Gravitational Lensing 2. 1

Section 2

2. DEVIATIONS FROM HOMOGENEITY: THE GRAVITATIONAL

LENSING

1. THE GRAVITATIONAL LENSING

We have seen during the third year course how to treat the space-time geometry of an expanding universe on the basis of very general considerations referring to the Cosmological Principle, and essentially making use of the symmetry properties of the space-time inherent the Principle. It has thus been possible to define a metrics that has allowed us to set up in a rigorous way the concept of distances and, based on this, all cosmological observables. The Cosmological Principle and the consequent Robertson-Walker metrics assume a rigorously homogeneous and isotropic universe. As we have seen in the previous Section, this is valid on the large scales (essentially on scales larger than at least a hundred Mpc), but obviously breaks down on smaller scales where we have seen a high degree of inhomogeneity. In this Section we discuss the phenomenon of gravitational lensing, originating from deformations of the space-time metrics due to the inhomogeneities of the gravitational field induced by cosmic structures. The gravitational lensing offers not only a solid verification of the General Relativity theory (indeed it made its first experimental test), but even a unique instrument for studying the gravitational matter (baryons + dark matter) distribution in the Universe (hence potentially bypassing the very serious problem of cosmological bias mentioned in Sect. 1.4) and, finally, a method for constraining the cosmological parameters, in particular the ΩΛ parameter that so far we have found difficult to quantify in a direct way. However, gravitational lensing has countless applications in astrophysics and cosmology. Introduction In agreement with the Einstein theory of General Relativity, an object of a given mass produces a curvature in the space-time with its gravitational field, and imposes a deflection to the propagation of the electromagnetic radiation, otherwise happening in a straight line. The first confirmation of the General Relativity theory was given in

Gravitational Lensing 2. 2

1919 during a total solar eclipse 1. Thanks to this astronomical phenomenon, Sir Arthur Eddington observed that stars close to the solar limb resulted shifted from their real position, proving that the light propagation was bended near the star. Although the uncertainty in this positional measurement was large (0.3 arcsec), the precision was enough to disentangle the validity of the relativistic prediction against the Newtonian one (Dyson, Eddington, & Davidson 1920). Indeed, the idea that light rays coming from stars could be deflected by the Sun was not new: it was already considered by Newton in his publication about Optics, and by Laplace. Following Newton, light rays should have experienced the gravity force. In 1804 Soldner attempted to apply the Newtonian gravity to calculate the deflection of the light ray by the Sun, under the assumption that light is composed by massive particles. He found that, according to the classical theory of gravity, the deflection angle α of a photon with respect to its unperturbed path due to the presence of a point-like body of mass M would be given by 2

2 ,GMbc

α = where G is the gravity

universal constant, b the impact parameter (distance of the light ray from the mass M) and c the velocity of light in vacuum. The argument to obtain such result in the Newtonian limit is quite simple and refers to the classical process of scattering on a centrally symmetric field force with inverse quadratic power-law with distance (essentially a very similar phenomenon ruling for example the scattering of electrons on protons, producing bremsstrahlung emission). Let us imaging to consider a particle of mass m small and moving with velocity v, impacting at distance b with a body of much larger mass M . If the impact does not perturb too much the orbit (b not too small), the particle m will acquire a moment only orthogonally to the direction of the incoming particle.

1 In 1912 an argentinian expedition to Brazil was dedicated to measure, by exploiting a total eclipse, the phenomenon of the deflection of stellar positions by the Sun; unfortunately, the observations did not take place due to bad atmospheric conditions. It is interesting to note that at that time the Einstein prediction for the total deflection, based on a simple application of the equivalence principle, was half the correct value. It would be interesting to consider what could have happened to the General Relativity theory if the observation could have really been made. Later in 1914, a German expedition to Crimea, leaded by the astronomer Freundlich, attempted to measure the effect during a total eclipse on August 21; but because of the beginning of the First World War many attendees returned home and others were even jailed, for which this expedition too failed.

r = ct

v=c θ b

M

Gravitational Lensing 2. 3

Having set the origin of time t=0 at the moment when the particle (photon) is at the minimal distance b, we will have r2=c2t2+b2 , with sin(θ)=b/r, so that the component of the acceleration orthogonal to v will be 2 2 2 3/2/ ( )a GMb c t b⊥ = + and the corresponding component of velocity that is acquired will be

2 2 2 3/22 2 2 3/2

( / )v ( )(1 / )

GM d ct ba dt GMb dt c t bcb c t b

+∞ −⊥ ⊥ −∞= = + = =

+∫ ∫ ∫

2 3/2 2 1/2

2

2 , so that(1 ) (1 )

2vnewton

GM dx GM x GMcb x cb x cb

GMcbc

α

+∞

−∞

⊥

= = =+ +

= =

∫ [2.0]

with αnewton being the deflection angle in the classical limit. In reality, this calculation suffers by many inconsistencies, first of all the fact that the light does not possess mass at all to be influenced by the Newtonian gravity force. Hence the classical theory simply cannot be applied. In spite of that, eq. 2.0 represents a remarkable intuition by Newton and Soldner and is referred to as the Newtonian prediction. If we insert the values of M and b for the Sun, the amount of angular deviation from this formula is 0.87 arcseconds for a star with a line-of-sight tangential to the solar border. Einstein himself re-obtained this result in 1911 and appeared to confirm the calculation of Soldner. It was only in 1916, after having published the General Theory of Relativity that Einstein, by applying the field equations of his new theory, discovered that the expected deflection that light rays would experience under the influence of a solar mass would be two times the value obtained by the classical result (or 1.7 arcseconds). Let us see this with the following simplified argument. In the flat space-time of Special Relativity the infinitesimal 4D interval ds has a temporal component dt and a spatial one dl from the relation

2 2 2 2 ,ds c dt dl= − with light rays following space-time paths defined by the geodesic equation 2ds =0, that is straight lines. On the contrary, in General Relativity the light paths are curved; in this case, gravity affects both the spatial and time component of the photon’s path, so that the actual bending is twice the value in [2.0]. Indeed the bending itself is due to a change of the velocity propagation of light and a consequent change of the refractive index in the lens vicinity compared to a local Minkowskian frame. Indeed around a centrally symmetric mass distribution of mass M the metrics becomes that of Schwarzschild. In the limit of weak field Φ :

Gravitational Lensing 2. 4

( )

2 2 2 2

2 2

2 2 2 2 2

2 2

2 21 1

2 21 1 .

ds c dt dlc cGM GMc d dyt dxbc bc

dz

Φ Φ = + − − + − −

=

= + +

[2.1]

having set GM bΦ = . Since the corrections 2GMbc

are small compared to unity, we

can resolve the equation 2ds =0 by simple expansion of the terms in parentheses 2. In conclusion, the total deflection angle from the Einstein theory becomes

2

4ˆ GMbc

α = , [2.2]

where the angle is in radiant. That is, the deflection angle amounts to two times the prediction of the simplified Newtonian theory. Einstein realized that also more massive bodies than the Sun, like a galaxy, light bending could take place and, under suitable conditions, one could even observe multiple images of a single source, producing the phenomenon of gravitational lensing, but concluded that the amplitude of the effect is too small to be measurable with the astronomical imagers of the epoch (Einstein 1936). A major contribution to the field was made by the Swiss astronomer Fritz Zwicky. In 1937 he elevated the gravitational lensing phenomenon from a curiosity to a field with large potential when he pointed out that galaxies can split images of background sources by a large enough angle to be observed. He also argued that not only this would be a dramatic proof of General Relativity, but also that the lensing effect could be exploited to observe astronomical sources too faint to be detected on their own. Zwicky (1937) even calculated the probability of lensing by galaxies and concluded that it is of order of one per cent for a source at reasonably large redshift. All these predictions turned out to be correct, but this was realized at a much later time. All these ideas about lensing would have remained a mere speculation until the advent of better astronomical imagers and, at the same time, the discovery of a new class of suitable, extremely distant sources. These are the quasars discovered in 1962 by Schmidt (1963). These are ideal to measure the effect because they are extremely luminous and distant, point-like, and very bright. Because they are distant, the probability that their light crosses strong enough gravitational fields to produce

2 If we set ds2=0, then we have a light speed in the gravitational field 2

2 21 2 211 2

dl cc c cdt c c

− Φ Φ ′ = = − + Φ , that is

smaller than in vacuum. This can be expressed e.g. in terms of a refraction index 2

21 GMbc

n c c′= +

, while the

Newtonian expectation based on the testing particle velocities would just be ( )21v vv M bn G+′= .

Gravitational Lensing 2. 5

lensing effects is significant. The resulting magnifications can therefore be very large, and multiple image components are well separated and easily detected. The first gravitational lens to be detected was observed just by chance in 1979 by D.Walsh, B. Carswell e R. Weymann (1979) using the 2-meter telescope of the Astronomical Observatory of Kitt Peak: the source discovered is still known today with the name of Twin Quasar (QSO 0957+561), see Fig. 17 below: we can notice in this object two identical point-sources, two quasars, separated by only 6 arcseconds: in reality they are the two images of the same single quasar. Evidence in favor of this interpretation of the source comes from (a) the similarity of the spectra of the two images, (b) the fact that the colors and flux ratios between the two images are independent on frequency, (c) the presence of a foreground galaxy between the images acting as a lens, and (d) VLBI observations which show detailed correspondence between various knots of emission in the two radio images. A few illustrations of the effects of gravitational lensing are reported in the figures below. They are mostly in the form of arcs around a massive elliptical galaxy, for reasons that we will see later. Note finally that nothing in eqs. [2.1] and [2.2] depends on the photon energy, so the lensing effect in completely achromatic.

Fig.1. SLACS: the Sloan Lens ACS Survey. This is an imaging survey of lensed sources performed with the Hubble Space Telescope, based on the Sloan Digital Sky survey.

Gravitational Lensing 2. 6

Fig. 3. A few cases of strong gravitational lensing. The blue parts of the images are background lensed sources.

As of today, several hundreds of gravitationally lensed sources are known, particularly after the onset of the HST as the ultimate astronomical imager. Thanks to the progress in imaging quality, this field has made tremendous progress and is one of the areas of great interest and impact for observational cosmology, as we will discuss in the rest of this Section.

Gravitational Lensing 2. 7

Fig. 4. Matter distributions in the galaxy cluster 1E 0657-56 (known as the "bullet cluster") are shown in this composite image. About 1000 Mpc away, the "bullet cluster's" individual galaxies are seen in the optical image data, but their total mass adds up to far less than the mass of the cluster's two clouds of hot x-ray emitting gas shown in red. Representing even more mass than the optical galaxies and X-ray gas combined, the two blue clouds show the distribution of dark matter in the cluster. Otherwise invisible to telescopic views, the dark matter was mapped by observations of gravitational lensing of background galaxies. In a textbook example of a shock front, the bullet-shaped cloud of gas at the right was distorted during the collision between two galaxy clusters that created the larger "bullet cluster" itself. But the dark matter present has not interacted with the cluster gas except by gravity. The clear separation of dark matter and gas clouds is considered direct evidence that dark matter exists and a triumph of the gravitational lensing theory.

Gravitational Lensing 2. 8

2.2 GRAVITATIONAL OPTICS 3

To discuss the phenomenon of gravitational lensing in a simplified way, we will make a few assumptions, as the general problem of determining the trajectories of light rays deflected by generic gravitational fields may be very complex. 1) We assume that the general geometry of the universe can be described by the Robertson-Walker (RW) metrics of an homogeneous and isotropic space-time. 2) The light is assumed to travel following the RW paths from the source to the observer, except in a tiny fraction of its trajectory where the effect of the lens is produced. The lens is then essentially approximated as a plane lens. At the distance of the lens, the light trajectory is deflected by a small angle (<< 1 rad) by the perturbed gravitational field of the lensing object. 3) The lens gravitational field is assumed not to be strong, such that the lens gravitational potential Φ is 2cΦ << . We can assume a locally flat, Minkowskian space-time which is weakly perturbed by the Newtonian gravitational potential of the mass distribution constituting the lens. 4) The lens moves with non-relativistic peculiar velocity v<<c with respect to the local fundamental observer. Under such conditions, the light total deflection angle at the lens can be computed from [2.2] as

2

2ˆ dlc

α ⊥= ∇ Φ∫ [2.3]

where the gradient of the Newtonian gravitational potential Φ is calculated orthogonal to the direction of propagation of light. The factor 2 comes from the general relativistic treatment in Chap. 2.1. Eq. [2.3] is easily proven in the following. If we define as b the impact parameter of the light ray with respect to the barycenter of the lens mass distribution and with z a coordinate axis along the light path, the potential can be expressed as

2 2 1/2

2 2 3/2

( , ) and( )

( , )( )

GMb zb zd GMbb zdb b z⊥

Φ = −+Φ

∇ Φ = =+

[2.4]

such that, by integration, we are back to our previous result

3 We mostly use for this Chapter the formalism by NARAYAN and BARTELMANN, Proceedings of the 1995 Jerusalem Winter School, A Dekel & J. Ostriker, 1995.

Gravitational Lensing 2. 9

2 2

2 4ˆ GMdzc c b

α ⊥= ∇ Φ =∫ , [2.5]

that demonstrates the validity of [2.3] by comparison with [2.2]. Figure 5 below illustrates the various relevant geometrical quantities in the lensing phenomenon. α is the total deflection at the lens, α is the reduced deflection angle as seen by the observer. S is for the (real) source position, L is the deflected position, C the lens center. Distances are angular diameter distances.

Fig. 5 Geometry of the gravitational lensing. A light ray travels from the source S to the observer in O, in the vicinity of the lens with impact parameter ξ . The point C is the projection of the lens centre on the sky. L marks the apparent position on sky of the lensed source image. Cosmological distances are all angular diameter distances. With the assumption (2) above of a thin lens, it is commonly useful to express the mass distribution in the lens in terms of its surface mass density, that is the mass projected onto a plane orthogonal to the line-of-sight and containing the barycenter of the lens. This plane is called the lens plane, and the surface density is defined as

( ) ( , )z dzξ ρ ξ+∞

−∞Σ = ∫

Gravitational Lensing 2. 10

in terms of the projected lens mass density. In terms of this, the ratio M/b in [2.5] can then be expressed as

2 22 2

( ) ( ) ( ) ( ) and 4ˆ ( )lens plane lens plane

M d db

Gc

αξ ξ ξ ξ ξξ ξ ξξ ξ ξ ξ

′ ′ ′Σ − Σ′ ′=′− ′−

=∫ ∫ [2.6]

as a function of the distance ξ at which the light ray passes with respect to the barycenter, in the lens plane. For a circularly symmetric mass distribution in the lens, the integral is immediately done:

2 0

( )( ) with ( ) 2 ( )4ˆ dGcM M ξξξ ξ π ξ ξ ξξ

α < ′ ′ ′< = Σ= ∫ .

Let us now come to the fundamental relation about the gravitational lensing phenomenon, the lens equation. With reference to the quantities defined in Fig. 5, the lens equation can be expressed in terms of either the projected sizes on the celestial sphere of the relevant segments or the angles in the observer frame, and is the very simple consideration that the reduced angle α plus the angle β between lens center and the unperturbed source position should equal the angle θ of the source lensed position to the lens center:

α β θ+ = [2.7]

Alternatively, in terms of the segments , ,LS SC LC on the celestial sphere (see also Fig. 5 for the various definitions), we have

ˆ that is S S DSLS SC LC D D Dθ β α+ = = + [2.8]

where SD and DSD are the angular diameter distances observer to source and source to lens (deflector), respectively. Keep in mind of course that we assume here all angles being very small, 1 rad . Still from Fig. 5 we can derive a second relation between the total deflection angle α and the reduced deflection angle α :

ˆ ˆsuch that DSS DS

S

DD LS DD

α α α α= = = [2.9]

As an application of the lens equation, let us consider the case of a lens with a constant surface mass density (and unlimited size), ( ) constξΣ = , that is a uniform lens with 2( )M ξ πξ< = Σ . From [2.6] and considering that DDξ θ= , we get

22 2

4 4( ) DS DDS

S S

D G G D DD c c D

πα θ πξ θξ

Σ= Σ = [2.9bis]

Gravitational Lensing 2. 11

This relation can be simplified by introducing a reduced distance

2

4 such that ( )DDS

S

D GD D DD c

πα θ θΣ≡ = [2.10]

This relation brings us immediately to define a critical surface density, such that ( )α θ θ= for any value of θ . This condition is achieved if the projected surface

mass density equals the critical density

2 2

*

4 4S

DS D

Dc cG D D GDπ π

Σ = = . [2.11]

When this happens, from the lens equation [2.7], this implies that 0β = for any value of θ : this is a perfect lens, bringing all the light rays emitted by the source in any direction to the same focus at the position of the observer. Lenses for which

*Σ > Σ are said to be strong lenses and typically produce multiple images of a source.

The above scheme illustrates the effects of a perfect lens, with a surface density equal to the critical density. In any direction a source is emitting a light ray, this light ray is received by the observer in his position. The Universe at large as a lens. There is an interesting application of the concept of a uniform lens to the whole Universe. How does the matter in the Universe operate with respect to the propagation of light rays? Let us calculate, by order of magnitude, the surface density of the projected mass distribution in the Universe:

22

0

2 0.2 /universe m cc Kg gr cm

H mρΣ = Ω ≈ [2.12a]

where mΩ is defined in eq. [0.4b] and 0 4200c H Mpc . This value can be compared with the critical surface density referred to the whole universe. If we adopt for the reduced distance D the Hubble radius c/H0, then:

Gravitational Lensing 2. 12

2 20 8

224

10 0.1 /4 6.6 4200 3 10lens

c gr cmGDπ

+

Σ ≈ ≈× ×

. [2.12b]

So the two numbers roughly coincide, which means that the effect of the gravitational lensing operated by the whole Universe on the propagation of light is significantly strong. This explains in particular the fact that we discussed in Sect. 0.4 that the relation of the angular apparent size of an object and its distance bends at about z∼1 (see eqs. [0.23-0.27]), and that the apparent angular size of objects keep roughly constant at z>1 for reasonable values of mΩ . Essentially, the lensing effect starts to operate when looking at objects far enough that the content of gravitating matter along the line-of-sight is large enough to achieve the condition [2.12]. As we have seen, this effect depends on the average amount of gravitating matter in the Universe (e.g. through the parameters 0 and m qΩ , which determine universeΣ ). At the same time, these considerations also imply that inhomogeneities along the line-of-sight can produce characteristic deformations of the images of background sources, as we will see below. Circularly-symmetric lenses and point-like lenses. Some interesting properties of the lensing can be immediately appreciated if we specify the lens equation by referring to the case of a circularly-symmetric mass distribution with a profile ( )M θ . The lens equation gets, from [2.2]:

2 2

4 ( ) 4 ( )ˆ( ) DS DS DS

S S S D

D D DGM GMD D c c D D

ξ θβ θ α θ θ α θ θξ θ

= − = − = − = −

with reference to the scheme below

After defining another reduced distance

S D

DS

D DDD

′ ≡ , [2.13]

the relation for a symmetric lens can be written as

Gravitational Lensing 2. 13

1/22

2 2

4 ( ) 4 ( ) where EE

GM GMc D c D

θθ θβ θ θ θθ θ

= − = − = ′ ′ . [2.14]

This obviously holds in particular for the simplest case of a point-like lens. The angle Eθ is called the Einstein radius: for a source exactly aligned ( 0β = ) with a lens of mass M, the lens equation reduces to 2 2

Eθ θ= : the lens produces an image for an observer which is a ring of angular radius Eθ on sky. In the case of a source not aligned with the lens, the lens equation can be written as

2 2 2 210, with solutions ( 4 )2E Eθ θβ θ θ β β θ±− − = = ± − [2.15]

Fig. 6. This Illustrates the geometry of the deflection of light by a deflector in the center of each image. The lens is the inner circle, the source is the off-centre circle. The dashed circle marks the Einstein ring. The shaded region shows the lensed image. From (a) to (d): the source gets closer and closer to the lens. Interesting to note that for an extended source, like shown here as a white circle, the two images have apparent sizes larger than the real one. These can be obtained from the two crossing straight lines centred in the lens barycentre, the two lines being aligned with the borders of the source in its unperturbed position and defining the borders of the lensed images. To demonstrate this consider that every point in the source and lensed images can be treated according to eq. [2.15], and particularly the points at the borders that are aligned that are aligned along the two lines.

The two solutions correspond to two images of the source lying on opposite sides of the lens. One image is inside the Einstein ring, the other outside it, as illustrated in Figure 6. As β decreases, the lensed image approaches in shape that of the Einstein ring, [case d]), while the more it increases, the further apart the two images split, and eventually one image will go to coincide with the real source external to the Einstein ring, and the counter-image disappears [case a]. While for point sources we would just have two point-like images as solutions of [2.15], the two shaded arcs in Fig. 6 correspond to the images in the case of extended sources. There is a nice theorem that can be very easily verified that the apparent sizes of the two extended images can be simply obtained by projecting two straight lines from the lens barycenter to the borders of the unperturbed source image, as

Gravitational Lensing 2. 14

illustrated and discussed in the figure. The two lines delimit also the borders of the lensed images. This theorem comes immediately from consideration of the symmetry of solutions [2.15] for every luminous point making the source.

2.3 LENSING MAGNIFICATION

We have seen that gravitational lensing may produce quite substantial modifications of the source images, and in particular may largely increase the source angular size (e.g. Fig. 6). This effect of size amplification can also be seen considering a pair of rays emitted from two sides of a source: the ray passing closer to the deflector will be bent more than the other, thus the source will appear stretched. Gravitational lensing distorts the images of the sources, and typically amplifies them. Indeed, we have seen that the unperturbed source image may be split into various lensed images, some of which will be larger; some will be smaller than the real source depending on details of the lens. However, if we sum over all lensed images, the total area is typically increased. The magnification of the apparent angular size of a source is defined as

or in terms of our defined usual variables, as

Fig. 6bis. Representation of the generation of an Einstein ring. A source S on the optical axis of a circularly symmetric lens is imaged as a ring with an angular radius given by the Einstein radius θE.

Gravitational Lensing 2. 15

2

2

d dd dθ θ θµβ β β

≡ = . [2.17]

This has an important effect to amplifying the apparent flux of the source, an effect that has relevant implications when attempting to characterize observationally the properties of faint distant sources. To understand it, we need to check how lensing affects the surface brightness of the lensed sources. As a matter of fact, gravitational lensing does not alter the surface-brightness of the images. This can be shown with the following simple consideration. Let us start to consider that the surface brightness is conserved in an Euclidean space and that a Doppler shift changes the surface brightness according to the special relativistic invariant

3

I constn

n= , [2.18]

so that the only variation of the brightness should follow a variation in frequency.

Now light propagation can be described to happen along a path that is locally Euclidean, but for which there is a gravitational redshift between the start and the end of any given segment of the ray. For example, for a Schwarzschild metrics (that would describe the lens gravitational field), the gravitational redshift as a function of the radial distance r to the centre would be 1 1 1 /g Sz r r+ = − , with

Schwarzschild radius 2 2Sr GM c= . In a weak-field limit, this simplifies to 2

gz GM rc . In our case, however, since the lens depth is completely negligible compared to the source-observer distances, any light ray passing through the lens is blue-shifted by falling into the potential well by the same amount that it is red-shifted when getting out of it. So n does not change and, as a consequence, the surface brightness In does not vary by the effect of lensing.

Gravitational Lensing 2. 16

In conclusion, gravitational lensing amplifies the total source flux (integral of In over the source extent) just because of the apparent size magnification, according to eq. [2.17].

2.4 THE GENERAL LENSING SOLUTION

Let us treat here the more general case of a lens without any particular symmetry, still under the hypotheses mentioned in Chap. 2.2, in particular of a thin depth with respect to the characteristic distances and D D′ . A most effective treatment of the lensing effect can be achieved in terms of a lensing potential ( )ψ θ defined as

21 2( ) ( , )Ddz D zD c

ψ θ θ+∞

−∞= Φ

′ ∫ [2.19]

where D′ is from [2.13] and ( , )DD zξ θΦ = is the Newtonian gravitational potential as a function of the coordinates and zξ , following the scheme

A treatment in terms of a lensing potential is useful because its first and second derivatives are immediately related to fundamental quantities like the deflection angle and lens surface mass density. This is easily shown. Because in the lensing plane

DDξ θ= and D DD Dθ ξθ ξ∂ ∂∇ = = = ∇∂ ∂ , and reminding from [2.9bis] and [2.3]

the expression of the reduced deflection angle

2 2

4 2( ) ( ) ( )DS DS

S S

D DG M dz zD c D c

α θ ξξ ⊥= < = ∇ Φ∫ , we have:

2 2

1 2 2( ) ( , ) ( , )DSD D D

S

DD dz D z dz D zD c D cθ ξψ θ θ θ α

+∞ +∞

⊥−∞ −∞

∇ = ∇ Φ = ∇ Φ = ′ ∫ ∫

[2.20]

Gravitational Lensing 2. 17

where the symbol ξ⊥∇ = ∇ is the gradient in the plane of the lens and orthogonal to the line-of-sight. So the angular gradient of the lens potential ( )ψ θ is just the reduced deflection angle α . Let us see now the meaning of the second spatial derivative of the lensing potential, that is the squared Laplacian. Still from [2.9bis] we have

2 22 2

22

2 2( ) ( , ) ( , )

2 ( , )

DS D DSD D

S S

D

D D Ddz D z dz D zD c D c

D dz D zc

θ θ θψ α θ θ θ

θ

+∞ +∞

⊥ ⊥−∞ −∞

+∞

⊥−∞

∇ = ∇ = ∇ ∇ Φ = ∇ Φ

= ∇ Φ

∫ ∫

∫The Laplacian in the integral can be expressed as

22 2

2z⊥∂ Φ

∇ Φ = ∇ Φ −∂

, but assuming

reasonable forward-backward symmetry in the lens this second derivative term will vanish when integrated in z (it is positive on one side of the lens plane and negative on the other side), so that 2 2

⊥∇ Φ = ∇ Φ . So from Poisson, 2 2 4 Gπ ρ⊥∇ Φ = ∇ Φ = , and finally

22 2

2 84D DS

S

D D G DG dzD c cθ

πψ π ρ+∞

−∞∇ = = Σ∫ . [2.21]

If we define the convergence K as

*( )( )K θθ Σ

≡Σ

[2.22]

the second derivative just becomes the two-dimensional version of the Poisson equation:

2 2 ( )Kθ ψ θ∇ = . [2.23]

This equation can be resolved in a similar way to the full 3-D equation: we need to find the Green function for the 2D divergence 2

θ ψ∇ , which turns out to be ln / 2θ θ π′− (while it is 1 θ θ ′− for the 3-D case; note that a point mass in 2D

gives a force 1 r∝ , instead of a force 21 r∝ of the 3D case). The lens potential can then be obtained from

21( ) ( ) lnK dψ θ θ θ θ θπ

′ ′ ′= −∫d d d d d

and for the deflection angle, from [2.20]

22

1( ) ( )K dθ θα θ θ θπ θ θ

′−′ ′=′−

∫d d

d d d

d

d d

[2.24]

Gravitational Lensing 2. 18

where we have indicated vector signs to indicate that the operations are performed in the 2D space of the lens plane. This is equivalent to the already derived eq. [2.6]. This relation is what you need to predict the deflection field from a model for the mass distribution in the lens as in ( )K θ

d

. Of course, as a cosmological application we would rather like to obtain the mass model from the observations, as discussed below. The linear solution.

The fundamental problem to be resolved by gravitational lensing analyses is that to produce a full mapping of the source plane onto the image plane, and vice-versa to obtain the total mass distribution in the lens via an inversion procedure. This mapping is performed through a matrix operator based on the lens convergence field [2.22] or the lensing potential [2.24]. With reference to the geometric setup in Fig. 5, the mapping should determine the image map in terms of the angle θ

d

as a function of the source map for β

d

, or viceversa βd

from θd

. This can be locally (i.e. point by point in the map) specified with the Jacobian matrix of the transformation of the two maps:

2

1( ) ( )i iij ij ij ij

j j i j

A Mβ α θ ψ θδ δθ θ θ θ

−∂ ∂ ∂= = − = − =∂ ∂ ∂ ∂

d d

[2.25]

which is immediately understood once we recall the lens equation [2.7]: β θ α= − and when we consider the relation [2.20], ( )θα ψ θ= ∇ . The indices i and j here indicate an orthogonal coordinate system, essentially two orthogonal directions in the lens plane. The Jacobian matrix 4 Aij of the transformation may be thought of as the inverse of a magnification tensor Mij , which comes from consideration of [2.17]. The local distortion of an image due to the lens is given by the determinant of A . The local magnification, as defined in [2.17], is just given by

4 In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. The Jacobian of a function describes the orientation of a tangent plane to the function at a given point. In this way, the Jacobian generalizes the gradient of a scalar valued function of multiple variables which itself generalizes the derivative of a scalar-valued function of a scalar. In other words, the Jacobian for a scalar valued multivariable function is the gradient and that of a scalar valued function of scalar is simply its derivative. Likewise, the Jacobian can also be thought of as describing the amount of "stretching" that a transformation imposes. For example, if 2 2 1 1( , ) ( , )x y f x y= is used to transform an image, the Jacobian of f , 1 1( , )J x y describes how much the image in the neighbourhood of 1 1( , )x y is stretched in the x and y directions. The Jacobian of the gradient has a special name: the Hessian matrix, which in a sense is the "second derivative" of the scalar function of several variables in question.

Gravitational Lensing 2. 19

[2.26]

Fig. 8. Effects of the strong gravitational distortions of a background source (Panel I) when it is located at different positions with respect to the axis of the gravitational lens. In this example, the lens is an squeezed ellipsoidal isothermal sphere. The ten positions of the source with respect to the critical inner and outer caustics are shown in the panel (S). The panels labelled (1) to (10) show the shapes of the images of the lensed source (from

Gravitational Lensing 2. 20

J.-P. Kneib, Ph.D. Thesis (1993)). Note the shapes of the images when the source crosses the critical caustics. Positions (6) and (7) correspond to cusp catastrophes and position (9) to a fold catastrophe (Fort and Mellier 1994).

Fig. 8bis. Same as in Fig.8, with an emphasis on the effects of fold catastrophes in the top panels and cusp catastrophes in the bottom. Depending on the relative positions of the source and the lens, one, three or five images may appear in the observer's sky.

Gravitational Lensing 2. 21

The fundamental operator in lensing is then the matrix 2 ( )

i j

ψ θθ θ

∂∂ ∂

d

, the gradient field of

a gradient field. Note that, in practice, it is custom to make a grid of ( )θΣ and to operate a Fourier transform of this with an FFT (fast Fourier transform) algorithm. Equation [2.25] shows that the matrix of second partial derivatives of the potential ψ (technically, the Hessian matrix of ψ ) describes the deviation of the lens mapping from the identity mapping.

Without entering into inessential mathematical details (however required for the quantitative analysis), the lensing produces a characteristic distortion in the shape of faint background sources which is illustrated in Fig. 9. Again, the mathematical treatment makes use of eqs. [2.25] and [2.26] and can be conveniently expressed in terms of a shear (or “distortion”) tensor.

Let us first consider that from [2.23] we can write

2 2

22 2

1 2

1 1 ( ) ( )( )2 2

K θψ θ ψ θθ ψθ θ

∂ ∂= ∇ = + ∂ ∂

d d

d

[2.30]

The elements of [2.25], 2 1( )ij ij i j ijA Mδ ψ θ θ θ −= − ∂ ∂ ∂ =d

, can then be used to construct a shear tensor:

Fig. 9. Illustration of the effects of weak gravitational lensing on the convergence and shear on a circular source. Convergence magnifies the image isotropically, and shear deforms it to an ellipse.

Gravitational Lensing 2. 22

[ ] [ ]

2 2

1 2 21 2

2

21 2

2 2 1 21 2 2 2 2 2

1 2 1 2

1 ( ) ( )( ) ( )cos ( )2

( )( ) ( )sin ( ) where

( ) ; cos ; sin

ψ θ ψ θγ θ γ θ φ θθ θ

ψ θγ θ γ θ φ θθ θ

γ γγ θ γ γ φ φγ γ γ γ

∂ ∂ ≡ − = ∂ ∂ ∂ ≡ = ∂ ∂

= + = =+ +

d d

d d d

d

d d d

d

[2.31]

With these definitions, the Jacobian matrix Aij of the transformation can be expressed synthetically as

[ ] [ ][ ] [ ]

cos sin1 0( ) [1 ( )] ( )

sin cos0 1A K

φ φθ θ γ θ

φ φ

= − ⋅ − ⋅

d d d

[2.32]

at any point in the lens plane. The meaning of the terms convergence and shear now becomes evident and intuitively clear from this expression. The first addendum of [2.32] is the convergence term: acting alone, this causes an isotropic focusing of light rays, leading to an isotropic magnification of a source. The source is mapped onto an image with the same shape but larger size (a circular source remains circular, though amplified and expanded).

The second addendum is the shear term. Shear introduces anisotropy (or astigmatism) into the lens mapping; the quantity γ , the factor of the second term, describes the magnitude of the shear (distortion), while φ describes its orientation (see Fig. 9). For a small circular source, like could be a distant background galaxy, its lensed image is an ellipse, with major and minor axes that can be demonstrated to be given by

( ) ( )1 11 ; 1a K b Kγ γ− −= − − = − +

and the magnification

( )2 2

1 1det( ) 1A K

µγ

= =− −

. [2.33]

Strong gravitational lensing. A strong [ *( )θΣ > Σ ] extended lens can produce many images, depending on the mass distribution in the lens plane, an odd number,

Gravitational Lensing 2. 23

typically 3 or 5 images, as can be demonstrated with a theorem. A point-like lens, instead, produces only 2 images, as we have seen. Various illustrations of the effects of strong lensing are reported in Figures 8 and 9. Caustics. For some strong lens configurations, the mapping between source and image plane may become singular somewhere in the map. This could happen whenever the Jacobian matrix Aij , that is a function of the image position θ , gets null values of the determinant, det(A)=0, and the magnification diverges according to eq. [2.33]. Then being A a function of θ , this null condition corresponds to a set of curves in the image plane, curves along which the source magnification gets formally infinite from [2.26]. These singularities are usually called caustics and they lead to interesting optical effects owing to the non-uniqueness of the mapping between image and source planes. This effect may happen with extended lenses. Basically a strong extended lens will generate a set of caustics in the source plane. When the source position in sky falls in correspondence to such lines, the image is (formally) infinitely magnified (µ →∞ ) and the stretching diverges ( /a b →∞ ).

Caustic lines are reported in the simulated images of Figs. 8 and 9. Near the caustics the shape of the images can be very complicated, producing giant arcs and multiple images. The flux magnification could be very large, up to a factor of a hundred in some cases (in a mathematical sense, the magnification effect is formally infinite at a caustic). The flux- (or luminosity-) conservation theorem. Note that, together with the flux amplification, we also have flux de-amplification by the same amount on average. Consider a source of luminosity L at the centre of a sphere: the energy flowing outside the sphere is independent if the matter inside is smooth or clumped. If clumped, the flux amplification caused by the lenses must compensate for the reduction in flux along under-dense lines of sight. Hence, the standard distance-redshift relation for a Friedmann model will be correct on average. Weak gravitational lensing. After having seen effects of strong lensing, we briefly mention here the weak lensing. This happens for values of the convergence field

( ) 1K θ < , that is for a sub-critical surface mass density.

An example of a simulated weak gravitational lensing is reported in Figure 10.

Gravitational Lensing 2. 24

For cosmological applications of all this, see Chap. 2.5.

Fig. 10. The principle of the weak gravitational lensing effect is illustrated here with a simulation. Due to the tidal component of the gravitational field in a cluster, the shape of the images (ellipses) of background galaxies are distorted and the galaxy images will be aligned tangentially to the cluster centre. By local averaging over the ellipticities of galaxy images, a local estimate of the tidal gravitational field can be obtained (the direction of the sticks indicates the orientation of the field, and their length is proportional to its strength). From this estimated tidal field the mass distribution can then be reconstructed. [From C. Seitz; and P. Schneider, Extragalactic Astronomy and Cosmology.]

Gravitational Lensing 2. 25

2.5 COSMOLOGICAL APPLICATIONS OF GRAVITATIONAL LENSING

The gravitational effects on light propagation offer us an ideal tool to measure the total gravitating mass content in cosmological objects and structures, which is of course a very uncertain task, as we know. We will exploit our analysis above of the phenomenon of gravitational lensing to discuss the following cosmological applications, in the order:

• Mapping the mass distribution of galaxy clusters • The case for strongly lensed high-redshift sources • The fraction of strongly lensed objects as a function of the cosmological

parameters ( ΛΩ )

• Time delays due to lensing and the estimate of H0 • Micro-lensing and the search for massive dark objects in the Milky Way • Variability of high-redshift quasars • Weak-lensing and the Large Scale Structure

2.5.1 Mapping the mass distribution in galaxy clusters One of the fundamental structures in the universe are the galaxy clusters, the largest existing virialized objects at the present time. Measuring their mass content and distribution is a key task and is of fundamental importance for our understanding of the nature and properties of dark matter, the dominant source of gravitation on the largest scales. The determination of the mass distribution in clusters can be achieved from measurements of the kinematics of galaxies and application of the virial theorem, however with the limitation of a large statistical noise (small number of "test particles"), and uncertainties related with the assumption that the virial theorem (that assuming relaxation, which may not be the case for many clusters, see for example the case in Fig. 4) holds valid. One alternative is to use X-ray observations of the hot intra-cluster plasma, which may be indeed of very high quality from existing observatories (XMM-Newton and Chandra). The estimate requires a physical modelling of the X-ray emission, that includes the thermo-dynamical status of the gas, outflowing or inflowing plasma components, cooling flows in the centre, plasma temperature gradients, etc. So, this one and the previous methods are rather model-dependent and uncertain.

Gravitational Lensing 2. 26

A deep HST image of the Abell cluster A2218, revealing a number of arcs and arclets.

A completely model independent estimate, instead, is provided by the lensing analysis of background galaxies. By order of magnitude, the expected radius of the Einstein ring for a typical rich galaxy cluster is

1/21/2

62 1/2

4 13 10 secEGpc

GM M arcc D M D

θ − = ×

[2.33]

Thus, clusters of galaxies with masses of 1015 Mo at cosmological distances can result in Einstein rings with angular radii of tens of arcseconds. Such rings were first reported by Lynds and Petrosian (1986) and Soucail et al. (1987). Of course, this technique requires the use of high-resolution imagers. With HST it has been possible to test the effect with the highest possible quality (see e.g. the famous image of the cluster Abell 2218 above, containing a number of blue and red giant arcs; the ellipticity and the incompleteness of these rings reflect the facts that the gravitational potential of the cluster is not precisely spherically symmetric and that the background galaxy and the cluster are slightly misaligned). A reconstruction of the lensing mass distribution based on the algorithms discussed in Chap. 2.4 and using the myriad of lensed background objects offers a highly precise and robust estimate of the total matter distribution in the most general situation, if only a good quality image is available. This is currently achieved and reported extensively in the literature, based on the strong lensing analysis in the inner cluster cores and on the weak lensing, both discussed in Sect. 2.4 (see also the result in Fig. 4). Here instead we gain further insight by considering the simplest case in which we presume to have some a-priori knowledge on the mass distribution in the cluster.

Gravitational Lensing 2. 27

This is the case for the most relaxed and regular galaxy clusters, the rich clusters, in which the spatial distribution of galaxies can be represented by the distribution of mass in an isothermal self-gravitating gas sphere (where galaxies are test particles). Isothermal means that temperature, or the mean kinetic energy of the particles, is about constant throughout the cluster. In physical terms, this means that the velocity distribution of the galaxies is Maxwellian with the same velocity dispersion (or temperature T) throughout the cluster. We may further assume that the cluster is a spherically symmetric structure in hydrostatic equilibrium, that is pressure gradients balance everywhere the gravity forces:

22 0

where 4rdp GMp M r dr

dr rρ π ρ∇ = = = ∫ [2.35]

2 2 22; ; 4 0r dp d r dp dM d r dpGM G Gr

dr dr dr dr dr drπ ρ

ρ ρ ρ

= − = − + =

and with the perfect gas law p kTρ µ= (where T is a measure of the characteristic velocity dispersion of particles):

2

24 0d r d G rdr dr kT

ρ π µ ρρ

+ =

[2.36]

(Lane-Emden eq.), a non-linear differential equation that in general must be solved numerically. For large values of the radial coordinate r, we have an analytic solution:

002

2( ) with 4

K kTr Kr G

ρπ µ

= = [2.37]

as can be immediately seen by inserting this into [2.36]. Solution [2.37] is not valid for small r, where the density decreases slower with r starting at r values corresponding to the cluster core radius rC , and in addition it has the unfortunate property that the density formally diverges at 0r → and the total mass linearly diverges at r →∞ (the density falls steeper at r greater than the tidal radius, Tr r> ). However it represents the matter distribution over a substantial fraction of the cluster volume. Now, we can express the pressure terms as either the temperature or velocity dispersion of particles: 2vkT µ= , so the constant K0 can be written in terms of the velocity:

2

0

v4 4

kTKG Gπ µ π

= = [2.38]

sin( ); cos( ) ; / cos( )z r dz r d pd r pθ θ θ θ θ= = = =

Gravitational Lensing 2. 28

where the velocity dispersion is easily measured with spectroscopic observations, or the temperature with X-ray observations of the plasma emission. It is now simple to integrate [2.37] to obtain the projected surface mass density:

so that the mass within the impact parameter p and the reduced deflection angle are:

This is a remarkable result, showing that the deflection angle is independent on the impact parameter p and is about

[2.39]

Fig. 13. Mass reconstruction in the cluster CL0024+17. On the left, the distribution of orientation of background galaxies, on the right the reconstructed mass density contours.

Arcs up to a few tens arcsec are then expected for lensed background galaxies close to the cluster cores, for rich clusters with line-of-sight rms velocities of 1000 Km/s. Eq. [2.39] is a rather robust expression for comparing the masses of clusters of galaxies, as measured by the gravitational deflection of background galaxies, with

Gravitational Lensing 2. 29

their internal velocity dispersions. For rich clusters with well-defined potential wells and relaxed structures, the two estimates of cluster mass agree within 10%. For more complex configurations, like in irregular clusters, the kinematic measure is practically impossible and the lensing measure remains as the most robust way. A very interesting example of total mass reconstruction in a peculiar cluster of galaxies, the "bullet cluster", with its relevant implications for the nature of dark matter, is reported and commented in Fig. 4.

Fig. 14. HST Planetary Camera imaging results on the z=2.28 IRAS galaxy IRASF10214. Results of an isothermal lens model are reported in panel e and f. The component #2 is the (elliptical) lensing galaxy at z∼1. Components #3 and #4 are galaxies in the field, possibly physically associated to IRASF10214 but not lensing counterparts.

Gravitational Lensing 2. 30

2.5.2 Strong lensing and the remarkable case of strongly lensed high-z galaxy IRASF10214+4724 Our abilities to explore the distant universe have experienced a dramatic improvement thanks to a number of new facilities with which we can explore deeply over essentially the whole e.m. spectrum. This means that as we go deep, the typical source distance and the chance of intervening objects along the line-of-sight increase. In 1991 a Nature paper reported the discovery in the 60 µm IRAS Faint source survey catalogue of an extremely luminous high-redshift (z=2.28) at the (relatively bright) flux limit of 200 mJy. The source luminosity (L = a few 1014 Lʘ) and distance where very untypical of the many thousands IRAS galaxies analysed at the time (whose redshifts turned out to be z ≈ 0.1 on average). The most important consequence was the enormous luminosity of the object and the lack of evidence for AGN nuclear activity, that meant that a large fraction of L was due to a huge rate of star-formation (SFR, several thousand Mʘ/yr), never found in any other galaxy. Also the redshift was remarkably high. The physical interpretation of this discovery remained a mystery for quite some time, until the first high quality HST observations revealed the system to be a very strongly lensed galaxy (µ ≈ 100), as shown in Fig. 14. These lensing results as in the figure clarified the nature of the source (the contribution by an hidden AGN cannot in fact be excluded but seems unlikely, the object is similar to the local ULIRG Arp 220), and brough the SFR to more reasonable values of several hundreds Mʘ/yr. On the other hand, this chance discovery of a very luminous phase of galaxies visible only in the far infrared set the scene for a new paradigm of dust-extinguished star formation, producing the most massive galaxies already at high-redshifts, otherwise undetectable at other (optical) wavelengths. This mode of galaxy formation was confirmed by all later long-wavelength observations from ground (with millimetric telescopes like ALMA) and from space (ISO, Spitzer, Herschel). 2.5.3 The fraction of strongly lensed objects at high-redshift It is of interest, now, to verify the probability of distant detected sources to experience strong lensing effects by structures along the line-of-sight. This has important consequences not only to understand, at least on a statistical sense, how much flux measurements for distant objects might be affected, but also to estimate cosmological parameters, as we will see. Let us start by considering the probability to obtain a strong lensing event along an arbitrary line-of-sight. This is given by an integral of the number density of lenses

( )n z times the strong lensing cross-section ( )zσ

Gravitational Lensing 2. 31

0

( ) ( ) ( ) ( )z

propP z n z z dr zσ′ ′ ′= ∫ [2.41]

where all factors here are in proper units. The strong lensing probability can also be viewed in terms of a lensing optical depth τ , that is the fraction of sky covered by the Einstein rings of the lenses. We can first discuss a zeroth-order heuristic calculation. We can take for ( )zσ the cross section corresponding to the Einstein ring ([2.14]) 1/224E GM c Dθ ′ = :

[ ]2 22 2 2

4 4 4DS D DSD E D

D S S

D D DGM GM GMDD Dc D D c D cπ π πσ π θ= = = = [2.42]

Note here the important fact that the factor of the reduced distance D DS SD D D D= peaks when D DSD D , that is the lensing is most effective when it is midway between the source and the observer. Indeed we can write that for a given M

0 for 0( ) 10 for

DD DS D S D DD

S S S D S

DD D D D D DD DD D D D D

σσ

σ → →−

∝ = ≈ ≈ − → →

In practical situations, the maximal effects of lensing on high-redshift objects happen for lenses falling at redshift 0.3 1z ≈ − and for background sources at 1 2z > − . In eq. [2.41] and [2.42] the probability or the optical depth are proportional to the product ( ) ( ) ( ) ( ) DP z n z z n z Mσ ρ∝ ∝ ∝ , with M the average lens mass and Dρ is the average cosmological mass density in lenses, and the probability is then proportional to it. We assume here for simplicity uniformly distributed point mass lenses contributing a value DΩ to the average density parameter, a constant comoving number density 0n [such that in proper units 3

0( ) (1 )n z n z= + ], and a strong lensing cross-section that corresponds to the Einstein ring ([2.42]). The integral [2.41] can be calculated after replacing the proper differential radial increment 0( ) (1 ) 1propdr z cdz H z z= + +Ω , to get:

0

3 1( ) ( )2 1

Sz D DSD

S

d d zz P z dzd z

τ Ω +=

+Ω∫ [2.44]

where we have defined 0D Dd H D c≡ as a re-scaled distance (and similarly for the other distances), and where Ω is the total density parameter. If we consider that

1D DS Sd d d ≈ , the integral can be simply resolved to give, for 1 and 3SzΩ = = ,

( 3) 0.5S Dzτ = ≈ Ω [2.45]

One of the most effective lensing structures are the cores of galaxies (particularly of early-type galaxies, always containing supermassive black-holes and high stellar concentrations). Indeed, the majority of strong lensing effects on high-redshift objects

Gravitational Lensing 2. 32

are associated with elliptical galaxies at z from 0.5 to 1. Since the visible cores of galaxies produce 1 310 10 (with 0.01)D gal gal

− −Ω Ω Ω ≈ , we might expect

roughly one in 3 410 − high-z quasars to be strongly lensed. A deeper more quantitative insight requires a more quantitative evaluation of the strong lensing probability as a function of the angular distance between the source and the lens. We first need to compute the total flux amplification for a source lying at a given angular separation from a point-mass lens as a function of source angular distance on sky. It is natural to refer again the calculation to the Einstein radius, every angle can be expressed a-dimensionally in this way. If we simplify the lens as a point-like mass, we can get immediate analytical results. The 2 solutions of the lens equation for the 2 images, as given in [2.15] are 2 20.5( 4 )Eθ β β θ± = ± − , considering the flux amplification definition [2.17] and differentiating [2.15], we have

2 22 2

1 2 2

2 2

2

4 )1 21 ( 4 )2 24

1 2 1 42 1 4

EE

E

dd

x xx

β β θθ θ βµ β β θβ β ββ θ

− −

−

+ − = = + + − = −

+ + +=

+

for the first solution, having defined E

x βθ

≡ and

2 2

2 2

1 2 1 42 1 4

d x xd x

θ θµβ β

− −

−

+ − += =

+ for the second solution. The sum of the two

makes

2 2

1 2 2 2

1 2 1 2( )2 1 4 1 4

x xxx x x

µ µ µ−

−

+ += + = =

+ + [2.46]

This simple result is a very useful formula for gravitational lensing, the basis of many practical calculations 5. The next step is to compute the lensing cross-section as a function of the amplification, that is the proper area around a given lens through which the un-deviated light ray would need to pass to cause an amplification greater than µ . In this case this cross section will be:

[ ]2( ) ( )DDσ µ π β µ> =

5 See Peacock "Physical Cosmology", 1999

Gravitational Lensing 2. 33

where ( )β µ is the source un-lensed angle on sky within which the amplification is larger than µ. We need to invert [2.46] to get this angle ( )β µ for a given amplification µ . It is just a question of some algebra:

2 22 2 2 2

2 2

1 1 11 4 ; (1 4) 1 ; 1 ( 1)2 4 4x x xx x

x x xµ µ µ

+ = + + = + + + − =

24 2 2

2 2

4 4 1 1/ ( 1)4 24 0 21 2 1

x x xµ µ

µ µ

− ± + −+ − = ⇒ = = −

− −

by taking only the positive solution, for x2 being positive, such that:

2

2 2

2

12

1Eµ µ

β θµ

− −=

−

With Eθ from [2.14], we finally have as the cross-section for a point-lens [see also 2.42]:

2

2 22 2 2

18 8 1( )1 1 1

GMD GMDc c

µ µπ πσ µµ µ µ µ

− −> = × = ×

− − + − [2.47]

(as it can be easily verified). This scales proportionally to 2µ− for large values of the amplification. For an isothermal sphere, whose central density is parametrized by the central velocity dispersion as in eq. [2.37], we have instead (Peacock 1999):

( )

222

2 2

222

22

4 v 4( ) ( 2)

4 v( ) ( 2)

1

GD

c

GD

c

π πσ µ µµ

π πσ µ µµ

> = >

> = < −

[2.48]

[note that the probability is symmetric around 1µ = ].

Now, the probability to obtain a strong lens event along an arbitrary line-of-sight with amplification factor >µ becomes

0

( , ) ( ) ( , ) ( )z

propP z n z z dr zµ σ µ′ ′ ′> = >∫ [2.45]

where ( ) and ( , )n z zµσ > are in proper units and where we set the previously defined proper differential radial increment. In all cases considered above, for high amplification values the scaling of the lensing cross-section and of the amplification probability are 2( ) ( )P µ σ µ µ−> ∝ > ∝ .

Gravitational Lensing 2. 34

More details about the strong lensing probability are reported in Fig. 15, where we see plotted the various dependences on the source redshift, and particularly on the cosmological parameters and MΛΩ Ω , for flat universes + 1MΛΩ Ω = . In particular, the increased lensing probability for Λ -dominated models is potentially an important signature of vacuum energy. This dependence on ΛΩ is explained by considering that the volumes enclosed and the scales within a given redshift distance in a given sky area increase at increasing ΛΩ . We see in particular that changing

ΛΩ from 0.1 to 0.9 increases the strong lensing probability by almost a factor 10.

Then we can obtain limits to the value of ΛΩ from the frequency and properties of gravitational lenses in complete samples of quasars and radio galaxies.

Fig. 15. The optical depth to gravitational lensing in various models. The lower curves deal with the observed galaxy population only; results are shown for zero vacuum energy (zero Λ ) with different Ω values (solid lines) and flat Λ-dominated models with

0.3 and 0.1=Ω (broken lines). The upper solid line shows a inhomogeneous universe with all dark matter as point-mass lenses. [Figure reported in Peacock 2005].

Strong lensing surveys and constraints on the parameter ΩΛ. From an observational point of view, the largest survey to date designed specifically to address this problem has been the Cosmic Lens All Sky Survey (CLASS) in which a very large sample of flat spectrum radio sources were imaged by the Very Large Array

Gravitational Lensing 2. 35

(VLA), Very Long Baseline Array (VLBA) and the MERLIN long-baseline interferometer. The sources were selected according to strict selection criteria and resulted in the detection of 13 sources which were multiply imaged out of a total sample of 8958 radio sources. More recently, the CLASS collaboration has reported the point-source lensing rate to be one per 690 ± 190 targets (Mitchell et al., 2005; see examples of a catalogue of similar images, the SLACS survey, in Fig.1). The analysis of these data required the information about the luminosity functions for different galaxy types found from the AAT 2dF survey, as well as models for the evolution of the population of flat-spectrum radio sources. The CLASS collaboration found that the observed fraction of multiply-lensed sources was consistent with flat world models, 1MΛΩ +Ω = , in which the vacuum density parameter ΛΩ at 68% confidence is

0.310.240.69+

Λ −Ω . [2.50]

2.5.4 Time delays due to lensing and the estimate of H0

One of the effects of gravitational lensing is to modify the time taken by a light signal emitted by the source to get to the observer. In fact, there are two independent effects at play. A first effect is a time delay between the light received by an observer from the lensed image and that from the direct image due to the geometrical configuration. This is associated with the second-order difference in the lengths of the two paths. This time difference 6 includes a term (1 )Dz+ converting the times from the lens to the observer ( Dz is the lens redshift) and due to the cosmological expansion (see Sect. 0.2):

2 2

(1 ) (1 )2 2

D Sg D D

DS

D Dc t z z DD

α α′D = + = + [2.52]

As we mentioned before, there is however a second significant term due to the effect of the gravitational potential over the time rates, or the light propagation velocity with respect to the observer. This can be immediately recognized by looking once again at the Schwarzschild metric solution in eq. [2.1] and is the same term responsible for the factor 2 between the Newtonian and the relativistic results in [2.0] and [2.2]. The smaller coordinate speed of light leads to a delay term equal to 6 The path difference from lens to observer is 2 2 2 2 2

1 ( 1 1) 2D D D D Dp D D D D Dα α αD = + − = + − because

1α << . From source to lens it is analogously 22 2 2DSp D αD = . 2 with D DSD Dα α= being the angle subtending the

lens in the source rest-frame, 2(from )D DSD Dα α= Summing the two we have: 2 2

1 2 2( )D D DSp p p D D DαD = D + D = + = 2 2(1 [ ]) 2 2D D S S D S DSD D D D D D Dα α= + − =

Gravitational Lensing 2. 36

2

2(1 )p Dc t z dlcΦ

D = − +∫ [2.53]

where the minus sign just accounts for the negative value of the gravitational potential, and in fact the two terms in [2.52] and [2.53] add to each other,

g pt t tD = D + D , and are of the same order of magnitude. Indeed, the second term is rather more model dependent, because it involves details of the lens gravitational potential. For distance scales of order of the Hubble radius (4200 Mpc), angles of few arcsec, and 1Dz ≈ , we expect delay timescales of about 1-2 years.

This time lag has been exploited to achieve an estimate of the Hubble constant completely independent from all classical distance indicators. The dependence of time delays of light signals at the observer on 0H is obvious (and is illustrated in the scheme below):

all the physical sizes of the source-lens-observer system scale with the inverse of 0H . Decreasing the Hubble parameter, all scales increase and time differences decrease by

01gt HD ∝ . If the time delay can be measured, and if one has a good model for the mass distribution of the lens, then the Hubble constant can be derived from measuring g pt t tD = D + D .

In practice, the measurement requires several critical measurements and calculations. We need to identify a bona-fide strong lens system with multiple images sufficiently bright, with measured redshifts (of course these need to coincide for all lens images). The lensing system (typically a galaxy or a cluster) has to be identified and characterized as much as possible, to best constrain the lensing solution with regard to the contribution from eq. [2.53]; the lens should have a redshift and possibly kinematic measurements of the velocity dispersion. The source should be variable on short timescales and light curves have to be precisely measured for all images. This technique has been applied to various systems, one of which is reported in Fig. 17. The corresponding light-curves are reported in Fig. 18. The analysis of this system has brought to an estimate of Hubble constant of

Gravitational Lensing 2. 37

Gravitational Lensing 2. 38

[2.54]

(Grogin & Narayan 1996), in terms of the central velocity dispersion of the lensing galaxy (this galaxy produces the lensing effect).

Fig. 18. The lightcurve of QSO 0957+561AB shows flux variations of about 0.1 mag. The upper curve shows variations delayed by 424.9+-1.2 days with respect to the bottom curve. Associated temporal structures are indicated in red.

2.5.5 Micro-lensing and the search for massive dark objects in the Milky Way 7

One of the great unknown in astrophysics and cosmology is the nature of dark matter. While its presence has been inferred from countless evidences on all astrophysical scales, its nature is still not quite understood. One of the strongest of all these evidences are the observed rotational curves of spiral galaxies and their flat shapes at large radial values, showing the presence of matter not emitting light. It has also been found that the structure of thin galactic disks would be highly unstable unless a massive halo hosts the galaxies, another (though indirect) evidence for spiral galaxies containing large amounts of dark matter in their halos.

7 We refer to Schneider “Extragalactic Astronomy & Cosmology”, 2005, as a useful reference for this section.

Gravitational Lensing 2. 39

The nature of dark matter is thus far unknown. We may expect two totally different kinds of dark matter candidates: • Astrophysical dark matter, consisting of compact objects – e.g., faint stars like white dwarfs, brown dwarfs, black holes, etc. Such objects were assigned the name of MACHOs (MAssive Compact Halo Objects). • Particle physics dark matter, consisting of elementary particles which have thus far escaped detection in accelerator laboratories. Although the origin of astrophysical dark matter would be difficult to understand (not least because of the baryon abundance in the Universe – as discussed later in Sect. 5 of this Course – and because of the metal abundance in the ISM), a direct distinction between the two alternatives through observations would be of great interest. Indeed, gravitational lensing offers a very interesting method (proposed by Paczynski in 1986) to probe if the dark matter in our Galaxy consists of MACHOs. The method exploits the so-called galactic micro-lensing, that is the lensing effect by dark collapsed objects in the halo of our Galaxy on background stars, particularly suited those in the Large and Small Magellanic Clouds (where millions of stars can be detected in a single wide-field image). Image separation. We need first to consider that micro-lensing by point-like masses produces double images that cannot be resolved with current standard instrumentation (like a wide-field telescope on ground): using the Einstein ring as a reference, we have from [2.14] that typical separations are only

1/21/2 1/21/23

2 24 4 0.9 10 sec

10DS

ES D

GMD GM M Darcc D D c D M Kpc

θ−

− ′ = = ≈ × ′

[2.55]

However, a lensing event, though not producing an observable split, can in principle amplify the total flux received by the observer by a significant amount. But how to recognize the effect among millions of background stars? The simple way is to consider the flux variation with time due to the relative motion of source, lens, and the observer. Therefore, the flux is a function of time, caused by the time-dependent magnification. How to recognize lensing events by dark objects against the numerous variable stars in the field? This is possible because the variable lensing flux amplification has a characteristic light-curve. Variability time-scale. The characteristic time-scale for micro-lensing variability is immediately calculated by considering a typical transverse velocity of the lens and assuming at rest the observer and the source:

1

3v v4 10 sec/200 / sec 10

D

D

Darc yrD Km Kpc

θ−

− = = ×

[2.56]

with a variability time-scale of

Gravitational Lensing 2. 40

1/2 1/21/2 1

v0.2 110 200 / sec

E D DE

S

D DMt yrsM Kpc Km D

θθ

− = −

[2.57]

or about a couple of months for typical parameter values. This is a quantity measurable, but again comparable to those of many variable stars. To distinguish the lensing effect from stellar variability one has to consider light curves at different wavelengths and compare one to the others: identical light-curves would be in strong favour of lensing, because the latter is completely achromatic. Light-curve shape. However, the smoking-gun in favour or not of a (transient) lensing event is the light-curve shape, because it has some unique features. First of all, it is symmetric with respect to the maximum, for normally behaved stars. Second, it is completely achromatic. Third it has a specific functional form. Let us see it. To good approximation, the relative motion of source, lens and observer can be considered linear, so that the position of the source in the source plane can be written as (we use here our usual symbol θ for the source position in the sky):

0 0( ) ( )t t tβ β β= + − [2.58]

and dividing it by the Einstein angle, we get the normalized angular distance source to lens:

22 2

2 20 02

( ) ( )( )( )E E E

t t t ttx t p pt

θβθ θ

− −≡ = + = +

[2.59]

where p is the normalized impact parameter of the lens trajectory in units of Eθ , that is the angular distance to the source when the lens is closest to the source, and 0t is the corresponding time instant, and Et from [2.57]. Putting [2.59] into our expression for the amplification factor for a point-mass lens in eq. [2.46], we get the light-curve:

( )2

0 0 2

1 2( )1 4

xS t S x t Sx x

µ += × =

+ [2.60]

where S is the time-dependent flux of the lensed source. The light-curve depends on only four parameters: the flux of the un-lensed source S0, the time of maximum magnification t0, the smallest distance of the source from the optical axis p, and the characteristic time-scale tE. All these values are directly measurable in the light curve. One obtains t0 from the time of the maximum of the light curve, S0 is the flux that is measured for very large and small times, S0 = S(t→±∞). Furthermore, p follows from the maximum magnification µmax = Smax/S0 and can be obtained by inversion of (2.46), and tE from the width of the light curve. An example of a measured light curve for a lensed star is reported in Fig. 20. Only tE contains information of astrophysical interest, because the time of the maximum, the un-lensed flux of the source, and the minimum separation p provide

Gravitational Lensing 2. 41

no information about the lens. Since , this time-scale contains the combined information on the lens mass, the distances to the lens and the source, and the transverse velocity: unfortunately, only the combination of these parameters can be derived from the light curve, but not the individual parameter values.

Fig. 19. Scheme of micro-lensing events and source-lens geometric configurations. P is impact parameter in units of the Einstein ring.

MACHOs observations. The programme of MACHO (indirect) observations started at the end of ‘80s. The idea was that if the halo of the Milky Way consists of compact objects, like isolated black-holes, neutron stars, low-mass stars, otherwise invisible. A distant compact source, like a background star, should occasionally be lensed by one of these MACHOs and thus show characteristic changes in flux, corresponding to a light-curve similar to those in Fig. 2.20. The number density of MACHOs is proportional to the probability or abundance of lens events, and the characteristic mass of the MACHOs is proportional to the square of the typical variation time-scale tE . What is needed is to measure the light curves of a sufficiently

Gravitational Lensing 2. 42

large number of background sources and extract all lens events from those to obtain information on the population of potential MACHOs in the halo. Typically the Galactic halo is modelled and the number of lensing events predicted to be compared with the observed numbers.

Fig. 20. The Light curve of the first observed microlensing event in the Large Magellanic Cloud, in two broad-band filters. The solid curve is the best-fit microlens light curve, with Amax = 6.86. The ratio of the magnifications in both filters is displayed at the bottom, and it is compatible with 1 [Alcock et al. 1993].

Unfortunately, the chance of such events to occur per unit time is extremely small and one needs an enormous number of background objects. The problem was resolved by taking repeated imaging of fields in the Magellanic Clouds, offering fields with huge number of stars per unit sky area, and subtracting one image from the other to look for variable objects in an automatic way. In the early 1990s, two collaborations (MACHO and EROS) among others began the search for micro-lensing events towards the Magellanic clouds. Another group (OGLE) started searching in an area of the Galactic bulge. All three reported MACHO detections in 1993. The statistical analyses to compute the areal density of lensing objects requires the definition of lensing cross section and optical depth as discussed in our previous 2.5.2 chapter. The conclusion of this investigation was that

Gravitational Lensing 2. 43

micro-lensing is consistent with 20% of the MW halo mass in the form of collapsed objects of mass about 0.5 M

(Alcock et al., see figure below), which is a large number in itself but implies that the bulk of dark matter is in diffuse form, likely non-baryonic dark matter. The interpretation of MACHO searches did not bring to conclusive results 8, particularly about the nature of the discovered objects.

As a final note, micro-lensing has been used for many other applications, for example to look for the presence of planets around lensed background stars, that would distort in some way the light-curves. A couple of moderate-mass planets (6 times the Earth mass) have been discovered in this way. Gravitational micro-lensing effects on high-redshift quasars. As an aside, we mention one further effect of lensing, concerning the observations of high-redshift quasars. This is probably a mere curiosity, but perhaps an interesting one to briefly report. What was considered (see Hawkins 1993) was the hypothesis that quasar variability is dominated by the effects of gravitational micro-lensing by a distribution of point-mass lenses in the Universe. It is based on a large-scale programme of monitoring quasar brightness variations in a homogeneous way over 17 years. Analysis of these observations, together with other surveys and monitoring programmes taken from the literature, were used to carry out a number of tests of the micro-lensing hypothesis, and to put constraints on possible models of intrinsic quasar variability. The results of this investigation, based on the analysis of high-redshift quasar light-curves and the application of eqs. [2.56] - [2.60], implied the possible existence of a population of 8 See Schneider “Extragalactic Astronomy & Cosmology”, for further discussion.

Likelihood contours obtained from the MACHO experiment for one specific model for the Milky Way halo. The abscissa is the fraction of the halo mass contained in MACHOs, the ordinate is the MACHO mass. The contours shown are the 60%, 90%, 95% and 99% confidence levels. [Bartelmann 2005]

Gravitational Lensing 2. 44

Jupiter-mass compact bodies producing lensing effects along almost every line of sight, and constitute sufficient material to make up the cosmological critical density. Constraints from primordial nucleo-synthesis require that at least 95% of this material be non-baryonic, and arguments are presented by Hawkins that the only plausible candidates for the compact objects are primordial black holes, probably formed during the quark/hadron transition (see Sect. 6 of this course). The suggestion thus was that the dark matter is composed of Jupiter-mass primordial black holes, with a mass density of about the cosmological critical density. However, this claim is not completely convincing. Quasars at any redshifts are known to be intrinsically variable on their own on timescales of about a year. Again the classic signature of micro-lensing is that the variability be achromatic and symmetric in time: even this is not known about the variability seen by Hawkins. 2.5.6 Weak-lensing and the Large Scale Structure

One of the ultimate uses of gravitational lensing is aimed at determining the mass distribution in the Large Scale Structure and the cosmological web in both the local as well as the distant Universe. As well known, such a mass estimate is virtually impossible by the standard methods used for inferring the mass distribution in galaxy clusters, e.g. via kinematical studies of galaxies and the virial theorem, or X-ray observations, because we are not in the presence here of anything like a dynamically stable and relaxed structure, but rather a slowly collapsing and evolving foam made of the gravitating dark-matter fluid. The technique exploits the mass reconstruction algorithms mentioned in Chap. 2.4, and looks for the ellipticity distributions of background galaxies to probe the integrated surface mass density along the line-of-sight, as illustrated in Fig. 20. The enormous potential of this technique is that it is completely unaffected by the uncertainties inherent in the bias parameter b (i.e. how galaxies or the hot plasmas map the dark matter distribution), because the determination of the mass distribution is not mediated by any assumptions in the case of gravitational lensing. The main limitation of this approach is that the signal is expected to be much lower than that generated by clusters of galaxies, for example. In addition, precise enough measurements of background galaxy ellipticity require a very good knowledge of the instrumental Point Spread Function. The approach to overcome these problems is a statistical one, that is to determine the distribution of ellipticity for large numbers of objects and reducing in this way the uncertainties.

Gravitational Lensing 2. 45

Fig. 20. Cartoon showing how light rays propagate through the Universe and are affected by the inhomogeneous matter distribution; they are deflected, and the shape and size of the background source images change. In particular, the image of a circular source get elliptical to a first approximation. Since the distribution of matter is structured on large scales, the image distortions of two neighbouring light beams is very similar, so that the observed ellipticities of neighbouring galaxies are correlated. The ellipticities of images of distant sources are closely related to the (projected) matter distribution (right panel) From a statistical analysis of the shapes of galaxy images, conclusions about the statistical properties of the matter distribution in the Universe can be drawn. [From CFHT Corporation].

Fig. 21. From the analysis of the data from the CFHT Legacy survey and comparison with model predictions, constraints on cosmological parameters can be derived. Confidence contours in the Ωm–σ8 parameter plane are shown, where ΩΛ = 0 was assumed. The effect of ΩΛ on the prediction of the shear is relatively small. The data suggest a Universe of low density. Presently the largest uncertainty in the quantitative analysis of shear data is the lack of knowledge of redshifts of the faint galaxies.

Gravitational Lensing 2. 46

A result of weak gravitational lensing in constraining the cosmological parameters in the local Universe is reported in Fig. 21. Significant constraints are derived from the Canada France Hawaii Telescope (CFHT) Legacy Survey of about 450,000 galaxy images on the 8 and m σΩ (the measure of ρ ρD on a scale of 8 Mpc, see eq. [1.45]) parameters. It is interesting to note that there are plans to exploit the weak lensing method to constrain the evolution of the Large Scale Structure with redshift (see below).

Weak-lensing and the Euclid space mission. Just a mention of the fact that one important astrophysical mission of the European Space Agency is Euclid, a survey observatory that will be dedicated in 2020 to an all-sky spectroscopic and imaging survey to determine the properties of cosmological dark-matter and particularly of dark-energy, and their time evolution. The space observatory will host a 1.2 m telescope, including in the focal plane a visual imager (550 to 900 nm, resolution 0.1 arcsec), a near-IR imager (Y, J, H bands) and a near-IR spectrograph (1100 to 2000 nm, spatial resolution 0.3 arcsec). The spectrograph will measure spectroscopic redshifts for at least 50 millions of galaxies over a wide redshift interval (0<z<2) and over half of the sky, plus about a billion of photometric redshifts 9 for photometrically selected galaxies. The optical imager will perform deep multiband imaging with the purpose to do a weak-lensing investigation based on techniques discussed in Chap. 2.4. Euclid will accurately measure the coherent pattern in galaxy ellipticities that gravitational lensing by large-scale structure imprints, which provides the most direct measurement of the (projected) matter power spectrum. Importantly, the availability of spectroscopic and photometric redshifts for the sources means that is possible to study the mass distribution in three dimensions, called weak lensing tomography. This requires that all the lensed background galaxies are sorted out and binned in redshift to estimate the redshift distance of the lensing structure.

9 Photometric redshifts are based on very low spectral resolution spectra that can be obtained with multiband photometric-imaging of a large sample of galaxies simultaneously in the same sky area.