Meta Theory of the Zeckendorf-Hilbert Universe

9.4 Geometric Nature of Dark Energy: Background Phase Drift Caused by Vacuum Density of States and Exponential Expansion

In the first three sections of Chapter 9, we established a cosmic evolution model based on information dissipation and adiabatic scattering. We proved that Hubble expansion is equivalent to evolution of the imaginary part of non-Hermitian Hamiltonian, while redshift is a consequence of phase drift. However, the greatest mystery in modern cosmology is: why is cosmic expansion accelerating?

In the standard model, this is attributed to “dark energy” or cosmological constant $\Lambda$. But in quantum field theory (QFT) calculations, vacuum zero-point energy leads to theoretical value of $\Lambda$ that is $10^{120}$ times larger than observed value. This “worst prediction in physics history” suggests a fundamental flaw in our understanding of vacuum.

This section will use QCA’s discrete ontology and unified time identity to propose a holographic geometric explanation of dark energy. We will prove that dark energy is not some mysterious fluid filling space but the temporal effect of vacuum density of states itself. The observed tiny $\Lambda$ value is precisely the “spectral residue” of microscopic information density at Planck scale under macroscopic holographic cutoff.

9.4.1 “Weight” of Vacuum: Absolute vs. Relative Density of States

When deriving unified time identity in Section 8.1, we used relative density of states ${\rho }_{\text{rel}}(E)=\rho (E)-{\rho }_0(E)$, where ${\rho }_0$ is the background density of states of free vacuum. For local scattering problems, this subtraction is reasonable, because we only care about differences caused by scatterers.

However, general relativity tells us that gravity couples to all forms of energy, including vacuum itself. In the QCA universe, vacuum is not “nothing” but a vast lattice network in ground state $|0{⟩}^{\otimes N}$.

According to finite information axiom, vacuum has extremely high absolute density of states ${\rho }_{\text{vac}}(E)$. At Planck scale, information capacity per Planck volume is 1 bit, meaning extremely high microscopic density of states.

Question: If ${\rho }_{\text{vac}}$ is so huge, according to $\kappa =\rho$, vacuum’s time flow rate (or gravitational effect) should be infinite, universe should collapse instantly. Why don’t we see this?

Answer: Because holographic principle limits the contribution of bulk degrees of freedom to macroscopic geometry.

9.4.2 Spectral Windowing and Holographic Cutoff: Eliminating the $10^{120}$ Factor Difference

Using PSWF spectral windowing theory established in Chapter 5, we know any macroscopic observer can only probe the universe through a finite “causal window.”

Let horizon radius be $R_H=c/H_0$. According to holographic principle, effective number of degrees of freedom $N_{\text{eff}}$ is not proportional to volume $V\sim R_H^3$ but proportional to area $A\sim R_H^2$.

Theorem 9.4.1 (Holographic Renormalization of Vacuum Energy)

In the QCA universe, although bulk density of states ${\rho }_{\text{bulk}}\sim V$ is huge, the effective density of states ${\rho }_{\text{eff}}$ that truly participates in long-range gravitational interactions (i.e., determines spacetime background curvature) is constrained by holographic bounds:

$${\rho }_{\text{eff}}(E)\approx \sqrt{{\rho }_{\text{bulk}}(E)}\sim \frac{A}{l_P^2}$$

Or more precisely, after smoothing microscopic vacuum spectrum using spectral windowing function $W(\omega )$, its low-frequency residual term (zero-mode residue) is no longer $k_{max}^4$ (quartic divergence) but $k_{max}^2{k}_{min}^2$ (infrared-ultraviolet mixing).

$${\rho }_{\Lambda }\sim {\Lambda }_{UV}^2{\Lambda }_{IR}^2\sim \frac{1}{l_P^2{R}_H^2}$$

This corrected energy density ${\rho }_{\Lambda }\sim (M_P{R}_H{)}^{-2}{M}_P^4\sim 10^{-123}{M}_P^4$ precisely matches current observations.

Physical Picture:

Dark energy is not the “bulk energy” of vacuum (that’s huge) but the “surface tension” or “boundary energy” of vacuum. It is the information cost that holographic horizon must pay to maintain causal connectivity.

9.4.3 Phase Drift and Exponential Expansion Mechanism

Now we explain why this non-zero vacuum energy causes accelerated expansion.

Recalling the redshift formula from Section 9.2, redshift originates from drift of scattering phase $\Phi$. For ordinary matter, $\Phi$ changes significantly with $E$ (${\partial }_E\Phi$ large), causing normal redshift.

But for vacuum, its density of states spectrum ${\rho }_{\text{vac}}(E)$ is flat (or has extremely high symmetry).

This leads to a special background phase drift rate ${\dot{\Phi }}_{\text{vac}}$.

Definition 9.4.2 (Vacuum Phase Flow)

Since vacuum is not dead but filled with quantum fluctuations (creation and annihilation of virtual particle pairs), this manifests in QCA evolution operator $U$ as an overall, energy-independent geometric phase factor $e^{-i\Lambda t}$.

This phase factor causes effective Hamiltonian to have a constant shift:

$$H_{\text{eff}}\to H_{\text{eff}}+{\Lambda }_{\text{vac}}\mathbb{I}$$

In gravitational field equations, this is equivalent to:

$$G_{\mu \nu }=8\pi G(T_{\mu \nu }^{\text{matter}}-{\rho }_{\Lambda }{g}_{\mu \nu })$$

where the negative sign originates from back-reaction of phase. Ordinary matter’s phase derivative ${\partial }_E\Phi >0$ (positive time delay), while vacuum’s overall phase drift manifests as a “negative time delay” or time advance tendency—space itself expands “out of nothing” to dilute this phase accumulation.

Corollary 9.4.3 (Equation of State $w=-1$)

From thermodynamic perspective, vacuum energy $E\sim \Lambda V$.

According to thermodynamic relation $dE=-PdV$ (for adiabatic expansion):

$$d(\Lambda V)=\Lambda dV=-PdV⟹P=-\Lambda =-{\rho }_{\Lambda }$$

This gives the equation of state for dark energy $w=P/\rho =-1$.

Therefore, exponential expansion (de Sitter space) is the natural ground state of QCA universe without matter load. Matter is just perturbations superimposed on this exponential expansion background.

9.4.4 Solution to Cosmological Constant Problem

Unified time theory solves the cosmological constant problem through the following three steps:

1. Microscopic Origin: Confirm $\Lambda$ originates from non-zero absolute density of states ${\rho }_{\text{vac}}$ of vacuum (based on finite information axiom).

2. Magnitude Correction: Using holographic principle and spectral windowing (PSWF), renormalize bulk energy $O(l_P^{-4})$ to boundary energy $O(l_P^{-2}{R}_H^{-2})$.

3. Dynamical Effect: Prove this boundary energy manifests as background phase drift, driving exponential expansion of metric.

Conclusion

Dark energy is geometric, not material. It is the manifestation of spacetime as a discrete information processing system, its background clock frequency.

The universe accelerates expansion because it tries to push the horizon far enough to accommodate continuously emerging quantum entanglement entropy in vacuum.

At this point, the cosmological part of Volume II: The Emergence of Time concludes. Starting from microscopic scattering time, we derived redshift, entropy increase, and dark energy. This proves that “time” as an emergent quantity has far greater explanatory power than traditional geometric parameter $t$.

In the upcoming Volume III: Entropic Origin of Gravity and Geometry, we will delve into the geometric details of this “entropic force,” deriving the tensor form of Einstein’s field equations.