Linear analysis and simulations of the cosmic-ray streaming instability: The importance of oblique waves

Cosmic-ray (CR) streaming instability (CRSI) is believed to play an important role in CR transport and CR feedback to galaxies. It drives the growth of magnetohydrodynamic (MHD) waves that scatter CRs, and leads to energy/momentum exchange between CRs and interstellar medium. Despite extensive research on CRSI, its dependence on the thermodynamic state of the gas and its multidimensional effects have not been systematically studied. In this study, we derive the dispersion relation of the CRSI for three types of MHD waves including their dependence on propagation direction and plasma $\beta$ (the ratio of thermal pressure to magnetic pressure). We verify the analytical dispersion relation with one-dimensional and two-dimensional magnetohydrodynamic particle-in-cell simulations. Furthermore, we use 2D simulations to investigate the role of oblique MHD waves in scattering CRs, and find that these waves are important in helping low-energy particles overcome the 90-degree pitch angle barrier. While magnetosonic waves tend to be damped by transit time damping under typical conditions, oblique Alfv\'en waves likely play an important role in low-$\beta$ plasmas.