Deriving the Exact Percentage of Dark Energy Using a Transfinite Version of Nottale ’ s Scale Relativity

In this paper Nottale’s acclaimed scale relativity theory is given a transfinite Occam’s razor leading to exact predictions of the missing dark energy [1,2] of the cosmos. It is found that 95.4915% of the energy in the cosmos according to Einstein’s prediction must be dark energy or not there at all. This percentage is in almost complete agreement with actual measurements.

Nottale's theory of scale relativity is a powerful Weyllike general gauge theory with applications in high energy particle physics as well as cosmology [3,7,8].In that respect it is quite similar to the mathematical and physiccal K and E-infinity theory [17][18][19].The main difference comes only from the systematic use of logarithmic scaling in Nottale's scale relativity [3,7,8] where as E-infinity is exclusively based on a transfinite Weyl scaling [4,5].In particular Nottale's theory gave up differentiability but not continuity [3,7,8].By contrast E-infinity gave up both differentiability as well as continuity but preserved the cardinality of the continuum [4,5] using the geometry of random elementary Cantor sets [3][4][5]18].Since the Hausdorff dimension of such random elementary Cantor sets is the golden mean and its powers, the scaling exponent of the theory are combinatorics of these infinitely many golden mean random Cantor sets [3][4][5][6][14][15][16][17][18].In this regard we stress that it is generally wrong to think that discontinuity of space-time introduces something unphysical or a-physical to a theory because an empty set is physical and present in nature as it is present in the fundamental axioms of set theory upon which our entire mathematical methods are based.
In the present work we use Nottale's theory to give first an accurate approximate solution to the problem of the missing dark energy in the cosmos [1,2].Subsequently we minimally deform Nottale's scale relativity [3,7,8] making it transfinitely almost exact.The so obtained results are in superb agreement with the cosmological measurements [1,2].

Scale Relativity-Preliminary Remarks
Scale relativity is a profound general theory which was developed toward the end of the eighties last century [3,7,8].The theory builds heavily on the tradition of Einstein-Minkowski geometrization of physics and simultaneously makes extensive use of what at the time was the new science of nonlinear dynamics and the great pioneering spirit of deterministic chaos and fractal geometry.That way scale relativity combined the great ideas of Einstein with those of H. Weyl's original gauge theory [4], R. Feynman and Garnet Ord's proposal for a fractal space-time [3].Scale relativity and fractal space-time sparked quite a revolution in the way we think about foundational problems and cutting edge research in theoretical physics and although it is not as visible as superstrings [9] or loop quantum mechanics [10], it is in no way less original or insightful.In fact, it may be more in a complimentary way as we will attempt to show in the present paper.scale relativity and E-infinity Cantorian space-time theory [3][4][5][6][7][8].Let us concentrate on the most important common aspect of the two theories.No doubt it is fractality and scaling.These in reverse order are the quintessence of the two theories from a conceptual view point.However when we look at the quantitative analytical treatment, then the two theories differ in slightly less than a minor way.This is because scale relativity employs logarithmic scaling more or less similar to the logarithmic scaling of the standard model of elementary particles and quantum field theory.One only needs to remember the logarithmic running of the coupling constant as a function of the energy scale used in the electroweak and strong interaction [9][10][11][12][13] to get the idea.Such logarithmic scaling is a powerful approximation to the unattainable non-perturbative exact solution but none the less, it is an approximation.By contrast E-infinity theory employs a golden mean based exact renormalization semi-group and exact golden mean scaling exponents [4][5][6]14].None the less, without Weyl and Nottale's insight into the interrelation between gauge theory and fractals, we could not have developed E-infinity theory in its present form which depends crucially on the many excellent results of not only non-commutative geometry [17], superstrings [9], M-theory [15,16] and loop quantum gravity [10] but also in a fundamental way on Nottale's scale relativity and of course Ord's fractal space-time [3].
A second important aspect of scale relativity is that it gives the Planck length the same status which only the velocity of light enjoys in Einstein's theory of relativity.Such a proposal seems at first sight to be controversial because unlike the speed of light, the Planck length cannot be measured experimentally in any realistic set up [7,12,13].Never the less, it seems to us that the theory of varying speeds of light [12,13] which was clearly influenced by Nottale's scale relativity and gives a convincing mathematical argument for Planck energy invariance while preserving the Lorentzian symmetry group invariance although velocity and energy could be made arbitrarily much larger than the light velocity and the Planck energy without violating both [12,13].
In the present work we will adopt scale relativity theory in substantially its original form to determine the dark energy content of the universe [1,2] which boils down to revising Einstein's energy mass relation in order to extend its applicability to the realm of quantum gravity [7,8,10].Subsequently we will show how Nottale's scale relativity could gradually be made transfinitely exact and obtain the same result obtained using E-infinity theory and the fractal 11 dimensional M-theory [15,16].We start in the next section with a few explicit examples to illustrate the main ideas of scale relativity in the light of the competing theories such as E-infinity theory and Het-erotic superstrings [9].

Calculus in
while the opposite or reciprocal operation, i.e. integration is replaced by the Dixmier trace [17]  .
Here we follow the standard notation used also in [17].

Scaling in E-Infinity Theory
In E-infinity Cantorian fractal space-time on the other hand we discovered quite early that the H. Weyl failed original gauge theory is valid in space with no scale at all such as the infinite dimensional but hierarchical Cantor set modelling E-infinity space-time [4,5].Consequently scaling down is analogous to differentiation while scaling up is analogous to the opposite operation namely integration.It is as simple a duality as that between adding and subtracting or multiplying and dividing.Maybe a simple example makes the idea clearer.Let us take the low energy electromagnetic coupling constant where 5  is Hardy's generic probability of quantum entanglement and We note that pure gravity     dimensions and the Riemann tensor The strong interaction is given by the compactified Lie symmetry group SL (2,7) and we could infer that scaling up "ergo pseudo integrate" of 0  results in obtaining all the 339 gluons as well as gravity.In fact we could be more accurate and write that [4,5]  where and are Hardy type transfinite quantum entanglement corrections [6,14].In addition we have for the quarks-like state of the strong force [4,5] and finally for the electroweak force.
To sum up the insight of this section we say that while in scale relativity calculus is replaced by non-standard analysis and logarithmic scaling, in our approach we need only the golden mean scaling operation down scaling replaces differentiation and up scaling replacing integration.Before the end of this part however we give a very important and instructive down scaling which starts from the number of the first level of massless particles like quantum states in a transfinite Heterotic string theory.We can determine the Ambjorn-Loll [20] extremely important spectral dimension of quantum gravity D s = 4.019999.
We know that the classical value of N o in Heterotic strings is found from (504) (16) = 8064.This is actually the multiplication of the holographic boundary   exactly as the value found by Ambjorn and Loll using an efficient computer [20].It is really curious to see that digital computers are far more accurate than calculus when it comes to high energy physics.However, golden mean computers are even far more accurate and efficient than digital computers [4].Next we look at a simple example of how Nottale's theory deals with scaling and non-standard analysis.

Some Examples from Scale Relativity Calculus
Let us start with the fundamental optimized coupling constant of scale relativity, namely unification coupling [3,7,8] 2 4π 39.4784176.
To bring this value in line with numerical experiments as well as E-Infinity's exact prediction of the non-super symmetric grand unification of all fundamental forces except for gravity, we must realize the following and change things accordingly: 1) The factor 4 stands in reality for the four topological dimensions of space-time.It must therefore be changed to the fractal-Hausdorff dimension of the core of spacetime, i.e. to that of a Hilbert 4D cube 4 + 3  = 4.2360679.
2) Second π 2 must be changed to one of its transfinite opposite numbers.In this case π 2 = 9.869604401 must be changed: 2 π 10  From the above we find   There is a very simple and elementary way to show that this is the exact value as well as how to obtain the super symmetric gs  which we know to be 26 The value of inverse electromagnetic coupling at low energy 0 should be divided equally among the number of fundamental equations at a certain energy scale.When gravity is out and the electrical force and the magnetic force are counted as one force, then we have only 3 fundamental forces with a fractal weight due to the fractality of space-time equal to .The common coupling or unification grand unification coupling is thus exactly as anticipated.Now if we admit gravity and count electrical force and magnetic force as two forces, then the number is 5 and the fractal weight is Finally in the case of only 4 fundamental forces the unification coupling is given by 0 3 32 2 32.18033989.4 k This 32.18033989.

 
is what we include approximately in our renormalization equation of unification using the logarithmic scaling as in Nottale's theory of scale relativity.We see this clearly from [18] 3 4 where 3 9, for super symmetric interaction [18] and exactly as anticipated.Here 10 16 GeV is the mass of the GUT monopole and 91 GeV is the mass of the electro-weak unification.In fact Nottale's scale relativity has generalized this logarithmic scaling and used Levy-Gillmann operators skillfully to achieve his result which although not exact, paved the way for our work and for the exceptionally beautiful work of Magueijo and Smolin [12,13] on varying speed of light theory (VSL).In the next section we will show how E-infinity as well as scale relativity can resolve the mysterious dark energy problem [1,2].

Resolution of the Missing Hypothetical Dark Energy Using Scale Relativity and E-Infinity
Scale relativity puts the running value of 0  at 10 16 GeV of scale relativity [3,7,8] for 105 Clearly at GUT  we have everything except gravity.Scaling 105 logarithmically and squaring it gives us now a measure for the error in Einstein's special relativity energy mass resolution when applied at ultra high energy and distances.That way we find the scaling exponent needed for , namely Einstein's energy-mass equation now reads as follows: Dividing through all the five interactions using the D T = 5 one finds [4] 22 22.18033989.5 This is of course the exact result and shows the high quality of accuracy in the Nottale method.Should we have used the fractal weight In the first case we look at an Einstein 4 dimensional space-time with 22 + k "dark" dimensions while in the second case we have a 5 dimensional Klein-Kaluza space-time with only 21 + k "dark" dimensions.Based on this analysis our tangible space is exactly four dimensional topologically and Hausdorffly.However it is the larger .
predicts that we have a missing dark energy of exactly E (dark) = 95.49150281%,almost the same as in the approximate scale relativity analysis following Nottale's theory.This reduction could be interpreted in a variety of intuitive ways which will be discussed in the conclusion of the paper.
It is instructive for a deep understanding of the present work to ponder the implication of a comparison between Nottale's theory of scale relativity and El Naschie's Einfinity theory which is summarized in Table 1.In Table 2, we give another instructive comparison between working in the bulk and working with the holographic boundary to derive the scaling which elevates Einstein's special relativity equation to an effective quantum gravity equation.

Discussion
Following the picture adopted by Heterotic string theory compactified on a Calabi-Yau manifold, every point in our spacetime is joined to a Calabi-Yau 6 dimensional real manifold containing internal symmetry and compactified dimensions [19].On this account we would have all in all (4)(6) = 24 dimensions and adding the string   world sheet to it arrives at the 24 + 2 = 26 Bosonic dimension.These dimensions move in the opposite direction of another 16 Fermionic dimensions from which one finds 26 -16 = 10 super symmetric dimensions.However in our transfinite version of Heterotic strings we do not need the 2 dimensional world sheet to arrive at 26.This is because the Hausdorff dimension of our core space is not 4 but and the 6 dimensions of the Calabi-Yau manifold [19] are not 6 but 6 + k = 6.18033898.Consequently the total dimension is given by


This is a wonderfully simple and intuitive picture and is numerically identical with our analysis which was based on superficially completely different theories such as Nottale's scale relativity [3,7,8] or E-infinity theory [3][4][5][6].It is now clear that must be scaled using which fully agrees with the measurement of WMAP and supernova analysis by predicting that exactly 95.4915028% of the energy of the cosmos must be dark energy [1,2].
To gain a deeper insight into the roots of scale relativity we should apply the original energy mass relation of scale relativity directly to the problem of dark energy.Even a fleeting glance at these equations reveal that they are in almost one to one correspondence with Einstein's equation and are also the inspiration to Magueijo-Smolin's beautiful energy-mass Planck length invariant equation.Following Nottale's notation we have [7,8]  Here  is the Plank length .From Sigalotti's analysis of the classical relativistic transition we know that when we set Exactly as in the previous analysis which means a reduction of 95.49150281% in energy which matches almost exactly the missing dark energy measurements [1,2].
There is even an outrageously simple way of arriving at 3) mass increase.All these classically feeble effects become noticeable only as the speed approaches the speed of light c [11].We handle this semi-classically, i.e. using common sense by introducing a boost 1  , where X is space coordinate, 2) , where t is ordinary time and 3) , where m 0 is non-relativistic mass [11].Thus in one stroke we reconciled and fused together classical mechanics with relativity and quantum mechanics via the non-classical geometry of fractals [3][4][5][6][7][8].This is magically beautiful.

Table 1 .
Comparison in calculating the Lorentz factor using scale relativity and E-infinity.

3
s energy-mass equation was based on a mere 4 dimensional flat non-fractal, non-fuzzy Euclidean manifold.Subtracting these 4 dimensions from D S = 26 + k we are left with 26 + k 4 = 22 + k hidden dimensions.

Non-Commutative Geometry, Scale Relativity and Cantorian Fractal Space-Time 3.1. Background
[17]classical measure theoretical methods, K-theory and categories[17].In non-commutative geometry A. Connes replaced differentiation of real or complex variables by a Poisson bracket of the form[17]

Table 2 . Comparison in predicting the Lorentz factor for dark energy step by step using the bulk and using the holo- graphic boundary [4,5].
[11]4] down scaling by a minimal Lorentz factor equal to half of the value of Hardy's quantum entanglement[6,14].In turn this causes a reduction of almost 95.5% of the classically predicted energy.This is what we call missing dark energy.It is the energy which would have been there if the space-time fabric were smooth, continuous and without holes.However actual space-time at quantum scales and surprisingly again at intergalactic scale displays a wild Cantorian fractal geometry and topology.It is a T-duality which we saw in the unification program at the Plank length, yet this time the surprising quantum effect of entanglement is showing its power at the Hubble length scale.The main equation obtained in the present work which is in the widest sense possible, meaning that it is almost invariant to the use of any mathematically and physically reasonably meaningful theory.It is a very robust result not affected by minor details of theoretical modeling.Thus we may show here in the conclusion what on reflection should have been presented in the introduction at the very beginning: Special relativity implies three strange effects[11]: 1) length contraction; 2) time delineation; c 