Abstract

In a previous paper (ref 8, AP4.7), a self-consistent theory called Model 1 was developed to answer ten major connected questions in astrophysics. The most important of these questions are:

• 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?

 Model 1 appears to successfully answer these questions. The unique features of this model are:

• There are two spaces in the universe, particle space and quantum vacuum space. A potential barrier separates them. One space contains visible matter, and the other space contains dark matter.

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

• Dark matter particles interact with each other and form a slowly building and moving bubble centered on a galaxy. Corridors of dark matter forming a cosmic web, which guide the development of new galaxies connect the bubbles of dark matter to each other.

• There, behind a potential barrier, the dark matter particles gain energy, build up in number and eventually exceed the ability of the barrier to contain them. They then explode back into particle space as a big bang. This process repeats to make a long series of new universes.

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

 There is reason to take Model 1 seriously. It quantitatively explains (see ref 15, AP4.7D):

• The origin, characteristics and operation of dark matter.
• The origin, characteristics and operation of dark energy.
• The origin, characteristics and operation of the Ultra High Energy Cosmic Rays (UHECR) that have been observed in the energy range beyond the GZK cutoff.
• The huge disparity in the different estimates of the vacuum potential energy.
• The large-scale cutoff and asymmetry in the Microwave Background Energy.

In this paper, Model 1 will be shown to predict the existing data on the origin, characteristics and operation of dark matter.  

 

The Problem

In ref 8, AP4.7, several problems were left for future efforts. In this paper, one of these problems, the details of the formation of the dark matter halo that surrounds galaxies, and its impact on the rotation of matter within galaxies have been singled out for exploration. AP 4.7 showed how a cloud of super particles would form after entering vacuum space. These super particles were identified with dark matter. Remember that this dark matter was needed to:

• Explain the dark matter cloud needed to allow the galaxies to coalesce into stable spirals.
• Explain the velocity distribution of visible matter around the galactic center of spiral galaxies. It doesn’t match the expected distribution using Newtonian gravity.
• Explain the strings, clumps and walls of galaxies observed by astronomers.

Some equations were developed in AP4.7 for the dark matter distribution, and solved to give a contained, dark matter halo similar to the one obtained from studying the rotation data of galaxies. That was a simplified, first level treatment. Here, we will move to the next level, and explain some problems raised in the earlier treatment as well as see how well it predicts the observations.

 

The Solution

The Dark Matter Cloud Needed to Coalesce Galaxies.

We note first that gravitational lensing probably caused by dark matter has been observed around galaxies (Peebles, 272), and velocities of galaxies and within groups (Peebles, 417) have been observed that imply the existence of dark matter. Also, the Dicke coincidences argument (Peebles, 273) has been used to imply the existence of dark matter. The argument for the existence of dark matter is strong. We ask if it is needed to form galaxies from dispersed matter particles.

Now, we are interested in the formation of an orbiting system consisting of smaller particles orbiting around a large group of particles and thus forming a proto-galaxy that would form the nucleus of the main galaxy. We start with a thick soup of hot particles (mostly hydrogen and helium) and photons that have just condensed out of hot matter particles (see ref 10 AP4.7C) and are beginning to combine from an ionized state into atoms. The particles have a Gaussian distribution (see the next section below) controlled by a complex diffusion and ionic attraction process. The soup is cooling as it expands into a space expanding because the pressure of space is still negative (see the next section), but the expansion is slowing. The vacuum potential energy is dwindling, as it forms into particles. Eventually the vacuum potential energy term falls below the kinetic energy term, and the accelerated expansion stops and turns into an inertial expansion against gravity. The visible matter then coalesces into a distribution of larger clusters of atoms that stick together through electrostatic attraction as noted recently by astronauts in orbit.

Next, we note that the smaller clusters are attracted to the larger clusters by gravity, but those not colliding directly with the larger are not easily captured into an orbit around a larger particle. It is well known that if a cluster falls toward a larger cluster and does not interact with another small cluster in the vicinity, it will gain enough kinetic energy in falling to enable it to escape, and it will not fall into orbit. The falling cluster must lose some kinetic energy in the vicinity of the larger cluster in order to fall below escape velocity and be captured into orbit. There are two ways for a cluster to lose energy:

• An inelastic collision with another small cluster moving slowly close to the large cluster.
• A gravitational interaction that deflects one (or more) small, slow particle in the vicinity of the larger cluster and thus causes turbulence and/or the ejection of other particles that results in energy loss to the falling cluster. 

Thus for a cluster to be captured into orbit and start a galaxy, there must be a cluster of atoms or a dense group of atoms to provide a massive center for a small cluster to fall toward, and a group of smaller, slow particles surrounding the massive center for the particle to interact with and lose energy to, so the particle can be captured. The particle field can be composed of visible matter and dark matter as long as the mass and velocity characteristics of the particle field are correct. The interaction characteristics are:

• There must be some low velocity massive clusters of atoms or groups of clusters to provide attractive centers to fall toward under the influence of gravity. These should have some sticky components, to build up into larger, attractive centers.
• There must be smaller, but still massive particles to be the orbiters. These don’t have to be all sticky- for example gas clouds can orbit.
• There must be a field of lower velocity particles in the vicinity to provide an energy-absorbing medium in the particle field. These should be less sticky, to avoid building up large particles too fast, and thus become a less efficient energy-absorbing medium.

Visible particles can obviously supply the first two characteristics because they have an electrostatic interaction. Dark matter can supply the last. Since it is dark, it has a low interaction cross-section with visible matter (see Appendix 1). Simulations have been tried with visible matter alone and found to form collections of interacting particles that don’t form stable orbiting groups and thus galaxies. Simulations with visible and dark matter were found to form galaxies that are stable. Thus we find that dark matter appears important in forming galaxies.

If dark matter is important in forming galaxies, we must ask where the dark matter came from to assist the formation of the first galaxies after the big bang in our universe. For the current generation of stars, Model 1 proposes that the dark matter comes from a shadow web of dark matter left over from a prior universe that existed before our big bang (ref 11, AP4.7F). As for the original dark matter after the first big bang, we address that problem in ref 14, AP4.7J)

The Velocity Distribution of Visible Matter in Spiral Galaxies.

Peebles (Peebles, 47) has described the problem of galactic rotation of spirals quantitatively in Figure 3.12. Rotating spiral galaxies can be divided into two parts. The central part of the galactic rotation can be quite well explained by Newtonian gravity. The outer part rotates too fast to be explained by Newtonian gravity. The outer mass would fly away from the galaxy. Peebles develops an expression for the mean total mass density ρ_t(r) at radius r that satisfies the observed rotational data as follows:

            ρ_t(r) = ν_c² / 4πGr²

  Where

            ν_c = circular rotation velocity as a function of radius.

            G = gravitational constant.

Visible matter alone as shown in Fig 3.12 cannot explain the mass density as described in the above equation. The difficulty occurs at radii beyond the rapid bend at ~5 KPS (~10²² cm). We will call this the critical radius. Model 1, however can explain it with a combination of visible matter spiraling in from intergalactic clouds, and dark matter moving out from the central black hole of the galaxy as follows:

We must start with the formation of dark matter. Dark matter starts with particles in an intense black hole generated gravitational field that are converted into super particles (see ref 13, AP4.7A for details) inside barrier shell bubbles by adding energy in a rate process (see ref 10, AP4.7C for details). The particles exist in particle space controlled by a single new real scalar field, φ. The energy density ρ and pressure p equations that result are as follows (Peebles, 396):

 ρ  = φ’ ²/2+ V; and p = φ’ ²/2 – V)

 Where:

            V = a potential energy density

            φ = a new real scalar field

           φ’ ²/2 = a kinetic energy term

Now, near the black hole center, the mass, and so the potential energy V is large. Thus V is large enough to make a significant contribution to the stress energy tensor. Also the potential energy V is a slowly varying function of φ because the black hole mass, which controls V does not change rapidly with time. Finally, the initial value of the time derivative of f is not expected to be large because Model 1 has the universe evolving smoothly from one state to another. Note that V being large and dependent on f does not imply that the space derivative of f is large compared to the time derivative of φ. Thus the kinetic energy term φ’ ²/2 can be small compared to V, and p can be negative. Thus the universe is dominated by a term that acts like a cosmological constant, and space expands in the zone where this condition applies. Thus we conclude that there is an r small enough that the pressure will turn negative in the surrounding volume, and space will expand there. Also, we conclude that this negative pressure condition will eventually fail, as r gets larger.

Therefore the value of V is high enough in the vacuum space zone near the black hole center to generate the expansion condition, and space expands. The expansion of space closer to the black hole than a particle near the black hole moves the particle away from the black hole center. The continued expansion of space away from the black hole center eventually reduces V below the kinetic energy term (φ’ ²/2), and the expansion stops. Meanwhile, the particle is dragged back through space by the gravitational attraction of the black hole mass. Thus the particles tend to concentrate in a Gaussian distribution around a radius where these two processes balance (see Appendix 3 for more details). At the same time, the enormous kinetic energy of the super particles (10¹⁷ GeV) is trying to force the super particles away from this radius of balance by diffusion. The concentration of particles at this balance radius constitutes a source of particles N_o(r) where diffusion controls, and moves the particles beyond it (see the diffusion equations, below). Recall that the super particles involved are in a thermal, Gaussian distribution, so a diffusing, thermally distributed band of super particles will form where the expansion field of space equals the gravitational field of the particles.  The expansion and then diffusion of super particles away from the black hole center will continue until the super particles reach the radius where the mean kinetic energy of the particles in the high end of the distribution is enough to carry them beyond the event horizon.

Beyond the radius of balance, the super particles diffuse away from the center of the black hole. 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 formed control, and cause the following equations to be valid. 

            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 smoothes the source output of the source. 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 1). 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 by the gravitational attraction of the galaxy. 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, and because the rotating visible matter cannot speed up the rotation velocity of the dark matter because the interaction cross section is small.

The visible matter comes from intergalactic space, and spirals in toward the center of mass of the galaxy forming on the web of dark matter left over from former stars. The web of dark matter interacts gravitationally with the incoming visible matter taking away energy, and allowing the visible matter to be captured into orbit. This capture gives the visible matter its circular velocity around the galactic center. According to Newtonian gravitation theory (Peebles, 47), then, the density in the galaxy will vary as:

            ρ_v(r) ~ ν_c² / 4πGr²

Here the circular velocity will increase to the critical radius and then fall off as:

            ν_c ~ r ^-½

Model 1 proposes that the total mass in the galaxy comes from the sum of the dark matter from the center of the galaxy and the visible matter from outside of the galaxy. Fig 3.12 shows that the visible matter will dominate the total density near the center of the galaxy. The visible matter density will start high and fall off rapidly (ν_c² / r²) toward the outer edge of the galaxy. However, the dark matter starts lower, but falls off more slowly  (~1/r²), so eventually they must meet. The visible matter comes down to meet the dark matter at the critical point, and causes the roll over observed by astronomers. At greater radii, the total density varies as (~1/r²) since dark matter dominates it. The circular velocity appears flat in this zone, then. Note that the interaction cross-section between dark and visible matter is small (see Appendix 1), so the visible matter does not speed the dark matter circular velocity up. Note also that the dark matter flows on into intergalactic space and a portion of this tunnels into particle space and is converted into ultra high-energy particles and potential energy (dark energy) there (see ref 12, AP4.7I for details).

In addition to the above relation, it has been observed that there is a relation between the mass of the central black hole of a galaxy and velocity dispersion of the stars of the bulge in those galaxies. This relation is called the M-sigma relation (Ferraese, 539). Note from above that the volume of the diffusion zone where shielded super particles are formed, and thus the number of shielded super particles formed, is directly dependent on the black hole mass (the number increases as the mass increases). Thus where the escaping shielded super particle mass equals the visible matter mass, matter with velocity greater than the escape velocity of visible plus dark matter will escape, causing the velocity curve to flatten, and the black hole mass to depend on the velocity dispersion. 

Model 1 may be able to explain another observation. Smolin notes that the critical radius identified above is found to be near a critical acceleration zone for most galaxies (Smolin, 210). This critical acceleration is ~1.2 x 10^-8 cm/sec², which is close to c²/R = 6.9 x 10^-8 cm/sec² (c is the velocity of light and R is the radius of the observable universe). This c²/R is a measure of the expansion for the accelerating, expanding universe. It is an upper limit of this acceleration, however, not the exact acceleration (see Appendix 4). We ask here if this closeness is significant. To do this, let us refer back to the equations:

             ρ  = φ’ ²/2+ V; and p = φ’ ²/2 – V

Recall that the intergalactic energy density ρ starts very high after the big bang with V dominant so the pressure is negative, causing space to expand. The V is used forming particles and anti particles with an excess of particles (see ref 3, AP4.7C for details of the excess), until f’ ²/2 dominates. Thus the pressure turns positive, and space stops accelerated expansion, but continues ordinary expansion with a low value of V. Then particle space begins to refill with potential energy from tunneling super particles as they break down into particles (see ref 10, AP4.7C for details of the breakdown). Since intergalactic space is refilling mostly with the potential energy V of the super particles and not f’ ²/2, the potential energy eventually dominates, and the pressure turns negative. Thus an accelerating, expanding intergalactic space with vacuum potential energy density of V ~10^-4 GeV/cc results (see ref 12, AP4.7I). The accelerated expansion of space that results is <c²/R (see Appendix 4). We have seen above that the visible and dark matter start to merge close to the edge of the galaxy and become equal at the critical radius where the circular velocity bends over and becomes flat. This is the boundary where the galaxy meets intergalactic space, so the acceleration there should be <c²/R on the intergalactic side of the boundary, and be roughly equal to that acceleration within the galaxy.

Model 1 appears to be capable of explaining one final measurement. Images of galaxies at very large red shifts usually show a wispy, disorganized structure unlike the tightly structured spiral galaxies closer to us in both space and time. We have shown that the inner spherical and outer spiral disk like structure can be explained by a combination of visible and dark matter. If the central black hole were less massive, it would generate fewer super particles (dark matter), and capture fewer visible particles into orbit, thus generating a wispy, disorganized structure. Now, Model 1 notes that the initial mass of the nodes and corridors of the cosmic web was very low as indicated in the section below, so we would expect this wispy, disorganized structure in the early galaxies. As the galaxies build up mass with time, the central black holes increase in mass, and the galactic structure tightens and improves. 

The Corridors of Dark Matter Between the Galactic Halos of Dark Matter.

Assume that two galaxies, Number 1 and Number 2, are close at a distance R₀. Both have central black holes. The total gravitational field from the two galaxies is diminished along the corridor between them as follows:

                         G = M₁G_o/R – M₂G_o/(R_o-R)  (R> R_o)

 In this case, assume a super particle starts moving away from black hole Number 1 due to the potential expansion field surrounding the black hole center faster than the gravitational field can attract it to the black hole. The gravitational field is less along the radial toward black hole 2 than it is along other radials, so a corridor of diminished field is formed. Thus the super particle moves further along that corridor than along other radials, and a buildup of super particles starts along that corridor. The presence of super particles in the corridor increases the density of particles there and thus increases the resistance to movement, and so particles preferentially accumulate along that corridor. Over time, a lattice of corridors (a cosmic web) forms using galaxies as nodes. This cosmic web acts as a nucleation net for dark matter along which dust and gas in particle space will be attracted. This dust and gas will then form galaxies preferentially along the web. This preferential formation of galaxies in strings and clumps and walls has been observed by astronomers.

 

Testing Model 1 with Data

In this paper, we have seen how Model 1 can quantitatively predict:

• The clouds of dark matter needed to explain the process of visible dust and gas coalescing into galaxies.
• The gravitational lensing due to dark matter around galaxies.
• The observed velocity distribution of visible matter within galaxies due to the dark matter halo coming from the black hole at the center of the galaxy.
• The corridors of dark matter that provide nucleation zones for galaxies to form groups, strings and walls.

These predictions form a strong argument for the existence of dark matter and the correctness of Model 1.

 In addition to the dark matter argument, there is a connected argument in favor of the acceptance of Model 1. To construct Model 1 we start with constants from quantum mechanics (Planck energy) and the standard model of particle physics (force unification energy), and choose two new constants (the spherical barrier radius and the barrier thickness). We can then quantitatively explain five observed phenomena that have yet to find a single consistent explanation, namely:

• The origin, characteristics and operation of dark matter (see above).
• The accelerating expansion of the universe (Dark energy)-see ref 8, AP4.7
• The UHECR (cosmic rays) that have been observed in the energy range beyond the GZK cutoff-see ref 12, AP4.7I.
• The huge disparity in the different estimates of the vacuum potential-see Kane, 112.
• The large-scale cutoff and asymmetry in the Microwave Background Energy-see ref11, AP4.7F.

We have used two equations to obtain two unknowns. The results were then used to satisfy the equations for three other phenomena. This is not possible unless the phenomena are connected and part of a single unified physical process.

 In addition to these data tests, there are some significant variations in the edges of the galaxies that should be watched for as tests of Model 1.

• If the black hole (or holes) in the center of a spiral galaxy is not feeding, the outer edge should stop showing evidence of new dark matter, but the evidence should not appear until after a delay time of ~ 10¹² sec (~ 10⁵ yrs) which is the time needed for the dark matter to move to the edge of the galaxy. This means that there may be some spiral galaxies without evidence of a dark matter halo.

•  There should be some variation in the amount of dark matter observed over time. This should show up as bumps in the velocity curve in the outer reaches of a spiral galaxy, and also bumps in the intensity of the spiral galaxy skirt..

 

Summary and Conclusions

A model (Model 1) has been developed in AP4.7 that predicts dark matter, dark energy and ultra high-energy cosmic rays. As part of this model, a set of super particles in a new space (vacuum space) was predicted that constitutes this dark matter. This dark matter tunnels into particle space and generates dark energy and the high-energy cosmic rays. The dark matter also explains

• The clouds of dark matter needed to explain the process of coalescing galaxies.
• The gravitational lensing of dark matter around galaxies.
• The velocity distribution of visible matter within galaxies.
• The corridors of dark matter that provide nucleation zones for galaxies to form groups, strings and walls.

Initial checks with existing data on galactic fringes have been made, and Model 1 has been found to be in agreement with the data. Possible variations in the fringes have been noted, and a program proposed to search for these variations. It was noted that Model 1 has been found to be valid as far as all of the current data checks can determine.

 

Appendix 1

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 = 4π ћ²/M [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, see reference 16, AP4.7E.

 

Appendix 2

A potential energy density of ~10⁵⁰ GeV/cc is found inside the barrier shell in vacuum space.  Inside the barrier shell itself, at a distance of ~ 10^-11 cm from the super particle, V has a value of ~10⁷³ GeV/cc. Beyond this distance is the vacuum value of particle space (10^-4 GeV/cc). As the particle approaches the center of the black hole, the high potential energy is used to form the barrier around the incoming particles, fill it with potential energy, accelerate the particles and thus make super particles and bounce them into vacuum space. Note that the super particles exist in dynamic equilibrium with particles in the high vacuum potential of vacuum space as shown in REF 10, AP4.7C.

 

 Appendix 3

A better and more accurate way to describe the situation of a particle near a black hole is to use general relativity language. The curvature of space-time increases as a test mass moves toward the black hole center. The test mass moves toward the center of the black hole by following the geodesic. Now if there is a potential energy near the center of the black hole, it will tend to null the curvature. If the potential energy is strong enough, it will give an opposite curvature, and there will be bulge around the center that reduces to a flat zone further out and then a cup further out. Thus we have a so-called Mexican Hat curvature. A test mass will then move along a geodesic to the zero curvature zone and circle in that zone. The strength and shape of the potential energy as a function of radius from the black hole center determine the size and shape of the zone of operation of test mass (the force balance zone). We note that further away from the black hole center, the volume increases. In this balance zone, the electromagnetic force, the centrifugal force and diffusion become important, and form the dark matter clouds. Further out, the up curve of the edge of the Mexican Hat helps contain the dark matter. This scalar field provides the only way to overcome the attraction of the test mass to the center of the black hole. Since both the mass and the potential energy come from the mass of the black hole, they can both become equally powerful.  

 

Appendix 4

Consider a universe expanding with acceleration A and a maximum speed c. Over a lifetime T, its size R would be:

             R = AT²

 We note that since:

            c = maximum possible speed    

Then:

            AT < c and so R< cT

Then it must be that:

            A < c² / R

 Note that Model 1 proposes that the universe’s acceleration is not constant, because the potential energy V is not constant. V starts high after the big bang, drops until it reaches the increasing V from tunneling particles, and then rises to its current value (see ref 10 AP4.7C for details). Thus we see that c² / R can be viewed as a kind of maximum average measure of the accelerated expansion of space, but it is not exact, and should not match the accelerated expansion of space at the edge of a galaxy exactly.

 

Appendix 5

More details are needed to show how the particles escape the black hole. We start by noting that in order to get the super particle to enter the barrier shell, it must gain an activation energy sufficient that its total energy is >10¹⁹ GeV (see ref 10, AP4.7C). Also, using the theory of granulated Planck space, we have shown that the velocities of particles in vacuum space with such high energy reach extreme values of velocity much higher than 2.99 x 10¹⁰ cm/sec (see ref 18, AP4.7M). Further, according to Newtonian gravitation theory, Peebles has shown (Peebles, 48), that for a spherically symmetric mass distribution, a particle orbiting the mass within a radius r [M(r)] will have a circular velocity ν_c of:

 

M(r) = ν_c²r /G

 

Now in particle space, when M(r) becomes large enough so that a particle has a velocity ν_c = 2.99 x 10¹⁰ cm/sec, an event horizon forms at that radius because an increase in ν_c is not possible to balance the force from an increase in M(r). Thus, a particle that moves inside that radius will continue to fall toward the black hole center. A more detailed argument requires general relativity, but the conclusion is the same. Now we have noted that in vacuum space, a super particle at the edge of the diffusing cloud of super particles, has a velocity ν_c greatly exceeding 2.99 x 10¹⁰ cm/sec, so it can maintain orbit at a much smaller radius. Thus super particles can maintain orbit at a radius well within the particle space event horizon. Thus when a super particle is formed and moved into vacuum space by increasing its energy to the excitation value, the resulting high velocity of the super particle places it outside the event horizon. Again, general relativity is needed for a more accurate treatment, but the result is the same. From there, the mean free path is so great that the super particles pass on out to the critical radius of the galaxy with a density dependence of 1/ r² as noted in the main text.

References

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

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

3.      J. Magueijo, New Varying Speed of Light Theories, arxiv.org/pdf/astro-ph/0305457.pdf

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

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

6.      J. D. Cobine, Gaseous Conductors, Dover publications, Inc., New York, 1958

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

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

9.      L. H. Wald, “AP4.7D HOW TO PROVE A THEORY’S CORRECTNESS” www.Aquater2050.com/2015/12/

10.  L. H. Wald, “AP4.7C DARK MATTER RATE EQUATIONS” www.Aquater2050.com/2015/11/

11.  L. H. Wald, “AP4.7F GATHERING DARK MATTER FOR THE BIG BANG AND ITS IMPACT ON MBR” www.Aquater2050.com/2015/12/

12.  L. H. Wald, “AP4.7I THE SUPER PARTICLE AS A COSMIC RAY” www.Aquater2050.com/2015/11/

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

14.  L. H. Wald, “AP4.7J CYCLE 1 AND THE CYCLICAL UNIVERSE” www.Aquater2050.com/2016/01/

15.  L. H. Wald, “AP4.7D HOW TO PROVE A THEORYS CORRECTNESS” www.Aquater2050.com/2015/12/

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

17.  L. H. Wald, “AP4.7E INTERACTION RATE OF DARK AND VISIBLE MATTER” www.Aquater2050.com/2015/12/

18.  D. Halliday, Introductory Nuclear Physics, John Wiley and Sons, New York, 1955.

19. L. H. Wald, “AP4.7M VARIABLE LIGHT SPEED IN MODEL 1” www.Aquater2050.com/2015/12/

20. Ferraese, L. and Merritt, D. “A Fundamental Relation Between Supermassive Black Holes and their Host Galaxies” The Astrophysical Journal The American Astronomical Society. 539 (1) (2000-08-10)