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We consider the relational approach to construct gauge-invariant observablesin cosmological perturbation theory using synchronous coordinates. We constructdynamical synchronous coordinates as non-local scalar functionals of the metricperturbation in the fully non-linear theory in an arbitrary gauge. We show thatthe observables defined in this dynamical coordinate system aregauge-independent, and that the full perturbed metric has the expected form inthese coordinates. Our construction generalises the familiar synchronous gaugein linearised gravity, widely used in cosmological perturbation theory, to thenon-linear theory. We also work out the expressions for the gauge-invariantEinstein equation, sourced either by an ideal fluid or a scalar field up tosecond order in perturbation theory, and give explicit expressions for theHubble rate -- as measured by synchronous observers or by observers co-movingwith the matter field -- up to that order. Finally, we consider quantisedlinear perturbations around Minkowski and de~Sitter backgrounds, and computethe two-point function of the gauge-invariant metric perturbation insynchronous coordinates, starting with two-point function in a general linearcovariant gauge. Although the gauge-fixed two-point function contains gaugemodes, we show that the resulting gauge-invariant two-point function onlycontains the physical tensor modes and it is thus positive, \ie, it has aspectral representation.

Synchronous coordinates and gauge-invariant observables in cosmological spacetimes