A new dark matter anomaly

There is an intense flood of exciting news from both biology, neuroscience, cosmology and particle physics which are very interesting from TGD point of view. Unfortunately, I do not have time and energy to comment all of them. Special thanks for Mark Williams and Ulla for sending links: I try to find time to write comments.

One of the most radical parts of quantum TGD is the view about dark matter as a hierarchy of phases of matter with varying values of Planck constant realized in terms of generalization of the 8-D imbedding space to a book like structure. The latest blow against existing models of dark matter is the discovery of a new strange aspect of dark matter discussed in the popular article Galaxy study hints at cracks in dark matter theories in New Scientist. The original article in Nature is titled as Universality of galactic surface densities within one dark halo scale-length. I glue here a short piece of the New Scientist article.

A galaxy is supposed to sit at the heart of a giant cloud of dark matter and interact with it through gravity alone. The dark matter originally provided enough attraction for the galaxy to form and now keeps it rotating. But observations are not bearing out this simple picture. Since dark matter does not radiate light, astronomers infer its distribution by looking at how a galaxy's gas and stars are moving. Previous studies have suggested that dark matter must be uniformly distributed within a galaxy's central region � a confounding result since the dark matter's gravity should make it progressively denser towards a galaxy's centre. Now, the tale has taken a deeper turn into the unknown, thanks to an analysis of the normal matter at the centres of 28 galaxies of all shapes and sizes. The study shows that there is always five times more dark matter than normal matter where the dark matter density has dropped to one-quarter of its central value.

In TGD framework both dark energy and dark matter are assumed to correspond to dark matter but with widely different values of Planck constant. The point is that very large value of Planck constant for dark matter implies that its density is in an excellent approximation constant as is also the density of dark energy. Planck constant is indeed predicted to be gigantic at the space-time sheets mediating gravitational interaction.

The appearance of number five as a ratio of mass densities sounds mysterious. Why the average mass in a large volume should be proportional to hbar at least if hbar is not too large? Intriguingly, number five appears also in the Bohr model for planetary orbits. The value of the gravitational Planck constant GMm/v₀ assignable to the space-time sheets mediating gravitational interaction between planet and star is gigantic: v₀/c ∼2^-11 holds true inner planes. For outer planets v₀/c is by a factor 1/5 smaller so that coresponding gravitational Planck constant is 5 times larger. Do these two fives represent a mere coincidence?

1. In accordance with TGD inspired cosmology suppose that visible matter and also the matter which is conventionally called dark matter has emerged from the decay and widening of cosmic strings to magnetic flux tubes. Assume that the string tension can be written as k×hbar/G, k a numerical constant.

2. Suppose that the values of hbar come as pairs hbar=n× hbar₀ and 5×hbar. Suppose also that for a given value of hbar the length of the cosmic string (if present at all) inside a sphere or radius R is given by L=x(n)R, x(n) a numerical constant which can depend on the pair but is same for the members of the pair (hbar,5×hbar). This assumption is supported by the velocity curves of distant stars around galaxies.

3. These assumptions imply that the masses of matter for a pair (hbar,5×hbar) corresponding to a given value of hbar in a volume of size R are given by M(hbar)= k× x(hbar)× hbar×R/G and M(5×hbar)= 5×M(hbar). This would explain the finding if visible matter corresponds to hbar₀, and x(n) is much smaller for pairs (n>1,5×n) than for the pair (1,5).

4. One can explain the pairing in TGD framework. Let us accept the earlier hypothesis that the preferred values of hbar correspond to number theoretically maximally simple quantum phases q= exp(i2π/n) emerging first in the number theoretical evolution having a nice formulation in terms of algebraic extensions of rationals and p-adics and the gradual migration of matter to the pages of the book like structure labelled by large values of Planck constant. These number theoretically simple quantum phases correspond to n-polygons drawable by ruler and compass construction. This predicts that the preferred values of hbar correspond to a power of 2 multiplied by a product of Fermat primes F_k=2^2k+1. The list of known Fermat primes is short and given by F_k, k=0,1,2,3,4 giving the Fermat primes 3,5,17,257, 2¹⁶+1. This hypothesis indeed predicts that Planck constants hbar and 5×hbar appear as pairs.

5. Why the pair (1, F₁=5) should be then favored? Could the reason be that n=5 corresponds also to the smallest integer making possible universal topological quantum computer: the quantum phase q=exp(i2π/5) characterizes the braiding coding for the topological quantum computer program. Or is the reason simply that this pair corresponds to the number theoretically simplest pair which must have emerged first in the number theoretic evolution?

6. This picture supports the view that ordinary matter and what is usually called dark matter are characterized by Planck costants hbar₀ and 5×hbar₀, and that the space-time sheets mediating gravitational interaction correspond to dark energy because the density of matter at these space-time sheets must be constant in an excellent approximation since Compton lengths are so gigantic.

7. Using the fact that 4 per cent of matter is visible this means that n=5 corresponds to 20 per cent of dark matter in standard sense. Pairs (n>1,5×n) should contribute the remaining 2 per cent of dark matter. The fractal scaling law

   x(n) proportional to 1/n^r

   allowing pairs defined by all Fermat integers not divisible by 5 would give for the mass fraction of conventional dark matter with n>1 the expression

   p = 6× ∑_k 2^-kr×[2^-r+ ∑ n_F^-r]× (4/100)= (24/100)× (1-2^-r)^-1×[2^-r+ ∑ n_F^-r] .

   Here n_F denotes a Fermat integer which is product of some Fermat primes in the set {3,17,257, 2¹⁶+1}. The contribution from n=2^k, k>0, gives the term not included to the sum over n_F. r=4.945 predicts p=2.0035 and that the mass density of dark matter should scale down as 1/hbar^r-1= 1/hbar^3.945.
8. The prediction brings in mind the scaling 1/a^r-1 for the cosmological mass density. a^-4 scaling for the radiation dominated cosmology is very near to this scaling. r=5 would predict p=1.9164 which is of course consistent with the data. This inspires the working hypothesis that the density of dark matter as function of hbar scales just like the density of matter as function of cosmic time at particular epoch. In matter dominated cosmology with mass density behaving as 1/a³ one would have r=4 and p=4.45. In asymptotic cosmology with mass density behaving as 1/a² (according to TGD) one would have r=3 and p=11.68.
9. Living systems would represent a deviation from the "fractal thermodynamics" for hbar since for the typical values of hbar associated with the magnetic bodies in living systems (say hbar= 2⁴⁴hbar₀ for EEG to guarantee the the energies of EEG photons are above the thermal threshold) the density of the dark matter would be extremely small. Bio-rhythms are assumed to come as powers of 2 in the simplest model for the bio-system: the above considerations raise the question whether these rhythms could be accompanied by 5-multiples and perhaps also by Fermat integer multiples. For instance, the fundamental 10 Hz alpha frequency could be accompanied by 2 Hz frequency and the 40 Hz thalamocortical resonance frequency by 8 Hz frequency.

Some further observations about number five are in order. The angle 2π/5 relates closely to Golden Mean appearing almost everywhere in biology. n=5 makes itself manifest also in the geometry of DNA (the twist per single nucleotide is π/5 and aromatic 5-cycles appear in DNA nucleotides). Could it be that electron pairs associated with aromatic rings correspond to hbar=5×hbar₀ as I have proposed? Note that DNA as topological quantum computer hypothesis plays a key role in TGD inspired quantum biology.

For the background see the chapter TGD and Astrophysics.