Dynamical Solution to the Eta Problem in Spectator Field Models

TL;DR

This work presents a dynamical mechanism where quantum corrections to the Hubble-induced mass of a spectator field create a flattened minimum and attractor behavior, naturally producing a slightly red-tilted curvature perturbation spectrum with ns ≈ 0.96 and a calculable running α_s. In curvaton setups with a quadratic vacuum potential, the model yields a predictive link between local non-Gaussianity f_NL^local and α_s, providing a falsifiable signature for forthcoming observations. The authors further extend the idea to complex scalar fields with an approximate U(1) symmetry, showing that the angular component can serve as the spectator field with similar spectral properties, while remaining natural in supersymmetric contexts. Overall, the framework relaxes eta-related fine-tuning, offers concrete observational predictions for f_NL and α_s, and connects SUSY-inspired scalar sectors to cosmological perturbation observables.

Abstract

We study a class of spectator field models that addresses the eta problem while providing a natural explanation for the observed slight deviation of the spectrum of curvature perturbations from scale-invariance. In particular, we analyze the effects of quantum corrections on the quadratic potential of the spectator field given by its gravitational coupling to the Ricci scalar and the inflaton energy, so-called the Hubble-induced mass term. These quantum corrections create a minimum around which the potential is flatter and to which the spectator field is attracted. We demonstrate that this attractor dynamics can naturally generate the observed slightly red-tilted spectrum of curvature perturbations. Furthermore, focusing on a curvaton model with a quadratic vacuum potential, we compute the primordial non-Gaussianity parameter $f_{\text{NL}}$ and derive a predictive relationship between $f_{\text{NL}}$ and the running of the scalar spectral index. This relationship serves as a testable signature of the model. Finally, we extend the idea to a broader class of models where the spectator field is an angular component of a complex scalar field.

Figures (9)

• Figure 1: The eta parameter of the spectator field as a function of $r_{\sigma} = \sigma/\sigma_0$ for different values of $b$, where $b \in [0.02, 0.1]$. The quantum corrections create a minimum $(r=1)$ around which the eta parameter is $O(0.01)$.
• Figure 2: The left panel shows the spectral index verses the number of inflationary e-folds for $b \in [0.02,0.1]$ and fixed $r_{\sigma,i} = 10^{-5}$. The solid curves show the spectral index computed using the slow-roll approximation as in Eq. \ref{['eq:n_s_analytical']}, while the dashed curves show the spectral index evaluated by numerically computing $r_{\sigma}(N)$. The blue-shaded region highlights areas where the scalar spectral index is $n_{s} = 0.965 \pm 0.004$ as measured by Planck at the $1\sigma$ level Planck:2018vyg, while the red-shaded region corresponds to areas where $|0.965-n_{s}| \le 0.02$. The right panel shows the range of the number of inflationary e-folds as a function of $b$, where the shaded regions correspond to the same ranges of $n_{s}$ as those in the left panel.
• Figure 3: The fine-tuning measure $F$ as a function of our model parameter $b$. The results show that when $b < 0.02$, $F<3$ meaning that the required fine-tuning is minimal. On the other hand, when $b > 0.1$, $F>10$, indicating that our model becomes increasingly unnatural for larger values of $b$.
• Figure 4: The red curve shows the running of the spectral index at $n_{s} = 0.965$ as a function of the potential parameter $b$. The blue-shaded region represents $n_{s}=0.965\pm0.004$, consistent with the $1\sigma$ constraints from Planck measurements. The gray-shaded region is excluded by Planck measurements of $\alpha_{s}$ at the $2\sigma$ level.
• Figure 5: The non-Gaussianity parameter $f_{\text{NL}}^{\text{local}}$ as a function of $b$ for $\Delta N \in \{50,60,70\}$ assuming the curvaton dominates before its decay (left) and $\alpha_s$ at $n_{s} = 0.965$ (right), where $\Delta N$ captures the duration of the non-harmonic curvaton evolution after horizon exit as defined in Eq. \ref{['eq:fNL_2']}. The gray-shaded regions are excluded by Planck$f_{\text{NL}}^{\text{local}}$ measurements at the $2\sigma$ level.