Tangential velocity of the dark matter in the bullet cluster from precise lensed image redshifts

We show that the fast-moving component of the "Bullet Cluster" (1E0657-56) can induce potentially resolvable redshift differences between multiply lensed images of background galaxies. This moving cluster effect, due to the tangential peculiar velocity of the lens, can be expressed as the scalar product of the lensing deflection angle with the tangential velocity of the mass components; the effect is maximal for clusters colliding in the plane of the sky with velocities boosted by their mutual gravity. The Bullet Cluster is likely to be the best candidate for the first measurement of this effect due to the large collision velocity and because the lensing deflection and the cluster fields can be calculated in advance. We derive the deflection field using multiply lensed background galaxies detected with the Hubble Space Telescope. The velocity field is modeled using self-consistent N-body/hydrodynamical simulations constrained by the observed X-ray and gravitational lensing features of this system. We predict that the triply lensed images of systems "G" and "H" straddling the critical curve of the bullet component will show the largest frequency shifts up to 0.5 km s^-1. These shifts are within the range of the Atacama Large Millimeter/Submillimeter Array for molecular emission, and are near the resolution limit of the new generation high-throughput optical-IR spectrographs. The detection of this effect measures the tangential motion of the subclusters directly, thereby clarifying the tension with ΛCDM, which is inferred from the gas motion less directly. This method may be extended to smaller redshift differences using the Lyα forest toward QSOs lensed by more typical clusters of galaxies. More generally, the tangential component of the peculiar velocities of clusters derived by our method complements the radial component determined by the kinematic Sunyaev-Zel'dovich effect, providing a full three-dimensional description of velocities.