For updated version—see www.Aquater2050.com/2015/11/
Abstract
In a previous paper (ref 1, AP4.7), a self-consistent theory called Model 1 was developed that appears to answer twelve 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 are observed at energies beyond the GZK cutoff come from?
In working out this model, some problems arose that are connected with it. The most important of the problems are on the detailed characteristics of the dark matter super particle that is the core of Model 1. Those problems were resolved in a companion paper AP4.7A (ref 2). The next most important of the problems have to do with the formation and shaping of the dark matter clouds. Those problems were addressed and resolved in a second companion paper AP4.7B (ref 3). The final important problem of Model 1 is explored and detailed here. 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.
The Problem
Model 1 (ref 1) 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), but were not detailed there to make the paper more 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 Conversion of Super Particles.
Consider the production of super particles S and 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 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.
Then the rate of production of S from P is assumed to be a first order reaction, and so we can write:
Rate = d[P]/d t = – kp[P]
Where:
[P] is the concentration of particles P
[S] is the concentration of super particles S
kp is the rate constant
The integrated rate production equation is:
[P] = [Po]exp(-kpt)
The rate of production of P from S is:
Rate = d[S]/d t = – ks[S]
And the integrated rate production equation is:
[S] = [So]exp(-kst)
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 Fg[r,M], which will be shortened to show only the principle dependence Fg[r].
The dependence of the rate constant kp on V for S production from P can be described as:
kp = Apexp(-Vg(ro) / Fg[r])
Where:
Vg(ro) = force unification potential = S particle excitation potential = 1017 GeV
ro = Black hole radius at which the Gaussian mean is ~ 1017 GeV
Also a quantum of potential energy is taken from particle space to facilitate this reaction. Note that when Fg[r] increases (more excitation potential is available), the rate constant increases toward Ap. When Fg[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 [Po]. Recall that the conversion rate does not go completely to zero because Fg[r] is a Gaussian and has high-energy tails.
The rate constant for [P] production from [S] is:
ks = Asexp(Vg(ro) / Fg[r])
Also a quantum of potential energy is given up into particle space in this process. Note that when Fg[r] increases (more excitation potential is available), the rate constant decreases toward As. When Fg[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.
This expression for k comes from the fact that we can expect that in a Gaussian distribution of potential energy quanta, the number with potential energy greater than Vo will be equal to exp(Vo(ro) / Fg[r]). The coefficient Ap 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. This charge occurs 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 barrier potential shell does not become functional until the super particle gains enough potential and kinetic energy to bounce against the Planck limit and then form and penetrate the barrier into vacuum space.
Rate of Passage into Vacuum Space and Return
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. Note that this again is a rate process in which the super particle state is changed, but the baryon is conserved. The Planck energy is not achieved, so there is not enough kinetic energy to disrupt the baryon into energy. While the black hole is feeding, the particle density builds up in the volume within ro. This process increases the mass and the potential energy of the black hole. Along with the density; the temperature builds up within a limited volume, which is constrained by gravity. Once the potential energy, and the kinetic energy (and thus the temperature) get close to the Planck energy, the shell forms, and the particle has a high probability of penetrating it. Then the rate of passage of the particle through the shell into vacuum space can be described by:
Rate = d[S]/d t = – kvs[S]
Where:
kvs is the rate constant for passage into vacuum space,
And:
kvs = Asexp(-Ea / RT)
Where:
Ea = 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 Ea to be proportional to exp(-Ea / 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 into vacuum space.
Once the super particle energy approaches the Planck energy, the production of the barrier potential shell is automatic, since the charge is already built into the super particle. When the shell is formed, it encloses the vacuum potential that exists at radius ro from the black hole center. The shell has an open space in it with a gap distance a (see Appendix 1, below). This space then captures the vacuum potential that exists in the location of formation near the black hole, namely, 1035 GeV/cc. The shell then forms a barrier wall of value 1019 GeV. This wall then captures a vacuum potential of somewhat lower value ~1033 GeV/cc due to the larger volume within the shell. This is the potential of vacuum space. The vacuum potential is contained by the barrier shell even if the super particle and its shell move away from the vicinity of the black hole, as it will, eventually, to become dark matter.
Once in vacuum space, the super particles expand away from the center of the black hole by virtue of the negative pressure due to the high potential (see step sequence, below). The super particles are ionized and the electric fields force the particles into the cloud of dark matter observed by astronomers (see AP4.7C, Ref 3). With expansion, the super ions cool and will recombine into neutral particles. Thus, with the barrier sphere, the super particles become the slowly moving, low cross section cloud of dark matter cloud that astronomers observe. Note that the super particles of the dark matter have a very low interaction and collision cross section (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.
On average, there will be more than one super particle inside a barrier potential shell radius. Thus when a super particle changes its state into a particle in dynamic equilibrium, the particle will still be in the vacuum space of another particle surrounded by high potential energy, and it can participate in the reverse reaction and produce a super particle again. Remember, a particle must be surrounded by a high potential field to produce a super particle (see the rate constant dependence above). Once the super particle passes beyond the barrier potential wall, and away from the black hole, it will have no high potential field, so the k for P production becomes large, and that for S production goes to zero. Once the super particle density gets low enough so that only one super particle is inside a barrier shell on average, the potential energy quanta begin to leak into particle space, and the super particles lose excitation potential energy and become particles.
Note that passage of super particles back into particle space is controlled by the passage through a barrier potential, which is described in Appendix 1 below. Note also that the activation energy used to move super particles into vacuum space also ionizes the super particles, so the dark matter that exists in vacuum space is an ionized gas. How this gas diffuses and forms in vacuum space is described in AP4.7B (ref 3).
Step Sequence from Particle Space into Vacuum Space and Back.
While in particle space inside ro, particles can become super particles because the potential energy due to the spatial distortion of the black hole is sufficiently high to allow super particle production. However, with the potential energy high, the pressure becomes negative according to the equation (see Appendix 3):
p = f’ 2/2 – V)
Then space expands, pumping super particles away from the black hole center. There, the potential energy is reduced, and the super particles convert into particles, giving up potential energy, with no 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 ro by gravitation. This process is repeated, thus keeping the density of super particles high inside ro.
Gradually, the potential and kinetic energy increases enough through energetic particle addition to a limited volume, to provide the potential and activation energy needed, and the super particles are bounced against the Planck energy into vacuum space inside the vacuum barrier shell, where the collision cross section is small. There, encased in the protective vacuum barrier shell, they are pushed away from the black hole center by the high potential energy (and thus negative pressure), and then form into the cosmic web observed as dark matter (see ref 3, AP4.7B). The volume is larger there, however, so the temperature goes down, and the average kinetic energy is less than the shell potential energy. The super particle ions recombine into neutral super particles. So a new heating process occurs, 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 reaches the shell potential energy. Then super particles flow over the barrier into particle space at a high rate. This flow we call a big bang.
The Rate of Production and Annihilation of Matter and Anti Matter.
Since the kinetic energy has actually reached and even temporarily exceeded the shell potential energy, the super particles have enough kinetic energy (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. Thus, the super particles will not only flow over the barrier, they will disintegrate into energy as they reach particle space according to the uncertainty principle:
δt = h/2πδE.
Now particle space expands. The temperature drops. The kinetic energy drops, and the potential energy in particle space dominates, so pressure turns negative. Space expands more rapidly. The kinetic energy from disrupted particles and the potential energy from space now generate new particles as 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 (Magueijo, 31). 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 particles annihilating each other, but not in equilibrium.
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 (ref 2, AP4.7A). This result can happen only if:
- 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 there. Since there is thermodynamic equilibrium, 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.
The number of energetic super particles behind the barrier drops rapidly as they flow out over the barrier, the most energetic particles first. The dark matter particles rush to the zone of barrier breach (the big bang site) at near infinite speed because of the near infinite light speed according to the equation:
c ~ (3 x 1010 cm/sec) / (1-E/Em )
Where:
Em = Planck energy (see ref 10 Magueijo, 31)
When the kinetic energy of the most energetic super particles 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. The big bang is over. Note that tunneling super particles are not at Planck energy, so they do not disrupt into energy and then form matter and anti matter. The super particles change state to particles through the rate process shown above. Specifically, they turn into extremely high-energy protons. We observe them as cosmic rays with energies between 1017 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 galaxies, the super particles in vacuum space begin slowly to be renewed through 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 (dark energy). The big bang vacuum potential is reduced by expansion and making particles until it is the same as that caused by the tunneling super particle buildup. Then the vacuum energy in particle space increases as the super particles convert to particles and give up their potential. This potential is the dark energy that causes the accelerated expansion of space, which we observe in our time (10-5 GeV/cc).
Testing Model 1C with Data
As with Model 1, Model 1A and Model 1B, Model 1C must be tested with data. The tests that support the composite Model 1 (Model 1 + Model 1A + Model 1B + Model 1C) are shown here. The composite Model 1 does the following.
- It correctly predicts interacting dark matter in and between galaxies, and it shows why the matter is dark.
- It describes a source for this dark matter, namely a black hole. This source does not violate the laws of physics as presently understood.
- It correctly describes the distribution of this dark matter with respect to the galaxies and shows why this distribution happens.
- It correctly predicts the observed fact that when galaxies collide, the visible matter interacts and coalesces while the dark matter passes right through the visible and dark matter.
- It details the characteristics of the dark matter particles to within our ability to measure them.
- It connects with a property predicted by the standard model of particle physics-namely the unification of forces, and predicts the impact of this unification.
- It provides an explanation for the difference in vacuum potential energies as measured (low) and as predicted (high).
- It provides a physically defensible procedure for describing what happens in a black hole.
- It describes the cause of the Big Bang, what triggers it, what stops it and where the energy causing it comes from. All of this description is in keeping with the data currently available.
- It correctly describes the immediate aftermath of the big bang; how early thermodynamic contact is maintained, why the expansion, why it stops, and where the energy causing it comes from.
- It correctly describes the later aftermath of the big bang; where the matter and anti matter came from, why we have cosmic background radiation and where our excess of matter over anti matter came from.
- It describes how quantum mechanical state details can be transmitted faster than the speed of light if coherence is maintained. Recent experiments have shown that this happens.
- It correctly predicts extremely high-energy cosmic rays beyond the GZK cutoff, and describes where they come from. These cosmic rays have been observed.
- It correctly predicts dark energy that accelerates the expansion of space, and describes where it came from, and the value of the dark energy. The accelerated expansion of space has been observed.
- It predicts the existence of a future new big bang, and estimates when it will happen.
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. The details of these super particles were explored in AP4.7A (ref 2). Next, the details of the shaping of these super particles into dark matter clouds were explored in AP4.7B (ref 3). Finally, 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 in this paper. Initial checks with existing data have been made, and Composite Model 1 (Model 1 + Model 1A + Model 1B) 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.
Appendix 1
The shell is described as follows (see also ref 1, AP4.7, Appendix 1):
[-(h2/2m d2/dr2 + V(r)]U(r) = EU(r)
Where
E = the energy of the particle.
V(r) = Vo[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.
ro = the radius of the barrier potential shell.
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 = 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 AP4.7 (ref 1, Appendix 1).
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:
s = h2π/M x 1/(Vo + E)
Where:
M = ms mp/ (ms + mp)
mp = mass of particle space baryons = 1 GeV.
ms = mass of super baryons = 1017 GeV.
Vo = potential of super baryons = 1019 GeV
E = kinetic energy of the particle space baryons = 1 GeV or less.
Then: s = 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,
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
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 = f’ 2/2 – V
Where:
V = a potential energy density
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. Then the particles expand 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 FUNDAMENTAL PROBLEMS IN ASTROPHYSICS” www.Aquater2050.com/2015/07/
2. L. H. Wald, “AP4.7A SUPER PARTICLE CHARACTERISTICS” www.Aquater2050.com/2015/07/
3. L. H. Wald, “AP4.7B SHAPING THE DARK MATTER CLOUD” www.Aquater2050.com/2015/07
5. Misner, Thorne, and Wheeler, Gravitation, New York, Freeman and Co., 1973.
6. J. Magueijo, New Varying Speed of Light Theories, arxiv.org/pdf/astro-ph/030545v3.pdf