasars contain Ininformation most \quadhttp bothsimulations : about / / thewww light . it l hyphen ivingreviews is \ emittingquad assumed source . org open that parenthesis the relative size of continuum positions region of or the mi − crolensesLiving Reviews is infixed Relativity and open the parenthesis lightcurves 1998 are hyphen produced 12 closing only parenthesis by the bulk motion betweenhttp : slash quasar slash www , galaxy period l and ivingreviews observer period . A org visualiz ation of a situation with chang − ing microlens positions can be found in Figure 1 2 for three different values of the surface mass density :

\ centerline { $ \ bullet \kappa = 0 . 2$in12a. }

\ centerline { $ \ bullet \kappa = 0 . 5$in12b. }

\ centerline { $ \ bullet \kappa = 0 . 8$in12c. }

The change of caustics shapes due to the motion of individual stars which can be looked at when clicking on one of the three panels of Figure 1 2 produces

\noindent additional fluctuations in the lightcurve $ [ ? , ? ] . $

\ hspace ∗{\ f i l l } This change of caustics shapes due to the motion of individual stars produces

\noindent additional fluctuations in the lightcurve $ [ ? , ? ] . $

\ hspace ∗{\ f i l l } Microlens − induced fluctuations in the observed brightness of quasars contain

\noindent information both about the light − emitting source ( size of continuum region or

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 29 .... Gravitational Lensing in Astronomy \noindentFigure 1 029 : ..\ Magnificationh f i l l Gravitational pattern in the Lensing s ource plane in comma Astronomy produced by a dense field of s tars in the lensing galaxy period .. The colo r reflects the magnification as a function \noindentof the quasarFigure position 1 :0 .. : the\quad s equenceMagnification b lue hyphen green pattern hyphen in red the hyphen s ource yellow planeindicates , in produced creasing by a dense field ofmagnification s tars29 in period the .. lensing Lightcurves galaxy taken along. \quad the yellowThe colo tracks r are reflects shown Gravitational in Figure the Lensing magnification 1 1 period in Astronomy as a function ofThe the microlensing quasarFigure 1position parameters 0 : Magnification were: \quad chosen patternthe according s in equence the to s aource model b plane lue for− image, producedgreen A of by− ared dense− fieldyellow of s indicatestars in in creasing magnificationthe quadruplethe lensing quasar . \quad galaxy Q 2237 .Lightcurves plus The 305 colo : kappa r reflects taken = 0 periodthe along magnification 36 the comma yellow as gamma a function tracks = 0 period of are the 44 quasar shown period position in Figure : 1 1 . TheLiving microlensing Reviewsthe s equencein Relativity parameters b lue open - green parenthesis were - red chosen - yellow 1998 hyphen indicates according 12 in closing creasing to parenthesisa model magnification for image . ALightcurves of thehttp quadruple : slashtaken slash along quasar www the period yellow Q l $ ivingreviews tracks 2237 are shown + period 305 in Figureorg : 1 1\ .kappa The microlensing= 0 parameters . 36 were , chosen\gamma = 0 . 44 .according $ to a model for image A of the quadruple quasar Q 2237 + 305 : κ = 0.36, γ = 0.44. Living Reviews in Relativity ( 1998 - 12 ) \ hspace ∗{\ f i l l } Living Reviews in Relativity (http 1998 :− / /12 www ) . l ivingreviews . org

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 30 \noindentFigure 1 1J : Microlensing . Wambsganss Lightcurve\ h f i l for l 30 th e ye l low tracks in Figure 1 0 period .. The s o lid and dashed lines indicate re latively small and large quasar s izes period .. The time axis \noindentis in unitsFigure of Einstein 1 radii 1 : divided Microlensing by unit velocity Lightcurve period for th e ye l low tracks in Figure 1 0 . \quad The s o l i d andLiving dashed ReviewsJ . linesWambsganss in Relativity indicate open re parenthesis latively 1998 small hyphen and 12 closing large parenthesis quasar s izes . \quad The30 time axis ishttp in : unitsslashFigure slash of 1 www Einstein 1 : Microlensing period l radii ivingreviews Lightcurve divided period for by th org e unit ye l low velocity tracks in .Figure 1 0 . The s o lid and dashed lines indicate re latively small and large quasar s izes . The time axis is in units of \noindent EinsteinLiving radii Reviews divided in by unitRelativity velocity . ( 1998 − 12 ) Living Reviews in Relativity ( 1998 - 12 ) \noindent httphttp : : / / / / www www . . l l ivingreviews ivingreviews . org. org 3 1 .... Gravitational Lensing in Astronomy \noindentFigure 1 23 : .. 1 Still\ h f pictures i l l Gravitational of microlensing caustics Lensing for in three Astronomy values of the surface mass density : .. a closing parenthesis kappa = 0 period 2 semicolon b closing parenthesis kappa = 0 period 5 semicolon c closing\noindent parenthesisFigure kappa 1 =2 0 : period\quad 8 periodStill .. pictures The s equences of microlensingare described caustics for three values of the surface massand analysed density3 1 quantitatively : \quad a in $open ) square\kappa bracket= ? closing 0 square . 2 bracket Gravitational ; $ period b $Lensing ) \ inkappa Astronomy= 0 . 5 ; $Living c Reviews $Figure ) \ inkappa 1 Relativity 2 : Still= openpictures 0 parenthesis of . microlensing 8 1998 . hyphen $ caustics\quad 12 for closingThe three parenthesis svalues equences of the surface are described mass density : andhttp analysed : slasha slash)κ = quantitatively www 0.2; b period)κ = 0l. ivingreviews5; c in)κ = $ 0.8 [period. The ? org s equences] . $ are described and analysed quantitatively in [?]. \ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998Living− Reviews12 ) in Relativity ( 1998 - 12 ) http : / / www . l ivingreviews . org \ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 32 \noindentbroad lineJ region . Wambsganss of the quasar\ commah f i l l ....32 brightness profile of quasar closing parenthesis .... and about the lensing objects open parenthesis masses comma density comma transverse velo city closing parenthesis period .... Hence from\noindent a comparibroad hyphen line region of the quasar , \ h f i l l brightness profile of quasar ) \ h f i l l and about the son betweenJ . observed Wambsganss and simulated quasar microlensing open parenthesis or lack of it closing parenthesis32 one can \noindentdraw conclusionsbroadlensing line about region objects the of density the ( quasar masses and mass , , scale density of brightness the microlenses , transverse profile period of quasar velo It is ) not city ) and . \ abouth f i l l theHence from a compari − trivial commalensing though objects comma ( masses to extract , density this , informationtransverse velo quantitatively city ) . period The Hence reason from is that a compari in - \noindentthis regimesonson of between optical between depth observed observed of order and simulated one and comma simulated quasar the magnification microlensing quasar ( is microlensing or not lack due of to it ) a one single can( or draw lack conclusions of it ) one can drawisolated conclusions microlensabout the comma density about but and the it rather mass density scale is a collective of and the microlenses mass effect scale of many . It of is stars not the periodtrivial microlenses ,This though means , to . extract It is this not trivialindividual ,information mass though determinations , quantitatively to extract are not . The this j ust reason impossible information is that from in this the quantitatively regime detection of optical of a depth . The of reason order one is , the that in thissingle regime causticmagnification hyphen of optical crossing is not depth duemicrolensing to a of single order event isolated comma one microlens , but the it magnification, does but notit rather even ismake a collective is sense not to effectdue try to of many a single isolateddo so commastars microlens since . This these means , events but individual are it not rather producedmass determinations is bya collectiveindividual are lenses not effect j toust the impossible power of many of from 8 period stars the detection Mass . This determi of means hyphen individuala single mass caustic determinations - crossing microlensing are not event j ust , but impossible it does not even from make the sense detection to try of a nations can only be done in a statistical sense comma by comparing good observations8 singleopen parenthesis causticdo so , frequently since− crossing these sampled events microlensing commaare not high produced photometric event by individual , but accuracy it lenses does closing. Mass not parenthesis evendetermi make with - nations simulations sense can to period try Inter- preting only be done in a statistical sense , by comparing good observations ( frequently sampled , high \noindentmicrolensedphotometricdo lightcurves so , since accuracy of multiply these ) with hyphen events simulations imaged are quasars not. Interpreting produced allows to microlensed determine by individual the lightcurves size $ of lenses multiply ˆ{ - 8 } . $ Mass determi − nationsof the continuum canimaged onlyemitting quasars be done allows region in to of determine thea statistical quasar the and size to learnof sense the even continuum , more by comparing about emitting region good of observationsthe quasar and (the frequently centralto engine learn sampled openeven more square , abouthigh bracket the photometric 56 central comma engine 64 accuracy comma [ 56 , 64 1 1 , 12 ) 1comma with 2 , 1 64 1 simulations 64 ] . closing square . Interpreting bracket period microlensedSo far the quotedblleftSo lightcurves far the best “ best quotedblright of ” multiplyexample example of− a microlensedimaged of a microlensed quasars quasar quasar is allows the is quadruple the to quadruple determine quasar quasar Q the 2237 size + ofQ the 2237 continuum plus305[? 305, ?, open83, emitting14 square, 164, 165 bracket, 171] region. ?In comma Figure of the ? 1 comma 3 quasar two images 83 andcomma of to this 1 learn 4 system comma even are 1 64 shown more comma which about 1 65 were comma t 1 7 1 closing squarethe bracketcentralaken period engine in 1 In 99 Figure 1 [ and 56 1 1 3 994, two 64 , respectively images , 1 1 of 2 this ,. 1system Whereas 64 ] . on the earlier observation image B ( top ) is are shownclearly which werethe brightest t aken in , 1three 99 1 years and 1 later 994 imagecomma respectively period .. Whereas on the Soearlier far the observationA ‘‘ ( bottom best image ) ’’ is B at example open least parenthesis comparable of a microlensed top in closingbrightness parenthesis . quasar is clearly is the the Since quadruple brightest the time comma quasardelay three in this years later image Q$2237A open parenthesissystem + is bottomonly305 a day closing [ or ?shorter parenthesis , ( because ? is at , of least the 83 comparable symmetric , 1 image in brightness 4 arrangement , period 1 ) 64 ...., any Since brightness , the 1 time 65 delay , in this 1 7system 1 is ] onlychange . a $ day on In orlarger shorter Figure time open scales 1 3 parenthesis two must images be becausedue to of microlensing of this the symmetric system . In image Figure arrangement 1 4 lightcurves closing parenthesis are comma areany shown brightnessshown which change for were the on four larger t images aken time of in scales Q 1 2237 99 must + 1 305 be and over due 1 ato 994 period microlensing , of respectively almost period a decade .. In .( from\quad [ 82Whereas ] ) . The on the earlierFigure .. observation 1changes 4 lightcurves of the are relative image shown brightnesses B for (the top four ) of isimages these clearly images of Q 2237 induced the plus brightest by305 microlensing over a , three are obvious years . later image period of almost4 . 3 a decade Einstein open parenthesis rings from open square bracket 82 closing square bracket closing parenthesis period ..\noindent The changesAIf of a ( point the bottom relative source ) brightnesses lies is exactly at least behind comparable a point lens ,in a ring brightness - like image . o\ ccursh f i l .l TheoSince - rists the had time delay in this of these imagesrecognized induced early by on microlensing [ 3 1 , 44 ]are that obvious such a periodsymmetric lensing arrangement would result in a ring \noindent4 period 3-system .. image Einstein , a is so rings only- called a “ day Einstein or - shorter ring ” . ( Can because we observe of Ein the - stein symmetric rings ? image There are arrangement two ) , anyIf a brightness pointnecessary source lies change requirements exactly on behind larger for a their point time occurence lens commascales : the a must ringmass hyphen distribution be due like to image of microlensing the o lens ccurs needs period to be. Theo\ axiallyquad hyphenIn Figurerists had\quad recognizedsymmetric1 4 early, lightcurves as seen on open from squarethe are observer shown bracket , and for 3 1 the comma the source four 44 must closing images lie square exactly of Qbracket on top$ 2237 thatof the such resulting + a symmetric 305 $ over lensing a arrangementperiod ofdegenerate almost a point decade - like caus ( from - t ic [ . 82 Such ] a ) geometric . \quad arrangementThe changes is highly of theunlikely relative for point brightnesses - ofwould these resultlike images in sources a ring induced . hyphen But astrophysical image by microlensing comma sources a so in hyphen the are real called obvious universe quotedblleft have. a finite Einstein extent hyphen , and ring it isquotedblright enough period .. Can we observeif a Ein part hyphen of the source covers the point caustic ( or the complete astroid caustic \noindentstein ringsin4 ? a ... case There 3 \ ofquad a are not twoEinstein quite necessary axial - rings requirementssymmetric mass for distribution their occurence ) in: order the mass to produce such an annular distributionimage of the . lens needs to be axially symmetric comma as seen from the observer comma \noindentand the sourceIfIn a must 1 point 988 lie , exactly the source first on example top lies of the exactly of resulting an “ Einstein behind degenerate ring a point ”point was hyphen discovered lens like, a [caus ring 60 ] hyphen .− like With highimage o ccurs . Theo − ristst ic period hadresolution Such recognized a geometric radio observations early arrangement on [ , 3 is the highly1 ,extended 44 unlikely ] that radio for pointsource such hyphen aMG symmetric 1131 like + sources 456 lensingturned period out But arrangement to be a wouldastrophysical resultring sources with in a a in diameter ring the real− ofimage universe about , 1have a. 75 so a arcsecfinite− called extent . The comma ‘‘ source Einstein and was itis identified enough− r i n g as ’ a ’ radio . \quad lobe atCan a we observe Ein − steinif a part rings ofredshift the source $ of ?z $S covers=\ 1quad.13 the, whereas pointThere caustic the are lens open two is a parenthesis necessarygalaxy or requirements the complete astroid for caustic their occurence : the mass distributionin a case of a8 notSimilarly of quite the, one axial lens cannot hyphen needs determine symmetric to the be temperature mass axially distribution of a symmetricblack body closing by measuring, parenthesis as seen the energy in from order of a the to single produce observer photon , andsuch the an annular sourceemitted image by must the black period lie body exactly , but one needs on topto measure of the a large resulting number of them degenerate and compare with point some underlying− l i k e caus − tIn ic 1 988. Such commatheory a . geometricthe first example arrangement of an quotedblleft is highly Einstein unlikely ring quotedblright for point was− discoveredlike sources open square . But bracket 60 closingastrophysical squareLiving bracket Reviews sources period in .. in RelativityWith the real( 1998 universe - 12 ) have a finite extent , and it is enough ifhigh a partresolutionhttp of radio the: / observationssource / www . covers l comma ivingreviews the .. the point extended . causticorg radio source ( or MG the 1 complete1 3 1 plus 456 astroid caustic turned out to be a ring with a diameter of about 1 period 75 arcsec period .. The source was \noindentidentified asin a a radio case lobe of at a redshift not quite of z sub axial S = 1− periodsymmetric 1 3 comma mass whereas distribution the lens is a galaxy ) in order to produce such8 sub an Similarly annular comma image one cannot . determine the temperature of a black body by measuring the energy of a single photon emitted by the black body comma but one needs to measure a large number of Inthem 1 988 and , compare the first with some example underlying of an theory ‘‘ periodEinstein ring ’’ was discovered [ 60 ] . \quad With highLiving resolution Reviews in Relativity radio observations open parenthesis , 1998\quad hyphenthe 12 extended closing parenthesis radio sourceMG $ 1 1 3 1 +http 456 : slash $ slash www period l ivingreviews period org turned out to be a ring with a diameter of about 1 . 75 arcsec . \quad The source was identified as a radio lobe at a redshift of $ z { S } = 1 . 1 3 ,$ whereasthe lens isagalaxy

$ 8 { S i m i l a r l y , }$ one cannot determine the temperature of a black body by measuring the energy of a single photon emitted by the black body , but one needs to measure a large number of them and compare with some underlying theory .

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 33 .... Gravitational Lensing in Astronomy \noindentFigure 1 333 : ..\ Twoh f i images l l Gravitational of the quadruple Lensing quasar Q in 2237 Astronomy plus 305 s eparated by three years period .. It is o bvious that th e re lative b rightnesses of the images change period .. Image B \noindentis clearly theFigure brightest 1 3 one : in\quad the leTwo f-t panel images comma of whereas the quadruple images A and quasar B are about Q $ 2237 + 305 $ s eparated by three yearsequally . bright\33quad in theIt right is o panel bvious period that .. open th parenthesis e re lative Credits b : rightnesses .. Gravitational Geraint Lewis Lensingof period the in closing images Astronomy parenthesis change . \quad Image B isFigure clearly .. 1Figure 4 the: .. Lightcurves1 brightest 3 : Two .. images of one th e in of four the the images quadruple le .. $ of fquasar Q−t 2237 $ Q plus panel2237 305 + over , 305 whereas as period eparatedof images by three A and years B . are about equallyalmost t enbrightIt years is o bvious period in the that .. The right th e changes re lative panel in b re rightnesses . lative\quad brightness( of Credits the imagesare very : change o\quad bvious .Geraint period Image .. openB Lewis is parenthesis clearly . ) the Credits : Geraint Lewisbrightest period one closing in the parenthesis le f − t panel , whereas images A and B are about equally bright in the right \noindentLiving ReviewspanelFigure in. Relativity\ (quad Credits1 open : 4 : parenthesis Geraint\quad LewisLightcurves 1998 . hyphen ) 12\quad closingof parenthesis th e four images \quad o f Q $ 2237 +http 305 : slash $Figure over slash www a 1 4period : periodLightcurves l of ivingreviews of periodth e four org images of Q 2237 + 305 over a period of almost almost tt en en years . . \ Thequad changesThe in changes re lative in brightness re lative are very brightness o bvious . are ( Credits very : o bvious . \quad ( Credits : Geraint Lewis . ) \noindent Geraint Lewis . ) Living Reviews in Relativity ( 1998 - 12 ) http : / / www . l ivingreviews . org \ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 34 \noindentat z sub LJ = .0 period Wambsganss 85 period\ h Recently f i l l 34 comma a remarkable observation of the Einstein ring 1 938 plus 666 was presented open square bracket 72 closing square bracket period The infrared HST image shows an almost perfectly circular\noindent at $ z { L } = 0 . 85 . $ Recently , a remarkable observation of the Einstein ring $ 1ring with 938 twoJ . + Wambsganss bright 666 parts $ plus the bright central galaxy period .. The contours agree 34 very well withat zL the= 0 MERLIN.85. Recently radio , amap remarkable open parenthesis observation see of Figure the Einstein 1 5 closing ring parenthesis 1938 + 666 period \noindentFigure 1 5waswas : .... presented presented Einstein [ ring 72 ] [ 1 . 938 72The plus ] infrared . 666 The open HST infrared parenthesis image shows HST from an image almost open shows square perfectly bracket an circular almost 72 closing ring perfectly with square two bracket circular closing parenthesisring with :bright .... two The parts bright le f-t plus panel parts the shows bright plus the central radio the galaxy bright . Thecentral contours galaxy agree . \quad The contours agree map as contourvery well superimposed with the MERLIN on the grey radio s cale map HST ( see slash Figure NiCMOS 1 5 ) . image semicolon th e right \noindentpanel is aFigure colorvery depiction 1well 5 : withEinstein of the the infrard ring MERLIN1938 HST + slash radio 666( NICMOSfrom map [ 72 ( image ] see ) : period Figure The .... le 1 Thef 5− t diameter )panel . shows of the radio the ring ismap a bout as contour0 period superimposed 95 arcs econds on period the grey .. open s cale parenthesis HST / NiCMOS Credits image : Neal ; thJackson e right period closing parenthesis \noindentBy now aboutpanelFigure a is half a 1 color dozen 5 : depiction cases\ h f i have l l ofEinstein thebeen infrard found HST ringthat /qualify NICMOS $1 as Einstein 938 image . + 666 ($ The diameter from [ of 72 ] ) : \ h f i l l The l e $ frings−t $ open panelthe square ring shows bracket is a bout the 1 0 2 .closing radio 95 arcs square econds bracket . ( period Credits : Neal Jackson . ) Their diameters vary between 0 periodBy now 33 about and about a half 2dozen arcseconds cases have period been All found of them that are qualify as Einstein \noindentfound in theringsmap radio [ as 1 regime2 contour ] . comma superimposed some have optical on or the infrared grey counterparts s cale HST as well / NiCMOS period image ; th e right Some of the EinsteinTheir diameters rings are vary not really between complete 0 . 33 rings and about comma 2 butarcseconds they are . quotedblleft All of them arebroken found quotedblright in the \noindentrings withradiopanel one or regime two is interruptions a , some color have depiction optical along the or infraredcircle of theperiod counterparts infrard The sources asHST well of / most . NICMOS Some Einstein of the image Einstein . \ ringsh f i l are l The diameter of rings havenot both really an extended complete and rings a compact, but they component are “ broken period ” rings .. The with latter one oris always two interruptions along the \noindentseen as a doublecirclethe ring. image The issources comma a bout of separated most 0 .Einstein by95 roughly arcs rings econds the have diameter both . an\quad of extended the Einstein( Credits and ring a compact period : Neal component Jackson . . ) In some casesThe monitoring latter is always of the seen radio as flux a double showed image that , the separated compact by source roughly is the diameter of the Einstein \ hspacevariable∗{\ periodringf i l l . This}By In some gives now casesthe about opportunity monitoring a half to of dozen measure the radio cases the flux t ime have showed delay been that and foundthe the Hubble compact that source qualify is variable as Einstein constant H. sub This 0 gives in these the systems opportunity period to measure the t ime delay and the Hubble constant H0 in these \noindentThe Einsteinsystemsr i n ring g s systems . [ 1 2 provide ] . some advantages over the multiply hyphen imaged quasar systems for the goalThe to determine Einstein ring the lens systems structure provide and some slash advantages or the Hubble over the multiply - imaged TheirLiving diameters Reviewsquasar in systems Relativityvary between for openthe goal parenthesis 0 to . determine 33 and 1998 aboutthe hyphen lens 2 12structure arcseconds closing and parenthesis / or . the All Hubble of them are foundhttp : inslashLiving the slash radio Reviews www period regime in Relativity l ivingreviews , some( 1998 have period - 12 optical org) or infrared counterparts as well . Some of thehttp Einstein : / / www rings . l areivingreviews not really . org complete rings , but they are ‘‘ broken ’’ rings with one or two interruptions along the circle . The sources of most Einstein rings have both an extended and a compact component . \quad The latter is always seen as a double image , separated by roughly the diameter of the Einstein ring . In some cases monitoring of the radio flux showed that the compact source is variable . This gives the opportunity to measure the t ime delay and the Hubble constant $ H { 0 }$ in these systems .

\ hspace ∗{\ f i l l }The Einstein ring systems provide some advantages over the multiply − imaged

\noindent quasar systems for the goal to determine the lens structure and / or the Hubble

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 35 .... Gravitational Lensing in Astronomy \noindentconstant period35 \ h First f i l l ofGravitational all the extended image Lensing structure in provides Astronomy many constraints on the lens period A lens model can be much better determined than in cases of just two or \noindentthree or fourconstant point hyphen . First like quasar of all images the period extended Einstein image rings thus structure help us to provides understand many constraints on thethe lensmass distribution35 . A lens of model galaxies can at moderate be much redshifts better period determined .. For the Gravitational than Einstein in ring cases Lensing of in Astronomyjust two or threeMG 1 or 654constant four plus 56 point 1. Firstit was− of foundlike all theopen quasar extended square images image bracket structure . ? Einstein closing provides square rings many bracket thus constraints that help the onradially us the to lens understandaveraged . 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However , many galaxy models predict a finite core in the mass models predictdistribution a finite of core a galaxy in the . mass The distribution non - detection of a galaxy of the period central .. images The non puts hyphen strong constraints on Thedetection third ofthe advantage the size central of the images is core that radii puts .since strong theconstraints emitting on the region size of the of core the radio flux is presumablyradii period4 . much 4 larger Giant than luminous that arcsof the and optical arclets continuum flux , \quad the radio lightcurves4 period 4Zwicky .. Giant of had the luminous pointed different arcs out and the images arcletspotential are use in not the affected 1 930 s , but by nobody microlensing had really followed . \quad up theHence the radioZwicky lightcurves hadidea pointed , not evenout between the after potential the the discovery use images in the of the1 should 930 lensed s comma agreequasars but with: Galaxiesnobody each had can other really be gravitationally very well lensed . followed upas the well idea . comma Since galaxies not even are after extended the discovery objects of , the the lensed apparent quasars consequences : Galaxies for them would be Anothercan be gravitationally interestingfar more dramatic lensed application than as well for period quasars is .. the : Since galaxies\quad galaxies should( are non be extended− heavily) detection deformedobjects comma once of they athe central are strongly image inapparent the Einstein consequenceslensed . rings for them . \quad wouldFor be far singular more dramatic lenses than for , \ quasarsquad there : should be no central image (galaxies the reason should be is heavily the discontinuity deformedIt came as once quite they a of surprise are the strongly deflection when lensed in 1 986 period angleLynds & ) Petrosian . However [ 87 ] , and many Soucail galaxy modelsIt came predict aset quite al . a [ a1 surprise 43 finite ] independently when core in 1 in 986 discovered the Lynds mass ampersand this distribution new gravitational Petrosian open of lensing a square galaxy phenomenon bracket . \ 87quad : closing magnifiedThe square non − bracket and Soucaildetection, distortedof thecentral and strongly images elongated puts images strong of background constraints galaxies on which the size happen of to the lie behind core r aet d i al i period . foreground open square clusters bracket of galaxies 1 43 closing( recent square HST images bracket ofindependently these discovered this new gravitational lensing phenomenon :two original arc clusters ( and others ) compiled by J . - P . Kneib can be found \noindentmagnifiedat4 comma [ . 74 4 ] distorted\ ,quad [ 75 ] . Giant and strongly luminous elongated arcs images and of arclets background galaxies which Rich clusters of galaxies at redshifts beyond z ≈ 0.2 with masses of order happen to lie14 behind foreground clusters of galaxies open parenthesis recent HST images of these \noindenttwo original10Zwicky arcM clusters are had very open pointed effective parenthesis lenses out theifand they others potential are centrally closing parenthesisuse concentrated in the .... . 1 compiled 930Their s Ein by , -J but stein period nobody radii hyphen are of had P period really Kneib canfollowed be found the up order the of idea 20 arcseconds , not even . Since after most the clusters discovery are not really of the spherical lensed mass quasars distributions : Galaxies and canat open be gravitationally squaresince bracket the alignment 74 closing lensed between squareas lens bracket well and source comma . \quad is open usuallySince square not galaxies bracket perfect 75, no closing are complete extended square Einstein bracket objects rings period , the apparentRich clustershave consequences of been galaxies found at aroundredshifts for them clus beyond - would ters z . thickapprox be But far there more 0 areperiod many dramatic 2 with examples masses than known of for order with quasars spectacularly : galaxies1 0 to thelong shouldpower arcs of which14 be M heavily sub areodot curved deformedare around very effective the once cluster lenses they center if arethey , with are strongly lengths centrally up lensed concentrated to about . 20period arcseconds .. Their . Ein hyphen stein radii are of the order of 20 arcseconds period .. Since mostLiving clusters Reviews are not in Relativity really ( 1998 - 12 ) \ hspacespherical∗{\ massf i l l distributions} It came asand quite since the a alignment surprise between whenhttp lens in : and1 / /986 source www Lynds . l ivingreviews $ \& $ Petrosian . org [ 87 ] and Soucail is usually not perfect comma no complete Einstein rings have been found around clus hyphen \noindentters periodet .. But al there . [ 1are 43 many ] independently examples known with discovered spectacularly this long new arcs which gravitational lensing phenomenon : magnifiedare curved around , distorted the cluster and center strongly comma with elongated lengths up images to about of 20 backgroundarcseconds period galaxies which happenLiving Reviews to lie in behind Relativity foreground open parenthesis clusters 1998 hyphen of galaxies 12 closing parenthesis ( recent HST images of these http : slash slash www period l ivingreviews period org \noindent two original arc clusters ( and others ) \ h f i l l compiled by J . − P . Kneib can be found

\noindent at[74] ,[75] .

\ hspace ∗{\ f i l l }Rich clusters of galaxies at redshifts beyond $ z \ thickapprox 0 . 2$ with masses of order

\noindent $ 1 0 ˆ{ 14 } M {\odot }$ are very effective lenses if they are centrally concentrated . \quad Their Ein − stein radii are of the order of 20 arcseconds . \quad Since most clusters are not really spherical mass distributions and since the alignment between lens and source is usually not perfect , no complete Einstein rings have been found around clus − t e r s . \quad But there are many examples known with spectacularly long arcs which are curved around the cluster center , with lengths up to about 20 arcseconds .

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 36 \noindentThe giantJ arcs . can Wambsganss be exploited\ inh f two i l l ways36 comma .. as is typical for many lens phenomena period Firstly they provide us with strongly magnified galaxies at open parenthesis very closing parenthesis \ hspacehigh redshifts∗{\ f i l period l }The .. giant These galaxies arcs can would be be exploited too faint to be in detected two ways or analysed , \quad in as is typical for many lens their unlensedJ . Wambsganss st ate period Hence with the lensing boost we can study these galaxies 36 \noindentin their earlyphenomena evolutionary . stFirstly agesThe comma giant they possiblyarcs provide can asbe infant exploited us with or proto in stronglytwo hyphen ways , galaxies magnified as is typical comma galaxiesfor relatively many lens at ( very ) highshortly redshifts afterphenomena the big . bang\ .quad Firstly periodThese they The providegalaxies other practical us with wouldstrongly application be magnified too of thefaint arcs galaxies to is to be at take ( detected very ) high or redshifts analysed in theirthem as unlensed tools. Theseto study st galaxies ate the potential . would Hence be and too with mass faint the distribution to be lensing detected of the boost or analysedlensing we galaxy can in their study unlensed these st ate galaxies . Hence incluster their periodwith early In the the evolutionary lensing simplest boost model we can st of a study ages spherically these , possibly galaxies symmetric in as their mass infant early distribution evolutionary or proto for st− agesgalaxies , possibly , as relatively shortlythe cluster afterinfant comma or the giant proto big arcs - galaxies bang form very . , relatively The close other to shortly the practical critical after curve the big application comma bang . which The other marks ofthe practical the arcs application is to take themEinstein as ring toolsof the period arcs to .. study is So to with take the the them potentialredshifts as tools of to the and study cluster mass the and potential distribution the arc and it is mass easy of distributionto the lensing of the galaxylensing clusterdetermine .galaxy a In rough the cluster estimate simplest . In of the the simplest model lensing model of mass a by of spherically a j sphericallyust determining symmetric symmetric the radius mass mass distribution distribution for the cluster for theof curvature cluster, giant and , interpretingarcsgiant form arcs very it asform close the to Einstein very the critical close radius curve to of the the, which lens critical system marks periodthe curve Einstein , ring which . So marks with the the EinsteinMore detailedredshifts ring modelling . of\quad the of cluster theSo lensing with and the clustersthe arc redshifts it which is easy allows toof determine for the the clusterasymme a rough hyphen estimate and the of thearc lensing it is mass easy to determinetry of theby mass a j roughust distribution determining estimate according the radiusof to the the of curvature lensing visible galaxies and mass interpreting plus by an j unknown ust it as determining the Einstein radius the of radius the lens ofdark curvature mattersystem component and . interpreting provides more accurate it as the determinations Einstein for radius the total of the lens system . cluster mass andMore its detailed exact distribution modelling of period the lensing More than clusters once which this detailedallows for modelling the asymme - try of the mass Morepredicted detaileddistribution additional modelling open according parenthesis of to the the counter lensing visible galaxies hyphen clusters plusclosing an which parenthesis unknown allows dark images matter for of thegiant component asymme arcs comma provides− which later were foundtry and of themore mass accurate distribution determinations according for the total to cluster the visible mass and its galaxies exact distribution plus an . unknown More than darkconfirmed matteronce spectroscopically component this detailed open providesmodelling square more bracket accurate 76 comma 39determinations closing square bracket for period the total clusterGravitational masspredicted lensing and additional itsis the exact third ( counter method distribution - for ) images the determination of . giant More arcs than of , which masses once later of this were founddetailed and modelling galaxy clustersconfirmed comma spectroscopically complementary [to 76 the , 39 mass ] . determinations by X hyphen ray analysis \noindentand the oldpredicted artGravitational of using additionalthe lensing virial theorem is the ( third counter and method the velo− for) city images the distribution determination of giant of ofmasses arcs , of which galaxy clusters later , were found and the galaxiescomplementary open parenthesis to the the mass latter determinations two methods use by X assumptions - ray analysis of hydrostaticand the old or art virial of using the virial \noindentequilibriumtheoremconfirmed comma and respectively the spectroscopically velo city closing distribution parenthesis of[ 76 period , 39 .. Although ] . there are still some discrepancies between the three methodsthe galaxies comma ( the .. latter it appears two methods that in relaxed use assumptions galaxy clusters of hydrostatic the agreement or virial Gravitationalbetween theseequilibrium different lensing , mass respectively is determinations the third) . Although method is very good there for open arethe square still determination some bracket discrepancies 8 closing of square between masses bracket the of three period galaxySome general clustersmethods results , , from complementaryit appears the analysis that in of relaxed giant to the arcs galaxy mass in galaxy clusters determinations clusters the agreement are : by X − ray analysis andClusters the of oldbetween galaxies art these areof dominated using different the mass by virial dark determinations matter theorem period is very ..and The good the typical [ 8velo ] ... quotedblleft city distribution mass hyphen of to hyphen light ratios quotedblrightSome general for clusters results obtained from the from analysis strong of giant open arcs parenthesis in galaxy and clusters weak are comma : Clusters see ofbelow galaxies closing parenthesis lensing\noindent analysesarethe dominated galaxies by (dark the matter latter . The two typical methods “ mass use - assumptions to - light of hydrostatic or virial are M slashratios L greater ” for clusters equal 1 0obtained M sub odot from slash strong L ( sub and odot weak open , see square below bracket ) lensing 65 analyses closing square bracket period \noindentThe distributionequilibrium of the dark , matter respectively follows roughly ) . the\quad distributionAlthough of the there are still some discrepancies between thelight three in the methodsgalaxies comma , \quad in particularit appears in the central that partin relaxed of the cluster galaxy period clusters .. The fact the agreement areM/L ≥ 10M /L [65]. that we see such arcs shows that the central surface mass density in clusters \noindent between these different mass determinations is very good [ 8 ] . must be highThe period distribution The radii of of the curvature dark matter of many follows giant roughly arcs is the comparable distribution to their of the light in the galaxies dist ance to, in the particular cluster centers in the semicolon central part this of shows the cluster that core . radii The of fact clusters that we endash see such the radii arcs shows that Someat which general the mass results profile from of the clusterthe analysis flattens towards of giant the center arcs endash in galaxy must be clusters are : Clustersthe of central galaxies surface are mass dominated density in clusters by dark must matter be high . .\ Thequad radiiThe of curvature t y p i c a l of\quad many giant‘ ‘ mass − to − l i g h t of order thisarcs distance is comparable or smaller to their period dist For ance stronger to the constraints cluster centers detailed ; this modelling shows that of core radii of clusters the mass distribution is required period \noindent –ratios the radii ’’ at which for clusters the mass profile obtained of the cluster from flattens strong towards ( and the weak center , – see must below be of order ) lensing analyses In Figuresthis 1 6 distanceand 1 7 two or smaller of the most . For spectacular stronger constraints cluster lenses detailed producing modelling of the mass distribution is arcs can be seen : Clusters Abell 22 1 8 and CL 24 plus 1 654 period Close inspection of the \ begin { a l i grequired n ∗} . HST image of Abell 22 1 8 revealsIn Figures that the 1 giant6 and arcs 1 7 twoare resolved of the most open spectacular parenthesis cluster Figure lenses 1 6 closing producing parenthesis comma arestructure M can / be seenL in\geq the individual1 0 components M {\odot .. open} square/L bracket{\ ?odot closing} square[ bracket 65 ]and used . for detailed \end{ a l i g narcs∗} can be seen : Clusters Abell 22 1 8 and CL 24 + 1654. Close inspection of the mass modelsHST of image the lensing of Abell cluster 22 1 period 8 reveals .. In that addition the giant to the arcs giant are resolved arcs comma ( Figure more 1 than 6 ) , 1 0 smallerstructure .. quotedblleft can be arclets seen in quotedblright the individual .. cancomponents be identified [?] in and Abell used 22 for 1 8 detailed period .. mass They models are farther of away Thefrom distribution the lens center of and the hence dark are not matter magnified follows and curved roughly as much the as the distribution of the light inthe the lensing galaxies cluster , . in In particular addition to the in giant the arcs central , more than part 1 0 of smaller the cluster “ arclets . ”\quad can The f a c t few giant arcsbe identified period .. in These Abell arclets 22 1 8 are . all They slightly are distorted farther away images from of backgroundthe lens center and hence are not thatgalaxies we period see such .. With arcs the cluster shows mass that model the it central is possible surface to predict mass the redshift density in clusters must be highmagnified . The and radii curved ofas much curvature as the few of giant many arcs giant . These arcs arclets is comparable are all slightly to distorted their Living Reviewsimages in of Relativity background open galaxies parenthesis . 1998 With hyphen the cluster 12 closing mass parenthesis model it is possible to predict the disthttp ance: slash to slash the www cluster period l ivingreviews centers ; period this org shows that core radii of clusters −− the r a d i i at whichredshift the mass profile of the cluster flattens towards the center −− must be of orderLiving this Reviews distance in Relativity or smaller( 1998 . For- 12 stronger) constraints detailed modelling of the masshttp distribution : / / www is . required l ivingreviews . . org

\ hspace ∗{\ f i l l } In Figures 1 6 and 1 7 two of the most spectacular cluster lenses producing

\noindent arcs can be seen : Clusters Abell 22 1 8 andCL $ 24 + 1 654 . $ Close inspection of the

\noindent HST image of Abell 22 1 8 reveals that the giant arcs are resolved ( Figure 1 6 ) ,

\noindent structure can be seen in the individual components \quad $ [ ? ] $ and used for detailed mass models of the lensing cluster . \quad In addition to the giant arcs , more than 1 0 s m a l l e r \quad ‘‘ arclets ’’ \quad can be identified in Abell 22 1 8 . \quad They are farther away from the lens center and hence are not magnified and curved as much as the few giant arcs . \quad These arclets are all slightly distorted images of background g a l a x i e s . \quad With the cluster mass model it is possible to predict the redshift

\noindent Living Reviews in Relativity ( 1998 − 12 )

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\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

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The properties statistical shows that incoherentof weak the lensing sources can be in both two a blessing different ways9 sub : A The welland hyphenangular a curse known for size astronomers exception of extended is: the The light statistical objects deflection incoherent is at changed the solar limb and comma the apparent where the difference bright − between the lensed and the unlensed9A well positions - known exception of stars is was the used light deflection to confirm at the General solar limb Relativity , where the comma difference \noindentsee Chapterbetweenness 2 period ofthe lensed a source and the unlensedis affected positions of , stars as was used first to confirm formulated General Relativity in 1 , 967 by Gunn [ 57 , 58 ] . Living Reviewssee Chapter in Relativity 2 . open parenthesis 1998 hyphen 12 closing parenthesis A weakhttp : lensingslash slash effect www period can l ivingreviews be a small period deformation org Living of the Reviews shape in Relativity of a cosmic( 1998 - 12 ) object , or a small modification of its brightnesshttp : / , / or www a . small l ivingreviews change of . its org position . In general the latter cannot be observed , since we have no way of knowing the

\noindent unaffected $ position ˆ{ 9 } . $

The first two effects −− slight shape deformation or small change in brightness −− in general cannot be determined for an individual image . Only when averaging over a whole ensemble of images it is possible to measure the shape distortion ,

\noindent since the weak lensing ( due to mass distributions of large angular size ) acts as the coherent deformation of the shapes of extended background sources .

The effect on the apparent brightness of sources shows that weak lensing can be both a blessing and a curse for astronomers : \quad The statistical incoherent

\ hspace ∗{\ f i l l } $ 9 { A }$ w e l l − known exception is the light deflection at the solar limb , where the difference

\noindent between the lensed and the unlensed positions of stars was used to confirm General Relativity ,

\noindent see Chapter 2 .

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 40 \noindentlens hyphenJ induced . Wambsganss change of\ theh f apparent i l l 40 brightness of open parenthesis widely separated closing parenthesis .... quot- edblleft standard \noindentcandles quotedblrightl e n s − induced endash like change type Ia of supernovae the apparent endash affects brightness the accuracy of of(the widely determination separated ) \ h f i l l ‘ ‘ standard of cosmologicalJ . 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The $[ first 53 attempts , go ? back , to the 1 years 63 ] .$ 1 978 slash 79The comma idea when to use Tyson the weak and distortionhis group triedand tangential to measure alignment the positions of faint and back orien - hyphenground galaxies Thet ations idea of toto the map use then the the newly mass weak discovered distribution distortion faint of blueforeground and galaxies tangential galaxies comma and which alignment clusters were hassuspected been of faint floating back around− for a groundto be at galaxies largelong distances timeto . 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There are ) sites a . number\ withquad very ofOnly reasons good with seeing for that smaller conditions : CCD period pixels , improved seeing con − ditions2 period ..at The the availability telecope1 . The of availability large sites high and resolution of astronomical improved cameras image sites with with qualityfields very of good view of seeing of the half conditions telescope . optics thea degree weak at lensing the moment effect open2 . parenthesis could The availability ultimately aiming of for large more be high measured closing resolution parenthesis . cameras period with fields of view of half 3 period .. The availability of methodsa degree to analyse at the these moment coherent ( aiming small for distortions more ) . period \ hspace4 period∗{\ ..f The i l l } awarenessWeak3 . lensing The of both availability observers is one of methods and of thetime to allotwo analyse cation sub these− committeesdisciplines coherent about small within distortions the . field of gravitational the potential of these weak lensing4 . The analyses awareness for extragalactic of both observers research and and time allo cation committees about \noindentcosmologylensing periodthe potential with of the these highest weak lensing rate analyses of growth for extragalactic in the research last couple and cosmology of years . ( along with Now we will brieflyNow we summarize will briefly the summarize technique the of techniquehow to use of the how weak to use lensing the weak lensing distortion in order \noindentdistortionto ingalactic getorder the to mass get microlensing the distribution mass distribution of ) the . underlying There of the underlyingare matter a number . matter of period reasons for that : 4 period 54 period . 5 . 1 1 .. Cluster Cluster mass mass reconstruction reconstruction \ centerlineThe first realThe{1 detection first . \ realquad ofdetection aThe coherent availability of a weak coherent lensing weak of signal lensing astronomical of distorted signal of back distorted sites hyphen back with - ground very good galaxies seeing was conditions . } ground galaxiesmeasured was in measured 1 990 around in 1 990 the around galaxy the clusters galaxy Abell clusters 1 689 Abell 1 689 \ hspaceand CL∗{\ 1409andf i plus l lCL}2 52 1409 . open\ +quad square52[152]The. bracketIt availability was 1 shown 52 closing that square the of orientation large bracket high period of background resolution It was shown galax that cameras - ies the – orientation the with angle fields of background of view of half galax hyphenof the semi - major axes of the elliptical isophotes relative to the center of the cluster – was \ centerlineies endashmore the{a angle likelydegree of to the be at semi tangentially the hyphen moment major oriented ( axes aiming relative of the for to elliptical the more cluster isophotes ) . than} relative radially to .For the an unaffected center of thepopulation cluster endash of background was more galaxies likely to one be would tangentially expect oriented no preferential relative direction to the . This analysis \ centerlinecluster thanis{ basedradially3 . \ onquad period the assumpThe For an availability - unaffected t ion that population the of major methods axesof background of to the analyse background galaxies these one galaxies coherent are intrinsically small distortions . } would expectrandomly no preferential oriented direction. period .. This analysis is based on the assump hyphen \ hspacet ion that∗{\ thef i lWith major l }4 the . axes\ elegantquad of theThe and background powerful awareness galaxiesmethod of developedare both intrinsically observers by Kaiser randomly and and Squires time [?] allo the weak cation lensing committees about oriented periodsignal can be used to quantitatively reconstruct the surface theWith potential the elegantLiving of Reviews and these powerful in weak Relativity method lensing developed( 1998 analyses - 12 by Kaiser) for and extragalactic Squires open square research bracket ? and closing square bracket cosmologythe weak lensinghttp . : signal / / can www be . used l ivingreviews to quantitatively . reconstruct org the surface Living Reviews in Relativity open parenthesis 1998 hyphen 12 closing parenthesis Nowhttp we : willslash slash briefly www period summarize l ivingreviews the technique period org of how to use the weak lensing distortion in order to get the mass distribution of the underlying matter .

\noindent 4 . 5 . 1 \quad Cluster mass reconstruction

\noindent The first real detection of a coherent weak lensing signal of distorted back − ground galaxies was measured in 1 990 around the galaxy clusters Abell 1 689

\noindent andCL $ 1409 + 52 [ 1 52 ] . $ It was shown that the orientation of background galax − i e s −− the angle of the semi − major axes of the elliptical isophotes relative to the center of the cluster −− was more likely to be tangentially oriented relative to the cluster than radially . For an unaffected population of background galaxies one would expect no preferential direction . \quad This analysis is based on the assump − t ion that the major axes of the background galaxies are intrinsically randomly o r i e n t e d .

With the elegant and powerful method developed by Kaiser and Squires $ [ ? ] $ the weak lensing signal can be used to quantitatively reconstruct the surface

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 4 1 .. Gravitational Lensing in Astronomy \noindentmass distribution4 1 \quad of theGravitational cluster period .. This Lensing method in relies Astronomy on the fact that the con hyphen vergence kappa open parenthesis theta closing parenthesis and the two components of the shear gamma 1 open parenthesis theta\noindent closing parenthesismass distribution comma gamma of 2 the open cluster parenthesis . theta\quad closingThis parenthesis method are relies linear oncom the hyphen fact that the con − binations of4 1 the Gravitational second derivative Lensing of the in effectiveAstronomy lensing potential Capital Psi open parenthesis theta closing parenthesis open\noindent parenthesismassvergence cf distribution period $ \kappa of the cluster( .\ Thistheta method)relies $ and on the the fact two that components the con - of the shear $ \gamma 1Equations ( \thetavergence open parenthesisκ),(θ) and 33 the comma\gamma two components 34 comma2 35 ( of closing the\theta shear parenthesisγ1(θ), γ $2( closingθ) are are parenthesis linear linear com com period - − After Fourier transforming the expressionsbinations for the conof the hyphen second derivative of the effective lensing potential Ψ(θ) ( cf . \noindentvergence andEquationsbinations the shear ( 33 one of , 34 obtains the , 35 )second ) linear . After relations derivative Fourier between transforming of the the transformed the effective expressions for lensing the con - potential vergence and $ \Psi ( \thetacomponents)the kappa-tilde shear ( $ one c f obtainssub . comma linear tilde-gamma relations between 1 comma the gamma-tilde transformed 2 components period .. Solvingκ ˜,γ˜1, γ˜2 for. tilde-kappaSolving and inverse Fourier transformingforκ ˜ and gives inverse an Fourier transforming gives an \noindentestimate forestimateEquations the convergence for the ( convergence33 kappa , 34 open ,κ 35( parenthesis details ) ) can . 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[ 89 , \ centerlineendash weak{ lensing$ \ bullet effect by$ large\quad scaleGoing structure further and slash . or\quad detectionWe of could dark present here only one weak lensing issue in } matter concentrations semicolon .. this considers both shear92, ?]). effects as well as \ centerline {some detail : the reconstruction of the mass distribution of galaxy clusters } magnification effectsAn open upcoming parenthesis comprehensive e period g review period on open weak square lensing bracket by Schneider 1 3 comma & Bartel 1 7 - comma 1 33 comma 1 68 closing squaremann bracket [ 1 closing 34 ] treats parenthesis both theory semicolon and applications of weak lensing in great depths . \ centerlineendash determination4{ .from 6 weakly Cosmological of the power distorted spectrum aspects background of the of matter ( strong images distribution ) . \quad lensing openMany parenthesis more e interesting period g period weak open square} bracket 2 1 closingGravitational square bracket lenses can closing be used parenthesis in two different semicolon ways to study the cosmo - logical parameters of \ centerline { lensing applications are under theoretical and observational investigation , } or the universe . The first is to explore a particular lens endash the weak lensing effects on theLiving cosmic Reviews microwave in Relativity background( 1998 open -parenthesis 12 ) e period g period open square bracket \ centerline {though . To name just a few : } 89 comma http : / / www . l ivingreviews . org 92 comma ? closing square bracket closing parenthesis period \ centerlineAn upcoming{−− comprehensiveConstraints review on on the weak distribution lensing by Schneider of the ampersand faint Bartel galaxies hyphen from weak lensing } mann open square bracket 1 34 closing square bracket treats both theory and applications of weak lensing in great \ centerlinedepths period{(e.g.,[51,86]); } 4 period 6 .. Cosmological aspects of open parenthesis strong closing parenthesis .. lensing \ centerlineGravitational{−− lensesgalaxy can be− usedgalaxylensing(e.g in two different ways to study $. the cosmo , hyphen [ ? ] ) ;$ } logical parameters of the universe period .. The first is to explore a particular lens \ centerlineLiving Reviews{−− inlensing Relativity by open galaxy parenthesis halos 1998 in hyphen clusters 12 closing ( e parenthesis . g . , [ 98 ] ) ; } http : slash slash www period l ivingreviews period org \ centerline{−− weak lensing effect by large scale structure and / or detection of dark }

\ centerline { matter concentrations ; \quad this considers both shear effects as well as }

\ centerline { magnification effects ( e . g . [ 1 3 , 1 7 , 1 33 , 1 68 ] ) ; }

\ hspace ∗{\ f i l l }−− determination of the power spectrum of the matter distribution ( e . g . [ 2 1 ] ) ;

\ centerline { or }

\ centerline{−− the weak lensing effects on the cosmic microwave background ( e . g . [ 89 , }

\ [ 92 , ? ] ) . \ ]

\ centerline {An upcoming comprehensive review on weak lensing by Schneider $ \& $ Bartel − }

\noindent mann [ 1 34 ] treats both theory and applications of weak lensing in great depths .

\noindent 4 . 6 \quad Cosmological aspects of ( strong ) \quad l e n s i n g

\noindent Gravitational lenses can be used in two different ways to study the cosmo − logical parameters of the universe . \quad The first is to explore a particular lens

\ centerline { Living Reviews in Relativity ( 1998 − 12 ) }

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Determine3 . matter Determine content the of relativeuniverse frequency Capital Omega of lensed sub objects compact , comma the distribution typical mass of scale M sub l ens comma f ocosmological l l o w i n g constant : Capital Omega sub Capital Lambda comma by comparison with theoretical models slash simulations splitting angles ∆θ as a function of lens and source redshifts zL/zS. period\end{ a l i g n ∗} 4 . Determine matter content of universe Ωcompact, typical mass scale Mlens, Since quasars are rare objects and lensing is a relatively rare phenomenon comma \ centerline {1 . \quad cosmologicalDo a systematic constant Ω studyΛ, by comparison of a sample with theoretical of high − modelsredshift / simulations objects . : quasar surveys . } steps 1 and 2 are quiteSince difficult quasars and are t imerare hyphen objects consuming and lensing period is a relatively Nevertheless rare phenomenon comma a number , of systematic quasar surveys with the goal to find open parenthesis many closing parenthesis lens systems with well \ centerlinesteps{2 1. and\quad 2 areIdentify quite difficult the and gravitational t ime - consuming lens . Nevertheless systems ,among a number them of systematic . } defined selectionquasar criteria surveys have with been the goal done to in find the past ( many and ) others lens systems are underway with well defined selection criteria right now open parenthesis e period g period open square bracket 28 comma ? comma 88 comma ? comma 1 73 closing \ centerlinehave{3 been . \quad done inDetermine the past and the others relative are underway frequency right now of ( e lensed . g .[28, objects?, 88, ?, ,173])\quad. the distribution of } square bracketThe closing largest parenthesis survey so period far , the CLASS survey , has looked at about 7000 radio sources at the The largestmoment survey so ( far the comma goal is 1 .. 0 the ) . CLASS In total survey CLASS comma found .. 1 has 2 new looked lens systems at about so 7000 far . Interestingly \ centerlineradio sources{ splitting at the moment angles .. open parenthesis $ \Delta the\ goaltheta is 1 0$ closing as a parenthesis function period of lens .. In total and CLASS source found redshifts 1 2 $ z { L },/ all the z lenses{ S have} . small $ separations} (∆θ < 3 arcsec ) , and all lensing galaxies are detected new lens systems[ 28 , 63 so ] far . That period leaves .. Interestingly little space comma for a population .. all thelenses of dark have objects small with separations masses of galaxies or open parenthesisbeyond Capital . A Delta theta less 3 arcsec closing parenthesis comma and all lensing galaxies are detected open square\ centerline bracket{ 284 comma . \quad 63 closingDetermine square bracketmatter period content That ofleaves universe little $ \Omega { compact } , $ typical mass scale $ M { ldetailed ens } discussion, $ } of lens surveys and a comparison between optical and radio surveys can be space for afound population in [?]. of dark objects with masses of galaxies or beyond period .. A detailed discussion of lens surveys and a comparison between optical and radio 2 \ hspace ∗{\Thef i l l idea} cosmological for the determination constant of the cosmological$ \Omega constant{\Lambda ΩΛ =} Λ/(3,H $0 ) from by comparison lens st atistics with theoretical models / simulations . surveys canisbe based found onin the open fact square that the bracket relative ? closinglens probability square bracket for multiple period imaging increases rapidly with The idea for the determination of the cosmological constant Capital Omega sub Capital Lambda = Capital Lambda slash increasing ΩΛ( cf . Figure 9 of [?]). This was first pointed out 1 990 [ 54 , 1 50 ] . The reason is open\ centerline parenthesis{ Since 3 H sub quasars 0 to the power are ofrare 2 closing objects parenthesis and lensing is a relatively rare phenomenon , } the fact that the angular di - ameter distances DS,DL,DLS depend strongly on the cosmological from lens st atistics is based on the fact that the relative lens probability for \noindent modelsteps . And1 and the 2 properties are quite that determine difficult the and probability t ime for− multipleconsuming lensing . ( Nevertheless i . e . the “ frac , a number of multiple imaging- tional increases volume ” rapidly that is withaffected increasing by a certain Capital lens Omega ) depend sub on Capital these Lambdadistances open [?]. parenthesis cf period Figure 9systematic of open square quasar bracket ? surveys closing square with bracket the goal closing to parenthesis find ( period many This ) lens systems with well defined selection criteria haveLiving been Reviews done in Relativity in the past( 1998 and - 12 others) are underway was first pointed out 1 990 openhttp square : bracket / / www 54 comma . l ivingreviews 1 50 closing square . org bracket period The reason is the fact that therightnow(e.g$. angular di hyphen [ 28 , ? , 88 , ? , 1 73 ] ) .$ ameter distances D sub S comma D sub L comma D sub LS depend strongly on the cosmological model period And \noindentthe propertiesThe that largest determine survey the probability so far for , \ multiplequad the lensing CLASS open survey parenthesis , \ iquad periodhas e period looked the quotedblleft at about frac 7000 hyphenradio sources at the moment \quad ( the goal is 1 0 ) . \quad In total CLASS found 1 2 newtional lens volume systems quotedblright so far that . \ isquad affectedInterestingly by a certain lens , closing\quad parenthesisall the depend lenses on have these small distances separations open square bracket$ ( ? closing\Delta square\theta bracket period< 3 $ arcsec ) , and all lensing galaxies are detected [ 28 , 63 ] . That leaves little spaceLiving for Reviews a population in Relativity open of dark parenthesis objects 1998 with hyphen masses 12 closing of parenthesis galaxies or beyond . \quad A http : slash slash www period l ivingreviews period org \noindent detailed discussion of lens surveys and a comparison between optical and radio surveys canbefound in $ [ ? ] . $

\noindent The idea for the determination of the cosmological constant $ \Omega {\Lambda } = \Lambda / ( 3 H ˆ{ 2 } { 0 } ) $ from lens st atistics is based on the fact that the relative lens probability for multiple imaging increases rapidly with increasing $ \Omega {\Lambda } ( $ cf . Figure 9 of $[ ? ] ) .$This was first pointed out 1 990 [ 54 , 1 50 ] . The reason is the fact that the angular di − ameter distances $ D { S } ,D { L } ,D { LS }$ depend strongly on the cosmological model . And the properties that determine the probability for multiple lensing ( i . e . the ‘‘ frac − tional volume ’’ that is affected by a certain lens ) depend on these distances $ [ ? ] . $

\ centerline { Living Reviews in Relativity ( 1998 − 12 ) }

\ centerline { http : / / www . l ivingreviews . org } J period Wambsganss .... 44 \noindentThis can beJ seen . Wambsganss comma e period\ h g f i period l l 44 when one looks at the critical surface mass density re hyphen quired for multiple imaging open parenthesis cf period Equation open parenthesis 1 6 closing parenthesis closing parenthesis which\noindent dependsThis on the can angular be seen , e . g . when one looks at the critical surface mass density re − diameter distJ . Wambsganss ances period 44 \noindentThe consequencesThisquired can of be for lensing seen multiple , studies e . g . when on imaging the one cosmological looks ( cfat the . constant Equation critical can surface be ( 1 mass 6 ) density ) which re - depends on the angular diametersummarizedquired dist as follows forances multiple period . imaging .. The analyses ( cf . Equation of the frequency ( 1 6 ) ) which of lensing depends are based on the on angular diameter dist lens systemsances found . in different optical and radio surveys period .. The main problem is Thestill consequences the smallThe number consequences of of lensing lenses ofperiod lensing studies Depending studies on on theon the the cosmological exact selection constant criteria constant can comma be summarized can only be as follows summarizeda few lens. systems as The follows analyses can be included of . the\quad frequency in theThe analyses ofanalyses lensing period are of .. based Nevertheless the on frequency lens systems comma of onefound lensing can in use different are optical based on lensthe existing systemsand samples radio found surveys to put in limits different. The on themain cosmological optical problem is and constantstill radio the small period surveys number .. Two of . different lenses\quad .The Depending main on problem the is stillstudies the foundexact small 95 selection percent number criteria hyphen of , lenses confidence only a few . limitslens Depending systems of Capital can on be Omega the included exact sub in Capital the selection analyses Lambda . criteria less Nevertheless 0 period , only66 open square bracketa few ? lens closing, one systems square can use bracket the can existing andbe includedCapital samples Omega to in put sub the limits Capital analyses on the Lambda cosmological . \ lessquad 0 period constantNevertheless 7 open . Twosquare different , bracket one can ? comma use 1 2 1 closingthe existing squarestudies bracket samples found period 95% This to− confidence put limits limits on of the ΩΛ < cosmological0.66[?] and ΩΛ < constant0.7[?, 121]. This . \quad Two different is based on the assumption of a flat universe (Ωmatter + ΩΛ = 1). Investigations is based on the assumption of a flat universe open parenthesis Capital Omega00 sub matter plus Capital Omega sub Capital Lambda\noindent = 1 closingonstudies the parenthesis matter found content period $ 95 of Investigations the\ universe% − $ fromconfidence ( both “ macro limits− and of $ “ micro\Omega - ” ){\ lensingLambda } < 0 .on 66 the matter [generally ? content conclude ] $ of theand that universe$ the\Omega fractional from open{\ matterLambda parenthesis in compact} both< form quotedblleft0 cannot . exceed 7 macro a [ few quotedblright ?percent , of the 1 hyphen 2 .. 1 and ] .. quotedblleft. $ This microcritical hyphen density quotedblright ( e . g .[?, ?, closing99, 132]) parenthesis. lensing generally conclude4 . 7 that Galactic the fractional microlensing matter in compact form cannot exceed a \noindentfew percentItis of has based the been critical known on density the for assumption more open than parenthesis two of decades ae period flat that g universe halos period of open galaxies $square ( must bracket\Omega contain ? comma{ matter ? comma} 99+ comma\Omega 1 {\Lambda } =32 closing 1 square )some . bracket unknown$ Investigations closing kind parenthesis of dark matter period . Many different particles / objects had been suggested as 4 period 7constituents .. Galactic microlensing of this halo dark matter . The candidates can be divided into the two broad \noindentIt has beencategorieson known the for matter more“ elementary than content two particles decades of the ” that and universe halos “ of astronomical galaxies from must ( bodies both contain ” ‘‘ . A macro conservative $ ’’ candidate{ − }$ \quad and \quad ‘ ‘ micro − ’ ’ ) l e n s i n g generallysome unknownfor conclude this kind dark ofdark matter that matter arethe brown period fractional dwarfs .. Many , objectsmatter different with particles in masses compact slash less than objects form 0.8 cannot hadM beenso that exceed the central a fewpercentofthecriticaldensity(e.gsuggestedtemperature as constituents is not of this high halo enough .. dark to start matter helium period fusion $. .. The . candidates [ These ? objects can , be are ? certain , to 99 exist , we 1 32 ] )divided . $ intojust the do two not broad know categories how many .. there quotedblleft are . elementary particles quotedblright .. and .. quotedblleft astronomical bodies quotedblright period AIn conservative 1 986 Paczy candidaten ´ ski [?] for suggested this dark a method matter are to test brown observationally dwarfs comma whether objects \noindentwith massesthe4 less . Milky 7 than\quad Way 0 period haloGalactic is 8 madeM sub of microlensing odot such so brown that the dwarfs central ( or temperature other astronomical is not high ob - enough to start heliumj ects fusion in roughly period this .. mass These range objects ) . are Subsequently certain toexist this type comma of dark we just matter do not candi know - \noindenthow manydateIt there has was are labelledbeen period known “ Macho for ” formore MAssive than Compact two decades Halo Object that [?] halos. If one of could galaxies continuously must contain In 1 986 Paczyobserve n-acute the brightness ski open square of stars bracket of our ? neighbouring closing square galaxy bracket Large suggested Magellanic a method Cloud to( test LMC observationally ) one whether \noindentthe Milkyshouldsome Way halo unknownsee is typical made kindfluctuations of such of brown dark in some dwarfs matter of open these . parenthesis stars\quad dueMany to or the other factdifferent astronomical that every particles now ob and hyphen then / one objects had been suggestedj ects in roughlyof as these constituents this compact mass range halo objects closing of this passes parenthesis halo in front\ periodquad of thedark Subsequently star and matter magnifies this . type\ itsquad brightness of darkThe matter candidates . The candi only hyphen can be divideddate was labelledintoproblem the quotedblleft with two this broad experiment Macho categories quotedblright is the low\ forquad probability MAssive‘‘ elementary forCompact such an Halo event particles Object : Only open about square ’’ \ onequad bracket outand of ? closing\quad square‘‘ astronomical bracketbodies period ’’three If . oneA million conservative could LMC stars wouldcandidate be significantly for this magnified dark atmatter any given are time brown . dwarfs , objects withcontinuously masses observeThe less underlying thethan brightness $0scenario of . is stars very 8 of simple our M neighbouring{\ : Dueodot to} the galaxy$ relative so Large that motion the of central ob - server temperature , lensing is not high enough toMagellanic startMacho helium Cloud and open fusion source parenthesis star . \quad the LMC projectedThese closing impactobjects parenthesis parameter are one should certain between see typicalto lens exist and fluctuations source , we changes just in some withdo of not these know howstars many due totime there the and fact are produces that . every a time now dependent and then one magnifi of these - cation compact . If halo the objects impact parameter is smaller than passes in frontan Einstein of the star radius and then magnifies the magnification its brightness is µ periodmin > ..1 The.34( only cf . Equation problem with ( 22 ) ) . \ hspacethis experiment∗{\ f i lFor l } isIn an the extended 1 low 986 probability Paczy source such for $ such\ aacute sequence an{ eventn} is$ : illustrated Only ski about $ in [ one Figure out ? 1 of 9 ] for $ five suggested inst ants of a time method to test observationally whether three million. The LMC separation stars would of the be significantly two images is magnified of order at two any Einstein given time radii period when they are of comparable \noindentThe underlyingmagnificationthe Milkyscenario Way , is which very halo corresponds simple is : made .. Due to only of to the aboutsuch relative a brown milliarcsecond motion dwarfs of ob . (hyphen Hence or other the two astronomical images cannot ob − server commabe resolved lensing individually Macho and source , we can star only the observe projected the impact brightness parameter of the combined between image pair . This \noindentlens and sourceisj i ects llustrated changes in in roughly with Figures time 20 and this and produces 2mass 1 which rangea time show dependent ) the . relative Subsequently magnifi tracks hyphen and thisthe respective type of light dark curves matter candi − cation periodLiving .. If Reviews the impact in Relativity parameter( is1998 smaller - 12 than) an Einstein radius then the \noindentmagnificationhttpdate is : mu was / min / labelled www greater . l1 period ivingreviews ‘‘ Macho 34 open ’’ .parenthesis for org MAssive cf period Compact Equation Halo open Object parenthesis $ [ 22 closing ? ]parenthesis . $ closingI f one parenthesis could period continuouslyFor an extended observe source such the a sequence brightness is illustrated of stars in Figure of 1 our 9 for neighbouring five galaxy Large Magellanicinst ants of time Cloud period ( LMC The separation ) one should of the two see images typical is of order fluctuations two Einstein in radii some of these starswhen they due are to of the comparable fact that magnification every now comma and which then corresponds one of tothese only about compact a halo objects passesmilliarcsecond in front period of .. theHence star the two and images magnifies cannot be its resolved brightness individually . comma\quad weThe can only problem with thisonly observeexperiment the brightness is the of low the combined probability image pair for period such .. an This event is i llustrated : Only in about one out of threeFigures million 20 and 2 LMC 1 which stars show would the relative be significantlytracks and the respective magnified light curves at any given time . Living Reviews in Relativity open parenthesis 1998 hyphen 12 closing parenthesis Thehttp underlying : slash slash scenario www period is l ivingreviews very simple period : org\quad Due to the relative motion of ob − server , lensing Macho and source star the projected impact parameter between lens and source changes with time and produces a time dependent magnifi − c a t i o n . \quad If the impact parameter is smaller than an Einstein radius then the magnification is $ \mu $ min $ > 1 . 34 ($ cf .Equation(22)).

For an extended source such a sequence is illustrated in Figure 1 9 for five inst ants of time . The separation of the two images is of order two Einstein radii when they are of comparable magnification , which corresponds to only about a milliarcsecond . \quad Hence the two images cannot be resolved individually , we can only observe the brightness of the combined image pair . \quad This is i llustrated in Figures 20 and 2 1 which show the relative tracks and the respective light curves

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 45 .... Gravitational Lensing in Astronomy \noindentfor five values45 of\ h the f i l minimum l Gravitational impact parameter Lensing u sub in minimum Astronomy period Figure 1 9 : .. Five snapshots of a gravitational lens s ituatio n : .. From left to right \noindentthe alignmentfor between five values lens and sof ource the ge minimum ts be tt er impact and better parameter comma until $ it u is perfect{\min } . $ in the rightmost45 panel period .. This results in the image of an .. quotedblleft Gravitational Einstein Lensing ring quotedblright in Astronomy period \noindentQuantitativelyforFigure five comma values 1 9 the of : total the\quad minimum magnificationFive impact snapshots mu parameter = mu 1of plusu amin mu gravitational. 2 of the two images lens open s ituatio parenthesis n cf : period\quad From left to right theEquation alignment openFigure parenthesis between 1 9 : Five 22 lens closing snapshots and parenthesis s of ource a gravitational closing ge ts parenthesis be lens tt s ituatio er entirely and n : better depends From , on left until the to impact right it is theparameter perfect u open parenthesisin the rightmost talignment closing parenthesis between panel lens = . r\quad openand s parenthesis ourceThis ge results ts t be closing tt er in and parenthesis the better image , untilslash of it R is suban perfect\ Equad be hyphen in‘‘ the Einsteinrightmost ring ’’ . tween thepanel lensed . star This and theresults lensing in the object image comma of an measured “ Einstein in the ring lens ” .plane open parenthesis here \ hspaceR sub∗{\ E isf the i l l Einstein} Quantitatively radiusQuantitatively of the lens , the comma , total the i total period magnification magnification e period theµ radius= $µ1\ + atmuµ which2 of= the a circular two\mu images image1 ( cf + . \mu 2 $ ofappears the two forEquation imagesperfect alignment ( (22 cf ) ) entirely . between depends source on comma the impact lens and parameter observeru comma(t) = r(t cf)/R periodE be Figure- 1 9 comma rightmosttween panel closingthe lensed parenthesis star and : the lensing object , measured in the lens plane ( here \noindentEquation:REquation openE is parenthesis the Einstein ( 22 43 radius) closing ) entirely of parenthesis the lens depends , i .. . mu e . open the on radius parenthesis the impact at which u closing parameter a circular parenthesis image $ appearsu = u to ( the for t power ) of 2 = plus 2r u radicalbig ( tperfect of ) open alignment / parenthesis R { betweenE u to}$ the source be power− , lens of 2 and plus observer 4 closing , cf parenthesis . Figure 1 sub 9 , period The time scalerightmost of such panel a quotedblleft ) : microlensing event quotedblright is defined as the time it takes the \noindentsource to crosstween the the Einstein lensed radius star : and the lensing object , measured in the lens plane ( here Equation: open parenthesis 44 closing parenthesis .. t sub 0 = R v sub bottom to the power of E thickapprox 0 period 2 1 µ(u) = u2 + 2up(u2 + 4) (43) 4\noindent yr radicalBigg$ of R M{ subE odot}$ to is the the power Einstein of M radicalbigg radius of of 1 0 theto the. lens power , of i D . sub e kp . thec to the radius power at of L which minus 1 a D circular image appears for perfect alignment between source , lens and observer , cf . Figure 1 9 , sub D sub S toThe the time power scale of ofL Row such 1 a v“ submicrolensing bottom Row event 2 ”200 is definedkm slash as sec the . time minus it 1 takes period the source to cross Here v subthe bottom Einstein is radius the open : parenthesis relative closing parenthesis transverse velocity of the lens period .... We parameterized\noindent rightmost the panel ) : time scale by quotedblleft typical quotedblright numbers for the distances of lensed and lensing star and \ begin { a l i g n ∗} q q   the relative transverse velocityE period Note alsoM that hereDL we used the simple relationv⊥ t = R 0.214yr M 10 − 1 D L − 1. (44) \mu ( u ) =0 uv⊥ ˆ{≈ 2 } + 2 { u kpc\ sqrt { D( u ˆ{ 2 } + 4 ) }} { . }\ tag ∗{$ ( D sub LS = D sub S minus D sub L open parenthesis which is notS valid200km for cosmological/ sec distances closing parenthesis period43 ) $} \endNote{ a l that i g nHere∗} fromv Equation⊥ is the ( open relative parenthesis ) transverse 44 closing velocity parenthesis of the lens it . is obvious that it We is not parameterized possible to thedetermine the mass oftime the scale lens from by “ one typical individual ” numbers microlensing for the distances event period of lensed .. The and duration lensing of star and the relative \noindentan event istransverseThe determined time velocity scale by three . Noteof unknown such also that a parameters ‘‘ here microlensing we : used .. the the mass simple event of the relation lens ’’ comma is defined as the time it takes the sourcethe transverse toD crossLS velo= DS city the− D andL Einstein( which the distances is not radius valid of lens for : and cosmological source period distances .. It is ) . impossible to disentangleNote these that for fromindividual Equation events ( 44 period ) it is .. obvious Only with that a it model is not forpossible the spatial to determine the mass of \ beginand velocity{ a l i gthe n distribution∗} lens from one of the individual lensing objectsmicrolensing and comparison event . The with duration .. quotedblleft of an event simulated is determined by t microlensing{ 0 } three= events unknown R quotedblright{ v parameters{\bot .. it :}} isˆ possible{ theE mass}\ to of obtainthickapprox the lens information , the transverse 0about . the velo masses 2 city 1 and of the 4 distances yr \ sqrt { M ˆ{ M } {\odot }}\ sqrt { 1 0 ˆthe{ D lensing}ˆ{of objectsL lens} { and andkp source their c .density}} It − is period impossible1 D to{ D disentangle ˆ{ L } these{ S for}}\ individuall e f t (\ begin events{ .array Only}{ withc} v {\bot }\\ 200What km seemeda / model to be\sec for an the impossible\end spatial{ array and task}\ velocity atright the t distribution) ime− endash1 of namely the . \ lensingtag determine∗{$ objects ( the and 44 comparison ) $} with “ \endbrightness{ a l i g nsimulated∗} of millions microlensing of stars on an events almost ” nightly it is possible basis endash to obtain was turned information into three about the masses of the Living Reviewslensing in objects Relativity and open their parenthesis density . 1998 hyphen 12 closing parenthesis \noindenthttp : slashHere slashWhat www $ seemed v period{\ tobot l be ivingreviews an}$ impossible is the period task ( org relative at the t ime ) transverse– namely determine velocity the brightness of the of lens . \ h f i l l We parameterized the millions of stars on an almost nightly basis – was turned into three \noindent time scale by ‘‘ typical ’’ numbers forLiving the Reviews distances in Relativity of lensed( 1998 and - 12 lensing) star and the relative transverse velocity . Note alsohttp that : here/ / www we . used l ivingreviews the simple . relation org

\noindent $ D { LS } = D { S } − D { L } ( $ which is not valid for cosmological distances ) .

Note that from Equation ( 44 ) it is obvious that it is not possible to determine the mass of the lens from one individual microlensing event . \quad The duration of an event is determined by three unknown parameters : \quad the mass of the lens , the transverse velo city and the distances of lens and source . \quad It is impossible to disentangle these for individual events . \quad Only with a model for the spatial and velocity distribution of the lensing objects and comparison with \quad ‘‘ simulated microlensing events ’’ \quad it is possible to obtain information about the masses of the lensing objects and their density .

What seemed to be an impossible task at the t ime −− namely determine the brightness of millions of stars on an almost nightly basis −− was turned into three

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 46 \noindentFigure 20 :J .... . Five Wambsganss .... re lative\ h .... f i tracks l l 46 .... between .... background s tar and foreground lens open parenthesis indicated as the central s tar closing parenthesis parametrized by the impact parameter u sub minimum period\noindent .... TheFigure 20 : \ h f i l l Five \ h f i l l re l a t i v e \ h f i l l t r a c k s \ h f i l l between \ h f i l l background s tar and foreground lens dashed linJ e . indicates Wambsganss th e Einstein ring for the lens open parenthesis after open square bracket ? closing46 square bracket closing\noindent parenthesisFigure( indicated period 20 : Five as the re lative central tracks s tar ) between parametrized background by thes tar impactand foreground parameter lens $ u {\min } . $big\ observationalh f i l( l indicatedThe campaigns as the central within s few tar years ) parametrized open parenthesis by the impact MACHO parameter comma EROSumin. comma OGLEThe exper hyphen iments closingdashed parenthesis lin e indicates period th .. e These Einstein groups ring looked for the at lens millions ( after of[?]) stars. in the LMC and towards the \noindentbulge of thebigdashed Milky observational Way lin comma e campaigns indicates and their within thfirst efew results Einstein years appeared ( MACHO ring essentially , for EROS the simultane , OGLE lens exper hyphen ( after - $ [ ? ] ) . $ously in theiments fall of ) . 1 993 These open groups square looked bracket at 5 millions comma of 1 stars 2 comma in the 1 LMC54 closing and towards square bracket the bulge period of the In the meantime more groups haveMilky j oined Way , and their first results appeared essentially simultane - \noindentthis effortously commabig inobservational some the fall of of them 1 993 with [campaigns 5 special , 1 2 , 1 emphases 54 within ] . In :the .. few emeantime period years g more period ( MACHO groups on covering have , EROS j ongoingoined , OGLE this mi effort hyphen exper , − crolensingsome events of open them parenthesis with special PLANET emphases comma : e . DUOg . on closing covering parenthesis ongoing mi comma - or on extending the microlensing search\noindent to crolensingiments ) events . \quad ( PLANETThese , DUO groups ) , or looked on extending at millions the microlensing of stars search in to the LMC and towards the bulgeunresolved ofunresolved the stars Milky open stars parenthesis Way ( “ , pixel and quotedblleft lensing their ” ) first in pixel the lensingAndromeda results quotedblright appeared galaxy ( AGAPE closing essentially parenthesis ) [36, ?] simultane, in the Andromeda− galaxy open parenthesisor to AGAPE cover the closing Magellanic parenthesis Clouds open completely square bracket around 36 the comma year ( ? MOA closing ) . squareHere bracket comma \noindentor to coverisously the a list Magellanic of in groups the Clouds currently fall completely of active 1 993 in around the [ 5 search , the 1 yearfor 2 microlensing , open 1 54 parenthesis ] . signatures In MOA the of meantime closing compact parenthesis objects more in groups period Here have j oined thisis a list effort ofthe groups halo , some currently of the of Milky active them Way in with the or elsewhere search special for : microlensing emphases signatures : \quad ofe . g . on covering ongoing mi − compact objects in the halo• ofMACHO the Milky ( WayMAssive or elsewhere Compact : Halo Object ) [ 6 , 7 , 1 67 ] . \noindentbullet .. MACHOcrolensing open• EROS parenthesis events ( Experience ( MAssive PLANET de Recherche Compact , DUO ) d Halo ’ , Objets or Object on Sombres extendingclosing ) parenthesis [ 1 0 , the 1 1 9 microlensing open , 45 ] square . bracket search 6 comma to 7 comma 1 67 closing square• bracketOGLE ( period Optical Gravitational Lens Experiment ) [ 1 56 , 1 53 , 1 49 ] . \noindentbullet EROSunresolved open parenthesis• starsAGAPE Experience ( ( ‘‘Andromeda pixel de Recherche lensing Galaxy and d ’’ quoteright Amplified ) in the Pixels Objets Andromeda Experiment Sombres closing galaxy ) [ 1 1 parenthesis , 1( 47 AGAPE ] . open $ ) square [ bracket36 , 1 0 comma ? ] 1 1 9 , comma $ • 45MOA closing ( MACHO square bracket Observations period in Astrophysics ) [ 2 , 148 ] . bullet .. OGLE open parenthesis• PLANET Optical ( Probing Gravitational Lensing Anomalies Lens Experiment NETwork closing ) [ 4 , 1parenthesis 1 1 ] . open square bracket 1 56 comma\noindent 1 53 commaor to 1 cover 49 closing the square Magellanic bracket• DUO period ( Clouds Disk Unseen completely Objects ) [around 3 ] . the year ( MOA ) . Here bullet .. AGAPE open parenthesis• GMAN Andromeda ( Global Galaxy Microlensing and Amplified Alert Network Pixels Experiment ) [ 1 9 ] . closing parenthesis open square bracket\noindent 1 1 commaLivingis a 1 Reviews list 47 closing of in groups Relativitysquare bracket currently( 1998 period - 12 active) in the search for microlensing signatures of compactbullet .. MOA objectshttp open : / in parenthesis / the www . halo MACHOl ivingreviews of the Observations Milky . Way org in Astrophysics or elsewhere closing : parenthesis open square bracket 2 comma 148 closing square bracket period \ centerlinebullet PLANET{ $ open\ bullet parenthesis$ \quad ProbingMACHO Lensing ( Anomalies MAssive NETworkCompact closing Halo parenthesisObject ) open [ 6 square , 7 , bracket 1 67 4 ] comma . } 1 1 1 closing square bracket period \ centerlinebullet DUO{ open$ \ parenthesisbullet $ Disk EROS Unseen ( Experience Objects closing de parenthesis Recherche open d square ’ Objets bracket Sombres 3 closing square ) [ 1 bracket 0 , 1 period 1 9 , 45 ] . } bullet .. GMAN open parenthesis Global Microlensing Alert Network closing parenthesis open square bracket 1 9 closing square\ centerline bracket{ period$ \ bullet $ \quad OGLE ( Optical Gravitational Lens Experiment ) [ 1 56 , 1 53 , 1 49 ] . } Living Reviews in Relativity open parenthesis 1998 hyphen 12 closing parenthesis \ hspacehttp :∗{\ slashf islash l l } www$ \ bullet period l ivingreviews$ \quad AGAPE period org ( Andromeda Galaxy and Amplified Pixels Experiment ) [ 1 1 , 1 47 ] . \ centerline { $ \ bullet $ \quad MOA ( MACHO Observations in Astrophysics ) [ 2 , 148 ] . }

\ centerline { $ \ bullet $ PLANET ( Probing Lensing Anomalies NETwork ) [ 4 , 1 1 1 ] . }

\ centerline { $ \ bullet $ DUO ( Disk Unseen Objects ) [ 3 ] . }

\ centerline { $ \ bullet $ \quad GMAN ( Global Microlensing Alert Network ) [ 1 9 ] . }

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 47 .... Gravitational Lensing in Astronomy \noindentFigure 2 147 : ..\ Fiveh f i microlensing l l Gravitational lightcurves Lensing for the tracks in Astronomy indicated in Figure 20 comma parametrized by the impact parameter u sub minimum period .. The verical axes is th e magnification \noindentin astronomicalFigure magnitudes 2 1 : \ requad lativeFive to the microlensing unlensed cas e comma lightcurves .. the horizontal for the axis tracks indicated in Figure 20 , parametrizeddisplays the47 time by in the .. quotedblleft impact parameter normalized quotedblright $ u {\min units} open Gravitational. $ parenthesis\quad LensingThe after vericalopen in Astronomy square axes bracket is ? th closing e magnification squarein astronomical bracketFigure closing 2 1 parenthesis magnitudes : Five microlensing period re lative lightcurves to the for the unlensed tracks indicated cas e in Figure, \quad 20 ,the parametrized horizontal by axis Living Reviewsthe impact in Relativity parameter openumin parenthesis. The verical 1998 hyphen axes is th12 e closing magnification parenthesis in astronomical magnitudes \noindenthttp : slashredisplays slash lative www to the the period unlensed time l ivingreviews cas in e\quad , theperiod‘‘ horizontal normalized org axis ’’ units ( after $ [ ? ] ) . $ displays the time in “ normalized ” units ( after [?]). \ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998Living− Reviews12 ) in Relativity ( 1998 - 12 ) http : / / www . l ivingreviews . org \ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 48 \noindentThe observationsJ . Wambsganss towards the Large\ h f i Magellanicl l 48 Cloud show that there are fewer microlensing events than one would expect if the halo of the Milky Way Thewas observations made entirely of towards these compact the objectsLarge period Magellanic .... The latest Cloud published show that results there from are fewerthe microlensing microlensingJ . Wambsganss experiments events that than monitor one stars would in the expect LMC indicate if the that halo the of the Milky Way 48 optical depthsThe toward observations the LMC towards is about the tau Large thickapprox Magellanic 3 times Cloud 1 0 show to the that power there of are minus fewer 7 microlensingperiod .... The observations are\noindent eventswas made than one entirely would expect of these if the halo compact of the Milky objects Way . \ h f i l l The latest published results from consistentwas with made 50 percent entirely of of the these Milky compact Way halo objects made . of compact objects The with latest most published results from \noindent thethe microlensing microlensing experiments experiments that monitor that stars monitor in the LMC stars indicate in thatthe theLMC indicate that the likely masses of 0 period 5 sub minus 0 period 2 to the power− of7 plus 0 period 3 M sub odot open square bracket 6 closing square bracketoptical period depths But the toward number the of LMC observed is about eventsτ ≈ is3 still× 10 small. The observations are \noindent consistentoptical with depths 50% oftoward the Milky the Way LMC halo is made about of compact $ \tau objects\ thickapprox with most 3 \times 1 0 ˆ{ − open parenthesis in this analysis+0.3 eight events were used closing parenthesis and hence the uncertainties are large semicolon 7 }in fact. $ commalikely\ h f iti massesl cannotl The of even observations 0.5− be0.2M excluded [6]. But that are the none number of the of observed observed events events is is due still to small an unknown( in halo this population analysis eight open events square were bracket used ) 55 and closing hence square the uncertainties bracket period are large ; in fact , it cannot \noindentThe sameeven typeconsistent be of experiment excluded with that open none $ parenthesis 50 of the\% observed$ searchingof events the for microlensing isMilky due to Way an unknown events halo closing made halo populationparenthesis of compact [ is 55 being objects ] . with most performed inThe the direction same type of of the experiment galactic bulge ( searching as well commafor microlensing the central events part ) of is the being performed in the \noindentMilky Waydirectionlikely period .. of masses By the now galactic more of bulge $0than 200as well. microlensing , 5ˆ the{ central+ events part0 have of . the been 3 } detected{ − in0 . 2 } M {\odot } [this 6 direction ]Milky . open $ Way parenthesis But . the By now number formore an example than of observed 200 see microlensing Figure events 22 closing events is parenthesis have still been small period detected .. Among in this themdirection are a number of quotedblleft( for binary an example lens quotedblright see Figure hyphen 22 ) . events Among open them parenthesis are a number which ofhave “ binary a very lenstypical ” - signature events ( of at least two caustic\noindent which( in have this a very analysis typical eightsignature events of at least were two used caustic ) crossings and hence , cf . the Figure uncertainties 23 ) . This is are large ; incrossings fact ,commaabout it cannot three cf period t imes even Figure as many be 23 excluded closing microlensing parenthesis that events none period as were of This expected the is about observed / predicted three t events imes . 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This type is ofevents have been detected in thismicrolensing directionto detect also holds ( planets for some an around promise example other forthe stars see future by Figure extending period 22 .. the) It .can sensitivity\quad be usedAmong of comma the binary them e period lensesare g perioda to number smaller to of ‘‘study binary theand frequency lens smaller ’’ of− companion binaryevents stars masses ( period which [ 1One 5 have , of 1 62 the a] most. very interesting typical possibilities signature is of at least two caustic crossingsto detect planets ,For cf arounda . recent Figure other comprehensive 23 stars ) by . extending This presentation is the about sensitivity of galactic three of microlensing the t imes binary as and many beyond microlensing see [?]. Various events aslenses were toexpected smalleraspects and of microlensing / smaller predicted companion in the . loSeveral masses cal group open groups are square reviewed trybracket in to detail 1 explain 5 comma . Another 1 this 62 closingreview ‘‘ over article square− on bracketabundance the period ’’ ofFor events a recentbasics to comprehensive have and the a results new presentation look of galactic at of the microlensing galactic stellar microlensing can content be found and and beyond in [ 1 the 23 ] dynamics . of the barsee /open bulge squareLiving of bracket Reviews the Galaxy? in closing Relativity square. The( bracket1998 latest - period 12 published) Various aspects results of microlensing can be found in the in lo cal [ group 7 ] . are reviewed in detail period http : / / www . l ivingreviews . org \ hspaceAnother∗{\ reviewf i l l } articleWith on these the basics microlensing and the results experiments of galactic microlensing gravitational can lensing has est ablished be found in open square bracket 1 23 closing square bracket period \noindentLiving Reviewsitself in Relativity as a new open tool parenthesis to study 1998 the hyphen structure 12 closing parenthesis of the Milky Way . \ h f i l l This type o f http : slash slash www period l ivingreviews period org \noindent microlensing also holds some promise for the future . \quad It can be used , e . g . to study the frequency of binary stars . One of the most interesting possibilities is

\noindent to detect planets around other stars by extending the sensitivity of the binary lenses to smaller and smaller companion masses [ 1 5 , 1 62 ] .

For a recent comprehensive presentation of galactic microlensing and beyond see $ [ ? ] . $ Various aspects of microlensing in the lo cal group are reviewed in detail . Another review article on the basics and the results of galactic microlensing can be found in [ 1 23 ] .

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 49 .... Gravitational Lensing in Astronomy \noindentFigure 22 :49 ....\ Observedh f i l l Gravitational Lightcurve of a microlensing Lensing ine vent Astronomy towards the bulge of the galaxy comma .... event OGLE hash 6 open square bracket 1 56 closing square bracket : .... The I hyphen band magnitude is\noindent p lo tt ed asFigure a function 22 : \ h f i l l Observed Lightcurve of a microlensing e vent towards the bulge of the of time open49 parenthesis Julian days closing parenthesis period .... In Gravitational the top panel Lensing comma in .... 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Extrapolating period .. Extrapolating from these thoughts from these , it thoughts should becomma possible it should to look be forward in t \noindentpossible toime( look long once forward again ) before in and t ime predict they once future again were andapplications actually predict future of observed gravitational applications . lensingAlthough of . \ h f i l l ‘‘ prediction ’’ \ h f i l l or predictabil − gravitational lensingHowever period , at any given time it requires very good intuition , some courage and maybe even \noindentHowever commaaity bit of is at ingenuity any considered given to time predict oneit requires qualitatively of the very important good new intuition phenomena criteria comma . someIt of does courage modern not need science much of either , many ( astro − and maybeto even envision a bit thatof ingenuity the known to predict lensing qualitatively phenomena new will phenomena become better period , sharper .. It , more . My \noindentdoes not needpredictions) physical much of for either thephenomena next to envision decade are that in this too the sense knowncomplicated are lensing humble phenomena andfor modest a minute : prediction ( j ust think of will become betterNo doubt comma there sharper will soon comma be more more determinations period .. My predictions of accurate for time the delays next in decade multiply in this - imaged \noindentsense are humblequasarthe weather systems and modest . forecast If : the models ) . will\ h getf i l lmoreThe precise reason as well why , the this value worked of the Hubble here constant is that gravitational No doubtH there0 determined will soon befrom more a number determinations of lens systems of accurate time delays in \noindentmultiply hyphenlensing imaged is quasar a simple systems geometrical period .. If the concept modelsLiving will which get Reviews more easily inprecise Relativity allows as well( qualitativecomma1998 - 12 ) estimates andthequantitative value of the Hubble calculations constant H sub . 0\ determinedquad Extrapolating fromhttp a number : / /from of www lens these. systems l ivingreviews thoughts , . it org should be possibleLiving Reviews to look in Relativity forward open in parenthesis t ime once 1998 again hyphen and 12 closing predict parenthesis future applications of gravitationalhttp : slash slash lensing www period . l ivingreviews period org However , at any given time it requires very good intuition , some courage and maybe even a bit of ingenuity to predict qualitatively new phenomena . \quad I t does not need much of either to envision that the known lensing phenomena will become better , sharper , more . \quad My predictions for the next decade in this sense are humble and modest :

No doubt there will soon be more determinations of accurate time delays in multiply − imaged quasar systems . \quad If the models will get more precise as well , the value of the Hubble constant $ H { 0 }$ determined from a number of lens systems

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 50 \noindentFigure 23 :J .. . Lightcurve Wambsganss of a b\ inaryh f i l lmicrolensing50 event towards the bulge of the galaxy comma .. event OGLE hash 7 open square bracket 1 55 closing square bracket : .. The I hyphen band the magnitude is\noindent plo tt ed overFigure time 23 : \quad Lightcurve of a b inary microlensing event towards the bulge of the galaxyopen parenthesis , J\quad . Wambsganss Julianevent days OGLE closing $ parenthesis\# 7 period [ 1.. In th 55 e top ] panel : th $ e ..\quad constantThe .. V I hyphen− band50 I hyphen the magnitude colo r of is plo tt ed over time th e s tar is shownFigure period 23 : Lightcurve of a b inary microlensing event towards the bulge of the galaxy , event \noindentThe maximumOGLE( Julian magnification#7[155] days : is)The more . \ Iquad - than band 2In the period thmagnitude e 5 mag top ishigher panel plo tt than ed th over the e \ unlensed timequad constant \quad V − I − colo r of th e s tar is shown . Theb rightness maximum( period Julian magnification .. days The ) duration . In is thof theemore top event panel than is th about 2 e . 80 5 constant daysmag period higher V .. - The I than - colo two the rins of eunlensed th ts e at s the tar is shown bleft rightness part show. The a. zoom\ maximumquad of thThe e magnification two duration peaks period is of more the.. The than event s tar 2 . had 5 is mag constant about higher80 brightness than days the unlensed in. \quad bThe rightness two . ins e ts at the leftthe year part precedingThe show duration a th zoom e microlensing of the of event th eevent is twoabout open peaks 80 parenthesis days . .\quadThe 1 992The two closing ins s tar e parenthesis ts at had the constant leftperiod part .. show A brightness model a zoom for this in event finds thea mass year ratioof preceding thof e1 twoperiod peaks th 2 between . e microlensing The th s e tar two had lensing constant event s tars brightness ( comma 1 992 .. in ) andthe . a\ yearquads eparation precedingA model of 1 th period e for microlensing 14this event finds aEinstein mass ratio radiievent period of( 1 9921 .. . open ) 2 . between parenthesis A model th for Credits e this two event : Andrzej lensing finds Udalski a smass tars period ratio , of closing\quad 1 . 2 parenthesisbetweenand a th s e eparation two lensing of 1 . 14 EinsteinLiving Reviewss radii tars in , Relativity . and\quad a s openeparation( Credits parenthesis of 1 : . 14 Andrzej 1998 Einstein hyphen Udalski radii 12 closing . . ( parenthesis ) Credits : Andrzej Udalski . ) http : slashLiving slash Reviews www period in Relativity l ivingreviews( 1998 period - 12 org) \noindent httpLiving : / Reviews / www . in l Relativityivingreviews ( . 1998 org − 12 )

\noindent http : / / www . l ivingreviews . org 5 1 .... Gravitational Lensing in Astronomy \noindentwill be accurate5 1 \ toh af i few l l percentGravitational or better and Lensing will probably in Astronomy turn out to be as reliable as H sub 0 values obtained with any other method open square bracket 1 69 closing square bracket period \noindentThe frequencieswill comma be accurate image separations to a few comma percent redshift or distributions better and of multiply will probably hyphen imaged turn out to be as quasars and5 1 their lenses will become a major tool in differentiating between Gravitational Lensing in Astronomy \noindentdifferent cosmologicalwillreliable be accurate models as to $ a period H few{ percent The0 } Sloan$ or valuesbetter Digital and Sky obtained will Survey probably comma with turn any e out period other to be g period as method comma [ 1 will 69 discover ] . a few hundredreliable new lensed as H0 values quasars obtained with very with well any defined other selection method [ criteria 1 69 ] . comma ideally Thesuited frequencies for thatThe purpose frequencies , image period separations, .. image Another separations angle , on redshift , the redshift cosmological distributions distributions model of and multiply the of multiply - imaged quasars− imaged and quasarsvalues of and Capitaltheir their lenses Omega will lenses and become Capital will a major Lambda become tool offer ain majordifferentiating the statistics tool of betweenin arcs differentiating period different .. The cosmological number between of high models redshift . differentgalaxies seenThe cosmological as Sloan arcs Digitaldepends Sky models crucially Survey on . , eThe the . g number . Sloan , will discover of Digital rich galaxy a few Sky hundred clusters Survey new ,lensed e . quasars g . , with will very discover a fewat intermediate hundredwell new defined redshifts lensed selection period quasars criteria .. And since, with ideally different very cosmologicalwell defined models selection predict very criteria , ideally different formationsuited for redshifts that purpose for clusters . Another comma angle this promising on the cosmological road should model be followed and the values of Ω and \noindentas well openΛsuited offersquare the bracketfor statistics that 1 9 closing ofpurpose arcs square . . The\ bracketquad numberAnother period of high angle redshift on galaxies the cosmological seen as arcs depends model and the valuesThe new o ffacilitiescrucially $ \Omega which on the become$ number and available of $ rich\Lambda galaxynow or clusters in$ the offer near at intermediate future the statisticsin the redshifts of. arcsAnd since . \quad differentThe number of high redshift galaxiesinfrared slashcosmological seen sub as hyphen arcs models mm depends slash predict mm very crucially domain different endash on formation like the SCUBA number redshifts comma of for rich clusters SIRTF galaxy comma , this promising FIRST clusters comma road IRAM endash willat open intermediateshould beredshifts followed . \quad And since different cosmological models predict very differenta completelyas formation well new [ window 1 9 ] . redshifts in these wavelength for clusters ranges comma , this with promisingsupposedly most road should be followed spectacular results in the arcs andThe cluster new facilities lensing regime which become period available now or in the near future in the \noindentQuasar microlensinginfraredas well / sub will [ 19- providemm /] mm . information domain – like on the SCUBA structure , SIRTF of the , FIRST quasars , IRAM – will open a completely and the interveningnew window clumped in these matter wavelength period Withranges the , with new supposedlyX hyphen ray most telescope spectacular AXAF results with in the arcs \ hspaceits high∗{\ spatialandf i l lcluster resolution}The lensing new it willfacilities regime become . possible which to become obtain X available hyphen ray lightcurves now or in the near future in the which due to the presumablyQuasar smaller microlensingemission region will will provide produce information dramatic on the structure of the quasars \noindentmicrolensingandinfrared events the intervening in /multiply sub clumped− hyphenmm / matter imagedmm domain . quasars With−− the period newl i k Xe .. - SCUBA Maybe ray telescope we , SIRTFcan AXAF quotedblleft , with FIRST its map high , IRAM quotedblright spatial−− w i lthe l open hot aspots completely of quasarsresolution new this it waywindow will period become in these possible wavelength to obtain X ranges- ray lightcurves , with which supposedly due to the most presumably spectacularThe largestsmaller number results emission of lensing in region the events will arcs in produce the and near dramatic cluster future will microlensing lensing doubtlessly regime events come in .multiply - imaged quasars . from the quotedblleftMaybe we can lo cal “ map quotedblright ” the hot spots microlensing of quasars experiments this way . monitoring galactic bulge stars period The \ hspaceart of t∗{\ akingf i lThe spectra l }Quasar largest of highly number microlensing magnified of lensing stars eventswill during providein themicrolensing near information future events will open doubtlessly on parenthesis the come structure as from the of “ lo the quasars pioneeredcal by ..” microlensing open square bracketexperiments 8 1 closing monitoring square galactic bracket bulge closing stars parenthesis . The will open up the fascinating possibility to\noindent investigateartand the of the t aking intervening spectra of highly clumped magnified matter stars during . With microlensing the new events X − ray ( as telescope AXAF with itsmetallicity highpioneered spatial of bulge st by resolution ars in[ 8 detail 1 ] ) or will it evenopen will resolve up become the the fascinating stellar possible surfaces possibility to and obtain study to investigate X − ray the lightcurves metallicity whichtheir center dueof to hyphen bulge the st to presumably ars hyphen in detail limb or variations smaller even resolve period emission the .. stellar In addition region surfaces of will being and producestudy an excellent their dramatic center tool to - study to - limb microlensingthe structurevariations of events the Milky . 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It is also imag- events ( as It is also imaginableinable that that before before too too long long such such microlensing microlensing events events could could be be detected directly by monitoring \noindentdetected directlyastrometricallypioneered by monitoring by the\quad position astrometrically[ of 8 the 1 ]star ) the very will position open of the up star the very fascinating possibility to investigate the metallicityaccuratelyaccurately open of square bulge [ 94 bracket ] .st ars 94 closing in detail square bracket or even period resolve the stellar surfaces and study theirIn due center courseIn we− due shouldto course− alsolimb we know should variations quantitatively also know . quantitatively how\quad muchIn dark addition how compact much dark of being compact an objects excellent contribute tool to study theobjects structure contributeto the mass of to the of the mass Milky halo of of the Way the halo Milky , of galactic the Way Milky , and Waymicrolensing what comma their mass and what rangewill their is also . The provide “ pixel lensing unbiased ” mass rangewill is period probe other .. The lines quotedblleft of sight through pixel lensing the Galactic quotedblright halo by will exploring probe other the Andromeda lines of sight galaxy through and the \noindentGalactic halootherstatistics by nearby exploring galaxies on the the Andromeda . This fraction will provide galaxy of andinformation binary other nearby stars on the galaxies ( three within - dimensional period certain mass relative distribution distances ) . Ex − tendingThis will the provideof the sensitivity halo information . on to the higher three hyphen mass dimensional ratios between mass distribution the binary of the components will halo period Weak lensing will be used to map not j ust the outskirts of massive galaxy clusters , but also \noindentWeak lensingtonaturally trace will be the used large lead to scale map to structure not the j ust detection bythe its outskirts effect of on of planets the massive back galaxy - around ground population stars ( of at galaxies distances . If of many clusters commawe find but good also ways to trace to discriminate the large scale between structure by its effect on the back hyphen \noindentground populationkiloparsecs of galaxies ! ) period . \ h .. f iIf l lweMicrolensing find good ways toLiving has discriminate the Reviews advantage between in Relativity compared( 1998 to- 12 all) other Earth − Living Reviews in Relativity open parenthesis 1998 hyphenhttp 12 closing : / / parenthesis www . l ivingreviews . org \noindenthttp : slashbound slash www planet period search l ivingreviews techniques period org that it is able to detect Earth − mass planets ! It is also imaginable that before too long such microlensing events could be detected directly by monitoring astrometrically the position of the star very

\noindent accurately [ 94 ] .

In due course we should also know quantitatively how much dark compact objects contribute to the mass of the halo of the Milky Way , and what their mass range is . \quad The ‘‘ pixel lensing ’’ will probe other lines of sight through the Galactic halo by exploring the Andromeda galaxy and other nearby galaxies . This will provide information on the three − dimensional mass distribution of the halo .

Weak lensing will be used to map not j ust the outskirts of massive galaxy clusters , but also to trace the large scale structure by its effect on the back − ground population of galaxies . \quad If we find good ways to discriminate between

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

\ hspace ∗{\ f i l l } http : / / www . l ivingreviews . org J period Wambsganss .... 52 \noindentsource galaxiesJ . at Wambsganss various redshifts\ h f comma i l l 52 this way we can ultimately produce a three hyphen dimensional map of the matter in the universe open parenthesis rather than a light map closing parenthesis ! .. This \noindentwill be ansource utmost useful galaxies thing for at the various understanding redshifts of structure , this formation way weand can ultimately produce a three − evolution semicolonJ . Wambsganss as an aside we will determine the matter content of the universe Capital Omega period52 \noindentSome othersourcedimensional possible galaxies applications at map various of of lensing redshifts the matter will , this be : way in .. 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Ω The. redshift of the most Somedist other ant object possibleSome will beother pushed applications possible beyond applications z = of 6 comma lensing of lensing and it will is quite be be : likely : The\quad that black itThe hole will blackin be the Galactic hole in center the Galacticmagnifiedaffects bycenter lensing all sourcesaffects period that .. The all are next nearsources generation or behind that theof experimentsare center near . Mapping or to map behind the this cosmic effect the will center be a complemen- . \quad Mapping thismicrowave effecttary background willdetermination be will a be complementary of sensititive the black enoughhole mass determination to and detect will the help gravitational to of study the the blacklens dynamics hole near mass the black and hole willsignature help of. to the The study matter redshift .. the quotedblleft of dynamics the most in dist Case near ant 1 object thequotedblright black will be hole pushedCase 2 . period beyond\quadzThe= 6, redshiftand it is quite of likely the most distWhat ant aboutthat object the it not will willhyphen be magnified be so hyphenpushed bypredictable lensing beyond . or The $ not z next hyphen = generation so 6 hyphen , of $ experiments easily and hyphen it is to predictable map quite the likelycosmic future of that lens hyphen it will be magnifieding ? .. Ultimatelymicrowave by lensing every background object . \quad in will theThe be sky sensititive isnext affected generation enough by open to detect parenthesis of the experiments gravitational ever so slight to lens closing map signature the parenthesis ofcosmic the lensing microwave background will00 be sensititive enough to detect the gravitational lens effects : ..matter this is the “ not in front hyphen yet hyphen reached regime of ultra weak lensing period .. I would like signatureto conclude of citing the two matter remarks\ thatquad. Bill‘ Press‘ in presented $\ l e f t in. f rhis o n lensing t \ begin outlook{ a l i g n e d } &’’ \\ &.at the IAU\end Symposium{Whata l i g n about e d 1}\ 73 theright in Melbourne not. -$ so - predictable open parenthesis or not 1 - 995 so closing- easily parenthesis - predictable period future .. Heof lens mentions - ing that? quotedblleft gravita hyphenUltimately every object in the sky is affected by ( ever so slight ) lensing effects : this is the Whatt ional about lensnot effectsthe - yet not period - reached− so period− regimepredictable period of ultra are present weak or lensing along not virtually− .so I would− everye a slike i line l y to− of conclude sightpredictable quotedblright citing two future remarks .. open of square lens bracket− 1ing 1 0 closing $ ? squarethat $ \ Billquad bracket PressUltimately periodpresented .. In in hisevery lensing object outlook in at the IAU sky Symposium is affected 1 73 in by Melbourne ( ever ( so 1 995 slight ) lensing e fa f note c t quite s :) so.\quad serious He mentionsthis extrapolation is that the “ gravita period not − .. - yet Press− pointsreached out that regime more and of moreultra as weak hyphen lensing . \quad I would l i k e totronomers concludet will ional citing open lens parenthesis effects two remarks. . have . are to presentclosingthat Bill parenthesis along Press virtually deal presented every with lensing line of in in sight the his ” next lensing [ decade 1 1 0 ] commaoutlook . In so a that lensing atwill the become IAUnot an Symposium quite .. quotedblleft so serious 1 73 ubiquitousextrapolation in Melbourne observational . Press ( 1 points technique 995 out ) . quotedblright that\quad moreHe and ..mentions more and hence as - tronomers that endash ‘‘ for will gravita better or − for worse :( .. have quotedblleft to ) deal Gravitationalwith lensing in lensing the next may decade well disappear , so that as lensing a unique will subbecome hyphen an specialty “ ubiquitous \noindentin astronomyobservationalt ional quotedblright lens technique period effects ” and . . hence . are – for present better or along for worse virtually : “ Gravitational every line lensing of may sight ’’ \quad [ 1 1 0 ] . \quad In aLiving not quite Reviewswell sodisappear in Relativity serious as a open extrapolationunique parenthesis sub - specialty 1998 . \ hyphen inquad astronomyPress 12 closing ” points. parenthesis out that more and more as − tronomershttp : slashLiving will slash Reviews www ( have period in to Relativity l ivingreviews ) deal( 1998 with period - lensing 12 org) in the next decade , so that lensing will becomehttp an :\ /quad / www‘‘ . ubiquitous l ivingreviews observational . org technique ’’ \quad and hence −− for better or f o r worse : \quad ‘‘ Gravitational lensing may well disappear as a unique sub − s p e c i a l t y in astronomy ’’ .

\noindent Living Reviews in Relativity ( 1998 − 12 )

\noindent http : / / www . l ivingreviews . org 53 .... Gravitational Lensing in Astronomy \noindent6 .. Acknowledgements53 \ h f i l l Gravitational Lensing in Astronomy It is a pleasure to thank Wes Colley comma .. Frederic Courbin comma .. Emilio Falco comma .. Henk \noindentHoekstra comma6 \quad NealAcknowledgements Jackson comma Tomislav Kundi acute-c sub comma Geraint Lewis comma and Andrzej Udalski for permission53 to use their figures period .. I would also like to thank Matthias Gravitational Bartel Lensing hyphen in Astronomy \noindentmann comma6It Emilio is Acknowledgements a Falco pleasure comma to Jean thank hyphen Wes Paul Colley Kneib comma , \quad BohdanFrederic Paczy n-acute Courbin ski comma , \quad SjurEmilio Refsdal comma Falco , \quad Henk RobertHoekstraIt , is Neal a pleasure Jackson to thank , Tomislav Wes Colley Kundi , Frederic $ \acute Courbin{c ,} Emilio{ , } Falco$ Geraint , Henk Lewis Hoekstra , , and Andrzej Udalski forSchmidt permission commaNeal Jackson Liliya to use Williams , Tomislav their comma Kundi figures andc ´, Geraint David . \quad Woods Lewis ,I for and would their Andrzej careful also Udalski reading like for to ofpermission open thank parenthesisto Matthias use their parts Bartel − mannof closing , Emilio parenthesisfigures Falco. I the would ,manuscript Jean also like− atPaul to various thank Kneib Matthias stages , and Bohdan Bartel their - usefulPaczy mann comments , Emilio $ \acute Falco period{ ,n Jean} ....$ Of - Paul ski particu Kneib , Sjur hyphen Refsdal , Robert lar help were, Bohdan the comments Paczyn ´ ofski J , u-dieresis Sjur Refsdal rgen , EhlersRobert and an unknown referee which \noindentimproved theSchmidtSchmidt paper , considerablyLiliya , Liliya Williams period Williams , and David , andWoods David for their Woods careful for reading their of ( careful parts reading of ( parts Living Reviewsof ) the in manuscript Relativity open at various parenthesis stages 1998 and hyphentheir useful 12 closing comments parenthesis . Of particu - \noindenthttp : slashlarof slash help ) the www were manuscript theperiod comments l ivingreviews atof J variousu ¨ rgen period Ehlers org stages and an and unknown their referee useful which comments improved the . \ paperh f i l l Of p a r t i c u − considerably . \noindent lar help were the comments of J $ \ddotLiving{u} Reviews$ rgen in Relativity Ehlers and( 1998 an - unknown 12 ) referee which improved the paper considerably . http : / / www . l ivingreviews . org

\ hspace ∗{\ f i l l } Living Reviews in Relativity ( 1998 − 12 )

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F )period . \quad commaFor .. and\quad a \quad r e l a t e d \quad o n l i n e \quad ver − s i o n \quad see : \quad C. \quad Renault , \quadLivinget Reviews\quad a in l Relativity . , \quad( 1998‘‘ -Observational 12 ) \quad l i m i t s \quad on \quad machos \quad in Brehin comma .. S period comma .. quotedblleft Observationalhttp : .. / limits / www .. . on l.. ivingreviews machos .. in .. . the org .. Galactic .. Halo quotedblrightthe \quad commaG a l a c t i c \quad Halo ’ ’ , \quad ( 1 997 ) , \quad [ Online \quad Journal ] : \quad c i t e d \quad on \quad 1 5 \quad Septem − berAstron 1 998 period , \ ..quad Astrophyshttp period : / / comma l ink .. 324. springer open parenthesis . de /L 69 l hyphenink / Ls 72 ervi closing c e parenthesis / j ournals comma / .. 230 open / parenthesisbibs / 7324003 1 997 closing / 2300 parenthesis l 69 period / small .. For . .. htm a .. related . \quad .. online4 . ..7 ver hyphen sion .. see : .. C period .. Renault comma .. et .. al period comma .. quotedblleft Observational .. limits .. on .. machos ..\noindent in [ 1 20 ] \quad Renn , J . , Sauer ,T . , and Stachel , J . , \quad ‘‘ The origin of gravitational lensing : Athe postscript .. Galactic .. to Halo Einstein quotedblright ’ s comma1 936 .. Science open parenthesis paper 1’’ 997 , closing\quad parenthesisSci ence comma , \quad .. open275 square , \quad bracket1 84 −− 1 86 , Online( 1 997 .. Journal ) . \ closingquad square2 , 1 bracket : .. cited .. on .. 1 5 .. Septem hyphen ber 1 998 comma .. http : slash slash l ink period springer period de slash l ink slash s ervi c e slash j ournals slash 230 slash\noindent [ 1 2 1 ] \quad Rix , H . − W., \quad ‘‘ Lensing Limits on the cosmological constant ’’ , 1 −− 6 , ( 1 996 ) . 4bibs . 6 slash 7324003 slash 2300 l 69 slash small period htm period .. 4 period 7 open square bracket 1 20 closing square bracket .. Renn comma J period comma Sauer comma T period comma and Stachel comma\noindent J period[ 1 comma 22 ] ..\ quotedbllefth f i l l Robinson The origin , L of gravitational . J . , \ h lensing f i l l ‘‘ : Early thoughts about Lenses in the $\ lA e f postscript t . Sky\ begin to Einstein{ array quoteright}{ c} ’’ s\\ 1 936, Science\end{ paperarray quotedblright}Sky\ right comma. $ and .. Science Tele − comma .. 275 comma .. 1 84 endash 1 86 comma \ centerlineopen parenthesis{ s cope 1 997 , closing 387 −− parenthesis389 , period ( November .. 2 comma , 1 1 983 ) . \quad 1 } open square bracket 1 2 1 closing square bracket .. Rix comma H period hyphen W period comma .. quotedblleft Lensing Limits\ hspace on∗{\ the cosmologicalf i l l } Living constant Reviews quotedblright in Relativity comma 1 endash ( 1998 6 comma− 12 open) parenthesis 1 996 closing parenthesis period 4 period 6 \ hspaceopen square∗{\ f i bracket l l } http 1 22 : closing/ / www square . l bracket ivingreviews .... Robinson . org comma L period J period comma .... quotedblleft Early thoughts about Lenses in the SkyRow 1 quotedblright Row 2 comma Sky and Tele hyphen s cope comma 387 endash 389 comma open parenthesis November comma 1 983 closing parenthesis period .. 1 Living Reviews in Relativity open parenthesis 1998 hyphen 12 closing parenthesis http : slash slash www period l ivingreviews period org J period Wambsganss .... 68 \noindentopen squareJ bracket . Wambsganss 1 23 closing\ h square f i l l bracket68 .... Roulet comma E period comma and Mollerach comma S period comma .... quotedblleft MicrolensingRow 1 quotedblright Row 2 comma Microlensing period Rep period comma 279 comma 67 endash 1\noindent 1 8 comma [ 1 23 ] \ h f i l l Roulet , E . , and Mollerach , S . , \ h f i l l ‘ ‘ $\ l e f t .Microlensing \ begin { array }{ c} ’’ \\ , \end{ arrayJ .} WambsganssPhys\ right . .$ Rep. ,279 ,67 −− 1 1 8 , 68 open parenthesis 1 997 closing parenthesis period .. 4 period 7 00 open square[ 1 bracket 23 ] Roulet 1 24 , closing E . , and square Mollerach bracket , S .. . , Sachs “ Microlensing comma R periodP hys K period. Rep . comma, 279 .., 67 quotedblleft – 1 1 8 , Gravitational , Waves\ centerline in General{( Relativity 1 997 ) VI . period\quad The4 outgoing . 7 } ( 1 997 ) . 4 . 7 radiation condition quotedblright comma Phys period Rev period Lett period comma 264 comma 309 comma open paren- [ 1 24 ] Sachs , R . K . , “ Gravitational Waves in General Relativity VI . The outgoing thesis\noindent 1 961 closing[ 1 24 parenthesis ] \quad periodSachs .. 1 ,R.K. , \quad ‘‘ Gravitational Waves in General Relativity VI . The outgoing radiation condition ” , Phys . Rev . Lett . , 264 , 309 , ( 1 961 ) . 1 radiationopen square condition bracket 1 25 ’’ closing , Phys square . bracket Rev . .. Lett Sanitt . comma , 264 N , period 309 , comma ( 1 961 .. quotedblleft ) . \quad Quasi1 hyphen stellar [ 1 25 ] Sanitt , N . , “ Quasi - stellar objects and gravitational lenses ” , Nature , 234 , 1 objects and gravitational lenses quotedblright comma Nature comma 234 comma 99 – 20 1 , ( 1 971 ) . 1 \noindent1 99 endash[ 20 1 1 25 comma ] \quad open parenthesisS a n i t t , 1N 971 . closing , \quad parenthesis‘ ‘ Quasi period− stellar .. 1 objects and gravitational lenses ’’ , Nature , 234 , [ 1 26 ] Schechter , P . , “ Distance Estimates From Time Delay Measurements Of Multiply 1open 99 −− square201 bracket , (1971) 1 26 closing . square\quad bracket1 .. Schechter comma P period comma .. quotedblleft Distance Estimates Imaged , Gravitationally Lensed QSOs : Observational Perspec - tive ” , ( 1 997 ) , [ Online From Time Delay Measurements Of Article from Strasbourg Meeting on Distance Scale Proceedings ] : cited on 1 5 September \noindentMultiply Imaged[ 1 26 comma ] \quad GravitationallySchechter Lensed , P QSOs . , :\quad .. Observational‘‘ Distance Perspec Estimates hyphen From Time Delay Measurements Of 1 998 , http : / / astro . u - strasbg . fr / howfar / s che chter . html . Multiplytive quotedblright Imaged comma , Gravitationally open parenthesis Lensed 1 997 closing QSOs parenthesis : \quad commaObservational open square Perspec bracket Online− Article from Editorial note : URL updated on 3 1 Aug Strasbourgtive ’’ Meeting , ( 1 on 997 Distance ) , [ Scale Online Article from Strasbourg Meeting on Distance Scale 200 1 . 7 , 4 . 6 ProceedingsProceedings closing ] : \ squarequad bracketcited : on .. cited 1 5 on September 1 5 September 1 998 1 998 , \ commaquad ..http http : : slash / / slash astro astro . u period− s t u r a hyphen s b g . [ 1 27 ] Schechter , P . L . , Bailyn , C . D . , Barr , R . , Barvainis , R strasbgfr / period howfar / s che chter . html . Editorial note : \quad URL updated on 3 1 \quad Aug . , and Becker , C . M . , “ The quadruple gravitational lens PG 1115 + 80 : fr slash howfar slash s che chter period html period Editorial note : .. URL updated on 3 1 .. Aug Time delays and models ” , 475 , L 85 – L 88 , ( 1 997 ) . For a related online \ centerline200 1 period{200 .. 7 comma 1 . \ 4quad period7 6 , 4 . 6 } version see : P . L . Schechter , et al . , “ The quadruple gravitational lens PG 1115 + 80 : open square bracket 1 27 closing square bracket .. Schechter comma .. P period L period comma .. Bailyn comma .. C Time delays and models ” , ( 1 997 ) , [ Artcle in Online Journal ] : cited on 1 5 September period\noindent D period[ 1comma 27 ] ..\ Barrquad commaSchechter .. R period , \ commaquad P.L., .. Barvainis comma\quad ..Bailyn R period , comma\quad ..C.D., and .. Becker\ commaquad Barr , \quad R., \quad Barvainis , \quad R., \quad and \quad Becker , 1 998 , http : / / www . j ournals . uchi cago . edu / ApJ / j ournal / i s C.M.,C period M\ periodquad comma‘ ‘ The .. quotedblleft\quad quadruple The .. quadruple\quad ..gravitational gravitational lens lens .. PG\ ..quad 1 1 1PG 5 plus\quad 80 : .. Time$ 1 .. delays1 1 sue s / ApJL / v 475 n 2 / 5649 / s c 0 . html . 7 5and + models 80 quotedblright : $ \quad commaTime ..\ 475quad commadelays .. L 85 endash L 88 comma .. open parenthesis 1 997 closing parenthesis [ 1 28 ] Schild , R . , and Thomson , D . J . , “ Twin QSO Q 957 + 561 time delay periodand models.. For a related ’’ , online\quad version475 ,see\quad : L 85 −− L 88 , \quad ( 1 997 ) . \quad For a related online version see : dataset ” , Astrophys . J . , 109 , 1 970 – 1 979 , ( 1 995 ) . For a related online ver - sion PP . period L . Schechter L period Schechter , et comma al . ,et\ alquad period‘‘ comma The quadruple.. quotedblleft gravitational The quadruple gravitational lens PG lens $ 1 PG 1 1 1 1 51 plus 5 see : R . Schild , et al . , “ Twin QSO Q 957 + 561 time delay dataset ” , ( 1 995 ) , [ +80 : 80 : $ ADS Astronomy Abstract Service ] : cited on 1 5 September 1 998 , http : / TimeTime delays delays and and models models quotedblright ’’ , ( comma 1 997 open ) , parenthesis [ Artcle 1 997 in closing Online parenthesis Journal comma ] : opencited square on 1bracket 5 Artcle / adsabs . h a − r vard . edu / cgi - bin / nph −bib query ?1995 AJ % inSeptember Online Journal 1 998 closing , square\quad brackethttp : : cited / / on www 1 5 . j ournals . uchiunderscore cago . edu / ApJ / j ournal / 2 E %2 E %2 E %2 E 19%2 E 1 970 S &db key = AST . 6 iSeptember s sue s 1 /ApJL/ 998 comma v .. 475 http n :slash 2 / slash5649 www / s period c 0 . junderscore ournals html . period\quad uchi7 cago period edu slash ApJ slash j ournal [ 1 29 ] Schild , R . , and Thomson , D . J . , “ The Q 957 + 561 time delay from op - slash tical data ” , Astrophys . J . , 1 13 , 1 30 – 1 35 , ( 1 997 ) . For a related \noindenti s sue s slash[ 1 ApJL 28 ] slash\ h vf i 475 l l nS c 2 h slash i l d 5649 , \ h slash f i l l sR., c 0 period\ h html f i l l periodand \ ..h 7 f i l l Thomson , \ h f i l l D.J., \ h f i l l ‘ ‘ Twin \ h f i l l QSO \ h f i l l Q online version see : R . Schild , et al . , “ The Q 957 + 561 time delay from optical data ” $ 957open square + bracket 56 1 1 28$ closing\ h f i l square l time bracket\ h f i .... l l Schilddelay comma .... R period comma .... and .... Thomson comma .... D , ( 1 995 ) , [ ADS Astronomy Abstract Service ] : cited on 1 5 September 1 998 , http : / period J period comma .... quotedblleft Twin .... QSO .... Q 957 plus 56 1 .... time .... delay / adsabs . h a − r vard . edu / cgi - bin / nph −bib query ?1997 AJ % datasetdataset quotedblright’’ , Astrophys comma . Astrophys J . , 109 period , 1 J 970period−− comma1 979 109 , comma ( 1underscore 995 1 970 ) endash . 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