Running of the running and entropy perturbations during inflation

In single field slow-roll inflation, one expects that the spectral index ns − 1 is first order in slow-roll parameters. Similarly, its running αs ¼ dns=d log k and the running of the running βs ¼ dαs=d log k are second and third order and therefore expected to be progressively smaller, and usually negative. Hence, such models of inflation are in considerable tension with a recent analysis hinting that βs may actually be positive, and larger than αs. Motivated by this, in this work we ask the question of what kinds of inflationary models may be useful in achieving such a hierarchy of runnings, particularly focusing on two– field models of inflation in which the late-time transfer of power from isocurvature to curvature modes allows for a much more diverse range of phenomenology. We calculate the runnings due to this effect and briefly apply our results to assess the feasibility of finding jβsj ≳ jαsj in some specific models.