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We study the linear stability of weakly magnetized differentially rotatingplasmas in both collisionless kinetic theory and Braginskii's theory ofcollisional, magnetized plasmas. We focus on the very weakly magnetized limitthat is important for understanding how astrophysical magnetic fields originateand are amplified at high redshift. We show that the single instability offluid theory - the magnetorotational instability mediated by magnetic tension -is replaced by two distinct instabilities, one associated with ions and onewith electrons. Each of these has a different way of tapping into the freeenergy of differential rotation. The ion instability is driven by viscoustransport of momentum across magnetic field lines due to a finite ion cyclotronfrequency (gyroviscosity); the fastest growing modes have wavelengthssignificantly longer than MHD and Hall MHD predictions. The electroninstability is a whistler mode driven unstable by the temperature anisotropygenerated by differential rotation; the growth time can be orders of magnitudeshorter than the rotation period. The electron instability is an example of abroader class of instabilities that tap into the free energy of differentialrotation or shear via the temperature anisotropy they generate. We brieflydiscuss the application of our results to the stability of planar shear flowsand show that such flows are linearly overstable in the presence of fluidgyroviscosity. We also briefly describe the implications of our results formagnetic field amplification in the virialized halos of high redshift galaxies.

Linear Instabilities Driven by Differential Rotation in Very Weakly Magnetized Plasmas