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Viscosity of neutron star matter and $r$-modes in rotating pulsars

We study viscosity of the neutron star matter and $r$-mode instability in rotating neutron stars. Contributions to the shear and bulk viscosities from various processes are calculated with account of in-medium modifications of the nucleon-nucleon interaction. A softening of the pion mode and a possibility of the pion condensation are included. The nucleon pairing is incorporated. In the shear viscosity we include the lepton contribution calculated with account of the Landau damping in the photon exchange, the nucleon contribution described by the medium-modified one pion exchange, and other terms, such as the novel phonon contribution in the 1S$_0$ superfluid neutron phase, and the neutrino term in the neutrino opacity region. The nucleon shear viscosity depends on the density rather moderately and proves to be much less than the lepton term. Among the terms contributing to the bulk viscosity, the term from the medium modified Urca reactions possesses strongest density dependence because of the pion softening. Contributions to the bulk viscosity arising from other reactions induced by charged weak currents are included. The radiative bulk viscosity induced by charged and neutral weak currents in the region of the neutrino transparency of the star is also calculated with account for in-medium effects. The computed frequency boundary of the $r$-mode stability for the stars with the mass $\ge 1.8 M_{\odot}$ proves to be above the frequencies of all rotating young pulsars. To explain the stability of rapid rotation of old pulsars in X-ray binaries we propose a novel efficient mechanism associated with appearance of condensates of low-lying modes of bosonic excitations with finite momentum and/or with an enhancement of the inhomogeneous pion/kaon condensates in some parts of the star, if the angular velocity exceeds a critical value.