引力透镜剪切和微扰的多极公式
Multipole Formulae for Gravitational Lensing Shear and Flexion
摘要：
用于会聚、势、剪切和屈曲的引力透镜方程在极坐标中是简单的，并且一旦剪切旋量和屈曲旋量旋转到"切向“基础上，就在多极展开下分离。我们用它来调查是否有用的孔径质量剪切公式推广到所有的多极和弯曲。我们重新推导了Schneider和Bartelmann的结果，即半径R处的剪切多极子m完全由R处的质量多极子加上R内部和外部的质量多极子的特定矩Q^m_in和Q^m_out决定。m>=0的多极子与Q_out无关。与m <0的多极子不同，m<0的多极子与Q_in无关。这些内部和外部质量矩可以通过平面的互补部分上的剪切（和/或屈曲）数据来确定，这对晶状体建模具有实际意义。我们发现，E/B分离的容易性在孔径矩中{ \em not}推广到m！0：内矩是从透镜观测可获得的{仅\em}非局部E/B判别式。我们也没有发现实际的本地E/B判别超出了ESTA，虽然他们可能存在。我们还表明，使用弱透镜的数据来约束附近的强透镜系统的恒定剪切项是不切实际的，没有强有力的事先约束相邻的质量分布。
英文摘要：
The gravitational lensing equations for convergence, potential, shear, and flexion are simple in polar coordinates and separate under a multipole expansion once the shear and flexion spinors are rotated into a ``tangential'' basis. We use this to investigate whether the useful monopole aperture-mass shear formulae generalize to all multipoles and to flexions. We re-derive the result of Schneider and Bartelmann that the shear multipole m at radius R is completely determined by the mass multipole at R, plus specific moments Q^m_in and Q^m_out of the mass multipoles internal and external, respectively, to R. The m>=0 multipoles are independent of Q_out. But in contrast to the monopole, the m<0 multipoles are independent of Q_in. These internal and external mass moments can be determined by shear (and/or flexion) data on the complementary portion of the plane, which has practical implications for lens modelling. We find that the ease of E/B separation in the monopole aperture moments does {\em not} generalize to m!=0: the internal monopole moment is the {\em only} non-local E/B discriminant available from lensing observations. We have also not found practical local E/B discriminants beyond the monopole, though they could exist. We show also that the use of weak-lensing data to constrain a constant shear term near a strong-lensing system is impractical without strong prior constraints on the neighboring mass distribution.
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