Meta Theory of the Zeckendorf-Hilbert Universe

27.3 Discrete Features in Cosmic Microwave Background (CMB): Planck-Scale Holographic Noise

In Sections 27.1 and 27.2, we respectively explored possibilities of detecting spacetime discreteness using gravitational waves and fast radio bursts. These two windows mainly focus on propagation effects (dispersion or scattering). However, there is a more fundamental detection pathway: Primordial Imprint.

Cosmic Microwave Background radiation (CMB) is earliest light of universe, recording snapshot of vacuum fluctuations during inflation. In standard inflation theory, these fluctuations originate from quantum field theory vacuum in continuous spacetime. But in QCA discrete ontology, horizon during inflation has finite information capacity. This means primordial perturbation spectrum cannot contain infinitely fine structure; there must exist Holographic Noise or Pixelation Artifacts. This section will prove that discrete structure of QCA causes specific modulations in CMB power spectrum at high multipoles (High-$l$), providing us a “negative” for directly observing Planck-scale geometry.

27.3.1 Holographic Information Bounds of Inflation Horizon

During inflation, universe is in approximate de Sitter state with constant Hubble parameter $H_{inf}$. This produces an event horizon with radius $R_H=c/H_{inf}$.

According to holographic principle we established in Chapter 9 (Section 9.4) and Chapter 15 (Section 15.2), this horizon limits maximum information (entropy) that can be physically encoded during inflation:

$$S_{max}=\frac{A_H}{4l_P^2}=\frac{\pi c^2}{G\hslash H_{inf}^2}$$

This means during inflation, universe does not have infinitely many independent quantum modes. Any mode with wavelength $\lambda <l_P$ is physically non-existent (or truncated by QCA lattice).

Definition 27.3.1 (Information Pixels During Inflation)

Inflation horizon surface can be divided into $N=S_{max}$ discrete Planck area elements (QCA horizon links). Each area element carries $O(1)$ bits of quantum fluctuation information.

When inflation occurs, these microscopic “pixels” are exponentially stretched:

$${\lambda }_{phys}(t)={\lambda }_0{e}^{Ht}$$

Once wavelength exceeds horizon scale, these discrete fluctuations are “frozen” into classical density perturbations.

Corollary: Macroscopic temperature fluctuations $\delta T/T$ we see on CMB today are essentially magnified images of microscopic QCA lattice fluctuations during inflation. If QCA lattice has specific geometric structure (such as non-commutative geometry or specific lattice symmetries), this structure will be imprinted on sky.

27.3.2 Holographic Noise and Power Spectrum Corrections

Standard inflation theory predicts a nearly scale-invariant power spectrum ${\mathcal{P}}_{\mathcal{R}}(k)\propto k^{n_s-1}$. Discreteness of QCA will introduce corrections on this basis.

Theorem 27.3.1 (Holographic Noise Spectrum)

Due to finiteness of holographic degrees of freedom, commutation relations $[{\varphi }_k,{\pi }_{k^{\prime }}]=i\delta (k-k^{\prime })$ of scalar perturbation modes ${\varphi }_k$ no longer hold exactly in discrete limit, but are corrected by generalized uncertainty principle (GUP) or non-commutative geometry.

This causes power spectrum $\mathcal{P}(k)$ to be superimposed with a holographic noise term:

$${\mathcal{P}}_{obs}(k)={\mathcal{P}}_{std}(k)\left[1+{\mathcal{E}}_{holo}(k)\right]$$

where ${\mathcal{E}}_{holo}(k)$ describes statistical errors brought by discreteness.

According to Hogan (2008) and subsequent development of holographic noise theory, this noise originates from holographic correlation $\Delta x_{\perp }\sim \sqrt{L{l}_P}$ between transverse position uncertainty $\Delta x_{\perp }$ and longitudinal distance $L$. In CMB, this manifests as anomalous noise or suppression in angular power spectrum $C_l$ at high $l$.

Estimated magnitude:

$$\frac{\delta C_l}{C_l}\sim {\left(\frac{H_{inf}}{M_P}\right)}^{\alpha }$$

where $\alpha$ depends on specific microscopic dynamics of QCA (usually $\alpha =1$ or $2$). Since inflation energy scale $H_{inf}$ may be as high as $10^{14}$ GeV, this effect is easier to detect than Lorentz violation in low-energy physics.

27.3.3 Non-Gaussianity: Fingerprints of Discrete Statistics

Besides corrections to power spectrum (two-point correlation function), most prominent feature of QCA discreteness may be Non-Gaussianity.

Standard slow-roll inflation predicts highly Gaussian fluctuations. However, in QCA networks, underlying microscopic states are discrete qubits, not continuous Gaussian fields.

According to central limit theorem, superposition of large numbers of bits tends toward Gaussian distribution, but in tails of distribution (extreme fluctuations) or higher-order correlation functions (three-point function $f_{NL}$), discreteness will reveal itself.

Prediction 27.3.2 (Granular Non-Gaussianity)

If spacetime is composed of discrete “spacetime atoms,” vacuum fluctuations cannot be perfect continuous random fields.

Temperature distribution map of CMB may contain tiny Granularity.

This manifests as specific shape factors (Shape Functions) in three-point correlation function (Bispectrum), with peaks at specific triangle configurations (such as folded or equilateral), corresponding to underlying geometric symmetries of QCA lattice.

$$f_{NL}^{QCA}\sim \frac{1}{\sqrt{N_{dof}}}\sim \frac{l_P}{R_H}$$

Although this value is extremely small for current cosmic horizon, during inflation horizon was smaller, this effect may be amplified and frozen.

27.3.4 Experimental Status and Prospects: Planck and LiteBIRD

Current observation data (such as Planck satellite) gives strict limits on non-Gaussianity ($f_{NL}\lesssim O(10)$), and power spectrum fits $\Lambda$CDM model extremely well.

This actually imposes smoothness constraints on QCA models:

1. Fine Structure: Microscopic structure of QCA must maintain high statistical isotropy and uniformity at this energy scale (see discussion on FRBs in Section 27.2).

2. Decoherence Suppression: Quantum-to-classical transition during inflation must very efficiently smooth out microscopic discrete phases.

Future Prospects:

Next-generation CMB polarization experiments (such as LiteBIRD, CMB-S4) will detect B-mode polarization produced by primordial gravitational waves.

• Holographic Gravitational Waves: QCA theory predicts there may be consistency relations between tensor power spectrum $r$ of primordial gravitational waves and scalar spectral index $n_s$ different from standard single-field inflation, because they both originate from fluctuations of same unified time parent scale $\kappa (E)$.

• Parity Violation: If non-zero signals are found in TB or EB cross-correlation spectra of CMB, this would be direct evidence of axion dynamics described in Section 17.3 or chiral structure of QCA networks (parity-violating gravity).

Conclusion

CMB observations provide most direct window for detecting Planck-scale structure of spacetime. Current “null results” impose strong constraints on QCA models, requiring them to exhibit extremely high symmetry and smoothness at macroscopic scales. Future high-precision polarization measurements may reveal subtle signatures of discrete geometry imprinted during earliest moments of universe.