The large-N running of the spectral index of inflation

TL;DR

The paper develops a large-$N$ (e-folds) framework, the $N$-formalism, to classify inflationary models into constant, perturbative, non-perturbative, and logarithmic universality classes based on the asymptotic behavior of the equation of state parameter $\epsilon(N)$. It derives the leading predictions for the scalar spectral index $n_s$, the tensor-to-scalar ratio $r$, and the running $\alpha_s$ for each class and assesses the accuracy of the leading $1/N$ terms, noting exceptions in the non-perturbative regime. A key result is that, across the viable models, the running $\alpha_s$ clusters around the permil level, $\alpha_s \sim \mathcal{O}(10^{-3})$, with $\log_{10}|\alpha_s|\approx-3.2$, which has important implications for future observational probes. The analysis provides a unifying perspective on inflationary predictions and guides interpretation of current Planck/BICEP2 constraints and future experiments aimed at measuring small-scale running.

Abstract

We extend previous classifications of inflationary models by means of their behaviour at large-N, where N is the number of e-foldings. In addition to the perturbative 1/N case, whose slow-roll parameters fall off as powers of 1/N, we introduce the constant, non-perturbative and logarithmic classes. This covers the large majority of inflationary models. Furthermore, we calculate the running of the spectral tilt for all these classes. Remarkably, we find that the tilt's runnings essentially cluster around the permil level. We comment on the implications for future experiments.

Figures (3)

• Figure 1: The plane $(n_s, \log_{10} r)$ with the different perturbative models: chaotic, inverse hilltop and hilltop (both with $\mu=1$), as well as the constant model. The colored bars correspond to the range of $N\in [50,60]$. The solid lines are exact while the dashed lines are the leading-$N$ contributions. Note that these essentially agree inside the Planck 2013 region. Also shown for reference is the model of Starobinsky.
• Figure 2: The plane $(n_s, \log_{10} r)$ with the different non-perturbative models: new and natural, as well as the logarithmic models: Moduli and Kähler. The solid lines correspond to the exact expressions, while the dashed lines are the leading $1/N$ contributions. Note that these differ significantly even inside the Planck 2013 region.
• Figure 3: The planes $(n_s, \log_{10} r)$ and $(n_s, \log_{10}(-\alpha_s)$ with all the models discussed in the text. The ranges of values correspond to the interval $N\in[50,60]$. The solid blue (black) lines indicate the Planck (BICEP2)constraints at one and two sigma. The dashed line corresponds to non-minimally coupled chaotic $\lambda\phi^4$ model, $\xi=0 \to 10$. The dot-dashed line corresponds to T-models in a range of values of $\alpha=0.1 \to 10$.