We present a careful analysis of non scale-invariant insocurvature perturbation produced in the power-law inflation. We first derive the exact form of the two-point function for a massless scalar field φ in a power-law background, a (t) ∝ t 1+n (n≫1). For generality, we allow the scalar field to have a small nonminimal coupling to gravity, ~ ξφ 2 R (|ξ|≪1). We then regard φ as an axion-like field whose quantum fluctuation gives rise to an isocurvature perturbation with its amplitude proportional to a trigonometric function of φ. As a concrete example, we consider the case when φ is a Majoron and the baryon density fluctuation is produced in proportion to sin (φ/f) where f is the symmetry breaking scale. We find the resulting spectrum of the (baryon) isocurvature perturbation depends very much on the sign of n*, where 1/n* = 1/n + ξ. For n* > 0, corresponding to an infrared unstable scalar field, the spectrum is white noise on large scales and almost scale-invariant on small scales. On the other hand, for n* < 0, corresponding to an infrared stable scalar field, the spectrum is almost scale-invariant on all the scales.
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Sasaki et al. (1991) studied this question.