Relativity and Cosmology

Logarithmic Periodic Modulation in the Primordial Power Spectrum from Discrete Scale Invariance

Authors: Tianliang Zhuang

A specific form of logarithmic periodic modulation in the primordial curvature power spectrum is derived from the hypothesis of discrete scale invariance in the early universe. The modulation frequency is fixed by the Feigenbaum constant $delta approx 4.669$ and an effective spectral dimension $d_s = 1.25$, yielding $omega = 2pi / ln(delta^{1/d_s}) approx 3.67$. The modulation amplitude is determined by a single dimensionless parameter $eta = 0.08931$, giving $B = eta^{1/2} approx 0.30$. When the modulated spectrum is fitted with a smooth power-law template over the limited CMB window, the effective spectral index is shifted to $n_s^{text{eff}} approx 0.966$, in close agreement with the Planck 2018 value $n_s = 0.965 pm 0.004$. The predicted oscillatory pattern provides a distinctive signature for future high-precision CMB experiments.

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[v1] 2026-04-21 18:16:16

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