Stochastic gravitational wave background from hydrodynamic turbulence in differentially rotating neutron stars

Hydrodynamic turbulence driven by crust-core differential rotation imposes a fundamental noise floor on gravitational wave observations of neutron stars. The gravitational wave emission peaks at the Kolmogorov decoherence frequency which, for reasonable values of the crust-core shear, \Delta\Omega, occurs near the most sensitive part of the frequency band for ground-based, long-baseline interferometers. We calculate the energy density spectrum of the stochastic gravitational wave background from a cosmological population of turbulent neutron stars generalising previous calculations for individual sources. The spectrum resembles a piecewise power law, \Omega_{gw}(\nu)=\Omega_{\alpha}\nu^{\alpha}, with \alpha=-1 and 7 above and below the decoherence frequency respectively, and its normalisation scales as \Omega_{\alpha}\propto(\Delta\Omega)^{7}. Non-detection of a stochastic signal by Initial LIGO implies an upper limit on \Delta\Omega and hence by implication on the internal relaxation time-scale for the crust and core to come into co-rotation, \tau_{d}=\Delta\Omega/\dot{\Omega}, where \dot{\Omega} is the observed electromagnetic spin-down rate, with \tau_{d}\lesssim 10^{7} yr for accreting millisecond pulsars and \tau_{d}\lesssim 10^{5} yr for radio-loud pulsars. Target limits on \tau_{d} are also estimated for future detectors, namely Advanced LIGO and the Einstein Telescope, and are found to be astrophysically interesting.