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The relativistic theory of structure formation in cosmology is based mainlyon linear perturbations about a homogeneous background. But we are now drivento understand the theory of higher-order perturbations in full detail, bothfrom observational and theoretical points of view. An important aspect of thislies in defining gauge-invariant perturbations at any order. We present a newcovariant approach to this based on a local separation of tensors into theirscalar, vector and tensor parts. Such a local decomposition necessarilyrequires a trio of mutually annihilating differential operators which form thebasis for defining gauge-invariant objects. It makes no use of non-localGreen's functions or pre-defined gauges, and can be used to define families ofscalar, vector and tensor modes at any order one chooses.

Local gauge-invariance at any order in cosmological perturbation theory