Effect of Background Evolution on the Curvaton Non-Gaussianity

TL;DR

This work investigates how the background evolution of the universe, characterized by $\rho_{\rm BG} \propto a^{-\alpha}$, and a curvaton potential with a small non-quadratic term influence the curvature perturbation and its non-Gaussianity, using the $\delta N$ formalism. It derives general expressions for the curvature perturbation $\zeta$ and the nonlinearity parameters $f_{\rm NL}$ and $g_{\rm NL}$ for a background with arbitrary equation of state and demonstrates that deviations from a purely quadratic potential amplify the sensitivity of these observables to the background parameter $\alpha$. The authors show that the background can significantly modify $f_{\rm NL}$ and $g_{\rm NL}$, providing quantitative examples (e.g., a dimension-6 term at 5% with $r_{\rm dec}=0.01$) where $\Delta f_{\rm NL} \sim \mathcal{O}(10)$ and $\Delta g_{\rm NL} \sim \mathcal{O}(10^4)$ between matter- and radiation-dominated backgrounds. They further highlight a $f_{\rm NL}$–$g_{\rm NL}$ relation that depends on the potential and background, implying that combined measurements of the bispectrum and trispectrum can probe both curvaton self-interactions and the background equation of state, potentially offering insights into inflaton dynamics at the end of inflation.

Abstract

We investigate how the background evolution affects the curvature perturbations generated by the curvaton, assuming a curvaton potential that may deviate slightly from the quadratic one, and parameterizing the background fluid density as ρ\propto a^{-α}, where a is the scale factor, and αdepends on the background fluid. It turns out that the more there is deviation from the quadratic case, the more pronounced is the dependence of the curvature perturbation on α. We also show that the background can have a significant effect on the nonlinearity parameters f_NL and g_NL. As an example, if at the onset of the curvaton oscillation there is a dimension 6 contribution to the potential at 5 % level and the energy fraction of the curvaton to the total one at the time of its decay is at 1 %, we find variations Δf_NL \sim \mathcal{O}(10) and Δg_NL \sim \mathcal{O}(10^4) between matter and radiation dominated backgrounds. Moreover, we demonstrate that there is a relation between f_NL and g_NL that can be used to probe the form of the curvaton potential and the equation of state of the background fluid.

Figures (8)

• Figure 1: Plots of $f_{\rm NL}$ and $g_{\rm NL}$ as a function of $n$ for several values of $\alpha$. The values of $s$ and $r_{\rm dec}$ are taken as $s=0.05$ and $r_{\rm dec}=0.01$.
• Figure 2: Plots of $\tilde{\zeta}_1 = \zeta_1 /\zeta_1^{\rm (quadratic)}$, which is normalized to $\zeta_1$ for the pure quadratic case, relative to that for the cases with $\alpha =4$ (left) and $\alpha =3$ (right). The values of $s$ and $r_{\rm dec}$ are taken as $s=0.05$ and $r_{\rm dec}=0.01$.
• Figure 3: Plots of $f_{\rm NL}$ relative to that for the cases with $\alpha =4$ (left) and $\alpha =3$ (right). The values of $s$ and $r_{\rm dec}$ are taken as $s=0.05$ and $r_{\rm dec}=0.01$.
• Figure 4: Plots of $g_{\rm NL}$ relative to that for the cases with $\alpha =4$ (left) and $\alpha =3$ (right). The values of $s$ and $r_{\rm dec}$ are taken as $s=0.05$ and $r_{\rm dec}=0.01$.
• Figure 5: Plots of $\tilde{\zeta}_1 = \zeta_1 /\zeta_1^{\rm (quadratic)}$, which is normalized to $\zeta_1$ for the pure quadratic case, relative to that for the cases with $\alpha =4$ (left) and $\alpha =3$ (right). The values of $n$ and $r_{\rm dec}$ are taken as $n=8$ and $r_{\rm dec}=0.01$.