ℓ(ℓ+1)C_ℓ ∝ [1 + B cos( (2π/ln q) ln ℓ + φ )]The angular power spectrum of the cosmic microwave background, denoted $C\ell$, measures the variance of temperature fluctuations at different angular scales $\theta \sim \pi/\ell$. In the standard $\Lambda$CDM model, $C\ell$ is a smooth function of $\ell$ with acoustic peaks at specific multiples of the sound‑horizon scale. The Syntactic Token Calculus predicts an additional log‑periodic modulation:
$$ \ell(\ell+1)C_\ell \propto \left[ 1 + B \cos\!\left( \frac{2\pi}{\ln q} \ln \ell + \varphi \right) \right], $$
where:
Log‑periodic oscillations are a signature of discrete scale invariance: the system is invariant under rescaling by a fixed factor $q$. If a pattern repeats when lengths are multiplied by $q$, then any observable that depends on scale will oscillate when plotted against the logarithm of the scale.
In the CMB context, the scale is the multipole moment $\ell$, which inversely corresponds to angular size. The prediction is that the power spectrum, after removing the smooth $\Lambda$CDM component, will show sinusoidal oscillations in $\ln \ell$ with period $\ln q$.
For a binary tree ($p=2$), $q=2$, and the period in $\ln \ell$ is $\ln 2 \approx 0.693$. That means the oscillation repeats every time $\ell$ increases by a factor of $e^{0.693} = 2$. In terms of $\ell$, peaks appear at $\ell, 2\ell, 4\ell, 8\ell, \dots$—a geometric progression.
The amplitude $B$ is expected to be small because the discrete scale invariance is broken by coarse‑graining (the Monna map) and by astrophysical foregrounds. However, even a tiny modulation ($B \sim 10^{-3}$) could be detectable with precise CMB data.
The log‑periodic oscillations arise directly from the hierarchical structure of the Bruhat‑Tits tree. The tree is self‑similar: scaling by factor $p$ maps the tree onto itself. This discrete scale symmetry is inherited by the syntactic patterns that represent cosmological perturbations.
Consider a density perturbation in the early universe. In the STC, this perturbation is a syntactic pattern on the tree. As the universe expands, the pattern is stretched along the tree’s depth. Because the tree has discrete branching, the stretching is not continuous; it proceeds in steps of factor $p$. This stepwise stretching imprints a periodicity in logarithmic scale on any correlation function derived from the pattern.
More formally, the two‑point correlation function of temperature fluctuations $\langle \delta T(\hat{n}1) \delta T(\hat{n}2) \rangle$ depends on the angular separation $\theta$. In the tree, angular separation corresponds to tree distance between the boundary points representing the directions $\hat{n}1$ and $\hat{n}2$. The tree distance is quantized in steps of $\ln p$ when measured in logarithmic coordinates. This quantization leads to log‑periodicity in the correlation function, which translates to log‑periodicity in $C_\ell$ after spherical‑harmonic transform.
The phase $\varphi$ is determined by the alignment of the tree with the observer’s vantage point. Different observers (at different locations in the tree) would measure different phases, but the period $\ln q$ is universal.
The amplitude $B$ is related to the depth of the tree that is probed by the CMB. The CMB photons last scattered at redshift $z \approx 1100$, corresponding to a comoving distance that maps to a certain depth in the tree. Deeper layers of the tree contribute higher‑frequency oscillations, but they are damped by Silk damping and projection effects. Thus, the observed oscillations are a low‑frequency remnant of the deep tree structure.
To test the prediction, we need to analyze the observed CMB power spectrum for log‑periodic oscillations. The following protocol can be used:
$$ r\ell = \frac{\ell(\ell+1)C\ell^{\text{obs}}}{2\pi} - \frac{\ell(\ell+1)C_\ell^{\text{theory}}}{2\pi}. $$
$$ r(x) = A \cos(2\pi f x + \varphi) + \text{noise}, $$
to estimate $f$, $A$, $\varphi$. Convert $f$ to $q$ via $q = e^{1/f}$.
Potential pitfalls:
Several independent analyses have reported hints of log‑periodic oscillations in the CMB power spectrum.
However, these hints are not yet statistically significant. The Planck collaboration’s own analysis did not find conclusive evidence for log‑periodicity, but they did not specifically search for it with an optimized template.
The South Pole Telescope (SPT) and Atacama Cosmology Telescope (ACT) data, which probe smaller angular scales (higher $\ell$), could provide a stronger test because the oscillation period in $\ln \ell$ is constant, so higher $\ell$ means more cycles within the observable range. A combined analysis of Planck (low‑$\ell$) and SPT/ACT (high‑$\ell$) could yield a definitive detection or exclusion.
If the STC is correct, the log‑periodic oscillations should be universal—they should appear not only in the CMB but also in large‑scale structure (galaxy clustering) and possibly in the primordial gravitational‑wave spectrum. Searching for these correlated signals across multiple datasets will be crucial.
Chapter 23 has presented the STC’s prediction of log‑periodic oscillations in the CMB angular power spectrum. These oscillations arise from the discrete scale invariance of the Bruhat‑Tits tree and can be detected via logarithmic resampling and Fourier analysis. Existing hints in the data encourage a dedicated search. If confirmed, log‑periodicity would be a smoking‑gun signature of a hierarchical, non‑Archimedean universe.
The next chapter explains how the continuous $R_h = ct$ model emerges as the coarse‑grained shadow of the discrete tree via the Monna map, resolving the apparent conflict between continuous and discrete cosmologies.