Evolution of newborn rapidly rotating magnetars: effects of r-mode and fall-back accretion

In this paper we investigate effects of the $r$-mode instability on a newborn rapidly-rotating magnetar with fall-back accretion. Such a magnetar could usually occur in core-collapse supernovae and gamma-ray bursts. We find that the magnetar's spin and $r$-mode evolution are influenced by accretion. If the magnetar is sufficiently spun up to a few milliseconds, gravitational radiation leads to the growth of the $r$-mode amplitude significantly. The maximum $r$-mode amplitude reaches an order $\sim 0.001$ when the damping due to the growth of a toroidal magnetic field balances the growth of the $r$-mode amplitude. If such a sufficiently spun-up magnetar was located at a distance less than 1\,Mpc, then gravitational waves would be detectable by the Einstein Telescope but would have an extremely low event rate. However, if the spin-up is insufficient, the growth of the $r$-mode amplitude is mainly due to the accretion torque. In this case, the maximum $r$-mode amplitude is of the order of $\sim 10^{-6}-10^{-5}$.