Running of the Scalar Spectral Index from Inflationary Models

TL;DR

This paper tackles the possibility of large negative running of the scalar spectral index $n$ within inflationary models. It shows that achieving substantial $dn/d\ln k$ typically requires a large third derivative of the potential, i.e., a pronounced feature or bump in the potential, and it introduces two constructive frameworks—the singular method and the index method—for engineering such potentials. The work also analyzes noncanonical kinetic terms as a way to realize large running, while highlighting the practical difficulty of sustaining many e-folds with strong running. Collectively, the results provide a structured approach to model-building under running constraints and clarify the observational implications for inflationary dynamics, including the potential need for features in the inflaton sector. The findings have broad relevance for interpreting cosmological data and for connecting inflationary phenomenology to underlying high-energy theories, such as supersymmetry and extra dimensions.

Abstract

The scalar spectral index n is an important parameter describing the nature of primordial density perturbations. Recent data, including that from the WMAP satellite, shows some evidence that the index runs (changes as a function of the scale k at which it is measured) from n>1 (blue) on long scales to n<1 (red) on short scales. We investigate the extent to which inflationary models can accomodate such significant running of n. We present several methods for constructing large classes of potentials which yield a running spectral index. We show that within the slow-roll approximation, the fact that n-1 changes sign from blue to red forces the slope of the potential to reach a minimum at a similar field location. We also briefly survey the running of the index in a wider class of inflationary models, including a subset of those with non-minimial kinetic terms.

Figures (2)

• Figure 1: The upper figure shows the potential as a function of the inflaton field $\phi$. The vertical long-dashed curve in the upper figure corresponds to a field value $63$ efolds before the end of inflation. In the lower figure, the solid curve corresponds to the behavior of $\eta$ while the short-dashed curve corresponds to the behavior of $\epsilon$. Note that inflation ends due to $\eta$ and not $\epsilon$. Also, one can clearly see that $\epsilon$ has a minimum near $\phi =1$ where $n-1$ changes sign. About 10 e-folds after the dashed line corresponds to $\phi\approx 1.1$.
• Figure 2: $n-1$ and $dn/d\ln k$ as a function of $N_{tot}-N$ where $N$ is the number of e-folds before the end of inflation and $N_{tot}\approx 63$ is the total number of efolds.