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We study the observation of stochastic gravitational-wave background (SGWB)made by pulsar-timing arrays in the spherical harmonic space. Instead of usingthe Shapiro time delay, we keep the Sachs-Wolfe line-of-sight integral for thetiming residual of an observed pulsar. We derive the power spectrum of thetiming residual, from which the overlap reduction functions and the bipolarspherical harmonics coefficients are constructed for the SGWB intensity andpolarization anisotropies. We have reproduced the previous results, noting thatwe have developed a fast algorithm for computing accurate overlap reductionfunctions and the bipolar spherical harmonics coefficients for thelinear-polarization anisotropy are worked out for the first time. Ourharmonic-space method is useful for future pulsar-timing-array observation on afew thousand pulsars and provides optimal estimators for testing thestatistical isotropy of the SGWB.

Timing-residual power spectrum of a polarized stochastic gravitational-wave background in pulsar-timing-array observation