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We investigate gravitational waves with sub-nanoHz frequencies ($10^{-11}$ Hz$\lesssim f_{\rm GW} \lesssim 10^{-9}$ Hz) from the spatial distribution of thespin-down rates of milli-second pulsars. As we suggested in Yonemaru et al.2018, gravitational waves from a single source induces the bias in the observedspin-down rates of pulsars depending on the relative direction between thesource and pulsar. To improve the constraints on the time derivative ofgravitational-wave amplitude obtained in our previous work (Kumamoto et al.2019), we adopt a more sophisticated statistical method called the Mann-WhitneyU test. Applying our method to the ATNF pulsar catalogue, we first found thatthe current data set is consistent with no GW signal from any direction in thesky. Then, we estimate the effective angular resolution of our method to be$(66~{\rm deg})^2$ by studying the probability distribution of the teststatistic. Finally, we investigate gravitational-wave signal from the GalacticCentre and M87 and, comparing simulated mock data sets with the real pulsardata, we obtain the upper bounds on the time derivative as $\dot{h}_{\rm GC} <8.9 \times 10^{-19} {\rm s}^{-1}$ for the Galactic Centre and $\dot{h}_{\rmM87} < 3.3 \times 10^{-19} {\rm s}^{-1}$ for M87, which are stronger than theones obtained in Kumamoto et al. 2019 by factors of 7 and 25, respectively.

Constraints on ultra-low-frequency gravitational waves with statistics of pulsar spin-down rates II: Mann-Whitney U test