Author’s note: Because the study is very long, it was not uploaded to the site. The continuing text is the summary report of the study.
Initial hypothesis:
Could there be a Cooperative Fractality Effect instead of Dark Matter, Dark Energy?
First Result
I wrote down the cosmic budget: 𝛀𝒃 = 𝟎. 𝟎𝟒𝟗 𝛀𝑫𝑴 = 𝟎. 𝟐𝟔𝟔 𝛀𝚲 = 𝟎. 𝟔𝟖𝟓 Total: 𝟏. 𝟎 From here: I found that an amplification like 𝜼 ≈ 𝟔. 𝟒 is required. This number appeared over and over again in all subsequent calculations. I think this might not be a coincidence.
Rotation Curves
I set this up: 𝑴𝒆𝒇𝒇(𝒓) = 𝑴𝒃(𝒓) (𝒓/𝒓𝟎)𝜶(𝒓) Then I moved to a stronger form: 𝚽𝑭 = 𝑨𝐥𝐧(𝟏 + 𝒓/𝒓𝒄) and 𝒈𝑭 = 𝑨 / (𝒓 + 𝒓𝒄) were obtained. As a result of this: 𝒗(𝒓) → 𝒄𝒐𝒏𝒔𝒕 happened. Meaning: Flat rotation curves emerged.
Milky Way
With 𝒗𝒇 ≈ 𝟐𝟑𝟎 𝒌𝒎/𝒔, I found 𝑨 = 𝟓𝟐𝟗𝟎𝟎 𝒌𝒎𝟐/𝒔𝟐. Then: 𝑴𝑭(𝟑𝟎 𝒌𝒑𝒄) ≈ 𝟑. 𝟕 × 𝟏𝟎𝟏𝟏𝑴⊙ came out. This amplified the visible matter by approximately 6 times. Again the same number: 𝜼 ≈ 𝟔
Tully-Fisher
I found this: If 𝑨(𝑴𝒃) ∝ (𝑴𝒃)1/2, then 𝒗𝟒 ∝ 𝑴𝒃 comes out. Result: Baryonic Tully-Fisher emerged naturally.
Fractal Halo
I derived this: 𝝆𝑲𝑭(𝒓) = ( 𝑨 / 𝟒𝝅𝑮 ) ( (𝒓 + 𝟐𝒓𝒄)/ (𝒓(𝒓 + 𝒓𝒄)𝟐) Its limits: 𝝆 ∼ 𝒓 -𝟏 at the center, 𝝆 ∼ 𝒓 -𝟐 on the outside. This produced a structure related to NFW.
Bullet Cluster
A very important point. Initially, that was the place everyone would attack. But when I built a synthetic map with two clusters, I saw that: If the cooperativity field 𝑪(𝒙, 𝒚) is a network carried along with the galaxies, the lensing peaks can be separated from the gas. Therefore: Bullet Cluster was not an automatic fatal blow. But it is not a proof yet either.
CMB
The first big surprise came here. Assuming the cooperative field behaves like: 𝛀𝑲𝑭,𝒎 = 𝟎. 𝟐𝟔𝟓(𝟏 + 𝒛)𝟑, at the time of recombination: 𝛀𝒎 𝒆𝒇𝒇 ≈ 𝟎. 𝟑𝟏𝟒 came out. And this is exactly close to the needed matter budget. Therefore: The sound horizon 𝒓𝒔 ≈ 𝟏𝟒𝟕 𝑴𝒑𝒄 can be preserved.
First Serious Crisis
𝒇𝝈𝟖 calculation. The first rough model: for 𝜿 = 𝟓. 𝟒𝒂𝟐, it yielded: 𝒇𝝈𝟖(𝟎) ≈ 𝟎. 𝟗𝟖. This was too large. But this failure produced a very important discovery: 𝜿 = 𝜿(𝒂) is not enough
The Main Discovery
After this: I moved to the idea of 𝜿 = 𝜿(𝑳, 𝒂). Meaning cooperativity became:
- time-dependent
- scale-dependent Right here, the study began to deepen.
RG Equation
I set up this flow equation: ( 𝝏𝜿 / 𝝏 𝐥𝐧 𝒂 ) + 𝒗 𝝏𝜿 / ( 𝝏 𝐥𝐧 𝑳 ) = 𝝁𝜿 ( 𝟏 − 𝜿 / 𝜿∗ ) In my eyes, this is the turning point of the study. Because for the first time:
- galaxy
- cluster
- cosmology entered into the same mathematics.
Analytical Solution
Then: 𝜿(𝑳, 𝒂) = 𝜿∗ / ( 𝟏 + 𝑨𝟎 (𝑳𝒄/𝑳)𝒒 𝒂 -(𝝁-𝒗𝒒)) came out. And this yielded: 𝑫𝒇(𝑳, 𝒂) = 𝟑 + 𝝐𝜿.
The Last Big Idea
In the very end, I arrived at this: Perhaps the main variable is not 𝜿. Perhaps the main physics is 𝑫𝒇(𝑳, 𝒂), meaning the effective fractal dimension. Stars: 𝑫𝒇 ≈ 𝟑. 𝟎𝟎 Galaxies: 𝑫𝒇 ≈ 𝟑. 𝟐 − 𝟑. 𝟑 Clusters: 𝑫𝒇 ≈ 𝟑. 𝟏 Cosmic scale: 𝑫𝒇 ≈ 𝟑. 𝟎𝟐 − 𝟑. 𝟎𝟖 In this framework: Dark matter = high 𝑫𝒇 regime Dark energy = low density large scale 𝑫𝒇 regime.
My Final Evaluation
Initial hypothesis: “Could cooperative fractality replace dark matter and dark energy?” The summary result of the study: At the galaxy scale yes, it is a surprisingly good candidate. It is compatible with Tully-Fisher. It produces a Halo profile. It tolerates the Bullet Cluster in the first order. It doesn’t have to ruin the sound horizon. ⚠ Structure formation and CMB peak heights are still the ultimate test. But in my opinion, the main achievement is this: I have now stepped out of the question “where is the missing mass?”.
Instead, I arrived at this question: 𝑫𝒇(𝑳, 𝒂) =?
If Fractal Relativity is ever going to truly work one day, I think at its heart there will not be a dark matter particle, but a scale and time-dependent effective fractal dimension field. This is the strongest result this study has given me.