Galaxy-galaxy flexion: Weak lensing to second order

In this paper, we develop a new gravitational lensing inversion technique. While traditional approaches assume that the lensing field varies little across a galaxy image, we note that this variation in the field can give rise to a flexion,'' or bending of a galaxy image, which may then be used to detect a lensing signal with increased signal-to-noise ratio. Since the significance of the flexion signal increases on small scales, this is ideally suited to galaxy-galaxy lensing. We develop an inversion technique based on the shapelets'' formalism of Refregier. We then demonstrate the proof of this concept by measuring a flexion signal in the Deep Lens Survey. Assuming an intrinsically isothermal distribution, we find from the flexion signal alone a velocity width of v(c) = 221 +/- 12 km s(-1) for lens galaxies of r < 21.5, subject to uncertainties in the intrinsic flexion distribution.