Linear polarization of the stochastic gravitational-wave background with pulsar timing arrays

Pulsar-timing collaborations have recently reported evidence for the detection of an isotropic stochastic gravitational-wave background (SGWB) consistent with one sourced by a population of inspiraling supermassive black hole binaries. However, a certain degree of anisotropy and polarization may be present. Thus, the characterization of the energy density and polarization of the background at different angular scales is important. In this paper, we describe the signatures of linear polarization in the stochastic gravitationalwave background on the timing residuals obtained with pulsar-timing arrays. We expand the linear polarization map in terms of spin-weighted spherical harmonics and recast it into the E-mode (parity even) and B-mode (parity odd) basis. We provide expressions for the minimum-variance estimators for the coefficients of that expansion and evaluate the smallest detectable signal as a function of the signal-to-noise ratio with which the isotropic GW signal is detected and the number of pulsars in the survey. We evaluate the covariance between the estimators for the spherical-harmonic coefficients of the linear polarization E modes and those for the intensity anisotropy. We also show that there is no covariance between the sphericalharmonic coefficients for the B modes of the linear polarization and those for the circular polarization, even though both have the same parity. Under the simplifying assumptions of our harmonic analysis, we show that detection of the lowest-order linear-polarization anisotropies is only possible if the isotropic intensity contribution is measured with a signal-to-noise ratio >= 300, with a pulsar-timing network comprised of more than 100 pulsars. We also show that, in this limit, the lowest order anisotropies in the E mode have a negligible covariance with corresponding anisotropies in the SGWB intensity. Our approach results in simple, elegant, and easily evaluated expressions for the overlap reduction functions for linear polarization.