b~
\hat{\alpha}=
4GM | |
c2b |
where G is the gravitational constant, M the mass of the deflecting object and c the speed of light. A naive application of Newtonian gravity can yield exactly half this value, where the light ray is assumed as a massed particle and scattered by the gravitational potential well. This approximation is good when
4GM/c2b
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~
\vec{\hat{\alpha}}(\vec{\xi})= | 4G |
c2 |
\intd2\xi\prime\intdz\rho(\vec{\xi}\prime,z)
\vec{b | |
^2 |
where
z
\vec{b}
d2\xi\primedz\rho(\vec{\xi}\prime,z)
(\vec{\xi}\prime,z)
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}\prime)=\int\rho(\vec{\xi}\prime,z)dz
where
\vec{\xi}\prime
\vec{\hat{\alpha}}(\vec{\xi})=
4G | |
c2 |
\int
(\vec{\xi | |
-\vec{\xi} |
\prime)\Sigma(\vec{\xi}\prime)}{|\vec{\xi}-\vec{\xi}\prime|2}d2\xi\prime
As shown in the diagram on the right, the difference between the unlensed angular position
\vec{\beta}
\vec{\theta}
\vec{\beta}=\vec{\theta}-\vec{\alpha}(\vec{\theta})=\vec{\theta}-
Dds | |
Ds |
\vec{\hat{\alpha}}(\vec{Dd\theta})
where
Dds~
Ds~
Dd~
In strong gravitational lensing, this equation can have multiple solutions, because a single source at
\vec{\beta}
The reduced deflection angle
\vec{\alpha}(\vec{\theta})
\vec{\alpha}(\vec{\theta})=
1 | |
\pi |
\intd2\theta\prime
(\vec{\theta | |
-\vec{\theta} |
\prime)\kappa(\vec{\theta}\prime)}{|\vec{\theta}-\vec{\theta}\prime|2}
where we define the convergence
\kappa(\vec{\theta})=
\Sigma(\vec{\theta | |
)}{\Sigma |
cr
and the critical surface density (not to be confused with the critical density of the universe)
\Sigmacr=
c2Ds | |
4\piGDdsDd |
We can also define the deflection potential
\psi(\vec{\theta})=
1 | |
\pi |
\intd2\theta\prime\kappa(\vec{\theta}\prime)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{\theta}-\vec{\beta}=\vec{\alpha}(\vec{\theta})=\vec{\nabla}\psi(\vec{\theta})
\kappa(\vec{\theta})=
1 | |
2 |
\nabla2\psi(\vec{\theta})
The deflection potential can also be written as a scaled projection of the Newtonian gravitational potential
\Phi~
\psi(\vec{\theta})=
2Dds | |
DdDsc2 |
\int\Phi(Dd\vec{\theta},z)dz
The Jacobian between the unlensed and lensed coordinate systems is
Aij=
\partial\betai | |
\partial\thetaj |
=\deltaij-
\partial\alphai | |
\partial\thetaj |
=\deltaij-
\partial2\psi | |
\partial\thetai\partial\thetaj |
where
\deltaij~
\gamma~
A=(1-\kappa)\left[\begin{array}{cc}1&0\ 0&1\end{array}\right]-\gamma\left[\begin{array}{cc}\cos2\phi&\sin2\phi\ \sin2\phi&-\cos2\phi\end{array}\right]
where
\phi~
\vec{\alpha}
The shear defined here is not equivalent to the shear traditionally defined in mathematics, though both stretch an image non-uniformly.
There is an alternative way of deriving the lens equation, starting from the photon arrival time (Fermat surface)
t=
zs | |
\int | |
0 |
{ndz\overc\cos\alpha(z)}
where
dz/c
1/\cos(\alpha(z)) ≈ 1+{\alpha(z)2\over2}
to get the line element along the bended path
dl={dz\overc\cos\alpha(z)}
\alpha(z),
ds2=0=c2dt2\left(1+{2\Phi\overc2}\right)-\left(1+{2\Phi\overc2}\right)-1dl2
\Phi\llc2
c'={dl/dt}=\left(1+{2\Phi\overc2}\right)c.
So the refraction index
n\equiv{c\overc'} ≈ \left(1-{2\Phi\overc2}\right).
The refraction index greater than unity because of the negative gravitational potential
\Phi
Put these together and keep the leading terms we have the time arrival surface
t ≈
zs | |
\int | |
0 |
{dz\overc}+
zs | |
\int | |
0 |
{dz\overc}{\alpha(z)2\over2}-
zs | |
\int | |
0 |
{dz\overc}{2\Phi\overc2}.
The first term is the straight path travel time, the second term is the extra geometric path, and the third is the gravitational delay. Make the triangle approximation that
\alpha(z)=\theta-\beta
\alpha(z) ≈ (\theta-\beta){Dd\overDds
{Dd\overc}{(\vec{\theta}-\vec{\beta})2\over2}+{Dds\overc}{\left[(\vec{\theta}-\vec{\beta}){Dd\overDds
(How? There is no
Ds
t=constant+{DdDs\overDdsc}\tau,~\tau\equiv\left[{(\vec{\theta}-\vec{\beta})2\over2}-\psi\right]
where
\tau
\psi(\vec{\theta})=
2Dds | |
DdDsc2 |
\int\Phi(Dd\vec{\theta},z)dz.
\tau
\vec{\theta}
0=\nabla\vec{\theta
which is the lens equation. Take the Poisson's equation for 3D potential
\Phi(\vec{\xi})=-\int
d3\xi\prime\rho(\vec{\xi | |
\prime |
)}{|\vec{\xi}-\vec{\xi}\prime|}
and we find the 2D lensing potential
\psi(\vec{\theta})=-
2GDds | |
DdDsc2 |
\intdz\int
d3\xi\prime\rho(\vec{\xi | |
\prime |
)}{|\vec{\xi}-\vec{\xi}\prime|} =-\sumi
2GMiDis | |
DsDic2 |
\left[\sinh-1{|z-Di|\overDi|\vec{\theta}-\vec{\theta}i|}\right]
Ds | |
| | |
Di |
+
0 | |
| | |
Di |
.
Here we assumed the lens is a collection of point masses
Mi
\vec{\theta}i
z=Di.
\sinh-11/x=ln(1/x+\sqrt{1/x2+1}) ≈ -ln(x/2)
\psi(\vec{\theta}) ≈ \sumi
4GMiDis | |
DsDic2 |
\left[ln\left({|\vec{\theta}-\vec{\theta}i|\over2}{Di\overDis}\right)\right].
One can compute the convergence by applying the 2D Laplacian of the 2D lensing potential
\kappa(\vec{\theta})=
1 | |
2 |
\nabla\vec{\theta
in agreement with earlier definition
\kappa(\vec{\theta})={\Sigma\over\Sigmacr}
\nabla21/r=-4\pi\delta(r)
\nabla\vec{\theta
We can also confirm the previously defined reduced deflection angle
\vec{\theta}-\vec{\beta}=\nabla\vec{\theta
where
\thetaEi
Mi
\vec{\theta}-\vec{\beta}=
2 | |
{\theta | |
E |
\over|\vec{\theta}|}.
The amplification matrix can be obtained by double derivatives of the dimensionless time delay
Aij={\partial\betaj\over\partial\thetai}={\partial\tau\over\partial\thetai\partial\thetaj}=\deltaij-{\partial\psi\over\partial\thetai\partial\thetaj}=\left[\begin{array}{cc}1-\kappa-\gamma1&\gamma2\ \gamma2&1-\kappa+\gamma1\end{array}\right]
where we have define the derivatives
\kappa={\partial\psi\over2\partial\theta1\partial\theta1}+{\partial\psi\over2\partial\theta2\partial\theta2},~\gamma1\equiv{\partial\psi\over2\partial\theta1\partial\theta1}-{\partial\psi\over2\partial\theta2\partial\theta2},~\gamma2\equiv{\partial\psi\over\partial\theta1\partial\theta2}
which takes the meaning of convergence and shear. The amplification is the inverse of the Jacobian
A=1/det(Aij)={1\over(1-\kappa)2
2 | |
-\gamma | |
1 |
2} | |
-\gamma | |
2 |
where a positive
A
A
For a single point lens, one can show (albeit a lengthy calculation) that
\kappa=0,~\gamma=
2 | |
\sqrt{\gamma | |
1 |
+
2} | |
\gamma | |
2 |
=
2 | |
{\theta | |
E |
\over|\theta|2},~
2= | |
\theta | |
E |
{4GMDds\overc2DdDs}.
So the amplification of a point lens is given by
A=\left(1-
4 | |
{\theta | |
E |
\over\theta4}\right)-1.
Note A diverges for images at the Einstein radius
\thetaE.
In cases there are multiple point lenses plus a smooth background of (dark) particles of surface density
\Sigma\rm\kappa\rm,
\psi(\vec{\theta}) ≈ {1\over2}\kappa\rm|\theta|2+\sumi
2 | |
\theta | |
E |
\left[ln\left({|\vec{\theta}-\vec{\theta}i|2\over4}{Dd\overDds}\right)\right].
To compute the amplification, e.g., at the origin (0,0), due to identical point masses distributed at
(\thetaxi,\thetayi)
A=\left[(1-\kappa\rm)2-\left(\sumi{
2 | |
(\theta | |
xi |
-
2 | |
\theta | |
yi |
)
2 | |
\theta | |
E |
\over
2 | |
(\theta | |
xi |
+
2) | |
\theta | |
yi |
2}\right)2-\left(\sumi{(2\thetaxi\thetayi)
2 | |
\theta | |
E |
\over
2 | |
(\theta | |
xi |
+
2) | |
\theta | |
yi |
2}\right)2\right]-1
This generally creates a network of critical curves, lines connecting image points of infinite amplification.
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 be written as a mapping from the unlensed angular position
\vec{\beta}
\vec{\theta}
\Phi~
\partial\betai | |
\partial\thetaj |
=\deltaij+
rinfty | |
\int | |
0 |
drg(r)
\partial2\Phi(\vec{x | |
(r))}{\partial |
xi \partialxj}
where
r~
xi~
g(r)=2r
rinfty | |
\int | |
r |
dr' \left(1-
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
Aij~
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.
In most fields of astronomy, the ellipticity is defined as
1-q~
q= | b |
a |
\phi~
\chi=
1-q2 | |
1+q2 |
e2i\phi=
a2-b2 | |
a2+b2 |
e2i\phi
\epsilon=
1-q | |
1+q |
e2i\phi=
a-b | |
a+b |
e2i\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 is 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|\cos2\phi,\left|\chi\right|\sin2\phi\}
\epsilon=\{\left|\epsilon\right|\cos2\phi,\left|\epsilon\right|\sin2\phi\}
(\bar{x},\bar{y})
qxx=
\sum(x-\bar{x | |
) |
2I(x,y)}{\sumI(x,y)}
qyy=
\sum(y-\bar{y | |
) |
2I(x,y)}{\sumI(x,y)}
qxy=
\sum(x-\bar{x | |
)(y-\bar{y}) |
I(x,y)}{\sumI(x,y)}
The complex ellipticities are then
\chi=
qxx-qyy+2iqxy | |
qxx+qyy |
\epsilon=
qxx-qyy+2iqxy | ||||||||||||
|
This can be used to relate the second moments to traditional ellipse parameters:
qxx=a2\cos2\theta+b2\sin2\theta
qyy=a2\sin2\theta+b2\cos2\theta
qxy=(a2-b2)\sin\theta\cos\theta
and in reverse:
a2=
| |||||||||||||
b2=
| |||||||||||||
\tan2\theta=
2qxy | |
qxx-qyy |
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:
qxx=
\sum(x-\bar{x | |
) |
2w(x-\bar{x},y-\bar{y})I(x,y)}{\sumw(x-\bar{x},y-\bar{y})I(x,y)}
qyy=
\sum(y-\bar{y | |
) |
2w(x-\bar{x},y-\bar{y})I(x,y)}{\sumw(x-\bar{x},y-\bar{y})I(x,y)}
qxy=
\sum(x-\bar{x | |
)(y-\bar{y}) |
w(x-\bar{x},y-\bar{y})I(x,y)}{\sumw(x-\bar{x},y-\bar{y})I(x,y)}
Here
w(x,y)~
Image moments cannot generally be used to measure the ellipticity of galaxies without correcting for observational effects, particularly the point spread function.[4]
Recall that the lensing Jacobian can be decomposed into shear
\gamma~
\kappa~
R~
a=
R | |
1-\kappa-\gamma |
b=
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|e2i\phi
as well as the reduced shear
g\equiv
\gamma | |
1-\kappa |
The lensing Jacobian can now be written as
A=\left[\begin{array}{cc}1-\kappa-Re[\gamma]&-Im[\gamma]\ -Im[\gamma]&1-\kappa+Re[\gamma]\end{array}\right] =(1-\kappa)\left[\begin{array}{cc}1-Re[g]&-Im[g]\ -Im[g]&1+Re[g]\end{array}\right]
For a reduced shear
g~
\chis~
\epsilons~
\chi=
| ||||||||||||
|
\epsilon=
\epsilons+g | ||||||
|
In the weak lensing limit,
\gamma\ll1
\kappa\ll1
\chi ≈ \chis+2g ≈ \chis+2\gamma
\epsilon ≈ \epsilons+g ≈ \epsilons+\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
\langle\epsilon\rangle=\langle\gamma\rangle
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=
\theta | |
\beta |
d\theta | |
d\beta |
In terms of convergence and shear
\mu=
1 | |
\detA |
=
1 | |
[(1-\kappa)2-\gamma2] |
For this reason, the Jacobian
A~
The reduced shear is invariant with the scaling of the Jacobian
A~
λ~
1-\kappa\prime=λ(1-\kappa)
\gamma\prime=λ\gamma
Thus,
\kappa
\kappa → λ\kappa+(1-λ)
\mu~
λ~
\mu\proptoλ-2