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Gravitational Lensing in Quasar Samples Astronomy and Astrophysics Review ManuscriptNr will b e inserted by hand later Gravitational Lensing in Quasar Samples JeanFrancois Claeskens and Jean Surdej Institut dAstrophysique et de Geophysique Universite de Liege Avenue de Cointe B Liege Belgium Received April Accepted Summary The rst cosmic mirage was discovered approximately years ago as the double optical counterpart of a radio source This phenomenon had b een predicted some years earlier as a consequence of General Relativity We present here a summary of what we have learnt since The applications are so numerous that we had to concentrate on a few selected asp ects of this new eld of research This review is fo cused on strong gravitational lensing ie the formation of multiple images in QSO samples It is intended to give an uptodate status of the observations and to present an overview of its most interesting p otential applications in cosmology and astrophysics as well as numerous imp ortant results achieved so far The rst Section follows an intuitive approach to the basics of gravitational lensing and is develop ed in view of our interest in multiply imaged quasars The astrophysical and cosmo logical applications of gravitational lensing are outlined in Section and the most imp ortant results are presented in Section Sections and are devoted to the observations Finally conclusions are summarized in the last Section We have tried to avoid duplication with existing and excellent intro ductions to the eld of gravitational lensing For this reason we did not concentrate on the individual prop erties of sp ecic lens mo dels as these are already well presented in Narayan and Bartelmann and on a more intuitive ground in Refsdal and Surdej Wambsganss prop oses a broad view on gravitational lensing in astronomy the reviews by Fort and Mellier and Hattori et al deal with lensing by galaxy clusters microlensing in the Galaxy and the lo cal group is reviewed by Paczynski and a general panorama on weak lensing is given by Bartelmann and Schneider The monograph on the theory of gravitational lensing by Schneider Ehlers and Falco also remains a reference in the eld Key words Gravitational Lensing Cosmology observations Quasars general Galaxies general Charge de Recherches du FNRS Belgium Directeur de Recherches du FNRS Belgium Correspondence to surdejastroulgacb e JF Claeskens J Surdej * I1 A) B) *I C) *I2 * S *S *S * I I3 Fig Light propagation in an euclidian space A following a straight line normal situation B following a curved tra jectory in this case the image p osition I is dierent from the source p osition S as seen by the observer C following multiple curved tra jectories with the formation of multiple images of the source S Gravitational lensing In this Section we rst prop ose a general denition of a cosmic mirage We then intro duce the basic formalism of the lensing equation the deection angle the amplication and the time delays Finally we explain how to nd cosmic mirages as eciently as p ossible What is a cosmic mirage People sometimes b elieve that mirages alike atmospheric mirages are synony mous of hallucinations This is absolutely not the case Common sense of reality is based on our everyday p erception of classical spacetime whose geometry is at and static and where light propagates along straight lines In this case we can only see one image of a given p oint source and it is precisely lo cated in the direction of the source p osition see Fig A The apparent source size ie its solid angle is then inversely prop ortional to the square of the source distance A very general denition of a mirage would b e a distorted view of the reality compared with what is exp ected by common sense Following that denition a mirage may arise in two situations if the light tra jectory is not straight then the observer sees the image in a direction dierent from that of the source Fig B multiple imaging o ccurs if more than one light tra jectory can reach the observer Fig C if space is expanding as in standard cosmologies the ob jects lo cated in the past lightcone of the observer lo ok bigger or closer than they actually are at the time of observation see app earance of sizes in Fig This cosmological mirage is corrected for by the use of cosmological angular distances which also take into account the eects due to the curvature of the Universe see Section In this review we are mainly interested in the rst kind of mirages ie when the light tra jectories are curved This happ ens when the light sp eed is not isotropic along the directions p erp endicular to the light path For example since the light sp eed in a material medium is v cn where n is the refractive index of that medium and c is the light sp eed in the vacuum the light rays are b ent when they travel into an inhomogeneous medium see Fig This happ ens in Gravitational Lensing in Quasar Samples 5 4 8 S 7 3 6 5 α 4 2 3 α S 2 1 1 O O 1 2 3 4 5 6 7 8 1 2 3 4 5 A) t = tem B) t = tobs Fig Eect due to the expansion of an euclidian space during light travel on the apparent size of distant ob jects see text Σ M Fig Left Deection of light in an inhomogeneous medium right euclidian representation of the gravitational deection of light refers to a p ortion of the wavefront The length of the arrows at the edge of the s is prop ortional to the apparent velo city of light the lower atmospheric air layers with strong temp erature or density gradients and atmospheric mirages may then b e formed for a more detailed analysis of atmospheric mirages see Refsdal and Surdej As a consequence of General Relativity spacetime is curved by the gravita tional p otential asso ciated with a massive ob ject In the weak eld approxi mation jjc its metric is simply eg Schutz c dt dx dy dz ds c c However this new spacetime can b e regarded as an euclidian one in which the velo city of light is apparently slowed down in the vicinity of the massive ob ject Indeed the light path b eing dened by the null geo desic ds in Eq it is easy to derive that formally acts on the light sp eed as a medium with an eective refractive index n where n c JF Claeskens J Surdej * I α α b * S θ b θ s O s DOD D DS DOS Fig Gravitational lensing fundamental scheme The gradient of the Newtonian gravitational eld in the vicinity of a massive ob ject is now resp onsible for the angular deection of the light tra jectories see Fig When such a deected ray reaches the observer a gravitational mirage is seen and the massive ob ject is called a gravitational lens For historical reviews on gravitational lensing see for example Schneiders Ehlers and Falco Refsdal and Surdej and Wambsganss The lens equation The lens equation connects the source p osition and the p ositions of the s images seen by an observer O given the deection angle pro duced by the gravitational lens The lens is assumed to b e thin and conned into the deector plane Fig shows the geometrical scheme of gravitational lensing Following rays backward from the observer to the source it is very easy to derive the lens equation as a relation b etween the very small angles and s D DS D s OD D OS By analogy with geometrical optics the observed images are in fact virtual images since they are seen through the lens The deection of the light ray is mainly pro duced around its closest approach to the lens b ecause astrophysical ob jects are much smaller than the total length of the light path Gravitational Lensing in Quasar Samples In order to correct for the cosmological mirage eect pro duced by the expansion and the curvature of the Universe D D and D must b e cos OD OS DS mological angular distances resp ectively b etween the observer and the deector the observer and the source and the deector and the source In Friedmann LematreRob ertsonWalker FLRW cosmological mo dels with present matter density and without cosmological constant the angular distance of a o o source at redshift z seen from an ob ject at redshift z is given by S D c D z z D S H z z o D S o h i p p z z z z o S o D o D o S For other FLRW cosmologies numerical calculations can b e eciently p erformed thanks to the ANGSIZ algorithm develop ed by Kayser et al The deection angle The deection angle can b e estimated in several dierent ways A rst metho d is based up on the weak eld metric Eq and the geo desic equation eg Weinb erg A second metho d consists in applying the Fermat Principle to null tra jectories in the metric to simultaneously derive the lens equation and the expression of This physical approach has rst b een derived by Schneider and is intuitively presented in Blandford Narayan The deection angle may also simply b e determined from the integration of the variations of the direction i of the ray along the unperturbed tra jectory see Fig st by making use of the well known Descartes law n cos i C along the light tra jectory and assuming small deection angles i since bc for astrophysical ob jects Dening b as the impact parameter in the deector plane p erp endicular to the unp erturb ed direction of motion x see Fig b D OD we get Z Z Z di dx r n dx r dx b b b dx n c where the last equality of Eq comes from relation The most basic deector corresp onds to the p oint mass M whose gravita tional p otential is simply GM GM p r b x The deection angle pro duced by a p oint mass is obtained by intro ducing Eq into Eq GM b b c b JF Claeskens J Surdej α i b r
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