Anomalous fluctuations of the microwave background and many-sheeted cosmology

Physicists are eagerly waiting for new WMAP results about the fluctuation spectrum of the cosmic microwave background. The analysis involves several difficult interpretational issues . The new results are highly interesting also from TGD point of view and therefore it deserves to make a brief summary about the situation.

Basic differences between inflationary and TGD inspired cosmology

Inflationary and TGD inspired cosmologies make remarkably similar predictions. There are also important differences following from quantum criticality of TGD Universe.

1. The flatness of 3-space during very early cosmology is the basic prediction of inflationary cosmology. TGD predicts a period of quantum critical cosmology with flat 3-space. There are two options. The critical period can follow cosmic string dominated period during which space-time sheets do not exist yet and end up with a transition to sub-critical cosmology. Alternatively, cosmic string dominated period can be followed by overcritical period followed by a critical phase transition period leading to sub-critical cosmology. Only subcritical cosmologies have global imbeddings and asymptotic cosmology is necesssary subcritical. The Robertson-Walker type metric during both critical and over-critical periods is fixed apart from the parameter fixing its duration and the metric component g_aa has exactly the same form for the two cases (a corresponds to lightcone proper time and the scale factor of RW cosmology denoted often by R). Hence TGD inspired early cosmology is extremely predictive.
2. In inflationary cosmology the temperature fluctuations reflect primordial density fluctuations amplified in scale during the exponential expansion. In TGD framework exponential expansion is replaced by an extremely sluggish logarithmic expansion during both overcritical phase (if present) and critical phase, and temperature fluctuations correspond to quantum critical long range fluctuations associated with the phase transition to subcriticality. The hierarchy of quantum coherent dark matters characterized by arbitrarily large values of hbar would be an essential piece of picture.

1. Fluctuations of microwave background as source of information about deviations from global homogenuity

The fluctuations of the microwave background temperature are due to the un-isotropies of the mass density: enhanced mass density induces larger red shift visible as a local lowering of the temperature. Hence the fluctuations of the microwave temperatures spectrum provide statistical information about the deviations of the geometry of the 3-space from global isotropy and homogenuity. The symmetries of the fluctuation spectrum can also provide information about the global topology of 3-space and for over-critical topologies the presence of symmetries is easily testable.

The first year Wilkinson microwave anisotropy probe observations allowed to deduce the angular correlation function. For angular separations smaller the 60 degrees the correlation function agrees well with that predicted by the inflationary scenarios and deriving essentially from the assumption of a flat 3-space (due to quantum criticality in TGD framework). For larger angular separations the correlations however vanish, which means the existence of a preferred length scale. The correlation function can be expressed as a sum of spherical harmonics. The J=1 harmonic is not detectable due to the strong local perturbation masking it completely. The strength of J=2 partial wave is only 1/7 of the predicted one whereas J=3 strength is about 72 per cent of the predicted. The coefficients of higher harmonics agree well with the predictions based on infinite flat 3-space.

Later some interpretational difficulties have emerged: there is evidence that the shape of spectrum might reflect local conditions. There are differences between northern and southern galactic hemispheres and largest fluctuations are in the plane of the solar system. In TGD framework these anomalies could be interpreted as evidence for the presence of galactic and solar system space-time sheets.

2. Dodecahedral cosmology?

The WMAP result means a discrepancy with the inflationary scenario and explanations based on finite closed cosmologies necessarily having Ω>1 but very near to Ω=1 have been proposed. J. P. Luminet (arXiv:astro-ph/0310253) has proposed that Poincare dodecahedral space, which is globally homogenous space obtained by identifying the points of S³ related by the action of dodecahedral group, or more concretely, by taking a dodecahedron in S³ (12 faces, 20 vertices, and 30 edges) and identifying opposite faces after 36 degree rotation, could explain the weakness of lowest partial waves. It was found to fit quadrupole and octupole strengths for 1.012<Ω <1.014 without an introduction of any other parameters than Ω .

However, according to Tegmark (arXiv:astro-ph/0310723), the quadrupole and octupole moments have a common preferred spatial axis along which the spectral power is suppressed so that dodecahedron model seems to be excluded. The analysis of Cornish et al (astro-ph/0310233) led to the same result. According to the recent Physics Web article of Luminet, the situation is however not yet completely settled, and there is even some experimental evidence for the predicted icosahedral symmetry of the thermal fluctuations.

The possibility to imbed also a very restricted family of over-critical cosmologies raises the question whether it might be possible to develop a TGD based version of the dodecahedral cosmology. The dodecahedral property could have two interpretations in TGD framework.

1. A space-time sheet with boundaries could correspond to a fundamental dodecahedron of S³. If temperature fluctuations are assumed to be invariant under the so called icosahedral group, which is subgroup of SO(3) leaving the vertices of dodecahedron invariant as a point set, the predictions of the dodecahedral model result.

2. An alternative interpretation is that the temperature fluctuations for S³ decomposing to 120 copies of fundamental dodecahedron are invariant under the icosahedral group. S³ would correspond to two space-time sheets containing 60 dodecahedrons on top of each other and glued along boundaries.

For neither option topological lensing phenomenon is present since icosahedral symmetry is not due to the identification of points of 3-space in widely different directions but due to symmetry which is not be strict. An objection against both options is that there is no obvious justification for the G invariance of the thermal fluctuations. The only justification that one can imagine is in terms of quantum coherent dark matter.

The finding of WMAP that the ratio Ω of the mass density of the Universe to critical mass density is Ω= 1+g_aa=1+ε , ε=0.02&plusm; 02. This is consistent with critical cosmology. If only slightly overcritical cosmology is realized, there must be a very good reason for this.

The WMAP constraint implies that the value of a which corresponds to the value of cosmic time a_s which characterizes the thermal fluctuations must be such that g_aa=ε holds true. The inspection of the explicit form of g_aa deduced in the subsection "Critical and over-critical cosmologies" of TGD and Cosmology requires that a_s is extremely near to the value a₀ of cosmic time for which g_aa=0 holds true: the deviation of a from a₀ should be of order (R/a₀)×R (R is CP₂ size about 10^-30 meters), and most of the thermal radiation should have been generated at this moment.

Since gravitational mass density approaches infinity at a→a₀ one can imagine that the spectrum of thermal fluctuations reflects the situation at the transition to sub-criticality occurring for Ω =1+ε. Thermal fluctuations would be identifiable as long ranged quantum critical fluctuations accompanying this transition and realized as a hierarchy of space-time sheets inducing the formation of structures. The scaling invariance of the fluctuation spectrum generalizes in TGD framework to conformal invariance. This means that the correlation function for fluctuations can have anomalous scaling dimension for which there is some evidence (astro-ph/9611208).

The transition k=1→0→-1 would involve the change in the shape of the S²⊂ CP₂ angle coordinate Φ as a function f(r) of radial coordinate of RW cosmology. The shape is fixed by the value of k=1,0,-1 . In particular, Φ would become constant in the transition to subcriticality. k=1→0 phase transition would be accompanied by the increase of the maximal size of space-time sheets to infinite in accordance with the emergence of infinite quantum coherence length at criticality. Whether this could be regarded as the TGD counterpart for the exponential expansion during inflationary period is an interesting question. In the transition to subcriticality also the shape of Θ as function of a necessarily changes since sin(Θ(a>a₀))>1 would be required otherwise.

3. Hyperbolic cosmology with finite volume?

Also hyperbolic cosmologies allow infinite number of non-simply connected variants with 3-space having finite volume. For these cosmologies the points of a=constant hyperboloid are identified under some discrete subgroup G of SO(3,1) . Also now fundamental domain determines the resulting space and it has a finite volume.

It has been found that a hyperbolic cosmology with finite-sized 3-space based on so called Picard hyperbolic space (astro-ph/0403597), which in the representation of hyperbolic space H³ as upper half space z>0 with line element ds²= (dx²+dy²+dz²)/z² can be modelled as the space obtained by the identifications (x,y,z)=(x+ma, y+nb,z). This space can be regarded as an infinitely long trumpet in z -direction having however a finite volume. The cross section is obviously 2-torus. This metric corresponds to a foliation of H³ represented as hyperboloid of M⁴ by surfaces m³=f(ρ) , ρ²= (m¹)²+(m²)² with f determined from the requirement that the induced metric is flat so that x,y correspond to Minkowski coordinates (m¹,m²) and z a parameter labelling the flat 2-planes corresponds to m³ varying from -∞ to +∞.

This model allows to explain the small intensities of the lowest partial waves as being due to constraints posed by G invariance but requires Ω=.95 . This is not quite consistent with Ω=1.02&plusm; .02 .

Also now two interpretations are possible in TGD framework. Thermal photons could originate from a space-time sheet identifiable as the fundamental domain invariant under G . Alternatively, a=constant hyperboloid could have a lattice-like structure having fundamental domain as a lattice cell with thermal fluctuations invariant under G . The shape of the fundamental domain interpreted as a surface of M⁴ is rather weird and one could argue that already this excludes this model.

Quantum criticality and the presence of quantum coherent dark matter in arbitrarily long length scales could explain the invariance of fluctuations. If Ω reflects the situation after the transition to subcriticality, one has Ω=g_aa-1=.95 . This gives g_aa=1.95 which is in conflict with g_aa<1 holding true for the imbeddings of all hyperbolic cosmologies. Thus Ω must correspond to the critical period and one should explain the deviation from Ω=1 . A detailed model for the temperature fluctuations possibly fixed by conformal invariance alone would be needed in order to conclude whether many-sheeted space-time might allow this option.

4. Is the loss of correlations due to the finite size of the space-time sheet?

One can imagine a much more concrete explanation for the vanishing of the correlations at angles larger than 60 degrees in terms of the many-sheeted space-time. Large angular separations mean large spatial distances. Too large spatial distance, together with the fact that the size of the space-time sheet containing the two astrophysical objects was smaller than now, means that they cannot belong to the same space-time sheet if the red shift is large enough, and cannot thus correlate. The size of the space-time sheet defines the preferred scale. The preferred direction would be most naturally defined by cosmic string(s) in the length scale of the space-time sheet. For instance, closed cosmic string would define an expanding 3-space with torus topology and thus having symmetries. This option would explain also the anomalies as effects due to galactic and solar space-time sheets.

Super-Conductivity in Many-Sheeted Space-Time

I have added a new chapter to "TGD and p-Adic Numbers". In the new chapter the description of super-conductivity in many-sheeted space-time is discussed. The notion of many-sheeted space-time alone provides strong motivation for this and I have developed various ideas about high T_c super-conductivity in parallel with ideas about living matter as a macroscopic quantum system. A further motivation and a hope for more quantitative modelling comes from the discovery of various non-orthodox super-conductors including high T_c superconductors, heavy fermion super-conductors and ferromagnetic superconductors. The standard BCS theory does not work for these super-conductors and the mechanism for the formation of Cooper pairs is not understood. There is experimental evidence that quantum criticality is a key feature of many non-orthodox super-conductors. TGD provides a conceptual framework and bundle of ideas making it possible to develop models for non-orthodox superconductors.

1. Quantum criticality, hierarchy of dark matters, and dynamical hbar

Quantum criticality is the basic characteristic of TGD Universe and quantum critical superconductors provide an excellent test bed to develop the ideas related to quantum criticality into a more concrete form.

The hypothesis that hbar is dynamical possessing quantized spectrum adds further content to the notion of quantum criticality. Phases with different values of hbar behave like dark matter with respect to each other in the sense that they do not have direct interactions. In large hbar phases various quantum time and length scales are scaled up which means macroscopic and macro-temporal quantum coherence.

The great idea is that the transition to large hbar phase occurs when perturbation theory based on the expansion in terms of gauge coupling constant ceases to converge: Mother Nature would take care of the problems of theoretician. The transition to large hbar phase obviously reduces gauge coupling strength alpha so that higher orders in perturbation theory are reduced whereas the lowest order "classical" predictions remain unchanged. A possible quantitative formulation of the criterion is that maximal 2-particle gauge interaction strength parameterized as Q₁Q₂α satisfies the condition Q₁Q₂α≈ 1.

A further hypothesis is that in the transition to large hbar phase the scaling hbar --> n×hbar/v₀, where n is integer and v₀≈ 2^-11 is expressible in terms of the ratio of Planck length to CP₂ length scale.

The only coupling constant strength of theory is Kähler coupling constant g_K² which appears in the definition of the Kähler function K characterizing the geometry of the configuration space of 3-surfaces (the "world of classical worlds"). The exponent of K defines vacuum functional analogous to the exponent of Hamiltonian in thermodynamics. The allowed values of g_K², which are analogous to critical temperatures, are determined by quantum criticality requirement and labelled by p-adic primes. hbar appears in the commutation and anticommutation relations of various superconformal algebras but not in the vacuum functional. For a given p-adic length scale space-time sheets with all allowed values of hbar are therefore possible. Hence the spectrum of quantum critical fluctuations could in the ideal case correspond to the spectrum of hbar coding for the scaled up values of Compton lengths and other quantal lengths and times. If so, large hbar phases could be crucial for understanding of quantum critical superconductors, in particular high T_c superconductors.

TGD actually predicts an infinite hierarchy of phases behaving like dark or partially dark matter with respect to the ordinary matter and the value of hbar is only one characterizer of these phases. These phases, especially so large hbar phase, seem to be essential for the understanding of even ordinary hadronic, nuclear and condensed matter physics. This strengthens the motivations for finding whether dark matter might be involved with quantum critical super-conductivity.

2. Many-sheeted space-time concept and ideas about macroscopic quantum phases

Many-sheeted space-time leads to obvious ideas concerning the realization of macroscopic quantum phases.

a) The dropping of particles to larger space-time sheets is a highly attractive mechanism of super-conductivity. If space-time sheets are thermally isolated, the larger space-time sheets could be at extremely low temperature and super-conducting.

b) The possibility of large hbar phases allows to give up the assumption that space-time sheets characterized by different p-adic length scales are thermally isolated. The scaled up versions of a given space-time sheet corresponding to a hierarchy of values of hbar are possible such that the scale of kinetic energy and magnetic interaction energy remain same for all these space-time sheets. For instance, for scaled up variants of space-time sheet having size scale characterized by L(151)=10 nm (cell membrane thickness) the critical temperature for superconductivity could be higher than room temperature.

c) The idea that wormhole contacts can form macroscopic quantum phases and that the interaction of ordinary charge carriers with the wormhole contacts feeding their gauge fluxes to larger space-time sheets could be responsible for the formation of Cooper pairs, have been around for a decade. The realization that wormhole contacts can be regarded as parton-antiparton pairs with parton and antiparton assignable to the light-like causal horizons accompanying wormhole contacts, opens the doors for more concrete models. The simplest idea is that em charged Cooper pairs can be modelled as a pair of charged particles at a space-time sheet X⁴_c topologically condensed to the background space-time sheet Y⁴ of condensed matter system. The Coulombic binding energy of charges particles with the quarks and antiquarks assignable to the wormhole throats feeding the em gauge flux to Y⁴ could be responsible for the energy gap.

d) Quantum classical correspondence has turned out be a very powerful idea generator. For instance, one can ask what are the space-time correlates for various notions of condensed matter such as phonons, BCS Cooper pairs, holes, etc... For instance, TGD predicts the existence of negative energy space-time sheets so that ordinary particles can and must exist in negative energy states (in cosmological scales the density of inertial energy is predicted to vanish). The question is whether holes could have quite concrete representation as negative energy space-time sheets carrying negative energy particles and whether the notion of Cooper pair of holes could have this kind of space-time correlate.

For details see the new chapter Super-Conductivity in Many-Sheeted Space-Time

Does Sun possess a solid surface?

For almost year ago I proposed a quantum model for planetary system based on the assumption that dark matter is in astroscopic quantum phase with a gigantic value of Planck constant and the quantum states of dark matter determine to a high degree the state of visible matter. Mercury corresponds in this model to n=3 Bohr orbit. A strange coindidence is that n=1 Bohr orbit corresponds in a reasonable approximation to solar radius. This raises the question whether solar surface could contain spherical shell representing a topological condensate of dense matter around dark matter, kind of spherical pre-form of planet below the photosphere.

Recently new satellites have begun to provide information about what lurks beneath the photosphere. The pictures produced by Lockheed Martin's Trace Satellite and YOHKOH, TRACE and SOHO satellite programs are publicly available in the web. SERTS program for the spectral analysis suggest a new picture challenging the simple gas sphere picture. The visual inspectation of the pictures combined with spectral analysis has led Michael Moshina to suggests that Sun has a solid, conductive spherical surface layer consisting of calcium ferrite. The article of Moshina provides impressive pictures, which in my humble non-specialist opinion support this view. Of course, I have not worked personally with the analysis of these pictures so that I do not have the competence to decide how compelling the conclusions of Moshina are. In any case, I think that his web article deserves a summary.

Before SERTS people were familiar with hydrogen, helium, and calcium emissions from Sun. The careful analysis of SERTS spectrum however suggest the presence of a layer or layers containing ferrite and other heavy metals. Besides ferrite SERTS found silicon, magnesium, manganese, chromium, aluminum, and neon in solar emissions. Also elevated levels of sulphur and nickel were observed during more active cycles of Sun. In the gas sphere model these elements are expected to be present only in minor amounts. As many as 57 different types of emissions from 10 different kinds of elements had to be considered to construct a picture about the surface of the Sun.

Moshina has visually analyzed the pictures constructed from the surface of Sun using light at wave lengths corresponding to three lines of ferrite ions (171, 195, 284 Angstroms). On basis of his analysis he concludes that the spectrum originates from rigid and fixed surface structures, which can survive for days. A further analysis shows that these rigid structure rotate uniformly.

The existence of a rigid structure idealizable as spherical shell in the first approximation could by previous observation be interpreted as a spherical shell corresponding to n=1 Bohr orbit of a planet not yet formed. This structure would already contain the germs of iron core and of crust containing Silicon, Ca and and other elements.

There is also another similar piece of evidence. A new planet has been discovered orbiting around a star in a triple-star system in the constellation Cygnus. The planet is a so-called hot Jupiter but it orbits the parent star at distance of .05 AU, which much less that than allowed by current theories of planetary formation. Indeed, the so called migration theory predicts that the gravitational pull of the two stars should have stripped away the proto-planetary disk from the parent star. If an underlying dark matter structure serves as a condensation template for the visible matter, the planetary orbit is stabilized by Bohr quantization.

For more details see the chapter TGD and Astrophysics of TGD.

Anomalous time dilation as a test distinguishing between TGD and GRT

TGD predicts the possibility of large anomalous time dilation effects due to the warping of space-time surfaces. In ordinary Euclidian three-space warping corresponds to an existence of larger imbeddings of planar surface as a flat surface obtained from planar surface by bending without stretching. Minkowski space indeed allows infinite number of warped flat imbeddings to M⁴×CP₂. Simple imbeddings of this kind are obtained by choosing a geodesic circle of CP₂ having radius R and allowings its angle coordinate φ to depend on M⁴ time coordinate t as φ=ωt. The metric is with respect to standard M⁴ coordinates given by g_tt=1-R²ω², g_ij=-δ_ij. The time dilation factor is g_tt^1/2 and can be quite large and due to warping rather than gravitation.

For the imbeddings of Scwartshild metric (as well as more general metrics) one obtains one parameter family of warped imbeddings and also now anomalous time dilation factor results. This effect manifests itself as anomalously large red shift as one compares the red shifts associated with different space-time sheets. The parameters of given space-time sheet can be adiabatically changed by external perturbations and lead to an adiabatic variation of the redshift. For instance, if the space-time sheet X⁴_c of clock moves "over" larger space-time sheet X⁴, it can form wormhole contacts with it inducing a change in warping and gradually increasing anomalous time dilation. When the clock is not "over" the larger space-time sheet anymore, the rate for the flow time as compared to the flow in reference system is suddenly increased to its original value.

The experimental findings of Russian physicist Chernobrov about anomalous changes in the rate of flow of time provide indirect support for this prediction. Chernobrov reports a slowing down of time by about 30 seconds per hour inside his experimental apparatus so that the average dilation factor during hour would be about Δ = 1/120. If the dilation is present all the time, the anomalous contribution to the gravitational potential would be by a factor ≈ 10⁷ larger than that of Earth's gravitational potential and huge gravitational perturbations would be required to produce this kind of effect.

The slowing of the time flow is reported to occur gradually whereas the increase for the rate of time flow is reported to occur discontinuously. Time dilation effects were observed in connection with the cycles of moon, diurnal fluctuations, and even the presence of operator.

Consider now the explanation of the basic qualitative findings of Chernobrov.

1. The gradual slowing of the time flow suggests that the parameter values of λ and ω change adiabatically. The formation of contacts indeed occurs with some finite rate.
2. Also the sudden increase of the rate of time flow is consistent with option a) since the splitting of contacts occurs immediately when the sheets X⁴_c and X⁴ are not "over" each other.
3. The occurrence of the effect in connection with the cycles of moon, diurnal fluctuations, and in the presence of operator support this interpretation. The last observation would support the view that intentional generation of almost vacuum space-time sheets is indeed possible.

For more details see the chapter TGD and GRT of TGD.

Dark matter revolution

The ideas related to dark matter that appeared already for year and half ago as I developed a model for topological quantum computation got a strong boost when I learned about the work of Nottale suggesting that the properties of astrophysical objects could be understood by assuming that dark matter corresponds to a phase in which hbar has a gigantic value, which can be fixed by applying Equivalence Principle. This lead to a revolution propagating through entire TGD (four books making about 5000 pages) and I have been carrying out the painful updating during this year and can now proudly declare that the first two books, TGD and p-Adic TGD are done! A good measure for the power of a new idea is the convergence of ideas and extinction of alternative options measured as a shortening of the text it induces. In many cases the reduction factor has been 1/2 or even more, which I regard really impressive. Also a lot of associative noise which is the unavoidable dark side of creative thinking has been cleaned out.

1. Interpretation of long ranged color and electro-weak gauge fields in terms of dark matter hierarchy

One of the basic predictions of TGD is the existence of long ranged classical color and electro-weak gauge fields. The interpretation of long ranged weak fields has been a longstanding challenge for TGD and I was more than a decade on a wrong track in this respect. The final solution of the problem came with realization that TGD unavoidably predicts infinite hierarchy of physics which are dark or more precisely, completely or partially dark, with respect to each other.

The observed elementary particles represent only a tip of an iceberg in TGD Universe although they contribute a sizable fraction to the energy density of the universe and determine to a high degree the properties of that portion of universe about which we have sensory perceptions. The dark physics corresponds typically to particle spectrum characterized by very light mass scales (say weak length scale of order atomic size or cell size). The fingerprints of dark matter are however visible already in nuclear physics and condensed matter physics and a rather detailed picture about the situation exists already now.

The contribution of these physics to mass density is not expected to be large: p-adic fractality suggests that the density of matter at a space-time sheets characterized by given p-adic prime scales as 1/L_p³, p≈2^k, k integer, preferably prime or power of prime. For instance, in cell length scale this would roughly mean density of few units of weak isospin per cell volume.

More generally, the hierarchies of physics are labelled by values of p-adic primes of particles, algebraic extensions of p-adic number fields, collection of conformal weights associated with a particle spectrum of a particular physics, and values of Planck constant labelled by generalized Beraha numbers B_q= 4cos²(π/q), q rational larger than 3. In particular, dark matter hierarchies with the values of hbar coming as hbar(n)= λ(m)^-nhbar(1), λ(m)=v₀/m≈ 2^-11/m are predicted. hbar(1) has discrete spectrum labelled by B_n, n integer, and varying in the range [1,2]×hbar, where hbar corresponds to the limit n goes to infinity.

One can say that TGD Universe is like a Mandelbrot fractal for which x--> 1/x transformation is performed. For Mandelbroot fractal the increase of resolution reveals completely new worlds endlessly, in TGD Universe the reverse operation does the same ad infinitum.

These physics couple only via the common bosons, such as graviton. The decay of particles of large hbar physics to those of smaller hbar physics is possible through de-coherence in which the Compton lengths and sizes of space-time sheets are reduced by 1/λ so that the particles do not anymore overlap in quantum sense and quantum coherence is lost. Even the de-coherence of ordinary laser beams could correspond to this process.

2. Number theoretical characterization of particles

Number theoretical vision leads to a vision in which elementary particles correspond to infinite primes, integers, or even rationals which in turn can be mapped to finite rationals. To infinite primes, integers, and rationals it is possible to associate a finite rational q=m/n by a homomorphism. q defines an effective q-adic topology of space-time sheet consistent with p-adic topologies defined by the primes dividing m and n (1/p-adic topology is homeomorphic to p-adic topology). The largest prime dividing m determines the mass scale of the space-time sheet in p-adic thermodynamics. m and n are exchanged by super-symmetry and the primes dividing m (n) correspond to space-time sheets with positive (negative) time orientation. Two space-time sheets characterized by rationals having common prime factors can be connected by a #_B contact (join along boundaries contact) and can interact by exchange of particles characterized by divisors of m or n.

This picture would dictate the selection rules for the interactions between particles belonging to the hierarchy of physics predicted by TGD. Particles would be characterized by a collection of p-adic primes defining as their product the integer characterizing the particle. Two particles would interact only if the integers characterizing them have common prime factors. Graviton should correspond to a product of primes common to all particles. Particles are completely dark relative to each other if they interact only via graviton exchange.