Updated gravitational-wave upper limits on the internal magnetic field strength of recycled pulsars

Recent calculations of the hydromagnetic deformation of a stratified, non-barotropic neutron star are generalized to describe objects with superconducting interiors, whose magnetic permeability \mu is much smaller than the vacuum value \mu_0. It is found that the star remains oblate if the poloidal magnetic field energy is \gtrsim 40% of total magnetic field energy, that the toroidal field is confined to a torus which shrinks as \mu decreases, and that the deformation is much larger (by a factor \sim \mu_0/\mu) than in a non-superconducting object. The results are applied to the latest direct and indirect upper limits on gravitational-wave emission from Laser Interferometer Gravitational Wave Observatory (LIGO) and radio pulse timing (spin-down) observations of 81 millisecond pulsars, to show how one can use these observations to infer the internal field strength. It is found that the indirect spin-down limits already imply astrophysically interesting constraints on the poloidal-toroidal field ratio and diamagnetic shielding factor (by which accretion reduces the observable external magnetic field, e.g. by burial). These constraints will improve following gravitational-wave detections, with implications for accretion-driven magnetic field evolution in recycled pulsars and the hydromagnetic stability of these objects' interiors.