_{0}/c=2

^{-11}. This velocity defines the rotation velocities of distant stars around galaxies. The presence of a parameter with dimensions of velocity should carry some important information about the geometry of dark matter space-time sheets.

Velocity like parameters appear also in other contexts. There is evidence for the Tifft's quantization of cosmic red-shifts in multiples of v_{0}/c=2.68× 10^{-5}/3: also other units of quantization have been proposed but they are multiples of v_{0} (see this).

The strange behavior of graphene includes high conductivity with conduction electrons behaving like massless particles with light velocity replaced with v_{0}/c=1/300. The TGD inspired model explains the high conductivity as being due to the Planck constant h(M^{4})= 6h_{0} increasing the delocalization length scale of electron pairs associated with hexagonal rings of mono-atomic graphene layer by a factor 6 and thus making possible overlap of electron orbitals. This explains also the anomalous conductivity of DNA containing 5- and 6-cycles (same reference).

** 1. Is dark matter warped?**

The reduced light velocity could be due to the warping of the space-time sheet associated with dark electrons. TGD predicts besides gravitational red-shift a non-gravitational red-shift due to the warping of space-time sheets possible because space-time is 4-surface rather than abstract 4-manifold. A simple example of everyday life is the warping of a paper sheet: it bends but is not stretched, which means that the induced metric remains flat although one of its components scales (distance becomes longer around direction of bending). For instance, empty Minkowski space represented canonically as a surface of M^{4}× CP_{2} with constant CP_{2} coordinates can become periodically warped in time direction because of the bending in CP_{2} direction. As a consequence, the distance in time direction shortens and effective light-velocity decreases when determined from the comparison of the time taken for signal to propagate from A to B along warped space-time sheet with propagation time along a non-warped space-time sheet.

The simplest warped imbedding defined by the map M^{4}→ S^{1}, S^{1} a geodesic circle of CP_{2}. Let the angle coordinate of S^{1} depend linearly on time: Φ= ω t. g_{tt}} component of metric becomes 1-R^{2}ω^{2} so that the light velocity is reduced to v_{0}/c=(1-R^{2}ω^{2})^{1/2}. No gravitational field is present.

The fact that M^{4} Planck constant n_{a}h_{0} defines the scaling factor n_{a}^{2} of CP_{2} metric could explain why dark matter resides around strongly warped imbeddings of M^{4}. The quantization of the scaling factor of CP_{2} by R^{2}→ n_{a}^{2}R^{2} implies that the initial small warping in the time direction given by g_{tt}=1-ε, ε=R^{2}ω^{2}, will be amplified to g_{tt}= 1-n_{a}^{2}ε if ω is not affected in the transition to dark matter phase. n_{a}=6 in the case of graphene would give 1-x≈ 1- 1/36 so that only a one per cent reduction of light velocity is enough to explain the strong reduction of light velocity for dark matter.

** 2. Is c/v _{0} quantized in terms of ruler and compass rationals?**

The known cases suggests that c/v_{0} is always a rational number expressible as a ratio of integers associated with n-polygons constructible using only ruler and compass.

- c/v
_{0}=300 would explain graphene. The nearest rational satisfying the ruler and compass constraint would be q= 5× 2^{10}/17≈ 301.18. - If dark matter space-time sheets are warped with c
_{0}/v=^{11}one can understand Nottale's quantization for the radii inner planets. For dark matter space-time sheets associated with outer planets one would have c/v_{0}= 5× 2^{11}. - If Tifft's red-shifts relate to the warping of dark matter space-time sheets, warping would correspond to v
_{0}/c=2.68× 10^{-5}/3. c/v_{0}= 2^{5}× 17× 257/5 holds true with an error smaller than .1 per cent.

** 3. Tifft's quantization and cosmic quantum coherence**

An explanation for Tifft's quantization in terms of Jones inclusions could be that the subgroup G of Lorentz group defining the inclusion consists of boosts defined by multiples η= nη_{0} of the hyperbolic angle η_{0}≈ v_{0}/c. This would give v/c= sinh(nη_{0})≈ nv_{0}/c. Thus the dark matter systems around which visible matter is condensed would be exact copies of each other in cosmic length scales since G would be an exact symmetry. The property of being an exact copy applies of course only in single level in the dark matter hierarchy. This would mean a delocalization of elementary particles in cosmological length scales made possible by the huge values of Planck constant. A precise cosmic analog for the delocalization of electron pairs in benzene ring would be in question.

Why then η_{0} should be quantized as ruler and compass rationals? In the case of Planck constants the quantum phases q=exp(imπ/n_{F}) are number theoretically simple for n_{F} a ruler and compass integer. If the boost exp(η) is represented as a unitary phase exp(imη) at the level of discretely delocalized dark matter wave functions, the quantization η_{0}= n/n_{F} would give rise to number theoretically simple phases. Note that this quantization is more general than η_{0}= n_{F,1}/n_{F,2}.

For more details see the chapter TGD and Astro-Physics.