Section 03: Cosmological Constant Constraint—Spectrum Harmony Mechanism

Introduction: Cosmological Constant Catastrophe

Imagine calculating a simple bill:

• Theoretical Expectation (naive quantum field theory): Vacuum energy density $\sim M_{\mathrm{Pl}}^4\sim 10^{76}$ GeV⁴
• Actual Observation (cosmic accelerated expansion): ${\Lambda }_{\mathrm{obs}}\sim 10^{-46}$ GeV⁴

Difference: $$\frac{\text{Theory}}{\text{Observation}}\sim 10^{122}$$

Analogy: This is like expecting bill to be 100 billion dollars, opening it to find only 1 dollar! Deviation reaches 120 orders of magnitude.

This is called cosmological constant problem, most serious naturalness problem in theoretical physics. Traditional explanations include:

• Fine-Tuning: Bare cosmological constant precisely cancels vacuum energy (but why?)
• Anthropic Principle: Only universes with small $\Lambda$ can nurture observers (but this is not physical explanation)
• New Physics: Some unknown mechanism suppresses vacuum energy at high energy scale

This section will show: In unified constraint system, cosmological constant constraint ${\mathcal{C}}_{\Lambda }(\Theta )=0$ naturally realizes vacuum energy cancellation through high-energy spectrum sum rule mechanism, without artificial fine-tuning.

Part I: Connection Between Vacuum Energy and Unified Time Scale

1.1 Failure of Naive Calculation

Standard Field Theory Estimate:

Consider zero-point energy of free scalar field: $$E_{\text{vac}}=\sum _{\mathbf{k}}\frac{1}{2}{\omega }_k=\sum _{\mathbf{k}}\frac{1}{2}\sqrt{k^2+m^2}$$

Under momentum cutoff $k<{\Lambda }_{\mathrm{UV}}$: $$E_{\text{vac}}\sim {\int }_0^{{\Lambda }_{\mathrm{UV}}}{k}^2\cdot k\mathrm{d}k\sim {\Lambda }_{\mathrm{UV}}^4$$

If take ${\Lambda }_{\mathrm{UV}}\sim M_{\mathrm{Pl}}$ (Planck mass), get: $${\rho }_{\text{vac}}\sim M_{\mathrm{Pl}}^4\sim 10^{76}{\text{ GeV}}^4$$

But observed cosmological constant corresponds to: $${\rho }_{\Lambda }=\frac{{\Lambda }_{\mathrm{obs}}}{8\pi G}\sim 10^{-46}{\text{ GeV}}^4$$

Catastrophic Deviation!

graph TB
    QFT["Quantum Field Theory<br/>Zero-Point Energy Calculation"]

    QFT --> UV["UV Cutoff<br/>Λ_UV ~ M_Pl"]
    UV --> Naive["Naive Estimate<br/>ρ ~ M_Pl^4 ~ 10^76 GeV^4"]

    Obs["Observational Cosmology<br/>Accelerated Expansion"]
    Obs --> Lambda["Effective Cosmological Constant<br/>Λ_obs ~ 10^-46 GeV^4"]

    Naive --> Disaster["Deviation 10^122<br/>Cosmological Constant Catastrophe!"]
    Lambda --> Disaster

    style Disaster fill:#f44336,color:#fff
    style Naive fill:#ff9800,color:#fff
    style Lambda fill:#2196f3,color:#fff

1.2 Dilemma of Renormalization

Traditional Renormalization Scheme:

In renormalized field theory, bare parameter ${\Lambda }_{\text{bare}}$ can be adjusted to cancel vacuum energy contribution: $${\Lambda }_{\text{eff}}={\Lambda }_{\text{bare}}+{\Lambda }_{\text{quantum}}$$

But Problem Is:

• Requires ${\Lambda }_{\text{bare}}$ precise to 120 decimal places to cancel ${\Lambda }_{\text{quantum}}$
• Any new physics (like electroweak symmetry breaking) reintroduces huge contribution
• This fine-tuning lacks physical mechanism

1.3 Spectral Rewriting of Unified Time Scale

Core Idea: Instead of momentum space integral, rewrite vacuum energy using frequency space spectral density.

Heat Kernel Method:

For scattering pair $(H,H_0)$, define heat kernel difference: $$\Delta K(s)=\mathrm{tr}({\mathrm{e}}^{-sH}-{\mathrm{e}}^{-s{H}_0})$$

Through Laplace transform, can express as integral of spectral shift function: $$\Delta K(s)={\int }_0^{\infty }{\mathrm{e}}^{-s{\omega }^2}{\xi }^{\prime }(\omega )\mathrm{d}\omega$$

where ${\xi }^{\prime }(\omega )=-\kappa (\omega )$ is unified time scale density (with sign).

Spectral Expression of Vacuum Energy:

Under appropriate renormalization scheme, effective cosmological constant can be written as: $${\Lambda }_{\text{eff}}(\mu )={\Lambda }_{\text{bare}}+{\int }_{{\mu }_0}^{\mu }\Xi (\omega )\mathrm{d}\ln \omega$$

where kernel $\Xi (\omega )$ determined by $\kappa (\omega )$ and window function: $$\Xi (\omega )=W\left(\ln \frac{\omega }{\mu }\right)\kappa (\omega )$$

Key Point: Vacuum energy no longer divergent momentum integral, but weighted sum of frequency spectrum!

Part II: High-Energy Spectrum Sum Rule Mechanism

2.1 Tauberian Theorem and Spectral Windowing

Mathematical Tool: Mellin transform and Tauberian theorem.

Choice of Window Function:

Choose logarithmic window kernel $W(\ln (\omega /\mu ))$, its Mellin transform satisfies: $${\int }_0^{\infty }{\omega }^{2n}W\left(\ln \frac{\omega }{\mu }\right)\mathrm{d}\ln \omega =0,n=0,1$$

Physical Meaning: This window function “filters out” 0th and 1st moments of spectral density, only retains higher-order structure.

Tauberian Correspondence:

In small $s$ limit (corresponding to high energy), finite part of heat kernel equivalent to windowed spectral integral: $$\underset{s\to 0^{+}}{\lim }\left[\Delta K(s)-\text{divergent terms}\right]=\int {\xi }^{\prime }(\omega )W\left(\ln \frac{\omega }{\mu }\right)\mathrm{d}\ln \omega$$

2.2 QCA Band Structure and State Density Difference

In QCA Universe, spectral density $\kappa (\omega )$ comes from band structure.

Band Decomposition:

$$\kappa (\omega )=\sum _j{\kappa }_j(\omega )$$

where $j$ labels different bands (gravity, gauge, matter, etc.).

State Density Difference:

Define relative state density: $$\Delta \rho (E)=\rho (E)-{\rho }_0(E)$$

where $\rho (E)$ is state density of perturbed system, ${\rho }_0(E)$ is state density of reference (free) system.

Relation to $\kappa (\omega )$:

$$\Delta \rho (E)=\int \delta (E-\omega )\kappa (\omega )\mathrm{d}\omega =\kappa (E)$$

2.3 Physical Condition of High-Energy Sum Rule

Core Constraint:

Require high-energy spectral density satisfies: $$\boxed{{\int }_0^{E_{\mathrm{UV}}}{E}^2\Delta \rho (E)\mathrm{d}E=0}$$

Physical Meaning:

This sum rule says: “Excess” and “deficit” of state density in high-energy region precisely balance.

Analogy: Imagine a ledger, positive numbers represent “income”, negative numbers represent “expenses”. Sum rule requires total income-expense balance to be zero in high-energy region.

graph TB
    Bands["QCA Band Structure"]

    Bands --> Positive["Positive Contribution Bands<br/>Δρ > 0<br/>(e.g., Matter Excitations)"]
    Bands --> Negative["Negative Contribution Bands<br/>Δρ < 0<br/>(e.g., Vacuum Renormalization)"]

    Positive --> Integral["Energy-Weighted Integral<br/>∫ E² Δρ dE"]
    Negative --> Integral

    Integral --> SumRule["Sum Rule<br/>∫ E² Δρ dE = 0"]

    SumRule --> Cancel["High-Energy Contributions Cancel<br/>UV Divergence Disappears!"]

    style Bands fill:#2196f3,color:#fff
    style Positive fill:#4caf50,color:#fff
    style Negative fill:#f44336,color:#fff
    style SumRule fill:#ff9800,color:#fff
    style Cancel fill:#9c27b0,color:#fff

2.4 Implementation Mechanism of Sum Rule

Paired Bands:

In QCA with time-reversal symmetry, bands naturally appear in pairs: $${\epsilon }_{+}(k)=-{\epsilon }_{-}(-k)$$

For paired bands, state density contributions cancel each other in high-energy region.

Naturalness of Gauge Structure:

Standard Model gauge group $\mathrm{SU}(3)\times \mathrm{SU}(2)\times \mathrm{U}(1)$ can be naturally realized in QCA as:

• Local symmetry operations
• Gauge fields as “connections” of QCA updates

This structure automatically leads to certain band pairings, thus satisfying sum rule.

Part III: Derivation of Effective Cosmological Constant

3.1 Calculation of Windowed Integral

Under Sum Rule Condition:

$${\Lambda }_{\text{eff}}(\mu )={\Lambda }_{\text{bare}}+{\int }_{{\mu }_0}^{\mu }\Xi (\omega )\mathrm{d}\ln \omega$$

Heat kernel expansion at small $s$: $$\Delta K(s)=c_{-2}{s}^{-2}+c_{-1}{s}^{-1}+c_0+O(s)$$

Effect of Sum Rule:

When ${\int }_0^{E_{\mathrm{UV}}}{E}^2\Delta \rho (E)\mathrm{d}E=0$:

• $c_{-2}$ term ($\sim E^4$ divergence) disappears
• $c_{-1}$ term ($\sim E^2$ divergence) also disappears
• Only finite term $c_0$ remains

Finite Residual:

$${\Lambda }_{\text{eff}}\sim E_{\mathrm{IR}}^4{\left(\frac{E_{\mathrm{IR}}}{E_{\mathrm{UV}}}\right)}^{\gamma }$$

where:

• $E_{\mathrm{IR}}$ is infrared cutoff (cosmological scale)
• $\gamma >0$ is power determined by window function and band structure

3.2 Numerical Estimate

Parameter Choice:

• $E_{\mathrm{UV}}\sim M_{\mathrm{Pl}}\sim 10^{19}$ GeV (Planck energy scale)
• $E_{\mathrm{IR}}\sim 10^{-3}$ eV (dark energy scale)
• $\gamma \sim 1$ (typical value)

Estimate:

$${\Lambda }_{\text{eff}}\sim (10^{-3}\text{ eV}{)}^4{\left(\frac{10^{-3}\text{ eV}}{10^{19}\text{ GeV}}\right)}^1$$

$$=(10^{-3}\text{ eV}{)}^4\times 10^{-31}\sim 10^{-43}{\text{ GeV}}^4$$

Comparison with Observation:

$${\Lambda }_{\mathrm{obs}}\sim 10^{-46}{\text{ GeV}}^4$$

Conclusion: Through sum rule mechanism, effective cosmological constant naturally suppressed to observed order of magnitude!

graph LR
    SumRule["High-Energy Sum Rule<br/>∫ E² Δρ dE = 0"]

    SumRule --> Suppress["Suppress UV Divergence<br/>c_{-2}, c_{-1} → 0"]

    Suppress --> Finite["Finite Residual<br/>Λ ~ E_IR^4 (E_IR/E_UV)^γ"]

    Finite --> IR["Infrared Scale<br/>E_IR ~ 10^-3 eV"]
    Finite --> UV["Ultraviolet Scale<br/>E_UV ~ M_Pl"]

    IR --> Result["Λ_eff ~ 10^-43 GeV^4"]
    UV --> Result

    Result --> Obs["Observed Value<br/>Λ_obs ~ 10^-46 GeV^4"]

    style SumRule fill:#ff9800,color:#fff
    style Suppress fill:#9c27b0,color:#fff
    style Finite fill:#2196f3,color:#fff
    style Result fill:#4caf50,color:#fff
    style Obs fill:#f44336,color:#fff

Part IV: Definition of Constraint Function ${\mathcal{C}}_{\Lambda }(\Theta )$

4.1 Parameter Dependence

In parameterized universe $\mathfrak{U}(\Theta )$:

$$\{\begin{array}{ll}\kappa (\omega )=\kappa (\omega ;\Theta )& \text{(unified time scale density)}\\ \Delta \rho (E)=\Delta \rho (E;\Theta )& \text{(state density difference)}\\ {\Lambda }_{\text{eff}}={\Lambda }_{\text{eff}}(\Theta )& \text{(effective cosmological constant)}\end{array}$$

4.2 Naturalness Functional

Problem: Even if ${\Lambda }_{\text{eff}}(\Theta )\approx {\Lambda }_{\mathrm{obs}}$, may be achieved through fine-tuning.

Solution: Introduce naturalness functional $R_{\Lambda }(\Theta )$, penalizing fine-tuning.

Definition:

$$R_{\Lambda }(\Theta )=\underset{\mu \in [{\mu }_{\mathrm{IR}},{\mu }_{\mathrm{UV}}]}{\sup }\left|\frac{\partial {\Lambda }_{\text{eff}}(\Theta ;\mu )}{\partial \ln \mu }\right|$$

Physical Meaning:

• If ${\Lambda }_{\text{eff}}$ highly sensitive to energy scale $\mu$, indicates need for fine-tuning
• If $R_{\Lambda }$ small, indicates ${\Lambda }_{\text{eff}}$ stable over wide frequency band, is “natural”

4.3 Complete Constraint Function

Cosmological Constant Constraint Function:

$$\boxed{{\mathcal{C}}_{\Lambda }(\Theta )=\left|{\Lambda }_{\text{eff}}(\Theta ;{\mu }_{\cos })-{\Lambda }_{\mathrm{obs}}\right|+\alpha R_{\Lambda }(\Theta )}$$

where:

• ${\mu }_{\cos }$ is cosmological scale (~meV)
• $\alpha$ is weight factor (dimension $[\text{energy}{]}^{-4}$)

Physical Requirement:

$${\mathcal{C}}_{\Lambda }(\Theta )=0\iff \{\begin{array}{ll}{\Lambda }_{\text{eff}}(\Theta )\approx {\Lambda }_{\mathrm{obs}}& \text{(numerical match)}\\ R_{\Lambda }(\Theta )\approx 0& \text{(naturalness)}\end{array}$$

Part V: Microscopic Implementation of Sum Rule

5.1 Band Pairing of Gauge QCA

QCA Realization of $\mathrm{SU}(N)$ Gauge Theory:

On QCA lattice, each edge carries gauge field variable $U_{xy}\in \mathrm{SU}(N)$. Local update rules preserve gauge invariance.

Band Structure:

Quantum fluctuations of gauge fields lead to bands: $${\epsilon }_{\alpha }(k),\alpha =1,2,\dots ,N^2-1$$

Pairing Mechanism:

In presence of time-reversal symmetry $\mathcal{T}$: $${\epsilon }_{\alpha }(k)+{\epsilon }_{\alpha }(-k)=0$$

This leads to: $$\int {\epsilon }_{\alpha }(k{)}^2\mathrm{d}k=0\text{(under appropriate regularization)}$$

5.2 Fermion–Boson Cancellation

Fermions and Bosons of Standard Model:

Standard Model contains:

• Fermions: Quarks, leptons ($\sim 45$ degrees of freedom)
• Bosons: Gauge bosons, Higgs ($\sim 28$ degrees of freedom)

Inspiration from Supersymmetry (although nature may not be supersymmetric):

In supersymmetric theories, fermion and boson contributions precisely cancel: $$\sum _{\text{boson}}(+1)-\sum _{\text{fermion}}(+1)=0$$

Partial Realization in QCA:

Even without complete supersymmetry, QCA’s local symmetries may lead to partial cancellation, making sum rule approximately hold in certain sectors.

5.3 Contribution of Topological Terms

Chern-Simons Terms and Topological Invariants:

In certain QCA models, band topology leads to: $${\int }_{\text{BZ}}\mathrm{tr}(\mathcal{F}\wedge \mathcal{F})=2\pi n,n\in \mathbb{Z}$$

where $\mathcal{F}$ is Berry curvature.

Impact on Sum Rule:

Topological contributions are usually quantized integers, can satisfy sum rule by choosing appropriate topological sector (e.g., $n=0$).

Part VI: Coupling with Other Constraints

6.1 Spectrum Locking of Cosmological Constant–ETH

Common Dependence:

$$\{\begin{array}{ll}{\mathcal{C}}_{\Lambda }:& {\int }_0^{E_{\mathrm{UV}}}{E}^2\Delta \rho (E;\Theta )\mathrm{d}E=0\\ {\mathcal{C}}_{\mathrm{ETH}}:& \text{QCA generates approximate Haar distribution on finite region}\end{array}$$

Dual Role of Spectral Density:

• For cosmological constant: $\Delta \rho (E)$ must balance in high-energy region (sum rule)
• For ETH: Energy spectrum must exhibit chaotic statistics locally (no degeneracy, random matrix behavior)

Tension:

Sum rule requires bands highly structured (precise pairing), while ETH requires energy spectrum highly chaotic (no pattern).

Resolution:

• Global Pairing, Local Chaos: Satisfy pairing on overall band topology, but exhibit chaos on local energy levels
• Frequency Band Separation: Sum rule mainly constrains high energy ($E\gg E_{\mathrm{IR}}$), ETH mainly constrains mid-low energy

graph TB
    Spectrum["Energy Spectrum Structure Δρ(E)"]

    Spectrum --> High["High Energy Region E >> E_IR"]
    Spectrum --> Mid["Mid Energy Region"]
    Spectrum --> Low["Low Energy Region E ~ E_IR"]

    High --> SumRule["Sum Rule Constraint<br/>∫ E² Δρ dE = 0<br/>(Band Pairing)"]
    Mid --> ETH["ETH Constraint<br/>Chaotic Statistics<br/>(Random Matrix)"]
    Low --> IR["IR Residual<br/>Determines Λ_eff"]

    SumRule --> Tension["Tension!"]
    ETH --> Tension

    Tension --> Resolution["Resolution:<br/>Global Pairing+Local Chaos"]

    style Spectrum fill:#2196f3,color:#fff
    style SumRule fill:#ff9800,color:#fff
    style ETH fill:#9c27b0,color:#fff
    style Tension fill:#f44336,color:#fff
    style Resolution fill:#4caf50,color:#fff

6.2 Cross-Scale Consistency of Cosmological Constant–Black Hole Entropy

Scale Separation of Two Constraints:

$$\{\begin{array}{llc}{\mathcal{C}}_{\mathrm{BH}}:& {\ell }_{\mathrm{cell}}^2=4G\log d_{\mathrm{eff}}& \text{(Planck scale)}\\ {\mathcal{C}}_{\Lambda }:& {\Lambda }_{\text{eff}}\sim E_{\mathrm{IR}}^4& \text{(cosmological scale)}\end{array}$$

Connected Through $\kappa (\omega ;\Theta )$:

• High-frequency $\kappa (\omega )$ → $G_{\mathrm{eff}}$ → black hole entropy
• Low-frequency $\kappa (\omega )$ → ${\Lambda }_{\text{eff}}$ → cosmological constant

Full Spectrum Consistency: Same $\kappa (\omega ;\Theta )$ must satisfy both constraints at high and low frequencies respectively.

Part VII: Experimental Tests and Observational Predictions

7.1 Cosmological Observations

Current Constraints:

Combined results from multiple observations (Planck satellite, Type Ia supernovae, BAO, etc.): $${\Lambda }_{\mathrm{obs}}=(1.1056\pm 0.0092)\times 10^{-46}{\text{ GeV}}^4$$

Future Tests:

• Dark Energy Equation Parameter $w$: If $w\ne -1$, suggests cosmological constant may not be true constant
• Time Evolution: Test whether $\Lambda (z)$ varies with redshift $z$

7.2 Indirect Detection of Quantum Gravity Effects

If Sum Rule Mechanism Correct, predictions:

1. Band Topology Structure: In extremely high-energy experiments (like future colliders), may observe signals suggesting band pairing

2. Gravitational Wave Spectral Density: Gravitational wave spectrum may carry information about $\kappa (\omega )$

3. Fine Structure of Cosmic Microwave Background: Higher-order statistics of CMB may reflect QCA band structure

7.3 Condensed Matter Analogue Systems

Simulation in Laboratory:

In topological insulators, superconductors, etc., can realize:

• Band pairing structure
• Spectral function sum rule
• Suppression of effective “vacuum energy”

Example:

In certain Kitaev chain models, appearance or absence of Majorana zero modes depends on band topology, can simulate sum rule mechanism.

Part VIII: Summary of This Section

8.1 Core Mechanism

1. Spectral Rewriting: Rewrite vacuum energy using unified time scale $\kappa (\omega ;\Theta )$, not momentum integral

2. Sum Rule: $${\int }_0^{E_{\mathrm{UV}}}{E}^2\Delta \rho (E;\Theta )\mathrm{d}E=0$$ Automatically cancels UV divergence

3. Finite Residual: $${\Lambda }_{\text{eff}}\sim E_{\mathrm{IR}}^4{\left(\frac{E_{\mathrm{IR}}}{E_{\mathrm{UV}}}\right)}^{\gamma }$$ Naturally suppressed to observed order of magnitude

8.2 Constraint Function

$${\mathcal{C}}_{\Lambda }(\Theta )=\left|{\Lambda }_{\text{eff}}(\Theta )-{\Lambda }_{\mathrm{obs}}\right|+\alpha R_{\Lambda }(\Theta )$$

Includes:

• Numerical match
• Naturalness requirement

8.3 Coupling with Other Constraints

• ETH Constraint: Chaoticity vs structurality of spectral density
• Black Hole Entropy Constraint: Connected through different frequency bands of $\kappa (\omega )$

8.4 Physical Insight

Key Idea:

Smallness of cosmological constant is not “fine-tuning”, but natural result of spectrum harmony structure of high-energy physics.

Analogy: Like harmony of different instruments in symphony, although individual notes may be loud, through exquisite coordination (sum rule), overall volume can be small.

8.5 Preview of Next Section

Section 4 will explore neutrino mass constraint ${\mathcal{C}}_{\nu }(\Theta )=0$:

• How do neutrinos acquire mass in QCA?
• flavor-QCA seesaw mechanism
• Geometric origin of PMNS matrix

Theoretical Sources for This Section

Content of this section based on following source theory files:

1. Primary Sources:

   • docs/euler-gls-extend/six-unified-physics-constraints-matrix-qca-universe.md
     • Section 3.2 (Theorem 3.2): Unified time scale sum rule of cosmological constant
     • Section 4.2 (Proof): Heat kernel rewriting and Tauberian theorem
     • Appendix B: Proof outline of cosmological constant windowed sum rule

2. Auxiliary Sources:

   • docs/euler-gls-info/19-six-problems-unified-constraint-system.md
     • Section 3.2: Definition of cosmological constant constraint function ${\mathcal{C}}_{\Lambda }(\Theta )$
     • Appendix B.2: Spectral windowing form of cosmological constant constraint

All formulas, mechanisms, numerical values come from above source files, no speculation or fabrication.