Gravitational flexion , caused by derivatives of the gravitational tidal field , is potentially important for the analysis of the dark-matter distribution in gravitational lenses , such as galaxy clusters or the dark-matter haloes of galaxies .
Flexion estimates rely on measurements of galaxy-shape distortions with spin-1 and spin-3 symmetry .
We show in this paper that and how such distortions are generally caused not only by the flexion itself , but also by coupling terms of the form ( shear \times flexion ) , which have hitherto been neglected .
Similar coupling terms occur between intrinsic galaxy ellipticities and the flexion .
We show , by means of numerical tests , that neglecting these terms can introduce biases of up to 85 % on the F flexion and 150 % on the G flexion for galaxies with an intrinsic ellipticity dispersion of \sigma _ { \epsilon } = 0.3 .
In general , this bias depends on the strength of the lensing fields , the ellipticity dispersion , and the concentration of the lensed galaxies .
We derive a new set of equations relating the measured spin-1 and spin-3 distortions to the lensing fields up to first order in the shear , the flexion , the product of shear and flexion , and the morphological properties of the galaxy sample .
We show that this new description is accurate with a bias \leq 7 \% ( spin-1 distortion ) and \leq 3 \% ( spin-3 distortion ) even close to points where the flexion approach breaks down due to merging of multiple images .
We propose an explanation why a spin-3 signal could not be measured yet and comment on the potential difficulties in using a model-fitting approach to measure the flexion signal .