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 ${\alpha }_s=\mathrm{d}{n}_s/\mathrm{d}\log k$ and the running of the running ${\beta }_s=\mathrm{d}{\alpha }_s/\mathrm{d}\log k$ 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 ${\beta }_s$ may actually be positive, and larger than ${\alpha }_s$. 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 assess the feasibility of finding $|{\beta }_s|\gtrsim |{\alpha }_s|$ in some specific models.

• Received 15 June 2016

DOI:https://doi.org/10.1103/PhysRevD.94.021301