  We consider the relational approach to construct gauge-invariant observables
in cosmological perturbation theory using synchronous coordinates. We construct
dynamical synchronous coordinates as non-local scalar functionals of the metric
perturbation in the fully non-linear theory in an arbitrary gauge. We show that
the observables defined in this dynamical coordinate system are
gauge-independent, and that the full perturbed metric has the expected form in
these coordinates. Our construction generalises the familiar synchronous gauge
in linearised gravity, widely used in cosmological perturbation theory, to the
non-linear theory. We also work out the expressions for the gauge-invariant
Einstein equation, sourced either by an ideal fluid or a scalar field up to
second order in perturbation theory, and give explicit expressions for the
Hubble rate -- as measured by synchronous observers or by observers co-moving
with the matter field -- up to that order. Finally, we consider quantised
linear perturbations around Minkowski and de~Sitter backgrounds, and compute
the two-point function of the gauge-invariant metric perturbation in
synchronous coordinates, starting with two-point function in a general linear
covariant gauge. Although the gauge-fixed two-point function contains gauge
modes, we show that the resulting gauge-invariant two-point function only
contains the physical tensor modes and it is thus positive, \ie, it has a
spectral representation.
