For updated version—see www.Aquater2050.com/2015/11/

Abstract

In a previous paper (ref 7, 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:1

• 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.  

There is one very strong argument in favor of the acceptance of Model 1. To construct it we start 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 then quantitatively explain five observed phenomena that have yet to find a single consistent explanation, namely:

• The accelerating expansion of the universe (Dark energy)-see AP4.7I.
• The process of creation and rotation of the galaxies (Dark matter)-see ref 10, 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 Kane, 112.
• The large-scale cutoff and asymmetry in the Microwave Background Energy-see AP4.7F.

Two variables cannot satisfy five equations without apriori connections in at least three of those equations. Not since the discovery of Planck’s constant have so many phenomena been explained by so few constants.

In working out this model, some problems arose that are connected with it. The most important of those problems are those associated with the detailed characteristics of the dark matter super particle that is the core of Model 1. Those problems will be addressed here and most will be resolved.

 

The Problem

In ref 10, AP4.7, several problems were left for future efforts. In this paper, one has been singled out for effort, namely the characteristics of super particles.

Super Particles. AP 4.7 postulated the existence of a set of super particles with characteristics similar to bosons, electrons and photons, but at much higher energy. These super particles were then shown to be consistent with the current data on dark matter and dark energy. As noted in AP4.7, the characteristics of the super particles were indicated briefly, but they were not defined in detail. This task will be accomplished here. The overall characteristics required of the super particles are:

1. They must generate a potential barrier to separate the high-energy space (vacuum space) from the low energy space (particle space), and ensure that the super particles interact with particles only on the level of an elastic scattering, which has a very low cross section.
2. They must unify the four forces of particle space into one force at about 10¹⁷ GeV (the known potential unification energy), and they must survive at extremely high kinetic energy (~ 10¹⁷ to 10¹⁹ GeV). When the super particles break down, they must generate the particles associated with the four forces in matter and anti matter forms.
3. The force exchange photons must be able to ignore each other’s barrier potential to generate interacting forces.
4. They must generate something equivalent to an exchange force in order to interact with each other and form a dark matter ionized cloud around a galaxy.
5. The super particle and its barrier shell must have a low interaction cross-section with ordinary particles to be undetectable and thus be known as dark matter.
6. The particles of particle space must use two steps to pass into vacuum space.
7. Eventually, the dark matter temperature increases enough that the super particle kinetic energy exceeds the potential energy. Then a new big bang starts.
8. The big bang must unfold in such a way as to generate an excess of matter.
9. The super particles that remain in vacuum space will start to tunnel into particle space. There they must decay into UHECR’s and give up potential energy. 

In this paper, we will define the super particles in some detail, and take them through their various incarnations to describe where matter and anti matter come from. For the purposes of this paper, the particles of interest will be a proton and an electron. A deuteron also appears possible. The super particles will be essentially a high-energy state of a proton and a high-energy state of an electron. We start with the results of ref 7, AP4.7.

The Solution

A Super Particle for Model 1

Model 1 is a basic, self-consistent physical model for a super particle that will fit the roll of dark matter and generate dark energy. The unique features of Model 1 are, then:

•  The super particle comes from a particle in particle space (baryon conservation), which passes through a black hole where it gains enough potential and kinetic energy to become a super particle, and then enters high-energy vacuum space where it operates as dark matter. Thus there are three zones in a super particle.

The innermost zone is the particle itself. Using the uncertainty relation and the known size of the proton in particle space (10^-13 cm with energy ~ 1 GeV), one can estimate, for a super proton with energy 10¹⁷ GeV, a size of ~ 3 x 10^-20 cm.

The next zone is vacuum space where the super particle operates, which has a radius of 10^-11 cm. This is a space for particles with potential energy ~ 10¹⁷ GeV-i.e. super particles. We note from the barrier equation that super particles with kinetic energy greater than 10¹⁷ GeV move freely in vacuum space. If the kinetic energy drops below 10¹⁷ GeV, however, the probability of operating in and passage through quickly attenuates toward zero. Note also that there is no force that requires the super particle to be centered in the barrier shell. It moves freely and bounces off the inside of the barrier shell.

The outermost zone is the vacuum space barrier with a thickness of ~ 10^-31 cm. This is a barrier shell with a potential energy of ~ 10¹⁹ GeV. We note that the super particle does not pass this barrier freely unless it has a kinetic energy of ~10¹⁹ GeV, but it can tunnel through.

Beyond is particle space.

The mathematical details of these zones are given in Appendix

• The coupling strengths for the electromagnetic force, the weak force and the strong force converge to a single super force at ~ 10¹⁷ GeV as shown in Appendix 3 (Kane, 281). The super particle components are assumed to provide the charges and potential energies needed to generate this super force. In order to remain stable, a super particle must operate in a high potential energy vacuum space with a density of ~10⁵⁰ GeV/cc as shown in Appendix 2. (see ref 8, AP4.7C for more details). As mentioned in 1 above, the equations show that the super particle operates well in the kinetic energy zone of ~ 10¹⁷ to ~ 10¹⁹ GeV. Below this, the super particles disappear (i.e. break down into ordinary particles), and within the zone, they operate above the barrier which has a potential energy of ~ 10¹⁷ GeV (10⁵⁰ GeV/cc). In this zone, they are super particles with a convergent, single super force. Note that the potential energy density matches that of the Higgs mechanism (see Appendix 5). When the super particles pass over the barrier, they break down, and generate the particles associated with the four forces in matter and anti matter forms.
• We note from the equations of Appendix 2 that since the rest mass of the force exchange photons is zero, these photons can pass right through the barrier, and allow the super force to operate. The ordinary photons can also pass through the barrier, but they do not interact with super particles.
• The unified force is the exchange force that operates in vacuum space. With high enough kinetic energy, this force can break, and the super particles are then ionized. If the super particle energy gets too low, the super particle breaks down into particles, and the super force becomes the standard model forces. This helps to account for the finite size of the dark matter clouds. At the edge of the cloud. The temperature has reduced enough for the particles to break down (see ref 10, AP4.7B for details).
• The interaction cross-section between particles in particle space and super particles in vacuum space is elastic and very low (see ref 11, AP4.7E). That is why we call it dark matter.
• The particles of particle space use a two-step process to pass into vacuum space. Step 1. The potential energy of the high gravitational field in a black hole provides the potential energy needed to make a particle into a super particle, and appears to be the only place in the universe that can provide it (see Appendix 4).
• Step 2. In order to pass the barrier of the shell, a super particle needs activation kinetic energy that boosts its total kinetic energy to ~10¹⁹ GeV, which is near the Planck energy. The addition of energetic particles through the black hole provides this energy. With this energy, the particle bounces on the Planck energy and enters vacuum space where it loses some average kinetic energy (temperature) in the larger volume there. At this point, it becomes an ionized dark matter gas particle where it generates a dynamic equilibrium between particles and super particles in the high vacuum potential environment (see ref 8, AP4.7C).
• As high-energy super particles are added to the dark matter cloud from the black hole, they form a kinetic equilibrium with a temperature, and the temperature increases. When the mean kinetic energy of the super particles nears the potential energy of the barrier shell, the particles in the high-energy tail flow over it and initiate a big bang. Note that in this high-energy environment, the speed of light is high (Magueijo, 31). The super particles rush to the principal (highest super particle energy) black hole to pass into particle space, leaving a feint, residual imprint of the old cosmic web (see ref 10, AP4.7B for details).
• When the super particles enter the even larger volume of particle space, the temperature drops. They cannot reenter vacuum space because the kinetic energy is below the barrier potential. They can no longer participate in the dynamic equilibrium of vacuum space, so they must decompose into particles. Since the kinetic energy has actually exceeded the shell potential energy, the super particles have enough kinetic energy (at the Planck energy) after flowing into particle space, that collisions can cause complete disruption of the super particle into energy. Note that both positive and negative ions from the ionized gas of super particles are temporarily converted to energy, so charge is conserved.
• The steps to generate matter are tricky. In the initial phase after the big bang, the temperature is still high, so the speed of light is high and the resulting large scale interactions produce thermodynamic equilibrium. As particles are formed from super particle potential energy, the CP violation that would normally produce an excess of matter, produces a dynamic equilibrium and forms both particles and anti particles in roughly equal numbers. The particles and anti particles annihilate each other and produce photons. The potential energy is still high, so the pressure is negative, and space expands. Potential energy loss, and the expansion of space reduce the vacuum potential energy, so the expansion slows. Temperature drops more, so light speed drops. The particles lose contact with each other, and the CP violation starts to produce an excess of matter. The particles and forces freeze out one sub particle and its force at a time.  We end up with a lot of photons and a significant amount of matter, but very little anti matter (see ref 8, AP4.7C for details).
• Super particles that remain in vacuum space will tunnel into particle space and show up as cosmic ray events that we observe in the energy range of ~ 10¹⁰ to 10¹⁹ GeV, which is beyond the GZK cutoff. They appear everywhere in the galaxies, as the observations show (see ref 12, AP4.7I for details).
• Potential energy from the tunneling super particles is dumped in particle space as the super particles break down. This potential energy builds up in particle space to become dark energy. This dark energy causes the accelerated expansion of particle space we observe.  

 Note that these processes involve a sequence of rate processes that will be quantified in ref 8, AP4.7C.

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 9, AP4.7D.

• Model 1 is constructed from elements of general relativity, quantum mechanics and classical physics in their appropriate energy realms, all of which are well accepted. It is self-consistent in each of those realms. 
• 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.
• 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.
• It predicts the existence of a cosmic ray with energy between ~ 10¹⁰ GeV and ~ 10¹⁹ GeV that can be observed beyond the GZK limit where it should not exist. These cosmic rays have been observed.
• 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 ref 7, AP4.7 that predicts dark matter and energy and extremely high-energy cosmic rays, which are observed beyond the GZK cutoff. As part of this model, a set of super particles was predicted in a new space (vacuum space), and generates this dark energy and the high-energy cosmic rays. The details of these super particles were not pursued in AP4.7, and so they have been pursued in this paper. The super particles have been found to be essentially ultra high-energy states of protons and electrons, where a spherical shell barrier that contains a high potential energy density surrounds the proton. Initial checks with existing data have been made, and Model 1 has been found to be in agreement with the data. Certain details of Model 1 are given in two other papers (ref 8, AP4.7C, and ref 10, AP4.7B). Model 1 has been found to be valid as far as the currently available data can determine.

Appendix 1

First, let us develop the energy balance from general relativity. It has a kinetic energy and potential energy term. In general relativity, the potential energy is in the curvature of space and shows up as pressure in the energy balance equation (Peebles, 394).

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

The pressure here comes from a single new scalar field, f. The energy density and pressure equations that result are as follows (Peebles, 396):

Here,

ρ  = f’ ²/2+ V; and p = f’ ²/2 – V

            Where:

            V = a potential energy density due to gravity

            f = a new real scalar field

            f’ ²/2 = the kinetic energy term          

Note that under the correct circumstances, the pressure p can be either positive (contracting) or negative (expanding), depending on the potential or kinetic energy terms.

Appendix 2 The Barrier Equations

Next, consider the following non-relativistic three-dimensional time independent Schrödinger equation (see ref 7, AP4.7).

[-(h²/8p²m)Ѳ+V(x, y, z)]Y(x, y, z) = EY(x, y, z)

Where:

            V(x, y, z) = barrier potential

            E = particle energy

If we convert to spherical coordinates, and let:

Y(r, q, f) = R(r) Y(q, f)

Where:

            Y(l, m) = Spherical Harmonics = (4p) ^-1/2  , if  l = 0 (spherical symmetry)

Now, let:           

R(r) = U(r)/r, then:

Y(r, q, f) = (4p) ^-1/2  U(r)/r   

Now, consider:

[-(h²/2m d²/dr² + V(r)]U(r) = EU(r)  

 Where:

             E = the energy of the particle.

             V(r) = V_o[Q(r) – Q(r-w)] = the vacuum potential.

And:

             Q(x) = the Heavyside step function of width a starting at x =0

             w = the potential zone width.

Note that any solution to the equation is unchanged if the step function is moved along the r axis to r_ob.  Then:

1.      One can think of starting at 0 and moving to r_ov. Think of this as moving within the super particle itself, so the complex kinetic and potential energy distributions applicable within the particle exist here.

2.      One then moves from r_ov to r_ob., so w = r_ob -r_ov .= v. This is vacuum space, so the vacuum potential is V(r) = V_ov[Q(r_ov) – Q( r_ov-v)] = the vacuum potential

3.      One then moves from r_ob to r_ob + a, so w = a. this is the space within the barrier itself, i.e. between vacuum space and particle space, so the vacuum potential is V(r) = V_ob[Q(r_ov) – Q( r_ov-v)] = the barrier vacuum potential.

Thus the equation governs passage from the super particle through vacuum space and then through the barrier and into particle space. For convenience, we will let r  = 0for solving the equation. Note that what we are describing is two concentric spherical shells with thicknesses v and a wrapped around a super particle. The shell with thickness v is vacuum space. The shell with thickness (a) is the barrier between vacuum space and particle space.

Note that what we will calculate is the transmission probability density (T = t²  = r) or probability of transmission. The solution to the equation is a combination of left and right moving wave functions that are continuous at the boundaries of the barrier (r = 0 and r = w) along with their derivatives.

The solution can be used to generate the transmission through the barrier T (see Ref 8), which is as follows: 

If E>V

            T = 1/(1+V₀²sin²(k₁w)/4E(E-V₀)          

If E<V,

            T = 1/(1+V_o² sinh²(k₁w)/4E(V₀-E),

Where

             k₁= (8p²m(V₀-E)/h²) ^1/2

Here we have set up the equations for either of two shells, a vacuum space shell and a barrier shell. The rate of passage of a particle through either spherical shell for a galaxy is:

R = T h/2p k_o n_sp / m r_ob particles/sec galaxy

The best fit for the vacuum space sphere and the barrier shell of the very rough data available is (see ref 13, AP4.7G for details):          

            r_ov = 10^-20 cm

            r_ob = 10^-11 cm

            a = 10^-31 cm

There are three important cases.

Case 1

            E > 10¹⁷ GeV

            V_ov = potential energy ~ 10¹⁷ GeV

            V_ov/cc = potential energy density ~ 10⁵⁰ GeV/cc

            N = number of super particles in vacuum space = 10⁶⁹ /galaxy

This is the vacuum space case for particles with potential energy ~ 10¹⁷ GeV-i.e. super particles. We note that super particles with kinetic energy greater than 10¹⁷ GeV move freely in vacuum space. If the kinetic energy drops below 10¹⁷ GeV, however, the probability of operating in and passage through quickly attenuates toward zero.

Case 2

            E ~ 10¹⁷ GeV

            V_ob = 10¹⁹ GeV

            V_ob/cc = potential energy density ~ 10⁷² GeV/cc

            N = number of super particles in vacuum space = 10⁶⁹ /galaxy

This is the case of leakage of super particles into particle space by tunneling through the barrier. In this case, T ~ 10^-50, and R is ~ 10³⁵ particles/sec galaxy. For all 10⁸ galaxies of particle space, R is ~10⁴³ particles/sec. When these particles reach particle space, they break down to ordinary particles and give up their phase change energy (10¹⁷ GeV/particle) into the particle space vacuum energy, or dark energy, which over 10¹⁰ years has made a dark energy of ~ 10^-4 GeV/cc. This is the same dark energy that causes the accelerated expansion of space that requires a potential energy of ~ 10^-4 GeV/cc.

Case 3

            E = 10¹⁹ GeV

            V_ob = 10¹⁹ GeV

This is the case of big bang passage over the barrier. In this case, T = 1, and R = 10⁹³ particles/sec. When these particles reach particle space, they break down into ordinary particles and give up their phase change energy into particle space vacuum energy, or expansion energy, which expands particle space.

Appendix 3

Now Kane (Kane, 279) shows that the couplings of the forces satisfy the equation:          

            1/α_i (M²) = 1/ α_i (μ²) + b_i /4π ln M²/ μ²

Where:

            α_i  are the couplings for the forces

            M is the mass scale at which we want to calculate α_i

            μ is a scale where the coupling is measured

            b_1, b_{2 ,} b₃are calculated for the U(1) (electromagnetic), SU(2) (weak) and SU(3) (strong) interactions.

Kane then develops the equation to show the coupling strengths as a function of the mass (energy) at which we want to calculate them. They are shown to converge close to a single value at a mass M_G  ~ 10¹⁷ GeV. The convergence is not exact, because it is sensitive to radiative corrections, but it is close. If they meet exactly, that would be taken as a very strong indicator of a grand unification into a single force (a GUT). There are several kinds of GUT’s. For our purposes here, a GUT with SU(5) symmetry will be chosen because one of its sub symmetries (U(1)) will be needed for the barrier potential (see below). Super symmetry is not necessary, and there are problems with its use (see ref 9, AP4.7D), so it will not be chosen. We should ask if the unification of forces implies a change in the detailed characteristics of its particles and forces. This will be explored more below, but generally, since baryons are conserved at this point, there does not seem to be any reason to change the detailed characteristics of the sub particles and forces involved. Also, we will need one of the symmetries later. Because the operating energy is high, the super particle is relativistic and so has spin and anti matter. Thus the particles that are in the standard model should exist in some form, and a super force for the super particle is still accompanied by the exchange of a photon. The potential energy involved in the function of these particles, of course, increases in the extreme energy of vacuum space.

Appendix 4

We must ask if a black hole has the ability to provide the required 10¹⁷ GeV to 10¹⁹ GeV in kinetic energy and 10¹⁹ GeV in potential energy needed to form super particles from particles and provide a barrier. We note that the kinetic and potential energy varies inversely with r (Misner, 911). Thus there is an r with the proper energies. However, the Planck energy must be reached before the Planck radius is reached. It was found that energy is reached easily by a 10⁶ solar mass black hole worked.  It was also noted that such a massive black hole was common for galactic centers. So it was used as the example for the work done in this paper and in AP4.7. It was also noticed that the potential energy necessary to obtain the super particle state is very high. To get an electron to move into orbit near a nucleus requires a potential energy ~ electron volts. To get an electron to merge with a proton and make a neutron in a neutron star requires a potential energy ~ GeV, which requires the gravitational potential obtained from ~ 2 solar masses. To get a proton to become a super proton in vacuum space requires a gravitational potential of ~ 10¹⁷ GeV, and a kinetic energy of ~ 10¹⁹ GeV which requires the gravitational potential obtained from a black hole mass in the range of 10⁴ < m < 10⁹ solar masses. This implies that the central black hole in a galaxy is the principle provider of dark matter in the universe, although black holes from a very large stars that might be able to generate such energetic particles do exist, and were common in the early universe

Appendix 5

Kane (Kane, 112) has estimated the contribution of spontaneous symmetry breaking to the vacuum energy density of the universe. This energy density estimate came from calculations based on the Higgs mechanism. He obtains a value of ~ 10⁴⁹ GeV/cc. This value is admittedly approximate. He must guess at the value of the Higgs self coupling. However, the vacuum energy density varies only linearly on this self-coupling value, so the energy density is not very sensitive to this value-perhaps an order of magnitude. Now Model 1 estimates the potential energy needed to generate the barrier wall as (see exercise 6, above) ~ 10¹⁹ GeV. This energy requires that the energy density within the wall is ~ 10⁷² GeV/cc, and the energy density within the spherical shell of vacuum space is ~10⁵⁰ GeV/cc. Thus it appears that there is about as much potential energy in spontaneous symmetry breaking as is required for super particle construction. This makes us wonder if there is a connection between the Higgs field and the new scalar field of general relativity described in Appendix 1. For more details on this issue, see ref 13, AP4.7G.

References

1.      L. Smolin, The Trouble with Physics, Boston, New York: Mariner Books, 2006.

2.      G. Kane, Modern Elementary Particle Physics, Ann Arbor, Michigan, Perseus Publishing.

3.      J. Magueijo, New Varying Speed of Light Theories, arxiv.org/pdf/astro-ph/030545v3.pdf

4.      J. Magueijo, Faster than the speed of light, Penguin Books, New York, New York, 2003.

5.      P. J. E. Peebles, Principles of Physical Cosmology, Princeton, New Jersey, Princeton University Press.

6.      Misner, Thorne, and Wheeler, Gravitation, New York, Freeman and Co., 1973.

7.      L. H. Wald, “AP4.7 DARK MATTER AND ENERGY-FUNDAMENTAL PROBLEMS IN ASTROPHYSICS” www.Aquater2050.com/2015/09/

8.      L. H. Wald, “AP4.7C DARK MATTER RATE EQUATIONS” www.Aquater2050.com/2015/09/

9.      L. H. Wald, “AP4.7D HOW TO PROVE A THEORY’S CORRECTNESS” www.Aquater2050.com/2015/09/

10.  L. H. Wald, “AP4.7B SHAPING THE DARK MATTER CLOUD” www.Aquater2050.com/2015/09/

11.  L.H. Wald, “AP4.7E “INTERACTION RATE OF DARK WITH VISIBLE MATTER” www.Aquater2050.com/2015/10/

12.  L.H. Wald, “AP4.7I “SUPER PARTICLES AS COSMIC RAYS” www.Aquater2050.com/2015/10/

13.  L.H. Wald, “AP4.7G “ORIGIN OF THE NEW REAL SCALAR FIELD” www.Aquater2050.com/2015/10/