  (abridged version) The separate universe conjecture states that in General
Relativity a density perturbation behaves locally (i.e. on scales much smaller
than the wavelength of the mode) as a separate universe with different
background density and curvature. We prove this conjecture for a spherical
compensated tophat density perturbation of arbitrary amplitude and radius in
$\Lambda$CDM. We then use Conformal Fermi Coordinates to generalize this result
to scalar perturbations of arbitrary configuration and scale. In this case, the
separate universe conjecture holds for the isotropic part of the perturbations.
The anisotropic part on the other hand is exactly captured by a tidal field in
the Newtonian form. We show that the separate universe picture is restricted to
scales larger than the sound horizons of all fluid components. We then derive
an expression for the locally measured matter bispectrum induced by a
long-wavelength mode of arbitrary wavelength. We show that nonlinear
gravitational dynamics does not generate observable contributions that scale
like local-type non-Gaussianity $f_{\rm NL}^{\rm loc}$, and hence does not
contribute to a scale-dependent galaxy bias $\Delta b \propto k^{-2}$ on large
scales; rather, the locally measurable long-short mode coupling assumes a form
essentially identical to subhorizon perturbation theory results, once the
long-mode density perturbation is replaced by the synchronous-comoving gauge
density perturbation. Apparent $f_{\rm NL}^{\rm loc}$-type contributions arise
through projection effects on photon propagation, which depend on the specific
large-scale structure tracer and observable considered, and are in principle
distinguishable from the local mode coupling induced by gravity. We conclude
that any observation of $f_{\rm NL}^{\rm loc}$ beyond these projection effects
signals a departure from standard single-clock inflation.
