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 ¼ dn s =d log k and the running of the running β s ¼ 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 twofield 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 jβ s j ≳ jα s j in some specific models. DOI: 10.1103/PhysRevD.94.021301 Constraining models of inflation is one of the most important goals of cosmology. By constraining or even ruling out models of inflation, cosmologists learn a great deal about model building in theories beyond the standard model. Even with the latest cosmological observations [1,2], there is still a plethora of inflationary models compatible with data [3]. It was recently pointed out that observations of the cosmic microwave background (CMB) radiation are consistent with a rather large running of the running of the spectral index, 1 [4,5]. The constraints on α s and β s given in [5] are α s ¼ 0.011 AE 0.010 and β s ¼ 0.027 AE 0.013 (fixing the pivot scale at k ¼ 0.05 Mpc −1 ), which implies β s > 0 at the 2σ confidence level and hints that the running of the running may be larger than the running itself. The running α s also appears to be positive (although less significantly), which leads to a slight tension with a wide range of inflationary models that predict a negative running [6]. Future CMB experiments are needed to determine these parameters more precisely, but given this first hint of what is potentially a powerful piece of evidence in early universe cosmology, it is interesting and worthwhile to consider the theoretical viability for such a hierarchy of runnings to be realized in inflation.In standard single field slow-roll models, the running is of second order in slow-roll parameters and the running of the running is third order [see Eqs. (5)- (7) below]. Thus, in such models of inflation, one would quite generally expect β s to be smaller than α s . These observational hints motivate the current work, in which we study predictions of α s and β s in single and two-field inflationary scenarios with the intention of understanding what kinds of inflationary models could be consistent with such a hierarchy of runnings.While almost all investigations of inflationary models make predictions for the spectral index, relatively few study the running [6][7][8][9], and almost none discuss the running of the running [4,10]. This is largely underst...