A Note on the Consistency Condition of Primordial Fluctuations

TL;DR

The work derives generalized consistency relations for primordial fluctuations by linking squeezed-limit $N$-point functions to the spatial variation of local observables, extending the classic single-clock inflation result to cases where short modes are inside the horizon and derivatives act on fluctuations. Through a local gauge reinterpretation of long-wavelength modes as a background rescaling, the authors show that leading squeezed-limit behavior is captured by derivatives of $Q_N$ with respect to $\ln k$ and the soft-mode power $P(k_L)$, with explicit tree-level formulas for external and internal soft legs and for multiple soft momenta. They also analyze derivative operators and quantify subleading corrections, arguing most linear-in-$k_L$ corrections arise from coordinate changes and are thus suppressed to $\mathcal{O}((k_L/k_S)^2)$; the framework extends to loop-level IR effects, where IR issues are coordinate artifacts that cancel in physical (distance) measurements. Overall, the paper provides a versatile, gauge-consistent toolkit for assessing infrared effects in inflation and for testing single-clock inflation, including complex squeezed configurations and internal soft limits.

Abstract

We show that the squeezed limit of (N+1)-point functions of primordial correlation functions in which one of the modes has a very small wavenumber can be inferred from the spatial variation of locally measured N-point function. We then show how in single clock inflation a long wavelength perturbation can be re-absorbed in the background cosmology and how in computing correlation functions the integrals of the interaction Hamiltonian are dominated by conformal times of order of the short wavelength modes, when the long mode is already outside of the horizon. This allows us to generalize the consistency condition for N-point functions to the case in which the short wavelength fluctuations are inside the horizon and derivatives acts on them. We further discuss the consistency condition in the soft internal squeezed limit in which in an (N+M)-point function with (N+M) short modes the sum of the first N modes is a very soft momentum. These results are very useful to study infrared effects in Inflation.

Figures (2)

• Figure 1: Consistency condition with one soft external momentum. Dashed red lines connecting the vertices or the vertices with the external operators lead to $\zeta^{cl}(k,t_a) \zeta^{cl*}(k,t_b)$ where $t_{a,b}$ are the time of evaluation of the operators $\zeta_{ki}(t)$ or of the Hamiltonian $H_I$. Because of the contour rotation, the wavefunctions of operators with momentum $k$ associated to the interaction Hamiltonians decay at early time with a time scale of order $\delta \eta\sim 1/k$. Therefore, all the vertices must be close in time to the final time, within a $\delta\eta\sim 1/k_S$. This means that the long wavelength fluctuation $\zeta_{k_L}$ is out of the horizon both at the final time and at the time of evaluation of the integrals. This means that at all times involved in the calculation $\zeta_{k_L}$ can be treated as a rescaling of the coordinates, implying the consistency condition. This also means that the result would have not changed if long wavelength fluctuation $\zeta(k_L)$, which in the plot is represented as evaluated at the same final time as the short wavelength fluctuations, had been evaluated at a different final time when the mode is still outside of the horizon.
• Figure 2: Consistency condition with one soft internal momentum: in this case, for the particular momenta configurations that we have chosen, there are some diagrams where one of the lines connecting two vertices has a very low momentum. These are the diagrams that dominate the $N$-point functions. Since all the vertices are connected to an external $\zeta$ which has a short momentum, the interaction has to happen at a time close to the final time by a time $\delta\eta\sim 1/k_S$, with $k_S$ being the typical momentum of the short fluctuations. This means that at the time of evaluation of the integrals the long mode is already outside of the horizon and can therefore be re-absorbed with a rescaling of the coordinates. This implies that the consistency condition holds.