For updated version—see www.Aquater2050.com/2015/12/

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 building and moving bubble 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 an endless 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.  

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 in 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 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 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

 

Assume there are conditions when the net pressure is 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. If the geometry is close to cosmologically flat, we can write the line element in the Robertson-Walker form,

 

            Ds² = dt² – a(t)² (dr² + r² dΩ).

 

            Then:

 

            √¯-g = a³.

 

In this limit, the energy density and the effective pressure in the field are:

 

            p = f’ ²/2 – V,           r = f’ ²/2 + V

 

            Where:

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

            f = a new real scalar field

 

Here, it is assumed that V is a slowly varying function of f and the initial value of the time derivative of f is not too large. Then the kinetic energy f’ ²/2 is small compared to V, and the pressure is negative, and depends on V. If in addition, V is large enough to make a significant contribution to the stress-energy tensor, then space expands under the expansion pressure of V.

 

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, space will expand there and push the particle space containing the particle away. 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. But the potential energy is inversely proportional to the radius from the black hole (r), i.e.:

 

            V ~ – GM/r

 

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 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 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. 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 strength and shape of the potential energy as a function of radius from the black hole center determine the size and shape of the zone of operation of 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 particles, so the potential energy density goes down. In this balance zone, the electromagnetic force, the centrifugal force and diffusion become important, and form the dark matter clouds (see ref 7, AP4.7B). Further out, the up curve of the edge of the Mexican Hat helps contain the dark matter. This scalar field appears to provide the only way to overcome the attraction of the test mass to the center of the black hole. Since both the mass and the potential energy come from the mass of the black hole, they can both become equally powerful. Note that the potential energy term can now be identified as the cosmological constant. 

 

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 three 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∂_μf ∂ ^μf – (½μ²f ²+ ¼ λf ⁴)

 

Note the reflection symmetry; f = –f. 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,

 

            f ² = – μ²/ λ = ν²

 

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:

 

            f(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) + constant

           

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, so the solutions do not. When this occurs, it is called “spontaneous symmetry breaking”.

 

Global symmetry

Now we take f to be a complex scalar.

 

            f = (f₁ + if₂)/2^1/2

 

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

 

            f  becomes  e ^iχ f

 

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,

 

            f₁ ² + f₂ ² = – μ²/ λ = ν ²

 

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

 

            f =[(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 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 1, AP4.7 Appendix A) 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 describes how V and f 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. Model 1 also estimates the vacuum energy of vacuum space by starting with the value of the barrier potential (10¹⁹ GeV) needed to form the barrier shell for isolating super particles in vacuum space away from particle space. The spherical barrier shell was estimated to have a radius of 10^-5 cm, so the vacuum energy density within the shell is ~ 10³³ GeV/cc. The shell also has a thickness of 10^-7 cm, so the energy density in the shell itself is ~ 10³⁵ GeV/cc. Now, pushing the super particle through the shell into vacuum space requires an activation kinetic energy of 10² GeV to go from super particle formation energy of ~10¹⁷ GeV to vacuum space entry energy of ~ 10¹⁹ GeV (see ref 2, AP4.7A). This activation energy is expected to end up in vacuum space as well, so the vacuum energy density is ~ 10⁵⁰ GeV/cc. Note that this is within the error of Kane’s estimate of the Higgs contribution of spontaneous symmetry breaking to the vacuum energy.

 

We have seen that three scalar fields are involved in the construction of 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 f. 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 f: Peebles (Peebles, 395) starts with a scalar field f that is fundamental to the spatial expansion that pushes the super particles away from the black hole center. Both are 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 (~ 10⁴⁹ GeV/cc) to within the error in calculation.      

 

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 then we answer the question.

 

Both 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):

 

           f ² = – μ²/ λ = ν²

 

Where mass m _η is defined by:

          m _η² = -2 μ²  

 

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

 

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

 

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

    

Where mass is implicitly defined in the definitions of:

            f = a new real scalar field

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

            r(m,f) = energy density

 

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

 

• We can think of m determining f (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 f (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 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 symmetry as well as potential energy spontaneously to the vacuum upon entering particle space. Each field is universally present in each space as required by Model 1 and the Standard Model.

 

Testing Model 1 with Data

This portion of Model 1 must be tested with data. A summary of the tests that support Model 1 including those supplied by this paper are shown here. A much more detailed description of the tests that support Model 1 is given in ref 20, AP4.7D.

1. Model 1 is constructed from elements of general relativity, quantum mechanics and classical physics in their appropriate energy realms. It is self-consistent in each of those realms. 
2. It satisfies all the physical data currently known in areas of dark matter and dark energy. No other model or theory is known that satisfies all these data.
3. It connects with the standard model of particle physics, a general relativity description of black holes, the theory of ionized gasses, the theory of the speed of light at extreme energies, the theory of the Planck high-energy limit of quantum mechanics and the theory of the Higgs field.
4. It predicts the existence of a cosmic ray with energy between ~ 10¹⁷ GeV and ~ 10¹⁹ GeV that can be observed beyond the GZK limit where it should not exist. This cosmic ray has been observed.
5. It predicts the existence of a new low cross section super particle with a barrier shield that can be directly observed. This super particle is now being searched for. Preliminary results are positive.

 

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 was predicted in a new space (vacuum space) that constitutes this dark matter, and generates this dark energy and the extremely high-energy cosmic rays. In this paper, we have asked how super particles gather from the enormous observable universe to the site of the big bang. We also asked what happens to the residual dark matter after the big bang ceases. We asked finally, how the residual dark matter influences the development of the microwave background radiation and the galaxy formation in particle space. We have obtained answers to all of these questions, and found that they are intrinsically imbedded in Model 1. Thus, Model 1 is compatible with all known data.

 

In this paper, we asked to determine the origin of the real scalar field that forms the basis of the expansion of space in a black hole. We found that the real scalar field is the Higgs field, and it causes mass in particle space, it controls expansion and contraction of space in vacuum space, and it controls spontaneous symmetry breaking in particle 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

1. L. H. Wald, “AP4.7 DARK MATTER AND ENERGY-FUNDAMENTAL PROBLEMS IN ASTROPHYSICS” www.Aquater2050.com/2015/08/
2. L. H. Wald, “AP4.7A SUPER PARTICLE CHARACTERISTICS” www.Aquater2050.com/2015/08/
3. L. H. Wald, “AP4.7C DARK MATTER RATE EQUATIONS” www.Aquater2050.com/2015/08/
4. L. H. Wald, “AP4.7D HOW TO PROOVE A THEORY’S CORRECTNESS” www.Aquater2050.com/2015/08/
5. P. J. E. Peebles, Principles of Physical Cosmology, Princeton, New Jersey, Princeton University Press.
6. G. Kane, Modern Elementary Particle Physics, Ann Arbor, Michigan, Perseus Publishing.
7. L. H. Wald, “AP4.7B SHAPING THE DARK MATTER CLOUD” www.Aquater2050.com/2015/08/