Gravitational Lensing in Astronomy

3.1 Lens equation

The basic setup for such a simplified gravitational lens scenario involving a point source and a point lens is displayed in Figure 2. The three ingredients in such a lensing situation are the source S, the lens L, and the observer O. Light rays emitted from the source are deflected by the lens. For a point-like lens, there will always be (at least) two images Si and S2 of the source. With external shear — due to the tidal field of objects outside but near the light bundles — there can be more images. The observer sees the images in directions corresponding to the tangents to the real incoming light paths.

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In Figure 3 the corresponding angles and angular diameter distances D_L, D_S, D_LS are indicated³. In the thin-lens approximation, the hyperbolic paths are approximated by their asymptotes. In the circular-symmetric case the deflection angle is given as

$$\tilde \alpha (\xi ) = \frac{{4GM(\xi )}}{{{c^2}}}\frac{1}{\xi },$$

(4)

where M(ξ) is the mass inside a radius ξ. In this depiction the origin is chosen at the observer. From the diagram it can be seen that the following relation holds:

$$\theta {D_{\rm{S}}} = \beta {D_{\rm{S}}} + \tilde \alpha {D_{{\rm{LS}}}}$$

(5)

(for θ, β, \({\tilde \alpha}\) ≪ 1; this condition is fulfilled in practically all astrophysically relevant situations). With the definition of the reduced deflection angle as α(θ) = (D _LS /D _S ) \({\tilde \alpha}\) (θ), this can be expressed as:

$$\beta = \theta - \alpha (\theta ).$$

(6)

This relation between the positions of images and source can easily be derived for a non-symmetric mass distribution as well. In that case, all angles are vector-valued. The two-dimensional lens equation then reads:

$$\vec \beta = \vec \theta - \vec \alpha (\vec \theta ).$$

(7)

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3.2 Einstein radius

3.3 Critical surface mass density

3.4 Image positions and magnifications

3.5 (Non-)Singular isothermal sphere

3.6 Lens mapping

The determinant of the Jacobian \({\mathcal A}\) is the inverse of the magnification:

$$\mu = \frac{1}{{{\rm{det}}\;{\mathcal A}}}.$$

(31)

Let us define

$${\psi _{ij}} = \frac{{{\partial ^2}\psi }}{{\partial {\theta _i}\partial {\theta _j}}}.$$

(32)

The Laplacian of the effective potential ψ is twice the convergence:

$${\psi _{11}} + {\psi _{22}} = 2\kappa = {\rm{tr}}{\psi _{ij}}.$$

(33)

With the definitions of the components of the external shear γ,

$${\gamma _1}(\vec \theta ) = \frac{1}{2}({\psi _{11}} - {\psi _{22}}) = \gamma (\vec \theta )\cos [2\varphi (\vec \theta )].$$

(34)

and

$${\gamma _2}(\vec \theta ) = {\psi _{12}} = {\psi _{21}}) = \gamma (\vec \theta )\sin [2\varphi (\vec \theta )]$$

(35)

(where the angle φ reflects the direction of the shear-inducing tidal force relative to the coordinate system), the Jacobian matrix can be written

$${\mathcal A} = \left( {\begin{array}{*{20}{c}} {1 - \kappa - {\gamma _1}}&{ - {\gamma _2}}\\ { - {\gamma _2}}&{1 - \kappa + {\gamma _1}} \end{array}} \right) = (1 - \kappa )\;\left( {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right) - \gamma \;\left( {\begin{array}{*{20}{c}} {\cos \;2\varphi }&{\sin \;2\varphi }\\ {\sin \;2\varphi }&{ - \cos \;2\varphi } \end{array}} \right).$$

(36)

The magnification can now be expressed as a function of the local convergence κ and the local shear γ:

$$\mu = {({\rm{det}}\;{\mathcal A}{\rm{)}}^{{\rm{ - 1}}}} = \frac{1}{{{{(1 - \kappa )}^2} - \gamma {}^2}}.$$

(37)

Locations at which det A = 0 have formally infinite magnification. They are called critical curves in the lens plane. The corresponding locations in the source plane are the caustics. For spherically symmetric mass distributions, the critical curves are circles. For a point lens, the caustic degenerates into a point. For elliptical lenses or spherically symmetric lenses plus external shear, the caustics can consist of cusps and folds. In Figure 4 the caustics and critical curves for an elliptical lens with a finite core are displayed.

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3.7 Time delay and “Fermat’s” theorem

4.1 Multiply-imaged quasars

In 1979, gravitational lensing became an observational science when the double quasar Q0957+561 was discovered. This was the first example of a lensed object [194]. The discovery itself happened rather by accident; the discoverer Dennis Walsh describes in a nice account how this branch of astrophysics came into being [193].

It was not entirely clear at the beginning, though, whether the two quasar images really were an illusion provided by curved space-time — or rather physical twins. But intensive observations soon confirmed the almost identical spectra. The intervening “lensing” galaxy was found, and the “supporting” cluster was identified as well. Later very similar lightcurves of the two images (modulo offsets in time and magnitude) confirmed this system beyond any doubt as a bona fide gravitational lens.

By now about two dozen multiply-imaged quasar systems have been found, plus another ten good candidates (updated tables of multiply-imaged quasars and gravitational lens candidates are provided, e.g., by the CASTLE group [57]).

Gravitationally lensed quasars come in a variety of classes: double, triple and quadruple systems; symmetric and asymmetric image configurations are known.

For an overview of the geometry of multiply-imaged quasar systems, see the collection of images found at [71].

A recurring problem connected with double quasars is the question whether they are two images of a single source or rather a physical association of two objects (with three or more images it is more and more likely that it is lensed system). Few systems are as well established as the double quasar Q0957+561; but many are considered “safe” lenses as well. Criteria for “fair”, “good”, or “excellent” lensed quasar candidates comprise the following:

• There are two or more point-like images of very similar optical color.

• Redshifts (or distances) of both quasar images are identical or very similar.

• Spectra of the various images are identical or very similar to each other.

• There is a lens (most likely a galaxy) found between the images, with a measured redshift much smaller than the quasar redshift (for a textbook example see Figure 5).

• If the quasar is intrinsically variable, the fluxes measured from the two (or more) images follow a very similar light curve, except for certain lags — the time delays — and an overall offset in brightness (cf. Figure 8).

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For most of the known multiple quasar systems, only some of the above criteria are fully confirmed. And there are also good reasons not to require perfect agreement with this list. For example, the lensing galaxy could be superposed to one quasar image and make the quasar appear extended; color/spectra could be affected by dust absorption in the lensing galaxy and appear not identical; the lens could be too faint to be detectable (or even a real dark lens?); the quasar could be variable on time scales shorter than the time delay; microlensing can affect the lightcurves of the images differently. Hence, it is not easy to say how many gravitationally lensed quasar systems exist. The answer depends on the amount of certainty one requires. In a recent compilation, Keeton and Kochanek [92] put together 29 quasars as lenses or lens candidates in three probability “classes”.

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 measurement of the time delay can be used to determine the Hubble constant). As an ensemble, the lens systems are also analysed statistically in order to get information about the population of lenses (and quasars) in the universe, their distribution in distance (i.e. cosmic time) and mass, and hence about the cosmological model (more about that in Section 4.6). Here we will have a close look on one particularly well investigated system.

4.1.1 The first lens: Double quasar Q0957+561

The quasar Q0957+561 was originally found in a radio survey, subsequently an optical counterpart was identified as well. After the confirmation of its lens nature [194, 193], this quasar attracted quite some attention. Q0957+561 has been looked at in all available wavebands, from X-rays to radio frequencies. More than 100 scientific papers have appeared on Q0957+561 (cf. [142]), many more than on any other gravitational lens system. Here we will summarize what is known about this system from optical and radio observations.

In the optical light, Q0957+561 appears as two point images of roughly 17 mag (R band) separated by 6.1 arcseconds (see Figure 6). The spectra of the two quasars reveal both redshifts to be z_Q = 1.41. Between the two images, not quite on the connecting line, the lensing galaxy (with redshift z_G = 0.36) appears as a fuzzy patch close to the component B. This galaxy is part of a cluster of galaxies at about the same redshift. This is the reason for the relatively large separation for a galaxy-type lens (typical galaxies with masses of 10^11–12M_⊙ produce splitting angles of only about one arcsecond, see Equation (10)). In this lens system, the mass in the galaxy cluster helps to increase the deflection angles to this large separation.

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A recent image of Q0957+561 taken with the MERLIN radio telescope is shown in Figure 7. The positions of the two point-like objects in this radio observation 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-quiet. But this also tells us that a possible third image of the quasar must be very faint, below the detection limit of all the radio observations⁶. In Figure 7, a “jet” can be seen emerging from image A (at the top). It is not unusual for radio quasars to have such a “jet” feature. This is most likely matter that is ejected from the central engine of the quasar with very high speed along the polar axis of the central black hole. The reason that this jet is seen only around one image is that it lies outside the caustic region in the source plane, which marks the part that is multiply imaged. Only the compact core of the quasar lies inside the caustic and is doubly imaged.

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As stated above, a virtual “proof” of a gravitational lens system is a measurement of the “time delay” Δt, the relative shift of the light curves of the two or more images, I_A(t) and I_B(t), so that I_B(t) = const × I_A(t + Δt). Any intrinsic fluctuation of the quasar shows up in both images, in general with an overall offset in apparent magnitude and an offset in time.

Q0957+561 is the first lens system in which the time delay was firmly established. After a decade long attempt and various groups claiming either of two favorable values [139, 146, 163, 192], Kundić et al. [106] confirmed the shorter of the two (cf. Figure 8; see also Oscoz et al. [103] and Schild & Thomson [164]):

$$\Delta {t_{Q0957 + 561}} = (417 \pm 3){\rm{days}}$$

(41)

With a model of the lens system, the time delay can be used to determine the Hubble constant⁷. In Q0957+561, the lensing action is done by an individual galaxy plus an associated galaxy cluster (to which the galaxy belongs). This provides some additional problems, a degeneracy in the determination of the Hubble constant [66]. The appearance of the double quasar system including the time delay could be identical for different partitions of the matter between galaxy and cluster, but the derived value of the Hubble constant could be quite different. However, this degeneracy can be “broken”, once the focussing contribution of the galaxy cluster can be determined independently. And the latter has been attempted recently [59]. The resulting value for the Hubble constant [106] obtained by employing a detailed lens model by Grogin and Narayan [73] and the measured velocity dispersion of the lensing galaxy [58] is

$${H_0} = (67 \pm 13)\;{\rm{km}}\;{\rm{se}}{{\rm{c}}^{{\rm{ - 1}}}}\;{\rm{Mp}}{{\rm{c}}^{{\rm{ - 1}}}},$$

(42)

where the uncertainty comprises the 95% confidence level.

• Going further. In addition to the above mentioned CASTLES list of multiple quasars with detailed information on observations, models and references [57], a similar compilation of lens candidates subdivided into “presently accepted” and “additional proposed” multiply imaged objects can be found at [71]. More time delays are becoming available for other lens systems (e.g., [23, 162]). Blandford & Kundić [26] provide a nice review in which they explore the potential to get a good determination of the extragalactic distance scale by combining measured time delays with good models; see also [161] and [205] for very recent summaries of the current situation on time delays and determination of the Hubble constant from lensing.

4.2 Quasar microlensing

Light bundles from “lensed” quasars are split by intervening galaxies. With typical separations of order one arcsecond between center of galaxy and quasar image, this means that the quasar light bundle passes through the galaxy and/or the galaxy halo. Galaxies consist at least partly of stars, and galaxy haloes consist possibly of compact objects as well.

Each of these stars (or other compact objects, like black holes, brown dwarfs, or planets) acts as a “compact lens” or “microlens” and produces at least one new image of the source. In fact, the “macro-image” consists of many “microimages” (Figure 9). But because the image splitting is proportional to the lens mass (see Equation (4)), these microimages are only of order a microarcsecond apart and can not be resolved. Various aspects of microlensing have been addressed after the first double quasar had been discovered [38, 39, 67, 90, 136, 171, 195].

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The surface mass density in front of a multiply imaged quasar is of order the “critical surface mass density”, see Equation (16). Hence microlensing should be occuring basically all the time. This can be visualized in the following way. If one assigns each microlens a little disk with radius equal to the Einstein ring, then the fraction of sky which is covered by these disks corresponds to the surface mass density in units of the critical density; this fraction is sometimes also called the “optical depth”.

The microlenses produce a complicated two-dimensional magnification distribution in the source plane. It consists of many caustics, locations that correspond to formally infinitely high magnification.

An example for such a magnification pattern is shown in Figure 10. It is determined with the parameters of image A of the quadruple quasar Q2237+0305 (surface mass density κ = 0.36; external shear γ = 0.44). Color indicates the magnification: blue is relatively low magnification (slightly demagnified compared to mean), green is slightly magnified and red and yellow is highly magnified.

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Due to the relative motion between observer, lens and source, the quasar changes its position relative to this arrangement of caustics, i.e. the apparent brightness of the quasar changes with time. A one-dimensional cut through such a magnification pattern, convolved with a source profile of the quasar, results in a microlensed lightcurve. Examples for microlensed lightcurves taken along the yellow tracks in Figure 10 can be seen in Figure 11 for two different quasar sizes.

In particular when the quasar track crosses a caustic (the sharp lines in Figure 10 for which the magnification formally is infinite, because the determinant of the Jacobian disappears, cf. Equation (31)), a pair of highly magnified microimages appears newly or merges and disappears (see [27). Such a microlensing event can easily be detected as a strong peak in the lightcurve of the quasar image.

In most simulations it is assumed that the relative positions of the microlenses is fixed and the lightcurves are produced only by the bulk motion between quasar, galaxy and observer. A visualization of a situation with changing microlens positions can be found in Figure 12 for three different values of the surface mass density:

• k = 0.2 in 12a.

• k = 0.5 in 12b.

• k = 0.8 in 12c.

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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 12 produces additional fluctuations in the lightcurve [105, 198].

This change of caustics shapes due to the motion of individual stars produces additional fluctuations in the lightcurve [105, 198].

Microlens-induced fluctuations in the observed brightness of quasars contain information both about the light-emitting source (size of continuum region or broad line region of the quasar, brightness profile of quasar) and about the lensing objects (masses, density, transverse velocity). Hence from a comparison between observed and simulated quasar microlensing (or lack of it) one can draw conclusions about the density and mass scale of the microlenses. It is not trivial, though, to extract this information quantitatively. The reason is that in this regime of optical depth of order one, the magnification is not due to a single isolated microlens, but it rather is a collective effect of many stars. This means individual mass determinations are not just impossible from the detection of a single caustic-crossing microlensing event, but it does not even make sense to try do so, since these events are not produced by individual lenses⁸. Mass determinations can only be done in a statistical sense, by comparing good observations (frequently sampled, high photometric accuracy) with simulations. Interpreting microlensed lightcurves of multiply-imaged quasars allows to determine the size of the continuum emitting region of the quasar and to learn even more about the central engine [69, 83, 147, 199].

So far the “best” example of a microlensed quasar is the quadruple quasar Q2237+0305 [79, 80, 110, 135, 199, 200, 207]. In Figure 13 two images of this system are shown which were taken in 1991 and 1994, respectively. Whereas on the earlier observation image B (top) is clearly the brightest, three years later image A (bottom) is at least comparable in brightness. Since the time delay in this system is only a day or shorter (because of the symmetric image arrangement), any brightness change on larger time scales must be due to microlensing. In Figure 14 lightcurves are shown for the four images of Q2237+0305 over a period of almost a decade (from [109]). The changes of the relative brightnesses of these images induced by microlensing are obvious.

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4.3 Einstein rings

If a point source lies exactly behind a point lens, a ring-like image occurs. Theorists had recognized early on [40, 55] that such a symmetric lensing arrangement would result in a ring-image, a so-called “Einstein-ring”. Can we observe Einstein rings? There are two necessary requirements for their occurence: the mass distribution of the lens needs to be axially symmetric, as seen from the observer, and the source must lie exactly on top of the resulting degenerate point-like caustic. Such a geometric arrangement is highly unlikely for point-like sources. But astrophysical sources in the real universe have a finite extent, and it is enough if a part of the source covers the point caustic (or the complete astroid caustic in a case of a not quite axial-symmetric mass distribution) in order to produce such an annular image.

In 1988, the first example of an “Einstein ring” was discovered [77]. With high resolution radio observations, the extended radio source MG1131+0456 turned out to be a ring with a diameter of about 1.75 arcsec. The source was identified as a radio lobe at a redshift of z_S = 1.13, whereas the lens is a galaxy at z_L = 0.85. Recently, a remarkable observation of the Einstein ring 1938+666 was presented [94]. The infrared HST image shows an almost perfectly circular ring with two bright parts plus the bright central galaxy. The contours agree very well with the MERLIN radio map (see Figure 15).

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By now about a half dozen cases have been found that qualify as Einstein rings [127].

Their diameters vary between 0.33 and about 2 arcseconds. All of them are found in the radio regime, some have optical or infrared counterparts as well. Some of the Einstein rings are not really 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. 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 time delay and the Hubble constant H₀ in these systems.

The Einstein ring systems provide some advantages over the multiply-imaged quasar systems for the goal to determine the lens structure and/or the Hubble constant. First of all the extended image structure provides many constraints on the lens. A lens model can be much better determined than in cases of just two or three or four point-like quasar images. Einstein rings thus help us to understand the mass distribution of galaxies at moderate redshifts. For the Einstein ring MG 1654+561 it was found [100] that the radially averaged surface mass density of the lens was fitted well with a distribution like Σ(r) ∝ r ^α , where α lies between -1.1 ≤ α ≤ -0.9 (an isothermal sphere would have exactly α = -1!); there was also evidence found for dark matter in this lensing galaxy.

Second, since the diameters of the observed rings (or the separations of the accompanying double images) are of order one or two arcseconds, the expected time delay must be much shorter than the one in the double quasar Q0957+561 (in fact, it can be arbitrarily short, if the source happens to be very close to the point caustic). This means one does not have to wait so long to establish a time delay (but the source has to be variable intrinsically on even shorter time scales. . . ).

The third advantage is that since the emitting region of the radio flux is presumably much larger than that of the optical continuum flux, the radio lightcurves of the different images are not affected by microlensing. Hence the radio lightcurves between the images should agree with each other very well.

Another interesting application is the (non-)detection of a central image in the Einstein rings. For singular lenses, there should be no central image (the reason is the discontinuity of the deflection angle). However, many galaxy models predict a finite core in the mass distribution of a galaxy. The non-detection of the central images puts strong constraints on the size of the core radii.

4.4 Giant luminous arcs and arclets

Zwicky had pointed out the potential use in the 1930s, but nobody had really followed up the idea, not even after the discovery of the lensed quasars: Galaxies can be gravitationally lensed as well. Since galaxies are extended objects, the apparent consequences for them would be far more dramatic than for quasars: galaxies should be heavily deformed once they are strongly lensed.

It came as quite a surprise when in 1986 Lynds & Petrosian [114] and Soucail et al. [178] independently discovered this new gravitational lensing phenomenon: magnified, distorted and strongly elongated images of background galaxies which happen to lie behind foreground clusters of galaxies (recent HST images of these two original arc clusters (and others) compiled by J.-P. Kneib can be found at [96], [97].

Rich clusters of galaxies at redshifts beyond z ≈ 0.2 with masses of order 10¹⁴M_⊙ are very effective lenses if they are centrally concentrated. Their Einstein radii are of the order of 20 arcseconds. 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 clusters. But there are many examples known with spectacularly long arcs which are curved around the cluster center, with lengths up to about 20 arcseconds.

The giant arcs can be exploited in two ways, as is typical for many lens phenomena. Firstly they provide us with strongly magnified galaxies at (very) high redshifts. These galaxies would be too faint to be detected or analysed in their unlensed state. Hence with the lensing boost we can study these galaxies in their early evolutionary stages, possibly as infant or proto-galaxies, relatively shortly after the big bang. The other practical application of the arcs is to take them as tools to study the potential and mass distribution of the lensing galaxy cluster. In the simplest model of a spherically symmetric mass distribution for the cluster, giant arcs form very close to the critical curve, which marks the Einstein ring. So with the redshifts of the cluster and the arc it is easy to determine a rough estimate of the lensing mass by just determining the radius of curvature and interpreting it as the Einstein radius of the lens system.

More detailed modelling of the lensing clusters which allows for the asymmetry of the mass distribution according to the visible galaxies plus an unknown dark matter component provides more accurate determinations for the total cluster mass and its exact distribution. More than once this detailed modelling predicted additional (counter-) images of giant arcs, which later were found and confirmed spectroscopically [99, 49].

Gravitational lensing is the third method for the determination of masses of galaxy clusters, complementary to the mass determinations by X-ray analysis and the old art of using the virial theorem and the velocity distribution of the galaxies (the latter two methods use assumptions of hydrostatic or virial equilibrium, respectively). Although there are still some discrepancies between the three methods, it appears that in relaxed galaxy clusters the agreement between these different mass determinations is very good [9].

Some general results from the analysis of giant arcs in galaxy clusters are: Clusters of galaxies are dominated by dark matter. The typical “mass-to-light ratios” for clusters obtained from strong (and weak, see below) lensing analyses are M/L ≥ 100M_⊙/L_⊙ [84].

The distribution of the dark matter follows roughly the distribution of the light in the galaxies, in particular in the central part of the cluster. The fact that we see such arcs shows that the central surface mass density in clusters must be high. The radii of curvature of many giant arcs is comparable to their distance to the cluster centers; this shows that core radii of clusters — the radii at which the mass profile of the cluster flattens towards the center — must be of order this distance or smaller. For stronger constraints detailed modelling of the mass distribution is required.

In Figures 16 and 17 two of the most spectacular cluster lenses producing arcs can be seen: Clusters Abell 2218 and CL0024+1654. Close inspection of the HST image of Abell 2218 reveals that the giant arcs are resolved (Figure 16), structure can be seen in the individual components [98] and used for detailed mass models of the lensing cluster. In addition to the giant arcs, more than 100 smaller “arclets” can be identified in Abell 2218. They are farther away from the lens center and hence are not magnified and curved as much as the few giant arcs. These arclets are all slightly distorted images of background galaxies. With the cluster mass model it is possible to predict the redshift distribution of these galaxies. This has been successfully done in this system with the identification of an arc as a star-forming region, opening up a whole new branch for the application of cluster lenses [50].

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In another impressive exposure with the Hubble Space Telescope, the galaxy cluster CL0024+1654 (redshift z = 0.39) was deeply imaged in two filters [33]. The combined picture (Figure 17) shows very nicely the reddish images of cluster galaxies, the brightest of them concentrated around the center, and the bluish arcs. There are four blue images which all have a shape reminiscent of the Greek letter Θ. All the images are resolved and show similar structure (e.g., the bright fishhook-like feature at one end of the arcs), but two of them are mirror inverted, i.e. have different parity! They lie roughly on a circle around the center of the cluster and are tangentially elongated. There is also another faint blue image relatively close to the cluster center, which is extended radially. Modelling reveals that this is a five-image configuration produced by the massive galaxy cluster. All the five arcs are images of the same galaxy, which is far behind the cluster at a much higher redshift and most likely undergoes a burst of star formation. This is a spectacular example of the use of a galaxy cluster as a “Zwicky” telescope.

In CL0024+1654 the lensing effect produces a magnification of roughly a factor of ten. Combined with the angular resolution of the HST of 0.1 arcsec, this can be used to yield a resolution that effectively corresponds to 0.01 arcsec (in the tangential direction), unprecedented in direct optical imaging. Colley et al. [42] map the five images “backward” to the source plane with their model for the cluster lens and hence reconstruct the unlensed source. They get basically identical source morphology for all arcs, which confirms that the arcs are all images of one source.

Recently, yet another superlative about cluster lenses was found: A new giant luminous arc was discovered in the field of the galaxy cluster CL1358+62 with the HST [62].

This arc-image turned out to be a galaxy at a redshift of z = 4.92. Up to a few months ago this was the most distant object in the universe with a spectroscopically measured redshift! In contrast to most other arcs, this one is very red. The reason is that due to this very high redshift, the Lyman-α emission of the galaxy, which is emitted in the ultra-violet part of the electromagnetic spectrum at a wavelength of 1216 Åis shifted by a factor of z + 1 ≈ 6 to the red part at a wavelength of about 7200 Å!

4.5 Weak/statistical lensing

In contrast to the phenomena that were mentioned so far, “weak lensing” deals with effects of light deflection that cannot be measured individually, but rather in a statistical way only. As was discussed above “strong lensing” — usually defined as the regime that involves multiple images, high magnifications, and caustics in the source plane — is a rare phenomenon. Weak lensing on the other hand is much more common. In principle, weak lensing acts along each line of sight in the universe, since each photon’s path is affected by matter inhomogeneities along or near its path. It is just a matter of how accurate we can measure (cf. [144]).

Any non-uniform matter distribution between our observing point and distant light sources affects the measurable properties of the sources in two different ways: The angular size of extended objects is changed and the apparent brightness of a source is affected, as was first formulated in 1967 by Gunn [74, 75].

A weak lensing effect can be a small deformation of the shape of a cosmic object, or a small modification of its brightness, or a small change of its position. In general the latter cannot be observed, since we have no way of knowing the unaffected position⁹.

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, 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: The statistical incoherent lens-induced change of the apparent brightness of (widely separated) “standard candles” — like type Ia supernovae — affects the accuracy of the determination of cosmological parameters [63, 88, 197].

The idea to use the weak distortion and tangential alignment of faint background galaxies to map the mass distribution of foreground galaxies and clusters has been floating around for a long time. The first attempts go back to the years 1978/79, when Tyson and his group tried to measure the positions and orientations of the then newly discovered faint blue galaxies, which were suspected to be at large distances. Due to the not quite adequate techniques at the time (photographic plates), these efforts ended unsuccessfully [185, 191]. Even with the advent of the new technology of CCD cameras, it was not immediately possible to detect weak lensing, since the pixel size originally was relatively large (of order an arcsecond). Only with smaller CCD pixels, improved seeing conditions at the telecope sites and improved image quality of the telescope optics the weak lensing effect could ultimately be measured.

Now we will briefly summarize the technique of how to use the weak lensing distortion in order to get the mass distribution of the underlying matter.

4.5.1 Cluster mass reconstruction

The first real detection of a coherent weak lensing signal of distorted background galaxies was measured in 1990 around the galaxy clusters Abell 1689 and CL1409+52 [186]. It was shown that the orientation of background galaxies — 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. This analysis is based on the assumption that the major axes of the background galaxies are intrinsically randomly oriented.

With the elegant and powerful method developed by Kaiser and Squires [87] the weak lensing signal can be used to quantitatively reconstruct the surface mass distribution of the cluster. This method relies on the fact that the convergence κ(θ) and the two components of the shear γ₁(θ), γ₂(θ) are linear combinations of the second derivative of the effective lensing potential Ψ(θ) (cf. Equations (33, 34, 35)). After Fourier transforming the expressions for the convergence and the shear one obtains linear relations between the transformed components ̃κ, ̃γ₁, ̃γ₂. Solving for ̃κ and inverse Fourier transforming gives an estimate for the convergence κ (details can be found in [87, 86, 128] or [179]).

The original Kaiser-Squires method was improved/modified/extended/generalized by various authors subsequently. In particular the constraining fact that observational data are available only in a relatively small, finite area was implemented. Maximum likelihood techniques, non-linear reconstructions as well as methods using the amplification effect rather than the distortion effect complement each other. Various variants of the mass reconstruction technique have been successfully applied to more than a dozen rich clusters by now.

Descriptions of various techniques and applications for the cluster mass reconstruction can be found in, e.g., [1, 16, 18, 20, 30, 34, 76, 85, 129, 170, 173, 174, 176, 206]. In Figure 18 a recent example for the reconstructed mass distribution of galaxy cluster CL1358+62 is shown [78].

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• Going further. We could present here only one weak lensing issue in some detail: the reconstruction of the mass distribution of galaxy clusters from weakly distorted background images. Many more interesting weak lensing applications are under theoretical and observational investigation, though. To name just a few:

• — Constraints on the distribution of the faint galaxies from weak lensing (e.g., [61, 113]);

• — galaxy-galaxy lensing (e.g., [32]);

• — lensing by galaxy halos in clusters (e.g., [130]);

• — weak lensing effect by large scale structure and/or detection of dark matter concentrations; this considers both shear effects as well as magnification effects (e.g. [13, 17, 168, 204]);

• — determination of the power spectrum of the matter distribution (e.g. [22]); or

• — the weak lensing effects on the cosmic microwave background (e.g. [119, 122, 175]).

An upcoming comprehensive review on weak lensing by Schneider & Bartelmann [21] treats both theory and applications of weak lensing in great depths.

4.6 Cosmological aspects of (strong) lensing

Gravitational lenses can be used in two different ways to study the cosmological parameters of the universe. The first is to explore a particular lens system in great detail, determine all possible observational parameters (image positions/brightnesses/shapes; matter/light distribution of lens; time variability etc.) and model both lens and source in as much detail as possible. This way one can in principle determine the amount of dark matter in the lens and -maybe even more importantly — the value of the Hubble constant. A reliable determination of the Hubble constant establishes the extragalactic distance scale, something astronomers have been trying to do for more than 70 years [126].

The second approach is of statistical nature: find out how many (what fraction of) quasars are multiply imaged by gravitationally lensing, determine their separation and redshift distributions [184] and deduce the value of (or limits to) Ω_compact — matter in clumps of, say, 10⁶ ≤ M/M_⊙ ≤ 10¹⁴ — and to Ω_Λ — the value of the cosmological constant.

Since quasars are rare objects and lensing is a relatively rare phenomenon, steps 1 and 2 are quite difficult and time-consuming. Nevertheless, a number of systematic quasar surveys with the goal to find (many) lens systems with well defined selection criteria have been done in the past and others are underway right now (e.g. [35, 116, 118, 201, 209]).

The largest survey so far, the CLASS survey, has looked at about 7000 radio sources at the moment (the goal is 10000). In total CLASS found 12 new lens systems so far. Interestingly, all the lenses have small separations (Δθ < 3arcsec), and all lensing galaxies are detected [35, 82]. That leaves little space for a population of dark objects with masses of galaxies or beyond. A detailed discussion of lens surveys and a comparison between optical and radio surveys can be found in [101].

The idea for the determination of the cosmological constant Ω_Λ = Λ/(3H ₀ ² ) from lens statistics is based on the fact that the relative lens probability for multiple imaging increases rapidly with increasing Ω_Λ (cf. Figure 9 of [37]). This was first pointed out 1990 [64, 183]. The reason is the fact that the angular diameter 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 “fractional volume” that is affected by a certain lens) depend on these distances [37]. This can be seen, e.g. when one looks at the critical surface mass density required for multiple imaging (cf. Equation (16)) which depends on the angular diameter distances.

The consequences of lensing studies on the cosmological constant can be summarized as follows. The analyses of the frequency of lensing are based on lens systems found in different optical and radio surveys. The main problem is still the small number of lenses. Depending on the exact selection criteria, only a few lens systems can be included in the analyses. Nevertheless, one can use the existing samples to put limits on the cosmological constant. Two different studies found 95%-confidence limits of Ω_Λ < 0.66 [102] and Ω_Λ < 0.7 [117, 156]. This is based on the assumption of a flat universe (Ω_matter + Ω_Λ= 1). Investigations on the matter content of the universe from (both “macro-” and “micro-”) lensing generally conclude that the fractional matter in compact form cannot exceed a few percent of the critical density (e.g. [36, 46, 131, 167]).

4.7 Galactic microlensing

It has been known for more than two decades that halos of galaxies must contain some unknown kind of dark matter. Many different particles/objects had been suggested as constituents of this halo dark matter. The candidates can be divided into the two broad categories “elementary particles” and “astronomical bodies”. A conservative candidate for this dark matter are brown dwarfs, objects with masses less than 0.08M_⊙ so that the central temperature is not high enough to start helium fusion. These objects are certain to exist, we just do not know how many there are.

In 1986 Paczyński [137] suggested a method to test observationally whether the Milky Way halo is made of such brown dwarfs (or other astronomical objects in roughly this mass range). Subsequently this type of dark matter candidate was labelled “Macho” for MAssive Compact Halo Object [72]. If one could continuously observe the brightness of stars of our neighbouring galaxy Large Magellanic Cloud (LMC) one should see typical fluctuations in some of these stars due to the fact that every now and then one of these compact halo objects passes in front of the star and magnifies its brightness. The only problem with this experiment is the low probability for such an event: Only about one out of three million LMC stars would be significantly magnified at any given time.

The underlying scenario is very simple: Due to the relative motion of observer, lensing Macho and source star the projected impact parameter between lens and source changes with time and produces a time dependent magnification. If the impact parameter is smaller than an Einstein radius then the magnification is μ_min > 1.34 (cf. Equation (22)).

For an extended source such a sequence is illustrated in Figure 19 for five instants 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. Hence the two images cannot be resolved individually, we can only observe the brightness of the combined image pair. This is illustrated in Figures 20 and 21 which show the relative tracks and the respective light curves for five values of the minimum impact parameter u_min.

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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. The duration of an event is determined by three unknown parameters: the mass of the lens, the transverse velocity and the distances of lens and source. It is impossible to disentangle these for individual events. Only with a model for the spatial and velocity distribution of the lensing objects and comparison with “simulated microlensing events” it is possible to obtain information about the masses of the lensing objects and their density.

The observations towards the Large Magellanic Cloud show that there are fewer microlensing events than one would expect if the halo of the Milky Way was made entirely of these compact objects. The latest published results from the microlensing experiments that monitor stars in the LMC indicate that the optical depths toward the LMC is about τ ≈ 3 × 10^-7. The observations are consistent with 50% of the Milky Way halo made of compact objects with most likely masses of 0.5 _-0.2 ^+0.3 M_⊙ [7]. But the number of observed events is still small (in this analysis eight events were used) and hence the uncertainties are large; in fact, it cannot even be excluded that none of the observed events is due to an unknown halo population [65].

The same type of experiment (searching for microlensing events) is being performed in the direction of the galactic bulge as well, the central part of the Milky Way. By now more than 200 microlensing events have been detected in this direction (for an example see Figure 22). Among them are a number of “binary lens”-events (which have a very typical signature of at least two caustic crossings, cf. Figure 23). This is about three times as many microlensing events as were expected/predicted. Several groups try to explain this “over-abundance” of events to have a new look at the stellar content and the dynamics of the bar/bulge of the Galaxy. The latest published results can be found in [8].

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With these microlensing experiments gravitational lensing has established itself as a new tool to study the structure of the Milky Way. This type of microlensing also holds some promise for the future. It can be used, e.g. to study the frequency of binary stars. One of the most interesting possibilities is to detect planets around other stars by extending the sensitivity of the binary lenses to smaller and smaller companion masses [115, 196].

For a recent comprehensive presentation of galactic microlensing and beyond see [138]. Various aspects of microlensing in the local group are reviewed in detail. Another review article on the basics and the results of galactic microlensing can be found in [158].