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HIGHER ORDER GALAXY SHAPES
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Outline
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Extending Weak Lensing Beyond First Order
We have seen in the Weak Lensing section the form of the distortion matrix. In full it can be written as
where κ is the convergence, and γ is the shear. ρ is a rotation, which is unobservable in gravitational lensing; however, it can be observed as a result of telescope rotation, see eg Bacon et al 2000.
We also saw in Weak Lensing that an image's surface brightness f_I at position thetai can be written in terms of the surface brightness f_S of the source:
Finally, we need the result from General Lensing that A can be written in terms of derivatives of the 2D potential psi:
and we can write the convergence as
and the shear as
Now we can extend all of these ideas: suppose that A varies across an image - this will lead to a circular source being distorted, not into an ellipse, but into some sort of arc. We could describe this by allowing A to be a function of theta in our formalism - this is the approach taken by strong lensing studies. Alternatively, we could adopt a mean A across the image, and write the corrections to this in a Taylor series to second order:
This is the move that leads to quantifying flexion, which we will now go on to examine.
Flexion
The D tensor introduced above contains information about how flexed an image is. It is given by
We can rewrite this in terms of flexion components, which can be conveniently defined as
These are the 1-flexion and 3-flexion respectively. 3-flexion has 120 degree rotational symmetry, while 1-flexion only has 360 degree rotational symmetry (ie it is a vector).
Untangling the various components of F and G and matching to D_ijk, we find that we can write
This gives us the description we need of how objects are lensed into weak arcs. We can use this to flex circular or elliptical objects and see what the effect of flexion looks like:
1-flexed 3-flexed 1-flexed 3-flexed
(Positive F_1) (Positive G_1) (Positive F_1) (Positive G_1)
Twist and Turn
This is not quite all there is to say at second order. We have found four degrees of freedom in D (ie F_1, F_2, G_1 and G_2). But counting elements, there initially appear to be eight degrees of freedom in D, and with some effort the remaining ones can be found (see Bacon and Schaefer 2008):
Here we call C the turn and T the twist. However, observationally we can always find a twist that gives the same observed distortion as a given turn. This is because a given source position beta_i is given by terms including (D_i12+D_i21), but never D_i12 or D_i21 on their own. Therefore there are only truly 6 measurable modes, not 8.
It turns out that twist and turn have no effect on circularly symmetric objects (in this, they are similar to rho). The effect of twist and turn on elliptical objects can be shown:
Twisted Turned
(Positive T_1) (Positive C_1)