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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, ΔΩ, 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, Ω_{gw}(ν)=Ω_αν^α, with α=-1 and 7 above and below the decoherence frequency respectively, and its normalisation scales as Ω_α\propto(ΔΩ)^{7}. Non-detection of a stochastic signal by Initial LIGO implies an upper limit on ΔΩand hence by implication on the internal relaxation time-scale for the crust and core to come into co-rotation, τ_{d}=ΔΩ/\dotΩ, where \dotΩ is the observed electromagnetic spin-down rate, with τ_{d}\lesssim 10^{7} yr for accreting millisecond pulsars and τ_{d}\lesssim 10^{5} yr for radio-loud pulsars. Target limits on τ_{d} are also estimated for future detectors, namely Advanced LIGO and the Einstein Telescope, and are found to be astrophysically interesting.