_{1}

^{*}

Condense matter methods and mathematical models used in solving problems in solid state physics are transformed to high energy quantum cosmology in order to estimate the magnitude of the missing dark energy of the universe. Looking at the problem from this novel viewpoint was rewarded by a rather unexpected result, namely that the gap labelling method of integrated density of states for three dimensional icosahedral quasicrystals is identical to the previously measured and theoretically concluded ordinary energy density of the universe, namely a mere 4.5 percent of Einstein’s energy density,
*i*
*.e. E*(
*O*) =
*mc*
^{2}/22 where
*E* is the energy,
*m* is the mass and
*c* is the speed of light. Consequently we conclude that the missing dark energy density must be
*E*(
*D*) = 1 －
*E*(
*O*) =
*mc*
^{2}(21/22) in agreement with all known cosmological measurements and observations. This result could also be interpreted as a strong evidence for the self similarity of the geometry of spacetime, which is an expression of its basic fractal nature.

It is well known that many models and mathematical techniques that proved to be valuable in the low energy domain of condense matter physics were found to be of considerable usefulness in clarifying basic questions in high energy physics [

It is well known that electronic band theory is a very useful and successful theory in the physics of solids that solved difficult problems connected to the design of solar cells and transistors as well as illuminating fundamental properties of solids such as optical absorption and electrical resistance [

Now with a somewhat unconventional but well motivated idea in the back of our minds, namely that of draw- ing an instructive analogy between the zero density inside a band gap and the geometry and topology of the crystal lattice on the one side and ordinary energy and dark energy density contained in the structure of our cosmos on the other side, we will start here by extending the above concepts and notions to quasi periodic crystalline [

Probably the simplest group of one dimensional systems to illustrate the theory at hand is an automatic sequences such as period doubling, the Rudin-Shapiro sequence and Thue-Morse sequence [

grated density of states having the same information as that of a higher dimensional model. Let the two letters alphabet be given by

Following Bellissard’s general exposition and his notation we find two matrices [

and

The largest Eigenvalue of the above is

and the integrated density of states (IDS) is given by [

where

As noted by Landi [

which is reminiscent of the previous recursive Fibonacci example. Proceeding in the usual way Landi can then prove the proposition that the c star algebra of the Penrose tiling gives rise to a group given by [

and

This result is again identical to that obtained by Connes and noting the one to one correspondence between the bijection formula of E-infinity Cantorian spacetime [

and von Neumann-Connes dimensional function it follows that k_{o}_{+} as well as [IDS] are simply mathematical tautology, albeit an extremely instructive one bringing various theories for the micro cosmos and the large structure of spacetime [

exactly as should be while the zero set is given clearly by [

so that

Next we discuss the vital physical and cosmological implication of the preceding results.

One of the most important conclusions arrived at from the preceding Section 2 is that what we called topological probability

Hardy’s quantum probability of entanglement [

Being a probability we could look upon it as being the inverse of a dimension, i.e. un-normed probability given by the bijection formula [

Therefore we have the normed probability [

Alternatively we could see P(Hardy) as living in a negative four dimensional space

Here we tacitly made use of the notion of the degree of emptiness of an empty set introduced first by the late inventor of the word fractals, B. Mandelbrot [

Let us return to

where the 22 may be viewed as the compactified dimensional subset of the 26 dimensions of the bosonic string theory [

There is one aspect of theoretical physics that is so incredibly beautiful that one cannot find the right words to describe it. This is a first hand experience of the present author which happens whenever he notices that two totally different fields can be directly connected and analogies established simply because the same stringent logic, i.e. the same mathematical pattern and schemes are obeyed by both fields. One such case is the connection between super conductivity and the high energy physics of elementary particles [

The present work reveals a similar situation where the extremely small and large ultra obeys basically the same subgroup of R generated by Z and the golden mean number

From all of the above we have considerable renewed confidence in our proposal made some two decades ago that the universe as a whole can be regarded as huge quasicrystals [