# Massimo Capaccioli » 20.Gravitational lensing

### What is a gravitational lens?

Gravitational lensing describes astrophysical phenomena due to the action of gravity on the light propagation. As you can see from the picture of the central region of the galaxy cluster Abell 2218, the perturbation of gravity produces spectacular phenomena. In contemporary astrophysics, these phenomena give very accurate information on the mass distribution of galaxies and galaxy clusters and on the intrinsic properties of very distant sources, the light of which is amplified. In the context of this course we will be interested in the use of gravitational lensing to measure the mass of galaxies and galaxy clusters.

Newton was the first one to speculate that masses should deflect light. He considered the light as made of massive particles, therefore such an action of gravity seemed natural to him: «Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action strongest at the least distance?» However, he never investigated in deeper detail this idea.

Galaxy cluster Abell 2218, with several giant gravitational arcs. Credit: NASA-HST.

### A short history of lensing

Cover of the first edition of Opticks or a treatise of the reflections, refractions, inflections and colours of light, 1704, where Netwon mentioned a possible action of gravity on light.

### A short history of lensing

Despite providing the first observational evidence of the new theory, light deflection was considered for many decades as just a curiosity. Einstein calculated once the effects of light deflection. He derived all the basic equations of the phenomenon now called microlensing and estimated its probability. He found so low a number to think that the phenomenon could never be observed; he could not predict the dramatic technological development which made this possible in the early 1990’s. Fritz Zwicky was the only astronomer who considered the light deflection by astronomical bodies something more than a theoretical curiosity.

In 1937 he published two short and seminal papers in which he proposed the gravitational lensing by the “extragalactic nebulae” as a tool to estimate the mass of galaxies and to investigate the problem of the “missing mass“. He wrote: «The gravitational fields of a number of foreground nebulae may therefore be expected to deflect light coming to us from certain background nebulae. The observations of such gravitational lens effects promises to furnish us with the simplest and most accurate determination of nebular masses. No thorough search for these effects has as yet been undertaken

Zwicky actually started an observational program, to discover gravitational lenses, with no success, as he reported in his book about Morphological astronomy in 1957 (ed. Springer).

Year 1919: page of the Times reporting the observations of the light deflection during the total eclipse.

### A short history of lensing

The seminal short paper by Zwicky on the ??nebulae? as gravitational lenses, published in the Astrophysical Joiurnal.

### A short history of lensing

The first observational evidence for gravitational lenses was obtained only 50 years later Zwicky’s intuition, when the double quasar 0957+561 was identified as a double image of the same distant source. About ten years later, the first gravitational arc was finally observed. Finally, the Hubble Space Telescope, launched in 1990, open a golden age for gravitational lensing, with the continuous discovery of new lensed galaxies, owing to its high spatial resolution.

It is not possible to detail further the interesting history of this branch of modern astrophysics. Students interested in a more detailed description of the history of gravitational lensing can refer to the introduction of the Springer book Gravitational Lenses by Schneider, Ehelers, and Falco (1993). We will now discuss the physical foundations of light deflection in General Relativity, without discussing (for lack of time) the interesting formal aspects.

Cover of Science magazine, reporting the discovery of the lensed quasar QSO 0957+561.

Image of the first gravitational arc, discovered in 1986 by G. Soucail and her team at the CFHT.

### A simple derivation via the equivalence principle

Using an argument based on the equivalence  principle, but still without using the full and complex equations of Relativity, Einstein realized that massive bodies must deflect light. The argument is as follows. The principle of equivalence states that gravity and acceleration cannot be distinguished. In other words, a free falling observer does not feel gravity and an accelerated observer can interpret the resulting inertial force as due to a gravitational field. Suppose that the observer is contained in a box with a hole on its left side (see figure). If the box is accelerated upwards, the observer interprets the inertial force on him as a gravitational force acting downwards.

Suppose now that a light ray enters the hole on the left side of the box and propagates towards right. As the box is moving upwards, the ray will hit the wall of the box on the opposite side at a lower point than it enter. As the box is accelerated the light ray appears curved. Then, based on the principle of equivalence, light must be deflected by gravity. Indeed, we can imagine of reversing the experiment: let the box be stationary and within the gravitational field whose intensity is such to resemble the previous acceleration. If light is not deflected by gravity, then the observer has the possibility to discriminate between gravity and acceleration, violating the principle of equivalence.

Gedanken experiment to show that the Equivalence Principle implies that gravity deflects light.

### A simple derivation

In order to derive the precise relation between the mass distribution of the gravitational lens and the deflection angle, we have to use the theory of General Relativity (GR; Einstein, 1916). According to GR, the deflection is described by geodesic lines following the curvature of the space-time. In a curved space-time, geodesic lines are lines which are as straight as possible, resembling straight lines in flat space-time. As a light ray follows the curvature, it is bent towards the mass, which causes the space-time to be curved. As we could imagine, the deflection angle is larger for light rays passing closer to the gravitational lens. This bending gives rise to several important observable phenomena, discussed in the following.

The amount of deflection is larger for light rays passing closer to the gravitational lens.

### GL produces multiple images

(1) A gravitational lens produces multiple images.

As geodesics in curved space-time are not unique, multiple paths around a single mass become possible: for instance, one around the left and one around the right side of the deflector, as in the figure. The observer will then see multiple images of a single source, where an image is seen along the backward tangent of each ray arriving at his position.

In the following we will see that a regular gravitational lens (that is, a gravitational which mass distribution has no singularity) always produces an odd number of images.

The amount of deflection is larger for light rays passing closer to the gravitational lens.

### GL produces multiple images

(2) A gravitational lens produces distorted images.

The amount of deflection of two neighbor rays is usually different close to a gravitational lens. Suppose a pair of rays, one from one side and one from the other side of a source, passes by a lensing mass distribution. The ray passing closer to the deflector will be bent more than the other, thus the source will appear stretched. It is thus expected that gravitational lensing will typically distort the different images of the source. By the same mechanism, they can appear larger or smaller than they originally are.

A gravitational lens: the arc is the distorted and elongated image of a background source. Credit: NASA-HST.

### GL produces multiple magnification

(3) Image magnification.

Since photons are not created, neither destroyed by the lensing effect the surface brightness of the source will remain unchanged. Since, as we said, the size is not conserved, this implies that the source can be either magnified or de-magnified by lensing (that is, total flux amplification will be less than 1). If it is enlarged it will appear brighter, otherwise fainter.

### Time delay

(4) Time delay.

In case that multiple light paths are possible between the source and the observer, since they will be characterized by different lengths, the light travel times will differ for the different images. One of the images will appear first, the others will be delayed. Before understanding in details the quantitative aspects of gravitational lensing, let’s see now some of its manifestations, according to the mass of the gravitational lens.

The figure at the RHS shows the light curves of the two images (inset) of the lensed quasar QSO 0957+561, shifted by 423 days and 0.06 mag to allow overlap, from Burud et al. (2001).

Adapted from Burud et al., A&A, 380, 805, 2001.

### Time delay

Quasars may change their luminosity also on short time scale (days or weeks). Therefore, multiply imaged quasars are the ideal target for measuring the time delay between the different images. As mentioned above, the variation of the source luminosity shows up at different times in the two images since the path of light travel is different for them. Since such a time delay (a phase shift in the observed light curves) can provide a direct measure of the distances involved, this quantity is of great importance. Hence, it is not surprising that many attempts have been made to estimate it. The Norwegian astrophysicist Sjur Refsdal (1964), while a PhD student, was the first one to suggest that a measurement of the time delay would provide a unique method to measure the Hubble constant. At the time of writing, about 20 lensed system are known with a measured time delay.

The figures sketches two ideal (continuous) light curves of two images of the same variable source shifted in time (and in flux) by the difference in the light paths (and in the amplification).

### Gravitational lensing scheme

Adapted from a sketch by Y. Mellier.

### Deflection angle by a point mass

We now calculate the deflection angle due to the simplest mass distribution, a point mass. While this model is not physical (that is, its mass distribution is not regular), it is particularly useful. First, it allows us to understand many aspects of lensing phenomena; the derived formula will be used when deriving with more complex models. Finally, the equation for the deflection angle holds true in the case of a spherically symmetric system acting as gravitational lens.

As a fact, the generalization of this simple model to spherical systems is obvious, as far as the light path is outside the system, as a consequence of the Gauss theorem. In other words, when the impact parameter b is larger than the radius of the source of the gravitational field, we can neglect its extension and consider all the mass M to be located at its center. The geometry of the lensing system is shown in the RHS figure. The light rays propagate along the direction z and has impact parameter b. As the gravitational field is weak (i.e., non relativistic), the total deflection will be small.

Geometry of a lensing system with a point mass M and a light ray with impact parameter b.

### Deflection angle by a point mass

Let’s calculate the deflection angle as the light travels in a flat space-time and its path curved by Newtonian gravity. A more coherent derivation, within the General Relativity, will be given later. The transverse component of the force exerted by the mass $M$ on a particle (photon) with an impact parameter $b$ is:

$\displaystyle{ a = \frac{G M b}{(z^2 + b^2)^{3/2}}} .$

The total integrated velocity is given by:

$\displaystyle{ v = \int a dt = \int a dz/c = \frac{2 G M}{b c}},$

where the integral is computed along the unperturbed light path (this approximation holds true as long as the deflection angle is small).

The resulting deflection angle is then the ratio between the tangential velocity variation and the velocity of the particle itself:
$\alpha=2GM / bc^2,$
a result which is wrong by a factor 2, as we will see later on. Let’s verify that deflection angle is indeed small. Considering a lensing mass with the mass of the Sun, size of the order of  $10^{-7}pc,$ we obtain a deflection angle of  $1\ arcsec.$

Geometry of a lensing system with a point mass M, and a light ray with impact parameter b.

### Light propagation in GR: Fermat Principle

We now derive the equation for the deflection angle within the context of General Relativity. First of all, we will consider the approximation of geometrical optics: light propagates along rays, and all wave aspects are neglected. The starting point is the Fermat’s Principle. In its simplest form, this Principle states that light takes the path between two points for which travel time is an extremal [expand].

The French mathematician Pierre Fermat (1601-1665).

### Light propagation in GR: Fermat Principle

The speed of light in a medium with refractive index $n$ is $c/n$, where the vacuum (i.e., no gravitational field) is characterized by $n=1$. Travel time along a generic path is given by:

$\displaystyle{ \int \frac{n}{c}dl }.$

The Fermat Principle states that the actual light path $\textbf{x}(l)$  is the one for which:

$\delta\int_A^B n\left(\textbf{x}(l)\right)dl=0,$

where the points A (starting) and B (ending) are kept fixed (see figure).The refractive index $n$ is related to the gravitational field. We assume the gravitational field is weak, that is: $U$ is the Newtonian gravitational potential. Such condition holds true for stars, galaxies and galaxy clusters, that is, all the observationally relevant found so far in gravitational lensing. [Exercise: find out the value  $U/c^2$ for a typical galaxy and galaxy cluster.]

Sketch for the Fermat Principle.

### Light propagation in GR: Fermat Principle

Let us consider the metric of space-time $g_{\mu \nu }$ around a gravitational lens. The metric of unperturbed space-time is the Minkowski metric $\eta_{\mu \nu}$ which is, in presence of a weak gravitational field, only weakly distorted. The diagonal terms are therefore:

$\displaystyle{{\rm diag} \, g_{\mu \nu } = \left[1+ \frac{2U}{c^2}, -\left(1-\frac{2U}{c^2}\right), -\left(1-\frac{2U}{c^2}\right), -\left(1-\frac{2U}{c^2}\right)\right] }.$

$\displaystyle{ ds^2 = g_{\mu \nu } dx^{\mu} dx^{\nu} = \left(1+ \frac{2U}{c^2}\right) c^2 dt^2 - \left(1-\frac{2U}{c^2}\right) d\textbf{x}^2 }.$

Light travel on paths where $ds=0;$; hence, its velocity, in presence of a weak gravitational field, is:

$c'=\frac{|d\textbf{x}|}{dt}=c\sqrt{\frac{1+2U/c^2}{1-2U/c^2}}\simeq c\left(1+\frac{2U}{c^2} \right),$

where we used the so-called weak field approximation. Finally, the refraction index in a weak gravitational field is:

$\displaystyle{ n = \frac{c}{c'} \simeq 1 - \frac{2 U}{c^2}}.$

### The refractive index of a gravitational field

The relation defining the refractive index of a gravitational field is the basis of gravitational lensing optics. As the Newtonian potential $U$ is always negative, the speed of light $c'$ in the presence of a gravitational field is always lower than in the vacuum (note: in this context, vacuum simply stands for a region where there is no appreciable gravitational field). In general the refractive index will depend both on space and time: in the following we will only consider astrophysical situations in which the there is no time dependence. In other words, the mass distribution within a gravitational lens can be considered as static while the light rays is deflected by it. To shown this, let us consider a galaxy acting as a gravitational lens. Its size is of order $10\ kpc$, that is  $\sim10^5\ lyr.$ Hence, the travel time for a light bundle across the galaxy is approximately $10^{12}{\rm s},$ much less of the time for any significant change in the galaxy mass distribution.

### Lens equation from a variational principle

We now show, without entering into details, how the equation for the deflection angle can be derived for a generic, static mass distribution acting as a gravitational lens. Derivation with full details is given in supplementary material, but every student with a basic knowledge of analytical mechanics can and should derive it. Let $\textbf{x}(l)$ be the path between fixed points A and B. The travel time is proportional to the quantity:

$\displaystyle{\int_A^B n[\textbf{x}(l)] {\rm d}l} .$

The actual path follows from the requirement:

$\delta \int_A^B n[\textbf{x}(l)] {\rm d}l = 0.$

This is a standard variational problem, that can be solved by introducing the affine parameter $\lambda,$ such that $\displaystyle{ \displaystyle{dl = \left| \frac{d\textbf{x}}{d \lambda} \right| \, d \lambda}},$ and defining the Lagrangian:

$\displaystyle{L(\dot{\textbf{x}}, \textbf{x}, \lambda) \equiv n[\textbf{x}(\lambda)] \, \left|\frac{d\textbf{x}}{d \lambda} \right|}.$

A generic curve characterized by the tangent vector x and the associated affine parameter ?.

### The refractive index of a gravitational field

The deflection angle due to a generic, static gravitational lens is:

$\vec{\alpha}=\frac 2 {c^2}\int_{\lambda_A}^{\lambda_B}\nabla_\perp U d \lambda$

This integral has to be computed on the actual path followed by the light, hence is no of great use in practice. However, as long as the gravitational field is weak (in terms mentioned above), the integral can be computed along the unperturbed path, that is the one followed by the light in absence of any influence by the deflecting gravitational field. This is similar to the well-known Born approximation used in scattering problems.

### The point mass, again

Let’s suppose a light ray passes a lens at $z=0.$ The deflection angle is then given by:

$\displaystyle{\vec{\alpha} = \frac{2}{c^2} \int_{-\infty}^{+\infty} \nabla_{\perp} U \, d\lambda }.$

It is a simple exercise to obtain now the deflection angle for a point mass, confirming the relation given before. The reader might recall the definition of the Schwarzschild radius:

$\displaystyle{ R_S = \frac{2 G M }{c^2}} .$

With this quantity in mind, the deflection assumes a very simple form:

$\displaystyle{\alpha = 2 \, \frac{R_S}{b}}.$

Actual path followed by the light, the unperturbed path and the deflection angle.

Actual path followed by the light, the unperturbed path and the deflection angle.

### The time delay

Let us now derive a relation for the time delay effect. The time delay has two contributions: a geometrical one (due to the fact that two different paths have intrinsic different lengths) and a gravitational one (due to the effect of the gravitational field).

The gravitational time delay, also known as the Shapiro time delay, is a direct consequence of the fact that the speed of light is a function of the gravitational refractive index, as shown above: $\displaystyle{ c' = \frac{c}{n}}.$

Hence, when considering a perturbed path, the time delay is:

$\displaystyle{\Delta t = \int \frac{dl}{c'} - \int \frac{dl}{c} = \int (n-1) dl = - \frac{2}{c^3} \int U dl} .$

Such relation was first found by Irwin I. Shapiro (1964) and observationally verified within the Solar System with radio waves.

### A quote from Albert Einstein

«In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuum, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light).» Albert Einstein, The General Theory of Relativity: Chapter 22. A Few Inferences from the General Principle of Relativity.

### The thin lens approximation

In realistic astrophysical situations, the lens mass distribution has length scale which is much smaller than the distances between source, lens and observer. Therefore, we can consider all the lens mass distribution to be located on a plane, the so-called lens plane. This greatly simplifies the formalism, allowing first introducing the surface density:

$\displaystyle{\Sigma(\vec{\xi}) = \int \rho(\vec{\xi}, z) dz },$

where $\vec{\xi}$ is two-dimensional vector on the lens plane.

[The student can now demonstrate that the following relation holds:

${\vec {\hat \alpha}} = \frac{4 G}{c^2} \,\int \frac{(\vec{\xi} - \vec{\xi'}) \Sigma(\vec{\xi'})}{|\vec{\xi} -\vec{\xi'}|^2} {\rm d}^2 \xi'.$

The lensing geometry for a generic, thin lens.

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