# Gravitational lensing as a powerful astrophysical tool Multiple quasars giant arcs and extrasolar planets.

код для вставкиСкачатьAnn. Phys. (Leipzig) 15, No. 1 – 2, 43 – 59 (2006) / DOI 10.1002/andp.200510169 Gravitational lensing as a powerful astrophysical tool: Multiple quasars, giant arcs and extrasolar planets Joachim Wambsganss∗ Zentrum für Astronomie der Universität Heidelberg (ZAH), Mönchhofstr. 12–14, 69120 Heidelberg, Germany Received 7 September 2002, revised 14 November 2002, accepted 15 December 2002 Published online 23 December 2005 In the 25 years since the discovery of the ﬁrst double quasar Q0957+561, gravitational lensing has established itself as a valuable tool in many branches of astronomy. Fields as different as galactic structure, cosmology, or extrasolar planets beneﬁt from the gravitational lensing effect. This article starts with a brief historic reﬂection, then the basics of light deﬂection are reviewed. Observable lensing effects and a few examples of strong lensing phenomena are shown. In the main part four applications of “strong” lensing will be presented and discussed: • The determination of the Hubble constant from time delay measurements in multiple quasars; it is argued that this method of determining H0 is competitive with other methods by now. The lensing-derived values of H0 are on the low side. • Microlensing of quasars – the effects of compact stellar-mass objects on the apparent brightness – allows us to constrain the quasar size and the occurrence of dark matter objects. • The frequency of giant luminous arcs strongly depends on the high mass end of the galaxy cluster distribution. Recent investigations show that arc statistics is in agreement with the concordance cosmological model. • Searching for extrasolar planets is one of the most recent applications of gravitational lensing. The ﬁrst detection shows that the method works well. This planet-search method is complementary to other programs and has the potential to detect exo-planets with lower masses than other ground-based techniques. An outlook is provided on the prospects of gravitational lensing in the next few years. In particular the magniﬁcation effect on faint high-redshift sources will be used for the investigation of the early universe, and the detection of low-mass extrasolar planets will provide a valuable sample for statistical evaluations of the frequency of exoplanets. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 A short history of light deﬂection Gravitational lensing is considered a relatively new ﬁeld in astrophysics. However, the history of light deﬂection is more than 200 years old (see in more detail in [30]). As early as 1784, Michell considered the deﬂection of light by the gravity of other bodies. In 1801, Soldner published a paper on light deﬂection, in which he determined – based on Newtonian mechanics – the deﬂection of a light ray just passing the solar limb to α,Soldner = 2GM c2 R = 0.84 arcsec (with G – gravitational constant, c – velocity of light, M – mass of the sun, R – radius of the sun). More than 100 years later, Einstein worked on the same problem and derived the same value [8]. Only after the General Theory of Relativity was ﬁnished, Einstein published the value of 4GM α,Einstein1915 = 2 = 1.74 arcsec, c R ∗ E-mail: jkw@ari.uni-heidelberg.de c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 44 J. Wambsganss: Gravitational lensing – powerful astrophysical tool for the light deﬂection at the solar limb, which was measured and proven to be correct in the famous solar eclipse expeditions led by Eddington in 1919 [7]. In the 1920s/1930s, there were a few papers dealing with lensing, e.g., Chwolson investigated the situation of double imaging. In particular he ﬁgured out that for perfect alignment between lens and source the result would be a ring-like image [5]. Einstein looked again into this issue and derived the magniﬁcations for the double images of a background star lensed by an intervening foreground star, but he was very sceptical about the possibility of observing this gravitational lensing effect [9]. Zwicky, on the other hand, was convinced that galaxies should and would act as gravitational lenses, for him this appeared to be an unavoidable consequence [43, 44] of the light deﬂection theory. In the 1960s there was another wave of theoretical investigations of the lensing effect. In particular, Refsdal showed that one can determine the Hubble constant from the time delay between the images of a multiply lensed quasar [25]. And ﬁnally in 1979, Walsh et al. [33] discovered the ﬁrst doubly imaged quasar Q0957+561. Although the deﬂection of light at the solar limb – hailed as the ﬁrst experiment to conﬁrm a prediction of Einstein’s theory of General Relativity – happened already in 1919, it took more than half a century to establish this phenomenon observationally in some other environment. 2 The basics of gravitational lensing The path, the size and the cross section of a light bundle propagating through spacetime in principle are affected by all the matter between the light source and the observer. For most practical purposes one can assume that the lensing action is dominated by a single matter inhomogeneity at some location between source and observer. This is usually called the “thin lens approximation”: all the action of deﬂection is thought to take place at a single distance. Here the basics of lensing will be brieﬂy derived and explained in the thin lens approxmation: lens equation, Einstein radius, image positions and magniﬁcations, time delay. More detailed reviews/introductions on lensing can be found in, e.g., [23, 30, 38]. 2.1 Lens equation The basic setup for such a simpliﬁed gravitational lens scenario involving a point source and a point lens is displayed in Fig. 1. 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 S1 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. In Fig. 1, the corresponding angles and angular diameter distances DL , DS , DLS are indicated. In the thinlens approximation the hyperbolic paths are approximated by their asymptotes. In the circular-symmetric case the deﬂection angle is given as α̃(ξ) = 4GM (ξ)1 , c2 ξ (1) where M (ξ) is the mass of the lens 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: θDS = βDS + α̃DLS (2) (for θ, β, α̃ 1; this condition is fulﬁlled in practically all astrophysically relevant situations). With the deﬁnition of the reduced deﬂection angle as α(θ) = (DLS /DS )α̃(θ), this lens equation can be expressed as: β = θ − α(θ). c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (3) www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 1 – 2 (2006) 45 Fig. 1 The relations between the various angles and distances involved in the lensing setup can be derived for α̃ 1 and formulated in eq.(3), the lens equation. The symbols ‘O’, ‘L’, and ‘S’ mean ‘observer’, ‘lens’, and ‘source’, respectively. ‘S1 ’and ‘S2 ’are the two apparent positions of the doubly imaged source. The angular diameter distances DL , DS , and DLS are between observer-lens, observer-source, and source-lens. All angles involved are small compared to one. 2.2 Einstein radius For a point lens of mass M the deﬂection angle is given by Eq. (1). Plugging it into Eq. (3) and using the relation ξ = DL θ (cf. Fig. 1) one obtains: β(θ) = θ − DLS 4GM . DL DS c2 θ (4) For the special case in which the source lies exactly behind the lens (β = 0), due to the symmetry a ring-like image occurs whose angular radius is called Einstein radius θE : 4GM DLS θE = . c2 DL DS (5) The Einstein radius deﬁnes the angular scale for a lens situation. For a massive galaxy with a mass of M = 1012 M at a redshift of zL = 0.5 and a source at redshift zS = 2.0 (here H = 50km sec−1 Mpc−1 is used as the value of the Hubble constant and an Einstein-deSitter universe) the Einstein radius is M θE ≈ 1.8 arcsec (6) 12 10 M (note that for cosmological distances in general DLS = DS − DL !). For a galactic microlensing scenario in which stars in the disk of the Milky Way act as lenses for bulge stars close to the center of the Milky Way, the scale deﬁned by the Einstein radius is θE ≈ 0.5 M milliarcsec. M (7) 2.3 Image positions and magniﬁcations The lens equation (3) can be re-formulated in the case of a single point lens: β =θ− www.ann-phys.org 2 θE . θ (8) c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 46 J. Wambsganss: Gravitational lensing – powerful astrophysical tool Solving this for the image positions θ, one ﬁnds that an isolated point source always produces two images of a background source. The positions of the images are given by the two solutions: 1 2 2 β ± β + 4θE . (9) θ1,2 = 2 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. Hence the magniﬁcation µ is given as µ= θ dθ . β dβ (10) In the symmetric case above the image magniﬁcation can be written as (by using the lens equation): 4 −1 θE u2 + 2 1 (11) = √ ± . µ1,2 = 1 − θ1,2 2u u2 + 4 2 Here u is deﬁned as the “impact parameter”, the angular separation between lens and source in units of the Einstein radius: u = β/θE . The magniﬁcation of one image (the one inside the Einstein radius) is formally negative. This means it has negative parity: it is mirror-inverted. For β → 0 the magniﬁcation diverges: in the limit of geometrical optics the Einstein ring of a point source has inﬁnite magniﬁcation1! The sum of the absolute values of the two image magniﬁcations is the total magniﬁcation µ: u2 + 2 µ = |µ1 | + |µ2 | = √ . u u2 + 4 (12) Note that this value is (always) larger than one2! The “sum” of the two image magniﬁcations is unity (considering that one value is negative) µ1 + µ2 = 1. (13) 2.4 Time delay and Fermat’s theorem The deﬂection angle is the gradient of an effective lensing potential ψ (see [29]). Hence the lens equation can be rewritten as or −∇ θψ = 0 (θ − β) (14) 2 − ψ = 0. θ 1(θ − β) ∇ 2 (15) The term in brackets appears as well in the physical time delay function for gravitationally lensed images: (1 + zL )DL DS 1 2 (16) τ (θ, β) = τgeom + τgrav = (θ − β) − ψ(θ) . c DLS 2 β), the gravitational potential ψ, and the This time delay surface is a function of the image geometry (θ, distances DL , DS , and DLS . The ﬁrst part – the geometrical time delay τgeom – reﬂects the extra path length 1 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. 2 This does not violate energy conservation, since this is the magniﬁcation relative to an “empty” universe and not relative to a “smoothed out” universe. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 1 – 2 (2006) 47 compared to the direct line between observer and source. The second part – the gravitational time delay τgrav – is the retardation due to the gravitational potential of the lensing mass (known and conﬁrmed as Shapiro delay in the solar system). From Eqs. (15) and (16) it follows that the gravitationally lensed images appear at locations that correspond to extrema in the light travel time, which reﬂects Fermat’s principle in gravitational-lensing optics. The (angular-diameter) distances that appear in Eq. (16) depend on the value of the Hubble constant [40]; 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 ψ of the lens (see below). 3 Effects of gravitational lensing The deﬂection and deformation of a light bundle due to the gravitational attraction of some intermediate matter distribution can lead to four different observational effects: • Shift of position The offset of the position of a background star due to some foreground star was the ﬁrst observational realization of gravitational light deﬂection lensing. The measurement of the positional shift of stars near the sun during the solar eclipse 1919 was in agreement with Einstein’s prediction that the angle of deﬂection at the solar limb is 1.75 arcseconds. In general, though, the shift of position of a background source due to gravitational lensing by some intervening object is not directly observable. The reason is simply that there is no way to determine “the true position” of an object, unless – as was the case in the eclipse – the lensing action is only temporary. A similar application, the shift of quasar positions due to relative motion of intermediate compact objects has been suggested by [20] as an observational test for intervening compact objects with the next generation astrometric satellite SIM (Space Interferometric Mission). This was worked out quantitatively and in more detail by [31] recently. • (De-)magniﬁcation Due to the change of the cross section of a light bundle by the attraction, the apparent brightness of an object is changed as well. Few objects are highly magniﬁed, most objects are slightly demagniﬁed. Due to photon conservation (except for the negligible contribution of the very few photons which are passing black holes too closely), the average magniﬁcation is one. This effect materializes differently for “point” sources (unresolved) and for “extended” sources. The former appear brighter or fainter, the latter appear bigger or smaller as a consequence of lensing. In both cases it is hardly possible in individual cases to tell whether and how strongly lensing affects the images, because the luminosity functions of point sources (stars, quasars) as well as the “shape” function of extended sources (galaxies) are very broad in general. In the case of astrophysical “standard candles” (e.g., supernovae type Ia), though, lensing may have an observable effect (cf. [35]). In time variable scenarios, due to the relative motion of lens, source and observer, the change of magniﬁcation as a function of time can be followed. This technique is used in the search for (dark?) massive compact halo objects (machos). It is called microlensing, because the splitting angle is too small to be observed directly (for a review see [24]). • Distortion A further consequence of lensing is the differential distortion of extended sources. When the shape of an object is dramatically affected by strong lensing, this is easily recognizable individually, like, e.g., giant luminous arcs which are images of ordinary background galaxies strongly distorted by galaxy clusters (e.g. [6]). If the mass concentration is weaker, slightly tangentially expanded arclets can be found at larger distances to the cluster center in a statistical way. The most striking distortion effect can be seen in the Einstein rings, where an extended source is situated perfectly behind a symmetric lens (e.g. [13]). www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 48 J. Wambsganss: Gravitational lensing – powerful astrophysical tool Fig. 2 Top left: example of distant source with some structure. Top right: two-dimensional magniﬁcation distribution; bright means high magniﬁcation. Bottom left: source superimposed on the magniﬁcation distribution. Bottom right: corresponding image conﬁguration which shows all the effects of lensing: shift of position, image distortion, magniﬁed/demagniﬁed and multiple images. • Multiple imaging The most dramatic lensing effect is the multiple imaging of background sources. More than one hundred cases of double and multiple quasars are now known today (cf. the CASTLEs Web page listed under [22]). And there are multiply imaged galaxies as well: an example of up to ﬁve images of a background galaxy is produced by the foreground galaxy cluster CL0024+1654 ( [6]). In Fig. 2, all four effects of lensing are visible. In the top left panel there is an example of an extended “source” with some internal structure. The top right panel shows a certain magniﬁcation distribution in the source plane due to a number of point lenses in the foreground. The different gray scales indicate the magniﬁcation as a function of position: bright means highly magniﬁed. In the bottom left panel the “source” is superimposed on the magniﬁcation pattern, and in the bottom right panel the image conﬁguration produced by this arrangement of lenses can be seen. Compared with the original source in the top left panel, it is obvious that the images are shifted in position, strongly distorted, (de-)magniﬁed, and multiple. Even the change of parity can be seen in the mirror-inverted letters. 4 Lensing phenomena: multiple quasars, luminous arcs, Einstein rings Quite a variety of spectacular lensing phenomena have been observed in recent years. In Fig. 3, three of the most dramatic examples are presented: Multiply-imaged quasars, Einstein rings and Giant luminous arcs: 4.1 Multiply-imaged quasars In Fig. 3a, Q0957+561, the ﬁrst double quasar to be discovered [33], is shown in a recent HST image. The two quasar images have identical redshift (zQ = 1.41), they are separated by 6.1 arcseconds, the core of the lensing galaxy (zG = 0.36) can be seen close to image B (top), as well as some extended emission of galaxy light. By now more than 100 multiply-imaged quasar systems have been found, some of them with up to ten images (three sources). Updated tables of multiply-imaged quasars and gravitational lens candidates with separation and redshift information as well as additional references are provided by the CASTLE group [22]. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 1 – 2 (2006) a) b) c) d) 49 Fig. 3 a) In this Hubble Space Telescope image of the double quasar Q0957+561A,B, the two images A (bottom) and B (top) are separated by 6.1 arcseconds. Image B is about 1 arcsecond away from the core of the galaxy, and hence seen “through” the galaxy [11]. b) Einstein ring B1938+666; the infrared HST/NICMOS image shows the ring with a diameter of about 0.95 arcseconds plus the central lensing galaxy [13]. c) Galaxy Cluster CL0024+1654 with multiple images of a blue background galaxy [7]. d) Galaxy Cluster Abell 2218 with many giant luminous arcs (Credits: E.E. Falco, N. Jackson, W.N. Colley, T. Kundić, HST, STScI and NASA). 4.2 Einstein rings If a point source lies exactly behind a point lens, a ring-like image occurs. Theorists had recognized early on [5, 9] that such a symmetric lensing arrangement would result in a ring-image, a so-called “Einsteinring”. Can Einstein rings be observed? There are two necessary requirements for their occurrence: the mass distribution of the lens needs to be roughly axially symmetric, as seen from the observer, and (part of) the source must lie exactly on top of the resulting degenerate point-like caustic. A remarkable example of an Einstein ring is B1938+666. The infrared HST image [13] shows an almost perfectly circular ring with two bright parts plus the bright central galaxy (see Fig. 3b). By now about a dozen cases have been found that qualify as Einstein rings [22]. Their diameters vary between 0.33 and about 2 arcseconds. Most of them were originally found in the radio regime, some have optical or infrared counterparts as well. 4.3 Giant luminous arcs and arclets Rich clusters of galaxies at redshifts beyond z ≈ 0.2 with masses of order 1014 M are very effective lenses if they are centrally concentrated. Their Einstein radii are of the order of 20 arcseconds. Since most clusters do not really have 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. In Fig. 3c one of the most spectacular cluster lenses producing arcs can be seen: CL0024+1654 (redshift z = 0.39). Most of the bright galaxies are part of the galaxy cluster. There are four tangentially stretched images which have a shape reminiscent of the Greek letter Θ. All the images are resolved and show similar structure (e.g., a 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 (see [6]). This is a spectacular www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 50 J. Wambsganss: Gravitational lensing – powerful astrophysical tool example of the use of a galaxy cluster as a “Zwicky” telescope. In Fig. 3d a second remarkable arc-cluster is shown, Abell 2218, yet another example of a massive galaxy cluster producing many dozens of strongly curved arc images of background galaxies. 5 Four applications of strong lensing In this section, four astrophysical/cosmological applications of strong gravitational lensing are presented. The ﬁrst one is the determination of the Hubble constant, the next addresses the question whether the dark matter in the halos of distant galaxies can be in the form of compact objects, the third one deals with the statistics of giant luminous arcs as a tool to judge the concordance cosmological model, and ﬁnally recent progress and success in the search for extrasolar planets with gravitational microlensing is being reported. 5.1 Time delay and Hubble constant More than 40 years ago, Refsdal pointed out that the differential time delay between two or more gravitationally lensed images of a background object establishes an absolute physical distance scale (c∆t) in the system [25]. Thus, the distance to a high–redshift object can be directly measured, once it is known what fraction of the total travel time the time delay establishes. Hubble’s constant H0 is inversely proportional to the time delay, the constant of proportionality depends on the cosmological model and on the details of the lens model. The main strengths of this lensing method for the determination of the extragalactic distance scale are: • It is a geometrical method based on the well understood and experimentally veriﬁed physics of General Relativity in the weak–ﬁeld limit. By contrast, most conventional astronomical techniques for measuring extragalactic distances rely either on empirical relationships or on our understanding of complex astrophysical processes, or both. • It provides a direct, single step measurement of H0 for each system and thus avoids the propagation of errors along the “distance ladder” which is no more secure than its weakest rung. • It measures distances to cosmologically distant objects, thus precluding the possibility of confusing the local with the global expansion rate. • Independent determinations of H0 in many lens systems with different source and lens redshifts provide an internal consistency check on the answer. The application of this method needs two ingredients: an accurate measurement of the time delay and a reliable theoretical model for the mass distribution of the lensing system. Due to the difﬁculty of fulﬁlling these two requirements, the practical application of Refsdal’s method for measuring H0 has proven quite challenging and has been long delayed. For the lens system 0957+561A,B – by far the best studied case – there were two basic problems: First, there has been sufﬁcient ambiguity in detailed models of the mass distribution in the lensing galaxy and the associated galaxy cluster to allow values of H0 different by a factor of two or more to be consistent with the same measured time delay. Fortunately, this problem has been much alleviated by recent theoretical and observational work. Second, despite extensive optical and radio monitoring programs extending over a period of more than 15 years, as of 1995, values of the measured delay discrepant by more than 30% were debated in the literature3. In particular, there were two sets of values favored: studies found delays either in the ranges 400–420 days or 530–540 days. These two rough values, the “short delay” and the “long delay” were obtained both by applying the same statistical techniques to different data sets and by applying different statistical techniques to the same data. Only relatively recently 3 There are a number of reasons why it took so long, among them poor coverage of the quasar’s lightcurve in the early years (it was very difﬁcult to get time on big telescopes for “monitoring” quasars), temporal lack of intrinsic variability of the quasar, and also the fact that different statistical methods resulted in diverging values for the time delay. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 1 – 2 (2006) 51 Fig. 4 Combined lightcurves of leading image A (blue, full symbols, lower dates: Dec 94 ... Apr 95) and image B (red, open symbols, upper dates: Dec 95 ... June 96) of the double quasar Q0957+561, the latter is shifted by the optimal time delay of ∆t = 417 days (after [18]). a robust determination of the time delay has been achieved [18] which resolved the controversy in favor of the short delay and which is summarized here: Continued photometric monitoring of the gravitational lens system 0957+561A,B in the g and r bands with the Apache Point Observatory 3.5 m telescope showed a sharp event in the (trailing) B image light curve in Jan/Feb 1996, at the time predicted for the “short value” of the differential time delay in a previously published paper [17]. The prediction was based on the observation of the event during 1995 in the (leading) A image. This success conﬁrmed the “short delay”, and the absence of any such feature at a delay near 540 days rejects the “long delay” for this system, thus resolving the long standing controversy. A series of statistical analyses of the light curve data yield a quite robust best ﬁt delay of (417 ± 3) days (2σ). The data points for both images in the g-band together are displayed in Fig. 4. For a complex gravitational lens system such as 0957+561, however, even a precise measurement of the differential time delay may still leave one far from the goal of a good determination of H0 (cf. [27]), simply because of large uncertainties and degeneracies in the lens mass distribution. The relation between time delay and distance is controlled primarily by the total projected mass of the lens within the circle deﬁned by the diameter connecting the two images M (< r) and by its derivative dM/dr. Moreover, changing Ωmatter from 1.0 to 0.1 increases H0 by only 7%. Introducing the cosmological constant while keeping the universe ﬂat results in an increase of just 4% in the Ωmatter = 0.25, ΩΛ = 0.75 model. There is one remaining degeneracy which cannot be removed by the observed properties of the lensing conﬁguration alone, namely how much of the lensing is contributed by mass associated with the galaxy G1 versus how much is supplied by the associated galaxy cluster [11]. Since these two must sum to a known value (in order to produce the observed splitting), the degeneracy may be parameterized by either the one or the other. One can adopt either σobs , the central line–of–sight velocity dispersion of G1, to measure its contribution, or κ, the dimensionless lensing surface mass density contributed by the cluster, to parameterize its effect. In order to derive an H0 value, one needs to independently measure one or the other parameter. Measuring both provides a good internal consistency check. www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 52 J. Wambsganss: Gravitational lensing – powerful astrophysical tool With a value of σcl = (720 ± 250) km/s adopted as the 95% conﬁdence region for the cluster velocity dispersion, and using r = 22 arcsec for the distance of G1 from the center of the cluster and rc = 5 arcsec for the cluster core radius, the estimate for the cluster surface density at the position of the lensing event is κ = (0.22 ± 0.14) (2σ) (details in [18]). This result is insensitive to the precise value of rc , but it could increase considerably if G1 were much closer to the center of the cluster. Thus, a value of 1−κ +12 −1 km s−1 Mpc−1 = 64+12 H0 = 64−13 Mpc−1 −13 km s 0.78 is obtained, where the errors reﬂect the total 95% conﬁdence interval. A measurement of the velocity dispersion in G1 from a high signal–to–noise Keck LRIS spectrum [10] showed a value roughly in the range σobs = (275 ± 30) km/s (2σ). Inserting this value of σobs and again propagating all the relevant errors, leads to 2 σobs −1 H0 = 64+7.5 km s−1 Mpc−1 = 64+13 Mpc−1 . −9.8 −14 km s −1 275 km s (2σ error intervals). A more detailed report on the data, the model and the interpretation can be found in [18]). By now, time delays have been measured in about a dozen multiple quasar systems, and values for the Hubble constant are determined for all of them. The uncertainties in H0 are still not as small as one would like them to be (in the other cases the lens models are often more accurate, but the time delays have larger errors), but the bottom line is that the lens-derived values for the Hubble constant are always “lowish” [26], though formally lensing determined values are still in agreement within 2σ with the slightly higher value of H0 determined by the Hubble Space Telescope Key Project. In a very recent paper [16], the time delays and mass distributions of ten quasar lens systems were analysed coherently. Modelling the lens mass distributions as isothermal spheres (which would correspond to ﬂat rotation curves), an average value of H0 = (48±3) km/s/Mpc was obtained. Using a mass distribution corresponding to a constant mass-to-light ratio, the result was H0 = (73 ± 3) km/s/Mpc. These two values bracket the range of values for the Hubble constant derivable from gravitational lensing, with a clear tendency towards the lower end, if we allow for the existence of dark matter in the outer parts of galaxies, for which there is overwhelming evidence. This result shows in fact, that one can also turn around the problem: If one assumes the Hubble constant is known, one can take it and use it to determine the (dark matter) mass distribution of galaxies this way, as was done by [15]. 5.2 Microlensing of quasars Two quasar systems have been explored in detail for microlensing, the effects of stellar mass objects along the line of sight to the quasar images. The most dramatic effects have been seen in the quadruple system Q2237+0305, where four quasar images are centered around the core of the lensing galaxy. Since its discovery, this system has shown uncorrelated ﬂuctuations between the images, which were interpreted as microlensing [12, 34]. The problem was, however, the poor coverage in time. Only with the dedicated telescope and the dedicated scientists of the OGLE team, a good time coverage could be reached, with more than 100 data points per year. This observing strategy resulted in a dramatic increase in data quality, as can be seen in Fig. 5a and [41]. The individual images ﬂuctuate by as much as a factor of two in a few months, and these ﬂuctuations are very well resolved. In this multiple quasar system there is no need to invoke potential dark matter objects as lenses: the four images are seen through the central part of the lensing galaxy, which is full of ordinary main sequence stars. The interpretation of the data is consistent with low-mass stars in the core of the galaxy being the lenses, and the size of the quasar continuum emission region to be of order 1015 cm or smaller (e.g., [42]). The other system that has been closely inspected for microlensing – and which is more interesting with respect to dark matter searches – is the double quasar Q0957+561. The lightcurves of images A and B c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 1 – 2 (2006) 53 Difference g light curves in 0957+561 700 750 800 850 image A interpolated 0.05 0 -0.05 Fig. 5 a) Flux measurements of the quadruple quasar Q2237+0305 by the OGLE team [41] clearly show dramatic microlens-induced changes of the four quasar images (top). b) Difference g-band light curves between images A and B of Q0957+561 (time-shifted and scaled), after [39]. Top panel: image A light curve linearly interpolated for the epochs of image B observations; bottom panel: vice versa (bottom). image B interpolated 0.05 0 -0.05 700 750 800 850 t/days (Julian Date - 2449000) (corrected for time delay and magnitude difference, see Fig. 4) are very similar. In a quantitative analysis of the difference light curve and by comparison with intensive numerical simulations, it is shown in [39], that the halo of the lensing galaxy cannot be made up of compact objects in the mass range between 10−6 M and 10−2 M , for quasar sizes between 1014 cm and 3 × 1015 cm. In the following, the method and the line of argument is explained: www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 54 J. Wambsganss: Gravitational lensing – powerful astrophysical tool Fig. 6 Top: Magniﬁcation patterns for machos of mass 10−3 , 10−5 , 10−7 M (from left to right). The line indicates a track corresponding to an observing time of 160 days (with a transverse velocity of v⊥ = 600 km/sec). Bottom: Light curves corresponding to the respective (white) tracks above. In Fig. 4 the two lightcurves of images A and B are shown, shifted by the best value of the time delay. They are very similar, which is a requirement for Refdal’s method to work properly. However, there is no a priori reason why the two lightcurves should be perfectly identical. They certainly do reﬂect the same intrinsic ﬂuctuations in the quasar, but along the lines of sight the two light bundles are differently affected by microlensing, and as a result one can expect to get (slightly) different light curves. In fact, compact objects in (the halo of) the lensing galaxy must produce some microlens-induced, uncorrelated ﬂuctuations in the two light curves. The absence of any such difference allows us to constrain the possible population of compact objects. The “difference” lightcurve between image A and the time shifted image B shows very little variations, as shown in Fig. 5b. The points with their errorbars are consistent with identical light curves. In particular, there is no gradient visible, which could be the ﬁrst-order indication of a long-term microlensing event. Conservatively, one can state that for this data set there is no observable difference between the two light curves A and B greater than |∆m| = 0.05 mag. In order to compare this observational result quantitatively with the effect potential “machos” in the halo of the lensing galaxy would have, various microlensing simulations were performed. The machos were distributed randomly, according to values of surface mass density κ and external shear γ of the two images obtained from models of the lens system 0957+561 (κA = 0.36, γA = 0.44; κB = 1.17, γB = 0.83). Then light rays were “shot” through these mass distributions (cf. [34]). The density of the deﬂected light rays in the source plane is proportional to the magniﬁcation. The magniﬁcation distribution is not smooth but contains sharp caustics (cf. Fig. 6). The light rays are collected in square ﬁelds covered by 25002 pixels. In order to cover a large enough dynamical range, simulations were made with sidelengths of 20θE , 200θE , and 2000θE . With an assumed effective transverse velocity of v⊥ = 600 km/sec, randomly oriented tracks were determined in these magniﬁcation patterns with lengths of 160 days (i.e. the overlap between the light curves A and B). This was done for different masses of the lensing objects, ranging from 10−8 ≤ mmacho /M ≤ c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 1 – 2 (2006) 55 Fig. 7 Display of the “exclusion probability” for speciﬁed macho masses and quasar sizes. The height of the “blocks” indicates the probability for microlens-induced changes of more than 0.05 mag for an observing period of 160 days, as obtained from simulations. 10−1 (with steps in factors of 10), and also for a range of quasar sizes. For each pair of parameters (macho mass and quasar size), 100,000 random light curves for images A and B were calculated and the fraction which had a maximum amplitude of greater than 0.05 mag was determined. In Fig. 7 the results of this study are displayed: for a speciﬁc macho mass and quasar size the “exclusion probability” is shown: the fraction of light curves which produce changes of more than the observational limit of 0.05 mag. The “white” blocks indicate exclusion probabilities of more than 99%. This means that a halo consisting of machos with masses 10−7 ≤ mmacho ≤ 10−3 M can be excluded for quasar sizes of 3 × 1014 cm or smaller from the lack of microlensing-induced variability. For larger masses, the time basis is not (yet) long enough for a conclusive answer but one would expect to see some microlensing ﬂuctuations in the near future. The numerical method is explained in detail in [28]. There also additional results are presented for cases in which the machos make up only 50% or only 25% of the total halo mass. In these cases the limits go down a bit, but there are no dramatic changes. 5.3 Lensing constraints on cosmology: arc statistics Gravitational lensing directly measures mass density ﬂuctuations along the lines of sight to very distant objects. No assumptions need to be made concerning bias, the ratio of ﬂuctuations in galaxy density to mass density. Hence, lensing is a very useful tool to study the universe at intermediate redshifts. This was done, e.g., regarding the frequency of giant luminous arcs predicted by various cosmological models. In [2] it was stated that a Lambda-dominated ﬂat cosmological model (Ωmatter = 0.3, ΩΛ = 0.7 known as “concordance cosmology”) would underpredict luminous arcs by about an order of magnitude. Recently it was shown by [36] that this result was mainly based on the assumption that the sources are all at redshift unity. The probability for arcs, however, is a steep function of source redshift. If one allows for sources at redshift two or three (some of the observed arcs are even at higher redshifts), then the discrepancy disappears, as can be seen at Fig. 8. So the frequency of giant arcs apparently is in concord with the concordance cosmology. 5.4 Searching for planets with microlensing In 1991, it was suggested that a fair fraction of stellar microlensing events towards the Galactic bulge should display signatures of binarity, and that even planetary companions should be detectable [21]. Starting in 1993, a number of teams (MACHO, EROS, OGLE, MOA) monitored of order 107 stars in the bulge in order to detect microlensing effects of intermediate stars or dark compact objects [1,19,32]. By now, more than 3000 microlensing events towards the galactic bulge have been found, currently over 500 events are www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 56 J. Wambsganss: Gravitational lensing – powerful astrophysical tool Fig. 8 Probability for giant arcs with total magniﬁcation (corresponding to length-to-width ratio) above 10, as a function of source redshift (from [36]). For comparison, the value for source redshift of unity obtained by [2] is indicated as well. detected per season. About 10% of them show the signature of binary lenses. This data set allows, among other things, to study the mass distribution of the Galactic disk with unprecedented accuracy. But one of the main goals of these monitoring experiments is still the detection of planets around the lensing stars. In addition to the groups mentioned above, there are two other teams (PLANET, MicroFUN) who specialized in following up of current stellar lensing events with good photometric accuracy and very high temporal coverage, in order to ﬁnd possible small deviations from the smooth single-lens lightcurve, which would be the signature of a planet. The signatures of planets are of short duration (of order hours) and typically have small amplitudes (a few percent), as was shown, e.g., in [3, 37]. But the main aspect are: such planetary deviations at stellar microlensing lightcurves are rare: even if all stars had planets, only a small fraction of the microlensing lightcurves would show their signatures, due to the geometric path of the background star with respect to the planetary caustic. Finally, in April 2004 the ﬁrst microlensing planet was announced: At a NASA press conference the MOA/OGLE/MicroFUN teams announced their detection of a microlensing event which can be explained only with a very low mass companion to the primary star: OGLE 2003-BLG-235 or MOA 2003-BLG-53. The result is published in [4], the lightcurve and the model are shown in Fig. 9. In the original words of the authors: “A short-duration (∼7 days) low-amplitude deviation in the light curve due to a single-lens proﬁle was observed in both the MOA and OGLE survey observations. We ﬁnd that the observed features of the light curve can only be reproduced using a binary microlensing model with an extreme (planetary) mass ratio of 0.0039+11 −07 for the lensing system. If the lens system comprises a main-sequence primary, we infer that the secondary is a planet of about 1.5 Jupiter masses with an orbital radius of ∼3 AU.” This ﬁrst unambiguous microlensing planet detection (Fig. 9; Bond et al. 2004) proves: Microlensing as a planet search technique has stepped out of its infancy. It is a viable method which is complementary to c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 1 – 2 (2006) 57 Fig. 9 a) Lightcurve of the planetary microlensing event OGLE 2003-BLG-235/MOA 2003-BLG53 (from [4]): open (ﬁlled) symbols are MOA (OGLE) data points, data points are shown individually in the top level, and binned in one-day intervals in the bottom panel (upper panel). b) Zoomed lightcurve: Data points and models covering about 18 days around the planetary deviation: long-dashed line – single lens case; short-dashed line – double lens with q ≥ 0.03; solid line – best ﬁt with q = 0.004 (lower panel). other techniques. Microlensing remains the most promising method for the detection of low-mass planets with ground-based techniques, in principle even Earth-mass objects are within the sensitivity range. 6 What is the future of lensing? Gravitational lensing is an exceptional ﬁeld in astronomy in the sense that its occurrence and many of its features – e.g. multiple images, time delays, Einstein rings, quasar microlensing, galactic microlensing, weak lensing – were predicted long before they were actually observed. Although “prediction” or predictability is considered one of the important criteria of modern science, many (astro-)physical phenomena are too complicated for a quantitative or even qualitative prediction. The reason why this worked here is that gravitational lensing is a simple geometrical concept which easily allows qualitative estimates and quantitative calculations. www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 58 J. Wambsganss: Gravitational lensing – powerful astrophysical tool Within the last 25 years, gravitational lensing has changed from being considered a geometric curiosity to a helpful and in some ways unique tool of modern astrophysics and cosmology. By now almost a dozen different realizations of lensing are known and observed, and surely more will show up. Gravitational Lensing is useful in many areas. To mention just a few: lensing is a good tool in the search for dark matter; in cosmology some of the important cosmological numbers can be determined with the help of lensing: the value of the Hubble constant H0 and the matter density Ωmatter as well as the contribution of the cosmological constant ΩΛ ; lensing helps determine the size/structure/physics of quasars, the structure of the Milky Way, evolution of galaxies, and even in the search for extra-solar planets. Extrapolating from these thoughts, it should be possible to look forward in time once again and predict future applications of gravitational lensing. It does not need much vision to predict that the known lensing phenomena will become better, sharper, more. Many more multiple quasar systems are being found, which will provide us with a very good statistics of image separations and redshift distributions, in order to study Ωcomp , the compact matter density of the universe in great detail, as well as the contribution of the cosmological constant ΩΛ (cf. [14]). There will be “good systems” among them, well suited for a very accurate measurement of the Hubble constant. Other studies will provide excellent data on galaxy cluster, from giant luminous arcs over arclets up to weakly distorted background galaxies. These data sets can be used to provide very accurate determinations of the cluster masses and their mass distributions. Maybe we get a handle on the dark matter distribution this way. And no doubt there will be many planets detected by lensing. The future of lensing appears to be bright, or even luminous. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] C. 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