  Cosmological perturbation theory is an example of a gauge theory, where gauge
transformations correspond to changes in the space-time coordinate system. To
determine physical quantities, one is free to introduce gauge conditions (i.e.
to work with specific space-time coordinates), and such conditions are often
used to simplify technical aspects of the calculation or to facilitate the
interpretation of the physical degrees of freedom. Some of the prescriptions
introduced in the literature are known to fix the gauge only partially, but it
is commonly assumed that the remaining gauge degrees of freedom can be fixed
somehow. In this work, we show that this is not necessarily the case, and that
some of these gauges are indeed pathological. We derive a systematic procedure
to determine whether a gauge is pathological or not, and to complete
partially-fixed gauges into healthy gauges when this is possible. In this
approach, the Lagrange multipliers (i.e. the perturbed lapse and shift in the
ADM formalism) cannot appear in the off-shell definition of the gauges, they
necessarily arise as on-shell consequences of the gauge conditions. As
illustrative applications, we propose an alternative, non-pathological
formulation of the synchronous gauge, and we show that the uniform-expansion
gauge (as well as any gauge ensuring vanishing lapse perturbations) can hardly
be made healthy. Our methodology also allows us to construct all
gauge-invariant variables. We further show that at the quantum level, imposing
that the constraints are satisfied necessarily leads to gauge-invariant quantum
states, hence any quantum treatment of cosmological perturbations precludes the
gauge from being fixed. We finally discuss possible generalisations of our
formalism.
