aa r X i v : . [ a s t r o - ph ] S e p Astronomy & Astrophysics manuscript no. November 9, 2018(DOI: will be inserted by hand later)
Weak lensing goes bananas: What flexion really measures
Peter Schneider , Xinzhong Er , Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, D-53121 Bonn, Germanye-mail: peter, [email protected] Max-Plank-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, D-53121 Bonn, GermanyReceived ; accepted
Abstract.
In weak gravitational lensing, the image distortion caused by shear measures the projected tidal grav-itational field of the deflecting mass distribution. To lowest order, the shear is proportional to the mean imageellipticity. If the image sizes are not small compared to the scale over which the shear varies, higher-order distor-tions 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. Rather, higher-order image distortions measure thereduced flexion, i.e., derivatives of the reduced shear. We derive the corresponding lens equation in terms of thereduced flexion and calculate the resulting relation between brightness moments of source and image. Assumingan isotropic distribution of source orientations, estimates for the reduced shear and flexion are obtained; these arethen 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 reducedflexion and source size. If this product increases above the derived threshold, multiple images of the source areformed locally, and the formalism breaks down. Finally, we show how a general (reduced) flexion field can bedecomposed 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; they can also be split up into corresponding E- andB-modes.
Key words. cosmology – gravitational lensing – large-scale structure of the Universe – galaxies: evolution – galaxies:statistics
1. Introduction
Weak gravitational lensing provides a powerful tool for studying the mass distribution of clusters of galaxies as well asthe large scale structure in the Universe (see Mellier 1999; Bartelmann & Schneider 2001; Refregier 2003; Schneider2006; Munshi et al. 2006 for reviews on weak lensing). It has led to constraints on cosmological parameters, such asthose 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 tidalgravitational field of the intervening matter distribution. Based on the assumption that the orientation of distantsources 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 linearizedlens equation. This yields a valid description of the mapping from the image to the source sphere, as long as theimages are small compared to the length-scale on which the shear varies. However, this linear approximation breaksdown for larger sources, or in regions of the lens plane where the shear varies rapidly. The most visible failure of thelinearized lens equation is the occurrence of giant arcs, which in most cases correspond actually to multiple imagesof a background source; to model them, the full lens equation needs to be studied. However, there is an intermediateregime where the linearized lens equation breaks down, although (locally) no multiple images are formed – the arcletsregime. Arclets are fairly strongly distorted images of background sources (Fort et al. 1988; Fort & Mellier 1994),though they do not correspond to multiple images.
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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. Itcorresponds to the derivative of the shear; in combination with a strong shear, this can deform round images intoarclets, 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 shearderivatives. 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 explicitlyconsider 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 κ of the form κ → λκ + (1 − λ ) leaves the shapeof all observed images invariant. In usual weak lensing, this is accounted for by recognizing that not the shear γ can beobtained from observations, but only the reduced shear g = γ/ (1 − κ ) (Schneider & Seitz 1995). The difference of shearand reduced shear is typically small, in particular in applications of cosmic shear, since along most lines-of-sight, thevalue of κ is very much smaller than unity. In applications of flexion, however, we expect that the surface mass densityno longer is very small; for instance, arclets occur in the inner parts of clusters where κ > ∼ .
1. Therefore, the differencebetween shear and reduced shear can no longer be neglected. Gradients of the shear are not directly observable; onlyderivatives of the reduced shear are, and thus we define the (reduced) flexion in terms of derivatives of g . In Sect. 2.1we briefly recall the irreducible tensor components which are defined in term of their behavior under rotations of thecoordinate system. It turns out that a complex notation for these tensor components is very useful. In Sect. 2.2 weexpand the lens equation to second order, before deriving the corresponding lens equation (and relation for the localJacobian) which is invariant under mass-sheet transformations. The second-order term in this lens equation is fullydescribed by our reduced flexion components G and G .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 κ field.Therefore, shear fields are decomposed into E- and B-modes (Crittenden et al. 2002; Schneider et al. 2002), wherethe former are due to a κ field, whereas the latter describes the remaining (“curl”) part. A similar situation occurs inflexion, which has four components. Therefore, in Sect. 3 we consider the decomposition of a general flexion field intocontributions due to the gradient of the shear and those not related to the shear field. The former one can then befurther subdivided into flexion resulting from an E- and B-mode shear field. We carry out this decomposition for theflexion as well as for the reduced flexion.In Sect. 4 we then define brightness moments of sources and images and derive the transformation laws betweenthem. This approach is very similar to the HOLICs approach developed by Okura et al. (2006) and later consideredby Goldberg & Leonard (2007), except that we explicitly write all relations in terms of the reduced shear and thereduced flexion. Generalizing the usual assumption that the expectation value of the source ellipticity is zero – dueto the phase averaging over source orientations – to the expectation values of all source shape parameters which arenot invariant under coordinate rotations (as appropriate for a statistically isotropic Universe), we obtain in Sect. 5estimates for the reduced shear and reduced flexion in terms of the brightness moments of the images. In Sect. 6 weperform a number of numerical experiments to test the validity of our approach and the accuracy of the estimatorsderived. 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 ofthe 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 – thecaustic introduces a preferred orientation into the source plane. In appendix B we provide a full classification of thecritical curves of the second-order lens equation and use these results in order to obtain the maximum source size (forgiven values of the reduced flexion) for which the flexion concept still makes sense. We discuss our results in Sect. 7.
2. Complex lensing notation
Like in many other instances in weak lensing, flexion is best described by using complex notation, which we shallbriefly introduce next and which will be used for vectors and tensor components throughout this paper.
For a two-dimensional vector x = ( x , x ), we define the complex number x = x +i x . Under rotations of the coordinatesystem by an angle ϕ , x gets multiplied by the phase factor e − i ϕ . For a tensor of second rank, whose Cartesiancomponents are Q ij , we define the complex numbers Q = Q − Q +i( Q + Q ) and Q = Q + Q +i( Q − Q ).A rotation of the coordinate systems by an angle ϕ multiplies Q by the phase factor e − ϕ , whereas Q remainsunchanged. This is most easily seen by considering that the prototype of a second rank tensor is Q ij = x i y j , where x and y are vectors; the foregoing statements are then obtained by noting that the complex numbers xy and x ∗ y eter Schneider, Xinzhong Er: What flexion really measures 3 are multiplied by e − ϕ and 1, respectively, under coordinate rotations. According to this transformation behavior, weshall loosely speak about Q as a spin-0 quantity, whereas x and Q are spin-1 and spin-2 quantities, respectively.We shall be dealing only with totally symmetric tensors. If Q ij is symmetric, then Q := Q − Q + 2i Q ; Q := Q + Q . (1)If T ijk is a symmetric third-rank tensor, we define its spin-3 and spin-1 components as T := T − T + i (3 T − T ) ; T := T + T + i ( T + T ) . (2)Furthermore, if F ijkl denotes a symmetric fourth-rank tensor, we decompose it into its spin-4, spin-2 and spin-0components, respectively, F := F − F + F +4i ( F − F ) ; F := F − F +2i ( F + F ) ; F := F +2 F + F . (3)Apart from notational simplicity, the complex lensing notation provides a check for the validity of equations. In a validequation, each term has to have the same spin. The product of a spin- m and a spin- n quantity has spin m + n . Thecomplex conjugate of a spin- n quantity has spin − n . In weak lensing, the lens equation is linearized locally by writing the relative source coordinate β in terms of theimage position θ as β i = θ i − ψ ,ij θ j , where ψ is the deflection potential, indices separated by a comma denote partialderivatives with respect to θ i , and summation over repeated indices is implied. Note that the form of this equationimplies that the origin of the lens plane, θ = 0, is mapped onto the origin of the source plane. The surface massdensity κ and the complex shear γ at the origin are given in terms of the deflection potential, κ = ( ψ , + ψ , ) / γ = ( ψ , − ψ , ) / ψ , , being spin-0 and spin-2 fields, respectively. In our complex notation, the locally linearizedlens equation reads β = (1 − κ ) θ − γθ ∗ . (4)We next generalize this result to a second-order local expansion of the lens equation, which in Cartesian coordinatesreads β i = θ i − ψ ,ij θ j − ψ ,ijk θ j θ k /2. The third-order derivatives of ψ are related to the gradient of κ and γ . To writethese derivatives also in complex form, we define the differential operators ∇ c := ∂∂θ + i ∂∂θ ; ∇ ∗ c := ∂∂θ − i ∂∂θ . (5)The differential operator ∇ c turns a spin- n field into a spin-( n + 1) field, whereas ∇ ∗ c reduces the spin by one unit.One finds, for example, ∇ c κ = 12 [ ψ , + ψ , + i ( ψ , + ψ , )] ; ∇ c γ = 12 [ ψ , − ψ , + i (3 ψ , − ψ , )] ; ∇ ∗ c γ = ∇ c κ , (6)and we recognize the combinations of third derivatives of ψ which form the spin-1 and spin-3 combinations defined in(2). The final relation in (6) is the relation between first derivatives of κ and γ found by Kaiser (1995), here expressedin compact form. It expresses the fact that the third-order derivatives of the deflection potential can be summarizedin the spin-3 field G ≡ ∇ c γ and the spin-1 field F ≡ ∇ ∗ c γ , where we introduced the usual notation for the two flexionquantities. The second-order lens equation in our complex notation then reads β = (1 − κ ) θ − γθ ∗ − F ∗ θ − F θθ ∗ − G ( θ ∗ ) . (7)Since this is no longer a linear equation, a source at β may have more than one image. In fact, up to four images ofa source can be obtained, as can be seen for the special case of γ = 0 = F and by placing the source at β = 0. Inthis case, if we set G = |G| e ζ , then one solution is θ = 0, and the other three are θ = 4(1 − κ ) / |G| e i ϕ , with ϕ = ζ , ϕ = ζ + 2 π/ ϕ = ζ + 4 π/
3. Of course, the origin for the occurrence of these solutions lies in the fact that G is aspin-3 quantity. We shall later need the Jacobian determinant det A of this lens equation, which isdet A = (1 − κ ) − γγ ∗ + θ · ∇ (cid:2) (1 − κ ) − γγ ∗ (cid:3) + O ( θ )= (1 − κ ) − γγ ∗ − θ (cid:20) (1 − κ ) F ∗ + γ ∗ F + γ G ∗ (cid:21) − θ ∗ (cid:20) (1 − κ ) F + γ ∗ G + γ F ∗ (cid:21) + O ( θ ) , (8)where the first expression is just the first-order Taylor expansion of the Jacobian around the origin, and in the secondstep we made use of the relation θ · ∇ = ( θ ∇ ∗ c + θ ∗ ∇ c ) /
2. We point out that (8) is not the full Jacobian of the lensequation (7), but only its first-order expansion; the full Jacobian contains quadratic terms in θ . We will return to thisimportant issue further below. Peter Schneider, Xinzhong Er: What flexion really measures
The observables of a gravitational lens system are unchanged if the surface mass density κ is transformed as κ ( θ ) → κ ′ ( θ ) = λκ ( θ ) + (1 − λ ) (Gorenstein et al. 1988). In the case of weak lensing, the shape of images is unchanged underthis transformation (Schneider & Seitz 1995). Because of this mass-sheet degeneracy, not the shear is an observable inweak lensing, but only the reduced shear g = γ/ (1 − κ ). In fact, since we expect that the most promising applicationsof flexion will come from situations where κ is not much smaller than unity, the distinction between shear and reducedshear is likely to be more important for flexion than for the usual weak lensing applications. Hence, at best we canexpect from higher-order shape measurements to obtain an estimate for the reduced shear and its derivatives. For thisreason, 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, wedivide (7) by (1 − κ ) to obtainˆ β ≡ β (1 − κ ) = θ − gθ ∗ − Ψ ∗ θ − θθ ∗ − Ψ ( θ ∗ ) with Ψ = 14 F (1 − κ ) ; Ψ = 14 G (1 − κ ) . (9)We will now express the coefficients in the lens equation (9) in terms of the derivatives of the reduced shear, G ≡ ∇ ∗ c g = F + g F ∗ (1 − κ ) ; G ≡ ∇ c g = G + g F (1 − κ ) . (10)The expression for F / (1 − κ ) in terms of the reduced shear and its derivatives has been derived by Kaiser (1995); inour notation it reads F (1 − κ ) ≡ −∇ c ln(1 − κ ) = G − gG ∗ − gg ∗ ⇒ Ψ = G − gG ∗ − gg ∗ ) . (11)The expression for the derivative of γ in terms of the reduced shear can be easily obtained from differentiating thedefinition γ = (1 − κ ) g , ∇ c γ (1 − κ ) = G (1 − κ ) = G − g ∇ c κ (1 − κ ) = G − g ( G − gG ∗ )1 − gg ∗ ⇒ Ψ = G − g ( G − gG ∗ )4 (1 − gg ∗ ) . (12)The derivatives G , of the reduced shear are those quantities we can hope to observe; to distinguish them from F and G , one might call G , the reduced flexion. The Jacobian determinant det ˆ A of the mapping between the image position θ and the rescaled source position ˆ β then becomesdet ˆ A = det A (1 − κ ) = 1 − gg ∗ − η ∗ θ − ηθ ∗ , where η = ∇ ∗ c g − g ( ∇ ∗ c g ) ∗ g ∗ ∇ c g G − gG ∗ g ∗ G θ . Note that a similar equation for the determinant wasobtained in Okura et al. (2007; their eq. A1), but they consider only the case of | g | ≪
1; this has also consequencesfor the relations between source and image brightness moments, to be derived further below.
3. Compatibility relations
Flexion has a total of four components, namely the real and imaginary parts of F and G . A measurement of flexionwill thus yield four components, and we might ask whether these components are independent. We recall a similarsituation in shear measurements. The shear has two components; on the other hand, the shear is defined as secondpartial derivatives of the deflection potential, which is a single scalar field. Therefore, the two shear componentscannot be mutually independent if they are due to a gravitational lensing signal. Of course, the measured shear isnot guaranteed to satisfy the condition that the two shear components can be derived from a single scalar deflectionpotential, 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 shearis 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 ψ ( θ ) = ψ E ( θ ) + i ψ B ( θ ) and a complex surface mass density κ = κ E + i κ B (Schneider et al. 2002). Each component of ψ satisfies its own Poisson equation, ∇ ψ E = 2 κ E , ∇ ψ B = 2 κ B . Making use of this decomposition, the shear becomes γ = γ + i γ = ( ψ , − ψ , ) / ψ , = (cid:20) (cid:0) ψ E , − ψ E , (cid:1) − ψ B , (cid:21) + i (cid:20) ψ E , + 12 (cid:0) ψ B , − ψ B , (cid:1)(cid:21) . (14) eter Schneider, Xinzhong Er: What flexion really measures 5 The distinction between E- and B-mode shear can be obtained by considering second partial derivatives of the shearcomponents. Taking the derivative of (14), one obtains F = ∇ ∗ c γ = (1 / (cid:0) ψ E , + ψ E , − ψ B , − ψ B , (cid:1) + (i / (cid:0) ψ E , + ψ E , + ψ B , + ψ B , (cid:1) = κ E , − κ B , + i (cid:0) κ E , + κ B , (cid:1) , (15)which can be expressed in more compact form as F = ∇ c (cid:0) κ E + i κ B (cid:1) = ∇ c κ . (16)A further derivative yields for the components F , = κ E , − κ B , ; F , = κ E , − κ B , ; F , = κ E , + κ B , ; F , = κ E , + κ B , . (17)However, it is easier to consider directly the complex derivative of F , from which we obtain ∇ ∗ c F = ∇ ∗ c ∇ ∗ c γ = ∇ (cid:0) κ E + i κ B (cid:1) = F , + F , + i ( F , − F , ) . (18)Thus, if the shear field is a pure E-mode field, ∇ ∗ c ∇ ∗ c γ is real. An imaginary part of ∇ ∗ c ∇ ∗ c γ is due to a B-mode field.This then yields the local distinction between E- and B-mode shear.Since the flexion has four components, whereas the lens can be described by a single scalar field, we expect thatthere are three constraint relations a flexion field has to satisfy if it is due to a lensing potential. In fact, even if weleave the shear field arbitrary (that is, even if we allow it to be composed of E- and B-modes), then we expect twoconstraint equations, since the flexion field has two components more than the shear field. These constraint equationsare easy to obtain. First, if the flexion field is due to a shear field, then we have ∇ c ∇ ∗ c γ = ∇ ∗ c ∇ c γ → H := ∇ c F − ∇ ∗ c G = 0 , (19)where we defined the spin-2 quantity H . It may describe contributions to the flexion which are not caused by ashear field, such as due to noise, intrinsic source alignments or higher-order terms (such as lens-lens coupling) in thepropagation equation for light bundles. As a spin-2 field, a non-zero H can be decomposed into its E- and B-modes. If H ≡
0, then the spin-3 flexion G is completely determined by the spin-1 flexion F up to an additive constant, as canbe best seen in Fourier space, for which (19) yields ˆ G ( ℓ ) = − iˆ γ ( ℓ ) ℓ = ( ℓ/ℓ ∗ ) ˆ F ( ℓ ). Second, if the flexion field is solelycaused by a gravitational lens effect, i.e., by a pure E-mode shear field, then ∇ ∗ c F is real, i.e., F i := ∇ ∗ c F − ∇ c F ∗ = 0 . (20)Thus, flexion from a pure E-mode shear field is characterized by the three constraint equations H ≡ F i ≡ H := ∇ c G − ∇ ∗ c G = 0 , (21)as follows from the definition (10) of the two flexion components in terms of the reduced shear. Again, if this equationis satisfied, G is completely determined by G , up to an additive constant. Second, if the flexion is caused by a pureE-mode shear, i.e., if the shear is due to a real surface mass density, then we employ the quantity ln(1 − κ ), which isreal and invariant under mass-sheet transformations, up to an additive constant. Therefore, K ≡ −∇ ∗ c ∇ c ln(1 − κ )must be real. We find: K = ∇ ∗ c (cid:18) ∇ c κ − κ (cid:19) = ∇ ∗ c (cid:20) − gg ∗ ( G − G ∗ g ) (cid:21) = (cid:2) ∇ ∗ c G − g ( ∇ c G ) ∗ (cid:3) − gg ∗ + (cid:0) G g ∗ + gG G ∗ − G G ∗ − g G ∗ G ∗ (cid:1) (1 − gg ∗ ) , (22)so that a flexion coming from an E-mode shear field satisfies K = K ∗ .We point out that the foregoing relation suggests a natural way to use flexion for finite-field mass reconstructionsin weak lensing. Seitz & Schneider (2001) formulated the finite-field mass reconstruction from measured reduced shearin terms of a von Neumann boundary value problem for K = − ln(1 − κ ), whose solution determines K up to anadditive constant. The ‘source’ for ∇ K was determined by the reduced shear and its derivatives, and is given by (22).In Seitz & Schneider (2001), the derivatives of the reduced shear were obtained by finite differencing of g . If flexion ismeasured, one can replace the ‘source’ for ∇ K by a weighted sum of the differentiated reduced shear field and thecombination ( K + K ∗ ) / Let H ( θ ) be any spin-2 field, and denote by H E and H B the E- and B-mode components of H . They can be obtained from H most easily in Fourier space, namelyˆ H E ( ℓ ) = 12 (cid:2) ˆ H ( ℓ ) + ˆ H ∗ ( − ℓ ) e β (cid:3) ; ˆ H B ( ℓ ) = 12 (cid:2) ˆ H ( ℓ ) − ˆ H ∗ ( − ℓ ) e β (cid:3) , as can be best seen by taking the shear field as a prototypical spin-2 field; here, β is the phase of the complex wave number ℓ . Peter Schneider, Xinzhong Er: What flexion really measures
4. Brightness moments of source and image
We consider an image of a source, and denote the brightness distribution of the source by I s ( β ). Since surface brightnessis conserved by lensing, the brightness distribution of the image is I ( θ ) = I s ( β ( θ )). Since the scaling of the sourceplane is unobservable, we shall only work in the following in terms of the scaled source plane coordinates, and thereforedrop the hat on β , as well as on A .We define the origin of the image (or lens) plane as the center-of-light of the image under consideration, i.e. werequire Z d θ θ I ( θ ) = 0 . (23)Let F ( β ) be a function of the source coordinate; we define the operator Mom[ F ( β )] asMom[ F ( β )] = Z d β F ( β ) I s ( β ) = Z d θ det A ( θ ) F ( β ( θ )) I ( θ ) ≈ Z d θ (1 − gg ∗ − η ∗ θ − ηθ ∗ ) F ( β ( θ )) I ( θ ) , (24)where here and in the following, we use the linear approximation for det A . In particular, setting F = 1, one finds thatMom[1] ≡ S = Z d β I s ( β ) = Z d θ (1 − gg ∗ − η ∗ θ − ηθ ∗ ) I ( θ ) = (1 − gg ∗ ) S = det A S , (25)since first-order moments of the light distribution in the lens plane vanish, due to our choice (23) of the coordinatesystem. Here, S is the flux of the lensed image, so that S = S / det A , as usual, where det A is the Jacobian at theorigin θ = 0. The origin of the coordinates in the source plane is the image of the origin in the lens plane as mapped with the lensequation. In particular, this does not coincide with the center-of-light of the source, which is given by ¯ β ≡ Mom[ β ] /S ,or¯ β = 1 S Z d β β I s ( β ) = 1 S (1 − gg ∗ ) Z d θ (1 − gg ∗ − η ∗ θ − ηθ ∗ ) (cid:2) θ − gθ ∗ − Ψ ∗ θ − θθ ∗ − Ψ ( θ ∗ ) (cid:3) I ( θ ) . (26)Expanding the integrand, we note that terms linear in θ vanish, due to (23). Defining the second-order brightnessmoments of the image in the form Q ≡ S Z d θ θ I ( θ ) ; Q ≡ S Z d θ θ θ ∗ I ( θ ) , (27)we obtain for the source centroid shift¯ β = 3 G g ∗ − G ∗ − gG ∗ − gg ∗ ) Q + 4 gG ∗ + g G ∗ − G g ∗ − G (3 + gg ∗ )2(1 − gg ∗ ) Q + 5 gG − g G ∗ − (1 − gg ∗ ) G − gg ∗ ) Q ∗ . (28)We now write these equations in a more compact form; for this, we define the matrix G by G T = ( G ∗ , G ∗ , G , G ),where the ‘T’ denotes the transpose of the matrix. Then,¯ β = BG , (29)where the coefficients of B = ( b , b , b , b ) are given by b = g Q − gQ − gg ∗ ) ; b = 8 gQ − Q − g Q ∗ − gg ∗ ) ; b = 3 g ∗ Q − gg ∗ ) Q + 5 gQ ∗ − gg ∗ ) ; b = (3 gg ∗ − Q ∗ − g ∗ Q − gg ∗ ) . (30)The centroid shift in the source plane is thus given by the product of the derivatives of the reduced shear (expressedby G and G ) and the area of the image, which is proportional to Q and Q . Of course, since the reduced shearand its derivatives are not directly observable, the centroid shift in unobservable as well. To get an order-of-magnitudeestimate of ¯ β , we assume that the source has a linear angular size Θ s , consider the reduced shear to be of order unity,and let Θ c be the angular scale on which the reduced shear varies. Then, G n = O (cid:18) c (cid:19) ; Q n = O (cid:0) Θ (cid:1) ⇒ ¯ β = O (cid:18) Θ Θ c (cid:19) . (31) eter Schneider, Xinzhong Er: What flexion really measures 7 Next we consider the second-order brightness moments of the source, defined as Q s2 = Mom[( β − ¯ β ) ] /S =Mom[ β ] /S − ¯ β and Q s0 = Mom[( β − ¯ β )( β − ¯ β ) ∗ ] /S = Mom[ ββ ∗ ] /S − ¯ β ¯ β ∗ . By defining the third-order brightnessmoments of the image through T ≡ S Z d θ θ I ( θ ) ; T ≡ S Z d θ θ θ ∗ I ( θ ) , (32)we obtain Q s2 = Q − gQ + g Q ∗ + 2 g ∗ G − G ∗ − gG ∗ − gg ∗ ) T + 8 gG ∗ − (4 + 3 gg ∗ ) G − g ∗ G + 2 g G ∗ − gg ∗ ) T + (7 + gg ∗ ) gG − g G ∗ + (3 gg ∗ − G − g G ∗ − gg ∗ ) T ∗ + (1 − gg ∗ ) gG − g G + 2 g G ∗ − gg ∗ ) T ∗ − ¯ β , (33) Q s0 = − g ∗ Q + (1 + gg ∗ ) Q − gQ ∗ + 6 g ∗ G ∗ + (3 gg ∗ − G ∗ − g ∗ G − gg ∗ ) T + 2 g ∗ G + (11 + 3 gg ∗ ) g ∗ G − (7 + 9 gg ∗ ) G ∗ − (1 + 3 gg ∗ ) gG ∗ − gg ∗ ) T + 2 g G ∗ + (11 + 3 gg ∗ ) gG ∗ − (1 + 3 gg ∗ ) g ∗ G − (7 + 9 gg ∗ ) G − gg ∗ ) T ∗ + 6 gG − g G ∗ − (1 − gg ∗ ) G − gg ∗ ) T ∗ − ¯ β ¯ β ∗ (34)Note that Q s0 is real. In a more compact notation, (33) reads Q s2 = Q − gQ + g Q ∗ + A G − ¯ β , (35)where the matrix A = ( a , a , a , a ) has coefficients a = − g T ∗ + 2 g T − gT − gg ∗ ) ; a = − g T ∗ + g (7 + gg ∗ ) T ∗ − (4 + 3 gg ∗ ) T + 2 g ∗ T − gg ∗ ) ; a = 2 g T ∗ − g T ∗ + 8 gT − T − gg ∗ ) ; a = g (1 − gg ∗ ) T ∗ − (1 − gg ∗ ) T ∗ − g ∗ T − gg ∗ ) . (36) We now define the third-order brightness moments of the source, separated into a spin-3 and a spin-1 component, T s3 = Mom[ (cid:0) β − ¯ β (cid:1) ] S = Mom[ β ] S − β Mom[ β ] S + 3 ¯ β Mom[ β ] S − ¯ β = Mom[ β ] S − βQ s2 − ¯ β , (37)where we used that Mom[ β ] /S = Q s2 + ¯ β and Mom[ ββ ∗ ] /S = Q s0 + ¯ β ¯ β ∗ . Similarly, we obtain T s1 = Mom[ (cid:0) β − ¯ β (cid:1) ( β ∗ − ¯ β ∗ )] S = Mom[ β β ∗ ] S − Q s0 ¯ β − Q s2 ¯ β ∗ − ¯ β ¯ β ∗ . (38)Defining the fourth-order brightness moments of the image by F = 1 S Z d θ ( θθ ∗ ) I ( θ ) ; F = 1 S Z d θ θ θ ∗ I ( θ ) ; F = 1 S Z d θ θ I ( θ ) , (39)where F n is a spin- n quantity, we obtain for the third-order moments of the source: T s = τ + C G + O ( ¯ β ) , (40)where the matrix T s is defined by its transpose T s , T = ( T s ∗ , T s ∗ , T s , T s ). The elements of τ are τ = T ∗ − g ∗ T ∗ + 3 g ∗ T − g ∗ T ; τ = − gT ∗ + (1 + 2 gg ∗ ) T ∗ − g ∗ (2 + gg ∗ ) T + g ∗ T ; τ = τ ∗ ; τ = τ ∗ , (41)where the last two relations are obvious. The 4 × C is given explicitly in Appendix A; each of its elementsconsists of a sum of terms proportional to fourth-order brightness moments, F n , and terms proportional to squaresof second-order brightness moments. Okura et al. (2007) and Goldberg & Leonard (2007) have derived expressions Peter Schneider, Xinzhong Er: What flexion really measures similar to (40), though using a number of simplifying assumptions (such as | g | ≪
1) and (in the latter paper), notconsidering the reduced flexion.We will now consider the order-of-magnitudes of the various terms appearing in (35) and (40). Assuming that thethird-order moments of the sources are small, then the third-order moments of the image are given by the product of C and G . With G = O (1 / Θ c ) and C = O (cid:0) Θ (cid:1) , we find that T = O (cid:0) Θ / Θ c (cid:1) = O (cid:0) Θ (cid:1) (Θ s / Θ c ). To get an estimate ofthe size of the various terms in (35), we note that the first three terms on the right-hand side (those proportional tothe Q n ) are of order O (cid:0) Θ (cid:1) , whereas AG = O (cid:0) Θ / Θ c (cid:1) O (1 / Θ c ) and ¯ β = O (cid:0) Θ / Θ (cid:1) . Hence, the last two terms areof equal magnitude in general, each of them being smaller than the first three terms by a factor (Θ s / Θ c ) . Only if thesource is of the same order as the scale over which the reduced shear varies do the last two terms in (35) contribute.In (40), we have neglected the terms ¯ β , since they are two powers of (Θ s / Θ c ) smaller than the terms written down.
5. Shear and flexion estimates
We see that (40) is a linear equation for G , which can thus be solved, G = C − ( T s − τ ) . (42)Inserting this into (35) then yields Q s2 = Q − gQ + g Q ∗ + A C − ( T s − τ ) − ( BG ) . (43)We are thus left with a single complex equation for g , which contains the observable brightness moments of the image,as well as the unobservable brightness moments of the source. This equation can be used to estimate the reduced shearif we make assumptions concerning the properties of the source brightness moments. We assume that the sources areoriented randomly, which implies that all quantities with spin unequal zero have a vanishing expectation value. Thatis, we set Q s2 = 0, T s = 0, to arrive at Q − gQ + g Q ∗ = A C − τ + (cid:0) BC − τ (cid:1) =: Y ( g ) , (44)where we have indicated that the right-hand side depends on the reduced shear (in fact it does so in a very complexmanner). However, since we have argued above that the terms on the left-hand are much larger than those on theright-hand side, an iterative solution of this equation is suggested. Assume the right-hand side is given, then we getthe solutions g = χ | χ | ± s − | χ | + Y χ ∗ Q ! , where χ = Q Q (45)is the complex ellipticity of the image. Obviously, there are two solutions g for a given value of Y . This situation issimilar to that of ‘ordinary’ weak lensing, where this ambiguity also occurs: as shown by Schneider and Seitz (1995),from shape measurements of background galaxies, and cannot distinguish locally between an estimate g and 1 /g ∗ = g/ | g | . The same occurs here; we therefore assume that we pick one of the two solutions, say the one correspondingto the ‘ − ’ sign; this then yields for small shear g ≈ χ/
2. It should be stressed that flexion impacts the determinationof shear from the second-order brightness moments, due to its impact on higher-order brightness moments; hence, ingeneral the determination of shear and flexion are coupled.We start the iteration by setting Y = 0. This yields a first-order solution for the estimate of g , g = χ | χ | (cid:16) − p − | χ | (cid:17) . (46)We then use the iteration equations Y n = Y ( g n − ) ; g n = χ | χ | − s − | χ | + Y n χ ∗ Q ! . (47)This procedure converges quickly to one of the two solutions ( g, G , G ); the other solution is obtained by taking the‘+’ sign in the above equations.Of course, our approach of setting Q s2 = 0 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 Q s2 vanishes, the resulting estimatorfor g is a non-linear function of χ s = Q s2 /Q s0 and thus biased. The bias depends on the ellipticity distribution of thesources. It should be stressed, however, that a modified definition of image ellipticity exist such that its expectationvalue is an unbiased estimate of the reduced shear (Seitz & Schneider 1997). eter Schneider, Xinzhong Er: What flexion really measures 9 The flexion estimator is given by (42). Since the matrix C contains many terms, this is a fairly complicated equationin general. A simpler estimate is obtained if we assume that the reduced shear is small, | g | ≪
1, in which case thematrix C simplifies considerably – see Appendix. Furthermore, if we assume that the brightness moments of spin = 0are much smaller than the corresponding ones with spin 0, then we find the simple relations T s1 ≈ T − F − Q G ; T s3 ≈ T − F G . (48)If we then set the T s n = 0, as would be true for the expectation value, then we obtain as estimates for the reducedflexion G ≈ F − Q T ; G ≈ F T . (49)Thus, the flexion is then given by the third-order brightness moments of the image, divided by a quantity that justdepends on the size of the image. Similar relations to (49) have been given in Goldberg & Leonard (2007), whereasOkura et al. (2007) obtain a different expression for G . We will check the accuracy of (49) in Sect. 6 below.A more accurate estimate is obtained if we consider the reduced shear as well as the ratios of non-zero spinbrightness moments to zero spin moments (such as | Q /Q | or | F , /F | ) to be of order δ , and then expand the flexionto first order in the (small) parameter δ to obtain G = 4 T F − Q + 4 [2 F ∗ + 3 F g ∗ − Q (2 g ∗ Q + Q ∗ )]9 F (4 Q − F ) T + 4 [3 F g − F − Q ( gQ − Q )]9(3 F − Q ) T ∗ + 4 F T ∗ F (4 Q − F ) ,G = 4 T F + 8(5 F − Q Q ) T F (4 Q − F ) + 28 F T ∗ F (4 Q − F ) . (50)
6. Numerical tests of flexion estimators
In this section we describe some simulations that we have performed in order to test the behavior of the estimatorsgiven in the previous section.
We model the sources as elliptical Gaussians, truncated at three times the scale ‘radius’ Θ s chosen such that thearea of a source is independent of its ellipticity. The ellipticity of the sources follows a Gaussian distribution, witha dispersion of χ s of R = 0 . | χ s | ≤ .
9. For eachsource, we map a grid of pixels from the lens plane to the source plane using the lens equation to obtain the brightnessdistribution in the lens plane. From this distribution, the brightness moments of the image are measured. A shift in thelens plane coordinates is applied as to satisfy (23). We then apply the shear and flexion estimators described above tothe resulting brightness moments Q n , T n and F n . The shear and flexion estimates are then averaged over the Gaussianellipticity distribution of the sources, in particular over their random orientation.It should be noted that flexion is a dimensional quantity ∝ Θ − . As can be checked explicitly from Sect. 4, the wayflexion appears in the equations is always with one order higher in the source (or image) size than the other terms inthe equations. As an example, we consider (40); the left-hand side and the first term on the right-hand side are ∝ Θ ,whereas the coefficients of the matrix C ∝ Θ . This then implies that the accuracy of the flexion estimates does notdepend on the magnitude of the flexion and the source size individually, but only on the product G n Θ s . Therefore,the following results are quoted always in terms of this product. As we mentioned before, the lens equation (7) can give rise to multiple images. As can be seen from the example givenafter (7), if the flexion is sufficiently small, all but one of these multiple images will be located at a large distancefrom the origin, and the central image of an extended source will be isolated. In this case, this central, or primary,image (the shape of which we measure here) is not crossed by a critical curve, and thus the source is not crossed bya caustic. The multiple images at large distances from the origin then result from the low-order Taylor expansion ofthe lens equation, which most likely breaks down at these image positions anyway; hence, these additional images areof no relevance. If, however, the flexion becomes sufficiently large – or if the source is large enough – this is no longer G Θ G Θ phase G ,G =0 00.050.10.20.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 G Θ G Θ phase G ,G = π /4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 G Θ G Θ phase G =0, phase G = π /2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 G Θ G Θ phase G = π /3, phase G = π /6 Fig. 1.
Constrains on the combination of source size and reduced flexion for the validity of the concept of flexion. Each curveshows the dividing line between a circular source of limiting isophote Θ being cut by a caustic (above the curve) or not (belowthe curve); in the former case, the assumptions underlying the flexion concept break down. The different curves in each panelare for different values of g , chosen as g = 0 . , . , . , . ,
0, as indicated by different line types. Without loss of generality, wechoose g to be real and non-negative. The four panels differ in the phase of the reduced flexion, as indicated. E.g., in the upperleft panel, the phases of G , G are the same as that of g the case, and the multiple images of an extended source will merge. If that happens, the whole method of determiningshear and derivatives thereof from brightness moments will break down. This can be most easily seen by consideringthe caustic curve cutting the source. Different parts of the source will be mapped onto a different number of imagepoints in the lens plane, and the caustic curve introduces a direction into the situation. Hence, the assumption ofan isotropic orientation of sources can no longer be employed. Mathematically, this can be seen from (24); there, thetransformation between source and image plane no longer is correct if multiple images do occur. More precisely, thetransformation between source and image coordinates in the calculation of the brightness moments implicitly assumesthat within the limiting isophote of the primary image, the lens equation is invertible. Owing to what was said above,the condition that the central image is isolated (so that locally no multiple images occur) can be expressed solelyby the products G n Θ s . These products approximately measure the fractional change of the reduced shear across theimage of a source.In our simulations we can check whether a critical curve crosses our central image, just by controlling the sign of theJacobian determinant (the true one, not the linear approximation eq. 13). If the source size becomes too large, somepoints in the image will have a negative Jacobian. In the Appendix B, we consider the critical curves and caustics ofthe lens equation (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. 1. Each panel shows the dividing line between parameter pairs ( G Θ , G Θ)for a circular source of limiting isophotal radius Θ; below the curves, no local multiple images occur, whereas forparameter pairs above the lines, the flexion formalism using moments necessarily breaks down. The different lines ineach panel correspond to different values of g . The occurrence of critical curves also is the reason why we truncated eter Schneider, Xinzhong Er: What flexion really measures 11 Fig. 2.
Accuracy of the estimates for reduced shear and flexion. The left panel shows contour of constant fractional error of5%, 10% and 15%, on the estimate of the reduced shear g , as a function of G i Θ s , where we chose g = 0 .
05 as input value, andassumed the phases of G , G to be the same as that of g . The estimate was obtained by solving the iteration equations givenin Sect. 5. The right panel shows the fractional error levels at 3, 5, and 10% for the reduced flexion, as quantified by (51), wherethe estimate was obtained again with the iterative procedure. In both cases, we assumed circular sources the intrinsic ellipticity distribution of the sources in the simulations, since in the limit of | χ s | →
1, keeping the sourcearea fixed, there will be orientation angles for which the source will hit a caustic.
We now present some results of our numerical simulation regarding the accuracy with which the reduced shear andflexion can be obtained with our moment approach. For given input values of g , G and G , we either measurethe brightness moments for a single circular source, or average the results over an ellipticity distribution, as describedabove. It should be noted that we have to deal with a 5-dimensional parameter space, namely the 3 complex parameters g , G and G , minus one overall phase that can be chosen, e.g., to make g real and positive. Thus, instead of samplingthe parameter space comprehensively, we only give a few selected results.We start by considering a circular source, and determine the effect of flexion on the determination of the reducedshear. The left-hand panel of Fig. 2 shows contours of constant fractional deviation ∆ g/g , in the flexion parameterplane. Here it is assumed that the phase of both flexion components is the same as that of g (as would be the case inan axially-symmetric lens potential). Errors of order 5% occur already for p | G + G | Θ s ∼ .
03, and the fractionalerror increases approximately linearly with the strength of flexion (or with the source size), although it does notscale equally with both flexion components. The reason for this effect has been mentioned before – flexion affects thetransformation between source and image quadrupole moments, as can be seen in (33).In Fig. 3, 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 describedbefore. The left and right panel show the behavior of the expectation value of G and G , respectively, where theother flexion component was set to zero. The dashed curve shows the identity, the plus symbols were obtained byusing the approximate estimator (49), whereas the crosses show the expectation values as obtained by employing thefull expression (42), where the corresponding value of g was obtained by the iterative process described in Sect. 5. Itis 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 (49) performs remarkably well. It is seen that the estimates for G behave better than those for G . This can also be seen from the right-hand panel of Fig. 2, where we plot contoursof constant fractional error∆ G := s(cid:12)(cid:12)(cid:12)(cid:12) ∆ G G (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∆ G G (cid:12)(cid:12)(cid:12)(cid:12) , (51)where ∆ G n is the deviation of the estimate of G n from its input value. For simplicity, we have assumed a circularsource. We see that the accuracy decreases much faster with increasing G than with increasing G . The reason forthat may be related to the fact that the estimator of G is more strongly affected by the non-linearity of the equations,as can also be seen in (50). Fig. 3.
Comparison of the reduced flexion estimators (49) with the full expression (42) and the input value. The horizontal andvertical axis show G i Θ s , i = 1 ,
3. For both panels, we take g = 0 .
05, and G = 0 ( G = 0) for the left (right) panel. The lineindicates the input value, the plus symbols show the simplified reduced flexion estimate (49), and the crosses result from thefull expression of reduced flexion (42). As can be seen from the left-hand panel, the full estimator for the reduced flexion yieldsa more biased result that the approximate expression (49); we have not found a reasonable explanation for this behavior
7. Conclusions and further work
In this paper, we have studied the effect of flexion in weak gravitational lensing. The main results are summarized asfollows: – Owing to the mass-sheet degeneracy, flexion itself cannot be determined, but only reduced flexion. We have thereforewritten the second-order lens equation (which contains the derivatives of the reduced shear, i.e., flexion) as well asthe relations between the brightness moments of source and image strictly in terms of the reduced shear and thereduced flexion. – We pointed out that a general flexion field can be decomposed into a pair of components which is due to a shearfield, i.e., its derivatives, and a pair of components not related to shear. The former pair can be further separatedinto flexion due to an E- and B-mode shear, with only the E-mode flexion expected to arise from gravitationallensing. For the second pair of components, no physical interpretation is available; if they arise in measurements,they are most likely due to noise or intrinsic shape effects of sources. General relations to separate these componentsare given. – We derive the relations between low-order brightness moments of source and image, taking into account thatthe presence of flexion leads to a centroid shift, and it also affects the relation between second-order brightnessmoments – and thus the estimate of the reduced shear. Hence, the presence of flexion has an impact on the shearmeasurements. Starting from these moment equations, we obtain approximate estimates for the reduced shear andflexion. – We point out a limit where the flexion formalism ceases to be valid, namely when the product of source size andflexion is sufficiently large that parts of the source are multiply imaged locally, i.e., where a caustic cuts throughthe source. We have quantified this with numerical simulations, and also have given a complete classification of thecritical curves of the second-order lens equation employed in flexion studies. – We have performed a number of numerical experiments to study the bias of the reduced shear and flexion estimators.However, due to the high dimensionality of parameter space, no comprehensive study has been presented here. Wealso point out that only the product of flexion and source size matters in the accuracy of estimates.The possible occurrence of critical curves in highly distorted images may provide a serious obstacle to applicationsof flexion. Perhaps the most promising application of flexion measurements are those in regions where the shear fieldvaries on small scales, i.e., close to galaxies (and thus can be used for galaxy-galaxy lensing) or in the inner regionsof clusters. However, if one finds a strongly distorted image of a background galaxy as in the case of the arclet A5 inAbell 370 (Fort et al. 1988), how can one be sure that it is not due to a merged double image of the source? Usingflexion for studying small-scale structure in mass distributions can therefore be affected by the occurrence of multipleimaging.Similar to the situation in shear measurements, the moment approach for flexion as presented here must bemodified in several ways to be applicable to real data. First, brightness moments must be weighted in order not to bedominated by the very noisy outer regions of the image. As is known from shear measurements, such a weighting affects eter Schneider, Xinzhong Er: What flexion really measures 13 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 have been successfully achieved for second-order brightness moments byKaiser et al. (1995; see also Luppino & Kaiser 1997). Goldberg & Leonard (2007) consider these effects in the contextof 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, thePSF effects are much more subtle, and we expect that a formalism in analogy to Kaiser et al. (1995) needs to bedeveloped – and that this formalism for higher-order brightness moments will be considerably more difficult than forthe shear case.But even disregarding these complications, the present paper only scratches the surface in investigating estimatorsfor reduced flexion and their properties. As mentioned before, the second-order lens equation contains five essentialparameters. The bias of an estimator for reduced shear and flexion will depend on these parameters, as well as on theintrinsic ellipticity (and higher-order moments) distribution of sources. One might ask whether it is possible to findan unbiased flexion estimator, such as was possible to construct for the reduced shear. Unfortunately, we have beenunable to make analytic progress: even for a circular Gaussian source, the brightness moments of the image cannotbe calculated analytically. Our ray-tracing algorithm with which we conducted our numerical simulations is almostcertainly sub-optimal; a more advanced method should be developed to reduce the numerical efforts in calculatingbrightness moments. Beside the bias, it would be interesting to calculate the variance of the various estimators, ormore precisely, their covariance.It may turn out that measurements of flexion, and PSF corrections, are more conveniently done with shapelets, aswas originally considered by Goldberg & Bacon (2005), Bacon et al. (2006) and Massey et al. (2006). Even if this turnsout to be the case (see Leonard et al. 2007 for an application of flexion measurements in the galaxy cluster A 1689), themoment approach provides a more intuitive picture of the effects of flexion. In addition, the weak lensing communityhas profited substantially from the existence of several different methods to measure shear (see Heymans et al. 2006;Massey et al. 2007 for the first results of a comprehensive Shear TEsting Programme, in which these various methodsare studied and compared); therefore, the development of different techniques for measuring flexion will certainly beof interest once the flexion method will be put to extensive use.
Acknowledgments
We thank Jan Hartlap and Ismael Tereno for useful comments on this paper. This work was supported by the DeutcheForschungsgemeinschaft under the project SCHN 342/6–1 and the TR33 ‘The Dark Universe’. XE was supported forthis research through a stipend from the International Max-Planck Research School (IMPRS) for Radio and InfraredAstronomy at the University of Bonn.
Appendix A: The matrix C In this Appendix, we list the coefficients of the matrix C which occurs in (40):4(1 − gg ∗ ) C = − gF ∗ + (9 gg ∗ − F + 6 g ∗ (1 − gg ∗ ) F + g ∗ (5 gg ∗ − F + 6 gQ ∗ Q − gg ∗ Q + (3 − gg ∗ ) Q ∗ Q + 6 g ∗ (4 gg ∗ − Q Q + 3 g ∗ (1 − gg ∗ ) Q − gg ∗ ) C = 5 gF ∗ − gg ∗ ) F ∗ + 9 g ∗ (3 + gg ∗ ) F − g ∗ (12 + gg ∗ ) F + 7 g ∗ F − gQ ∗ + 6(3 + 4 gg ∗ ) Q ∗ Q − g ∗ (3 + gg ∗ ) Q − g ∗ (5 + 3 gg ∗ ) Q ∗ Q + 6 g ∗ (8 + gg ∗ ) Q Q − g ∗ Q − gg ∗ ) C = − F ∗ + 26 g ∗ F ∗ − g ∗ F + 22 g ∗ F − g ∗ F + 15 Q ∗ − g ∗ Q ∗ Q + 48 g ∗ Q + 24 g ∗ Q ∗ Q − g ∗ Q Q + 9 g ∗ Q − gg ∗ ) C = − g ∗ F ∗ + 6 g ∗ F ∗ − g ∗ F + 2 g ∗ F + 6 g ∗ Q ∗ − g ∗ Q ∗ Q + 12 g ∗ Q + 6 g ∗ Q ∗ Q − g ∗ Q Q − gg ∗ ) C = 2 g F ∗ − g g ∗ F + [4 gg ∗ (1 + gg ∗ ) − F + 2 g ∗ (1 − gg ∗ ) F − g Q ∗ Q + 4 g (1 + 2 gg ∗ ) Q + 6 g g ∗ Q ∗ Q + [2 − gg ∗ (3 + 2 gg ∗ )] Q Q + 2 g ∗ (4 gg ∗ − Q − gg ∗ ) C = − g F ∗ + 2 g (7 + 4 gg ∗ ) F ∗ − gg ∗ (8 + gg ∗ )] F + 2 g ∗ (8 + 5 gg ∗ ) F − g ∗ F + 9 g Q ∗ − g (13 + 8 gg ∗ ) Q ∗ Q + 4[3 + gg ∗ (8 + gg ∗ )] Q + [5 + gg ∗ (16 + 3 gg ∗ )] Q ∗ Q − g ∗ (16 + 11 gg ∗ ) Q Q + 15 g ∗ Q − gg ∗ ) C = 7 gF ∗ − gg ∗ ) F ∗ + 3 g ∗ (7 + 5 gg ∗ ) F − g ∗ (9 + 2 gg ∗ ) F + 5 g ∗ F − gQ ∗ + (16 + 38 gg ∗ ) Q ∗ Q − g ∗ (7 + 5 gg ∗ ) Q − g ∗ (13 + 11 gg ∗ ) Q ∗ Q + 2 g ∗ (17 + 4 gg ∗ ) Q Q − g ∗ Q − gg ∗ ) C = (3 gg ∗ − F ∗ − gg ∗ F ∗ + 3 g ∗ (1 + gg ∗ ) F − g ∗ F + (1 − gg ∗ ) Q ∗ + 2 g ∗ (2 + 7 gg ∗ ) Q ∗ Q − g ∗ (2 + gg ∗ ) Q − g ∗ (1 + gg ∗ ) Q ∗ Q + 6 g ∗ Q Q -30 -15 0 15 30 45 60-45-30-150153045-30 -15 0 15 30 45 60-45-30-150153045 -60 -45 -30 -15 0 15 30-60-45-30-1501530-60 -45 -30 -15 0 15 30 -60-45-30-1501530 Fig. A.1.
The critical curves (left-hand panel) and caustics (right-hand panel) of the lens equation (9) for the cases of hyperboliccritical curves, as described in Sect. B.2. The parameters chosen here are g = 0 . G = 0 .
07 + 0 . G = 0 .
03 + 0 . -20 -15 -10 -5 0 5 10 15 20-20-15-10-505101520-20 -15 -10 -5 0 5 10 15 20-20-15-10-505101520 -20 -15 -10 -5 0 5 10 15 20-20-15-10-505101520-20 -15 -10 -5 0 5 10 15 20-20-15-10-505101520 Fig. A.2.
Same as Fig. A.1, but for the parabolic case, with parameters g = 0 . G = − . G = 0 . The other eight elements follow trivially from the foregoing ones, since the second half of the matrix is just the complexconjugate one of the first half, i.e., C = C ∗ , C = C ∗ etc., or in general, C ij = C ∗ − i, − j . Appendix B: Critical curves and caustics
In this Appendix we consider the critical curves of the lens equation (9). For this, we need to derive the full Jacobian,which can most easily be obtained from considering θ and θ ∗ as independent variables, and then use ∂/∂θ = ∂/∂θ + ∂/∂θ ∗ , ∂/∂θ = i ( ∂/∂θ − ∂/∂θ ∗ ), which can be inverted to yield ∂/∂θ = ∇ ∗ c / ∂/∂θ ∗ = ∇ c /
2. With these relations,one finds that det A = ( ∂β/∂θ )( ∂β ∗ /∂θ ∗ ) − ( ∂β/∂θ ∗ )( ∂β ∗ /∂θ ) = ( ∇ ∗ c β ∇ c β ∗ − ∇ c β ∇ ∗ c β ∗ ) /
4. Carrying out thesederivatives, the Jacobian becomesdet A = 1 − gg ∗ − η ∗ θ − ηθ ∗ + A ∗ θ + Bθθ ∗ + A ( θ ∗ ) , (B.1) eter Schneider, Xinzhong Er: What flexion really measures 15 -20 -15 -10 -5 0 5 10 15 20-40-30-20-1001020-20 -15 -10 -5 0 5 10 15 20-40-30-20-1001020 -25-20-15-10 -5 0 5 10 15 20 25-25-20-15-10-50510152025-25-20-15-10 -5 0 5 10 15 20 25-25-20-15-10-50510152025 Fig. A.3.
Same as Fig. A.1, but for the elliptical case, with parameters g = 0 . , G = 0 .
015 + 0 . , G = 0 .
19 + 0 . with A = 4 (cid:0) Ψ − Ψ ∗ Ψ (cid:1) ; B = 4 (Ψ Ψ ∗ − Ψ Ψ ∗ ) ; η = 4Ψ + 2 g Ψ ∗ + 2 g ∗ Ψ . (B.2)Note that A is a spin-2 quantity, whereas B is a real scalar, i.e., has spin-0. In the generic case, the critical curves(det A = 0) are conical sections, which may be degenerate, though. We will now perform a complete classification ofcases that can occur, as well as to derive the critical curve(s) in parametric form. As we shall see, the type of conicalsection is determined, amongst other parameters, by the discriminant∆ = B − AA ∗ . (B.3) B.1. Zero discriminant
We start with the case that ∆ = 0, which implies B = 4 AA ∗ , or B = ± | A | . The case A = 0 = B either impliesthat Ψ = 0 = Ψ , in which case also η = 0 so that no critical curves occur, or that Ψ = Ψ / Ψ ∗ , for which η = 0in general. In this latter case, the critical curve is a straight line, satisfying η ∗ θ + ηθ ∗ = 1 − gg ∗ . As can be seen byinspection, it reads θ = 1 − gg ∗ η ∗ + i λη , −∞ < λ < ∞ . (B.4)If A = 0, the phase of A is defined. Since it is a spin-2 quantity, we write A = | A | e ϕ A . Furthermore, we introducethe rotation θ = x e i ϕ A . Then the equation for the critical curve reads( x ± x ∗ ) = ν ∗ x + νx ∗ + gg ∗ − | A | , with ν = η e − i ϕ A | A | , (B.5)and the sign on the left-hand side of the equation depends on the sign of B , where we used B = ± | A | . The parametricform of the critical curve, which takes the form of a parabola, can then be written as θ = 2 e i ϕ A ( ν ∗ − ν ) (cid:18) λ − λν + 1 − gg ∗ | A | (cid:19) ; θ = 2 e i ϕ A ( ν ∗ + ν ) (cid:18) − gg ∗ | A | − i λν − λ (cid:19) , (B.6)where the first (second) equation applies for B >
B < ν is real (for B >
0) or purely imaginary (for
B <
B.2. Non-zero discriminant
If ∆ = 0, we can perform a translation to eliminate the linear term in det A . Hence we define θ = θ + ϑ and choose θ such that terms linear in ϑ vanish. We then obtain for θ and for the critical curve condition θ = Bη − Aη ∗ ∆ ; A ∗ ϑ + Bϑϑ ∗ + A ( ϑ ∗ ) = C , (B.7) with C = Bηη ∗ − A ( η ∗ ) − A ∗ η ∆ + gg ∗ − −
1∆ ( gA ∗ + g ∗ A + B ) =: − V , (B.8)where the second step was obtained by inserting the expression for η in terms of the Ψ’s, and in the final one wedefined V as the expression in the parenthesis.As the first case, we consider A = 0 and B = 0 (the case A = 0 = B was treated above), which implies that Ψ = 0and B = − Ψ ∗ <
0. The equation for the critical curve then reduces to B | ϑ | = C . Furthermore, ∆ = B , and C = −
1. Thus, the critical curve is a circle of radius 1 / (2 | Ψ | ) and center θ , or θ = θ + e i λ / (2 | Ψ | ), 0 ≤ λ < π .We now consider the case A = 0; then the phase ϕ A of A is defined, as used before. Introducing a rotation bydefining ϑ = x e i ϕ A , the equation for the critical curve becomes | A | h x + ( x ∗ ) i + Bxx ∗ = ( B + 2 | A | ) x + ( B − | A | ) x = C . (B.9)The presence and topology of critical curves now depends on the signs of ∆ and C . We first consider the case C = 0;then, if ∆ >
0, no critical curves occur, except for the isolated point x = 0. If ∆ <
0, the critical curves are two straightlines, as can be obtained from (B.7): inserting the ansatz ϑ = λ e i ζ , one obtains e ζ − ϕ A ) = ( − B ± i √− ∆) / (2 | A | ).Thus, the critical curves are parametrized as θ = θ + λ e i ϕ A s − B ± i √− ∆2 | A | ; −∞ < λ < ∞ . (B.10)For the case of C = 0, the consideration of (B.9) yields the result that for ∆ <
0, the critical curves consist of twohyperbolae. From (B.8) we see that negative ∆ implies
C >
0. Also note that ∆ < | A | − B > | A | + B >
0. The critical curves then read θ = θ + e i ϕ A V √− ∆ ± cosh λ p | A | + B + i sinh λ p | A | − B ! ; −∞ < λ < ∞ . (B.11)For the other case, ∆ >
0, we find from (B.8) that
C <
0. If B ± | A | >
0, we then see from (B.9) that no criticalcurves exist. If B ± | A | <
0, which in particular implies
B <
0, the critical curve is an ellipse parametrized as θ = θ + e i ϕ A V √ ∆ cos λ p − | A | − B + i sin λ p | A | − B ! ; 0 ≤ λ < π . (B.12)This concludes the classification of critical curves of the lens equation (9). The caustics are obtained by inserting theparametrized form of the critical curves into the lens equation. In order to see whether a critical curves cuts throughthe primary image of a circular source of outer isophotal radius Θ, we calculate the minimum value β min of | β ( λ ) | alongthe caustics. If β min > Θ, the image is not cut by a critical curve. For an elliptical critical curve, the maximum sourcesize allowed is β min ; these values are plotted in Fig. 1. In the cases where two critical curves exist (e.g., two straightlines or hyperbolae), the situation is slightly more complicated. Consider, e.g., the case of two straight critical curves.Only those sections of them that are closer to the origin are relevant for this consideration, since if the primary imageof the source is not cut by these closer sections of critical curves, it will still be an isolated image; the caustics comingfrom the outer sections of the critical curves correspond to multiply imaged source sections of secondary images.Accounting for this complication, the maximum sources size have been obtained, as plotted in Fig. 1. References
Bacon, D.J., Goldberg, D.M., Rowe, B.T.P. & Taylor, A.N. 2006, MNRAS, 365, 414Bartelmann, M. & Schneider, P. 2001, Phys. Rep., 340, 291Crittenden, R.G., Natarajan, P., Pen, U.-L. & Theuns, T. 2002, ApJ, 568, 20Falco, E.E., Gorenstein, M.V. & Shapiro, I.I. 1985, ApJ, 289, L1Fort, B., Prieur, J.L., Mathez, G., Mellier, Y. & Soucail, G. 1988, A&A, 200, L17Fort, B. & Mellier, Y. 1994, A&AR, 5, 239Goldberg, D.M. & Bacon, D.J. 2005, ApJ, 619, 741Goldberg, D.M., & Leonard, A. 2007, ApJ, 660, 1003Gorenstein, M.V., Shapiro, I.I., & Falco, E.E. 1998, ApJ, 327, 693Leonard, A., Goldberg, D.M., Haaga, J.L. & Massey, R. 2007, astro-ph/0702242Kaiser, N. 1995, ApJ, 439, 1Kaiser, N., Squires, G., & Broadhurst, T. 1995, ApJ, 449, 460eter Schneider, Xinzhong Er: What flexion really measures 17Luppino, G.A. & Kaiser, N. 1997, ApJ, 475, 20Massey, R., Rowe, B., Refregier, A., Bacon, D.J. & Berg´e, J. 2006, astro-ph/0609795Mellier, Y. 1999, ARA&A, 37, 127Munshi, D., Valageas, P., Van Waerbeke, L.& Heavens, A. 2006, astro-ph/0612667Okura, Y., Umetsu, K. & Futamase, T. 2007, ApJ, 660, 995.Refregier, A., 2003, ARA&A, 41, 645Schneider, P. 2006, in: