Abstract
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 spatial 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 Green's 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. We explain how the Slavnov-Taylor identity is modified for these more general states.
Highlights
Both sides of this expression are evaluated at a late time t∗ when the wavelengths responsible for the correlations seen in the cosmic microwave background and large-scale structure have been inflated to a size larger than the inflationary horizon
Within a wide class of inflationary models—those with a single inflaton field whose potential obeys a set of ‘slow-roll’ conditions, and where the fluctuations are in the BunchDavies state—this relation says that the amplitude of the three-point function in this ‘soft’ limit should be small, since the basic properties of the power spectrum are already known through the observations made by Planck and its predecessors
The symmetry that remains even after we have made a general choice for the quantum fluctuations about an inflationary background leads to relations amongst their correlation functions
Summary
We begin by briefly reviewing the classical background used in inflation. The simplest models of inflation contain a single scalar field φ(t, x) with a potential V (φ). We use our freedom to choose our coordinates so that they have two useful properties: first, that the inflaton field has no fluctuations at all, and is given entirely by its background value, φ(t, x) = φ(t), and second, that the spatial part of the metric remains proportional to δij. The fact that the change of coordinates can be related to an overall change in the factor multiplying the flat spatial metric means that ξi(t, x) is generating a conformal transformation. We can make this property still more explicit by taking the trace of this equation, which allows us to solve for δζ directly, δζ ξ k ∂k ζ. A fuller analysis of these residual symmetries, including a treatment of what additional time-dependent information can be gleaned by using the adiabatic properties [18] of the fluctuations at larger scales, is found in [5, 6]
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