User:TallJimbo/WeakLensing/Mathematics

In General Relativity, a point mass deflects a light ray with impact parameter $$b~$$ by an angle $$\hat{\alpha} = \frac{4GM}{c^2b}$$. A naïve application of Newtonian gravity yields exactly half this value, although Newtonian gravity does not formally say anything about the bending of light.

In situations where General Relativity can be approximated by linearized gravity, the deflection due to a spatially extended mass can be written simply as a vector sum over point masses. In the continuum limit, this becomes an integral over the density $$\rho~$$, and if the deflection is small we can approximate the gravitational potential along the deflected trajectory by the potential along the undeflected trajectory, as in the Born approximation in Quantum Mechanics. The deflection is then

$$\vec{\hat{\alpha}}(\vec{\xi})=\frac{4 G}{c^2} \int dz \int d^2\xi^{\prime} \frac{(\vec{\xi}-\vec{\xi}^{\prime})\rho(\vec{\xi}^{\prime},z)}{|\vec{\xi}-\vec{\xi}^{\prime}|^2} $$

where $$\vec{\xi}$$ is the vector impact parameter, and $$z$$ is the line-of-sight coordinate.

Thin lens approximation
In the limit of a "thin lens", where the distances between the source, lens, and observer are much larger than the size of the lens (this is almost always true for astronomical objects), we can define the projected mass density

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

where $$\vec{\xi}$$ is a vector in the plane of the sky. The deflection angle is then

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



As shown in the diagram on the right, the difference between the unlensed angular position $$\vec{\beta}$$ and the observed position $$\vec{\theta}$$ is this deflection angle, scaled by a ratio of distances

$$\vec{\beta}=\vec{\theta}-\vec{\alpha}(\vec{\theta}) = \vec{\theta} - \frac{D_{ds}}{D_s} \vec{\hat{\alpha}}(\vec{D_d\theta})$$

where $$D_{ds}~$$ is the distance from the lens to the source $$D_s~$$ is the distance from the observer to the source, and $$D_d~$$ is the distance from the observer to the lens. For extragalactic lenses, these must be angular diameter distances.

In strong gravitational lensing, this equation can have multiple solutions, because a single source at $$\vec{\beta}$$ can be lensed into multiple images.

Convergence and deflection potential
It is convenient to define the convergence

$$ \kappa(\vec{\theta}) = \frac{\Sigma(D_d\vec{\theta})}{\Sigma_{cr}} $$

and the critical density (not to be confused with the critical density of the universe)

$$ \Sigma_{cr} = \frac{c^2 D_s}{4\pi G D_{ds}D_d} $$

The scaled deflection angle $$\vec{\alpha}(\vec{\theta})$$ can now be written as

$$ \vec{\alpha}(\vec{\theta}) = \frac{1}{\pi}\int d^2 \theta^{\prime} \frac{(\vec{\theta}-\vec{\theta}^{\prime})\kappa(\vec{\theta}^{\prime})}{|\vec{\theta}-\vec{\theta}^{\prime}|^2} $$

We can also define the deflection potential

$$ \psi(\vec{\theta}) = \int d^2 \theta^{\prime} \kappa(\vec{\theta}) \ln |\vec{\theta}-\vec{\theta}^{\prime}| $$

such that the scaled deflection angle is just the gradient of the potential and the convergence is half the [Laplacian]] of the potential:

$$ \vec{\alpha} = \vec{\nabla} \Psi $$

$$ \kappa = \frac{1}{2} \nabla^2 \Psi $$

The deflection potential can also be written as a scaled projection of the Newtonian gravitational potential $$\Phi~$$ of the lens

$$ \psi(\vec{\theta}) = \frac{2 D_{ds}}{D_d D_s c^2} \int \Phi(D_d\vec{\theta},z) dz $$

Lensing Jacobian
The Jacobian between the unlensed and lensed coordinate systems is

$$A_{ij}=\frac{\partial \beta_i}{\partial \theta_j}=\delta_{ij} - \frac{\partial \alpha_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}$$

where $$\delta_{ij}~$$ is the Kronecker delta. Because the matrix of second derivatives must be symmetric, the Jacobian can be decomposed into a diagonal term involving the convergence and a trace-free term involving the shear $$\gamma~$$

$$A=(1-\kappa)\left[\begin{array}{ c c } 1 & 0 \\ 0 & 1 \end{array}\right]-\gamma\left[\begin{array}{ c c } \cos 2\phi & \sin 2\phi \\ \sin 2\phi & -\cos 2\phi \end{array}\right]$$

where $$\phi~$$ is the angle between $$\vec{\alpha}$$ and the x-axis. The term involving the convergence magnifies the image by increasing its size while conserving surface brightness. The term involving the shear stretches the image tangentially around the lens, as discussed in Weak Lensing Observables.

The shear defined here is not equivalent to the shear traditionally defined in mathematics, though both stretch an image non-uniformly.



General weak lensing
In weak lensing by large scale structure, the thin-lens approximation may break down, and low-density extended structures may not be well approximated by multiple thin-lens planes. In this case, the deflection can be derived by instead assuming that the gravitational potential is slowly varying everywhere (for this reason, this approximation is not valid for strong lensing). This approach assumes the universe is well described by a Newtonian-perturbed FRW metric, but it makes no other assumptions about the distribution of the lensing mass.

As in the thin-lens case, the effect can written as a mapping from the unlensed angular position $$\vec{\beta}$$ to the lensed position $$\vec{\theta}$$. The Jacobian of the transform can be written as an integral over the gravitational potential $$\Phi~$$ along the line of sight

$$ \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} + \int_0^{r_\infty} dr  g(r) \frac{\partial^2  \Phi(\vec{x}(r))}{\partial x^i \partial x^j} $$

where $$r~$$ is the comoving distance, $$x^i~$$ are the transverse distances, and

$$ g(r) = 2 r \int^{r_\infty}_r \left(1-\frac{r^\prime}{r}\right)W(r^\prime) $$

is the lensing kernel, which defines the efficiency of lensing for a distribution of sources $$W(r)~$$.

The Jacobian $$A_{ij}~$$ can be decomposed into convergence and shear terms just as with the thin-lens case, and in the limit of a lens that is both thin and weak, their physical interpretations are the same.

Weak lensing observables
In weak gravitational lensing, the Jacobian is mapped out by observing the effect of the shear on the ellipticities of background galaxies. This effect is purely statistical; the shape of any galaxy will be dominated by its random, unlensed shape, but lensing will produce a spatially coherent distortion of these shapes.

Measures of ellipticity
In most fields of astronomy, the ellipticity is defined as $$1-q~$$, where $$q=\frac{b}{a}$$ is the axis ratio of the ellipse. In weak gravitational lensing, two different definitions are commonly used, and both are complex quantities which specify both the axis ratio and the position angle $$\phi~$$:

$$ \chi = \frac{1-q^2}{1+q^2}e^{2i\phi} = \frac{a^2-b^2}{a^2+b^2}e^{2i\phi} $$

$$ \epsilon = \frac{1-q}{1+q}e^{2i\phi} = \frac{a-b}{a+b}e^{2i\phi} $$

Like the traditional ellipticity, the magnitudes of both of these quantities range from 0 (circular) to 1 (a line segment). The position angle is encoded in the complex phase, but because of the factor of 2 in the trigonometric arguments, ellipticity is invariant under a rotation of 180 degrees. This is to be expected; an ellipse unchanged by a 180° rotation. Taken as imaginary and real parts, the real part of the complex ellipticity describes the elongation along the coordinate axes, while the imaginary part describes the elongation at 45° from the axes.

The ellipticity is often written as a two-component vector instead of a complex number, though it is not a true vector with regard to transforms:

$$ \chi = \{\left|\chi\right|\cos 2\phi, \left|\chi\right|\sin 2\phi\} $$

$$ \epsilon = \{\left|\epsilon\right|\cos 2\phi, \left|\epsilon\right| \sin 2\phi\} $$

Real astronomical background sources are not perfect ellipses. Their ellipticities can be measured by finding a best-fit elliptical model to the data, or by measuring the second moments of the image about some centroid $$(\bar{x},\bar{y})$$

$$ q_{xx} = \frac{\sum (x-\bar{x})^2 I(x,y)}{\sum I(x,y)} $$

$$ q_{yy} = \frac{\sum (y-\bar{y})^2 I(x,y)}{\sum I(x,y)} $$

$$ q_{xy} = \frac{\sum (x-\bar{x})(y-\bar{y}) I(x,y)}{\sum I(x,y)} $$

The complex ellipticities are then

$$ \chi = \frac{q_{xx}-q_{yy} + 2 i q_{xy}}{q_{xx}+q_{yy}} $$

$$ \epsilon = \frac{q_{xx}-q_{yy} + 2 i q_{xy}}{q_{xx}+q_{yy} + 2\sqrt{q_{xx}q_{yy}-q_{xy}^2}} $$

The unweighted second moments above are problematic in the presence of noise, neighboring objects, or extended galaxy profiles, so it is typical to use apodized moments instead:

$$ q_{xx} = \frac{\sum (x-\bar{x}^2 w(x-\bar{x},y-\bar{y}) I(x,y)}{\sum w(x-\bar{x},y-\bar{y}) I(x,y)} $$

$$ q_{yy} = \frac{\sum (y-\bar{y})^2 w(x-\bar{x},y-\bar{y}) I(x,y)}{\sum w(x-\bar{x},y-\bar{y}) I(x,y)} $$

$$ q_{xy} = \frac{\sum (x-\bar{x})(y-\bar{y}) w(x-\bar{x},y-\bar{y}) I(x,y)}{\sum w(x-\bar{x},y-\bar{y}) I(x,y)} $$

Here $$w(x,y)~$$ is a weight function that typically goes to zero or quickly approaches zero at some finite radius.

Image moments cannot generally be used to measure the ellipticity of galaxies without correcting for observational effects, particularly the point spread function.

Shear and reduced shear
Recall that the lensing Jacobian can be decomposed into shear $$\gamma~$$ and convergence $$\kappa~$$. Acting on a circular background source with radius $$R~$$, lensing generates an ellipse with major and minor axes

$$a = \frac{R}{1-\kappa-\gamma}$$

$$b = \frac{R}{1-\kappa+\gamma}$$

as long as the shear and convergence do not change appreciably over the size of the source (in that case, the lensed image is not an ellipse). Galaxies are not intrinsically circular, however, so it is necessary to quantify the effect of lensing on a non-zero ellipticity.

We can define the complex shear in analogy to the complex ellipticities defined above

$$ \gamma = \left|\gamma\right| e^{2i\phi} $$

as well as the reduced shear

$$ g \equiv \frac{\gamma}{1-\kappa} $$

The lensing Jacobian can now be written as

$$ A=\left[\begin{array}{ c c } 1 - \kappa - \mathrm{Re}[\gamma] & -\mathrm{Im}[\gamma] \\ -\mathrm{Im}[\gamma] & 1 -\kappa + \mathrm{Re}]\gamma]\end{array}\right] =(1-\kappa)\left[\begin{array}{ c c } 1-\mathrm{Re}[g] & -\mathrm{Im}[g] \\ -\mathrm{Im}[g] & 1+ \mathrm{Re}[g]\end{array}\right] $$

For a reduced shear $$g~$$ and unlensed complex ellipticities $$\chi_s~$$ and $$\epsilon_s~$$, the lensed ellipticities are

$$ \chi = \frac{\chi_s+2g+g^2\chi^*}{1+|g|^2 - 2\mathrm{Re}(g\chi^*)} $$

$$ \epsilon = \frac{\epsilon_s+g}{1-g^*\epsilon} $$

In the weak lensing limit, $$\gamma \ll 1$$ and $$\kappa \ll 1$$, so

$$ \chi \approx \chi_s+2g \approx \chi_s+2\gamma $$

$$ \epsilon \approx \epsilon_s+g \approx \epsilon_s+\gamma $$

If we can assume that the sources are randomly oriented, their complex ellipticities average to zero, so $$ \langle \chi \rangle = 2\langle \gamma \rangle $$ and $$\langle \epsilon \rangle = \langle \gamma \rangle$$. This is the principal equation of weak lensing: the average ellipticity of background galaxies is a direct measure of the shear induced by foreground mass.

Magnification
While gravitational lensing preserves surface brightness, as dictated by Liouville's theorem, lensing does change the apparent solid angle of a source. The amount of magnification is given by the ratio of the image area to the source area. For a circularly symmetric lens, the magnification factor μ is given by

$$ \mu = \frac{\theta}{\beta} \frac{d\theta}{d\beta} $$

In terms of convergence and shear

$$ \mu = \frac{1}{\det A} = \frac{1}{[(1-\kappa)^2-\gamma^2]} $$

For this reason, the Jacobian $$A~$$ is also known as the "inverse magnification matrix".

The reduced shear is invariant with the scaling of the Jacobian $$A~$$ by a scalar $$\lambda~$$, which is equivalent to the transformations $$ 1-\kappa^{\prime} = \lambda(1-\kappa) $$ and $$ \gamma^{\prime} = \lambda \gamma $$.

Thus, $$\kappa$$ can only be determined up to a transformation $$\kappa \rightarrow \lambda \kappa+(1-\lambda)$$, which is known as the "mass sheet degeneracy." In principle, this degeneracy can be broken if an independent measurement of the magnification is available because the magnification is not invariant under the aforementioned degeneracy transformation. Specifically, $$\mu~$$ scales with $$\lambda~$$ as $$\mu \propto \lambda^{-2}$$.