Running of the Running and Entropy Perturbations During Inflation

In single field slow-roll inflation, one expects that the spectral index $n_s -1$ is first order in slow-roll parameters. Similarly, its running $α_s = dn_s/d \log k$ and the running of the running $β_s = dα_s/d \log k$ 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 $β_s$ may actually be positive, and larger than $α_s$. 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 assessing the feasibility of finding $|β_s| \gtrsim |α_s|$ in some specific models.