Local gauge-invariance at any order in cosmological perturbation theory

The relativistic theory of structure formation in cosmology is based mainly on linear perturbations about a homogeneous background. But we are now driven to understand the theory of higher-order perturbations in full detail, both from observational and theoretical points of view. An important aspect of this lies in defining gauge-invariant perturbations at any order. We present a new covariant approach to this based on a local separation of tensors into their scalar, vector and tensor parts. Such a local decomposition necessarily requires a trio of mutually annihilating differential operators which form the basis for defining gauge-invariant objects. It makes no use of non-local Green's functions or pre-defined gauges, and can be used to define families of scalar, vector and tensor modes at any order one chooses.