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We describe an alternative approach to the analysis of gravitational-wavebackgrounds, based on the formalism used to characterise the polarisation ofthe cosmic microwave background. In contrast to standard analyses, thisapproach makes no assumptions about the nature of the background and so has thepotential to reveal much more about the physical processes that generated it.An arbitrary background can be decomposed into modes whose angular dependenceon the sky is given by gradients and curls of spherical harmonics. We derivethe pulsar timing overlap reduction functions for the individual modes, whichare given by simple combinations of spherical harmonics evaluated at the pulsarlocations. We show how these can be used to recover the components of anarbitrary background, giving explicit results for both isotropic andanisotropic uncorrelated backgrounds. We also find that the response of apulsar timing array to curl modes is identically zero, so half of thegravitational-wave sky will never be observed using pulsar timing, no matterhow many pulsars are included in the array. An isotropic, unpolarised anduncorrelated background can be accurately represented using only three modes,and so a search of this type will be only slightly more complicated than thestandard cross-correlation search using the Hellings and Downs overlapreduction function. However, by measuring the components of individual modes ofthe background and checking for consistency with isotropy, this approach hasthe potential to reveal much more information. Each individual mode on its owndescribes a background that is correlated between different points on the sky.A measurement of the components that indicates the presence of correlations inthe background on large angular scales would suggest startling new physics.

Mapping gravitational-wave backgrounds using methods from CMB analysis: Application to pulsar timing arrays