Abstract

There are currently eight 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 in the energy range 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 summarized and defended with data from astronomers in this paper.

 

The Most Significant Current Questions in Astrophysics

There are eight connected major unanswered questions in astrophysics.

1. What is dark matter and where does it come from? Are particles and forces unified?
2. Why is there such a huge disparity between different estimates of vacuum energy (the vacuum catastrophe)? Can a model of the particles of the universe be constructed that allows for this disparity?
3. Can the distribution of dark matter into bubble halos connected by a web that is connected to galaxy distributions be explained?
4. Is it possible to explain what happened before the big bang and what initiated it and what happened afterward? Can the fine-tuning of the universe that produces life be explained?
5. Is it possible to explain the accelerated expansion of space and dark energy?
6. Is it possible to explain the large-scale cutoff and asymmetry in the Microwave Background Energy?
7. Is there a practical procedure for obtaining answers on questions that arise in the area of quantum gravity-especially for questions involving black holes?
8. Ultra high-energy cosmic rays (UHECR’s) have been observed in the range 10¹⁰ GeV to 10¹⁹ GeV. They should interact with the microwave background, and disappear above the BZK cutoff. Can this be explained?

Model 1 described below addresses all of these questions and answers them in a way that agrees with the existing data. Only a simplified version of these answers will be given here. The details will be given in connected papers referred to in the references. Together, they appear to give a complete and self-consistent answer to these questions.

Model 1. A Basic Physical Model Describing Dark Matter and Dark Energy.

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. 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 bubble halo centered on a galaxy. Corridors of dark matter (the cosmic web) connect the bubble halos of dark matter to each other. This web guides the development and organization of new galaxies,

• The dark matter particles in vacuum space gain energy and eventually exceed the ability of the barrier to contain them. They then explode back into particle space as a big bang (Type 1). This process repeats to make a series of new universes, each with higher entropy.  Eventually, the entropy is high enough that a Type 1 big bang cannot occur, and the balance of pressure and density in particle space shifts, causing a complete collapse and a different type of big bang (Type 2). This Type 2 big bang restarts the Type 1 cycles. 

• After a Type 1 big bang exhausts itself, super particles continue to tunnel through the barrier into particle space. The super particles are unstable and break down into particles (protons) with extreme kinetic energy. In doing so, they give up potential energy into particle space. The potential energy gradually builds up to become the dark energy that we observe as the cause of our accelerating, expanding universe. The extreme energy protons are observed as cosmic rays with energy between the energy of force unification and the Planck energy, which is beyond the GZK cutoff.  

 

Model 1 Answers to the Significant Questions

The answers to the significant questions of astrophysics as given by Model 1 are shown in this section. The equations and calculations for Model 1 are summarized in the Appendices.

1). According to Model 1, baryons exist at two energy levels, one above the unification energy (~10¹⁷ GeV), and one below. The two baryons exist in two spaces, one with high potential energy and one with low. A potential energy 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. We live in particle space, and it contains the low energy particles. In vacuum space, the potential energy density is extremely high, and the particles are in a high-energy form (above the force unification energy) and they will be called super particles hereafter. Many symmetry groups and attendant theories have been proposed to describe the unified force particle, but SO(10) appears to be the favorite at present (Appendix 2). The potential barrier consists of a spherical potential energy shell that surrounds each super particle (see Appendix 1, and also Appendix 2). Four forces act on particles in particle space. Super particles act on each other and on particles through the gravitational force, and this interaction has great importance (see section 3, below). Super particles don’t interact with particles through the weak and the strong forces in vacuum space primarily because the exchange particles are short range. Particles with rest mass can reflect off the super particle barrier shell as shown in Appendix 1. The case of the electromagnetic force is more complex (see also Appendix 10 for more details).

• For the case of moderate energy (E<10¹⁷ GeV) photon penetration of the barrier shell, we note that the barrier shell has no electric charge, and no magnetic moment, so it will not interact with photons and thus impede the passage of photons through the shell. Thus moderate-energy photons can pass through the barrier shell, and can interact with the super particle, but unless they can generate energy greater than 10¹⁷ GeV in the super particle, they will not cause a change in its state.
• For the case of high-energy photons interacting with shielded super particles, the high-energy exchange photons pass freely through the shell to the super particle, so the exchange photons can generate electromagnetic forces between super particles through the barrier shell. 

Thus particles interact primarily with the super particle potential barrier, but have a low interaction cross-section (s ~10^-45 cm² – Appendix 3). So super particles would be extremely difficult to detect, but maybe not impossible (see ref 17, AP4.7E). So super particles will not interact well with the particle detectors we normally use, and so we call the super particles dark matter.  

As a curious byproduct of Model 1, it can be used to show that the speed of light does not remain constant over the full range of particle and photon energies because of the graininess of the Planck vacuum. At extremely high and extremely low energies, the speed is higher than and lower than 2.99 x 10¹⁰ cm/sec respectively (see Appendix 6). Also as a byproduct, Model 1 can also be used to show how instantaneous transfer of state for entangled particles can occur even though it violates the speed of light limit on signal transfer (see Appendix 6).

2). In vacuum space, three of the four forces become unified in the high energy (both kinetic and potential) that exists there. This unification has been predicted for a long time (Kane, 281) The energy of unification is ~ 10¹⁷ GeV as determined by extrapolating the four coupling strengths to high energy. Vacuum space has a high energy density. Kane estimates ~ 10⁴⁹ GeV/cc (Kane, 112). The cosmological constant carries its own vacuum potential energy in particle space. It has been calculated to be ~ 10^-4 GeV/cc from the acceleration of the expansion of the universe. Clearly, this difference in potential energy in the two spaces demands a barrier between them. The potential barrier for Model 1 is the spherical potential energy shell mentioned above with radius r_ob = 10^-11 cm and thickness a = 10^-31 wrapped around the super baryon, and it has a potential energy value of ~ 10¹⁹ GeV (see Appendix 1). 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 4). Then, the potential energy is used to make super particles, and the expansion stops. At this radius, the spatial expansion due to potential energy roughly balances the particle attraction due to gravity, so the diffusion and electromagnetic forces and centrifugal force can be effective.

3). In Model 1, the generation of the dark matter halo occurs as follows. Particles increase in kinetic and potential energy as they descend toward the black hole center.  As the particle kinetic energy, and the spatial potential energy approach the unification energy (10¹⁷ GeV), the particles can convert to super particles. At the same time, potential energy barrier shells are being formed. The super particles then increase their kinetic energy to nearly 10¹⁹ GeV, where they can penetrate the shell and become stabilized as shielded super particles (see Appendix 5). The shielded super particles can penetrate the black hole event horizon in the following way. The speed of light is determined by the granularity of Planck space (see Appendix 6). When the kinetic energy nears 10¹⁹ GeV, the speed of light is increased above 2.99 x 10¹⁰ cm/sec, and the energetic particle finds that its velocity is higher than the moderate energy speed of light, and so it is above the escape velocity of the event horizon formed at the moderate energy speed of light (Appendix 5). Beyond the event horizon, the density of super particles is low enough that the mean free path is greater than the distance to the edge of the galaxy, and the super particles fly free to intergalactic space creating a dark matter density that reduces as ~1/r² from the galactic center. Now Peebles (Peebles, 47) shows that the matter density in a galaxy at a radius r is:

             ρ_t(r) = ν_c² / 4πGr²

             Where

             ν_c = circular rotation velocity as a function of radius.

             G = gravitational constant.

The astronomical data for spiral galaxies show that ν_c starts lowtoward the center of the galaxy and rises to a maximum and then flattens out to a constant value. Newtonian gravity requires that ν_c reduce beyond the maximum, or the matter beyond the maximum would escape from the galaxies gravity. Since ν_c is constant beyond the maximum, the density must increase and vary as ~1/r² out there. Now if the density of the dark matter that is decreasing at a slower rate (~1/r²) becomes equal to the density of the visible matter (~ν_c² / 4πGr² ) at the maximum ν_c , then the density of the dark plus the visible matter would vary as ν_c ~1/r². This, of course is what we observe, so the dark matter halo appears to be due to the escaping super particles. Note that Model 1 shows dark matter bubble halos centered on galaxies’ central black hole and connecting corridors of dark matter between the bubbles. The dark matter moves away from the black holes in each galaxy preferentially in the direction of other black holes. Thus they form a net like structure with bubble halos around galaxies as nodes.(the cosmic web). This structure is not visible in particle space but it is a nucleation zone for galaxies to coalesce on through gravity. So intergalactic gas and dust tend to form galaxies around this structure in groups, strings and walls. More details on this dark matter halo are given in reference 15, AP4.7B.

4). In order to explain the operation of the big bang, we must first describe the existence of two types of mass energy recycling through big bangs, namely:

• Type 1 recycle–A comprehensive but temporary rotation of mass energy from its state as particles within a galaxy in particle space to super particles formed in a black hole, to super particles in vacuum space contained in a barrier shell operating in intergalactic space, and then to super particles in a barrier shell, which are hot enough to spill over the barrier from vacuum space to particle space in a big bang, and then finally cycle back to particles in a galaxy in particle space. This recycle is temporary because the gradual buildup of low energy shielded super particles in intergalactic particle space eventually halts the recycle process.
• Type 2 recycle–A comprehensive and eternal rotation of mass energy after a complete collapse (big bang) caused by a mass buildup in intergalactic particle space from its low entropy temporary rotation through particle and vacuum space to a comprehensive rotation through a high entropy state to a renewed low entropy state so it can go back to its temporary rotation. 

We note also that two processes determine which recycle type will happen, namely:

• The super particle heating process in intergalactic particle space.
• The process of the increase in the mass density of intergalactic particle space.

If the heating of the super particles to near Planck energy occurs first, the Type 1 recycle occurs. If heating is slowed by the buildup of low energy super particles from temporary mass rotations, the mass buildup reaches a critical state and causes a complete collapse. This process is described in more detail in Appendix 7.

We note finally that when a new eternal universe starts, it must define certain basic constants (see Appendix 7). Of these constants, perhaps the most significant are the symmetry groups and charges. The symmetry groups and charges are determined by natural selection in a series of big bangs that will:

• Provide a particle energy organization principle (a symmetry group) to fit the particles within the maximum energy (E_pl). 

• Order the charges for each symmetry group in a way that ensures that the universe can recycle and extend its life.

This process also maximizes the probability of the development of intelligent life even though that goal is not the primary determining factor (see ref 28, AP4.7Q for more details).

5). 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, over 10¹⁰ years, this increasing vacuum potential passes the decreasing big bang vacuum potential and increases to the value we observe now (10^-4 GeV/cc). This (according to Model 1) is the potential that in our time we call dark energy, which causes the accelerated expansion of space that we observe (See Appendix 9). Note that as a result, the acceleration we observe should go through a minimum and then start gradually increasing (see ref 27, AP4.7I for more details).

6). 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 not compatible with the normal inflation theory (see Smolin, 206, and also ref 20, AP4.7D). 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 (see ref 18, AP4.7F for details). Thus the reduction is compatible with Model 1.

7). In doing this work, it was necessary to answer certain basic questions about physics in black holes. This requires the use of Schrödinger’s equation of quantum physics, the results of the Standard Model of particle physics, and the kinetic theory of gasses in a zone close to a black hole. Some of these results (Smolin, 250) are already available from Loop Quantum Gravity, and they appear to be 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 high-energy environment. However, the Loop Quantum Gravity basis is not complete, so more work should be done.

8). Model 1 predicts the existence of extremely high cosmic ray energy events (up to ~ 10²⁸ EV), which are beyond the GZK cutoff. A cosmic ray is essentially a very high-energy proton that reacts with the earth’s atmosphere and causes a gamma shower that we observe. It was found, using the standard model of particle physics, that a proton with energy above a certain energy (the GZK cutoff) interacts with the microwave background in particle space to form other particles. Since microwave background is everywhere, protons with extremely high energy (i.e. from super nova etc.) should not last very long in particle space.  Yet such cosmic rays are observed (Magueijo, 33). Now Model 1 predicts super particles tunneling through the potential barrier into particle space. These super particles are unstable in particle space, and break down into protons with energy in the range of ~ 10¹⁷ to ~ 10¹⁹ GeV. They occur anywhere there is dark matter-i.e. anywhere within a galaxy, and at lesser density, beyond. Thus these protons will appear too close to the earth to react with microwaves before interacting with the earth’s upper atmosphere to cause a cosmic ray gamma shower. Thus we expect ultra high-energy cosmic rays (UHECR’s) and find that they are indeed observed (see ref 27, AP4.7I for more details).           

 

Gaining Acceptance for Model 1

Model 1 must be tested with data. A summary of the tests that support Model 1 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 omni directional cosmic rays with energy between ~10¹² and 10¹⁹ GeV that can be observed beyond the GZK limit. These cosmic rays have 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.

In addition to the supporting evidence shown above, it may be possible to extract a super particle from behind its barrier so it can be studied. This possibility is described in Appendix 10.

 

Summary and Conclusions

A model has been developed that predicts dark matter and energy and the extremely high-energy protons, which exist 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 (ref 14, AP4.7A,  ref 15, AP4.7B, and ref 16, AP4.7C). Experiments that would check the accuracy of Model 1 have been proposed (ref 20, AP4.7D). The model has been found to be valid as far as the current checks can determine.

 

 

Appendix 1. The Basic Equation for the Barrier

Consider the following 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  , for  l = 0 (spherical symmetry)

Now, let:

            R(r) = U(r)/r, then:

            Y(r, q, f) = (4p) ^-1/2  U(r)/r   

Then, consider:

            [-(h²/2m d²/dr² + V(r)]U(r) = EU(r)

            Where

            E = the energy of the particle = p²/2m for a super particle.

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

            And 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 movingthrough 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 vice versa. 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.

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 either of two shells, a vacuum space shell and a barrier shell. The rate of passage of a particle through either spherical shell for a galaxy is:

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

The best fit for the vacuum space sphere and the barrier shell of the very rough data available is (see ref 19, AP4.7G for details):    

            r_ov = 10^-20 cm which defines the inner boundary of the vacuum shell

            r_ob = 10^-11 cm which defines the outer boundary of the vacuum shell

            a = 10^-31 cm which defines the outer boundary of the barrier shell

There are three important cases

Case 1

            E > 10¹⁷ GeV

            V_ov = potential energy ~ 10¹⁷ GeV

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

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

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

Case 2

            E > 10¹⁷ GeV

            V_ob = 10¹⁹ GeV

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

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

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

Case 3

            10¹⁹ < E < 1.19 x 10¹⁹ GeV

            V_ob = 10¹⁹ GeV

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

It is important to note that if we use the relativistic Schrödinger equation, we end with a description of the exchange force (see Kane, 29), where forces are achieved by the exchange of particles (including massless particles-photons). These exchange particles (including photons) will be blocked by the potential barrier in the same way that is shown above. Thus the forces will be inoperative unless the exchange particles have enough energy to exceed the barrier potential energy. This high energy for exchange particles will exist for super particles in vacuum space (energy > 10¹⁷ GeV), but not for particles in particle space (energy < 10¹⁷ GeV). So we see that forces (especially electromagnetic forces) will operate through the barrier between super particles in vacuum space (energy > 10¹⁷ GeV), but not between particles and super particles in particle space (energy < 10¹⁷ GeV). Therefore super particles will appear “dark” to matter detectors.

For more details on the super particle and its barrier, see ref 14, AP4.7A.

 

Appendix 2. Super Particle Symmetry and its Barrier Shield

The fact that gauge coupling strengths seem to meet at an energy of ~10¹⁷ GeV and that the neutrinos oscillate in mass suggest that the Standard Model is incomplete. For this reason, many Grand Unification Theories (GUTs) and related Lie symmetry groups that describe them have been generated (ref 25). The smallest simple Lie group that contains the Standard Model at lower energy is SU(5), but it predicts a proton decay which has been searched for but not found. Consequently, it has fallen out of favor. Group SO(10) (see ref 25) predicts a longer proton decay time, so it appears to be the current favorite. Model 1 does not require unification with SO(10). The only things required of the symmetry group are that the super particle couplings are strong enough to form a finite lifetime super particle, and that it contain the three standard model groups SU(3) X SU(2) X U(1). Clearly, SO(10) fits these criteria. For more details on this subject, see reference 26, AP4.7P. It is shown in reference 26, AP4.7P, that the universe chooses the correct symmetry group and its charges from the many available through a process of natural selection.

 

Appendix 3. Particle-Super Particle Interaction Cross Section

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 because they are hidden behind the potential energy shell. 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 proton off of 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^-70 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). The sum of these contributions would not be expected 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. For more details on this scattering cross section, see reference 16, AP4.7E.

 

Appendix 4. Potential Energy and Expansion Pressure

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

• A contraction due to mass operating toward the center of the black hole competing with a space expansion or contraction due to a real scalar field.
• 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 diffusion force due to the motions and collisions of particles due to their high temperature.
• 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.

Inside the black hole event horizon, the most important is the first. The expansion and contraction of space in this zone is controlled by the following field density (ρ_f) and pressure (p) equations from general relativity (see Peebles, 396)

             ρ_f = φ’ ²/2+ V = field density

             p = φ’ ²/2 – V = pressure

            Where:

            V = a potential energy density

            φ = a new real scalar field

            φ’=the time rate of change of the field

            φ’ ²/2 = a field kinetic energy term

Also, the cosmological equation for the time evolution of the expansion parameter (a(t)) due to mass density (ρ_m) is (see Peebles, 75):

ä/a = -4/3πG (ρ_m+ 3p) = acceleration of the cosmological expansion parameter        

Note from the field pressure equation, that if the potential energy exceeds the field kinetic energy, the field pressure is negative. Also if the field pressure is large enough, it can dominate the pressure. Thus if the negative field pressure term is large enough to exceed the mass density term, the acceleration of the cosmological expansion parameter ä/a turns positive, and space expands. If the potential energy V is small compared to the field kinetic energy term, field pressure is positive, the acceleration ä/a is negative, and space contracts.

Model 1 notes that V increases as the position approaches the center of the black hole (Meisner, 910).  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 φ’ ²/2 is small compared to V, so the pressure is negative. As position approaches the black hole center, -p will eventually decrease enough to turn ä/a positive and space will expand under the expansion pressure of V. The space closer to the black hole center than the particle will then expand and push the particle away from the black hole center.  

Then, as V decays with time into super particles, and the particle is pushed further from the black hole center, the kinetic energy term exceeds the potential energy, the pressure turns positive, and space contracts. Then the particle forces drag the particles toward the black hole center. Thus we have an expanding space in dynamic equilibrium with particle forces surrounding the black hole center (see ref 15, AP4.7B for more details). The particles will then form into a Gaussian distribution at radius r. At this radius, the particles are forced into a potential energy barrier shell to form a super particle (see ref 16, AP4.7C). The effect of the diffusion, and electromagnetic forces on the super particle ion cloud is explored in Appendix 8.

Another way of looking at this process involving the divergence of the Robertson-Walker line element and thus the expansion of spatial distances is described in Peebles, 395 .

 

Appendix 5 Formation and Escape of Super Particles in a Black Hole

Particles from particle space fall into black holes, and the kinetic energy increases as the particles are falling (see Misner, 910). The particles are moving into a smaller volume as well, so the potential energy density increases dramatically. The particles collide and reach equilibrium and so have an average energy and a temperature. As the particle kinetic energy, and the spatial potential energy approach the unification energy (10¹⁷ GeV), the particles can convert to super particles. This potential energy comes from the gravitational potential energy of the black hole. The super particles eventually increase their potential energy to nearly 10¹⁹ GeV. They then generate the spherical potential barrier, which has a strength of ~10¹⁹ GeV. The kinetic energy increases as well to ~10¹⁹ GeV. This kinetic energy gives the activation energy needed to enter into vacuum space inside the shell. The vacuum potential energy density of particle space near the black hole center is ~10⁵⁰ GeV/cc. This high potential energy density generates a negative pressure in space, and space expands (see Appendix 4), and forces the super particles away from the black hole center. At some radius from the black hale center, the expansion of space and the attraction of the particles toward the black hole are roughly equal, and the diffusion and electromagnetic forces become important (see Appendix 7), and the super particles move toward the event horizon. At a further radius from the black hole center, the enormous kinetic energy of the super particles (KE near10¹⁹ GeV) is sufficient to move the super particle beyond the event horizon, and super particles begin to escape and form the dark matter halo around the galaxy of section 3 above.

The escape of super particles from black holes can be understood as a result of the increase of light speed at extreme energies. The event horizon can be seen as a result of the mass of a black hole becoming large enough so that the escape velocity is higher than the speed of light (2.99 x 10¹⁰ cm/sec).  When the kinetic energy of a super particle nears 10¹⁹ GeV, however, the speed of light is increased above 2.99 x 10¹⁰ cm/sec (see appendix 6), and the energetic particle finds that its velocity exceeds that of the moderate energy event horizon, so it escapes.

 

Appendix 6. The Speed of Light at Extreme Energy

In order to make constant speed of light and constant Planck length compatible, Amelino-Camelia and Magueijo (Amelino-Camelia, 6 and ref 11, Magueijo, 251 and ref 10, Magueijo, 31) developed a modified light speed relation. This relation shows that when the energy increases enough, the speeds of particles and photons increase to values greater than the speed of light in a vacuum at moderate energy (2.99 x 10¹⁰ cm/sec). This relationship has been developed further, and incorporated into Model 1 in reference 24, AP4.7M. There it was found, using the theory of granulated Planck space, that the velocities of particles in vacuum space reach extreme values much higher than 2.99 x 10¹⁰ cm/sec, when the super particle kinetic energy (E_sp) approaches E_pl. A summary of that condition is given here.

Consider the condition where super particles have a potential energy V_sbo, which is only slightly less than the Planck energy. Particles and photons are roughly the same size as Planck granules, and moving in and out of each granule is important. For example, when super particles gain enough energy to operate freely in barrier space, they are operating in the zone 10¹⁹< E < 1.22 x 10¹⁹ GeV. Thus since the length of travel for a particle is n Planck lengths long, and N is the number of disruptions along the travel length, then the distance traveled is nl_p, and the time used for this travel is just Nt_p, for these disruptions, then the speed of light v_o as it moves in and out of each Planck granule is:

            v_o = nl_p/ Nt_p ½[(1 + (E_smo+ E_ske)/(E_smo+E_ske– V_sbo)]

                  = 2.99 x 10¹⁰n / N ½[(1 + (E_smo+ E_ske)/(E_smo+E_ske– V_sbo)] cm/sec

            Where:

            E_ske = super particle kinetic energy

            E_smo = super particle rest mass energy

Now, as the super particle energy moves up from the lower limit (V_sbo) we see that:

            (E_smo+E_ske) > V_sbo

Then:

            ½[(1 + (E_smo+ E_ske)/(E_smo+E_ske– V_sbo)] ~ 1

And since the energy is still removed from the Planck energy, N = n, so:

            v_o = nl_p/ Nt_p = c = 2.99 x 10¹⁰ cm/sec

Now recall that the potential energy of a Planck granule is the Planck energy, so as the super particle kinetic energy moves higher still toward the Planck energy, the super particle can pass right through the potential energy of the Planck granule just like a super particle passes through the super particle barrier shell when the kinetic energy equals the barrier potential energy. Thus the particle cannot detect the potential energy edge of a Planck granule. Then the number of disruptions along the travel length (N) is reduced from n to the large edges like disruptions in space such as the shock at an edge of a solar system, a galaxy, a boundary of a corridor in the cosmic web or an edge of the observable universe. This is a much smaller number than n, so n/N >> 1, and:

            v_o = nl_p/ Nt_p >> 2.99 x 10¹⁰ cm/sec

Note that the same argument holds for photon velocities, except E_smo is zero. Note also that v_o is never infinite.

Thus we are able to conclude that according to Model 1, the super particle (and an energetic photon) velocity can be large enough to allow for escape from the event horizon of a black hole, and for the gathering of particles to the big bang from the entire observable universe when the super particle energy approaches the Planck energy

Another important case of extreme light speed occurs with entangled particles. When two particles have separated a large distance, and still have maintained entanglement, a special condition occurs. We recall that two-particle entanglement means that the physical state of the two particles is precisely the same, even the phase and the spin, so the physical laws governing them are precisely the same. They also remain in contact through exchange photons, so the exchange photons must also have precisely the same phase and spin, and they too are entangled along their path of travel. Since the exchange photons maintain entanglement throughout their travel, the physical laws remain the same in each of the Planck vacuum granules throughout the travel length. Thus the state of the photon in the vacuum granules is conserved (Noethers theorem). This granule entanglement and state conservation means that an exchange photon cannot detect a discontinuity at the boundary of each grain so n/N >>1, and

             v_o = nl_p/ Nt_p >> 2.99 x 10¹⁰ cm/sec 

Thus state can be transferred at superluminal speeds (see ref 29, AP4.9R for details).

 

Appendix 7. Big Bang Operation

Refer back to the equations of Appendix 4. Note from the field pressure equation, that if the potential energy exceeds the field kinetic energy, the field pressure is negative. If the field pressure is large enough, it can dominate the mass pressure. If the negative field pressure term is large enough to exceed the density term, the acceleration of the cosmological expansion ä/a turns positive, and space expands. If the potential energy V is small compared to the field kinetic energy term, density controls, the acceleration is negative, and space contracts.

Which type of cycle occurs depends on which of the above processes progresses fastest in intergalactic particle space, because a Type 1 or a Type 2 recycle must occur in intergalactic space to involve all of particle space. These processes will now be compared.

1). The Heating of the Super Particles in Intergalactic Particle Space

The shielded super particles inside the event horizon of a black hole are in a broad energy distribution. Those super particles in the high end of this distribution have a high enough kinetic energy to pass through the event horizon, but those in the medium to low end do not. They must gain energy to pass out into the galaxy. Thus the super particles that reach intergalactic particle space have high kinetic energy. These new hot super particles will heat the distribution in intergalactic particle space. If the average kinetic energy of the super particles approaches the Planck energy, they achieve the speed (greater than 3×10¹⁰ cm/sec see ref 3, AP4.7M) necessary to reach the first principal black hole during the contraction phase before a Type 1 big bang. Also, such hot super particles have the energy needed to flow over the potential barrier. So a Type 1 big bang happens for nearly the whole universe. However, if the density of low average kinetic energy intergalactic super particles is large, as will happen after several Type 1 big bangs, it will take a long time to raise the temperature of the super particles, and another scenario can occur first

2). The Increase in the Mass Density of Intergalactic Particle Space

In the middle epoch of the life cycle of the universe, the super particle density is low because most of them have passed over the barrier, and the potential energy has mostly been converted into particles, and the kinetic energy has been given to these particles. Gradually, new shielded super particles are formed by new black holes, and start tunneling into particle space and leaving their potential energy there as dark energy. Thus the potential energy density in particle space goes through a minimum and starts to rise and cause the expansion of space. The shielded super particle and particle density is rising too, but it is below the threshold where it dominates the expansion parameter equation. In the late epoch of the life cycle of the universe, the mass density r_m in intergalactic particle space is increasing rapidly due to the rapid increase in the number of black holes and the increase in super particles coming from each black hole. Also the potential energy V is increasing from disrupting super particles that have tunneled through the barrier shell but at a much slower rate, because tunneling through the barrier shell is a very slow process compared to super particles escaping from black holes. Thus the pressure term in the cosmological expansion equation is decreasing at a much slower rate than the mass density is building, and eventually the acceleration of the cosmic expansion parameter will turn negative. If the temperature increase of the super particles is delayed by buildup of residual low energy super particles from repeated Type 1 big bangs, the mass density buildup will eventually dominate in the expansion parameter equation, and the acceleration of the cosmic expansion parameter will turn negative. Space will contract, and a Type 2 big bang will result.

We note finally that when a new eternal universe starts, it must define certain basic constants of two types (see ref 26, AP4.7P):

• The primary fundamental constants, which are the Planck energy E_pl, the Planck time t_pl, the Planck length l_pl, and the total mass-energy of the universe E_o
• The secondary derivative constants, which are the constant c, the Planck mass m_p, the Planck constant ђ, the gravitational constant G, and the symmetry groups and charges for the electromagnetic, weak and strong forces.

The symmetry groups and charges are determined by natural selection in a series of big bangs. The groups and charges that extend the life of each cycle and ensure recycling will maximize the probability of existence. Thus the natural selection process should:

• Provide a particle energy organization principle (a symmetry group) to fit the particles within the maximum energy (E_pl). 
• Order the charges for each symmetry group in a way that ensures that the universe can recycle and extend its life.

This process incidentally maximizes the probability of the development of intelligent life even though that goal is not the primary determining factor (see ref 28, AP4.7Q for more details.

 

Appendix 8. Diffusion Zone

In Appendix 4, the formation of a zone where expansion forces due to potential energy, and contraction forces due to gravity become roughly equal. This radius of balance supports an energy high enough to ionize the super particles. Therefore, at this radius, the high concentration gradient and the electric fields cause an ionized diffusion zone to form where the following equations are valid (Cobine, 51). 

            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 under the influence of electric or gravitational force 

            V = Ion velocity

            n = Ion concentration

            E = Electric field

            G = Gravitational field tempered by centrifugal force

            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 these equations, and get

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

            Where:

            D = (D⁺ K^– + D^– K⁺ ) / ( K⁺+ K^– – 2 K⁺ K^– / K^g ) = total diffusion coefficient

                  r= Radius from black hole source

            t= Time

            N_o(r) = particles diffusing from an “instantaneous” point source  

It should be noted that an “instantaneous” point source was chosen because it represents the time span that the black hole was feeding, which is short compared to the lifetime of a galaxy. There are expected to be many short feeding episodes at higher intensities with lower, longer, smoother episodes in between. The solution shows that the relaxation time (the time needed for the bubble to flatten out) can be long for intense matter intakes, which make large, short-lived density increases. Thus a large input (N_o) into the black hole will take a long time to diffuse out into the galaxy. So the procedure for formation and distribution of super particles tends to smooth out the unevenness in the distribution rate of super particles from the black hole. Also, due to the speed limit on the ion velocity, it would take a long time for a change in feed rate of a black hole to show up in the outer fringes of the galaxy where it can be observed

The principal portion of the diffusion coefficient can be approximated by the equation

            D = v L = v / 3nπd²

            Where:

            v = mean velocity of the super particles

            d = effective diameter of the super particle.

            L = mean free path = 1 / 3nπd²

            n =  super particle density

Within the event horizon of the black hole, the mean free path of the super particles is small enough to support diffusion, and according to Model 1, the super particles within their barrier shell beyond the event horizon become dark matter (n = ρ_d(r)). These super particles within their barrier shell are called dark matter because they are difficult to detect with visible matter detectors (Appendix 3). Using the total matter density expression obtained by Peebles for matter just beyond the event horizon (say r ~ 10⁵ cm), and using an effective super particle diameter of d < 10^-11cm, the mean free path of a super particle just outside the event horizon is >10²⁶ cm. This is further than the distance from the black hole center to the critical radius (~ 10²² cm), so super particles exiting the black hole event horizon will fly straight to intergalactic space with few collisions. Furthermore, they have a large enough kinetic energy (>10¹⁷ GeV) that they will not be stopped at the event horizon (see Appendix 5). Thus the super particle (dark matter) density ρ_d(r) will vary as:

             n = ρ_d(r) ~ K / r².

When the super particles reach intergalactic space, the density will flatten out to the residual value of intergalactic space. This intersection point is somewhat beyond the critical radius. The rotation of these super particles around the galactic center is small because they originate from a diffusing particle cloud near the center of the central black hole of the galaxy.

 

Appendix 9. Dark Energy

Using the upper limit of the cosmological constant, the vacuum energy density of particle space is estimated to be ~ 10^-4 GeV/cc. This is the particle space vacuum potential energy causing the accelerated expansion of space we observe now.  An estimate of the vacuum potential energy based on the potential energy of the Higgs particle (Kane, 112), is ~ 10⁴⁹ GeV/cc. Note the equality of this value (within the error of the estimate) to the value of the vacuum space potential energy density (see Appendix 1). This equality will be explored again elsewhere (ref 19, AP4.7G). Note finally, that the current vacuum space energy density needed to give the observed accelerated expansion of space (~ 10^-4 GeV/cc), results from the particles tunneling from vacuum space over 10¹⁰ years (see Appendix 1, Case 2).

 

Appendix 10. Photon Penetration of and Super Particle Extraction from the Barrier

We note that the barrier shell has no electric charge, and no magnetic moment, so it will not interact with electromagnetic photons and thus impede the passage of photons through the shell to the super particle where interaction can take place. On the other hand, particles with rest mass such as the fermions and some of the bosons of a super particle will be impeded by the barrier shell as shown in Appendix 1. There are two important cases of interest here.

Case 1

            E > 10¹⁷ GeV

            V_ov = potential energy ~ 10¹⁷ GeV

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

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

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

Case 2

            10¹⁷ < E < 1.19 x 10¹⁹ GeV

            V_ob = 10¹⁹ GeV

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

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

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

Thus exchange photons can reach super particles in vacuum space, and participate in a force interaction such as electromagnetic attraction and repulsion in an ionized gas of super particles if the photon energy is high enough. However, although a low energy exchange photon from a particle in particle space can reach through the barrier to a super particle in vacuum space, it doesn’t have enough energy to impact the super particle’s function and thus be observed, unless an energy buildup condition can be obtained that pushes the super particle out of the barrier. Here, we summarize the conditions for such a buildup. The details of this process are given in reference 30, AP4.7L.

In order to address this problem, we must set up the rate of passage equations for super particles to particle space. Here these equations are valid for either a vacuum space sphere or a barrier shell. The rate of passage of a particle through either of these spaces is:

            R = T ћ k_o n_sp / m_spr particles/sec

            Where:

            k₁= (2m(V₀-E)/ђ²) ^½ = momentum of the super particle in vacuum space

            k_o = (2m(E)/ ђ²) ^½ = momentum of the super particle in particle space

            m_sp = super particle mass

            T = 1/(1+V₀² sinh²(k₁w)/4E(V₀-E) = transmission through high energy spac

Now if:

            n_sp = N_g = number density of super particles in a galaxy, R will be rate of passage into a galaxy.

            r = r_op tor_ob = r_ov then the rate is for travel in and through vacuum space.

            r = r_ob to (r_ob + a) then the rate is for travel in and through the barrier.           

The super particle is trapped in the barrier shell bouncing back and forth in vacuum space with a resonant frequency determined by the shell diameter as it is reflected from the barrier walls. If we send in electromagnetic photons (gammas) with the wrong energy, it will be absorbed but there will be interference, and the energy will be dissipated. But if we send in gamma photons with just the right energy, they will find shielded super particles with the correct phase, and tap the super particle at each reflection cycle in a building resonance. We see that if each reinforcing tap of energy is a quantum of gamma ray energy (hυ_r) at the resonant frequency υ_r, then the super particle position is pushed a bit further into the barrier shell before being reflected, and the super particle maximum potential energy V is raised a bit more. Also the maximum kinetic energy of the super particle in vacuum space barrier is raised a bit as well. The resonant frequency is:

             ν_r = [V_ov / r_ov² m]^½

                  = 10²¹ 1/sec

This is the same as the oscillating frequency of a super particle in its barrier shell.

There is some damping in this resonance caused by the energy loss due to tunneling. This is small, however because the transmission T is small as long as the kinetic energy is much less than the potential energy. As the resonance builds, the super particle kinetic energy reaches the barrier potential energy, and it passes through the shell.

We note that there is an analog in nuclear theory of a gamma stimulating a super particle inside a barrier shell followed by emission of a super particle. This analog is called the compound nucleus hypothesis. In the compound nucleus hypothesis (Halliday, 327), a particle enters a nucleus, stimulates a resononce and becomes subject to strong internal forces. Its direction is changed and its excitation energy dissipated, so its identity is lost. A compound nucleus is thus formed, and its dominant feature is its long life. Sooner or later, a nucleon comes close to the nuclear surface with enough energy and it escapes. If this process takes too long, a photon is emitted to relieve the excess energy. This hypothesis can be expressed as follows.

            σ(a,b) = σ(c)P_b(ε)

Where:

σ(a,b) is the cross section for the complete reaction X (a,b)Y

σ(c) = cross section for the formation of compound nucleus with excitation energy ε by absorbing the particle a

P_b(ε) = the normalized probability that the nucleus will decay by emission of b

Here it is assumed that P_b(ε) is independent of the mode of formation of the compound nucleus. This theory has been used to calculate the probabilities of gamma absorption and super particle emission for a shielded super particle (see ref 30, AP4.7L). The results show that such a process is feasible.

Thus we appear to be able to boost the super particle energy toward the barrier potential by many small taps of the resonant gammas. If enough taps are absorbed before the shielded super particle moves out of the influence of the gamma beam exciting it (a dwell time), the super particle energy exceeds the barrier energy and breaks through the barrier to particle space where we can observe it. An experiment is proposed in reference 30, AP4.7L for the extraction of super particles from their shells.

 

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