Recursive definition of roulette amplitudes

A key element of the Roulettes formalism is recursive expressions developed in Normann & Clarkson (2020) for the amplitudes $\alpha_s^m$ and $\beta_s^m$ . As before, using angular coordinates $\theta_i$ in the lens plane, such that

\partial_i=\frac{\partial}{\partial\theta_i}

(1)

one may show that the lowest-order amplitudes are given by

\alpha_1^0 = -\partial_1 \psi^\textrm{R} \quad\text{and}\quad \beta_1^0 = -\partial_2 \psi^\textrm{R}

The higher-order ones may now be obtained through recursive relations, and are given as $$

\alpha_{s+1}^{m+1} = (C_+^+)_{s+1}^{m+1} (\partial_1 \alpha_s^m - \partial_2 \beta_s^m)

\beta_{s+1}^{m+1} = (C_+^+)_{s+1}^{m+1} (\partial_1 \beta_s^m + \partial_2 \alpha_s^m)

\alpha_{s-1}^{m+1} = (C_-^+)_{s+1}^{m+1} (\partial_1 \alpha_s^m + \partial_2 \beta_s^m)

\beta_{s-1}^{m+1} = (C_-^+)_{s-1}^{m+1} (\partial_1 \beta_s^m - \partial_2 \alpha_s^m)

with $$

(C_+^+)_s^m = 2^{\delta_{0(s-1)}} \frac{m + 1}{m + 1 + s} \quad\text{and}\quad (C_-^+)_s^m = 2^{-\delta_{0s}} \frac{m + 1}{m + 1 - s}

The astute reader may notice that amplitudes for even sums $s+m$ cannot be found through these relations. However, the contribution from these terms are equal to zero. In other words, one can calculate all the amplitudes needed from the aforementioned relations.

References¶

Normann, B. D., & Clarkson, C. (2020). Recursion relations for gravitational lensing. General Relativity and Gravitation, 52(3). 10.1007/s10714-020-02677-z