Gravitational lensing formalism

Gravitational lensing formalism

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 can yield exactly half this value, where the light ray is assumed as a massed particle and scattered by the gravitational potential well.

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 ho~, 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^2xi^{prime} frac{(vec{xi}-vec{xi}^{prime}) ho(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 [ cite journal | last = Bartelmann| first = M. | coauthors = Schneider, P. | year = 2001 | month = January | title="Weak Gravitational Lensing"| journal = Physics Reports D | volume=340 | issue = 4-5 |pages = 291-472 | bibcode = 2001PhR...340..291B] .

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 ho(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{eta} and the observed position vec{ heta} is this deflection angle, scaled by a ratio of distances, described as the lens equation

vec{eta}=vec{ heta}-vec{alpha}(vec{ heta}) = vec{ heta} - frac{D_{ds{D_s} vec{hat{alpha(vec{D_d heta})

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{eta} can be lensed into multiple images.

Convergence and deflection potential

It is convenient to define the "convergence"

kappa(vec{ heta}) = frac{Sigma(D_dvec{ heta})}{Sigma_{cr

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

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

The scaled deflection angle vec{alpha}(vec{ heta}) can now be written as

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

We can also define the "deflection potential"

psi(vec{ heta}) = frac{1}{pi}int d^2 heta^{prime} kappa(vec{ heta}^{prime}) ln |vec{ heta}-vec{ heta}^{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{ heta}) = vec{ abla} Psi(vec{ heta})

kappa(vec{ heta}) = frac{1}{2} abla^2 Psi(vec{ heta})

The deflection potential can also be written as a scaled projection of the Newtonian gravitational potential Phi~ of the lens [ cite journal | last = Narayan| first = R. | coauthors = Bartelmann, M. | year = 1996 | month = June | title="Lectures on Gravitational Lensing"| journal = eprint arXiv:astro-ph/9606001 | bibcode ]

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

Lensing Jacobian

The Jacobian between the unlensed and lensed coordinate systems is

A_{ij}=frac{partial eta_i}{partial heta_j}=delta_{ij} - frac{partial alpha_i}{partial heta_j}= delta_{ij} - frac{partial^2 psi}{partial heta_i partial heta_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 [egin{array}{ c c } 1 & 0 \ 0 & 1 end{array} ight] -gammaleft [egin{array}{ c c } cos 2phi & sin 2phi \ sin 2phi & -cos 2phi end{array} ight]

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 be written as a mapping from the unlensed angular position vec{eta} to the lensed position vec{ heta}. The Jacobian of the transform can be written as an integral over the gravitational potential Phi~ along the line of sight [cite book |title=Modern Cosmology|last=Dodelson|first=Scott|year=2003|publisher=Academic Press|location=Amsterdam|isbn=0-12-219141-2]

frac{partial eta_i}{partial heta_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} ight)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^{2iphi} = frac{a^2-b^2}{a^2+b^2}e^{2iphi}

epsilon = frac{1-q}{1+q}e^{2iphi} = frac{a-b}{a+b}e^{2iphi}

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 ight|cos 2phi, left|chi ight|sin 2phi}

epsilon = {left|epsilon ight|cos 2phi, left|epsilon ight| sin 2phi}

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 (ar{x},ar{y})

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

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

q_{xy} = frac{sum (x-ar{x})(y-ar{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} + 2sqrt{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-ar{x})^2 w(x-ar{x},y-ar{y}) I(x,y)}{sum w(x-ar{x},y-ar{y}) I(x,y)}

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

q_{xy} = frac{sum (x-ar{x})(y-ar{y}) w(x-ar{x},y-ar{y}) I(x,y)}{sum w(x-ar{x},y-ar{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. [ cite journal | last = Bernstein| first = G. | coauthors = Jarvis, M. | year = 2002 | month = February | title="Shapes and Shears, Stars and Smears: Optimal Measurements for Weak Lensing"| journal = Astronomical Journal | volume=123 | issue = 2 |pages = 583-618 | bibcode = 2002AJ....123..583B]

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 ight| e^{2iphi}

as well as the "reduced shear"

g equiv frac{gamma}{1-kappa}

The lensing Jacobian can now be written as

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

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

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

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+2gamma

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 angle = 2langle gamma angle and langle epsilon angle = langle gamma angle.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.


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{ heta}{eta} frac{d heta}{deta}

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 transformations1-kappa^{prime} = lambda(1-kappa)andgamma^{prime} = lambda gamma.

Thus, kappa can only be determined up to a transformation kappa ightarrow 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}.


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