The constancy of \zeta in single-clock Inflation at all loops
  Studying loop corrections to inflationary perturbations, with particular
emphasis on infrared factors, is important to understand the consistency of the
inflationary theory, its predictivity and to establish the existence of the
slow-roll eternal inflation phenomena and its recently found volume bound. In
this paper we show that \zeta-correlators are time-independent at large
distances at all-loop level in single clock inflation. We write the n-th order
correlators of \dot\zeta\ as the time-integral of Green's functions times the
correlators of local sources that are function of the lower order fluctuations.
The Green's functions are such that only non-vanishing correlators of the
sources at late times can lead to non-vanishing correlators for \dot\zeta\ at
long distances. When the sources are connected by high wavenumber modes, the
correlator is peaked at short distances, and these diagrams cannot lead to a
time-dependence by simple diff. invariance arguments. When the sources are
connected by long wavenumber modes one can use similar arguments once the
constancy of \zeta\ at lower orders was established. Therefore the conservation
of \zeta\ at a given order follows from the conservation of \zeta\ at the lower
orders. Since at tree-level \zeta\ is constant, this implies constancy at
all-loops by induction.
