The consistency equation hierarchy in single-field inflation models

TL;DR

This paper shows that single-field slow-roll inflation entails an infinite hierarchy of consistency relations linking the scalar and tensor spectra across all scales. It derives explicit leading- and next-to-leading slow-roll expressions for the entire hierarchy and clarifies how these results subsume prior approximate relations, including showing that the CST proposal is equivalent to the second consistency equation. The work provides a cohesive framework tying together $A_S(k)$, $A_T(k)$, $n_S$, $n_T$, and their derivatives, with explicit differentiation rules that generate higher-order relations. While testing beyond the first, canonical relation is observationally challenging, the hierarchy offers a complete specification of how scalar and tensor observables must relate if single-field slow-roll inflation in Einstein gravity holds. The analysis also clarifies the status of proposed approximate relations such as constant running and scale-coincidence tests, showing when they align with or depart from the full consistency-hierarchy predictions.

Abstract

Inflationary consistency equations relate the scalar and tensor perturbations. We elucidate the infinite hierarchy of consistency equations of single-field inflation, the first of which is the well-known relation A_T^2/A_S^2 = -n_T/2 between the amplitudes and the tensor spectral index. We write a general expression for all consistency equations both to first and second-order in the slow-roll expansion. We discuss the relation to other consistency equations that have appeared in the literature, in particular demonstrating that the approximate consistency equation recently introduced by Chung and collaborators is equivalent to the second consistency equation of Lidsey et al. (1997).