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 $\alpha_s =
dn_s/d \log k$ and the running of the running $\beta_s = d\alpha_s/d \log k$
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 $\beta_s$ 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 assessing the
feasibility of finding $|\beta_s| \gtrsim |\alpha_s|$ in some specific models.
