Kinetic simulations of the cosmic ray pressure anisotropy instability: cosmic ray scattering rate in the saturated state

Cosmic ray (CR) feedback plays a vital role in shaping the formation and evolution of galaxies through their interaction with magnetohydrodynamic waves. In the CR self-confinement scenario, the waves are generated by the CR gyro-resonant instabilities via CR streaming or CR pressure anisotropy, and saturate by balancing wave damping. The resulting effective particle scattering rate by the waves, {\nu}eff, critically sets the coupling between the CRs and background gas, but the efficiency of CR feedback is yet poorly constrained. We employ 1D kinetic simulations under the Magnetohydrodynamic-Particle-In-Cell (MHD-PIC) framework with the adaptive {\delta}f method to quantify {\nu}eff for the saturated state of the CR pressure anisotropy instability (CRPAI) with ion-neutral friction. We drive CR pressure anisotropy by expanding/compressing box, mimicking background evolution of magnetic field strength, and the CR pressure anisotropy eventually reaches a quasi-steady state by balancing quasi-linear diffusion. At the saturated state, we measure {\nu}eff and the CR pressure anisotropy level, establishing a calibrated scaling relation with environmental parameters. The scaling relation is consistent with quasi-linear theory and can be incorporated to CR fluid models, in either the single-fluid or p-by-p treatments. Our results serve as a basis towards accurately calibrating the subgrid physics in macroscopic studies of CR feedback and transport.