We extend our previous work on mapping gravitational-wave backgrounds using techniques borrowed from the analysis of cosmic microwave background data to backgrounds which have non-general-relativity (non-GR) polarisations. Our analysis and results are presented in the context of pulsar timing array observations, but the overarching methods are general, and can be easily applied to LIGO or eLISA observations using appropriately modified response functions. Analytic expressions for the pulsar timing response to gravitational waves with non-GR polarisation are given for each mode of a spin-weighted spherical-harmonic decomposition of the background, which permit the signal to be mapped across the sky to any desired resolution. We also derive the pulsar timing overlap reduction functions for the various non-GR polarisations, finding analytic forms for anisotropic backgrounds with scalar-transverse ("breathing") and vector-longitudinal polarisations, and a semianalytic form for scalar-longitudinal backgrounds. Our results indicate that pulsar timing observations will be completely insensitive to scalar-transverse mode anisotropies in the polarisation amplitude beyond dipole, and anisotropies in the power beyond quadrupole. Analogous to our previous findings that pulsar timing observations lack sensitivity to tensor-curl modes for a transverse-traceless tensor background, we also find insensitivity to vector-curl modes for a vector-longitudinal background.
Gair, Jonathan R., Joseph D. Romano, and Stephen R. Taylor. "Mapping gravitational-wave backgrounds of arbitrary polarisation using pulsar timing arrays." Physical Review D 92, no. 10 (2015): 102003. https://doi.org/10.1103/PhysRevD.92.102003
Physical Review D