Recursion relations for gravitational lensing

The weak lensing formalism can be extended to the strong lensing regime by integrating the nonlinear geodesic deviation equation. The resulting ‘roulette’ expansion generalises convergence, shear and flexion to arbitrary order. By formulating this into a compact complex notation we derive a family of recursion relations between the various coefficients of this expansion, generalising the Kaiser-Squires relations to strong lenses.


Introduction
In 1993 Kaiser & Squires wrote an important paper [1] where they found expressions for the mass in terms of the shear field measured in galaxy clusters. Attempts had been given also previously, but these attempts involved models of the lens. Kaiser & Squires' work were further developed by Schneider and Seitz, and paved the way for an array of papers describing the implementation of Kaiser and Squires' work to obtain real data. Note, however, the recent works by Fleury et al. [2], where the Kaiser-Squires theorem between shear and convergence is found to be violated due to the finiteness of the beam. In subsequent papers, they further develop the theory of extended sources [3], showing that one must go beyond shear [4].
There are two main routes to calculating the effect of lensing of a source by a gravitational lens. One of them is to calculate all the null geodesics converging at the point of observation. The other route is to start by the geodesic deviation equation (GDE).
More accurate information about the lens is obtained by starting from the Bazanski equation rather than the linear GDE. The GDE originates from a first order Taylor expansion in the Ricci rotation coefficients. The final result contains one power of Riemann;ξ + R a kbk ξ b = O(ξ,ξ) 2 1st order GDE. (1) We follow the notation of [5]: A dot (˙) denotes derivative w.r.t. the null geodesic †. The Bazanski equation takes into account terms to second order in the deviation ξ and its derivativeξ. In the context of weak lensing, this was investigated in [5]. It was generalised to arbitrary order in screen space derivatives (leading order terms) in [6,7] ‡. In these works it is shown that if one only keeps leading order screen space derivatives the equation to solve becomes the GDE with the replacement From here, by applying the same solution algorithm as that found in [5], it is shown how the general solutions may be written as a sum over 'roulettes'. Hence we shall refer to this as the Roulette formalism for gravitational lensing.
The scope of this work is all within the weak field approximation (perturbed Minkowski spacetime in Poisson gauge). First, we recast the previously developed Roulette formalism to complex variables, and use this to find recursion relations for the Roulette modes.
Second, we show a very neat formula for the Roulette modes. We explicitely write down the derivatives of the lensing potential in terms of roulette modes α m s and β m s , up to fourth order. This is done generally, without making any assumptions about symmetries of the screen space.

Roulette modes in the weak field approximation
For a general thin lens in the weak field and flat sky approximation the Roulette modes [7] take the form respectively. Here 0 ≤ s ≤ m + 1 and the roulette modes may be non-zero only if m + s is odd. Also, dθ sin k θ cos m−k+1 θ cos sθ (5) † Further notation throughout: a, b, c · · ·denote spacetime indices, A, B, C · · · are tetrad indices in screen space. k or ξ as index denotes projection of that index onto k or ξ, respectively.
‡ The fully general GDE [8], valid to all order in all derivatives (not only screen space), is not dealt with in this work.
The equations (3) and (4) are horrendous, but it is easy enough to calculate α 0 1 and β 0 1 . We find By the recursion relations introduced in this paper, these two 'lowest modes' of expansion will suffice. The rest may be obtained by recursion relations, which we derive below. The recursion relations we find are: and here (A + + ) and (B + − ) are algebraic coefficients given by We also provide a more compact notation for these relations (and the inverses) in Section 3. In order to do so, however, we must first introduce complex variables.

Complex notation
Define first the complex variables By use of Pascal's triangle we now, upon a bit of straight forward algebra, obtain from equations (3) and (4) the expression where we have defined Define similarly the differentiation operator Note that complex derivatives may be further explored through de Wittinger's calculus. Furthermore, let * denote the complex conjugate.
A particularly useful expression for γ m s may be found by use of the recursion relations explained in the next section. The result is Here δ − = (m + 1 − s)/2, and Γ m s are numerical coefficients given by Refer to Appendix E for an outline of the proof. Note that the expression for γ m s given here does not involve any sums or integrals. Hence it is computationally much more economic then the foregoing expression (14). Below we have calculated the first few complex roulette modes. Mode γ m s is given in row s, column m, starting with s = 0, m = 0 in the upper left corner.

Recursion relations
We define in this section a set of raising and lowering operators that will serve as recursion relations between the different modes. In want of a better name we call the identity operator D, † and define The connection to the Roulette formalism is made by defining two base operators D + + and D + − as follows ‡:

Base operators
Here A + + and B + − are numerical factors given by (12). Note that A + + and B + − will depend on the particular value of m and s. The recursion relations are now given by applying the base operators to γ m s in the following manner: The + and − signs are hence there to indicate whether we add or subtract to the number m (upper index) and s (lower index). Figure 1 illustrates the procedure. The above † The reason for the root name D is that these operators will become differentiation operators. ‡ Note that in stead of explicitly writing the indices m s all the time, we work with operator identities relations are the same as the relations given in the previous section, Eqs. (8)-(11). Note that the coefficients are evaluated at the end points (see the example below). A proof of the recursion relations can be found in Appendix A. The two recursion relations are independent of each other, and all non-zero modes α m s and β m s can therefore be obtained by these relations by starting from the lowest modes γ 0 1 = α 0 1 + iβ 0 1 , (given in eq. (7)).
We shall note in passing that any higher order operator D a+ b+ such that γ +b +a = D a+ b+ γ may be constructed from the base operators. Refer to Appendix C, where the operators are studied in more detail, for an explanation. As an example we may for instance calculate γ 3 2 from γ 1 0 in the following manner.
In the last line we inserted for γ 0 1 from (18), just to show that we obtain the correct expression, as compared with γ 4 3 in (19)).

Inverse operators
We define inverses with natural notation in the following way. Recalling that D is the identity element we have and similarely This is a natural notation, as one thinks of the '+' s and .'−' s as canceling against each other when they are on the same script level. Indeed the notation is right in suggesting that for instance Refer to Appendix C for mathematical details. By application to γ, we find explicit expressions for the inverses as follows. By (23) and (28) we find Endoving the inverse A + + with the same natural notation, A − − ≡ (A + + ) −1 (and similarly for B + − ) we hence find the corresponding operator identities.

Inverse base operators
Here we have invoked (17). Note that −1 here should be taken simply as the inverse operator to †.The index form of A − − and B − + must be handled with care, however. † That is; The key lies in the fact that the inverse is taken at a different position in {m, s} -space. This is easy to see when we write it out, as follows.
Comparing with (12), and recognizing that the inverse in the last step is just the functional inverse, we necessarily have The inverses of the recursion relations (24) are now given as Figure 1 illustrates the procedure.

Comparing with other works
It is instructive at this point to compare with previous works. Kaiser & Squires described a method for inverting the measured shear caused by the lens to obtain the surface mass density of a lens [1]. To do so they started with the lensing equation. Their results are very well described by Schneider, who extended their work in two papers [9,10]. Schneider finds a way to use only local data to reconstruct the lens mass from the shear field, and also points out the need for a generalized 'Kaiser-Squires' inversion procedrue, to account for stronger lensing effects. The standard procedure when starting from the lensing equation, is to define the lensing potential θ is a vector of the two angles parametrizing the lensing 2-sphere, andκ is the dimensionless surface mass density. We use a tilde (˜) to distinguish the notation from our own, where we have taken a GDE-like approach and hence defined the lensing potential differently. From the linearized lens mapping β = θ − ∇ψ one defines now dβ = A(θ) dθ, where β is the angular position on the source 2-sphere. One finds then Therefore one must havẽ where ,i refer to derivative w.r.t. θ i . From this, Kaiser & Squires obtained an inversion formula, yielding an expression for the surface mass density. Definingγ =γ 1 + iγ 2 one Kaiser-Squires inversion formula (42) where D = (θ 2 1 − θ 2 2 + 2iθ 1 θ 2 )/|θ| 4 . As mentioned, this formula gives the convergence (surface mass density) in terms of the shear field generated by the lens.

Connection to roulette modes
This section is just to show that we obtain similar expressions. Indeed, calculating the first few modes from (14) we find These equations have precisely the same structure as (39)-(41) †. This means that the convergence-to-shear map that Kaiser & Squires have found, correspond to the first few terms in the series expansion of roulette modes. The higher order terms may in the same way be obtained from (14). In [7] the expressions (43)-(45) were inverted in a Kaiser-Squires like fashion to obtain the mass of a spherically symmetric (thin) lens in terms of the roulette modes. In this work, we do the same, except we restrict attention to expressing the derivatives of the potential (and not the mass) in terms of the roulette modes. This is done in the next section.

Derivatives of the potential in terms of roulette modes
By equation (3) and (4) we can, as mentioned, calculate all the higher order modes. Unfirtunately we have not yet succeeded in writing down an explicit expression in closed form for the derivatives of the potential. We do think this is possible. It is, however, easy to invert the equations with a computer. To illustrate, we give in the following all the derivatives of the potential up to 4th order in terms of the roulette modes.
† Note that we have used a different definition of the lensing potential ψ, and different notation.
For brevity we used in the above notation such that ∂ n x ∂ m y ψ ≡ ψ nxmy , where n, m are positive integers.

Conclusion
In this work we found the recursion relations of the Roulette formalism in the weak field and thin lens approximation. Also, we have found a very simple expression for the roulette modes, invoking no sums or integrals.

Appendix A. Proof of recursion relations
In this appendix we show how to prove that the recursion relations (24) are correct.
Take first the definition of γ m+1 s+1 , equation (14). According to it one finds that the term with ∂ m−k+1 x ∂ k y ψ is given by Similarly, starting from the recursion relation γ m+1 s+1 = (A + + ) m+1 s+1 ∂ c instead, one finds the corresponding terms Equating equation (A.2) with (A.3) and inserting for (A + + ) m+1 s+1 gives, after some straight forward algebraic manipulation, the equation Following the same procedure for γ m+1 s−1 similarely gives the equation (A.5) The two relations (A.4) and (A.5) must hold for the recursion relations to be correct. In Appendix B they are proven to hold. Hence our proof is complete.

Additional relations
One may also note relations like which follow from manipulation with trigonometric identities. In this paper we omit the (straight forward) proofs, as we make no use of these relations. Take now the definition (B.14)