We investigate gravitational waves with sub-nanoHz frequencies ( 10 ^ { -11 } Hz \lesssim f _ { GW } \lesssim 10 ^ { -9 } Hz ) from the spatial distribution of the spin-down rates of milli-second pulsars .
As we suggested in \citet yon18 , gravitational waves from a single source induces the bias in the observed spin-down rates of pulsars depending on the relative direction between the source and pulsar .
To improve the constraints on the time derivative of gravitational-wave amplitude obtained in our previous work \citep kum19 , we adopt a more sophisticated statistical method called the Mann-Whitney U test .
Applying our method to the ATNF pulsar catalogue , we first found that the current data set is consistent with no GW signal from any direction in the sky .
Then , we estimate the effective angular resolution of our method to be ( 66 ~ { } { deg } ) ^ { 2 } by studying the probability distribution of the test statistic .
Finally , we investigate gravitational-wave signal from the Galactic Centre and M87 and , comparing simulated mock data sets with the real pulsar data , we obtain the upper bounds on the time derivative as \dot { h } _ { GC } < 8.9 \times 10 ^ { -19 } ~ { } { s } ^ { -1 } for the Galactic Centre and \dot { h } _ { M 87 } < 3.3 \times 10 ^ { -19 } ~ { } { s } ^ { -1 } for M87 , which are stronger than the ones obtained in [ \citeauthoryear Kumamoto et al.2019 ] by factors of 7 and 25 , respectively .