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

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