  In this work, we perform the first systematic investigation of effects of the
presence of dark matter on $r$-mode oscillations in neutron stars (NSs). Using
a self-interacting dark matter (DM) model based on the neutron decay anomaly
and a hadronic model obtained from the posterior distribution of a recent
Bayesian analysis, we impose constraints on the DM self-interaction strength
using recent multimessenger astrophysical observations. We also put new
constraints on the DM fraction for this model of DM. The constrained DM
interaction strength is then used to estimate DM self-interaction cross section
and shear viscosity resulting from DM, which is found to be several orders of
magnitude smaller than shear viscosity due to hadronic matter. Assuming
chemical equilibrium among DM fermions and neutrons, we estimate the bulk
viscosity resulting from the dark decay of neutrons considering different
scenarios for the temperature dependence of the reaction rate and investigate
the effect on the $r$-mode instability window. We conclude that DM shear and
bulk viscosity may significantly modify the $r$-mode instability window
compared with the minimal hadronic viscosities, depending on the temperature
dependence of the process. We also found that for the window to be compatible
with the X-ray and pulsar observational data, the rate for the dark decay
process must be fast.
