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

TL;DR

This paper develops a formalism to characterize linear polarization in the stochastic gravitational-wave background using pulsar timing arrays, including estimators and covariance analysis for polarization modes.

Contribution

It introduces a method to analyze linear polarization signatures in gravitational-wave backgrounds, expanding polarization maps with spin-weighted spherical harmonics and deriving estimators.

Findings

Derived minimum-variance estimators for polarization coefficients.

Evaluated detectability of polarization signals based on SNR and pulsar count.

Established covariance properties between polarization and intensity modes.

Abstract

Pulsar-timing collaborations have recently reported evidence for the detection of an isotropic stochastic gravitational-wave background consistent with one sourced by a population of inspiralling 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 gravitational-wave 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 spherical-harmonic coefficients for the $B$-modes of the linear polarization and those for the circular polarization, even though both have the same parity. Our approach results in simple, elegant, and easily evaluated expressions for the overlap reduction functions for linear polarization.