Implications of a Running Spectral Index for Slow Roll Inflation

TL;DR

This paper investigates whether a large negative running of the scalar spectral index, as suggested by WMAP data, can arise from a single-field slow-roll inflation model. Using the Hubble slow-roll (HSR) hierarchy and its flow equations, it expresses $n_s$, $r$, and $\alpha$ in terms of $\epsilon$, $\eta$, and $\xi$ and tracks the number of e-folds $N$ after horizon exit for given initial conditions. The main result is that a sizable negative $\alpha$ drives $N$ below $30$ in all viable single-field slow-roll trajectories, effectively ruling out such models unless higher-order slow-roll terms are included or multiple fields are involved; this would imply non-minimal or non-slow-roll inflation. The study highlights that a confirmed large running would challenge the simplest inflationary paradigm, while Planck-era data are expected to clarify whether $\alpha$ truly deviates from zero and constrain the viable inflationary scenarios.

Abstract

We analyze the weak (2 sigma) evidence for a running spectral index seen in the three-year WMAP dataset and its implications for single field, slow roll inflation. We assume that the running is comparable to the central value found from the WMAP data analysis, and use the Hubble Slow Roll formalism to follow the evolution of the slow roll parameters. For all parameter choices consistent with a large, negative running, single field, slow roll inflation lasts less than 30 efolds after CMB scales leave the horizon. Thus, a definitive observation of a large negative running would imply that any inflationary phase requires multiple fields or the breakdown of slow roll. Alternatively, if single field, slow roll inflation is sources the primordial fluctuations, we can expect the observed running to move much closer to zero as the CMB is measured more accurately at small angular scales.

Figures (2)

• Figure 1: We show the regions of the $(\epsilon,\eta)$ plane excluded by the assumed bounds on the spectral parameters. As $\xi$ increases, the portion of the plane for which $\alpha>0$ shrinks.
• Figure 2: We show the regions of the $(\epsilon,\eta)$ plane excluded by the assumed bounds on the spectral parameters for the same cuts displayed in Figure \ref{['pl:cuts1']}. We include contours (for $N=15,30,45,60$) showing the number of e-foldings occuring after the fiducial mode $k_0 =0.002\hbox{Mpc}^{-1}$ leaves the horizon. For $\xi =0.03$ the only regions of the $(\epsilon,\eta)$ plane with an acceptable value of $n_s$ have $N<15$. The running is a weak function of $\epsilon$ and $\eta$ -- the values of $\alpha$ above are taken from $\xi$ alone.