We describe an alternative approach to the analysis of gravitational-wave backgrounds, based on the formalism used to characterise the polarisation of the cosmic microwave background. In contrast to standard analyses, this approach makes no assumptions about the nature of the background and so has the potential to reveal much more about the physical processes that generated it. An arbitrary background can be decomposed into modes whose angular dependence on the sky is given by spin-weighted spherical harmonics. For the plus and cross polarisation states predicted by general relativity, we require spin-2 harmonics, which can also be written as (second) gradients and curls of scalar spherical harmonics. Backgrounds with non-GR polarisation states can be decomposed in a similar way, using spin-1 spherical harmonics for the two possible vector polarisations (or first gradients and curls of scalar spherical harmonics) and scalar spherical harmonics for the scalar transverse and scalar longitudinal polarisations.
We will describe how this approach can be used to map backgrounds with pulsar timing arrays (PTAs). We will derive the pulsar timing overlap reduction functions for each individual mode, which are given by combinations of scalar spherical harmonics evaluated at the pulsar locations, multiplied by weights that are functions of the distance to the pulsars in the array. The response of a pulsar timing array to the (tensor and vector) curl modes is identically zero, so a portion of the gravitational-wave sky will never be observed using pulsar timing, no matter how many pulsars are included in the array. An isotropic, unpolarised and uncorrelated background of GR polarisation can be accurately represented using only three modes, and so a search of this type will be only slightly more complicated than the standard cross-correlation search using the Hellings and Downs overlap reduction function. However, by measuring the components of individual modes of the background and checking for consistency with isotropy, this approach has the potential to reveal much more information. Each individual mode on its own describes a background that is correlated between different points on the sky. A measurement of the components that indicates the presence of correlations in the background on large angular scales would suggest startling new physics. While the focus of this talk will be on pulsar timing arrays, we will also describe extensions of this analysis to the mapping of gravitational-wave backgrounds with ground-based interferometers.