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

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Ann. 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
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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)
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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−
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2
θE
.
θ
(8)
c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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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
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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]).
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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
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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
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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
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Ann. Phys. (Leipzig) 15, No. 1 – 2 (2006)
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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.
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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
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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:
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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
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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
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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
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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.
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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.
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