Running of the Running and Entropy Perturbations During Inflation

Abstract

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 $\hat{I}{\pm }_s=d{n}_s/d\log k$ and the running of the running ${\hat{I}}_s^2=d\hat{I}{\pm }_s/d\log k$ 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 ${\hat{I}}_s^2$ may actually be positive, and larger than $\hat{I}{\pm }_s$. 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 $|{\hat{I}}_s^2|\gtrsim |\hat{I}{\pm }_s|$ in some specific models.

6 pages, 2 figures. Fixes typos present in published version. Corrected numerical parameters in example. General results remain unchanged