Constraints on ultra-low-frequency gravitational waves with statistics of pulsar spin-down rates

We probe ultra-low-frequency gravitational waves (GWs) with statistics of spin-down rates of millisecond pulsars (thereafter MSPs) by a method proposed in our previous work. The considered frequency range is 10−12 Hz ≲ fGW ≲ 10−10 Hz . The effect of such low-frequency GWs appears as a bias to spin-down rates that has a quadrupole pattern in the sky. We use the skewness of the spin-down rate distribution and the number of MSPs with negative spin-down rates to search for the bias induced by GWs. Applying this method to 149 MSPs selected from the ATNF pulsar catalogue, we derive upper bounds on the time derivative of the GW amplitudes of $\dot{h} \lt 6.2 \times 10^{-18}~{\rm s}^{-1}$ and $\dot{h} \lt 8.1 \times 10^{-18}~{\rm s}^{-1}$ in the directions of the Galactic Centre and M87, respectively. Approximating the GW amplitude as $\dot{h} \sim 2 \pi f_{\rm GW} h$, the bounds translate into h < 3 × 10−8 and h < 4 × 10−8, respectively, for fGW = 1/(1000 yr). Finally, we give the implications to possible supermassive black hole binaries at these sites.