Parametric decay instability of circularly polarized Alfv\'en wave in magnetically dominated plasma

We investigate parametric decay instability (PDI) of circularly polarized Alfv\'en wave into daughter acoustic wave and backward Alfv\'en wave in magnetically-dominated plasma, in which the magnetization parameter $\sigma$ (energy density ratio of background magnetic field to matter) exceeds unity. We analyze relativistic magnetohydrodynamics (MHD), focusing on wave frequencies sufficiently lower than the plasma and cyclotron frequencies. We derive analytical formulae for the dispersion relation and growth rate of the instability as a function of the magnetization $\sigma$, wave amplitude $\eta$, and plasma temperature $\theta$. We find that PDI persists even in high magnetization $\sigma$, albeit with a decreased growth rate up to $\sigma\to\infty$. Our formulae are useful for estimating the decay of Alfv\'en wave into acoustic wave and heat in high magnetization $\sigma$ plasma, which is a ubiquitous phenomenon such as in pulsars, magnetars, and fast radio bursts.