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.