15.3. Magnification and shear¶
15.3.1. Introduction¶
Nearby light rays that pass by a gravitational lens are deflected in a similar, but not identical manner and this causes lensed objects to be deformed and magnified. This deformation is described by the Jacobian of the transformation from the source plane position \(\boldsymbol{\beta}\) to the observed position \(\boldsymbol{\theta}\). This Jacobian is known as the magnification matrix with elements \begin{align} \mathcal{M}_{ij} = {\partial \theta_i \over \partial \beta_j}\,. \end{align} The overall increase \(\mu\) in the size of a lensed source is given by the determinant of \(\mathcal{M}\) and the parameter \(\mu = |\mathcal{M}|\) is called the magnification. The magnification depends on the position of the image relative to the lens.
As an example, we can compute the magnification for lensing by a singular isothermal sphere. We can write the solution of the lensing equation for the singular isothermal sphere from Section 15.1.3 as \begin{align} \boldsymbol{\theta} = \boldsymbol{\beta} \pm \theta_\mathrm{E}\,{\boldsymbol{\beta}\over |\boldsymbol{\beta}|}\,, \end{align} where the negative-sign solution is only a solution when \(|\boldsymbol{\beta}| < \theta_\mathrm{E}\) (\(\theta_\mathrm{E}\) is defined in Equation 15.14). The magnification matrix is then given by \begin{equation} \mathcal{M} = \begin{pmatrix} 1\pm {\theta_\mathrm{E} \beta_2^2\over |\boldsymbol{\beta}|^3} & \mp{\theta_\mathrm{E}\,\beta_1\,\beta_2\over |\boldsymbol{\beta}|^3}\\\mp{\theta_\mathrm{E}\,\beta_1\,\beta_2\over |\boldsymbol{\beta}|^3} & 1\pm {\theta_\mathrm{E} \beta_1^2\over |\boldsymbol{\beta}|^3}\end{pmatrix}\,, \end{equation} where \((\beta_1,\beta_2)\) are the components of the source vector. The magnification is \begin{align}\label{eq-gravlens-magnification-sis-direct} \mu = 1 \pm {\theta_\mathrm{E} \over |\boldsymbol{\beta}|} = {|\boldsymbol{\theta}| \over |\boldsymbol{\theta}| - \theta_{\mathrm{E}}}\,. \end{align} This example demonstrates a few important general aspects of the magnification produced by gravitational lensing. Firstly, the total magnification of all of the images—the sum of \(\mu\) for all produced images—is larger than 1: in this example, the total magnification is equal to \(\mu_{1} = 1 +\theta_\mathrm{E} / |\boldsymbol{\beta}|\) in the case of a single image or \(\mu_{1+2}=2\) when there are two images (excepting \(\boldsymbol{\beta} = 0\)). Thus, the total size of the source on the sky is increased. Secondly, there is a source location where the magnification diverges: at \(\boldsymbol{\beta} = 0\) for the singular isothermal sphere. Thirdly, at large separations between the source and the lens, the magnification tends to one, that is, the unlensed size.
15.3.2. The conservation of surface brightness and the magnification¶
The magnification is called the magnification, because it corresponds to the increase in the observed flux density \(F_\nu\) at a given frequency \(\nu\). The reason for this is as follows: for unresolved objects, modern CCD detectors measure the flux density—the amount of energy passing through a unit surface element per unit time and per unit wavelength—by essentially counting the number of photons of a given (range of) frequency incident on the detector per unit time. The flux density is related to the specific intensity or surface brightness \(I_\nu\) through \begin{equation} F_\nu = \int \mathrm{d}\Omega\,\cos \theta\,I_\nu \approx \Delta\Omega\,I_\nu\,\cos \theta\,, \end{equation} where the integral is over the entire angular extent of the observed object (\(\Omega\) is solid angle and the factor \(\cos\theta\) accounts for the angle between the line of sight to the object and the detector). For all but a small number of astronomical sources, the object is so small that we can approximate the integral \(\int \mathrm{d}\Omega\cdot\) as \(\Delta \Omega\cdot\), where \(\Delta \Omega\) is the angular size of the source; \(\cos \theta\) is typically one. If the wavelength of light remains the same during propagation (an important caveat that we will return to below), then surface brightness is conserved along the light path. This is because the surface brightness is related to the phase-space density \(f\) of photons through \begin{equation} I_\nu = {2h\nu^3 \over c^2}\,f\, \end{equation} where \(h\) is the Planck constant. The phase-space density is conserved for light in the absence of absorption, emission, or scattering, because of the optics equivalent of Liouville’s theorem in the Hamiltonian form of geometrical optics (derivation similar to that in Chapter 5.3; this conservation law is also known as the conservation of etendue). The conservation of phase-space density itself is a consequence of the conservation of phase-space volume and of the number of photons (as no photons are assumed to be created or destroyed); this continues to hold in the general theory of relativity because the phase-space volume is Lorentz invariant (e.g., Misner et al. 1973). Because surface brightness is conserved, the ratio of the lensed vs. unlensed flux density of an unresolved source is therefore the magnification \(\mu\): lensing increases the flux density (and thus the apparent magnitude) of background sources, because it increases their angular size at constant surface brightness. As the singular-isothermal-sphere example above demonstrates, lensing thus typically increases the magnitude of unresolved sources regardless of their relative orientation with respect to the lens (\(\mu \geq 1\ \forall \boldsymbol{\beta}\)). Thus, lensing typically increases the magnitude of a source for any observer in the Universe. At first glance, this seems counterintuitive!
Physically, what happens during lensing is that the momentum density of photons at the detector decreases (the momentum volume is \(\propto \nu^2\mathrm{d}\nu\mathrm{d}\Omega\)), while the spatial density increases to make up for this to satisfy Liouville’s theorem. Therefore, lensing effectively moves the observer closer to the source, where it appears larger on the sky and the momentum density is, thus, lower, and where the flux is higher. This explains the apparent paradox that lensing magnifies all background sources. One might be tempted to think that the increased brightness of a background object due to lensing needs to be counteracted by a decreased brightness as observed by another observer somewhere else in the Universe, but this is not borne out by the equations. That this is possible is because lensing effectively moves all observers closer to the source.
For resolved objects, detectors record the surface brightness directly rather than the flux density and because surface brightness is conserved, resolved objects therefore do not appear brighter than their unlensed counterparts (the same thing happens with any telescope, which does not make resolved objects brighter, just larger). However, the increased size of resolved objects increases both their total flux and their footprint on a detector and this therefore makes it easier to determine their internal structure, for example, with detailed photometric or spectroscopic observations.
Finally, as discussed above, the conservation of surface brightness requires that light does not change wavelength as it propagates. This is the case for observations in our Galaxy or in the nearby Universe, but once objects are located at cosmological distances, the expansion of the Universe causes the frequency of light to decrease as \(\nu \propto (1+z)^{-1}\). Surface brightness is then no longer conserved, but instead surface brightness is \(\propto (1+z)^{-4}\), because surface brightness is \(\propto \nu^3\) and there is an additional factor of \((1+z)\) due to time dilation. Thus, objects at cosmological distances do appear less bright than their local counterparts. Gravitational lensing does not change this: because lensing by galaxies happens in such a small region of space with \(\delta z \lll 1\), surface brightness is conserved as light passes through the lens and for cosmological sources, lensing therefore simply maintains the \((1+z)^{-4}\) dilution of surface brightness.
15.3.3. Convergence and shear¶
Much can be learned about the magnification matrix from considering its inverse \(\mathcal{A} = \mathcal{M}^{-1}\), because this matrix can be expressed in terms of the second derivative of the lensing potential \(\psi(\boldsymbol{\theta})\). Using the lensing equation (15.7) and the relation between the reduced deflection angle and the lensing potential from Equation (15.15), we have that \begin{align} \mathcal{A}_{ij} = {\partial \beta_i \over \partial \theta_j} & = \delta_{ij} -{\partial \alpha_i \over \partial \theta_j} = \delta_{ij} -{\partial^2 \psi \over \partial \theta_i\partial \theta_j}\,.\label{eq-gravlens-inverse-mag-as-d2lensing} \end{align} This form makes it immediately clear that \(\mathcal{A}\), and thus its inverse \(\mathcal{M}\) as well, is a symmetric matrix. It therefore only has 3 independent components. It is convenient to decompose \(\mathcal{A}\) into a diagonal matrix and a trace-free part as \begin{align} \mathcal{A}_{ij} & = \begin{pmatrix} 1 - \kappa - \gamma_1 & -\gamma_2 \\ -\gamma_2 & 1-\kappa + \gamma_1\end{pmatrix}\label{eq-gravlens-inverse-mag} = \begin{pmatrix} 1 - \kappa & 0 \\ 0 & 1-\kappa \end{pmatrix} -\begin{pmatrix} \gamma_1 & \gamma_2 \\ \gamma_2 & -\gamma_1\end{pmatrix}\,, \end{align} where \(\kappa\) is the convergence through Equation (15.21) and the parameters of the trace-free part are \begin{align}\label{eq-gravlens-shear-and-lensing-pot} \gamma_1 & = {1\over 2}\left({\partial^2 \psi \over \partial \theta_1\partial \theta_1} - {\partial^2 \psi \over \partial \theta_2\partial \theta_2}\right)\,;\quad \gamma_2 = {\partial^2 \psi \over \partial \theta_1\partial \theta_2}\,. \end{align} The eigenvalues of the trace-free part are \begin{equation} \pm\gamma = \pm\sqrt{\gamma_1^2 + \gamma_2^2}\,. \end{equation} The trace-free part can also be written in terms of an angle \(\phi\) in which case the full inverse magnification matrix can be written as \begin{align}\label{eq-gravlens-Aij-inverse-mag-gamma} \mathcal{A}_{ij} & = (1-\kappa)\,\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} +\gamma\,\begin{pmatrix}\cos 2\phi & \sin 2\phi \\ \sin 2\phi & -\cos 2\phi\end{pmatrix}\,. \end{align} (Note here that the trace-free part is \(-\gamma\) times the angle matrix.) The eigenvectors of this matrix (and of the trace-free part alone) are \((\sin \phi,-\cos\phi)\) and \((\cos\phi,\sin\phi)\) and the corresponding eigenvalues are \begin{equation}\label{eq-gravlens-shear-eigenvalues} \lambda_{_{t}^{r}}= 1-\kappa\pm\gamma = 1-\kappa\mp\sqrt{\gamma_1^2 + \gamma_2^2}\,, \end{equation} where the \(\lambda_t = 1-\kappa-\gamma\) eigenvalue has eigenvector \((\sin\phi,-\cos \phi)\), while the \(\lambda_r =1-\kappa+\gamma\) eigenvalue has eigenvector \((\cos \phi,\sin\phi)\). It is left as an exercise to show that for an axially-symmetric lens, the angle \(\phi\) is the angle between the reduced deflection angle and the \(x\) axis. For an axially-symmetric lens, the deflection angle points along \((\cos\phi,\sin \phi)\), and \(\lambda_t\), therefore, gives the distortion tangential to the deflection angle and tangential to a circle around the lens’ center, while \(\lambda_r\) gives that radial to it. This is the origin of the tangential and radial descriptors.
The image distortion by lensing is therefore the combination of an isotropic scaling of the source and a shear. Notably absent is a rotation component; this is absent because the magnification matrix is symmetric. Lensing thus scales and shears images, but it does not rotate them. The shear can be thought of in two ways: either by the amount the image is squashed or stretched along the \(x=0\) (with \(\gamma_1\)) and \(x=y\) (with \(\gamma_2\)) directions or by the amount the image is squashed or stretched along the line that makes an angle \(\phi\) with the \(x\) axis. Adding the isotropic scaling effect of the convergence, all of these image distortions are illustrated in Figure 15.7 (where non-labeled parameters are zero).
[15]:
def transform_circle(kappa,gamma1,gamma2,radius=1.,sgn=1.):
A= numpy.array([[1.-kappa-gamma1,-gamma2],[-gamma2,1.-kappa+gamma1]])
M= numpy.linalg.inv(A)
circle_x= numpy.linspace(-radius,radius,301)
circle_y= sgn*numpy.sqrt(radius**2.-circle_x**2.)
return numpy.dot(M,numpy.vstack((circle_x,circle_y)))
kappas= [0.2,0.,0.,-0.2,0.,0.]
gamma1s= [0.,0.2,0.,0.,-0.2,0.]
gamma2s= [0.,0.,0.2,0.,0.,-0.2]
phis= [0,0,45,0,90,0-45]
figure(figsize=(12,6))
for ii,(kappa,gamma1,gamma2,phi) in enumerate(zip(kappas,gamma1s,gamma2s,phis)):
subplot(2,3,ii+1)
fill_between(*transform_circle(0.,0.,0.,sgn=-1),
transform_circle(0.,0.,0.,sgn=1)[1],
facecolor='0.8')
plot(*transform_circle(kappa,gamma1,gamma2,sgn=1),'k-')
plot(*transform_circle(kappa,gamma1,gamma2,sgn=-1),'k-')
xlim(-1.5,1.5)
ylim(-1.5,1.5)
gca().set_aspect('equal')
gca().axis('off')
if kappa != 0.:
label= rf'$\kappa = {kappa:+.1f}$'
elif gamma1 != 0.:
label= rf'$\gamma_1 = {gamma1:+.1f};\ \phi= {phi:.0f}^\circ$'
else:
label= rf'$\gamma_2 = {gamma2:+.1f};\ \phi= {phi:+.0f}^\circ$'
galpy_plot.text(label,title=True,fontsize=18.)
tight_layout();
Figure 15.7: The effect of convergence \(\kappa\) and shear \((\gamma_1,\gamma_2)\).
The effect of the convergence and shear on a circular image with radius \(r\) is to transform it to an ellipse with semi-major and semi-minor axes \begin{align}\label{eq-gravlens-lensed-axis-ratio} a = {r \over 1-\kappa-\gamma}\,,\quad b = {r \over 1-\kappa+\gamma}\,. \end{align} Note that the shear in gravitational lensing does not correspond to the typical definition of shear in mathematics, where shear is a mapping that moves each point along a fixed direction by an amount proportional to the distance between the point and the line that is parallel to the shear direction and goes through the origin. Nevertheless, qualitatively, the shear in gravitational lensing is similar to the mathematical definition.
The magnification is \(\mu = |\mathcal{M}| = |\mathcal{A}|^{-1}\) and therefore \begin{equation}\label{eq-gravlens-magnification-from-kappa-gamma} \mu = {1 \over |(1-\kappa)^2-\gamma^2|} = {1 \over |\lambda_r\,\lambda_t|}\,. \end{equation}
To illustrate the inverse-magnification matrix formalism, let’s revisit the example of the singular isothermal sphere that we considered at the beginning of this section. The potential of the singular isothermal sphere is \(\Phi(R,z) = \sigma^2\,\ln\left(R^2+z^2\right)+\mathrm{constant}\) from Equation (5.73) and we can compute the lensing potential \(\psi(\boldsymbol{\theta})\) from Equation (15.16) by fixing the constant such that \(\psi(0) = 0\). Using \(R \equiv D_{\mathrm{L}}|\boldsymbol{\theta}|\), we find that \begin{align} \psi(\boldsymbol{\theta}) & = 2\,{D_{\mathrm{LS}}\over D_{\mathrm{S}}\,D_{\mathrm{L}}}\,{\sigma^2\over c^2}\,\int_{-\infty}^{\infty} \mathrm{d} z\,\left[\ln\left(R^2+z^2\right)-\ln z^2\right]\\ & = 2\,{D_{\mathrm{LS}}\over D_{\mathrm{S}}\,D_{\mathrm{L}}}\,{\sigma^2\over c^2}\,\int_{-\infty}^{\infty} \mathrm{d} z\,\int_0^R\mathrm{d} \tilde{R}^2\,{1 \over \tilde{R}^2+z^2}\\ & = \theta_\mathrm{E}\,|\boldsymbol{\theta}|\,, \end{align} using the definition from Equation (15.14) for the Einstein angle in the case of the singular isothermal sphere. Because \(\boldsymbol{\alpha} = \nabla_{\boldsymbol{\theta}} \psi\), this leads to the same reduced deflection angle as in Equation (15.13). We can then compute the convergence \(\kappa\) and shear components \(\gamma_1\) and \(\gamma_2\) using Equations (15.21) and (15.61), and we find that \begin{align}\label{eq-gravlens-kappa-gammas-sis} \kappa & = {1 \over 2}\,{\theta_\mathrm{E}\over|\boldsymbol{\theta}|}\,;\quad \gamma_1 = {1 \over 2}\,{\theta_\mathrm{E}\left(\theta_2^2-\theta_1^2\right)\over |\boldsymbol{\theta}|^3}\,;\quad \gamma_2 = -\,{\theta_\mathrm{E}\theta_1\theta_2\over |\boldsymbol{\theta}|^3}\,, \end{align} and therefore \begin{equation}\label{eq-gravlens-sis-gamma-kappa} \gamma = \sqrt{\gamma_1^2 + \gamma_2^2} = \kappa\,. \end{equation} The eigenvalues of the inverse magnification matrix are therefore \(\lambda_r = 1\) and \(\lambda_t = 1-2\kappa\): images are only distorted in the tangential direction, leading to circular arcs surrounding the lens. This gives a first indication of why extended lensed images are generally arcs around the center of the lens. The magnification is \begin{equation} \mu = {1\over|1-2\kappa|}\,. \end{equation} Using Equation (15.70), it is easy to show that this is equivalent to Equation (15.56) that we computed directly at the beginning of this section. This expression once more shows why the convergence \(\kappa\) is an important quantity in gravitational lensing. If \(\kappa \ll 1\)—or, equivalently, \(\Sigma \ll \Sigma_{\mathrm{crit}}\)—the magnification is \(\mu \approx 1\); however, for \(\kappa \approx 1\), \(\mu\) becomes large.
Lensing potentials with constant convergence or shear are of some interest. Such potentials may seem unphysical, because galactic lens mass distributions should have \(\kappa, \gamma \rightarrow 0\) at large separations, but as we saw in Section 15.2.4 above, a constant-convergence mass sheet is an important degeneracy in lens modeling and can be re-arranged into a more physical disk distribution. Similarly, a lensing potential with constant shear can be used to approximately model the contribution to the shear from the environment of a lens. Because the convergence and shear are second derivatives of the lensing potential, constant convergence or shear is obtained from a lensing potential that is quadratic in \((\theta_1,\theta_2)\) \begin{equation} \psi(\theta_1,\theta_2) = a\theta_1^2 + b\theta_1\,\theta_2 + c\theta_2^2\,, \end{equation} and from Equations (15.21) and (15.61) we then have that \(\kappa = a+c\), \(\gamma_1 = a-c\), and \(\gamma_2 = b\). A lensing potential with zero shear has \(a=c=\kappa/2\) and \(b=0\) and is thus \begin{equation}\label{eq-gravlens-masssheet-zero-shear} \psi(\theta_1,\theta_2) = {\kappa \over 2}\,\left(\theta_1^2 + \theta_2^2\right) = {\kappa \over 2}\,|\boldsymbol{\theta}|^2\,. \end{equation} This is the lensing potential of an infinite mass-sheet. It is no surprise that an infinite, homogeneous mass-sheet has zero shear, because shear distorts images in specific directions and there is no preferred direction anywhere on a homogeneous mass-sheet. An external convergence \(\kappa_\mathrm{ext}\) is often included in strong-lensing analyses to account for mass along the line of sight that is not part of the main lens. A pure-shear lensing potential (with \(\kappa = 0\)) has \(a = -c = \gamma_1/2\) and \(b = \gamma_2\) and therefore \begin{equation} \psi(\theta_1,\theta_2) = {\gamma_1 \over 2}\left(\theta_1^2 - \theta_2^2\right) + \gamma_2 \theta_1\,\theta_2\,. \end{equation} Such an external shear contribution is often included in lens models to account for the shear coming from the larger-scale environment of the lens.
15.3.4. Galaxy masses from weak gravitational lensing¶
We discussed in Section 15.2.4 above how strong-lensing observations can determine the mass within the Einstein radius of a lens well, with the main complication being the mass-sheet degeneracy. Equations (15.48) and (15.49) show that for the typically observed \(\theta_\mathrm{E}\sim 1''\), strong lensing constrains the mass of massive galaxies close to their centers at \(\mathcal{O}(1-10\,\mathrm{kpc})\). To determine the mass profile at larger distances, for example, to map the dark matter distribution out to the virial radius, we therefore have to venture beyond the strong-lensing regime where \(\kappa \approx 1\) to the weak lensing regime where \(\kappa, \gamma \ll 1\).
In the weak lensing regime, background sources are not multiply imaged, so the observables used in strong lensing analyses that we discussed in Section 15.2.4—relative image positions and relative arrival times—are no longer available. When only a single image of a background source is produced, its position does not contain any information about the lens, because all of the information is degenerate with the unobservable source position. Similarly, even if the source is variable, there is no information in the arrival time, because we cannot know when the signal would have arrived without the intervening lens. However, the single image is affected by the second derivatives of the lensing potential in the form of magnification and shear. In the weak lensing regime \(\kappa, \gamma \ll 1\), the magnification and shear are small, but by combining measurements from many background sources, they can be constrained.
While magnification by gravitational lensing is highly useful for making background objects easier to study, its use for determining the lens’ mass distribution is limited. The reason is that it is difficult to know the intrinsic size (for a resolved source) or flux (for an unresolved source) that provides the baseline from which magnification is measured. For resolved galaxies, there is essentially no standard ruler on galactic scales that is robust enough to be used as a baseline (that is, we cannot, for example, assume that a certain type of galaxy has a size of \(X\,\mathrm{kpc}\) between isophotes of \(Y\,\mathrm{mag\,arcsec}^{-2}\), which would then appear as \(\approx\mu X\,\mathrm{kpc}\)). Similarly, standard candles among unresolved extragalactic sources are rare. The best standard candles on cosmological scales are the type Ia supernovae discussed in Chapter 11.3.1, but they are so rare that only a single multiply-imaged type Ia supernovae is known at the time of writing (Goobar et al. 2017) and few weakly-lensed type Ia supernovae have been observed (e.g., Nordin et al. 2014). However, magnification can be detected statistically by its effect on the background population of unresolved sources. The most obvious effect is that the lensing magnification lowers the effective magnitude limit, such that a survey that is sensitive to unlensed sources down to a given flux limit is sensitive to a factor \(\mu\) fainter behind the lens. However, a competing effect is that the effective volume that is observed is smaller, because the lens magnifies the entire background area as well; therefore, the sources behind the lens come from an area of the sky that is effectively \(\mu\) smaller than the unlensed population. Whether or not more or less background sources are observed behind a lens is therefore set by how steeply the number of sources increases as a function of magnitude, the luminosity function of the sources. Generally, the luminosity function increases fast enough that the overall effect is that of more background sources. Such magnification measurements have been done using quasars (e.g., Ménard et al. 2010) and high-redshift Lyman-break galaxies (e.g., Hildebrandt et al. 2009). By combining such measurements for many lenses, the typical mass profile of massive galaxies out to multiple Mpc can be determined.
Because magnification is so hard to determine from a set of background sources, shear is typically the quantity constrained by weak gravitational lensing through its effect on the shape of background galaxies. In Equation (15.65), we already saw that the effect of weak gravitational lensing is to deform a circular background source into an ellipse, but to focus on the effect on the shape of a background source, it helps to re-write the inverse magnification matrix \(\mathcal{A}\) from Equation (15.60) as \begin{equation}\label{eq-gravlens-inversemag-asonemkappa-g} \mathcal{A}_{ij} = (1-\kappa)\begin{pmatrix} 1 - g_1 & -g_2 \\ -g_2 & 1 + g_1\end{pmatrix}\,, \end{equation} where \begin{equation}\label{eq-gravlens-reduced-shear} g_i = {\gamma_i \over 1-\kappa}\,, \end{equation} is the reduced shear (similarly, \(g = \gamma/[1-\kappa]\)). Equation (15.76) shows that \((1-\kappa)\) simply performs an overall scaling of the background source and that it is only the reduced shear components that change the shape. Thus, only the reduced shear can be determined from galaxy shapes. In the weak-lensing regime, \(\kappa \ll 1\) and, therefore, \(g_i \approx \gamma_i\). For this reason, it is often said that weak-lensing can directly constrain the shear, but technically it is always the case that it is really the reduced shear that is constrained by shape measurements. In terms of the reduced shear, we can re-write Equation (15.65) for the effect of weak lensing on a circular background source as \begin{equation} a = {r/[1-\kappa] \over 1-g}\,,\quad b = {r/[1-\kappa] \over 1+g}\,, \end{equation} and the axis ratio \(a/b = (1+g)/(1-g)\) is a function of \(g\) alone.
Background sources are not circular, but to a good approximation they can typically be represented as ellipses. We can characterize these using their eccentricity (or ellipticity in this context), defined in the usual way as \(e = (a-b)/(a+b)\) with \(a\) and \(b\) the semi-major and semi-minor axis. But the effect of the shear on elliptical background sources is most easily expressed as a change to the two-dimensional eccentricity defined as \begin{equation}\label{eq-gravlens-twoecc} \vec{e} = [e\cos 2\theta,e\sin 2\theta] = e\,e^{2i\theta}\,, \end{equation} where \(\theta\) is the position angle of the ellipse (the angle between the ellipse’s major axis and the \(x\) axis) and where in the second equality we have equivalently expressed this as a complex number (the angle appears as \(2\theta\) here, because an ellipse is symmetric under a rotation by \(180^\circ\); we avoid calling this two-dimensional quantity a vector, because it does not transform as a vector under rotations). We can similarly express the reduced shear as \begin{equation}\label{eq-gravlens-twod-reducedshear} \vec{g} = [g\cos 2\phi,g\sin 2\phi] = [g_1,g_2] = g\,e^{2i\phi}\,. \end{equation} The shear \(\boldsymbol{\gamma}\) can be represented similarly. The effect of the magnification matrix on the two-dimensional eccentricity is most easily expressed using the complex-number representation and in the limit \(g < 1\) is given by (Kochanek 1990; Miralda-Escude 1991; Seitz & Schneider 1997) \begin{equation} \vec{e} \rightarrow {\vec{e} +\vec{g} \over 1+\vec{g}^*\,\vec{e}}\,, \end{equation} where \(^*\) denotes complex conjugation. In the weak-lensing regime, this simplifies to \begin{equation}\label{eq-gravlens-twode-weaklensing} \vec{e} \rightarrow \vec{e} +\vec{g} \approx \vec{e} +\boldsymbol{\gamma}\,, \end{equation} where the final expression approximates \(\vec{g} \approx \boldsymbol{\gamma}\). Thus, the effect of weak-lensing shear is to change the two-dimensional eccentricity of background sources by the two-dimensional (reduced) shear. Background sources may have an unknown distribution of eccentricities \(e\), but their orientations are random and therefore the average \(\langle \vec{e}_\mathrm{int}\rangle\) of the intrinsic two-dimensional eccentricities is zero. Thus, we find that the average \(\langle \vec{e}_\mathrm{obs}\rangle\) of the observed two-dimensional eccentricities at a given location behind a lens is \begin{equation} \langle \vec{e}_\mathrm{obs}\rangle = \vec{g} \approx \boldsymbol{\gamma}\,. \end{equation} The average of the observed eccentricities is therefore a direct measurement of the (reduced) shear!
Because it is easier to determine observationally, weak lensing analyses typically use a different definition of the ellipticity, \(\chi = (a^2-b^2)/(a^2+b^2)\), with a similar definition of the two-dimensional ellipticity as in Equation (15.79). In terms of this ellipticity, Equation (15.82) becomes \begin{equation}\label{eq-gravlens-twodchi-weaklensing} \boldsymbol{\chi} \rightarrow \boldsymbol{\chi} +2\,\vec{g} \approx \boldsymbol{\chi} +2\,\boldsymbol{\gamma}\,, \end{equation} and thus similarly \(\langle \boldsymbol{\chi}_\mathrm{obs}\rangle = 2\,\vec{g} \approx 2\,\boldsymbol{\gamma}\).
In practice, the width of the unknown eccentricity distribution means that we have to average over quite a few background sources for the mean intrinsic eccentricity to be much less than the shear (especially because the shear is small itself). Thus, we require a large population of background sources that is furthermore dense on the sky. For individual galaxies, the shear itself and the number of background sources is too small to directly perform this analysis. However, in galaxy clusters both the shear itself is larger (because of the more extended mass distribution compared to a galaxy) and the number of background sources is larger because of the larger sky footprint of galaxy clusters, which are \(\gtrsim 100\) times larger than massive galaxies. Thus, in galaxy clusters, one can determine the two-dimensional shear \(\boldsymbol{\gamma}\) over the entire area of the cluster. This shear field can then be inverted to obtain the convergence. That this is possible is clear from the fact that the shear is given by second derivatives of the lensing potential, while the lensing potential itself can be computed as a two-dimensional integral over the convergence (Equation 15.24). This can be done in various ways, e.g., directly using the shear in a two-dimensional integral that relates the convergence to the shear (Kaiser & Squires 1993) or relating the convergence to an integral over the gradient of the shear (Kaiser 1995). However, as we saw in Equation (15.74), a constant-density mass-sheet does not give rise to shear and shear-based weak lensing is therefore subject to the mass-sheet degeneracy: shear-based weak lensing can only determine the convergence up to an unknown constant. This degeneracy can be broken, e.g., by observing a single magnification, because a mass-sheet with convergence \(\lambda\) changes the magnification by \(1/(1-\lambda)^2\), or using non-lensing information such as velocity dispersions or X-ray constraints on the cluster mass.
For galaxies, even massive ones, the induced shear is too small and the available populations of background sources are generally too small to detect the lensing signature (e.g., Bartelmann & Schneider 2001). However, it is still possible to detect the subtle effect of gravitational lensing and constrain the mass profile by combining measurements from many different galaxies of a certain type and determining their typical mass profile. Because it is difficult to combine the direct shear \(\boldsymbol{\gamma}\) measurements directly for many different galaxies, this is often done by looking for the effect of lensing on the orientation of background galaxies rather than on their shape (Brainerd et al. 1996). This is known as galaxy-galaxy lensing. We can calculate the weak-lensing effect on background-source orientations by considering how the background distribution of two-dimensional eccentricities \(\vec{e}\) is modified by lensing. From Equation (15.82), we know that the intrinsic distribution \(p_{\mathrm{int}}(\vec{e})\) is transformed to the observed distribution \begin{equation}\label{eq-gravlens-pobse-pinte} p_{\mathrm{obs}}(\vec{e}) = p_{\mathrm{int}}(\vec{e}-\vec{g}) \approx p_{\mathrm{int}}(\vec{e}) - \sum_i{\gamma_i\cdot{\partial p_{\mathrm{int}}(\vec{e}) \over \partial e_i}}\,, \end{equation} where in the second approximation we have assumed the weak-lensing regime \(\vec{g} \approx \boldsymbol{\gamma}\) and that \(|\boldsymbol{\gamma}| \ll |\vec{e}|\) (when using the other ellipticity definition \(\boldsymbol{\chi}\), \(\vec{g}\) and \(\gamma_i\) simply require an additional factor of two, see Equation 15.84). For the two-dimensional eccentricity, we again use the complex-number representation \(\vec{e} = e\,e^{2i\theta}\). Assuming that the intrinsic orientations are fully random and that the intrinsic distribution of \(\theta\) is therefore uniform, we have that \(p_{\mathrm{int}}(\vec{e}) \propto p_{\mathrm{int}}(e)\). The background galaxy is located at an angle \(\eta\) in the coordinate frame used for the lens, but we can rotate the coordinate system such that this angle is zero. In this coordinate system, the shear is given by \begin{equation} \begin{pmatrix} \gamma_1' & \gamma_2' \\ \gamma_2' & -\gamma_1' \end{pmatrix} = \begin{pmatrix} \gamma_1\cos 2\eta + \gamma_2\sin 2\eta& -\gamma_1\sin 2\eta +\gamma_2\cos 2\eta \\ -\gamma_1\sin 2\eta +\gamma_2\cos2\eta & -\gamma_1\cos 2\eta - \gamma_2\sin 2\eta\end{pmatrix}\,, \end{equation} because the shear matrix \(\vec{G} = \begin{pmatrix} \gamma_1 & \gamma_2 \\ \gamma_2 & -\gamma_1 \end{pmatrix}\) transforms under rotation matrices \(\vec{R}\) as \(\vec{R}\vec{G}\vec{R}^T\). In Equation (15.85) we then have that \begin{equation} \sum_i{\gamma_i\cdot{\partial p_{\mathrm{int}}(\vec{e}) \over \partial e_i}} = -\gamma_T\cos 2\theta\,{\mathrm{d} p_{\mathrm{int}}(e) \over \mathrm{d} e}\,, \end{equation} where we have defined \begin{equation}\label{eq-gravlens-tangential-shear-galaxy-galaxy} \gamma_T = -\left(\gamma_1\cos 2\eta +\gamma_2\sin 2\eta\right)\,. \end{equation} The observed distribution of orientations \(\theta\) is then given by \begin{align} p_{\mathrm{obs}}(\theta) & = \int \mathrm{d} e\,e\,\int\mathrm{d}\theta\,p_{\mathrm{obs}}(\vec{e}) = {2\over\pi}\,\left(1-\gamma_T\cos 2\theta\int \mathrm{d} e\,p_{\mathrm{int}}(e)\right) = {2\over\pi}\,\left(1-\gamma_T\cos 2\theta\,\left\langle {1\over e}\right\rangle\right)\,, \end{align} using that \(\mathrm{d}\vec{e} = e\,\mathrm{d}e\,\mathrm{d}\theta\) and where we have used integration by parts assuming that the surface terms are zero (they are for typical eccentricity distributions). The factor of \(2/\pi\) comes from the fact that the orientation can be constrained to \(0 \leq \theta \leq \pi/2\) for this purpose, because of the \(\cos 2\theta\) dependence. For typical values of the shear \(\gamma_T \approx 0.01\) and the average inverse eccentricity \(\langle 1/e\rangle \approx 8\) (Brainerd et al. 1996), the intrinsic and lensed orientation distributions are shown in Figure 15.8.
[16]:
oneovere= 8.
gammat= 0.01
thetas= numpy.linspace(0.,numpy.pi/2.,101)
figure(figsize=(5,3.5))
plot(thetas*180./numpy.pi,1./180.*thetas**0,label=r'$\mathrm{intrinsic}$')
plot(thetas*180./numpy.pi,1./180.*(1.-gammat*numpy.cos(2.*thetas)*oneovere),
label=r'$\mathrm{lensed},\, \gamma_T = 0.01$')
ylim(0.,0.008)
xlabel(r'$\theta\,(\mathrm{deg})$')
ylabel(r'$p(\theta)$')
legend(frameon=False,fontsize=18.);
Figure 15.8: The effect of weak lensing on the orientation of background sources.
Thus, we see that the effect of lensing is to subtly skew the distribution of orientation angles towards \(90^\circ\). Because we defined the coordinate system such that the background galaxy lies along \(y=0\), this means that the background source’s major axis is skewed towards being perpendicular to the line connecting the source to the lens; that is, the preference is towards tangential alignment.
By examining the distribution of orientation angles behind lenses of a certain type, one can therefore infer the radial dependence of the shear \(\gamma_T\) defined in Equation (15.88). For general lens mass distribution, this provides a constraint on the two-dimensional shear. But generally, observational analyses measure the azimuthally-averaged tangential shear \(\langle\gamma_T\rangle\) and this is related to the azimuthally-averaged convergence \(\langle \kappa \rangle\) as (Kaiser 1995) \begin{equation} \langle\gamma_T(\xi)\rangle = \langle \kappa (< \xi)\rangle - \langle \kappa (\xi)\rangle\,, \end{equation} where \(\langle \kappa (< \xi)\rangle\) is the average convergence inside of \(\xi\) (equal to the enclosed mass divided by \(2\pi\Sigma_{\mathrm{crit}}\)) and \(\langle \kappa (\xi)\rangle\) that at \(\xi\). Thus, a measurement of \(\langle\gamma_T\rangle\) directly constrains the mass profile, specifically, the difference between the average convergence at smaller radii and the local convergence (because of the mass-sheet degeneracy, it is again only possible to constrain the convergence up to a constant). Because the preferential tangential alignment of background galaxies can be measured out to the virial radius and beyond, these measurements allow the dark matter halos of massive galaxies to be measured (e.g., Hoekstra et al. 2004). Alternatively, it is possible to perform an analysis of the tangential modification of the eccentricities themselves along similar lines (e.g., Sheldon et al. 2004).
Because weak-lensing analyses must combine shape measurements from many background sources, weak lensing is a field in which all of the observational and interpretational details matter greatly. Much of the weak-lensing literature is therefore concerned with the proper determination of galaxy shapes from images of galaxies, with an especially large role played by the point-spread function that blurs all observations of galaxies. Another important issue is that of intrinsic alignments between background sources, which is important at large distances from the lenses where the intrinsic shear is \(\approx 10^{-4}\). We will not discuss these issues here, but instead refer to reader to excellent reviews on this topic (e.g., Mandelbaum 2018). In the context of our discussion, however, we point out one complication: we have phrased the entire weak-lensing formalism in this section in terms of the dimensionless quantities \(\kappa\) and \(\gamma_i\), but these are related to the physical mass distribution using the critical surface density, which depends on the distances involved as \(\Sigma_\mathrm{crit} \propto D_{\mathrm{S}}/[D_{\mathrm{LS}}\,D_{\mathrm{L}}]\). Thus, when using a single (cluster) lens with many background sources, or multiple (galaxy) lenses with many background sources, we have to accounted for the fact that each lens-source pair has a different value of \(D_{\mathrm{S}}/[D_{\mathrm{LS}}\,D_{\mathrm{L}}]\). While spectroscopic redshifts are often available for the lenses, the large number of background sources means that it is not typically feasible to obtain spectroscopic redshifts for the background sources, but instead that one has to rely on photometric redshifts. However, for two reasons this doesn’t pose a very large problem: (i) for low-redshift lenses and high-redshift background sources, \(D_{\mathrm{S}}/[D_{\mathrm{LS}}\,D_{\mathrm{L}}] \approx 1/D_{\mathrm{L}}\) and the lens redshift is therefore more valuable than the source redshifts, and (ii) the intrinsic noise in the measurement coming from the eccentricity distribution is much larger than the noise coming from the photometric redshifts. Of course, for the most sensitive lensing measurements, it is important to properly account for the redshifts of the lenses and sources.
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