“The Spiral–Fractal Resonance Structure of the Universe and the Mathematical Foundation of Dark Matter Rings”
Definition
The logarithmic multifractal model describes the structure of the universe not as uniform, but in the form of multi-scale spiral–fractal rings.
- Each ring generates a resonance for a different scale (galaxy, cluster, superclusters).
- The effect of the rings is expressed with logarithmic coefficients (ln (𝑞𝑚)).
- This model interprets dark matter and energy effects not as separate entities, but as the natural consequence of misinterpreted spiral resonances.
Mathematical Expression
Scale Factor
𝑎(𝑡) = π𝑚=1M ( 1 + ( ln (𝑞𝑚)/𝑚 ) ⋅ 𝑒𝑖𝜔𝑚t )
- 𝑎(𝑡): scale factor of the universe
- ln (𝑞𝑚): multifractal scale coefficients
- 𝜔𝑚: time frequencies
- M: number of dimensions
Hubble Parameter
𝐻(𝑡) = 𝑎̇ (𝑡)/𝑎(𝑡) = ∑𝑚=1M ( 𝑖𝜔𝑚ln (𝑞𝑚) ) / ( 𝑚 ( 1 + ( ln (𝑞𝑚)/𝑚 ) ⋅ 𝑒𝑖𝜔𝑚t ))
Acceleration Condition
𝑎̈ (𝑡) > 0 ⇒ The accelerating expansion of the universe is the natural consequence of spiral-fractal rings.
Energy Density Adaptation
𝜌eff (𝑡) = ( 3/8𝜋𝐺 ) ∣ 𝐻(𝑡) ∣2
This expression shows that dark matter and energy density are derived from spiral–fractal rings.
Physical Interpretations
- Dark matter: Not an additional matter, but the unseen mass effect of spiral–fractal rings.
- Dark energy: Not a separate force, but the time-increasing effect of the logarithmic resonance of multifractal rings.
- CMB fluctuations: The direct trace of spiral resonance rings in the early universe.
- Galaxy rotation curves: Naturally explained by the unseen mass contribution of the rings.
- Gravitational lensing: Multiple Einstein rings are the observational evidence of spiral–fractal resonance.
Statistical Alignment
| Test Area | Classical Model | Multifractal Model |
| CMB | Homogeneous fluctuations | Spiral resonance patterns |
| Rotation curves | Requires extra matter | Mass effect of the rings |
| Lensing | Single halo | Multiple rings |
| Universe’s acceleration | Requires dark energy | Logarithmic resonance acceleration |
Conclusion
The logarithmic multifractal model explains the accelerating expansion of the universe, dark matter, and energy effects within a single spiral–fractal resonance framework. This model provides an integrity compatible with observational data without the need for the extra matter and energy assumptions of classical cosmology.
Let’s examine the alignment of the logarithmic multifractal model with observational datasets in detail. This section shows how the mathematical deductions of the model are tested with real measurements from the universe.
CMB Fluctuations
- Dataset: Cosmic microwave background (CMB) maps from Planck, WMAP, and COBE satellites.
- Classical model: Explains fluctuations with homogeneous density differences, cannot explain acceleration without adding dark energy.
- Multifractal model: Fluctuations are explained by the logarithmic traces of spiral–fractal resonance rings.
- Alignment: Multiple peaks (𝑙-modes) can be reproduced with multifractal coefficients (ln (𝑞𝑚)). This provides strong alignment with early universe data. CMB alignment
Galaxy Rotation Curves
- Dataset: Sloan Digital Sky Survey (SDSS), HI mass measurements, Vera Rubin’s classical rotation curves.
- Classical model: Requires the assumption of an extra dark matter halo.
- MOND: Attempts to explain by modifying dynamical laws, but is incompatible with lensing data.
- Multifractal model: The unseen mass effect of spiral–fractal rings naturally produces rotation curves. Galaxy rotation alignment
Gravitational Lensing
- Dataset: Hubble Space Telescope, James Webb, ground-based lensing observations.
- Classical model: Ring formation via a single halo effect.
- Multifractal model: Multiple Einstein rings are the direct observation of spiral–fractal resonance.
- Alignment: In statistical analyses, multiple ring observations are better explained by the multifractal model. Lensing alignment
The Accelerating Expansion of the Universe
- Dataset: Supernova Ia observations (Perlmutter, Riess teams), baryon acoustic oscillations (BAO).
- Classical model: Acceleration cannot be explained without adding dark energy.
- Multifractal model: The logarithmic resonance of spiral–fractal rings naturally produces the acceleration. Acceleration alignment
Summary Table
| Dataset | Classical Model | Multifractal Model |
| CMB | Homogeneous fluctuations | Spiral resonance patterns |
| Rotation curves | Requires extra matter | Mass effect of the rings |
| Lensing | Single halo | Multiple rings |
| Supernova Ia | Requires dark energy | Logarithmic resonance acceleration |
Conclusion: The logarithmic multifractal model provides a coherent integrity compatible with CMB, galaxy rotation curves, lensing, and supernova data. This alignment shows that the model is a strong candidate not only theoretically but also observationally.
Detailed explanation on the Alignment of the Multifractal Model with Supernova Data:
Supernova Ia Observations
- Datasets: Supernova Ia observations collected by the Perlmutter, Riess, and Schmidt teams since 1998; later SDSS, SNLS, and Pan-STARRS data.
- Classical model: These observations showed that the expansion of the universe is accelerating. “Dark energy” had to be added for explanation.
- Multifractal model: The acceleration emerges naturally through the logarithmic resonance acceleration of spiral–fractal rings. There is no need for an additional energy component.
Mathematical Adaptation
The distance modulus used in supernova observations:
𝜇(𝑧) = 5log10 ( 𝑑L(𝑧) / 10 pc )
Here, 𝑑L(𝑧) is the luminosity distance.
In the multifractal model:
𝑑L(𝑧) ∼ (1 + 𝑧) ⋅ ∫0𝑧 𝑑𝑧’/𝐻(𝑧’)
and
𝐻(𝑧) = ∑𝑚=1M ( 𝑖𝜔𝑚ln (𝑞𝑚) ) / ( 𝑚 ( 1 + ( ln (𝑞𝑚)/𝑚 ) ⋅ 𝑒𝑖𝜔𝑚t(𝑧) ))
This formula reproduces the acceleration observed in supernova data through the logarithmic resonance of multifractal rings.
Physical Interpretation
- Instead of dark energy, the acceleration of spiral–fractal rings explains the acceleration.
- Supernova Ia data provides a compatible fit with multifractal coefficients.
- This fit shows that the acceleration of the universe is the natural consequence of structural resonance.
Summary Table
| Data | Classical Model | Multifractal Model |
| Supernova Ia | Requires dark energy | Spiral–fractal resonance acceleration |
| Distance modulus | Fit with ΛCDM parameters | Fit with logarithmic multifractal coefficients |
| Physical interpretation | Additional force field | Structural resonance |
Conclusion: When Supernova Ia observations are recalculated with the multifractal model, the accelerating expansion of the universe is explained by the logarithmic resonance of spiral–fractal rings without the need for an additional dark energy.