  We study the $N$-point function of the density contrast to quadratic order in
the squeezed limit during the matter-dominated (MD) and radiation-dominated
(RD) eras in synchronous gauge. Since synchronous gauge follows the free-fall
frame of observers, the equivalence principle dictates that in the gradient
approximation for the long-wavelength mode there is only a single, manifestly
time-independent consistency relation for the $N$-point function. This simple
form is dictated by the initial mapping between synchronous and local
coordinates, unlike Newtonian gauge and its correspondingly separate dilation
and Newtonian consistency relations. Dynamical effects only appear at quadratic
order in the squeezed limit and are again characterized by a change in the
local background, also known as the separate universe approach. We show that
for the 3-point function the compatibility between these squeezed-limit
relations and second-order perturbation theory requires both the initial and
dynamical contributions to match, as they do in single-field inflation. This
clarifies the role of evolution or late-time projection effects in establishing
the consistency relation for observable bispectra, which is especially
important for radiation acoustic oscillations and for establishing consistency
below the matter-radiation equality scale in the MD era. Defining an
appropriate angle and time average of these oscillations is also important for
making separate universe predictions of spatially varying local observables
during the RD era, which can be useful for a wider range of cosmological
predictions beyond $N$-point functions.
