Abstract

In a previous paper (ref 1, AP4.7), a self-consistent theory called Model 1 was developed to answer ten major connected questions in astrophysics. The most important of these questions are:

• How can dark matter be explained and described?
• How can dark energy be explained and described?
• Where do the extremely high-energy cosmic rays that occur beyond the GZK cutoff come from?

Model 1 appears to successfully answer these questions. The unique features of this model are:

• There are two spaces in the universe, particle space and quantum vacuum space. A potential barrier separates them. One space contains visible matter, and the other space contains dark matter.

• There is a cycling of mass-energy between these spaces through the black holes that connect them. Particles pass from our space through black holes where they are converted into super particles that operate with unified force. They then pass into the high-energy vacuum space where they become dark matter operating behind the potential barrier.

• Dark matter particles interact with each other and form a slowly moving halo centered on a galaxy. The bubbles of dark matter are connected to each other by corridors of dark matter forming a cosmic web, which guides the development of new galaxies.

• There, behind a potential barrier, the dark matter particles gain energy, build up in number and eventually exceed the ability of the barrier to contain them. They then explode back into particle space as a big bang. This process repeats to make a series of new universes.

• After the big bang exhausts itself, super particles continue to tunnel through the barrier into particle space. The super particles are unstable and break down into particles (protons) with extreme kinetic energy. In doing so, they give up potential energy into particle space. The potential energy gradually builds up to become the dark energy that we observe as the cause of our accelerating, expanding universe. The extreme energy protons are observed as cosmic rays with energy between the energy of force unification and the Planck energy, which is beyond the GZK cutoff.  

There is reason to take Model 1 seriously. It quantitatively explains (see ref 12, AP4.7D):

• The origin, characteristics and operation of dark matter.
• The origin, characteristics and operation of dark energy.

• The origin, characteristics and operation of the Ultra High Energy Cosmic Rays (UHECR) that have been observed in the energy range beyond the GZK cutoff.

• The huge disparity in the different estimates of the vacuum potential energy.
• The large-scale cutoff and asymmetry in the Microwave Background Energy.

In working out this model, some problems arose that are connected with it. One of the most fundamental is to determine the origin of the real scalar field that forms the basis of the expansion of space near a black hole. This problem will be addressed here. 

 

The Problem

There are three fields that are used to develop Model 1.

• The real scalar field used for expansion of space near a black hole in general relativity.
• The real and complex scalar field used to describe the Higgs mechanism.
• The real scalar field of Model 1 used to describe the operation of super particles in vacuum space.

We will describe each of them in some detail, and use the descriptions to draw conclusions about the origin of the Model 1 real scalar field.

 The expansion field from general relativity.

The local energy conservation equation from general relativity (Peebles, 395) is,

            r’ = -3 (r + p) a’/a

            Where:

            p = pressure

            r = energy density

            r’ = rate of change of energy density

            a = space expansion factor

            a’ = rate of change of space expansion factor

Now according to Peebles, there are conditions when the net pressure can be negative, i.e.:        

            p < –r/3,

The negative pressure comes from a single new scalar field f. Suppose the field is close to spatially homogeneous, so we can ignore space derivatives compared to time derivatives. Under these conditions, the energy density and the effective pressure in the field are:

            p = ø’ ²/2 – V(ø),           r = ø’ ²/2 + V(ø)

            Where:

            V(fø) = a potential energy density, with the black hole mass as a source.

            ø = a new real scalar field

Now, near the black hole center, the mass, and so the potential energy V is large. Thus V is large enough to make a significant contribution to the stress energy tensor. Also the potential energy V is a slowly varying function of f because the black hole mass, which controls V does not change rapidly with anything. Finally, the initial value of the time derivative of f is not expected to be large because Model 1 has the universe evolving smoothly from one state to another. Thus the kinetic energy term f’ ²/2 can be small compared to V, and p can be negative. When all this happens, the universe is dominated by a term that acts like a cosmological constant, and space expands in the zone where this condition applies.

Consider a particle close to a black hole. It will move through space toward the center of the black hole. If V is large compared to the kinetic energy term nearer the center of the black hole as expected, space will expand there and push the particle space containing the particle away. In addition, diffusion will tend to disperse particles away from the center of the black hole as follows.

            V = -D/n dn/dx

If we think of gravity as a Newtonian type attractive force, it attracts a particle through expanding space. If the space is expanding faster than gravity pulls it, the particle will move away from the black hole center. But the potential energy is inversely proportional to the distance from the black hole center, so the potential energy drops as the particle moves away from the black hole center. Thus, the particle moves away from the influence of the black hole until the expansion due to potential energy and diffusion, and the attraction due to gravity balance.

A better and more accurate way to describe this situation is to use general relativity language. The curvature of space-time increases as a test mass moves toward the black hole center. The test mass moves toward the center of the black hole following the geodesic. Now if there is a high potential energy near the center of the black hole, it will tend to null the curvature. If the potential energy is strong enough, it will give an opposite curvature, and there will be bulge around the center that reduces to a flat zone further out and then a cup further out still. Thus we have a so-called Mexican Hat curvature. A test mass will then move along a geodesic to the zero curvature zone around the black hole and circle in that zone. The kinetic energy of the test masses in the zone will try to carry the test mass up the sides of the cup and away from the flat in a diffusion action. The shape of the potential energy as a function of radius from the black hole center determines the shape of the zone of operation of the test mass (the force balance zone). We note that further away from the black hole center, the volume increases, and potential energy is used to generate super particles, so the potential energy density goes down. In this balance zone, the electromagnetic force, the centrifugal force and diffusion become important, and form a source for super particles moving out (see ref 7, AP4.7B).

Note that the potential energy term can now be identified as the source acting like a cosmological constant that controls the expansion of the universe. The field comes from the potential energy deposited in particle space by super particles tunneling into particle space and decaying into particles, thus leaving their excess potential energy there (see ref 8, AP4.7I)

The Higgs field

The Higgs mechanism is based on the assumption that there is a universal spin zero field f that is a doublet in SU(2) space and carries a non-zero U(1) hypercharge, but is a singlet in color space. Kane (Kane, 98) describes spontaneous symmetry breaking for four cases, reflective symmetry, global symmetry, the Albelian Higgs Mechanism and the full Standard Model. We will explore them here.

Reflective symmetry

We start with the Lagrangian:

            ½∂_μø∂ ^μø – (½μ²fø²+ ¼ λø ⁴)

Note the reflection symmetry; ø= –ø. To find the ground state and the excitations, it is necessary to expand the field around the minimum and establish the perturbation terms. In field theory, it is conventional to call the minimum the ground or vacuum state, and the perturbation terms are excitations, which are the particles. The mass is determined by the form of the Lagrangian.

To get the minimum of the potential, we take the derivative, set it equal to zero, and get,

            ø ² = – μ²/ λ = ν

The value ν is the vacuum expectation value, and f is called the Higgs field. We have to work with one value, so we choose the positive root. To expand the function, we set

            ø(x) = ν + η(x)

We then get for the Lagrangian:

            ½∂_μ η ∂ ^μ η – (λ ν² η ²+ λ ν η ³ + ¼ λ η ⁴) + constan          

This Lagrangian represents the description of a particle with mass

            m _η² = 2 λ ν² = -2 μ²  

The reflection symmetry is gone here. The symmetry was broken when a specific vacuum state was chosen, so the vacuum does not have the symmetry of the original Lagrangian, and therefore the solutions do not. When this occurs, it is called “spontaneous symmetry breaking”.

Global symmetry

Now we take f to be a complex scalar.

            ø = (ø₁ + iø₂)/2^1/2

Then the Lagrangian is invariant under a global gauge transformation where:

            ø becomes  e ^iχ ø

Again we expand around the minimum. To get the minimum of the potential, we take the derivative, and set it equal to zero, and get,

           ø₁ ² +ø₂ ² = – μ²/ λ = ν ²

Then we expand the Lagrangian around the minimum as above by setting:

            ø =[(ν + η(x) + i ρ(x)]/2^1/2

We find that the expansion represents the description of a μ field particle with mass         

            m _η²  =  -2| μ² |            

Note that the term with ρ² has gone, so the ρ field particle has zero mass. The continuous global symmetry (U(1) invariance under rotation) was spontaneously broken when we chose a particular vacuum or ground state. As a result, the spectrum will contain a massless spin-zero boson. 

 The Albelian Higgs mechanism

Here, we will make the Lagrangian invariant under local gauge transformations. We do this by introducing a massless vector field A_μ. Then we go through the same minimization and expansion. As a result, a mass term for the Gauge boson shows up in the Lagrangian. The expansion represents the description of a particle with mass

            M_A = g ν  

This mass is non-zero only when the gauge symmetry is spontaneously broken by the Higgs field acquiring a vacuum expectation value and becoming:

            ν  = – μ² / λ.

The spectrum now has a single real Higgs boson with mass

            M_A  = (2 λ ν ²) ^½ = 2^½ μ

The symmetry was spontaneously broken when the Higgs field acquired the specific vacuum state value shown. Note that we have a condition that the Lagrangian is gauge invariant, but the vacuum is not (Kane, 104). This point is important because the equation used to describe the vacuum barrier between particle space and vacuum space (see ref 3, AP4.7A Appendix 2) does not have the gauge massless vector field A_μ.

The Standard Model Higgs mechanism

The Higgs mechanism provides mass and spontaneous symmetry breaking for the weak isospin force with its W and Z bosons, and the strong force with its gluons in particle space. These forces show their effects in particle space in radioactive decay and unusual “high” energy scattering results (kinetic energy up to 1 GeV). For the extreme potential energy of vacuum space, the situation changes. The Model 1 super particle potential energy (V > 10¹⁷ GeV) is high enough to stabilize the particles and unify the forces even at extreme kinetic energy. The potential V ~ 10¹⁷ GeV is estimated to be the unification mass energy (Kane, 281).  However, super particles have a finite lifetime as well (ref 3, AP4.7C), and must be renewed by the potential energy of vacuum space (see Higgs vacuum potential below).

The Higgs vacuum potential

Kane (Kane, 112) estimated the contribution of spontaneous symmetry breaking to the vacuum energy of the universe (particle space for Model 1). He found it to be roughly 2 x 10⁴⁹ GeV/cc, with an error of perhaps an order of magnitude based on uncertainty in the value of the Higgs self coupling.

Thus we have described the mechanism using the Higgs field that allows for the particle symmetry breaking in steps from continuous global and reflective symmetry through U(1) gauge symmetry to Standard Model symmetry. The super particles in vacuum space apparently operate with U(1) gauge symmetry because they use electromagnetic force in forming the dark matter clouds (see Ref 7, AP4.7B). It is important to note that a significant amount of potential energy exchange with vacuum is involved in moving from the high symmetry of vacuum space to the lower symmetries of particle space.

Model 1

Model 1 is based on a scalar field like the expansion field of general relativity. It is a spin zero field like the Higgs field. Further, the particles carry a non-zero U(1) hypercharge, as does the Higgs. Also, Model 1 describes how V and f of Model 1 vacuum space work together to move super particles away from the center of the black hole into a zone where electromagnetic, centrifugal and diffusion forces control to form the dark matter clouds. The procedure used to choose the constants and form details of Model 1 will be given here.

1.      The two key energy values were first chosen. We start with the value of the vacuum space potential (~10¹⁷ GeV) needed to form the unified force super particles in vacuum space. Kane (Kane, 281) estimates the energy at which the forces unify. It is the energy of the most significant particle change as energy increases. Thus it seems a logical starting point for the range of the super particle. The ending point must be the Planck energy (~10¹⁹ GeV) because it is the theoretical maximum possible, and so is expected to be the entry kinetic energy into vacuum space.  

2.      The spherical barrier shell is then estimated to have a radius of 10^-11 cm, so the vacuum energy density within vacuum space (volume ~10^-33 cc) needed to maintain the super particle formation energy of ~10¹⁷ GeV is ~10⁵⁰ GeV/cc (see ref 3, AP4.7C for equilibrium equations). Note that this is within the margin of error of Kane’s estimate of the Higgs contribution of spontaneous symmetry breaking to the vacuum energy (2 x 10⁴⁹ GeV/cc).

3.      The accelerated expansion of the universe recently measured by astronomers is apparently caused by a particle space vacuum potential energy of 10^-4 GeV/cc. Now the observable universe has been estimated to have a volume of 10⁸¹cc. This means ~10⁷⁷ GeV must be accounted for in particle space. It can be accounted for by super particles (sp’s) tunneling through the vacuum space barrier into particle space, converting to particles, and depositing their phase change energy of ~10¹⁷ GeV/sp in particle space over ~10¹⁰ years (the lifetime of the universe). This is expected to be done by an increasing number of black holes in galaxies. Thus we start with 1 soon after the big bang and build to ~10⁸ in our time, after ~10¹⁰ years. This would require an sp tunneling rate per galaxy of R ~ 10³⁵ sp/s. Now, the rate of tunneling through the barrier for each galaxy is controlled by the following equation for the (see ref 2, AP4.7A) rate of passage of all sp’s through their spherical shell barriers.

            R = T h/2p k_o n_sp / m r = 10³⁴ particles/sec

            Where

            k_o= (8p²m(E)/h²) ^½  = 10³³ /cm

            m = super particle mass ~ 10^-5 gm (average sp mass ~10¹⁸ GeV).

            n_sp = number of super particles of dark matter ~10⁶⁹/galaxy.

            r = radius of the spherical barrier shell ~10^-11 cm.

And (see ref 2, AP4.7A-Appendix2)

            T = 1/(1+V_o² sinh²(k₁a)/4E(V₀-E) = 10^-50.

             Where

             k₁= (8p²m(V₀-E)/h²) ^1/2 = 10³⁴ /cm

             V_o = potential energy/sp ~10¹⁹ GeV/sp

             E = kinetic energy/sp ~10¹⁸ GeV/sp

             a = barrier thickness.

  This can only happen if

             a = 10^-31 cm                   

This in turn makes the barrier shell volume ~10^-53 cc, so since the barrier potential energy is ~10¹⁹ GeV the barrier shell potential energy density is ~10⁷² GeV/cc

4.      Note that the super particle also makes a proton with kinetic energy in the range ~10¹⁰ to 10¹⁹ GeV/cc when it passes through the barrier from vacuum space and sheds its phase change potential energy. This would show up as ultra high-energy cosmic rays (UHECRs-see ref 8, ap4.7I), which have been observed.

We see here that by starting with two basic constants from quantum mechanics and the standard model of particle physics (force unification energy and the Planck energy), and choose two new constants (the spherical barrier radius and the barrier thickness), we can quantitatively explain five observed phenomena that have yet to find a single consistent explanation, namely:

• The accelerating expansion of the universe (Dark energy)-see ref 1, AP4.7.
• The process of creation and rotation of the galaxies (Dark matter)-see ref 7, AP4.7B.
• The UHECR (cosmic rays) that have been observed in the energy range beyond the GZK cutoff-see ref 8, AP4.7I.
• The huge disparity in the different estimates of the vacuum potential-see 3 and 4 above.
• The large-scale cutoff and asymmetry in the Microwave Background Energy-see ref 9, AP4.7F.

We have seen that three scalar fields and attendant potential energies are involved in the construction of the super particles, the barrier shell, and the Higgs mechanism. We must ask how they are related to each other.

 

The Solution

Now we need to see if the three real scalar fields we are talking about can be identified as the same one.

Kane (Kane, 98) starts with a scalar field ø_H . He shows that it is fundamental to forming mass, breaking symmetry and yielding potential energy to the vacuum, and it is the Higgs field. This field is used to generate a potential energy term symmetric in ø:

Peebles (Peebles, 395) starts with a scalar field ø that is fundamental to the spatial expansion that pushes the super particles away from the black hole center. Both can be generated by the potential energy of the black hole.

The only thing that gives us pause in identifying them as the same scalar field, is that one (the Higgs field) is described by quantum mechanics, and the other (the Expansion field) is described by general relativity. There is an argument, however, that justifies calling the fields the same, Loop Quantum Gravity. This argument is described in Appendix 1.

Finally, we have shown that that the total potential energy density found in the vacuum space inside the barrier shell by super particle considerations (~ 10⁵⁰ GeV/cc) is equal to that found in vacuum space by Higgs field considerations (~ 2×10⁴⁹ GeV/cc).       

Thus we appear to be justified in identifying the three fields as the same field.

 

Significance of the Unification of the Fields

Kane shows (Kane, 97) that in the normal development of the Standard Model of particle physics, one does not ask about the sources of the Higgs field, one merely assumes its existence. Also Peebles (Peebles, 395) does not ask about the source of the new scalar field that interacts with space and under some circumstances gives a negative pressure from gravity. Here we ask what Model 1 says the source is and we find the question is answerable.

All fields can be thought of as coming from the mass charges of massive super particles, just as the electromagnetic fields that fill all space can be thought of as coming from the electrical charges of electrically charged particles.

The Higgs field comes from the two parameters that define the potential energy quantum mechanically (see above):

            ø_H ² = – μ²/ λ = ν²

Where mass m _η is defined by:

            m _η² = -2 μ²  

So ø_H can be thought of as coming from mass as the charge.

The scalar field of general relativity comes from (see above):

            p(ø ) = ø’ ²/2 – V(ø) ,   r(m,ø) = ø’ ²/2 + V(m,ø)

Where mass is implicitly defined in the definitions of:

            ø = a new real scalar field

           V(m,ø)  = potential energy density, a function of m and f.

            r(m,ø) = energy density

So ø can be thought of as coming from the same mass charge and:

• We can think of m determining ø (the expansion field) which determines the potential energy (curvature of space) in general relativity. Then the curvature of space determines the dynamics of the particles in macroscopic space.
• We can think of m determining ø_H (the Higgs field), which determines the potential energy (mass) of particles in the Standard Model. Then the mass of exchange particles determines the behavior of exchange forces, which determine the dynamics of the particles in microscopic space.
• The connector between macroscopic space and microscopic space is the black hole, which is the only thing that can operate in both spaces.

We note that the vacuum potential energy of particle space is small and generates a small field and a small mass (~ GeV) in particle space. The vacuum potential energy of vacuum space is large and generates a large field and a large mass (~ 10¹⁷ GeV) in vacuum space. The field does indeed come from super particles that lose potential energy spontaneously to particle space upon entering particle space. Each field is universally present in each space as required by Model 1 and the Standard Model.

 

 Summary and Conclusions

A model (Model 1) has been developed in AP4.7 that predicts dark matter, dark energy and extremely high-energy cosmic rays, which are observed in the energy range beyond the GZK cutoff. As part of this model, a set of super particles (dark matter) was predicted in a new space (vacuum space), and generates this dark energy and the extremely high-energy cosmic rays. All this is based on a scalar field. A scalar field also forms the basis of the Higgs mechanism of the standard model, and the space expansion mechanism of general relativity. 

 In this paper, we found that all of these fields appear to be the same field. Thus the real scalar field of general relativity is the Higgs field and the field of Model 1. It causes mass in particle space, it controls expansion and contraction of space in vacuum space, it controls spontaneous symmetry breaking in particle space, and it generates super particles in vacuum space.

 

Appendix 1

In working on dark matter and dark energy, it was necessary to answer certain basic questions about physics in black holes. Loop Quantum Gravity theory provided a way to accomplish this aim. A group of justifiable results (Smolin, 250) are already available from Loop Quantum Gravity that is compatible with Model 1. For example, Loop Quantum Gravity is finite. It is background independent. It fits into the notation used for the Standard Model (a quantum mechanical theory}, and for the Expansion field (a theory from General Relativity) as well. It predicts gravitons at low energy. Especially, it predicts a Newtonian type gravitational force at moderate energies. It can also be used to predict some important states in black holes. For example, it shows particles sinking into black holes, bouncing at the Plank energy and expanding into a new space. These are fundamental steps in this paper, and so they provide a defensible, background independent basis for constructing this model in such a high-energy environment.

 

References

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