_{1}

The frequency spectrum of the cosmical Zero Point Energy (ZPE) and its total density are so far unknown in their details. In the present complementary investigation, a revised theory forms the basis for studies of this concept in two respects. It first applies to the observable universe considered as an entity, as well as to included subregions such as the galaxies with supermassive black holes. Second, experiments are proposed on the maximum Casimir force arising between two metal plates of different materials and with a vanishing air gap in their spacing. This serves the purpose of making an indirect determination of the ZPE energy density in the laboratory, i.e. at the Earth’s orbit. The ZPE energy density is interpreted as dark matter density and its pressure gradient as dark energy force density.

In his pioneering studies of the harmonic oscillator Planck [

The low-frequency part of the ZPE fluctuations has to be accepted as an experimental fact, but a problem arises with its high-frequency part. The earlier performed conventional analysis leads namely to a spectrum having an infinite total energy density, as demonstrated by Terletskii [

To overcome this difficulty, the author [

Here

quency

The purpose of the complementary investigations presented in this paper is twofold. First, the theory of the ZPE spectrum will be used in models on dark matter and dark energy, applied both to the observable universe considered as an entity, and to galaxies including supermassive black holes at their centra. Second, an ex- perimental determination is desirable which aims at the so far unknown value of the average frequency

In the present revised theory on the ZPE spectrum [

This results in a finite integrated energy density

where the average frequency

The self-consistent spectrum of Equations (1) and (2) has been proposed as the basis for a new interpretation of dark matter and dark energy [

This represents the local contribution to dark energy.

The integrated relativistic mass due to ZPE further becomes

as being contained within a sphere of radius r. Here cases will also be considered where there exists a super- massive black hole of mass

where

A local balance between the expansive and contracting forces is given by

So far local forms of

In the case

and

The condition of an average balance can still be obtained by substituting

at the boundary

The condition

having a maximum

at

The energy density (11) has the following properties as a function of R:

・ A balance is only possible for

・ The density

・ For values

The present theoretical concepts and interpretations can here be applied to two examples:

・ The observable universe considered as an entity, with mean properties at its largest scale corresponding to an averaged ZPE energy density.

・ Subregions of the universe, on a smaller scale such as that of the galaxies within which there can exist gravitationally contracted local parts of higher ZPE energy density.

The radius of the observable universe is estimated to about 10^{26} m by astronomers. In its present stage of expansion it contains about 4% of normal matter, about 23% of dark matter, and about 73% of dark energy as given by Linder and Permutter [^{−26} kg∙m^{−3} according to Linde [

Turning first to the imagined case of local balance between dark matter and dark energy in absence of normal matter, this leads to Equations (6)-(8) for

in local equilibrium. With the radius R in the range from ^{−3}, respectively.

We next consider the present state of a certain deviation from equilibrium. As an approximation, the average density

With R in the range from ^{−3}, respectively. The density obtained from Equation (15) is thus within the same range as the estimated one. This can be taken as a support of the present theory.

As subregion we now take the Milky Way as an example, having a radius of about ^{−7} of the solar volume, corre- sponding to a radius of about

Turning to the average balance of Equations (9)-(13), we notice the following results obtained with

・ In the balance between the expansive and contracting forces, relation (11) permits several possible values of

・ There is a smallest possible radius equal to

・ The maximum of

・ The branch of Equation (11) at large

There is no exact theoretical indication so far about the magnitude of the dark ZPE energy density at the position of the Earth’s orbit. Here a possible experimental procedure will be presented for determination of it.

The first investigations on the Casimir force were performed on a small but nonzero air gap between two metal plates. The largest available force of this kind would on the other hand arise in the case of a vanishing air gap [

Starting from the spectrum of Equations (1) and (2), the Casimir force arises from the difference in pressure on the in- and outsides of the metal plates. Whereas the full ZPE pressure acts on their outsides, there is a reduced pressure on their insides due to the boundary condition which sorts out all frequencies below a certain limit

Normalizing by the introduction of

where

and

We first consider the case of a nonzero air gap of the width d, being much larger than the electromagnetic skin depth of the plates at relevant frequencies. Then frequencies lower than

as shown by Casimir [

We next turn to the maximum Casimir force of a vanishing air gap. Then the width d is replaced by the sum of the skin depths _{1}) and (_{2}) consisting of different metals, the choice of which will be described later. This leads to a total skin depth

where the effective electrical conductivity becomes

with

Since

We now consider a given total energy density u and a total pressure

・ The left-hand part of the figure relates to large values of

・ The right-hand part of the same figure corresponds on the other hand to small

due to Equations (17)-(19), where the frequency limit

A vanishing air gap has the advantage of a maximum Casimir force. The latter may even become recordable by means of a simple lever balance. This is gained at the expense of the following questions:

・ To avoid effects from the air pressure, the measurements should take place in vacuo.

・ To avoid microscopic matching between the metal structures, plate pairs of different metals should be chosen, as pointed out by Abramson [

・ Even with a maximum Casimir force, other surface and sticking mechanisms may interfere with the measurements, such as the Van der Waals forces. To minimize this problem, many independent measure- ments with various plate combinations have to be made, to sort out the special behaviour on the skin depths represented by

The possible relation between the Casimir force and the magnitude of the ZPE energy density at the orbit of the Earth, can be used in first estimations of this density:

・ From the results of the previous Section 2.3.2, the density

・ In earlier experiments with an air gap as small as^{14} s^{−1}.

A detailed research on the total ZPE energy density in the laboratory is proposed here, and may be performed in a series of measurements of the Casimir pressure

・ If all obtained values of

・ If the measured values of

The frequency spectrum of the zero Point Energy and its total density on cosmical scale are unknown in their details. The present complementary investigation includes a revised theory, forming the basis for determinations of this energy density. First, this concerns the values of the latter within the observable universe considered as an entity, as well as in subregions such as the galaxies including supermassive black holes. Second, experiments are proposed on the maximum Casimir force between metal plates with vanishing gap distance, with the purpose of determining the ZPE energy density in the laboratory, i.e. at the orbit of the Earth. These problems thus concern the ZPE on cosmical scale, with its interaction with gravity.

In the present approach the ZPE energy density is interpreted as a dark matter density, and its pressure gradient as a dark energy force density. The lack of emitted radiation is reconcilable with this picture. Thereby the crucial coincidence problem of equal orders of magnitude of dark matter and dark energy cannot be ex- plained by the cosmological constant. This problem is instead resolved by the present variable concepts originating from the same ZPE photon gas balance.

An additional and different effect due to ZPE arises on the microscopical scale of elementary particles, as explained earlier by the author [

Bo Lehnert, (2016) On the Cosmical Zero Point Energy Density. Journal of Modern Physics,07,1112-1119. doi: 10.4236/jmp.2016.710100