Systematics of adiabatic modes: flat universes

Abstract

Adiabatic modes are cosmological perturbations that are locally indistinguishable from a (large) change of coordinates. At the classical level, they provide model independent solutions. At the quantum level, they lead to soft theorems for cosmological correlators. We present a systematic derivation of adiabatic modes in spatially-flat cosmological backgrounds with asymptotically-perfect fluids. We find several new adiabatic modes including vector, time-dependent tensor and time-dependent scalar modes. The new vector and tensor modes decay with time in standard cosmologies but are the leading modes in contracting universes. We present a preliminary derivation of the related soft theorems. In passing, we discuss a distinction between classical and quantum adiabatic modes, we clarify the subtle nature of Weinberg's second adiabatic mode and point out that the adiabatic nature of a perturbation is a gauge dependent statement. 2 The new scalar adiabatic mode survives only if the scalar field is shift symmetric and leads to soft theorems for Ultra Slow Roll inflation [27]. 3 We use the notation of [25].

Soft theorems for shift-symmetric cosmologies

“…The adiabatic profiles ΔδN and Δψ are similarly determined from (10) and (9), the latter of which is needed for finding the homogeneous part of Δψ. This mode can further be shown to exist for a generic perfect fluid [12]. The addition of the second term in (12) ensures that this profile solves the ζ equations of motion in the soft q → 0 limit.…”

“…This mode can further be shown to exist for a generic perfect fluid [12]. The addition of the second term in (12) ensures that this profile solves the ζ equations of motion in the soft q → 0 limit. We have checked this explicitly to lowest order for a pure PðXÞ theory: the nonlinear shift in ζ must solve the linear equations of motion at finite momentum…”

“…Though the spatially constant part of ψ is immaterial at q ¼ 0, it is required for correctly extracting the physical q ≠ 0 mode (see Ref. [12]). We now consider the perturbations generated by ξ μ acting on an unperturbed FLRW,…”

Spatial curvature at the sound horizon

“…This first result has been extended to higher n-point functions for primordial scalar, tensor and vector perturbations [8,9,10,11,12,13,14,15,16,17,18,19]. Soft theorems are conveniently interpreted as the consequence of residual, non-linearly realized symmetries associated with adiabatic modes [20,9,21,18,22], namely physical perturbations that are indistinguishable from a change of coordinates in the neighborhood of a point in spacetime. In the presence of additional symmetries beyond diffeomorphism invariance, new adiabatic modes and new soft theorems can be derived.…”

“…The difference is only of order ΩK , much smaller than the ΩK /c 2 s effect that we are after here. 18 For cs 1, the linear combination αf eq N L + βf orth N L is an unbiased estimator for z = derived only at linear order in K and so they should be trusted as long as…”