Abstract

In a previous paper (ref 1, AP4.7), a self-consistent theory called Model 1 was developed to answer eight 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 connect 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 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 those problems has to do with the details of the rate equations that govern the production of the super particles and move them from particle space through vacuum space, through a big bang, and back into particle space. Those rate equations will be explored and their characteristics resolved here. Also some details of the ultra high cosmic rays and dark energy generation will be detailed.

 

The Problem

Model 1 (ref 1, AP4.7) is dependent on a series of rate equations that make the steps outlined therein occur in the proper sequence, and result in the proper equilibrium concentrations. Those rate equations were alluded to in AP4.7A (ref 2, AP4.7A), but were not detailed there to make the paper short enough to be readable. In this paper, the rate equations will be described, and the rate constants needed to ensure the proper step sequence for forming super particles and moving them through particle space and vacuum space will be established. For the purposes of this paper, the particles of interest will be a proton or deuteron (called a baryon) and an electron. The super particle will be essentially an excited state of a proton or deuteron and an excited state of an electron.

The rate processes and sequences that need detailing are:

• The rate of production and re-conversion of super particles..
• The rate of passage into vacuum space and return from it.
• The step sequence that moves particles from particle space into super particles in vacuum space.
• The rate of production and annihilation of matter and anti matter ending with a final excess of matter in particle space.

In each case, the dependence of the rates on kinetic and potential energy will be included. Also, the temperature dependence of the rate constants will be noted.

 

The Solution

Now, as indicated above, there are three reaction rates and a step sequence that are of interest here. We will investigate each of these separately.

Rate of Production and Re-Conversion of Super Particles.

Consider the production of super particles S and re-conversion back to particles P. Note that this is a conservative process in which the baryon and its components are conserved. Only the symmetry and potential energy and therefore the internal dynamics of the baryon components are changed. Thus, S may be considered an excited state of P. These processes can therefore be described in a way similar to chemical reactions as a rate process. We start with the equation:

            P + V  = S

Where:

            P is a particle

            S is a super particle

            V is the potential energy required for the conversion from P to S.

            The = sign shows that the process can go in either direction.

The equation indicates that the rate of production of S from P should be a first order reaction, and so we can write.          

            Rate = d[P]/d t = – k_p[P]

Where:

            [P] is the concentration of particles P

            [S] is the concentration of super particles S

            k_p is the rate constant         

The integrated rate production equation is:

            [P] = [P_o]exp(-k_pt)

The rate of production of P from S is:

             Rate = d[S]/d t = – k_s[S]

And the integrated rate production equation is:

            [S] = [S_o]exp(-k_st)       

Super particles are produced from particles and the potential energy V that comes from the distortion of vacuum space by mass in a black hole (see ref 2, AP4.7A). The potential energy V[r] is a function of r, which is the distance from the center of the black hole, and M, which is the mass of the black hole (see Misner, 911). This is not a single valued function of r, but a distribution dependent on mass, electromagnetic field, and rotation distributions within the black hole. This distribution can be approximated by a Gaussian function F_g[r,M], which will be shortened to show only the principle dependence F_g[r].

The dependence of the rate constant k_p on V for S production from P can be described as

            k_p = A_pexp(-V_g(r_o) / F_g[r])

Where:          

            V_g(r_o) = force unification potential = S particle excitation potential = 10¹⁷ GeV

            r_o = Black hole radius at which the Gaussian mean is ~ 10¹⁷ GeV

Also an appropriate amount of potential energy is taken from particle space to facilitate this reaction. Note that when F_g[r] increases (more excitation potential is available), the rate constant increases toward A_p. When F_g[r] decreases (less excitation potential is available), the rate constant decreases toward zero, the [P] conversion rate is reduced toward zero, and [P] remains near the initial value of [P_o]. Recall that the conversion rate does not go completely to zero because F_g[r] is a Gaussian and has high-energy tails.

The rate constant for [P] production from [S] is:

            k_s = A_sexp(V_g(r_o) / F_g[r]) 

Also an appropriate amount of potential energy is given up into particle space in this process. Note that when F_g[r] increases (more excitation potential is available), the rate constant decreases toward A_s. When F_g[r] decreases (less excitation potential is available), the rate constant increases, and [S] rapidly converts to [P], giving up its potential energy to particle space.

These expressions for k come from the fact that we can expect that in a Gaussian distribution of potential energy quanta, the number with potential energy greater than V_g will be equal to exp(V_g(r_o) / F_g[r]). The coefficient A_p is a frequency factor.

When the particles P are converted into super particles S, the super particles gain a charge for the barrier potential shell, and it forms. This charge exists because the continuity of the wave function and its derivative are maintained in passage of a super particle through the shell, so Noether’s theorem requires it (see Appendix 1). However, the super particle does not pass beyond the barrier into vacuum space until it gains enough kinetic energy (activation energy) to bounce against the Planck limit and then penetrate the barrier into vacuum space.

Rate of Passage into and Return from Vacuum Space

Passing into vacuum space and getting behind the barrier shell has to do primarily with getting both the kinetic and the potential energy near the Planck energy. When the particle gets close enough to the black hole center that the potential energy density is 10⁷² GeV/cc (the value needed for barrier shell production), the production of the barrier potential shell is automatic, since the charge guiding its production is already built into the super particle. Inside the shell, the potential energy of the field is greater than the kinetic energy of the field, so the pressure is negative, and the space expands until the potential energy density reaches 10⁵⁰ GeV/cc (the value needed for vacuum space production) and at this density, the potential energy density equals the kinetic energy density, and the pressure turns positive, so the expansion stops. The barrier shell is now formed, but the particle has not entered it. Meanwhile, the particle has been falling toward the black hole center and gaining kinetic energy. When the particle reaches the barrier potential energy value, it passes through the shell wall and the formation of the super particle inside its shell is complete. The rate of passage of the particle through the shell into vacuum space can be described by:

            Rate = d[S]/d t = – k_vs[S]

Where:

            k_vs is the rate constant for passage into vacuum space,

            k_vs = A_sexp(-E_a / RT)

            E_a = activation kinetic energy needed to push super particles into vacuum space

            R = super particle gas constant

            T = super particle temperature

Here, at temperature T, the super particles have energies given by a Boltzmann distribution, so we can expect the number of collisions of the shell with energy greater than E_a to be proportionalto exp(-E_a / RT). The activation energy is the energy between the super particle formation energy (10¹⁷ GeV) and the Planck energy (10¹⁹ GeV), where the super particles pass through the shell wall into vacuum space..

At this point, we see that the shell has an open space in it with a gap distance a (see Appendix 1, below). This space has captured the vacuum potential that exists in the location of formation near the black hole, namely, 10⁷² GeV/cc. The shell then forms a barrier wall of value 10¹⁹ GeV. This wall has also captured a vacuum potential of ~10⁵⁰ GeV/cc which is the vacuum space value within the shell. The particles and super particles change back and forth in a dynamic equilibrium in vacuum space. As long the particles are in the high potential energy of vacuum space, the particles can change into super particles and back. If they get outside the barrier, however, the potential energy is lower, and the super particles give up their potential energy to particle space and cannot become super particles again except near a black hole. The vacuum potential is maintained within the barrier shell even if the super particle and its shell move away from the vicinity of the black hole, as it will eventually do. These super particles constitute dark matter.

Once in vacuum space, the super particles are expanded away from the center of the black hole by virtue of the negative pressure due to the high potential in particle space near the black hole (see step sequence, below). The super particles are ionized in the vicinity of the black hole and the electric fields between the ions force the particles into the cloud of dark matter observed by astronomers (see ref 3, AP4.7B). With expansion, the super ions cool as they expand away from the black hole center and eventually will recombine into neutral super particles. Somewhat beyond the event horizon, the negative pressure has turned positive, and the mean free path of the super particles is long enough that they are in free flight to the edge of the galaxy. When the velocity of the super particles exceeds that of the visible matter of the galaxy, the super particles dominate the mass density of the galaxy. At this distance, the dark matter halo edge forms (see ref 3, AP4.7B for more details). Note that the super particles of the dark matter have a low interaction and collision cross section at the velocities of galactic motion (see Appendix 2). Thus, when galaxies collide, the luminous matter particles with high interaction cross-sections will interact and coalesce, but the cold dark matter particles with low interaction cross sections will pass right through as seen by astronomers when observing colliding galaxies. Here, the cold dark matter is observed through its gravitational lensing effects.

The Step Sequence from Particle Space into Vacuum Space and Back.

While in particle space inside r_o, particles can become super particles because the potential energy density of the black hole is sufficiently high to allow super particle production. However, with the potential energy high, the pressure also becomes negative according to the equation (see Appendix 3):

            p = ø’ ²/2 – V)

Then the space between the super particle and the black hole (with a higher potential energy) expands, pushing super particles away from the black hole center. There, the potential energy density is reduced, and the super particles convert into particles, giving up potential energy, with a reduced reverse process to replenish them. At the same time, the expansion of space stops, the pressure becomes positive, and the particles are dragged back inside r_o by gravitation. This process is repeated, thus keeping the density of super particles high inside r_o, and rapidly reducing outside.

Gradually, the kinetic energy increases enough through energetic particle addition to a limited volume, to provide the activation energy needed, and the super particles are bounced against the Planck energy into vacuum space inside the vacuum barrier shell. There, encased in the protective vacuum barrier shell, they are pushed away from the black hole center beyond r_o by the high potential energy (and thus negative pressure), and diffuse away from r_o, there to form into the galactic halos and a cosmic web observed as dark matter (see ref 3, AP4.7B). 

Far away from the black hole center, the super particle ions can form a stable equilibrium within the shell, converting from super particles to particles and reverting back into super particles because the vacuum potential energy is high enough to make this transition possible. The higher the temperature, the faster this rotation happens. If the temperature gets low enough, the ions can even recombine into neutral super particles. Now, a new heating process occurs in vacuum space, where super particles are heated by the addition of energetic super particles from particle space near the black hole center until the average kinetic energy in vacuum space comes close to the shell potential energy. Then super particles in the high-energy tail begin to flow over the barrier into particle space at a high rate. This flow we call a big bang. Super particles are gathered from all over the universe for this event (see ref 7, AP4.7F for more details of this process).

The Rate of Production and Annihilation of Matter and Anti Matter.

Once 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 in the energy cloud. Thus, the super particles will not only flow over the barrier, they will disintegrate into energy for a short time as they reach particle space according to the uncertainty principle:

            δt = h/2πδE.

This principle is essentially the rate equation for energy change. Now the energy expands into particle space. The temperature of the energy drops. The kinetic energy drops, so the potential energy in particle space dominates, and pressure is negative. Space expands more rapidly.  The potential energy from disrupted particles now generates new particles of matter and anti matter. Again charge is conserved, but baryons are not. All the particles and forces of the standard model gradually freeze out as the mixture cools. The matter and anti matter particles annihilate each other as they freeze out. The speed of light, which started high in the early, high-energy phase of the expansion, reduces as the energy drops according to the equation:

            v = nl_p/ Nt_p > c = 2.99 x 10¹⁰ cm/sec. 

Where the length of travel for a particle is n Planck lengths long, and N is the number of disruptions in space along the travel length. Then the distance traveled is nl_p, and time used for this travel including that for all disruptions is Nt_p. Now if the energy is low, then N=n, and v=c. If the energy is near the Planck energy, then N<<n, and v>c (see ref 9, AP4.7N for more details). Thus, the material in particle space, which started at equilibrium as a mixture of matter and anti matter particles annihilating each other, now becomes matter and anti matter particles, but not in equilibrium because they cannot be in contact at the lower light speed. 

The details of the process that generates particles are important now. In the formation of matter and anti matter, our current particle data show that there is roughly one matter baryon excess for 10¹⁰ photons in particle space. This result can happen only if (Kane, 290):

• Conservation of baryon number is violated.
• Charge-parity (CP) is violated.
• Particle space is not in thermodynamic equilibrium while the above conditions are satisfied.

As shown above, the conservation of baryon number was violated as soon as the super particles flowed over the potential barrier. Thus, the energy from super particles has reformed as matter and anti matter particles while still in thermodynamic equilibrium due to the high light speed at the high energy existing then. Since there is thermodynamic equilibrium at this time, a roughly equal number of particles and anti particles are produced. The matter and anti matter particles annihilate each other producing photons that eventually become our microwave background. As this process continues, the loss of thermodynamic equilibrium along with the continued CP violation results in an excess of matter particles. Particle space is then left with a large number of photons, and a small excess of matter, from which galaxies are formed. This condition is what we observe now.  Note that as long as the temperature is high enough, potential energy will be converted into particles, and the potential energy will drop.  As soon as the temperature is low enough, the potential energy will not be converted into particles.  

The energetic super particles behind the barrier drops rapidly in number and lose energy as they flow out over the barrier, the most energetic particles first (see ref 7, AP4.7F). The dark matter particles rush to the zone of barrier breach (the big bang site) at extremely high speed because of the high energy (close to the Planck energy). When the kinetic energy of the least energetic super particles in the distribution drops below the barrier potential, the super particle flow dies down to tunneling through the barrier of the super particles that remain in vacuum space. At this point, the big bang is over. Note that tunneling super particles are not near Planck energy, so they do not disrupt into energy and then form matter and anti matter. The vacuum potential energy continues to make particles until the temperature drops too low. At this low temperature,  the vacuum potential energy no longer forms into particles.

The super particles that continue to tunnel into particle space change state to particles through the rate process shown above while not in thermodynamic equilibrium (see ref 8, API). Specifically, they turn into ultra high-energy protons (UHECR’s), and vacuum potential energy. We observe the protons as cosmic rays (protons) with energies between 10¹⁰ and 10¹⁹ GeV. Some of this energy range is beyond the GZK cutoff (see Magueijo, 33), but it ends at the Planck energy. After the universe goes through the process of forming matter particles (see above) and then black holes, the super particles in vacuum space begin slowly to be renewed through these new black holes in a renewed particle space. The tunneling super particles give up their potential energy into particle space, which causes an increasing level of particle space vacuum potential energy. Eventually, the vacuum potential energy exceeds the kinetic energy, and the pressure turns negative, and the field becomes expansive again. This potential energy is the dark energy that causes the accelerated expansion of space, which we observe in our time (10^-4 GeV/cc). The vacuum kinetic energy term increases from the tunneling super particles as well. In fact it increases faster than the potential energy term. Eventually, the vacuum kinetic energy term will exceed the potential energy term, the pressure will turn positive, and the field will stop its accelerated expansion. For more details on this stage, see ref 9, AP4.7N.

 

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, the details of the rate equations that produce the super particles and move them from particle space through vacuum space, through a big bang, and back into particle space are explored. Initial checks with existing data have been made, and Model 1 has been found to be in agreement with the data. Possible problems with the model have been analyzed, and a theoretical program proposed as a fix. Experiments that would check the accuracy of the model have been proposed as well. 

 

Appendix 1

The barrier shell and vacuum space are described as follows (see also ref 2, AP4.7A, Appendix 2):

            [-(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-a)] = the vacuum barrier potential.

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

            a = the barrier potential width.

            r_ob = the radius of the barrier potential shell.

            r_ov = the inner radius of vacuum space.

And

            r_ov = 10^-20 cm

            r_ob = 10^-11 cm

            a = 10^-31 cm

The solution can be used to generate the transmission through the barrier T , 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= (8Π²m(V₀-E)/h²) ^1/2

            w = r_ob – r_ov  = for vacuum space

            w = a = for barrier

Note that what we have calculated 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 = a) along with their derivatives. This continuity means that the particles are unchanged in passing through the barrier, and by Noether’s theorem, a charge exists for this process. This solution is detailed in AP4.7A (ref 2, AP4.7A, Appendix 2).

Note also that the equation demands that the super particles move freely in vacuum space if the kinetic energy is greater than 10¹⁷ GeV where the potential is 10¹⁷ GeV. They stop (except for tunneling) when they hit the barrier if the kinetic energy is less than 10¹⁹ GeV where the potential is 10¹⁹ GeV.

 

Appendix 2

Here, we explore the characteristics of super particles to see if they can be observed in particle space- i.e. are they dark? First, super particles do not show charges associated with the electromagnetic, weak, and strong forces. They are combined into one super charge and hidden behind the barrier potential. The super particle spin, if any, would not show beyond the barrier as well. They have only the super charge associated with the unified force. Thus they will not interact with the detectors we normally use. Particles in particle space will scatter off the potential barrier surrounding the super particle, however, so it is necessary to calculate this scattering cross section. This scattering cross section is like the scattering of a proton off a neutron, but with different energies. This scattering cross-section has been calculated (Halliday, 47), and is as follows:

            σ = h²π/M x  1/(V_o + E)

Where:

M = m_s m_p/ (m_{s +} m_p)

m_p = mass of particle space baryons = 1 GeV.

m_s = mass of super baryons = 10¹⁷ GeV.

V_o = potential of super baryons = 10¹⁹ GeV

E = kinetic energy of the particle space baryons = 1 GeV or less.

            Then   σ = 10^-62 cm² 

This calculation is for high-energy scattering (S scattering), i.e. scattering of particles with kinetic energy that is of the order of the potential energy of the target particles (super particles with barrier shells). Halliday notes the possibility that there may be other scattering contributions due to the spin of the particles involved (Halliday, 48). However the sum of these contributions does not appear to exceed 10^-45 cm² 

Clearly, this scattering cross section would be difficult if not impossible to detect. So matter is dark or difficult to detect in particle space. Also, the collision cross section of the dark matter particles is low enough that in colliding galaxies, the dark matter particles would pass right through each other, while the visible matter would interact and coalesce, as seen by astronomers.

 

Appendix 3

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

            ρ’ = -3 (r + p) a’/a

Where:

            p = pressure

            σ = energy density

            σ’ = rate of change of energy density

            a = space expansion factor

            a’ = rate of change of space expansion factor

There are conditions when the net pressure is negative,     

            p < –r/3

Then the Robertson-Walker line element and thus the spatial distances diverge. The divergent condition applies when:

            p = ø’ ²/2 – V

 Where:

            V = a potential energy density

            ø = a new real scalar field

Close to a black hole center and in the early phase of the big bang, it is found that V is a slowly varying function of f and the initial value of the time derivative of ø is not too large. Then the kinetic energy ø’ ²/2 is small compared to V, and the pressure is negative, and depends on V. Then space expands under the expansion pressure of V, and the attractive pressure of gravity in the black hole is broken

           

References

1.      L. H. Wald, “AP4.7 DARK MATTER AND ENERGY-FUNDAMENTAL PROBLEMS IN ASTROPHYSICS” www.Aquater2050.com/2015/11/

2.      L. H. Wald, “AP4.7A SUPER PARTICLE CHARACTERISTICS” www.Aquater2050.com/2015/11/

3.      L. H. Wald, “AP4.7B SHAPING THE DARK MATTER CLOUD” www.Aquater2050.com/2015/11/

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

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

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

7.      L. H. Wald, “AP4.7F GATHERING DARK MATTER FOR THE BIG BANG AND ITS IMPACT ON THE CMB” www.Aquater2050.com/2015/07/

8.      L. H. Wald, “AP4.7I THE SUPER PARTICLE AS A COSMIC RAY” www.Aquater2050.com/2015/07/

9.      L. H. Wald, “AP4.7N A NEW CONCEPT OF MASS FOR MODEL 1” www.Aquater2050.com/2016/01/