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Negative spectral index of $f_{NL}$ in the axion-type curvaton model

We derive the spectral index of $f_{NL}$ and its running from isocurvature single field and investigate the curvaton models with a negative spectral index of $f_{NL}$ in detail. In particular, a numerical study of the axion-type curvaton model is illustrated, and we find that the spectral index of $f_{NL}$ is negative and its absolute value is maximized around $σ_*=πf/2$ for the potential $V(σ)=m^2f^2(1-\cos{σ\over f})$. The spectral index of $f_{NL}$ can be ${\cal O}(-0.1)$ for the axion-type curvaton model. A convincing detection of a positive $n_{f_{NL}}$ will rule out the axion-type curvaton model. In addition, we also give a general discussion about the detectable parameter space for the curvaton model with a polynomial potential.