Consistency equation hierarchy in single-field inflation models

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 $A_{\mathrm{T}}^2/A_{\mathrm{S}}^2=-n_{\mathrm{T}}/2$ 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, demonstrating in particular that the approximate consistency equation recently introduced by Chung and collaborators is equivalent to the second consistency equation of Lidsey et al. (1997).