  We show that a sizable running spectral index suggested by the recent SPT
data can be explained in the axionic curvaton model with a potential that
consists of two sinusoidal contributions of different height and period. We
find that the running spectral index is generically given by d ns/dlnk ~
(2pi/dN)(n_s - 1), where dN is the e-folds during one period of modulations. In
the string axiverse, axions naturally acquire a mass from multiple
contributions, and one of the axions may be responsible for the density
perturbations with a sizable running spectral index via the curvaton mechanism.
We note that the axionic curvaton model with modulations can also accommodate
the red-tilted spectrum with a negligible running, without relying on
large-field inflation.
