Spin wandering of an accreting neutron star near torque balance: implications for gravitational wave searches

Abstract

Torque balance between gravitational radiation reaction and magnetocentrifugal accretion is one reason why accreting neutron stars rotate slower than centrifugal break-up. Random processes in the accretion disk, such as Rayleigh-Taylor instabilities and flicker noise, drive the spin frequency $f_\ast$ and gravitational wave frequency $f_{\rm gw}\propto f_\ast$ to wander stochastically around their torque balance values. Here it is shown theoretically, in terms of an idealized Ornstein-Uhlenbeck model, that the standard deviation of $f_{\rm gw}$ is given by $σ_{f_{\rm gw}}(h_0) \propto f_{\rm gw} h_0 D βT^{1/2}$, or equivalently $σ_{f_{\rm gw}}(F_{\rm X}) \propto f_{\rm gw}^{1/2} F_{\rm X}^{1/2} D βT^{1/2}$, where $h_0$ is the characteristic wave strain, $F_{\rm X}$ is the X-ray flux, $D$ is the source distance, and $β$ and $T$ are the fractional amplitude and autocorrelation time-scale of the stochastic torque. One can use $σ_{f_{\rm gw}}(h_0)$ and $σ_{f_{\rm gw}}(F_{\rm X})$ to check the physical consistency of a detection candidate in a gravitational wave search by comparing with $σ_{f_{\rm gw}}$ observed. Spin wandering impacts search design, because it sets the maximum time interval $\max(T_{\rm coh})$, during which the signal may be treated as coherent. The Ornstein-Uhlenbeck calculation predicts $10^{-1} \lesssim \max(T_{\rm coh}) / (1\, {\rm day}) \lesssim 10^3$, with $3\times 10^{-2} \lesssim βT^{1/2} / (1 \, {\rm s^{1/2}} ) \lesssim 4$, for Rayleigh-Taylor instabilities and flicker noise. The predicted $\max(T_{\rm coh})$ is shorter typically than the year-long observation runs of audio-band interferometers operating today, emphasizing the utility of semi-coherent search algorithms, which divide the data into multiple coherent segments of duration $\max(T_{\rm coh})$.