Non-Gaussianity Consistency Relation for Multi-field Inflation

While detection of the “local form” bispectrum of primordial perturbations would rule out all single-field inflation models, multi-field models would still be allowed. We show that multi-field models described by the $\delta N$ formalism obey an inequality between $f_{\mathrm{N}\mathrm{L}}$ and one of the local-form {\it trispectrum} amplitudes, ${\tau }_{\mathrm{N}\mathrm{L}}$, such that $\tau_{\rm NL}>\frac12(\frac65f_{\rm NL})^2$ with a possible logarithmic scale dependence, provided that 2-loop terms are small. Detection of a violation of this inequality would rule out most of multi-field models, challenging inflation as a mechanism for generating the primoridal perturbations.