Sound-Wave Instabilities in Dilute Plasmas with Cosmic Rays:
  Implications for Cosmic-Ray Confinement and the Perseus X-ray Ripples
  Weakly collisional, magnetised plasmas characterised by anisotropic viscosity
and conduction are ubiquitous in galaxies, halos and the intracluster medium
(ICM). Cosmic rays (CRs) play an important role in these environments as well,
by providing additional pressure and heating to the thermal plasma. We carry
out a linear stability analysis of weakly collisional plasmas with cosmic rays
using Braginskii MHD for the thermal gas. We assume that the CRs stream at the
Alfv\'en speed, which in a weakly collisional plasma depends on the pressure
anisotropy ($\Delta p$) of the thermal plasma. We find that this $\Delta
p$-dependence introduces a phase shift between the CR-pressure and gas-density
fluctuations. This drives a fast-growing acoustic instability: CRs offset the
damping of acoustic waves by anisotropic viscosity and give rise to wave growth
when the ratio of CR pressure to gas pressure is $\gtrsim \alpha \beta^{-1/2}$,
where $\beta$ is the ratio of thermal to magnetic pressure, and $\alpha$,
typically $\lesssim 1$, depends on other dimensionless parameters. In
high-$\beta$ environments like the ICM, this condition is satisfied for small
CR pressures. We speculate that the instability studied here may contribute to
the scattering of high-energy CRs and to the excitation of sound waves in
galaxy-halo, group and cluster plasmas, including the long-wavelength X-ray
fluctuations in \textit{Chandra} observations of the Perseus cluster. It may
also be important in the vicinity of shocks in dilute plasmas (e.g., cluster
virial shocks or galactic wind termination shocks), where the CR pressure is
locally enhanced.
