Running Spectral Index from Inflation with Modulations

TL;DR

The paper investigates whether a large negative running of the scalar spectral index can arise from inflaton potentials with periodic modulations superimposed on smooth large-field potentials. By analyzing linear and power-law potentials with modulations, it derives analytic expressions for the power spectrum, spectral index, and running that include oscillatory contributions driving $n_s$ and $d n_s/d\ln k$ through $V'_{\rm mod}$ while preserving overall slow-roll dynamics. The results show that such modulations yield a spectrum with oscillations and enhanced small-scale power, capable of matching central WMAP7 values at the pivot and producing distinctive signatures potentially detectable by future 21 cm cosmology. These findings connect microphysical structures in the inflaton potential—plausible in string-inspired landscapes—to observable features in the primordial perturbations and offer a framework to reconcile certain inflationary models with data, while suggesting new observational probes of early-universe physics.

Abstract

We argue that a large negative running spectral index, if confirmed, might suggest that there are abundant structures in the inflaton potential, which result in a fairly large (both positive and negative) running of the spectral index at all scales. It is shown that the center value of the running spectral index suggested by the recent CMB data can be easily explained by an inflaton potential with superimposed periodic oscillations. In contrast to cases with constant running, the perturbation spectrum is enhanced at small scales, due to the repeated modulations. We mention that such features at small scales may be seen by 21 cm observations in the future.

Figures (6)

• Figure 1: A sketch of the inflaton potential with superimposed periodic oscillations. The dashed line is a smooth potential without modulations. The modulations are amplified for visualization purposes. The resulting curvature perturbation and its tilt at scales exiting the horizon as the inflaton approaches the markers are shown in Fig. \ref{['fig:im_sp']} and Fig. \ref{['fig:im_ti']}. The observed CMB scales are considered to lie within a half oscillation period.
• Figure 2: The curvature perturbation spectrum generated from the inflaton potential in Fig. \ref{['fig:im_pot']}.
• Figure 3: Spectral index and its running from the inflaton potential in Fig. \ref{['fig:im_pot']}.
• Figure 4: Curvature perturbation spectrum (black solid line) from the inflaton potential (\ref{['linpot']}), realizing central values of (\ref{['WMAP7']}) at the pivot scale $k_0 = 0.002 \mathrm{\, Mpc^{-1}}$ which is denoted by the filled circle. Spectrum with constant running (\ref{['simpleex']}) extrapolated from values at the pivot scale is shown as the black dotted line. Spectrum from a linear potential without any modulation is shown as the black dashed line for comparison.
• Figure 5: Variation of the spectral index $n_s$ and its running $dn_s / d\ln k$ as inflation proceeds along the modulated potential (\ref{['linpot']}). The plot starts from 70 e-folds before the end of inflation (end point of the innermost orbit), up to $\sim$ 10 e-folds before the end of inflation where most part of the orbit goes beyond the plotted region. Filled circle corresponds to the pivot scale (about $50$ e-folds before the end of inflation). The scale which exited the horizon 15 e-folds before (after) the pivot scale is indicated by an open (filled) star. Superimposed contours are constraints from 5-year WMAP (blue) and from 5-year WMAP+BAO+SN (red) (68% and 95% CL), when tensor moder perturbations, spectral tilt, and running index are allowed to vary Komatsu:2008hk.