The Linear Instability of Dilute Ultrarelativistic $\textit{e}$$^{±}$ Pair Beams

The annihilation of TeV photons from extragalactic TeV sources and the extragalactic background light produces ultrarelativistic $\textit{e}$$^{±}$ beams, which are subject to powerful plasma instabilities that sap their kinetic energy. Here we study the linear phase of the plasma instabilities that these pair beams drive. To this end, we calculate the linear growth rate of the beam-plasma and oblique instability in the electrostatic approximation in both the reactive and kinetic regimes, assuming a Maxwell Jüttner distribution for the pair beam. We reproduce the well-known reactive and kinetic growth rates for both the beam-plasma and oblique mode. We demonstrate for the oblique instability that there is a broad spectrum of unstable modes that grow at the maximum rate for a wide range of beam temperatures and wave-vector orientations relative to the beam. We also delineate the conditions for applicability for the reactive and kinetic regimes and find that the beam-plasma mode transitions to the reactive regime at a lower Lorentz factor than the oblique mode due to a combination of their different scalings and the anisotropy of the velocity dispersions. Applying these results to the ultrarelativistic $\textit{e}$$^{±}$ beams from TeV blazars, we confirm that these beams are unstable to both the kinetic oblique mode and the reactive beam-plasma mode. These results are important in understanding how powerful plasma instabilities may sap the energy of the ultrarelativistic $\textit{e}$$^{±}$ beams as they propagate through intergalactic space.P.C. is supported by the UWM Research Growth Initiative, the NASA ATP program through NASA grant NNX13AH43G, and NSF grant AST-1255469. A.E.B. and M.S. receive financial support from the Perimeter Institute for Theoretical Physics and the Natural Sciences and Engineering Research Council of Canada through a Discovery Grant. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. C.P. gratefully acknowledges support by the European Research Council under ERC-CoG grant CRAGSMAN-646955 and by the Klaus Tschira Foundation. E.P. gratefully acknowledges support by the Kavli Foundation. Support for A.L. was provided by an Alfred P. Sloan Research Fellowship, NASA ATP Grant NNX14AH35G, and NSF Collaborative Research Grant #1411920 and CAREER grant #1455342. G.V. acknowledges support from the Australian Research Council, project number DE140101960