EFT beyond the horizon: stochastic inflation and how primordial quantum fluctuations go classical

Abstract

We identify the effective field theory describing the physics of super-Hubble scales and show it to be a special case of a class of effective field theories appropriate to open systems. Open systems are those that allow information to be exchanged between the degrees of freedom of interest and those that are integrated out, such as would be appropriate for particles moving through a fluid. Strictly speaking they cannot in general be described by an effective lagrangian; rather the appropriate ‘low-energy’ limit is instead a Lindblad equation describing the time-evolution of the density matrix of the slow degrees of freedom. We derive the equation relevant to super-Hubble modes of quantum fields in de Sitter (and near-de Sitter) spacetimes and derive two of its implications. We show that the evolution of the diagonal density-matrix elements quickly approach the Fokker-Planck equation of Starobinsky’s stochastic inflationary picture. This allows us both to identify the leading corrections and provide an alternative first-principles derivation of this picture’s stochastic noise and drift. (As applications we show that the noise for massless fields is independent of the details of the window function used, and also compute how the noise changes for systems with a sub-luminal speed of sound, cs< 1.) We then argue that the presence of interactions drive the off-diagonal density-matrix elements to zero in the field