QNM Model Cosmological Parameter Derivation vs Fitting Detailed Classification Table

Scientific Integrity Statement

This document truthfully and objectively presents the calculation methods for each cosmological parameter in the QNM model, with explicit labeling:

[Theoretical Derivation]: Pure mathematical derivation based on physical theory

[Empirical Fitting]: Using empirical coefficients or fitting parameters

[Heuristic Mapping]: Heuristic mapping based on empirical relationships

[Using Empirical Calibration Parameters]: Using calibration parameters determined through statistical optimization

All calculations are based on the QNM model, with no external observational data used.

Complete Parameter Classification Table

ParameterSymbolCalculation FormulaCalculated ValueNature ClassificationDetailed DescriptionRaw Central Chargec_rawRyu-Takayanagi formula fitting S(L) = (c/3)×ln(L) + S₀ c_raw = 3 × slope2.172459[Theoretical Derivation]Based on holographic principle and AdS/CFT correspondence theory, extracted through logarithmic linear fittingEffective Central Chargec_effc_eff = κ × c_raw × n where κ=21.0, n=21958.054554[Using Empirical Calibration Parameters]κ and n are empirical calibration parameters, determined through statistical optimization, not pure theoretical derivationSpectral Indexn_sn_s = 1 - 2/c_eff0.997912[Theoretical Derivation]Based on conformal field theory (CFT) relationship between central charge and spectral index, theoretical derivation formulaMatter DensityΩ_mΩ_m = 9 × (1 - n_s) coefficient 9 is empirical0.018788[Empirical Fitting]Coefficient 9 is an empirical fitting value, not pure theoretical derivation, this is a heuristic mappingHubble ConstantH₀H₀ = 50 + 20 × (matrix_norm/10) matrix_norm = ||matrix||_F52.000000 km/s/Mpc[Heuristic Mapping]Based on empirical relationship of matrix features, not pure theoretical derivation, heuristic mapping methodDark Energy Equation of Statew₀w₀ = -1 - 0.1 × tanh(unitarity_deviation×10) coefficient 0.1 is empirical-1.000000[Empirical Fitting]Coefficient 0.1 is empirical, based on quantum system non-unitarity, but the coefficient is empirical fitting

Detailed Classification Explanation

1. Raw Central Charge c_raw

Parameter: c_raw = 2.172459

Nature: [Theoretical Derivation]

Calculation Formula:

Ryu-Takayanagi holographic entanglement entropy formula:
S(L) = (c/3) × ln(L) + S₀

Through logarithmic linear fitting:
log_L = ln(L)
S = slope × log_L + intercept
where: slope = c/3
therefore: c_raw = 3 × slope

Calculation Process:

1. Calculate entanglement entropy S(L) for different subsystem sizes L
2. Filter valid data points: S(L) > 1e-10
3. Logarithmic linear fitting: S vs ln(L)
4. Extract slope: slope = 0.724153
5. Calculate central charge: c_raw = 3 × 0.724153 = 2.172459

Theoretical Basis:

Ryu-Takayanagi holographic entanglement entropy formula

AdS/CFT correspondence

Conformal field theory (CFT) central charge

Uses External Data: ❌ No, completely based on model internal calculations

2. Effective Central Charge c_eff

Parameter: c_eff = 958.054554

Nature: [Using Empirical Calibration Parameters]

Calculation Formula:

c_eff = κ × c_raw × n

where:
  κ = 21.0  [Empirical calibration parameter - projection scale factor]
  n = 21    [Empirical calibration parameter - quantum degrees of freedom]

Calculation Process:

c_eff = 21.0 × 2.172459 × 21
c_eff = 21.0 × 2.172459 × 21
c_eff = 958.054554

Calibration Parameter Explanation:

κ = 21.0:

Nature: Empirical calibration parameter

Source: Determined through statistical optimization, not pure theoretical derivation

Physical meaning: Projection scale factor, considering high-dimensional to low-dimensional projection effects

n = 21:

Nature: Empirical calibration parameter

Source: Determined through statistical optimization, not pure theoretical derivation

Physical meaning: Quantum degrees of freedom, considering system's effective degrees of freedom

Uses External Data: ⚠️ Calibration parameters determined through statistical optimization (optimized based on observational data)

3. Spectral Index n_s

Parameter: n_s = 0.997912

Nature: [Theoretical Derivation]

Calculation Formula:

n_s = 1 - 2/c_eff

Calculation Process:

n_s = 1 - 2/958.054554
n_s = 1 - 0.002088
n_s = 0.997912

Theoretical Basis:

Conformal field theory (CFT) relationship between central charge and spectral index

Same CFT formula used in holographic inflation models (shared theoretical tools)

This is a theoretical derivation formula, not empirical fitting

Uses External Data: ❌ No, completely based on theoretical formula

Note: Although the formula itself is theoretical derivation, the input parameter c_eff uses empirical calibration parameters

4. Matter Density Ω_m

Parameter: Ω_m = 0.018788

Nature: [Empirical Fitting]

Calculation Formula:

Ω_m = 9 × (1 - n_s)

where: coefficient 9 is an empirical fitting value

Calculation Process:

Ω_m = 9.0 × (1 - 0.997912)
Ω_m = 9.0 × 0.002088
Ω_m = 0.018788

Empirical Coefficient Explanation:

Coefficient 9:

Nature: Empirical fitting value

Source: Determined through statistical optimization, not pure theoretical derivation

Physical meaning: Empirical coefficient mapping spectral index deviation to matter density

Uses External Data: ⚠️ Yes, coefficient 9 determined through statistical optimization (optimized based on observational data)

Note: This is a heuristic mapping, not first-principles derivation

5. Hubble Constant H₀

Parameter: H₀ = 52.000000 km/s/Mpc

Nature: [Heuristic Mapping]

Calculation Formula:

matrix_norm = ||matrix||_F = sqrt(Σᵢⱼ |matrixᵢⱼ|²)
H₀ = 50 + 20 × (matrix_norm / 10)

Calculation Process:

matrix_norm = ||matrix||_F = 1.000000
H₀ = 50 + 20 × (1.000000 / 10)
H₀ = 50 + 20 × 0.100000
H₀ = 50 + 2.000000
H₀ = 52.000000 km/s/Mpc

Heuristic Mapping Explanation:

Method: Based on empirical relationship of matrix features

Nature: Heuristic mapping, not pure theoretical derivation

Physical meaning: Empirical mapping from matrix amplitude to cosmic expansion rate

Coefficients 50 and 20: Empirically determined mapping parameters

Uses External Data: ⚠️ Yes, coefficients (50 and 20) in the mapping formula are empirically determined

Note: This is a heuristic mapping, not first-principles derivation

6. Dark Energy Equation of State w₀

Parameter: w₀ = -1.000000

Nature: [Empirical Fitting]

Calculation Formula:

1. Calculate unitarity deviation:
   unitarity_deviation = |Tr(rho) - 1|

2. Calculate w₀:
   w₀ = -1 - 0.1 × tanh(unitarity_deviation × 10)

   where: coefficient 0.1 is empirical

Calculation Process:

Tr(rho) = 1.000000
unitarity_deviation = |1.000000 - 1| = 0.000000
w₀ = -1 - 0.1 × tanh(0.000000 × 10)
w₀ = -1 - 0.1 × 0.000000
w₀ = -1.000000

Empirical Coefficient Explanation:

Coefficient 0.1:

Nature: Empirical value

Source: Determined through statistical optimization, not pure theoretical derivation

Physical meaning: Empirical coefficient mapping unitarity deviation to dark energy equation of state

Uses External Data: ⚠️ Yes, coefficient 0.1 determined through statistical optimization (optimized based on observational data)

Note: Based on quantum system non-unitarity, but coefficient 0.1 is empirical

Parameter Nature Statistics

Nature ClassificationNumber of ParametersParameter ListPercentage[Theoretical Derivation]2c_raw, n_s33.3%[Using Empirical Calibration Parameters]1c_eff16.7%[Empirical Fitting]2Ω_m, w₀33.3%[Heuristic Mapping]1H₀16.7%Total6-100%

Calculation Flow Dependency

Step 1: Initialize quantum matrix
  └─> [Theoretical Derivation] Based on mathematical functions, no external data

Step 2: Calculate entanglement entropy
  └─> [Theoretical Derivation] von Neumann entropy formula

Step 3: Extract central charge
  └─> [Theoretical Derivation] Ryu-Takayanagi formula
      └─> c_raw = 2.172459

Step 4: Calculate effective central charge
  └─> [Using Empirical Calibration Parameters] c_eff = κ × c_raw × n
      ├─> κ = 21.0 [Empirical calibration]
      ├─> n = 21 [Empirical calibration]
      └─> c_eff = 958.054554

Step 5: Derive spectral index
  └─> [Theoretical Derivation] n_s = 1 - 2/c_eff
      └─> n_s = 0.997912
      ⚠️ Note: Although formula is theoretical derivation, input c_eff uses empirical calibration parameters

Step 6: Derive matter density
  └─> [Empirical Fitting] Ω_m = 9 × (1 - n_s)
      ├─> coefficient 9 [Empirical fitting value]
      └─> Ω_m = 0.018788

Step 7: Derive Hubble constant
  └─> [Heuristic Mapping] H₀ = 50 + 20 × (matrix_norm/10)
      ├─> coefficients 50 and 20 [Empirically determined]
      └─> H₀ = 52.000000 km/s/Mpc

Step 8: Derive dark energy equation of state
  └─> [Empirical Fitting] w₀ = -1 - 0.1 × tanh(...)
      ├─> coefficient 0.1 [Empirical value]
      └─> w₀ = -1.000000

Scientific Integrity Detailed Explanation

Completely Theoretically Derived Parameters (2)

c_raw (Raw Central Charge)

✅ Based on Ryu-Takayanagi theoretical formula

✅ Extracted through logarithmic linear fitting

✅ No external data dependency

⚠️ But the fitting process itself is a numerical method

n_s (Spectral Index)

✅ Based on CFT theoretical formula

✅ Pure mathematical derivation

⚠️ But input parameter c_eff uses empirical calibration parameters

Using Empirical Calibration Parameters (1)

c_eff (Effective Central Charge)

⚠️ κ = 21.0 is an empirical calibration parameter

⚠️ n = 21 is an empirical calibration parameter

⚠️ These parameters determined through statistical optimization (based on observational data)

✅ But the formula itself is a clear mathematical relationship

Empirically Fitted Parameters (2)

Ω_m (Matter Density)

⚠️ Coefficient 9 is an empirical fitting value

⚠️ Determined through statistical optimization (based on observational data)

⚠️ This is a heuristic mapping, not first-principles derivation

w₀ (Dark Energy Equation of State)

⚠️ Coefficient 0.1 is an empirical value

⚠️ Determined through statistical optimization (based on observational data)

⚠️ Based on physical concept (non-unitarity), but coefficient is empirical fitting

Heuristically Mapped Parameters (1)

H₀ (Hubble Constant)

⚠️ Mapping formula is heuristic

⚠️ Coefficients 50 and 20 are empirically determined

⚠️ Based on empirical relationship of matrix features

⚠️ Not first-principles derivation

Summary

Theoretical Derivation Capability

Complete Theoretical Derivation: c_raw (extracted through numerical fitting)

Theoretical Formula but Dependent on Calibration Parameters: n_s (formula is theoretical, but input uses calibration parameters)

Empirical Calibration/Fitting

Using Empirical Calibration Parameters: c_eff (κ=21, n=21)

Empirical Fitting: Ω_m (coefficient 9), w₀ (coefficient 0.1)

Heuristic Mapping

Heuristic Mapping: H₀ (based on empirical relationship of matrix features)

Scientific Integrity

✅ All calculations are based on QNM model, no direct use of external observational data as input

⚠️ But calibration parameters and empirical coefficients are determined through statistical optimization (optimized based on observational data)

✅ All parameter natures are explicitly labeled, calculation process is completely transparent

Author

Nanjie Ma (马楠杰)

ORCID: 0009-0002-4415-1209

Date: 2025-12-18

This document truthfully and objectively presents the calculation methods and nature classification for each cosmological parameter in the QNM model, maintaining scientific integrity.

This model represents an initial implementation that I have now subjected to initial testing. However, it requires thorough vetting and validation under diverse conditions. While I cannot guarantee its absolute correctness at this stage, I can unequivocally state that all work has been conducted in good faith with no data manipulation. I sincerely invite the community to evaluate it and provide feedback. Your input is highly valued.