Plot of Eq. (87), showing small-scale perturbations in the Γ = 0 limit of a polytropic DM model (red solid line). The straight dashed lines represent Eq. (88) for β = 1 (green dashed line) and β = 0.60 (blue dashed line), each one of slope α = 3.970. It is evident that as long as Δ²(φ) ~ k⁰, polytropic perturbations with exhibit an effective power-law behavior with a scalar spectral index equal to .