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

Abstract

There are currently twelve connected major unanswered questions in astrophysics. The most important of these 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?

A self-consistent theory called Model 1 has been developed that answers these questions quantitatively. This model will be described and defended with data from astronomers in this paper.

 

 

The Most Significant Current Questions in Astrophysics

There are twelve connected major unanswered questions in astrophysics.

1. What is dark matter and where does it come from?
2. Are particles and forces unified?
3. Why is there such a huge disparity between different estimates of vacuum energy (the vacuum catastrophe)?
4. Can the distribution of dark matter into bubbles and lattices that are connected to galaxies be explained?
5. Does the speed of light change with energy? Specifically, explain the data that show that the fine structure constant changes as it travels across the universe.
6. Is it possible to explain what happened before the big bang and what initiated it and what happened afterward?
7. Is it possible to explain the accelerated expansion of space and dark energy?
8. Is it possible to project what will happen to the universe into the future?
9. Is it possible to explain the large-scale cutoff and asymmetry in the Microwave Background Energy?
10. Can the experiments that show an instantaneous transfer of state at long distances in conflict with the speed of light maximum on the transfer of information be explained?
11. Is there a practical procedure for obtaining answers on questions that arise in the area of quantum gravity-especially for questions involving black holes?
12. The above questions must be explained using a very few fixed constants (such as h) and parameters that work for all the questions. Having all the above questions correctly answered by a few fixed parameters and constants would constitute proof of this theory. However, an even more convincing proof of the validity of Model 1 would be to predict and observe an operating component (a particle) of Model 1. This prediction should be a final goal of the proof of validity and is included in this paper.

 

Model 1 described below addresses all of these questions and answers them with apparent success. Note that prior experimental and theoretical results that are satisfied by Model 1 are given in Appendix 7. The limitations of Model 1 will also be described in this paper.

 

Model 1. A Basic Self Consistent Physical Model Aimed at Addressing the Most Significant Current Questions in Astrophysics

The unique features of this model are:

• There are two spaces (particle space (our space) and quantum vacuum space) in the universe separated by a potential barrier.

• One space contains visible matter, and the other space contains dark matter.

• Coherence requirements form the boundary conditions for passage of particles through the potential barrier between the two spaces with certain interesting consequences.

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

• Dark matter particles interact with each other and form a slowly building and moving bubble centered on galaxies. The bubbles of dark matter connect with each other in corridors of dark matter forming a lattice on which other new galaxies are formed.

• There, behind a potential barrier they gain energy, build up in number and eventually exceed the ability of the high-energy barrier to contain them. They then explode back into our space as a big bang. This process recycles in a repetitive process.

• The high-energy vacuum space leaks super particles that tunnel through the barrier into our space. The super particles are unstable and break down into particles with extreme kinetic energy. In doing so, they give up potential energy. The potential energy gradually builds up in particle space to become the dark energy that we observe as the cause of our accelerating, expanding universe. The extreme energy particles are observed as cosmic rays with energy beyond the GZK cutoff.

 

Model 1 Characteristics

The primary characteristics of Model 1 are given in this section. The details, equations and calculations for Model 1 are given in the 7 Appendices.

1. There are several particles in two spaces of the universe that carry the energy. The structure of the universe also carries energy. A potential barrier separates the two spaces (see Appendix 1). One space will be called Quantum Vacuum space (vacuum space for short), and the other will be called particle space hereafter. We live in particle space. The structure that carries energy is the structure of the universe as described in the theory of General Relativity (Thorne). The primary entities in the two spaces that carry energy are wave packets or photons and particles such as electrons and baryons (combined quarks). In vacuum space, the energy density is extremely high, and the photons, electrons and baryons have a high energy form that will be called super photons, super electrons and super baryons hereafter. The potential barrier consists of a spherical shell that surrounds the super particles (see Appendix 1). In particle space, photons, electrons and baryons have a low energy form that will be called photons, electrons and baryons hereafter. Four forces act on these entities in particle space. Super particles do not show charges associated with the electromagnetic, weak, and strong forces. The charges are combined into one super charge and hidden behind the potential barrier. The super particle spin, if any, would not show beyond the barrier either. Super particles only have the super charge associated with the unified force. Thus they will not interact with the detectors we normally use. It is important to point out that super particles travel in particle space behind their barrier, which they carry with them, as if particle space was vacuum space. The super particles have blocking potentials just like particle space gauge potentials that nullify the barrier potential of other super particles and allow the super particle to maintain the continuity of its local phase as it moves, so the local phase is conserved. Then as a result of Noether’s theorem, super particles have a super charge.  Thus a super particle and a particle see the barrier between them, but super particles pass through each other’s barriers as if they are not there and allow the super particles to interact directly. A particle, however, sees the barrier potential of a super particle as a smooth spherical shell, and can scatter off of it. This scattering cross section is like billiard ball scattering such as that of a proton off a neutron, but with different energies. The scattering cross section has been calculated, and the value obtained (s = 10^-45 cm² – see Appendix 6) is very small, so super particles would be extremely difficult to detect in particle space. So we call super particles dark matter. But the particles do see each other through the gravitational force.
2. In vacuum space, the four forces become unified in the high energy (both kinetic and potential) that exists there. This force acts on the super electrons and super baryons through super photons and so it will be called the super force. This unification has been predicted for a long time (Kane, 281) The energy of unification is ~ 10¹⁷ GeV as determined by extrapolating the four separate forces. The potential barrier is a spherical shell of radius r_o and thickness a around the super baryon, and it has a value of ~ 10¹⁷ GeV (see Appendix 1). Note that super particles can become ionized-an important point as will be seen later, The structure of particle space is expanding, and is shown by the expansion rate equation (see Appendix 2), which has three terms, the density term, the curvature term and the cosmological constant term. Each term carries energy. The density term carries energy in the moving particles. The curvature term carries it in the pressure. The cosmological constant carries its own potential energy. It has been calculated to be ~ 10^-5 GeV/cc from the acceleration of the expansion of the universe. As a particle descends into a black hole, it gains both kinetic energy (due to increased particle velocity) and potential energy (due to increased curvature) until the potential energy provides the expansion necessary to break the grip of the gravitational attraction (see Appendix 5). Then, the potential energy is used to make super particles, and the expansion stops and we are left with high-energy dark matter in vacuum space.
3. The flow of energy (and particles) in these spaces is as follows. Particles from particle space fall into black holes, and the kinetic energy increases because the particles are falling. The particles are moving into a smaller volume as well, so the energy per particle and energy density increases dramatically (see Appendix 5). The particles collide and reach equilibrium and so have an average energy and a temperature. At the same time, the gravitational potential energy increases due to the increasing curvature of space. As the particle potential and kinetic energy approach the unification energy (10¹⁷ GeV), the particles can convert to super particles by increasing their potential energy by 10¹⁷ GeV. This potential energy comes from the gravitational potential energy of the black hole (primarily the central black hole of the galaxy). The super particles then generate the spherical potential barrier, which has a strength 10¹⁹ GeV, The potential energy density of vacuum space is the potential energy density behind the potential energy barrier shell. Thus the vacuum potential energy density of vacuum space is 10³⁵ GeV/cc because the potential energy is high (10¹⁹ GeV) and the volume of the shell in which it operates is small (10^-16 cc). This potential energy density is large compared to the vacuum potential of particle space (~ 10^-5 GeV/cc) because it is in a separate small space, and shielded from particle space by the barrier potential Part of the problem of seeing this difference as a catastrophe, is that there seems to be no clear agreement on the definition of the vacuum potential (see Appendix 6-Dark Energy). Here, the definition is clear, and the difference in values is explained. The vacuum potential given here is not the same as the vacuum potential given in other sources, however as shown in Appendix 6. 
4. According to Smolin (Smolin, 250) the particles that near the Planck energy bounce, and enter a different space (vacuum space). Note that the particles in vacuum space are super particles in order to remain stable. The super particles (including super photons), at extreme energy, are ionized, and interact with each other through an electromagnetic super force using exchange super photons. The super particles also feel other forces, namely, the gravitational force, the centrifugal force and the kinetic energy pressure of a gas at high temperature (see Appendix 4). Thus the super particles, as they enter vacuum space, first experience an expansion that breaks the grip of gravity due to the gravitational potential. Then, as the gravitational potential is used up making particles, the gravity starts to reassert itself, and the centrifugal, kinetic energy and electromagnetic forces become important (see Appendix 4). These forces allow us to lay out the equations of motion (see Appendix 5) for dark matter. The solution for the resulting equations show a dark matter bubble centered on the galaxy’s central black hole and connecting corridors of dark matter between the bubbles. They diffuse away from the black holes primarily in the direction of other black holes. The dispersion pressure, electromagnetic force, gravitational force and centrifugal force interact, and slow the particles down, so they do not keep spreading and thinning too rapidly. Thus they form a net like structure not visible in particle space that is a nucleation zone for galaxies to coalesce on through gravity. So particles tend to form galaxies around this structure in groups, strings and walls. The dark matter diffusing from the central black hole of a galaxy also forms a bump or bubble in mass centered on the central black hole, which tends to control the amount of mass gathered into a galaxy centered on that black hole. Note that gravity operates in both spaces, but the super particle force (other than the gravity portion) does not operate in particle space because of the barrier. This fits generally with the data on dark matter obtained by observing galaxies.  
5. Note that in order to have the Diffusion coefficient needed for such huge bubbles as are observed with galaxies, the particles involved must travel faster than 3 x 10¹⁰ cm/sec. This fits well with the experimental data on the varying fine structure constant, and the calculations that predict higher light speed at higher energies in order to keep the Planck length constant (see Appendix 3).
6. As the very energetic super baryons build up in vacuum space, they increase the average kinetic energy (and thus temperature) in vacuum space until most of the particles approach the Planck energy. Thus the kinetic energy of the particles approaches the potential energy of the vacuum barrier, and the transmission probability goes to unity (see Appendix 1).  When this happens, the super baryons flow rapidly through the barrier into particle space where the kinetic energy and density is low. This is the Big Bang. There, the super baryons and super electrons expand rapidly under the influence of the gravitational potential (see Appendix 4), and start to initiate a series of phase changes. The expansion potential energy comes from the phase change energy of the super particles (10¹⁷ GeV/super particle). This is enough potential energy to generate the rapid initial expansion of the universe. The kinetic energy is initially high, so the speed of light is high, and the universe is in thermal contact. Then the potential energy is converted into particles over ~ 5 billion years, and the expansion of the universe and light speed slows down.  The order of appearance of the particles appears to be the order of particle energy-i.e. Higgs, quarks, and then electrons. Each carries its appropriate force mediating boson. They appear in matter and anti matter forms, and annihilate to form photons with a slight excess of matter, which is the matter we see. These particles bring with them the charges and potentials that generate the four forces that operate at the lower energy of particle space. At the same time, the matter and anti matter particles are annihilating leaving a slight excess of matter. Thus there is roughly one baryon excess for 10¹⁰ photons in particle space. This result can happen only if:

–         Conservation of baryon number is violated. The recycling of energy in the universe through black holes in this model must violate conservation of baryon number. (See below)

–         Charge-parity (CP) is violated. (See below)

–         Particle space is not in thermodynamic equilibrium while the above conditions are satisfied.

For the conservation of baryon number violation, the matter entering the black holes has an excess of matter over anti matter. The super particles that come out of a big bang create equal numbers of particles and anti particles, so there must be a process in vacuum space that eliminates the excess matter to generate the equality of matter and anti matter that exits vacuum space. Thus baryon number is violated somewhere in vacuum space or the initial cooling phase. Note that this issue brings up a problem in vacuum space. Can super particles have matter and anti matter forms? See the problem section below for further analysis of this problem.

For the CP violation, it was originally thought that although C and P were separately not conserved by weak interactions, the combined CP would be a valid symmetry. However, it was found that although the processes and their conjugates that produce baryons occur, the probabilities of occurrence are slightly different-about one part in one thousand. This is the CP violation, and it has been found by experiment to be valid.

For Model 1, the particles and antiparticles annihilate each other as they are cooled to particle space temperatures. Light speed is reduced to c very rapidly in particle space and so equilibrium is lost just as rapidly. Thus this theory allows for an excess of baryons over anti baryons. 

Note here that the Big Bang flow ends when the vacuum space kinetic energy falls below the barrier potential. This cut off happens over a span of time because the super particles are in a Gaussian distribution, and so the particles will not all reach the barrier potential at the same time.

7. The residual super particles behind the barrier leak super particles even after the big bang is over. These tunnel through the barrier at a low rate (see Appendix 1). When these particles reach particle space, they break down into ordinary particles and give up their potential energy into an increasing particle space vacuum potential. Eventually, this increasing potential passes the decreasing big bang potential (at about 5 billion years from the big bang) and increases to the value we observe now (10^-5 GeV/cc). This is the potential that in our time we call dark energy, which causes the accelerated expansion we observe (See Appendix 4). Note that as a result, the acceleration we observe should go through a minimum and then start gradually increasing.
8. The resultant expansion thins out Particle space. Meanwhile, black holes in galaxies are re-filling vacuum space from particle space with hot super particles, and forcing the kinetic and potential energy in vacuum space up toward the barrier potential limit of the Planck energy. In a dense group of galaxies where the density and size of black holes generate dark matter, this energy is especially high. Thus, the kinetic energy of Vacuum space will eventually exceed that of the barrier potential near one or more of these galaxies, and a new Big Bang will start. Diffusion in vacuum space will feed vacuum space particles very rapidly to the Big Bang start point until the super particles in a very large area are depleted. Thus the life cycle of the universe is complete. A calculation assuming the rate constant for refilling vacuum space does not change beyond now, gives the time of the next big bang to be ~ 10¹² years from now (Appendix 5). This calculation is very rough and requires many assumptions, however.
9. There is a reduction in the spatial spectrum of the microwave background energy of the universe at a distance R (R~10²⁷ cm), which is close to the radius of the visible universe. This reduction is coincident with the edge of the dark matter bubble that marks the edge of the visible universe. This universe bubble consists of the dark matter bubbles of each galaxy along with the lattice of dark matter corridors that stretch between them. This bubble provides a gravitational edge for our local portion of the universe and so provides an edge to the background expansion zone, and also the microwave background.
10. There is a set of experiments that show an instantaneous transfer of state information over distances too large to allow for transfer of information at the speed of light. In order for this transfer to take place, the particles involved must remain coherent. This can be explained in terms of transfer of state through the barrier potential. In the development of the barrier transfer equations, a boundary condition for the transfer was that the particle wave function be continuous along with its derivative. This is equivalent to saying that particles being transferred must remain coherent. If coherence is conserved in both spaces, there must be a charge common to both spaces from Noether’s theorem. This means that there is a fundamental connection between the super charge of vacuum space and the separate charges of particle space. This conclusion was hinted at by the convergence of the four forces at ~ 10¹⁷ GeV. Thus if a particle in particle space can maintain its coherence to separate particles with connected states, it can be connected as a photon through the barrier to vacuum space, and then move as a super photon in vacuum space at a very high speed because of the effect of the high energy of vacuum space on the speed of light there. Thus, the photon moves to a location near the second particle in particle space where it passes through the barrier to the particle in particle space again. Since the super particles communicate faster than 3 x 10¹⁰ cm/sec because of the high energy there, the state in particle space can be transferred indirectly faster than 3 x 10¹⁰ cm/sec by this method. Note that neither a massive particle nor a photon has to pass through the barrier in order to carry this information, so the vacuum potential energy difference does not have to be paid in this process.
11. In doing this work, 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 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, and for General Relativity as well. It predicts gravitons at low energy. Especially, it predicts a Newtonian type gravitational force. 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 difficult environment.
12. Clearly, this model must be tested with data to establish its correctness. This turns out to be quite feasible. The first level test is for the general reasonableness of the results and the consistency with the known parameters of the universe (see Appendices 1 and 5). First, the known four-force merge point (~ 10¹⁷ GeV) was used as the potential needed to form a super particle, which is the constituent of dark matter. Then, the energy gained by a particle in falling into a galactic center black hole with ~ 10⁶ suns mass (reasonable for such black holes) was found to be enough to provide super particle conversion energy.  Then ~ 10¹⁹ GeV along with r_o = 10^-5 cm, and a = 10^-7 cm, for the barrier shell parameters was found to give reasonable answers for two key barrier passage cases. For E = 10¹⁷ GeV the leakage of particles and vacuum potential through the barrier due to tunneling gives a vacuum potential (or dark energy) for particle space equal to that observed as causing the accelerated expansion of particle space (10^-5 GeV/cc). In addition, it predicts the existence of extremely high cosmic ray events (up to ~ 10²⁸ EV), which are beyond the GZK cutoff. Such events are difficult to explain except with the leakage events described here (see ref 10 pg 33). For E = 10¹⁹ GeV, the rate of passage over the barrier is enough to provide the expansion energy and particles needed to explain the initial expansion and time of operation of the big bang. The potential energy provided is enough for the expansion needed and the time of operation is short enough to be reasonable and still provide the particles we observe. Further, a diffusion coefficient of ~10¹⁷ cm²/sec will provide the dark matter bubble found (by observing galactic motion) around each major galaxy. This will require a light speed ~ 10¹² cm/sec and a super particle scattering size of ~ 10^-5 cm which is consistent with the barrier size. The light speed is consistent with the increased light speed at high energy required to ensure constant Planck length (see ref 10). This model predicts a new big bang and allows for calculating the time before the new big bang. The estimate is ~ 10¹² yrs, which is reasonable. Thus first estimates are self-consistent and match our current data on the universe. The next steps to confirm Model 1 are given below in the section Further Proof of Model 1. 

 

 

Problems with Model 1

Certain problems with Model 1 were discovered in the course of this work. Those problems will be discussed here.

• Matter and Anti-matter. As noted in section 6, the particles start in the black hole as matter. They end after the big bang, as equal numbers of matter and anti matter particles. They then annihilate each other and reduce down to matter particles again. There are two ways this can happen. Either would be compatible with the theory.

O The matter remains matter after conversion to super particles. Thus super particles are matter. With this process, baryons are conserved. When the super particles flow over the top of the barrier, they convert completely to energy for a very short time as they reach particle space according to the uncertainty principle, and then come back into particle space as particles-both in matter and anti matter forms. Here, baryons are not conserved. From there, they convert to photons with a remainder of matter using CP violation. 

O The matter is converted directly to super particles, which combine the matter and anti matter into one form. It can still be ionized, so it has the equivalent of quarks and electrons, but no anti quarks and anti electrons. It would be a new form of matter, and baryons (combined quarks) would not be conserved. Then, when they flow over the top of the barrier and cool, they break down into anti matter and matter forms of the quarks and electrons. Afterward, they would convert to photons, and generate matter using CP violation as before.

Note tunneling particles should still give off an extremely energetic proton as the particle breaks down, but there may be a difference in the proton that would betray which type of reaction happened. The first procedure seems to be the most likely method, because it allows both matter and anti-matter particles at all times.

• The results given here appear to explain the existence of dark matter and dark energy, and how they are formed and shaped, and why galaxies form the way they do. The results also explain the very high-energy protons observed from cosmic rays. However, the results were put together from several theories that are defendable by themselves, but were connected in a somewhat ad hock fashion rather than with a seamless, single theory as one would like. It is possible that problems can arise in connecting them together that cause errors. Although Model 1 currently fits the average data, there is still a possibility for hidden problems. There are three ways to overcome this issue.

O Work out a seamlessly connected theory. This is difficult since we are working in the area of quantum gravity where progress is only made in small pieces.

O Check the details for several different galaxies with data supplied by astronomers. At the present time, this appears to be the most useful.

O Obtain more data on the high-energy protons. The details may betray their origins.

The most important of these problems have been explored and resolved in two companion papers-AP4.7A and AP4.7B.

 

Further Proof of Model 1

Theoretical Proof

Certain theoretical aspects need further development before Model 1 can be considered complete, and would add support to this model. These are:

• Work on a seamlessly connected theory that overcomes the arbitrariness used here in putting the components of this theory together.
• Work out a theory of extremely high proton generation that results from the exit of particles tunneling through the barrier.
• Work out the details of the theory of the super particle including its gauge nature.

 

Experimental Proof

Although Model 1 appears to satisfy all of the previous experimental results as shown above, it should also predict and satisfy one or more unique experimental results listed below.

• Several different detailed scenarios for real galaxies and dark matter bubbles (not the average ones used in this paper) should be worked out and compared. If they agree using the same diffusion coefficients, particle and barrier sizes and super photon speeds and ion and electron charges, that will be a success, and provide major support for Model 1.
• A detailed calculation of time to next big bang including the diffusion step for several galaxies should be made. If the results are compatible, and yield a significant amount of matter in the new universe, that would be a success, and provide support for Model 1.
• Even though the scattering cross section is extremely low, it may be possible to detect the super particle behind its barrier with billiard ball type collisions. An attempt should be made, if possible.
• If possible, a state transfer experiment that operates as near to across the universe as possible should be tried. This experiment may show if there is a limit to the state transfer speed, which would provide support for Model 1.

• It is important to search for and categorize extremely high-energy gamma ray events that can be connected to the particles tunneling through the barrier. It is known that such events exist (see Ref 10-Magueijo, 32). These extremely high-energy events beyond the GZK cutoff are difficult to explain except with the leakage events described here. Leakage events should go clear to 10¹⁹ GeV (10²⁸ EV) and then cut off. Please see the graph of these events in reference 10 on pg 33.

 

Summary and Conclusions

A model has been developed that predicts dark matter and energy and the extremely high-energy protons, which operate beyond the GZK cutoff. Initial order of magnitude checks with existing data have been made, and the model is found to be in agreement with the data. Possible problems with the model have been analyzed. The most important of the problems have been analyzed and resolved in companion papers AP4.7A and AP4.7B. Experiments that would check the accuracy of the model have been proposed. The model has been found to be valid as far as the current checks can determine. In addition, it agrees with all of the prior experimental and theoretical results listed in Appendix 7. A summary of the results that support this model is given here.

• It correctly predicts interacting dark matter in and around a galaxy, 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 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 defendable procedure for describing what happens in a black hole other than “a singularity forms”.
• 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.

 

 

Appendix 1. The Basic Equation

Consider the following non-relativistic three-dimensional time independent Schrödinger equation.

 

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

Where:

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

             a = the barrier potential width.

 

Note that if any solution to the equation is unchanged if the step function is moved along the r axis to r_o.  Then one can think of starting at 0 and moving in vacuum space to r_o, then moving through the barrier potential for a distance a, and then for x>a, we move in particle space. Thus the equation governs passage from vacuum space through the barrier into particle space, and back. For convenience, we will let r_o = 0for solving the equation. Note also, that what we are describing is a spherical shell of radius r_o and thickness a around a super particle.

 

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.

 

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(k₁a)/4E(E-V₀)

           

            If E<V,

 

            T = 1/(1+V_o² sinh²(k₁a)/4E(V₀-E),

 

Where k₁= (8p²m(V₀-E)/h²)) ^1/2

 

Here we have set up the equations for a super particle with spherical vacuum barrier of radius r_o and thickness a. Inside the radius is vacuum space. Outside the radius is the barrier shell of thickness a, and outside that is particle space. The rate of passage of a particle through the spherical vacuum barrier is:

 

             R = T h/2p k₁

 

The best fit of the very rough data available to the writer is:

 

             r_o = 10^-5 cm

             a =  10^-7 cm

 

There are two important cases.

 

            Case 1

            E = 10¹⁷ GeV

            V₀ = 10¹⁹ GeV

            N = number of super particles in vacuum space = 10⁶⁰ super particles

 

            In this case, T = 10^-50, and R = 10⁷⁴ GeV/cm² sec. This is the case of leakage of super particles for the whole universe by tunneling through the barrier. 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^-5 GeV/cc. This is the same dark energy that causes the accelerated expansion of space that requires a potential energy of ~ 10^-5 GeV/cc (see Appendix 4). Note that the number of particles N that has built up in a thermal equilibrium at energy E and remained there, did so because the gravitational energy of contraction in the black hole was used up in converting particles into super particles at this cross over energy for making them.

 

            Case 2

            E = 10¹⁹ GeV

            V₀ = 10¹⁹ GeV

 

In this case, T = 1, and R = 10¹²⁴ GeV/cm² sec. This is the case of big bang passage over the barrier. 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 inflation energy, which causes inflation. The amount needed is estimated to be ~10⁹⁴ GeV/cc (see ref 9), where the volume of space at the beginning of inflation is estimated to be (for one super particle barrier shell) 10^-14 cc. Thus the inflation potential energy required is ~ 10⁸⁰ GeV. The phase change energy (~10¹⁷ GeV/particle) from the ~10⁶⁰ particles in particle space including the dark matter is ~ 10⁷⁷ GeV. The discrepancy seen is not surprising considering the roughness of the calculation, and the fact that the theories used to obtain the estimates have not been matched for assumptions and conditions.

 

Appendix 2. The Expansion Rate of the Universe

The expansion rate equation of the universe (Peebles, 76) is,

 

            H² = (a’/a)² = 8/3 p Gr_b+ 1/ a²R²+L/3

  Where:

            H = Hubble’s constant

            R = Radius of curvature

            G = Gravitational constant

            a = Expansion parameter

            r = Density

            a’ = Rate of change

 

Appendix 3. The Speed of Light at Extreme Energy

In order to make constant speed of light and constant Planck length compatible, Amelino-Camelia (Amelino-Camelia, 6) develops a modified dispersion relation for photons as follows.

 

            E² ~   c²p² + L_p c E p²  

  Where:

            E= energy

            P= momentum

            L_p= Planck length

            c= velocity of light in vacuum

               

This corresponds to a deformed speed of light law:

 

            v_g (p) = c (1+L_p ! p! / 2)

 

This law is approximately valid when L_p <1.

For higher energies, the following can be used:

            c ~ (3 x 10¹⁰ cm/sec) / (1-E/E_m ) 

Where:

            E_m = Planck energy

(see ref 10 Magueijo, 31, where some assumptions have been made)

Note that when the energy increases, the speed of light increases. Magueijo (Magueijo, 251) describes it thus, “It was as if the speed of light became larger and larger as we approached the border between classical and quantum gravity. At the border, the speed of light seemed to become infinite and absolute space and time could be recovered, not in general, but for one specific length and time- L_p, and t_p …”.

 

Appendix 4. Potential Energy and Expansion Pressure

Particles around and in a black hole operate under the influence of four forces.

• A central force operating toward the center of the black hole due to gravitational attraction or repulsion.
• A central force operating away from the center of the black hole due to the centrifugal force of the particle’s rotation around the black hole’s center.
• A dispersive force due to the motions and collisions of particles due to their high temperature (kinetic energy pressure).
• An electromagnetic force pulling the components of an ionized particle cloud together due to the different diffusion rates of different particles (positive and negative particles} because of different diffusion coefficients.

 

As a particle falls into a black hole, gravitational attraction first controls. Loop quantum gravity has established the existence of this attraction in background independent terms. When the particle nears Planck energy, it bounces and expands into a new space, where gravitational repulsion takes over and the attraction of the black hole center can finally be balanced by a new set of forces as follows.

The gravitational force. The 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.

 

Then Model 1 shows that V is decaying with time into super particles. Also the kinetic energy is increasing due to the addition of new energetic particles through the black hole. Eventually, the kinetic energy term exceeds the potential energy, and the gravitational force turns attractive, balanced by diffusion and electromagnetic forces and tempered by the centrifugal force. The effect of these forces on the particle ion cloud is explored in Appendix 5.

 

At the same time, according to Model 1, super particles are tunneling slowly through the vacuum barrier between vacuum space and particle space and then breaking down into low energy particles, and thus adding phase change potential energy to the vacuum potential V in particle space. This is a potential energy rate that increases with time along with the number and size of black holes. Recall that the potential energy from the big bang is decreasing as the vacuum energy is used up to make particles since there is no input from over the barrier because the big bang has stopped. Thus, the potential energy density reached a minimum (at about 5 billion years ago according to the data), as the two rates become equal, and then begin to increase. This increase continues until it reaches the value that exists at this epoch (10^-5 GeV/cc). At the same time, particles (protons) are being added to particle space with extremely high kinetic energy (up to 10¹⁹ GeV). They are entering distributed in space, but preferentially in the vicinity of  galaxies, where the dark matter is in highest concentration. They are distributed in energy because the vacuum space dark matter is a Gaussian energy distribution of particles.

Appendix 5. Energy Density Estimates

The Model 1 description of the ion cloud unfolds as follows.

1.  An object, or a cloud of particles that enters a black hole trades gravitational potential energy for kinetic energy. At the same time the potential energy increases due to increasing curvature of space. The objects are torn apart into particles by shear forces in the black hole, and as they gain kinetic energy, they come into thermal equilibrium with other particles, and therefore have an increasing temperature. When the particles gain enough energy (on average), to be beyond the energy of unification of all four forces (~ 10¹⁷ GeV-see Kane, 281); but before they reach the Planck energy limit (1.22 x 10¹⁹ GeV), and the potential energy is also beyond the unification limit, they do the following. The particle ion cloud uses gravitational potential energy to expand (see Appendix 4), then the particles start to convert into super particles using up that same potential, so the expansion slows and the particles form into an ionized cloud of super particles under the influence of gravity, centrifugal force, kinetic energy pressure and electromagnetic force (see Appendix 4 above). Once the super particles form, they generate the unified force and the spherical potential energy barrier (see Appendix 1 above) that isolates Particle space from Vacuum space. The super particles will then operate between 10¹⁷ and 10¹⁹ GeV and be ionized at this temperature. As an example, a particles that descend to ~ 10^-6 cm from the center of a 10⁶ sun black hole will gain enough kinetic and potential energy to generate a super particle.
2. After the super particles enter Vacuum space and expand away from the black hole, they become dark matter to Particle space (our space). In Vacuum space, this cloud of particles becomes an ionized gas cloud having both super baryons and super electrons as well as super photons. The temperature is much too high for the super baryons and super electrons to combine into super atoms, but it is assumed that the super force is strong enough so that the super baryons will not break down into super quarks. It is above the ~10 ¹⁷ GeV threshold that makes the super force possible. It thus appears possible to analyze this ionized gas with the tools developed for ionized gasses in Particle space (Cobine, 50). The method given here is different because of the extra forces, but closely related to the method given there. We begin with the diffusion velocities of the ionized gaseous components.

 

            V⁺ = -D⁺/n⁺ dn⁺/dx + K⁺E + K⁺G

 

             V^– =-D^–/n^– dn^–/dx – K^–E + K^–G

 

             V^g = K^gG

 

             Where:

                    D = Diffusion coefficient

                    K = Ion or gravitational mobility 

                    V = Ion velocity

                    n = Ion concentration

                    E = Electric field

                    G = Gravitational field tempered by centrifugal force (see Appendix 4)

                    n^g = source of particles from the black hole

           

            We set:

 

                        V+ = V- = V^g = V,  n⁺ = n ^– = n^g = n,  dn⁺/dx = dn^–/dx = dn/dx

 

            We solve this equation, and get:

 

                        n= (N_o/4pDt)^3/2 exp(–r²/4Dt)  

 

            Where:

                        D = (D⁺ K^– + D^– K⁺ ) / ( K⁺ + K^– –  2 K⁺ K^– / K^g ) 

                                    r= Radius from black hole source

                        t= Time

                        N_o = particles diffusing from an “instantaneous”  point source  

           

            From basic kinetic theory we find:

 

                        D = Lc/3

 

            Where:

                        L = 1/npd² and d is the diameter of the shielded  super ion.

                        And c = the average velocity = 1.128c_o

                        And c_o ~ c or less (c_o is limited by the speed of light).

                        And c ~ (3 x 10¹⁰ cm/sec) / (1-E/E_m)

                        And E_m = maximum energy = Planck energy

 

            Note also that:

                        D/K  = kT/e and,

                       

                        E = 3kT/2

           

            Where:

                        T = Temperature of the plasma.

                        E = Energy of the ions.     

 

            We note from the above equations that:

a) The solution is the equation for a peaked function (bubble) that centers on the black hole in the galaxy, and sags with time just as the dark matter appears to do. Note that n falls off rapidly for r² > 4Dt.

b) The shape of the function is normally spherical, but can have tails if D Which is proportional to K is different in different directions. This is especially important for K^g (gravitational mobility). Any local increase in n in a radial direction will make a decrease in D, which will start to build up n in that corridor. Thus there is a tendency for lattices of dense dark matter corridors to build between bubbles.

c) In order to make the center dense part of the bubble have a diameter of the order of a galactic core (~10²⁰ cm for example), D must be large (~10²⁸ cm²/sec), and t must be large, (~10¹² sec, ~ the lifetime of a galaxy). In order to make D this large, d must be ~10^-5 cm, and c must be ~ 1 x 10¹² cm/sec and n is estimated as ~ 10^-6 particles/cc. This d is the diameter of the potential barrier shell, so collisions with the barrier shell apparently dominate diffusion. This value for c is clearly greater than the standard light speed as proposed by Amelino-Camelia, and Magueijo (see esp. Ref 10, Magueijo, 31). To quote from (Ref 11, Magueijo, 251), “It was as if the speed of light became larger and larger as we approached the border between classical and quantum gravity. At this border, the speed of light seemed to become infinite, and absolute space and time could be recovered, not in general, but for one specific length and time-L_p and t_p-so that everyone could agree on what belonged to classical and to quantum gravity.”  

d) Note that this super fast light throughout the bubble is also required to maintain the conditions for the validity of kinetic theory that has been assumed to make the equations above. Certainly the energy of the particles is high enough since we are operating near the Planck energy.

e) Note that N_o is an impulse source in the above equation, and acts like the black hole fed for a time short compared to the life of the galaxy and then stopped. Later, it started again. It is the average of a series of these feeding impulses that is convenient to use to calculate the characteristics of the dark matter bubble. This, of course, is how the black holes have been observed to operate. This impulse behavior can also explain how donut shaped bubbles of dark matter can form.  

f) Eventually, these super particles escape into intergalactic space to build up a uniform base of super particles in the universe.

g) Note that this explanation is only order of magnitude based on average galactic characteristics. More accurate calculations based on detailed dark matter data are necessary to confirm the model.  

3. The equations of Appendices 1, 4 and 5 above make it possible to estimate the time between big bangs. As mentioned above, the leakage of super particles into intergalactic space is a leakage into the base of super particles in Vacuum space. This base has the galactic bubble of super particles superimposed on it. A galactic bubble is expected to reach the spillover point first, so galactic parameters will be used for this calculation. At the present time, an average galaxy is estimated to have the value ~10⁶⁰ GeV which was built up over ~10¹⁰ years (the lifetime of our universe). The rate of buildup started at 0 because there were no black holes. As the number of black holes increased, the rate increased in vacuum space according to the formula

 

             R_vt_i² 

 

Where:

             R_v = vacuum space rate constant = 10¹⁷ GeV/sec²

              t_i  = time fromthe big bang to now = 10¹⁰ years= 3 x 10¹⁷ sec

 

Now the vacuum potential energy barrier has an estimated value of 10¹⁹ GeV. Also, the super particles build energy from E = 10¹⁷ to 10¹⁹ GeV. Assuming the same buildup rate continues in the future, the time required for the kinetic energy to reach the vacuum potential, and start spilling out into a new big bang is 10¹⁹ sec or 10¹² years.

       4.  It is important to note that dark matter is a result of two competing effects, namely:

O The ingestion of matter from particle space into black holes at a certain average rate.

O The leakage of dark matter from the dark matter bubble into interstellar vacuum space.

The buildup of dark matter into the dark matter bubble around a galaxy means that more matter is being ingested by black holes than is being leaked into interstellar space. The difference goes into the buildup around the galaxy. The buildup rate calculated in 3 above is the net difference between these rates. Eventually, all dark matter will find its way into the base in interstellar vacuum space and contribute to the kinetic energy behind the vacuum potential, but the bubble around the galaxy will hit the limit first, so its value is most important. Thus the calculation of the dark matter buildup rate to the big bang is only roughly valid. It is useful, however to give a description for what will happen.

5. It is also important to note that the use of kinetic theory from particle space in the very high-energy zone of vacuum space may introduce errors in the results.  Its use is justified only if the theory can later be justified and/or the experimental results match the experimental results (see Problems with Model 1, above).

 

Appendix 6. Dark Energy and Dark Matter

There are two kinds of material we are attempting to account for, dark matter and dark energy. They are separate problems, and will be handled separately.

• Dark Matter. It has been shown in Appendix 5 how two spaces connected by the black holes of Model 1 can account roughly for the observed distribution characteristics of dark matter. 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/(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   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.

 

• Dark Energy. Using the upper limit of the cosmological constant, the vacuum energy of particle space is estimated to be ~ 10^-5 GeV/cc. This is the vacuum potential energy we observe now causing the accelerated expansion we observe now. (see experimental results 8).

During inflation, the potential energy of particle space is estimated as ~ 10⁷¹ gm/cc or ~ 6 x 10⁹⁴ GeV/cc (see ref 9). It is believed that as particles were formed, this large potential was gradually used up.  

It is difficult to determine the quantum expectation value of a vacuum due to the interacting particles and anti particles exactly, but the value is not expected to be zero. It is estimated to be roughly one particle in every volume equal to the Compton wavelength of the particle cubed. Using the Planck mass (the highest) as the particle mass, one obtains a value of ~ 10⁹¹ gm/cc or ~ 10¹¹⁵ GeV/cc. A calculation using the Planck volume gives ~ 10¹²⁵ GeV/cc. Other estimates have been published, but most are in the range of ~ 10¹¹⁵ to 10¹²⁵ GeV/cc. It should be noted that not all of the particles and anti particles interacting in vacuum have even close to Planck mass. Indeed, one estimate based on the potential of the Higgs particle (Kane, 112), is ~ 10⁴⁹ gm/cc. It might be expected that an average value including other particles might be even lower. Thus the value used for the purpose of this model will be  ~ 10³⁵ GeV/cc.      

One of the results of the dark matter of Modle1 (see Appendix 1 Case 1) is that there is a rate of particles that tunnel from vacuum space through the vacuum barrier into particle space, and thus add phase change energy to the vacuum energy in particle space until it reaches ~ 10^-5 GeV/cc in our time as expected.   

• The results of these estimates show that it is possible to explain the dark matter and the dark energy in the sense that one can see roughly where they came from what they are doing now, and predict what will happen to them in the future.

  

Appendix 7. Prior Experimental and Theoretical Results

It is necessary to describe the prior experimental and theoretical results that Model 1 must satisfy in order to be a successful theory. Note by way of definition, that “accepted” means that some data exist that support the results given. Note also that “proved” means that the data supporting the results given have been cross-checked by different people and re-checked by different methods, and are thus accepted as valid.

 

The Basic Experimental Results.

Model 1 must satisfy the following basic experimental results.

1. Energy is conserved throughout the universe including its parts and its hierarchy. This is accepted, and most would consider it proved.
2. All current experiments confirm the Standard Model of particle physics. This is accepted and proved.
3. There is a microwave background radiation that spectrally Gaussian and is mostly spatially uniform, but has a measurable oscillating spatial spectrum that is not symmetric in its large-scale spatial modes. It also has a preferred direction. This is accepted and proved.
4. There is a reduction in the spectrum of the microwave background energy of the universe at a distance R (R~10²⁷ cm), which is close to the radius of the visible universe. A cutoff time, R/c, exists, which is roughly the age of the universe. An oscillation cycle, c/R, exists, which is roughly one oscillation per lifetime of the universe. This is accepted and proved.
5. There is a relationship between the size of a galaxy (diameter and number of stars) and the size (mass) of its central black hole. This is accepted, but not proved.
6. The galaxies in the visible universe form in groups, strings and walls. Gravitational lensing also shows a bubble of dark matter within the galactic structure. This structure acts like there is a net of dark matter connecting the bubbles of dark matter at the galaxies upon which other galaxies are strung. This is accepted but not proved.
7. Recent observations suggest that dark matter particles interact with each other and slow themselves down. Since the interactions did not affect the normal matter, they must have occurred through some force other than gravity that influences only dark matter. (Scientific American, 15) This is accepted, but not proved.
8. There is an expansion of the space between the galaxies of the universe, and an acceleration of the expansion as well. The acceleration started about 5 billion years ago. This is accepted, and proved.
9. The acceleration, c²/R, is roughly the rate at which the expansion of the universe is accelerating due to the cosmological constant-about 10^-8 cm/sec². This is accepted and proved.
10. In a galaxy, stars moving outside a certain orbital acceleration limit at less than 10^-8 cm/sec², accelerate more slowly than they should to be compatible with Newton’s law (the dark matter problem). This is accepted, and proved.
11. The fine structure constant changes over distances roughly equal to R, and has c in the denominator. This is accepted, but not proved.
12. Pioneers II and I have been calculated to be under the influence of an additional acceleration toward the sun of 8 x 10^-8 cm/sec/sec over that caused by the sun, a value close to c²/R. This is accepted and proved.
13. There is an instantaneous transfer of state for certain coherent experiments. This is accepted and proved.
14. Matter and anti matter particles come from nothing for a very short time period, and then recombine according to the uncertainty principle (ΔEΔt = ћ). This is accepted and proved.
15. Strength of each force (charge) is precisely fixed in a way that allows for life and stability over the long term. This is accepted and proved.
16. The universe has a very low curvature. This is accepted and proved.
17. There are four forces, and they appear to merge into one super force at a very high energy. This unification of forces is accepted, but not proved.
18. There are extremely high-energy cosmic ray events beyond the GZK cutoff that are difficult to explain (see Ref 10-Magueijo, 32). They should be absorbed by reactions with the CMB for energies beyond 10²⁰ EV. However, several higher energy events have been observed (see graph on pg 33).

 

The Basic Theoretical Results

Model 1 should satisfy the following basic theoretical results

19. Noether’s theorem is valid throughout the universe and its sub parts if they exist. This is accepted, but not proved.
20. Dramatically disparate vacuum energy estimates exist called the vacuum catastrophe. This estimate is accepted, but not proved. (See Appendix 6)
21. Plank energy and size marks the energy and size limit of the universe. This is accepted, but not proved.
22. Energy is contained in charges, fields and the structure of space. There is no evidence for “pure disembodied” energy. This is controversial, but reasonable. Many, but not most physicists accept it. It is certainly not proved.
23. Loop Quantum Gravity gives a theory that allows for some calculations to be made at very high energies. This is accepted, but not proved.
24. The big bang starts the expansion and particle formation of the universe and after a while the particle formation stops. This is accepted, but not proved.
25. According to one experiment, the fine structure constant changes over times of the order of the life of the universe. This can be explained by postulating that extremely high-energy photons travel faster than the lower energy photons. This result has been put into a theory called DSR, which allows c to be constant at low energies, and the Plank length to be constant at low and high energies. This is accepted, but not proved.
26. Using general relativity, the visible matter alone does not account for the flatness of space (~7%). Dark matter is postulated to account for certain dynamic features of galaxies and gravitational lensing (~23%). That still does not account for the flatness of space. Dark energy is postulated to account for the accelerated expansion of space (~70%). This total mass can account for the near flatness of space (see result 16, above)  Note, the existence of these three forms of energy is accepted, but not proved.
27. All known charges, including some that have not been confirmed, are in the E8 symmetry group. This is accepted, but not proved.
28. The arrangement of the galaxies (see result 6, above) in space implies the existence of a lattice of dark matter that encourages the formation of galaxies in strings, clumps, and walls.

 

 

References

 

1. L. Smolin, The Trouble with Physics, Boston, New York: Mariner Books, 2006.
2. B. A. Schumm, Deep Down Things, Baltimore, Johns Hopkins University Press, 2004.
3. Thorne, Miguel and Wheeler, Gravitation, New York, Freeman and Co., 1973.
4. G. Kane, Modern Elementary Particle Physics, Ann Arbor, Michigan, Perseus Publishing.
5. P. J. E. Peebles, Principles of Physical Cosmology, Princeton, New Jersey, Princeton University Press.
6. “Dark Matter Drops a Clue”, Pg 35, Scientific American, New York, June 2015.
7. I. G. Amelino-Camelia, “Testable Scenario for Relativity with Minimum-Length” hep-th/0012238. 
8. HTTPS://en.wikipedia.org/wiki/Rectangular_potential_barrier. 
9. WWW.astro.edu/~wright/cosmo_constant.html
10. J. Magueijo, New Varying Speed of Light Theories, arxiv.org/pdf/astro-ph/030545v3.pdf
11. J. Magueijo, Faster than the Speed of Light, Penguin Books, New York, New York, 2003.
12. J. D. Cobine, Gaseous Conductors, Dover publications, Inc., New York, 1958