We develop a method for treating the consistency relations of inflation that includes the full time-evolution of the state. This approach relies only on the symmetries of the inflationary setting, in particular a residual conformal symmetry in the sp
atial part of the metric, along with general properties which hold for any quantum field theory. As a result, the consistency relations that emerge, which are essentially the Slavnov-Taylor identities associated with this residual conformal symmetry, apply very generally: they are true of the full Greens functions, hold largely independently of the particular inflationary model, and can be used for arbitrary states. We illustrate these techniques by showing the form assumed by the standard consistency relation between the two and three-point functions for the primordial scalar fluctuations when they are in a Bunch-Davies state. But because we have included the full evolution of the state, this approach works for a general initial state as well and does not need to have assumed that inflation began in the Bunch-Davies state.
It is possible to define a general initial state for a quantum field by introducing a contribution to the action defined at an initial-time boundary. The propagator for this theory is composed of two parts, one associated with the free propagation of
fields and another produced by the operators of this initial action. The derivation of this propagator is shown for the case of a translationally and rotationally invariant initial state. In addition to being able to treat more general states, these techniques can also be applied to effective field theories that start from an initial time. The eigenstates of a theory with interacting heavy and light fields are different from the eigenstates of the theory in the limit where the interactions vanish. Therefore, a product of states of the noninteracting heavy and light theories will usually contain excitations of the heavier state once the interactions are included. Such excitations appear as nonlocal effects in the effective theory, which are suppressed by powers of the mass of the heavy field. By appropriately choosing the initial action, these excitations can be excised from the state leaving just effects that would be produced by a local action of the lighter fields.
We use the in-in or Schwinger-Keldysh formalism to explore the construction and interpretation of effective field theories for time-dependent systems evolving out of equilibrium. Starting with a simple model consisting of a heavy and a light scalar f
ield taken to be in their free vacuum states at a finite initial time, we study the effects from the heavy field on the dynamics of the light field by analyzing the equation of motion for the expectation value of the light background field. New terms appear which cannot arise from a local action of an effective field theory in terms of the light field, though they disappear in the adiabatic limit. We discuss the origins of these terms as well as their possible implications for time dependent situations such as inflation.
These notes present a detailed introduction to Maldacenas calculation of the three-point function generated by the simplest class of inflationary models: those with a single inflaton field whose potential satisfies the slow-roll conditions and whose
quantum fluctuations start in the asymptotic Bunch-Davies vacuum state. The three-point function should be the most readily observed evidence for non-Gaussianities amongst the primordial fluctuations produced by inflation. In these inflationary theories the non-Gaussianities are predicted to be extremely small, being naturally suppressed by the small slow-roll parameters.
