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We use a dual Einstein-Kaluza spacetime to calculate the exact energy density of dark energy and dark matter using a novel topological computation method. Starting from the said spacetime and ‘tHooft’s topological renormalon as well as the corresponding symmetry group, we show how the zero set quantum particle and the empty set quantum wave interact with the vacuum and give rise to pure dark energy and pure dark matter all along with ordinary energy density of the cosmos. The consistency of the exact calculation and the accurate observations attests to the reality of ‘tHooft’s renormalon dark matter, pure dark energy and accelerated cosmic expansion.

Let us stand by outlining the motivation and background leading to the present paper using the mathematical principles of E-infinity Cantorian fractal spacetime theory [

The present work goes much further in the above direction and formalizes the handling of what may be called the renormalon topological computation machinery [

Let us go directly to the formal handling of the renormalon topological computation without much ado. From von Neumann-Connes’ dimensional function, E-infinity postulates that a pre-quantum particle may be identified by the zero set

to be [

for ordinary energy density

for dark energy density. Finally for pure dark matter we have

all in excellent agreement with cosmic measurements [

The E-infinity hierarchy [

1) for ordinary energy we have

2) for dark energy we have

Now completing the to pologizing physics program of E-infinity, it was reasoned in numerous previous publications that the topological speed of light is equal

(a):

for the ordinary energy density and

(b):

for the dark energy density of the universe where

This result was of course noted long ago to be in astounding agreement with all modern measurements and cosmic observations [

Remembering that the zero set is given by 0 and

This alone should dispel any lurking doubt about our method. Now we can return to our main task, namely case (c) of dark matter density. This is consequently given in the uncoupled approximation by [

(c):

Consequently the pure dark energy must be

(d):

where k is the ‘tHooft renormalon given by

As mentioned in the introduction, the model we are using here for the entire universe is that of ‘tHooft’s holographic boundary [

Seen that way we can say the share of each unit particle-like state of the overall effect of the holographic boundary is the inverse of

1) The exact coupled expression for the attractive dark matter energy is

exactly as found using entirely different methods [

2) For the exact repulsive pure dark energy on the other hand one finds

Again this agrees exactly with our previous exact analysis using Heterotic string theory in conjunction with a fuzzy Kähler manifold [

It is fair to say that most if not all physicists think that D = 4 Einstein spacetime and D = 5 Kaluza spacetime are two alternatives of which one has to be chosen depending on what we want to determine and understand. In that respect the present Author was until quite recently no exception. However, it has dawned on us that our fractal version of Kaluza-Klein theory with

Consequently the total dimension is of course not 5 + 4 = 9 but rather

where E(D) is the total energy density of the dark sector and E(O) is the ordinary energy density. Now we can dissect E(D) into its two components namely dark matter and pure dark energy. It is here that we discover the beauty of D = E(Einstein) + 1 = D(Kaluza) = 5 and the reality of ‘tHooft’s renormalon k [

From the last expansion we see that our ordinary energy density is the exact one namely [

in full agreement with all previous results and measurements [

This leads to a density of

which is extremely close to the exact value of 22.18033% [

Again this is an excellent approximation to the exact value 0.733111 [

It is by now obvious to the reader that we did not obtain the exact value of the Dark matter energy density only because we ignored it unlike ordinary energy this two subdivision of the energy are weakly coupled. The coupling constant in this case is

which is the exact value. For pure dark energy on the other hand, one finds [

which is again the exact value obtained in the present paper using different methods. The advantage of using the novel dual Einstein-Kaluza spacetime formulation is a definite deeper insight and understanding of the various components of energy and the role of ‘tHooft’s quasi particle which we accept it to be a real elementary particle [

To have a chance to answer deep questions about the very nature of physical existence and probe troubling questions about the nature of physics, we need equally deep and sometimes illusively abstract mathematical tools and philosophical notions and concepts. In particular deep mathematics, it is both very simple and equally very hard to come by and to comprehend because it is remote from our mundane classical world in which we are born, cared for and nourished long before we can utter a single intelligible word as infants let alone thinking of ourselves as infallible geniuses.

In the present work, one of the simplest tools used may be the hardest to understand, namely setting the speed of light to be a constant equal

Let us recall that we have decided to start building our universe from scratch and began with the obvious, namely spacetime just as Einstein started for instance [

We are not at the end of our explanation but rather at its very beginning and the complete explanation can be found from the consistency of the entire present paper and previous papers plus the agreement of the so obtained results with painstaking experiments and cosmological measurements [

sionality, namely

charge is on the other hand equal to the topological K-K dimension, namely 5 which also acts on the vacuum and gives us

where

As a final word or a final question one may be permitted to ask what did we better understand, physically speaking, thanks to the present analysis? The answer is probably that we now know almost for sure that ordinary energy is basically zero measure fractals and by contrast the dark section resembles a five dimensional Kaluza-Klein massive object which clearly could not be observed directly by 3 + 1 dimensional creatures like us. In turn, this five-dimensional object is in reality two parts, coupled together. One part is fractal-like strings which are basically the dark matter part of the cosmos while the rest, i.e. the second part is a ‘tHooft object resembling an almost four dimensional fractal spacetime structure that is slightly smaller than our more familiar Einstein four dimensional spacetime. This is definitely not the last word in understanding but we feel it is a good step forward and the proof for the correctness of this statement is the incredibly simple solution for the problem of dark matter, dark energy and ordinary energy presented in Section 5 of the present paper.

I must admit that without Gerard ‘tHooft’s pioneering work and dimensional regularization my present work could not have been possible.

El Naschie, M.S. (2017) From a Dual Einstein-Kaluza Spacetime to ‘tHooft Renormalon and the Reality of Accelerated Cosmic Expansion. Journal of Modern Physics, 8, 1319-1329. https://doi.org/10.4236/jmp.2017.88085