TGD diary: About the problem of two Hubble constants

The usual formulation of the problem of two Hubble constants is that the value of the Hubble constant seems to be increasing with time. There is no convincing explanation for this. But is this the correct way to formulate the problem? In the TGD framework one can start from the following ideas discussed already earlier (see this).

Would it be better to say that the measurements in short scales give slightly larger results for H₀ than those in long scales? Scale does not appear as a fundamental notion neither in general relativity nor in the standard model. The notion of fractal relies on the notion but has not found the way to fundamental physics. Suppose that the notion of scale is accepted: could one say that Hubble constant does not change with time but is length scale dependent. The number theoretic vision of TGD brings brings in two length scale hierarchies: p-adic length scales L_p and dark length scale hierarchies L_p(dark)=nL_p, where one has h_eff=nh₀ of effective Planck constants with n defining the dimension of an extension of rationals. These hierarchies are closely related since p corresponds to a ramified prime (most naturally the largest one) for a polynomial defining an extension with dimension n.
I have already earlier considered the possibility that the measurements in our local neighborhood (short scales) give rise to a slightly larger Hubble constant? Is our galactic environment somehow special?

Consider first the length scale hierarchies.

The geometric view of TGD replaces Einsteinian space-times with 4-surfaces in H=M⁴\times CP₂. Space-time decomposes to space-time sheets and closed monopole flux tubes connecting distant regions and radiation arrives along these. The radiation would arrive from distant regions along long closed monopole flux tubes, whose length scale is L_H. They have thickness d and length L_H. d is the geometric mean d=(l_PL_H)^1/2 of Planck length L_P and length L_H. d is of about 10^-4 meters and size scale of a large neuron. It is somewhat surprising that biology and cosmology seem to meet each other.
The number theoretic view of TGD is dual to the geometric view and predicts a hierarchy of primary p-adic length scales L_p ∝ p^1/2 and secondary p-adic length scales L_2,p =p^1/2L_p. p-Adic length scale hypothesis states that p-adic length scales L_p correspond to primes near the power of 2: p ≈ 2^k. p-adic primes p correspond to so-called ramified primes for a polynomial defining some extension of rationals via its roots.
One can also identify dark p-adic length scales
L_p(dark) =nL_p ,
where n=h_eff/h₀ corresponds to a dimension of extension of rationals serving as a measure for evolutionary level. h_eff labels the phases of ordinary matter behaving like dark matter explain the missing baryonic matter (galactic dark matter corresponds to the dark energy assignable to monopole flux tubes).
p-Adic length scales would characterize the size scales of the space-time sheets. The Hubble constant H₀ has dimensions of the inverse of length so that the inverse of the Hubble constant L_H∝ 1/H₀ characterizes the size of the horizon as a cosmic scale. One can define entire hierarchy of analogs of L_H assignable to space-time sheets of various sizes but this does not solve the problem since one has H₀ ∝ 1/L_p and varies very fast with the p-adic scale coming as a power of 2 if p-adic length scale hypothesis is assumed. Something else is involved.

One can also try to understand also the possible local variation of H₀ by starting from the TGD analog of inflation theory. In inflation theory temperature fluctuations of CMB are essential.

The average value of h_eff is < h_eff>=h but there are fluctuations of h_eff and quantum biology relies on very large but very rare fluctuations of h_eff. Fluctuations are local and one has <L_p(dark)> = <h_eff/h₀> L_p. This average value can vary. In particular, this is the case for the p-adic length scale L_p,2 (L_p,2(dark)=nL_2,p), which defining Hubble length L_H and H₀ for the first (second) option.
Critical mass density is given by 3H₀²/8πG. The critical mass density is slightly larger in the local environment or in short scales. As already found, for the first option the fluctuations of the critical mass density are proportional to δ n/n and for the second option to -δ n/n. For the first (second) option the experimentally determined Hubble constant increases when n increases (decreases). The typical fluctuation would be δ h_eff/h ∼ 10^-5. What is remarkable is that it is correctly predicted if the integer n decomposes to a product n₁=n₂ of nearly identical or identical integers.
For the first option, the fluctuation δ h_eff/h_eff=δn/n in our local environment would be positive and considerably larger than on the average, of order 10^-2 rather than 10^-5. h_eff measures the number theoretic evolutionary level of the system, which suggests that the larger value of <h_eff> could reflect the higher evolutionary level of our local environment. For the second option the variation would correspond to δn/n≤ 0 implying lower level of evolution and does not look flattering from the human perspective. Does this allow us to say that this option is implausible?
The fluctuation of h_eff around h would mean that the quantum mechanical energy scales of various systems determined by <h_eff>=h vary slightly in cosmological scales. Could the reduction of the energy scales due to smaller value of h_eff for systems at very long distance be distinguished from the reduction caused by the redshift. Since the transition energies depend on powers of Planck constant in a state dependent manner, the redshifts for the same cosmic distance would be apparently different. Could this be tested? Could the variation of h_eff be visible in the transition energies associated with the cold spot?
The large fluctuation in the local neighbourhood also implies a large fluctuation of the temperature of the cosmic microwave background: one should have δT/T ≈ δn/n≈ δ H₀/H₀. Could one test this proposal?