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Massimo Capaccioli » 21.Gravitational lensing - Part II

The lens equation: the cosmological distances

Once we have in hands the equation for the deflection angles, we can calculate the images produced in the lensing event. This can be done from the geometry of the system given in figure of the previous slide. However, we need first to consider attentively the notion of distance involved. As gravitational lensing is at work on cosmological scales, we cannot use a naive Euclidean definition of distance. Having to deal with angular quantities, it is natural to use in the following the so-called angular-diameter equation D_{\rm A}, defined as: D_{\rm A} = \theta \times d , where d is the proper length of a source observed under the angle \theta. In a curved space-time, angular-diameter distances are not additive: therefore, the sum of the distance from the observer to the lens, D_{\rm L}, and the distance from the lens to the source, D_{\rm LS}, is not equal in general to the distance between the observer and the source, D_{\rm S}. We will not demonstrate the following relation for the angular-diameter distance between two redshifts i and j in an assumed cosmological model: \displaystyle{D_{ij} = \frac{c}{H_0} \frac{1}{\sqrt{1 - \Omega_m - \Omega_{\Lambda}}}\int_i^j \frac{\sqrt{1 - \Omega_m - \Omega_{\Lambda}}}{[(1+z)^2(1 + \Omega_m  z)-z(2+z)\Omega_{\Lambda}]^{1/2}}}.

Deriving the lens equation

Let us first define an optical axis (the dashed line in the RHS figure), connecting the lens center of mass to the observer. This definition is arbitrary and just helps in keeping equations simple. This axis is perpendicular to lens and source planes, and all angles are measured with respect to this direction. A source is located at angle \theta.  The following relation holds: \xi=D_L\times\theta.  As the gravitational field is weak, all the angles considered here are small, and we can write the following relation, simply derived on geometric grounds (assuming that all distances are the angular-diameter distances):

\vec{\theta} D_{\rm S} = \vec{\beta} D_{\rm S} +  \hat{\vec \alpha} D_{\rm LS}

Introducing the so-called reduced deflection angle, \vec{\alpha} = \frac{D_{LS}}{D_S} \hat{\vec{\alpha}}  , the above equation reads simply: \vec{\beta} = \vec{\theta} - \vec{\alpha} .

This deceptively simple equation is the famous lens equation: in the general case, it is nonlinear. Hence, it is possible to have multiple images \vec{\theta} corresponding to a single source position \vec{\beta}. The lens equation is trivial to derive and requires merely that the following Euclidean relation should exist between the angle enclosed by two lines and their separation:

separation = angle x distance

Sketch of a typical gravitational lensing system

Sketch of a typical gravitational lensing system

The lens equation (1)

Let us consider in details the lens equation, sometimes called also the ray-tracing equation: \vec{\beta} = \vec{\theta} - \vec{\alpha} .

Introducing a reference lengh on the lens plain \xi_0 and the corresponding quantity on the source plane,

\displaystyle{ \eta_ 0 = \xi_0 \frac{D_S}{D_L}},

we can introduce the following dimensioless vectors on the lens and source planes, respectively:

\displaystyle{ {\vec x} = \frac{\vec \xi}{\xi_0}},     \displaystyle{  {\vec y} = \frac{\vec \eta}{\eta_0}},

and the so-called scale deflection angle:

\vec \alpha( \vec x)= \frac{D_L D_{LS}}{\xi_0 D_S} \hat{\vec{\alpha}}(\xi_0 \vec{x}).

We can now write a dimensionless form of the lensing equation:

\vec y = \vec x - \vec{\alpha} (\vec x)

The axially symmetric lens

As a simple exercise, we derive the equations in the special case of an axially symmetric lens. It is important to stress that in most cases of astrophysical interest (galaxies, clusters of galaxies), the angular structure of the matter distribution and hence of the gravitational potential cannot be neglected when inverting the lens equation. However, an axially symmetric lens provides a basic introduction to many of the elements which are essential to realistic models without the need for numerical calculations. While in general the deflection angle is a two-dimensional vector, in the case of an axially symmetrical gravitational lens it is a one-dimensional function, as the light rays from the source to the observer lie in the plane determined by the center of the lens, the observer and the source. This is a consequence of the fact that when we have a circular lens, the mass distribution is a function only of the distance from the lens center.

[We leave to the reader the demonstration of the following equation for the deflection angle of a circular lens:

\hat \alpha=\frac{4GM(\xi)}{c^2\xi}

and the lens equation becomes a simple scalar equation:

y = x - {\hat \alpha} (x) .

The lensing potential

The so-called lensing potential is an important quantity in the characterization of a gravitational lens. The lensing potential \hat{\Psi}(\vec{\theta}) is obtained by projecting the three-dimensional Newtonian potential \Phi on the lens plane and by properly rescaling it:

\displaystyle{ \hat{\Psi}(\vec{\theta}) = \frac{D_{LS}}{D_S D_L} \frac{2}{c^2} \int \Phi(D_L \vec{\theta},z) {\rm d} z}.

The useful dimensionless expression reads:

\displaystyle{ \Psi = \hat{\Psi} \frac{D_L^2}{\xi_0^2} }.

The lensing potential is an important quantity, related to the deflection angle and to the mass distribution of the gravitational lens. The relation with the deflection angles reads simply:

\vec{ \nabla}_x \Psi(\vec{x}) = \vec{\alpha}(\vec x) .

The derivation of this equation is left to the reader.

The lensing potential (2)

The other important property of the effective lensing potential is the link with the mass distribution of the lens, \Sigma (\vec{x}). In other words, we will write a sort 2D counterpart of the Poisson equation of the gravitational field. It is useful now to introduce the dimensionless counterpart of the mass surface density, defined as follows: \displaystyle{ \kappa(\vec{x}) = \frac{\Sigma(\vec{x}) }{\Sigma_{\rm cr}} }, where we have used the critical density, so defined: \displaystyle{ \Sigma_{\rm cr} = \frac{c^2}{4 \pi G} \frac{D_S}{D_L D_{L S}}}. The critical density is a quantity which characterizes the lens system and which is a function of the angular diameter distances of lens and source. The quantity \kappa is better known as the convergence. Again, we leave to the attentive student to show the following equation: \Delta_x \Psi(\vec{x}) = 2 \kappa({\vec x}) . This can be derived from the well-known Poisson equation, having introduced the two-dimensional Laplacian: \Delta_{\theta} = \frac{\partial^2}{\partial \theta_1^2} + \frac{\partial^2}{\partial \theta_1^2} =(\Delta - \frac{\partial^2}{\partial z^2}) \, D_{\rm L}^2

The lensing potential (3)

The formal solution of the 2-D Poisson equation can be obtained by integrating it. Hence, the effective lensing potential can be written in terms of the convergence (i.e., the adimensional surface density): \displaystyle{ \Psi (\vec{x}) = \frac{1}{\pi} \, \int \kappa(\vec{x'}) {\rm ln} (|\vec{x}- \vec{x'}|) {\rm d}^2 x'} , where the integral is performed over the whole lens plane. It is not difficult to derive the link between the deflection angle and the convergence: \displaystyle{ \vec{\alpha} (\vec{x}) = \frac{1}{\pi} \, \int  {\rm d}^2 x' \, \kappa(\vec{x'}) \frac{\vec{x}- \vec{x'}}{|\vec{x}- \vec{x'}|}}.

The Jacobian matrix

Distortion effects due to convergence and shear on a circular source (adapted from Narayan & Bartelmann, 1995). One of the main features of gravitational lensing is the distortion introduced in the shape of the sources. This is particularly evident when the source has no negligible apparent size. For example, background galaxies can appear as very long arcs in galaxy clusters, while quasars’ images show no distortion as they are never resolved. [Why the image of a quasar is not resolved?] The distortion arises from the fact that light bundles are detected differentially. Ideally the shape of the images can be determined by solving the lens equation for all the points within the extended source.
In particular, when the source is much smaller than the angular size on which the physical properties of the lens change, the relation between source and images can be (locally) linearized, by means of the Jacobian matrix: A=\frac {\partial \vec y}{\partial \vec x}=\biggl(\delta_{ij}-\frac{\partial \alpha_i}{\partial x_j}\biggr). The elements of the Jacobian matrix can now be written in terms of the second derivatives of the lensing potential introduced above.

Distortion effects due to convergence and shear on a circular source (adapted from Narayan & Bartelmann, 1995).

Distortion effects due to convergence and shear on a circular source (adapted from Narayan & Bartelmann, 1995).

Shear and convergence (1)

\displaystyle{ A = \left(\delta_{ij} - \frac{\partial^2 \Psi}{\partial x_i \partial x_j} \right) }.

In the following, for sake of brevity, we will use the shorthand notation:

\displaystyle{ \Psi_{ij} = \frac{\partial^2 \Psi}{\partial x_i \partial x_j} }

The Jacobian matrix characterize the local behavior of the the lensing potential, that is, at each location on the lens plane. We can immediately split it in an isotropic and antisymmetric part. The isotropic part is simply given by the trace of the matrix, and it is related to the convergence:

\displaystyle{ \frac{1}{2} {\rm Tr} A = (\delta_{ij} ) \, ( 1 - \kappa) }.

The antisymmetric part is obtained simply by subtracting the isotropic component:

\displaystyle{ \left(A - \frac{1}{2}I_{ij} {\rm Tr}A \right)_{ij}} .

We leave to the student the exercise to compute the 4 components of the antisymmetric part of the Jacobian matrix. The antisymmetric part is called the shear matrix, and plays an important role in gravitational lensing.

Shear and convergence (2)

Let’s have a deeper look at the shear matrix. It is useful to introduce the pseudo-vector \vec{\gamma} , such that its components read:

\displaystyle{ \gamma_1 = \frac{1}{2} \, (\Psi_{11} - \Psi_{22})},

\displaystyle{ \gamma_2 = \Psi_{12} = \Psi_{21}}

The pseudo-vector: \gamma = (\gamma_1, \gamma_2), is called the shear. Its modulus is:

\gamma = \sqrt{\gamma_1^2 + \gamma_2^2}.

The Jacobian matrix can be written now as follows:

A = \left[ \left( \begin{array}{cc}1 - \kappa - \gamma_1 & -\gamma_2 \\-\gamma_2 & 1 - \kappa + \gamma_1 \end{array} \right)\right] .


Shear and convergence (2)

As there exists a coordinate rotation by an angle \phi such that:\gamma_1 = \gamma {\rm cos} 2\phi and \gamma_2 = \gamma {\rm sin} 2 \phi, the jacobian matrix reads now:

A = (1 - \kappa) \, \delta_{ij} \, - \gamma \left[ \left( \begin{array}{cc} {\rm cos} 2 \phi & {\rm sin} 2\phi \\{\rm sin} 2 \phi& - {\rm cos} 2 \phi \end{array} \right)\right] .

This last expression makes clear the geometrical and physical meaning of the convergence and the shear. The distortion induced by the convergence is isotropic, that is the images are only rescaled by a constant factor in all directions. On the other hand, the shear stretches the intrinsic shape of the source along a privileged direction. For this reason, a circular source, which is small enough compared to the scale of the lens is mapped into an ellipse when \kappa and \gamma are both non-zero. The semi-major and semi-minor axes are:

a = \frac{r}{1- \kappa - \gamma} , b = \frac{r}{1- \kappa + \gamma}

where r is the radius of the circular source.


Magnification (1)

An important consequence of the lensing distortion is the magnification. Through the lens equation, the solid angle element \delta \beta^2 (or equivalently the surface element \delta y^2) is mapped into the solid angle \delta \theta^2 (or in the surface element \delta x^2).

Since the Liouville theorem and the absence of emission and absorbtion of photons in gravitational light deflection ensure the conservation of the source surface brightness, the change of the solid angle under which the source is seen implies that the flux received from a source is magnified (or demagnified).

The magnification is quantified by the inverse of the determinant of the Jacobian matrix. For this reason, the matrix M = A^{-1} is called the magnification tensor. We therefore define:

\mu \equiv det M = \frac{1}{det A} = \frac{1}{(1- \kappa)^2 - \gamma^2}


Magnification (2)

The magnification is formally infinite when  {\rm det}A = 0, that is when the 2-D function \vec{y}(\vec{x}) is singular or not invertible. However, you will never find in the sky an infinitely amplified astronomical source. [Can you explain why?] In literature you can often find the so-called magnification tensor, M, which is simply the inverse of the Jacobian matrix. The eigenvalues of the magnication tensor measure the amplication in the tangential and in the radial direction. We leave to the student to show that they are:

\displaystyle{ \mu_t = \frac{1}{1 - \kappa - \gamma} }


\displaystyle{\mu_r = \frac{1}{1 - \kappa + \gamma} }.

[What is the meaning of these two quantities?]

Critical lines

The singularities of the mathematical application which defines the lensing, \vec{y}(\vec{x}), play an essential role in the modeling of the gravitational lenses. These singularities are defined as the the set of points at which the determinant {\rm det} A vanishes. These points form the so-called critical line. An image placed close to a critical line is very distorted and highly amplified. In more detail, the two conditions: \mu_t = \infty  and  \mu_r = \infty, define two curves in the lens plane, called the tangential and the radial critical line, respectively. An image forming along the tangential critical line is strongly distorted tangentially to this line. As you can imagine, any image forming close to the radial critical line is stretched in the direction perpendicular to the line itself.

The Einstein radius

We now consider an important quantity of any gravitational lens: the so-called Einstein radius. We start with a simple circular lens, already seen a few slides above. For a circular lens, the ray-tracing equation is one-dimensional and reads: \displaystyle{ \beta(\theta) = \theta - \frac{D_{\rm LS}}{D_S D_L} \, \frac{4 G M(\theta)}{c^2 \theta}} , where, as before, \beta is the unobservable source position, and \theta the multiple images position. Because of the rotational symmetry of the lensing system, when the source lies exactly on the optic axis (that is, \beta = 0 ), it is imaged into a ring. Indeed, if we set \beta=0, we derive from the previous equation the radius of that ring:

\displaystyle{  \theta_{\rm E} = \sqrt{\frac{4GM (\theta_{\rm E})}{c^2} \, \frac{D_{\rm LS}}{D_S D_L}}} .

This is referred to as the Einstein radius. The image observed on the lens plane (that is, the sky plane) has a circular shape, the so-called Einstein ring, with angular radius \theta_{\rm E}. It is important to note that the Einstein radius is not just a property of the lens, but depends also on the various distances in the whole lensing configuration (i.e., observer, lens and source). In order to observe a perfect Einstein ring, we need both a symmetric lens and a perfect alignment of the lens with the source, as observed from the observer location. As you can imagine this is very rare, and most of the time a slightly irregular ring can be observed, as for instance, in the next figure.


The Einstein radius

Schematic view of the Einstein ring produced by a circular gravitational lens.

Schematic view of the Einstein ring produced by a circular gravitational lens.

The Einstein radius

The Gravitational Lens 0038+4133, showing an almost perfect Einstein ting observed in the survey COSMOS, with the Hubble Space Telescope. Credit: NASA (HST).

The Gravitational Lens 0038+4133, showing an almost perfect Einstein ting observed in the survey COSMOS, with the Hubble Space Telescope. Credit: NASA (HST).

The Einstein radius

Einstein Ring around the gravitational gens SDSS J162746.44-005357.5, observed with the Hubble Space Telescope. Credit: NASA (HST).

Einstein Ring around the gravitational gens SDSS J162746.44-005357.5, observed with the Hubble Space Telescope. Credit: NASA (HST).

Lens models and dark matter

Why is the Einstein radius so important? Were we lucky enough to observe a full Einstein ring, we could derive an accurate estimate of the total mass of the lensing galaxy (within the Einstein radius) simply from knowledge of the source and lens redshifts. As the angular measurement of the Einstein radius and spectroscopic evaluation of the redshift are routine measurements done with great accuracy, a precise measurement of the total mass can be immediately obtained. The major limitation, of course, is given by lack of symmetry in the observed configuration, hence this direct estimate can rarely have an accuracy larger than 10-15%. When the luminous information can be used to obtain a determination of the stellar mass, the amount of dark matter can be estimated.

This is probably the most important application of gravitational lensing, already suggested about 80 years ago by Fritz Zwicky. It is important to note that the Einstein radius provides a natural angular scale to describe the lensing geometry. Even in the case of multiple imaging (that is, when a single perfect ring is missing), the typical angular separation of images is of order 2 \times \theta_{\rm E}. Hence, the curvature radius of giant arcs usually observed in the central regions of galaxy clusters or around massive elliptical galaxies, is very close to the Einstein radius.

[As an exercise, we leave to the student to estimate the total mass in the core of the galaxy cluster Abell 2667; see next figure. The redshift of the galaxy cluster is z = 0.23, the redshift of the giant gravitational arc is z = 1.03, and the Einstein radius is approximately 20 arcsec.]

Lens models and dark matter

Deriving the total mass within the Einstein radius. See text for details. Image of the galaxy cluster Abell 2667, obtained with the Hubble Space Telescope. Credit: NASA (HST).

Deriving the total mass within the Einstein radius. See text for details. Image of the galaxy cluster Abell 2667, obtained with the Hubble Space Telescope. Credit: NASA (HST).

A simple lens model: the point mass

A point mass is obviously not a physical model, but it is very useful in order to understand the basic principles of lensing and it is also widely used when dealing with stars acting as gravitational lenses, like in the case of Galactic microlensing, that it is addressed in this course. We leave to the reader the easy task to show the following relations. The Einstein radius simply reads:

\theta_E=\sqrt{\frac {4GM}{c^2}\frac{D_{LS}}{D_SD_L}}.

The lens equation is: \displaystyle{\beta = \theta - \frac{\theta_E^2}{\theta}}. As this equation allows two solutions, such a lens always produces two images. They are located on opposite sides with respect to the mass M. [Why? And what is their angular separation? Estimate it in the case of star as a lens.] The magnification of the two images is: \displaystyle{ \mu_{\pm} = \frac{1}{2} \pm \frac{y^2+2}{2y \sqrt{y^2+4}} }.


Schematic view of a point mass lens, with the positions of the two lensed images.

Schematic view of a point mass lens, with the positions of the two lensed images.

A Singular Isothermal Sphere as a lens

As you already know, a singular isothermal sphere (SII, hereafter) is a very good, zeroth-order approximation to the total mass distribution in an elliptical galaxy. Therefore, it is very important to consider here its lensing properties. Its density profile follows the relation:
\displaystyle{ \rho( r ) = \frac{\sigma^2_v }{2 \pi G r^2 } },

where \sigma_v is the velocity dispersion of the galaxy and r is the distance from the sphere center. By projecting the three-dimensional density along the line of sight, we obtain the corresponding surface density \Sigma:

\displaystyle{ \Sigma( \xi ) = \frac{\sigma^2_v}{2 G \xi}},

where as usual, \xi is the radial coordinate in the lens plane.

A Singular Isothermal Sphere as a lens

This model has two divergences: the density goes to infinity when \xi \rightarrow 0 and the total mass is infinite as well. However, the SIS is widely used. By choosing the following length scale in the lens plane:

\displaystyle{\xi_0 = 4 \pi \left(\frac{\sigma_v}{c}\right)^{2} \frac{D_{\rm L} D_{\rm LS}}{D_{\rm S}}}

it easy to see that:

\displaystyle{\Sigma (x) = \frac{1}{2 x} \Sigma_{\rm cr}}.

Hence, the convergence for the singular isothermal prole reads: \kappa=1/(2x), and the SIS lens equation has a simple form:

\displaystyle{ y = x- \frac{x}{|x|}}.

As the attentive reader might easily see, the length scale is exactly the Einstein radius of the system. The reader should also show that when the source is located at y<1, two images are produced, with separation equal to twice the Einstein ring, and when y>1, only one is produced. In the next two movies, the images configuration are shown for a SIS and a singular isothermal ellipsoid, that is when the axial symmetry is lost.


Time delay

As mentioned above, the deflection of light rays causes a delay in the time between the emission of light by the source and the signal reception by the observer. This time delay has two components: a geometrical one (due to the two different path lengths of the detected light rays) and a gravitational one, due to the fact that the effective refraction index n is lower when a gravitational field is present or, in other words, photons traveling through the gravitational field of the lens slow down. This component of the time delay is related to the lensing potential. Considering as reference the unperturbed light path in absence of the lens:

t = t_{\rm geom} + t_{\rm grav},

where the term t_{\rm geom} is proportional to the squared angular separation between the intrisic position of the source and the location of its image.

Time delay

Let’s consider a gravitational lens at redshift z_L, the total time delay introduced by gravitational lensing at the position \vec{x} on the lens plane is given by the following expression:

t(\vec x)=\frac {1+z_L}c \frac{D_L D_S}{D_{LS}}\biggl[\frac 1 2 (\vec x - \vec \beta)^2-\Psi(\vec x)\biggr].

As the light propagation follows the Fermat Principle, the lens equation can be written simply as: \nabla t (\vec{x}) = 0 . [Show this last equation.]


Gauging the Hubble constant

The value of the Hubble constant is directly related to the scale of the whole lensing configuration.

The value of the Hubble constant is directly related to the scale of the whole lensing configuration.

Time delay and the Hubble constant

As you might have noticed, all the relevant quantities in the description of the gravitational lensing phenomena are pure numbers (magnification, Einstein radius), but the time delay. Indeed, the lens equation is dimensionless, and the positions of images as well as their magnications are dimensionless numbers. Hence, any information on the image configuration alone does not provide any constraint on the scale of the lens geometry, and hence, on the Hubble constant. As mentioned above, Refsdal (1964) realized that the time delay, however, is proportional to the absolute scale of the system and does depend on H_0 .

Let us show this important relation. First, we note that the geometrical time delay is simply proportional to the path lengths of the rays which scale as H_0^{-1}. The gravitational time delay also scales as H_0^{-1}.

Time delay and the Hubble constant

Therefore, for a fixed gravitational lens, the quantity H_0 \times \delta t depends only on the lens model and the geometry of the system. A good lens model which reproduces the positions and magnications of the images provides the scaled time delay H_0 \times \delta t between the lensed images: a measurement of the time delay will yield the value of Hubble constant. For a SIS model, the expected time delay between the two images A and B is: \displaystyle{ \Delta t_{SIS} = \frac{1}{2} \frac{D_{\rm L} D_{\rm S}}{c D_{\rm LS}} (\theta_A^2 - \theta_B^2) }. Measurements of the time delay are not simple: you need a good photometry of overlapping sources over a very long time period (even longer than a decade). In recent years, many teams have succeeded in obtaining such remarkable measurements, deriving In general estimates for H_0 lower than the value obtained by the HST Key Project, about 73\ km/s/Mpc. The weak point in the lensing approach is that a very accurate model is necessary for the radial profile of the mass distribution of the lensing galaxy.

Time delay and the Hubble constant

Likelihood distributions for the Hubble constant obtained from a set of 3 lenses, under two different assumptions for the lensing galaxy mass model, with a comparison with the value obtained by the HST Key Project.

The figure on the RHS is taken from Kochanek and Schechter, Carnegie Obs. Astrophys. Series, Vol. 2, Measuring and Modeling the Universe, Cambridge Univ. Press, 2004.

Kochanek and Schechter, 2004 (see text for full reference).

Kochanek and Schechter, 2004 (see text for full reference).

Refsdal’s (1964) seminal paper

Incipit of the seminal paper by the young Sjur Refsdal, introducing the idea to measure the Hubble constant via the time delay.

Incipit of the seminal paper by the young Sjur Refsdal, introducing the idea to measure the Hubble constant via the time delay.

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