M⁸-H duality, implies fractal generalization of Hubble's law, predicts correct mass density of the Universe and resolves Hubble tension

The article by Thomas Brown (I am grateful to Esa-Juhani Ruoho for the link and also for inspiring discussions) discusses critically the interpretation of the Pollack effect. The vertex figure of ITT (IVF) (see this and this) is rhombicosidodecahedron (RID). I have proposed that the ITT is behind the genetic code and to be associated also with the hydrogen bonded water molecule clusters. Surprisingly, RID is identical with the third shell of the so-called icosahedral supercluster (ISC). This inspired the proposal about the duality between sensory representations realized by ISC at the level of the water and ITT realized at the level of the field body of the ISC.

The challenge is to understand how the complement of IVF, which should be outside RID, can correspond to the first and second shell of the ISC which are below the third shell. The obvious guess is ITT-ISC correspondences stating that the ITT realized at the field body of the ISC is related by inversion to ISC. M⁸-H duality, as the TGD counterpart of the momentum position duality, involves inversion in M⁴⊂ M⁸, having interpretation as momentum space, mapping it to M⁴ × CP₂. Is M⁸-H duality involved?

This question led to completely unexpected developments suggesting deep connections between fundamental physics (M⁸-H duality and the notions of gravitational and electric Planck constant as implications of number theoretic vision), physics of water (hydrogen bonded water clusters), consciousness theory (field body as controller of biological body forming sensory representations of biological body), biology (ITT view of the genetic code) and cosmology (generalization of Hubble's law to all scales).

In turned out that M⁸-H duality for the gravitational Planck constant leads to a fractal generalization of Hubble's law holding for causal diamonds CDs (CD=cd×CP₂) interpreted as counterparts of perceptive fields and analogous to cosmologies with big bang followed by a big crunch. This reflects the Russian doll cosmology predicted by TGD. Hubble constant appears as a scale dependent parameter charactrerizing the size scale of the cd. There is a single velocity type parameter β₀ parameter involved. This leads to a prediction for the Hubble length identified as gravitational Compton length interpreted as gravitational quantum coherence length at the gravitational field body. The large mass M, appearing in the expression of gravitational Planck constant, corresponds to the mass of a larger region containing the the quantum coherene region.

1. The proposed picture predicts that the mass of the visible Universe inside CD using solar mass as a unit is given by

   M(CD₁)/M_Sun= 2β₀(CD)L_H(CD)/r_S(Sun).

   Here r_S(Sun) equals 3 km. Assume that the radius of CD₁ can be expressed as L_H(CD₁)= xL_H(CD) so that one has V(CD₁)= x³V(CD). >

2. This gives an estimate for the mass within the Hubble radius

   M(CD₁)∼ 2× 10²² × β₀⁴(CD)×x^-3 M_Sun .

   Here the mass M(CD₁) corresponds to the mass within CD₁. M_Sun∼ 1.88 × 10⁵⁷ m_p, where m_p is proton mass. This predicts the average density

   ρ ∼ β₀⁴x^-3× 12× 10² m_p/m³ .

3. The density of baryons is estimated to be 5.9-6 protons per cubic meter (see this). The density ρ_B of ordinary (baryonic) matter is believed to be about p= 1/20 that is 5 percent of the total density: ρ∼ ρ_B/p= 20ρ_B∼ 120×m_p/m³. This gives β₀⁴x^-3∼ 1/10.
4. p-Adic length scales are good candidates for the size scales of CDs and seem to correspond to octaves p∼ 2^2k so that minimal scaling relating the sizes of CD and CD₁ containing CD should correspond to x=2. For β₀=1 the Universe would be a blackhole-like object with L_H= r_s/2β₀=r_s/2. For (p= 1/20,x=2) would predict β₀∼ .95. (β₀=1,x=2) would predict p∼ 6.1 per cent.

1. This requires that in short scales β₀ is 5-10 per cent smaller than in long scales. By β₀≤ 1 β₀=1 cannot be true in long scales (β₀=1,x=2) could be true in short scales (the rough estimate for ℏ_gr,E gives β₀∼ 1) and (β₀=.95,x=2) in long scales would predict difference 7.5 per cent Δ H₀/H₀ and resolve the Hubble tension.
2. β₀=1 in short scales as opposed to β₀=.95 in long length scales would require the scaling of baryon fraction from 5 percent in short scales to 6.1 percent in long scales. One would have L_H= r_s/2 and the Universe could be seen as a blackhole-like system for which the quantum coherence region would have radius L_H=r_s/2. This would give a p-adic fractal hierarchy of blackhole-like objects, which are quantum coherence regions of blackhole-like objects.
3. Why should the fraction of baryons be smaller in short scales than in long scales? A possible explanation is the transfer of baryons to dark baryons at monopole flux tubes, reducing the fraction of baryons in short scales (recent universe) from 6.1 percent to 5 percent. The cosmic evolution as an unavoidable increase of algebraic complexity would generate large h_eff phases and would also manifest as the formation of gravitational bound states such as galaxies, stars and planets.