Gauge-Invariant Variables on Cosmological Hypersurfaces

We show how gauge-invariant cosmological perturbations may be constructed by an unambiguous choice of hypersurface-orthogonal time-like vector field (i.e., time-slicing). This may be defined either in terms of the metric quantities such as curvature or shear, or using some matter field. As an example, we show how linear perturbations in the covariant fluid-flow approach can then be presented in an explicitly gauge-invariant form in the coordinate based formalism.