The supposedly missing dark energy of the cosmos is found quantitatively in a direct analysis without involving ordinary energy. The analysis relies on five dimensional Kaluza-Klein spacetime and a Lagrangian constrained by an auxiliary condition. Employing the Lagrangian multiplier method, it is found that this multiplier is equal to the dark energy of the cosmos and is given by where E is energy, m is mass, c is the speed of light, and λ is the Lagrangian multiplier. The result is in full agreement with cosmic measurements which were awarded the 2011 Nobel Prize in Physics as well as with the interpretation that dark energy is the energy of the quantum wave while ordinary energy is the energy of the quantum particle. Consequently dark energy could not be found directly using our current measurement methods because measurement leads to wave collapse leaving only the quantum particle and its ordinary energy intact.
The present work proposes a resolution for the missing dark energy of the cosmos [1-3], which according to the latest highly sophisticated cosmic measurements and analyses, constitutes one of, if not the most important problems facing not only cosmology but also theoretical physics at large [1-13].
There are in the meantime very strong founded feelings that the discrepancy between theory and measurement is due to the pressing need for a theory of quantum relativity [
There are several strategies and corresponding theories and methods to tackle the problems of computing the density of the dark energy of the cosmos [1-12,31,32]. Roughly speaking, these could be divided globally into two categories. Either one attempts to calculate the ordinary energy and then subtract it from the total theoretical value given by Einstein’s theory [1-12] or one could try to determine the dark energy density directly without determining the corresponding magnitude of ordinary energy first. At the end the final result must of course remain the same [
In earlier work [9,16,31] it was established that ordinary energy density is given by where which is approximately equal to in excellent agreement with measurements [1-4]. Consequently dark energy must be given by
where. This is approximately equal to [
In the present work we proceed directly to calculating E(D) from first principles without any reference to ordinary energy:
We mention in anticipation of the following sections that our main tools which we use to achieve our goal will be a combination of Noether’s theorem [
The main idea behind the present strategy of calculating the dark energy is to calculate a density ratio representing the degree of emptiness or what is equivalent the sparseness of elementary messenger particles with respect to empty space [5,9]. To do this we have to quantify the relevant inherent values of particles and space. It is self-evident that the first is given directly by the number of elementary messenger particles of the standard model [5,34], while the second is slightly more involved and requires us to think deeply about the compactified dimensions of the bosonic space of the strong interactions as well as the Kaluza-Klein D = 5 unification of gravity and electromagnetism [26,27,29,30]. We start with the messenger particles of the standard model.
At the time when Einstein and others considered the problems leading to the special theory of relativity around 1905 [5,35-37], nothing was then known about the elementary particles except for the electron and the photon and Maxwell’s electromagnetic field theory [
i.e., twelve massless gauge bosons [9,19].
To unify gravity with electromagnetism [
In Heterotic string theory we have 16 extra dimensions and 26 running in the opposite direction [
We note that this G will play a fundamental role in our analysis.
Noether’s theorem [5,19] shows us clearly the vital connection between symmetry, quantum particles and the conservation of energy. Thus the 12 – 1 = 11 messenger particles which are materialized by symmetry breaking bifurcation must be taken into account via the Lagrangian multiplier method to account for the constraint on energy in contrast to freely variating Newton’s kinetic energy when we let the velocity v tend to the velocity of light c. The same applies to the 26 – 5 = 21 compactified bosonic dimensions [
Next we write down a simple energy function playing the role of a Lagrangian in terms of a state variable “a” which need not be specified for the present purpose [
The leading positive term in the bracket is clearly nothing more than Newton’s kinetic energy while is a Lagrangian multiplier and G is the constraining auxiliary condition [
Letting the velocity the energy function of (5) becomes
where c is the speed of light.
Finally for a stable steady state solution given by vanishing of the second variation one obtains
Solving for one finds
Thus dark energy is a scaling of Einstein’s famous equation using a Weyl-Nottale scaling [7,19] exponent equal to.
Furthermore, we gain a new and marvellous equation resulting from summing up ordinary energy E(O) and dark energy E(D) and finding Einstein’s energy
exactly as we hoped and quite honestly as we expected from the outset [9,16].
To arrive at the exact expression for dark energy density which we know from the set-theoretical E-Infinity quantum mechanics to be [16,31]
and derive this formula without resorting to our fundamental result, namely that ordinary energy is the energy of the quantum particle while dark energy is the energy of the quantum wave as per the wave-particle duality, we proceed as follows:
From fractal logic [
In addition the photon is given by the fractal weight number rather than the classical unity [19, 34]. Consequently one finds [
where is the inverse theoretical electromagnetic fine structure constant, and is Hardy’s probability of quantum entanglement [20,23,25]. Similarly, the 26 bosonic string dimensions are actually 26 + k where which translate to [
Proceeding as in the integer analysis but using our new transfinite fractal weight values, we find the corresponding Lagrangian
Consequently gives
In other words the dark energy density is given by
in full harmony with our previous integer approximation analysis of Section 3.3. The result given by (16) is confirmation of the fact that dark energy is the energy of the Schrödinger quantum wave [
A method for determining the dark energy density of the cosmos is presented based on the concept of gauge boson density in a D = 5 Kaluza-Klein spacetime combined with 26 dimensional bosonic string space and using Noether’s theorem. It was found that the multiplier constraining the corresponding Lagrangian is equal to the dark energy density
This is the complementary energy of the quantum particle and refers exclusively to the absolute value of the negative energy of the quantum wave [