The lessons from the running of the tensor-to-scalar ratio

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We derive a simple consistency relation from the running of the tensor-to-scalar ratio. This new relation is first order in the slow-roll approximation. While for single field models we can obtain what can be found by using other observables, multi-field cases in general give non-trivial contributions dependent on the geometry of the field space and the inflationary dynamics, which can be probed observationally from this relation.