A&A 485, 363-376 (2008)
DOI: 10.1051/0004-6361:20078631
P. Schneider^{1} - X. Er^{1,2}
1 - Argelander-Institut für Astronomie, Universität Bonn,
Auf dem Hügel 71, 53121 Bonn, Germany
2 -
Max-Planck-Institut für Radioastronomie,
Auf dem Hügel 69, 53121 Bonn, Germany
Received 7 September 2007 / Accepted 27 February 2008
Abstract
In weak gravitational lensing, the image distortion caused by shear measures the projected tidal gravitational field of the deflecting mass distribution. To lowest order, the shear is proportional to the mean image ellipticity. If the image sizes are not small compared to the scale over which the shear varies, higher-order distortions occur, called flexion. For ordinary weak lensing, the observable quantity is not the shear, but the reduced shear, owing to the mass-sheet degeneracy. Likewise, the flexion itself is unobservable. Instead, higher-order image distortions measure the reduced flexion, i.e., derivatives of the reduced shear. We derive the corresponding lens equation in terms of the reduced flexion and calculate the resulting relation between brightness moments of source and image. Assuming an isotropic distribution of source orientations, estimates for the reduced shear and flexion are obtained and then tested with simulations. In particular, the presence of flexion affects the determination of the reduced shear. The results of these simulations yield the amount of bias of the estimators as a function of the shear and flexion. We point out and quantify a fundamental limitation of the flexion formalism in terms of the product of reduced flexion and source size. If this product increases above the derived threshold, multiple images of the source are formed locally, and the formalism breaks down. Finally, we show how a general (reduced) flexion field can be decomposed into its four components. Two of them are due to a shear field, carrying an E- and B-mode in general. The other two components do not correspond to a shear field, and they can also be split up into corresponding E- and B-modes.
Key words: gravitational lensing - galaxies: evolution - galaxies: statistics - cosmology: diffuse radiation
Weak gravitational lensing provides a powerful tool for studying the mass distribution of clusters of galaxies, as well as the large scale structure in the Universe (see Mellier 1999; Bartelmann & Schneider 2001; Refregier 2003; Schneider 2006; Munshi et al. 2006, for reviews on weak lensing). It has led to constraints on cosmological parameters, such as those characterizing structure formation and the mass density of the Universe.
In weak lensing, one employs the fact that the image ellipticity of a distant source is modified by the tidal gravitational field of the intervening matter distribution. Based on the assumption that the orientation of distant sources is random, the ellipticity of each image yields an unbiased estimate of the line-of-sight integrated tidal field, usually called shear in lensing. The shear thus carries information about the properties of the mass distribution. Formally, the shear is described in terms of a first-order expansion of the lens equation, i.e., the locally linearized lens equation. This yields a valid description of the mapping from the image to the source sphere, as long as the images are small compared to the length scale on which the shear varies. However, this linear approximation breaks down for larger sources, or does so in regions of the lens plane where the shear varies rapidly. The most visible failure of the linearized lens equation is the occurrence of giant arcs, which in most cases correspond actually to multiple images of a background source. To model them, the full lens equation needs to be studied. However, there is an intermediate regime where the linearized lens equation breaks down, although (locally) no multiple images are formed - the arclets regime. Arclets are fairly strongly distorted images of background sources (Fort et al. 1988; Fort & Mellier 1994), though they do not correspond to multiple images.
Arclets are the most natural application for flexion. Flexion has been introduced by Goldberg & Bacon (2005) and Bacon et al. (2006), and describes the lowest-order deviation of the lens mapping from its linear expansion; it has also been termed ``sextupole lensing'' and been treated by Irwin & Shmakova (2005, 2006, and references therein). It corresponds to the derivative of the shear; in combination with a strong shear, this can deform round images into arclets, giving rise to images which resemble the shape of a banana. In their original paper, Goldberg & Bacon (2005) considered only a single component of flexion which, however, only provides an incomplete description of shear derivatives. In Bacon et al. (2006), the need for a second flexion component was recognized.
In the first part of this paper, we present the general theory of flexion; in contrast to earlier work, we explicitly consider the quantities that can be actually observed, by accounting for the mass-sheet degeneracy (Falco et al. 1985; Gorenstein et al. 1988). That is, a change of the surface mass density of the form leaves the shape of all observed images invariant. In usual weak lensing, this is accounted for by recognizing that not the shear can be obtained from observations, but only the reduced shear (Schneider & Seitz 1995). The difference of shear and reduced shear is typically small, in particular in applications of cosmic shear, since along most lines-of-sight, the value of is very much smaller than unity. In applications of flexion, however, we expect that the surface mass density no longer is very small; for instance, arclets occur in the inner parts of clusters where . Therefore, the difference between shear and reduced shear can no longer be neglected. Gradients of the shear are not directly observable; only derivatives of the reduced shear are, and thus we define the (reduced) flexion in terms of derivatives of g. In Sect. 2.1 we briefly recall the irreducible tensor components which are defined in term of their behavior under rotations of the coordinate system. It turns out that a complex notation for these tensor components is very useful. In Sect. 2.2 we expand the lens equation to second order, before deriving the corresponding lens equation (and relation for the local Jacobian) which is invariant under mass-sheet transformations. The second-order term in this lens equation is fully described by our reduced flexion components G_{1} and G_{3}.
As is known from usual weak lensing studies, a measured shear is not necessarily accounted for by an (equivalent) surface mass density. Since the shear is a two-component quantity, it has one degree of freedom more than the field. Therefore, shear fields are decomposed into E- and B-modes (Crittenden et al. 2002; Schneider et al. 2002), where the former are due to a field, whereas the latter describes the remaining (``curl'') part. A similar situation occurs in flexion, which has four components. Therefore, in Sect. 3 we consider the decomposition of a general flexion field into contributions due to the gradient of the shear and those not related to the shear field. The former one can then be further subdivided into flexion resulting from an E- and B-mode shear field. We carry out this decomposition for the flexion as well as for the reduced flexion.
In Sect. 4 we then define brightness moments of sources and images and derive the transformation laws between them. This approach is very similar to the HOLICs approach developed by Okura et al. (2007a,b) and later considered by Goldberg & Leonard (2007), except that we explicitly write all relations in terms of the reduced shear and the reduced flexion. Generalizing the usual assumption that the expectation value of the source ellipticity is zero - due to the phase averaging over source orientations - to the expectation values of all source shape parameters which are not invariant under coordinate rotations (as appropriate for a statistically isotropic Universe), we obtain in Sect. 5 estimates for the reduced shear and reduced flexion in terms of the brightness moments of the images. In Sect. 6 we perform a number of numerical experiments to test the validity of our approach and the accuracy of the estimators derived. In particular, we point out that there is a fundamental limit where the theory of flexion has to break down - the second-order lens equation is non-linear and will in general have critical curves, leading to multiple images of the source (or parts of it). If the source is cut by a caustic, different parts of it will have different numbers of images, and the assumption of random source orientation (which underlies all weak lens applications) will break down - the caustic introduces a preferred orientation into the source plane. In Appendix B we provide a full classification of the critical curves of the second-order lens equation and use these results in order to obtain the maximum source size (for given values of the reduced flexion) for which the flexion concept still makes sense. We discuss our results in Sect. 7.
We shall be dealing only with totally symmetric tensors. If Q_{ij}is symmetric, then
The observables of a gravitational lens system are unchanged if the surface mass density is transformed as (Gorenstein et al. 1988). In the case of weak lensing, the shape of images is unchanged under this transformation (Schneider & Seitz 1995). Because of this mass-sheet degeneracy, the shear is not an observable in weak lensing, but only the reduced shear . In fact, since we expect that the most promising applications of flexion will come from situations where is not much smaller than unity, the distinction between shear and reduced shear is likely to be more important for flexion than for the usual weak lensing applications. Hence, at best we can expect from higher-order shape measurements to obtain an estimate for the reduced shear and its derivatives. For this reason, we shall rewrite the foregoing expressions in terms of the reduced shear.
The mass-sheet transformation is equivalent to an isotropic scaling of
the source plane coordinates. Hence, we divide (7) by
to obtain
The Jacobian determinant
of the mapping between the image
position
and the rescaled source position
then
becomes
Flexion has a total of four components, namely the real and imaginary parts of and . A measurement of flexion will thus yield four components, and we might ask whether these components are independent. We recall a similar situation in shear measurements. The shear has two components; on the other hand, the shear is defined as second partial derivatives of the deflection potential, which is a single scalar field. Therefore, the two shear components cannot be mutually independent if they are due to a gravitational lensing signal. Of course, the measured shear is not guaranteed to satisfy the condition that the two shear components can be derived from a single scalar deflection potential, since observational noise or intrinsic alignments of galaxies may affect the measured shear field. Therefore, one has introduced the notion of E- and B-modes in shear measurements (Crittenden et al. 2002). The E-mode shear is the one that can be written in terms of a deflection potential, whereas the B-mode shear cannot.
Formally, the E- and B-mode decomposition can be written in terms of a
complex deflection potential
and a complex surface mass
density
(Schneider et al. 2002). Each component of
satisfies its own Poisson
equation,
,
.
Making use of this decomposition, the shear
becomes
= | |||
= | (15) |
(16) |
= | |||
= | (17) |
(18) |
Since the flexion has four components, whereas the lens can be
described by a single scalar field, we expect that there are three
constraint relations a flexion field has to satisfy if it is due to a
lensing potential. In fact, even if we leave the shear field arbitrary
(that is, even if we allow it to be composed of E- and B-modes), then
we expect two constraint equations, since the flexion field has two
components more than the shear field. These constraint equations are
easy to obtain. First, if the flexion field is due to a shear field,
then we have
(20) |
= | |
= | (21) |
Figure 1: The four different flexion fields discussed in the text. The upper left (right) panel shows the flexion corresponding to an axially-symmetric E-mode (B-mode) shear field, where arrows indicate the spin-1 flexion and the skeletons the spin-3 flexion component. In the lower left (right) panel, the flexion fields are displayed which are not due to a shear field, but a non-zero E-mode (B-mode) field. | |
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As an explicit example, we consider the isothermal case. For
a singular isothermal sphere with Einstein radius
,
one then has
To obtain a similar example for the case that flexion is not derived
from a shear field, we choose first a pure E-mode spin-2 field for
,
(24) |
(25) |
We point out that the foregoing relation suggests a natural way to use flexion for finite-field mass reconstructions in weak lensing. Seitz & Schneider (2001) formulated the finite-field mass reconstruction from measured reduced shear in terms of a von Neumann boundary value problem for , whose solution determines K up to an additive constant. The ``source'' for was determined by the reduced shear and its derivatives, and is given by (26). In Seitz & Schneider (2001), the derivatives of the reduced shear were obtained by finite differencing of g. If flexion is measured, one can replace the ``source'' for by a weighted sum of the differentiated reduced shear field and the combination (K_{2}+K_{2}^{*})/2 of the flexion field, with the weights chosen according to the estimated noise properties of both contributions.
We consider an image of a source, and denote the brightness distribution of the source by . Since surface brightness is conserved by lensing, the brightness distribution of the image is . Since the scaling of the source plane is unobservable, we shall only work in the following in terms of the scaled source plane coordinates, and therefore drop the hat on , as well as on .
We define the origin of the image (or lens) plane as the
center-of-light of the image under consideration, i.e. we require
= | T_{3}^{*}-3 g^{*} T_{1}^{*}+3 g^{*2} T_{1}- g^{*3} T_{3} ; | ||
= | -g T_{3}^{*} +(1+2 g g^{*})T_{1}^{*} - g^{*}(2 +g g^{*})T_{1} +g^{*2} T_{3} ; | ||
= | (45) |
We will now consider the order-of-magnitudes of the various terms appearing in (39) and (44). Assuming that the third-order moments of the sources are small, then the third-order moments of the image are given by the product of and . With and , we find that . To get an estimate of the size of the various terms in (39), we note that the first three terms on the right-hand side (those proportional to the Q_{n}) are of order , whereas and . Hence, the last two terms are of equal magnitude in general, each of them being smaller than the first three terms by a factor . Only if the source is of the same order as the scale over which the reduced shear varies do the last two terms in (39) contribute. In (44), we have neglected the terms , since they are two powers of smaller than the terms written down.
We see that (44) is a linear equation for ,
which can thus be solved,
(47) |
(49) |
We start the iteration by setting Y_{0}=0. This yields a first-order
solution for the estimate of g,
(50) |
(51) |
Of course, our approach of setting yields a biased estimator for g; this is true even in the absence of flexion (e.g., Schneider & Seitz 1995). The reason is that, although the expectation value of vanishes, the resulting estimator for g is a non-linear function of and thus biased. The bias depends on the ellipticity distribution of the sources. It should be stressed, however, that a modified definition of image ellipticity exist such that its expectation value is an unbiased estimate of the reduced shear (Seitz & Schneider 1997).
(52) |
A more accurate estimate is obtained if we consider the reduced shear
as well as the ratios of non-zero spin brightness moments to zero spin
moments (such as |Q_{2}/Q_{0}| or
|F_{2,4}/F_{0}|) to be of order
,
and then expand the flexion to first order in the (small)
parameter
to obtain
In this section we describe some simulations that we have performed in order to test the behavior of the estimators given in the previous section.
It should be noted that flexion is a dimensional quantity . As can be checked explicitly from Sect. 4, the way flexion appears in the equations is always with one order higher in the source (or image) size than the other terms in the equations. As an example, we consider (44); the left-hand side and the first term on the right-hand side are , whereas the coefficients of the matrix . This then implies that the accuracy of the flexion estimates does not depend on the magnitude of the flexion and the source size individually, but only on the product . Therefore, the following results are quoted always in terms of this product.
Figure 2: Constraints on the combination of source size and reduced flexion for the validity of the concept of flexion. Each curve shows the dividing line between a circular source of limiting isophote being cut by a caustic (above the curve) or not (below the curve); in the former case, the assumptions underlying the flexion concept break down. The different curves in each panel are for different values of g, chosen as g=0.4,0.2,0.1,0.05,0, as indicated by different line types. Without loss of generality, we choose g to be real and non-negative. The four panels differ in the phase of the reduced flexion, as indicated. E.g., in the upper left panel, the phases of G_{1}, G_{3} are the same as that of g. | |
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Figure 3: Accuracy of the estimates for reduced shear and flexion. The left panel shows contour of constant fractional error of , and , on the estimate of the reduced shear g, as a function of , where we chose g=0.05 as input value, and assumed the phases of G_{1}, G_{3} to be the same as that of g. The estimate was obtained by solving the iteration equations given in Sect. 5. The right panel shows the fractional error levels at 3, 5, and 10% for the reduced flexion, as quantified by (55), where the estimate was obtained again with the iterative procedure. In both cases, we assumed circular sources. | |
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Figure 4: Comparison of the reduced flexion estimators (53) with the full expression (46) and the input value. The horizontal and vertical axis show . For both panels, we take g=0.05, and G_{3}=0 (G_{1}=0) for the left (right) panel. The line indicates the input value, the plus symbols show the simplified reduced flexion estimate (53), and the crosses result from the full expression of reduced flexion (46). As can be seen from the left-hand panel, the full estimator for the reduced flexion yields a more biased result that the approximate expression (53); we have not found a reasonable explanation for this behavior. | |
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In our simulations we can check whether a critical curve crosses our central image, just by controlling the sign of the Jacobian determinant (the true one, not the linear approximation Eq. (13)). If the source size becomes too large, some points in the image will have a negative Jacobian. In the Appendix B, we consider the critical curves and caustics of the lens Eq. (9), which allows us to determine the regions in flexion space where no local multiple imaging occurs. Some examples of this are plotted in Fig. 2. Each panel shows the dividing line between parameter pairs for a circular source of limiting isophotal radius ; below the curves, no local multiple images occur, whereas for parameter pairs above the lines, the flexion formalism using moments necessarily breaks down. The different lines in each panel correspond to different values of g. The occurrence of critical curves also is the reason why we truncated the intrinsic ellipticity distribution of the sources in the simulations, since in the limit of , keeping the source area fixed, there will be orientation angles for which the source will hit a caustic.
We start by considering a circular source, and determine the effect of flexion on the determination of the reduced shear. The left-hand panel of Fig. 3 shows contours of constant fractional deviation , in the flexion parameter plane. Here it is assumed that the phase of both flexion components is the same as that of g (as would be the case in an axially-symmetric lens potential). Errors of order 5% occur already for , and the fractional error increases approximately linearly with the strength of flexion (or with the source size), although it does not scale equally with both flexion components. The reason for this effect has been mentioned before - flexion affects the transformation between source and image quadrupole moments, as can be seen in (37).
In Fig. 4, we show the expectation value of the reduced
flexion components, as a function of the input flexion. The
expectation value has been determined by averaging over an isotropic
ensemble of elliptical sources, as described before. The left and
right panel show the behavior of the expectation value of G_{1} and G_{3}, respectively, where the other flexion component was set to
zero. The dashed curve shows the identity, the plus symbols were
obtained by using the approximate estimator (53), whereas
the crosses show the expectation values as obtained by employing the
full expression (46), where the corresponding value of gwas obtained by the iterative process described in
Sect. 5. It is reassuring that the expectation value
closely traces the input value, i.e., that the estimates have a fairly
small bias. Furthermore, we see that the approximate estimator (53) performs remarkably well. It is seen that the
estimates for G_{3} behave better than those for G_{1}. This can also
be seen from the right-hand panel of Fig. 3, where we
plot contours of constant fractional error
In this paper, we have studied the effect of flexion in weak gravitational lensing. The main results are summarized as follows:
Similar to the situation in shear measurements, the moment approach for flexion as presented here must be modified in several ways to be applicable to real data. First, brightness moments must be weighted in order not to be dominated by the very noisy outer regions of the image. As is known from shear measurements, such a weighting affects the relation between source and image brightness moments. Secondly, one needs to account for the effects of a point-spread function. Both of these modifications were successfully achieved for second-order brightness moments by Kaiser et al. (1995; see also Luppino & Kaiser 1997). Goldberg & Leonard (2007) consider these effects in the context of flexion. It should be noted, though, that their consideration of the PSF effects is restricted to unweighted moments, for which these effects are given by a simple convolution. In the case of weighted brightness moments, however, the PSF effects are much more subtle. Okura et al. (2007b) indeed developed a PSF correction scheme similar to that of Kaiser et al. (2005), now accounting for higher-order brightness moments of the images and the PSF; their application of this method to synthetic data and to images of the cluster A1689 is encouraging.
But even disregarding these complications, the present paper only scratches the surface in investigating estimators for reduced flexion and their properties. As mentioned before, the second-order lens equation contains five essential parameters. The bias of an estimator for reduced shear and flexion will depend on these parameters, as well as on the intrinsic ellipticity (and higher-order moments) distribution of sources. One might ask whether it is possible to find an unbiased flexion estimator, such as was possible to construct for the reduced shear. Unfortunately, we have been unable to make analytic progress: even for a circular Gaussian source, the brightness moments of the image cannot be calculated analytically. Our ray-tracing algorithm with which we conducted our numerical simulations is almost certainly sub-optimal; a more advanced method should be developed to reduce the numerical efforts in calculating brightness moments. Beside the bias, it would be interesting to calculate the variance of the various estimators, or more precisely, their covariance.
It may turn out that measurements of flexion, and PSF corrections, are more conveniently done with shapelets, as was originally considered by Goldberg & Bacon (2005), Bacon et al. (2006) and Massey et al. (2007a). Even if this turns out to be the case (see Leonard et al. 2007, for an application of flexion measurements in the galaxy cluster A1689), the moment approach provides a more intuitive picture of the effects of flexion. In addition, the weak lensing community has profited substantially from the existence of several different methods to measure shear (see Heymans et al. 2006; Massey et al. 2007b, for the first results of a comprehensive Shear TEsting Programme, in which these various methods are studied and compared); therefore, the development of different techniques for measuring flexion will certainly be of interest once the flexion method is put to extensive use.
Acknowledgements
We thank Jan Hartlap and Ismael Tereno for useful comments on this paper. This work was supported by the Deutsche Forschungsgemeinschaft under the project SCHN 342/6-1 and the TR33 ``The Dark Universe''. X.E. was supported for this research through a stipend from the International Max-Planck Research School (IMPRS) for Radio and Infrared Astronomy at the University of Bonn.
Figure A.1: The critical curves ( left-hand panel) and caustics ( right-hand panel) of the lens Eq. (9) for the cases of hyperbolic critical curves, as described in Sect. B.2. The parameters chosen here are g=0.05, , . A circular source is mapped onto two images, as indicated. If the source size were increased, it would hit the caustic, the two images would merge, and the flexion concept would break down. The unit of the reduced flexion is the inverse of the unit in which coordinates are measured. | |
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Figure A.2: Same as Fig. A.1, but for the parabolic case, with parameters g=0.05, G_{1}=-0.04, G_{3}=0.112. | |
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In this Appendix, we list the coefficients of the matrix which occurs in (44):
4(1-gg^{*}) C_{11} | = | -2g F_{2}^{*} +(9g g^{*}-3) F_{0} +6 g^{*}(1-2 g g^{*}) F_{2} | |
+g^{*2}(5 g g^{*}-3) F_{4} +6 g Q_{2}^{*} Q_{0} -12 g g^{*} Q_{0}^{2} | |||
+(3 -9 g g^{*}) Q_{2}^{*} Q_{2} + 6 g^{*} (4 g g^{*}-1) Q_{0} Q_{2} | |||
+3 g^{*2}(1- 3 g g^{*}) Q_{2}^{2} | |||
4(1-gg^{*}) C_{12} | = | 5 g F_{4}^{*} -2(5+6 g g^{*}) F_{2}^{*} +9 g^{*}(3+g g^{*}) F_{0} | |
-2 g^{*2}(12+g g^{*}) F_{2} +7 g^{*3} F_{4} -9 g Q_{2}^{*2} | |||
+6(3 +4 g g^{*}) Q_{2}^{*} Q_{0} -12 g^{*}(3+g g^{*}) Q_{0}^{2} | |||
-3 g^{*}(5+3 g g^{*}) Q_{2}^{*} Q_{2} +6 g^{*2}(8+g g^{*}) Q_{0} Q_{2} | |||
-15 g^{*3} Q_{2}^{2} | |||
4(1-gg^{*}) C_{13} | = | -7 F_{4}^{*} +26 g^{*} F_{2}^{*} -36 g^{*2} F_{0} +22 g^{*3} F_{2} -5 g^{*4} F_{4} | |
+ 15 Q_{2}^{*2} -54 g^{*} Q_{2}^{*} Q_{0} +48 g^{*2} Q_{0}^{2} +24 g^{*2} Q_{2}^{*} Q_{2} | |||
-42 g^{*3} Q_{0} Q_{2} +9 g^{*4} Q_{2}^{2} | |||
4(1-gg^{*}) C_{14} | = | -2 g^{*} F_{4}^{*} +6 g^{*2} F_{2}^{*} - 6 g^{*3} F_{0} +2 g^{*4} F_{2} | |
+6 g^{*} Q_{2}^{*2} -18 g^{*2} Q_{2}^{*} Q_{0} +12 g^{*3} Q_{0}^{2} | |||
+6 g^{*3} Q_{2}^{*} Q_{2} -6 g^{*4} Q_{0} Q_{2} | |||
4(1-gg^{*}) C_{21} | = | 2 g^{2} F_{2}^{*} -6 g^{2} g^{*} F_{0}+[4 g g^{*}(1+g g^{*})-2] F_{2} | |
+2 g^{*}(1-2 g g^{*}) F_{4} -6 g^{2} Q_{2}^{*} Q_{0} | |||
+4 g (1+2 g g^{*}) Q_{0}^{2} +6 g^{2} g^{*} Q_{2}^{*} Q_{2} | |||
+[2-4 g g^{*}(3+2 g g^{*})] Q_{0} Q_{2}+2 g^{*} (4 g g^{*}-1) Q_{2}^{2} | |||
4(1-gg^{*}) C_{22} | = | -5 g^{2} F_{4}^{*} +2 g (7+4 g g^{*}) F_{2}^{*} | |
-3[3+g g^{*}(8+g g^{*})] F_{0}+2 g^{*}(8+5 g g^{*}) F_{2} | |||
-7 g^{*2} F_{4} + 9 g^{2} Q_{2}^{*2} -2 g (13+8 g g^{*}) Q_{2}^{*} Q_{0} | |||
+4[3+g g^{*}(8+g g^{*})] Q_{0}^{2} | |||
+[5+g g^{*}(16+3 g g^{*})] Q_{2}^{*} Q_{2} | |||
-2 g^{*}(16+11 g g^{*}) Q_{0} Q_{2} +15 g^{*2} Q_{2}^{2} | |||
4(1-gg^{*}) C_{23} | = | 7g F_{4}^{*} -2(4 +9 g g^{*}) F_{2}^{*} +3 g^{*}(7+5 g g^{*}) F_{0} | |
-2 g^{*2}(9+2 g g^{*}) F_{2}+5 g^{*3} F_{4} -15 g Q_{2}^{*2} | |||
+(16+38 g g^{*}) Q_{2}^{*} Q_{0} - 4 g^{*}(7 +5 g g^{*}) Q_{0}^{2} | |||
- g^{*}(13+11 g g^{*}) Q_{2}^{*} Q_{2} +2 g^{*2}(17+4 g g^{*}) Q_{0} Q_{2} | |||
-9 g^{*3} Q_{2}^{2} | |||
4(1-gg^{*}) C_{24} | = | (3 g g^{*}-1) F_{4}^{*}-6 g g^{*2} F_{2}^{*} +3 g^{*2}(1 + g g^{*}) F_{0} | |
- 4 g^{*2}(2+g g^{*}) Q_{0}^{2}-3 g^{*2}(1+g g^{*}) Q_{2}^{*} Q_{2} | |||
+6 g^{*3} Q_{0} Q_{2}. |
Figure A.3: Same as Fig. A.1, but for the elliptical case, with parameters . | |
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In this Appendix we consider the critical curves of the lens Eq. (9), with the goal of finding the pricipal limits
of the applicability of the flexion formalism - which necessarily
breaks down if parts of the source are multiply imaged. For this, we
need to derive the full Jacobian,
which can most easily be obtained from considering
and
as independent variables, and then use
,
,
which can be inverted to yield
,
.
With these relations, one
finds that
.
Carrying out these
derivatives, the Jacobian becomes
(B.2) |
(B.3) |
(B.4) |
(B.5) |
(B.6) |
As the first case, we consider A=0 and (the case A=0=Bwas treated above), which implies that and . The equation for the critical curve then reduces to . Furthermore, , and C=-1. Thus, the critical curve is a circle of radius and center , or , .
We now consider the case ;
then the phase
of A is defined,
as used before. Introducing a rotation by defining
,
the equation for the critical curve becomes
(B.10) |
(B.11) |
(B.12) |