Measuring the gravitomagnetic distortion from rotating halos I: methods
  Source galaxy images are distorted not only by a static gravitational
potential, but also by frame-dragging induced by massive rotating objects like
clusters of galaxies. Such effect is well understood theoretically, it is
therefore of great interest to estimate its detectability for future surveys.
In this work, we analyze the lensing convergence $\kappa$ around rotating dark
matter halos. The rotation of the massive objects generates a gravitomagnetic
potential giving rise to an anisotropic contribution to the lensing potential.
We construct an estimator $\delta \kappa$ to describe the difference between
the symmetric enhancement and reduction of $\kappa$ around the halo rotation
axis, finding that it is well approximated by a function proportional to the
halo velocity dispersion squared times a dimensionless angular momentum
parameter. Using simulation mocks with realistic noise level for a survey like
LSST, we test our estimator, and show that the signal from frame-dragging of
stacked rotating lenses is consistent with zero within $1\sigma$. However, we
find that the most massive cluster in SDSS DR7 spectroscopic selected group
catalog has a line-of-sight rotation velocity of 195.0km/s and velocity
dispersion of 667.8km/s, which is at $1.2\times 10^{-8}$ odds according to the
angular momentum probability distribution inferred from N-body simulations. By
studying SDSS DR7 spectroscopic selected group catalog, we show how rotating
clusters can be identified, and, finding that fast rotating clusters might be
more abundant than in estimates based on simulations, a detection of
gravitomagnetic distortion may be at reach in future surveys
