Abstract

In a previous paper (Wald, Model 1-A), 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?
• How can the theories of symmetry and the Higgs field be used to identify the origin of the Higgs field and calculate the masses of the fundamental particles?

A self-consistent theory called Model 1 has been developed that answers these questions quantitatively. The unique features of this model are:

• There are two spaces in the universe, particle space and quantum vacuum space. A potential barrier composed of Higgs field 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 visible matter space through black holes where they gain energy and are converted into super particles (energy ~10¹⁷GeV) that operate with unified force behind the potential barrier (potential energy ~10¹⁹GeV), and become shielded super particles of dark matter.

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

• While in particle space, the super particles tunnel through the barrier shield into particle space. The super particles without their shield are unstable and break down into cosmic rays (protons) with ultra high kinetic energy (UHECR’s). 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 ultra high energy protons are observed as cosmic rays with energy between the energy of force unification (~10¹⁷GeV) and the barrier energy (~10¹⁹GeV), which is beyond the GZK cutoff. 

In a prior paper (Wald, Model 1-A), a type of dark matter called a shielded super particle was predicted by the mass-energy determination portion of Model 1 that predicts the observed mass-energy of the particles of the standard model. In this paper, the dark matter of Model 1 will be shown to satisfy the existing data on the characteristics and operation of the observable dark matter.  

 

The Problem

A prior paper (Wald, Model 1-A) predicted the formation of shielded super particles along with the particles of the standard model. It also showed how a cloud of shielded super particles would form by entering vacuum space through a black hole. In this paper a detailed description of the distribution, formation and rotation of the dark matter particles that threads through the universe like a web and surrounds galaxies like a halo will be described and shown to be needed in order to:

• Explain the dark matter needed to allow the galaxies to coalesce into stable spirals.
• Explain the velocity distribution of visible matter around the galactic center of spiral galaxies.
• Explain the strings, clumps and walls of galaxies observed by astronomers.

Model 1 is proposed as a means to show how dark matter is formed, how it operates, and to show the need for it to explain the existing body of observed data on galaxies in the universe.

 

The Solution

The Dark Matter Cloud Needed to Coalesce Galaxies.

We recall from the paper (Wald, Model 1-A) that there is theoretical evidence that extremely heavy particles (super particles) exist and that they are shielded (see Appendix 1). We note also that there is evidence that the shielded super particles are dark-that is they have a low interaction cross-section with matter particles (Appendix 2). Finally, there is evidence that both high energy (vacuum space) and low energy (Particle space) vacuums exist (Appendix 3). Also, we noted that there is experimental evidence that dark matter exists. We see first that gravitational lensing, which can be explained by dark matter has been observed around galaxies (Peebles, 272). Also the velocities of galaxies within groups (Peebles, 417) have been observed that are large enough to imply the existence of dark matter. In addition, the Dicke coincidences argument (Peebles, 364) can be used to imply the existence of dark matter.  Thus, the argument for the existence of dark matter is strong. The implication is that it may be needed to form galaxies from the dispersed matter particles generated by the big bang. In this section we ask if dark matter is a requirement for galaxy formation.

In order to answer this question, we must describe the equations that control the expansion and contraction of space. In order to provide for both expansion as well as contraction, Peebles postulates the existence of a new real scalar field, and develops the equations for the field density ρ, and the pressure p in space from general relativity.The expansion and contraction of space is controlled by the following field density  (ρ) and pressure (p) equations from general relativity (see Peebles, 396):

 

            ρ = φ’ ²/2+ V = field density        (1)

             p = φ’ ²/2 – V = pressure             (2)

 

            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, from the field equation of general relativity, Peebles develops the cosmological equation for the time evolution of the expansion parameter (a(t)) due to mass-energy density ( ρ_m) and pressure (p) (see Peebles, 75):

 

              ä/a = -4/3πG ( ρ_m+ 3p)                  (3)

                      = acceleration of the cosmological expansion parameter

           

Note from the field pressure equation (2), that if the potential energy exceeds the field kinetic energy, the pressure is negative. Then, if the negative pressure term in the field equation is large enough to exceed the mass density term in equation (3), 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, however, the field pressure term is positive, the acceleration ä/a is negative, and space contracts.

Finally, it has been suggested that mass is the conserved charge for the field for gravity by Schumm (Schumm, 10) and others. If this suggestion is true, it will result in the following equation for the field emanating from a massive particle, which we will call φ_h.

 

            φ = K_ho m /r²                          (4)             

 

It is important to understand that the field described above as a real, scalar field, is associated with spatial curvature as is mass-energy, and has the general characteristics of the real part of the Higgs field. So we will assume that the real part of the Higgs field accounts for the expansion and contraction of space, and so we conclude f  = f_h. But we must remember that the total Higgs must be complex as will be seen in Appendix 1.

In order to understand the formation of galaxies, we start at the big bang with matter formation by conversion of potential energy into a thick soup of cooling massive particles (mostly hydrogen and helium ions, electrons and atoms-see Wald, Model 1D) and photons that have just condensed from the hot mixture of the starting universe. In addition, we would expect some shielded super particles (dark matter) mixed in with the standard particles (see Wald, Model 1-A). The potential energy is high, so space expands. As space expands, the particles contained in it fly apart, and the relative velocity of the particles is reduced. Thus the particles collide with smaller and smaller velocity, and so the average kinetic energy (temperature) becomes less and less. At the same time, the vacuum potential energy is dwindling, as it is formed into particles.

The particles are not formed into a uniform mixture. There are large and small clusters of particles and gas. It is important to note that the smaller clusters are attracted to the larger clusters by gravity, but they are not swept immediately into orbit. It is well known that as a small cluster of particles falls toward a larger cluster, it will gain enough kinetic energy in falling to enable it to escape from the gravity of the larger cluster, and so it will not enter 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. There are two ways for a cluster to lose energy:

• An inelastic collision with another particle or small cluster moving close to the large cluster.
• A gravitational interaction that deflects a few third particles in the vicinity of the larger cluster thus causing turbulence and/or the ejection of the third particles that results in energy loss in the falling small cluster. 

So we see that to form galaxies, we need:

• Massive, slow moving or cold particles (dark matter) or clusters of particles to provide a center of attraction.
• Less massive groups falling toward the massive particle or cluster to provide the orbiting mass.
• Small mass groups of particles close to the centers of attraction to absorb energy from the falling mass.

Thus we conclude that dark matter is a requirement to form a universe with galaxies.

There are two initial conditions for the universe that are possible and important in determining the characteristics of a universe with galaxies.

·       An Initial Universe without Dark Matter structure from a Prior Universe

Here there would be some shielded super particles formed because the energy of the big bang would be high enough to form them, but there would be no “shadow web” of shielded super particles in a structure from a prior universe, so the universe that results would be thin and irregular, but gradually form structure with time. We would expect several recyclings to be needed to get significant structure.

·       A Recycled Universe containing Remnants of Dark Matter from a Prior Universe

Here there would be a “shadow web” of shielded super particles left over from a prior universe that could form a nucleation structure for new galaxies. The remnant of shielded super particles would provide a set of massive centers for new galaxy formation. These centers would be formed into a set of structures such as corridors, walls and groups reminiscent of the prior universe. The new particle groups formed by the new big bang would fall toward the massive centers, and collect into stars. When the density of new particles is high enough so that turbulence can siphon energy from falling particle groups, galaxies will form. The universe that forms would have many corridors, walls and groups 

Our universe has significant web structure in that it contains groups, clusters, corridors and walls of galaxies, so we conclude that our universe is likely to be a recycled universe. Thus it probably started with a “shadow web” of shielded super particles.

 

The Velocity Distribution of Visible Matter in Spiral Galaxies.

Peebles (Peebles, 47) has described the problem of galactic rotation of spirals quantitatively in his Figure 3.12. Rotating spiral galaxies can be divided into two parts. The rotation of the central part of the galaxy can be quite well explained by Newtonian gravity. The outer part rotates too fast, however, to be explained by Newtonian gravity; it would fly away from the central part. Peebles developed 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              (4)

 

                Where

            ν_c = circular rotation velocity as a function of radius.

            G = gravitational constant.

 

Thus visible matter alone cannot explain the stability of spiral galaxys 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:

• Dust, gas and some stars spiral down toward the center of the galaxies and increase the mass there. The mass in the central zone of each galaxy determines the velocity v_c needed to remain in orbit. Particles with velocity < v_c fall toward the center of the galaxy. Where v_c = c all particles (including photons) fall toward the center of the galaxy, and cannot have higher velocity to escape, so an event horizon is formed. In this way, a central black hole is formed (see Appendix 4 for more details)
• Massive particles fall until the density of particles is high enough to create a diffusion zone within the event horizon formed by a contraction zone layer with an expansion zone layer beneath it. Massive particles fall through the contraction layer and then are reflected by the expansion layer, and then fall back through the contraction layer again thus forming a circulation of particles within the zone that generates the conditions for diffusion (see Appendix 5 for details). The diffusion moves the highest energy super particles outward toward the event horizon (see Appendix 6 for the diffusion equations).
• Super particles are formed in the contraction zone, and covered in a barrier shell in the expansion zone to form a shielded super particle (see Appendix 1 for details).
• The highest energy shielded super particles have energy close to the Planck energy, and so have a velocity greater than 2.99×10¹⁰ cm/sec, (see Wald, Model 1-D Appendix A). Since this velocity is greater than v_c, they escape through the event horizon into the galaxy, and then into intergalactic space. Beyond the event horizon of the central black hole, the density of shielded super particles (dark matter) 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 the edge of the galaxy (see Appendix 5) creating a dark matter halo with a density that reduces as ~1/r² from the black hole at the galactic center to the edge of the galaxy.
• Now as shown in equation (4) above, 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 galaxy’s 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.
• In addition to this 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 that the volume of the diffusion zone where shielded super particles are formed, and thus the number of shielded super particles formed increases as the black hole mass increases. So when the 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 dependence of the black hole mass on the velocity dispersion.

 

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 shielded 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 shielded super particle moves preferentially 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. Astronomers have observed this preferential formation of galaxies in strings and clumps and walls.

 

Summary and Conclusions

A model (Model 1) has been developed that predicts dark matter, dark energy and ultra high-energy cosmic rays. As part of this model, a shielded super particle in a new space (vacuum space) was predicted that constitutes this dark matter. The dark matter explains

• The matter needed to explain the process of coalescing galaxies.
• The velocities of galaxies within groups that have been observed to be large enough to imply the existence of dark matter.
• The gravitational lensing of dark matter around galaxies.
• The velocity distribution of the visible matter within galaxies that imply the existence of a dark matter halo.
• The cosmic web that provides nucleation zones for the formation of galaxies in groups, strings and walls.

Initial checks with existing data on galaxies have been made, and Model 1 has been found to be in agreement with the data for all of the above phenomena.

 

Appendix 1

Here we give a more complete description of the quantization of the fermion and Higgs fields. We will use the same procedure to develop the mass of a particle from the Higgs field that is used to develop the energy of a photon from the electric field of an atom, and and compare it with the quantum electron case. Kane starts with the Lagrangian (see Kane, 98) of a Higgs field of a particle passing through an ambient Higgs field caused by many particles in a vacuum.

 

            T – V = ½∂_μφ ∂ ^μφ – (½μ²φ ²+ ¼ λφ ⁴)

 

Note that the potential energy V is related to the Higgs field as follows

 

            V = ½μ²φ ²+ ¼ λφ⁴

 

Where:

            λ = self interaction coefficient

            φ  = Higgs field – a complex quantity

 

The first term appears to be the interaction of the particle Higgs field with the ambient Higgs field. The second term appears to be the interaction of the particle field with itself and the ambient Higgs field. To find the excitation energies, and thus the masses, we must find the minimum of the potential and expand around the minimum to get excitations, which are the particles. In field theory, it is conventional to call the minimum the ground or vacuum state, and the perturbation terms are excitations. The form of the Lagrangian determines the mass of the particles. If these operations are performed, the result is:

 

m _η²  = -2 μ² = mass               

 

Note m _η is a complex quantity, so to get particle mass m_p (real), we must use m _η² to get the real quantity. Then, V can be written:

           

V = – m _η²φ ²+ ¼ λφ ⁴= -(m _η²– λ_tφ ²)φ ²         (1)

 

And if we note that particle mass energy is quantized as m_pη, then if the quantum state of the Higgs potential energy shifts, a massive particle is formed as follows:

 

ΔV_η = m_pη = m_η²φ ² (1- λ_tφ ²/m _η²) = particle mass energy        (2)

 

            Where:

            m _η² = mass charge

           

Now we note that as with the electron, each charged particle is moving in a reentrant pattern determined by its symmetry, so the mass energy is low due to destructive interference unless the particle orbit distance traveled is exactly n particle wavelengths long (n = 1,2,3…). This interference causes the quantization of the potential energy. Then the particle interferes constructively, and the potential energy wave function of a particle as it moves around its orbit in its symmetry pattern is the product of factors (see Kane, 90) as shown in the equation:

 

            V_n= m _η²φ² = (symmetry factor)(charge factor)(field factor)

 

The value of each factor is as follows.

 

The Symmetry factor specifies which symmetry is active in a particle, and because of constructive interference, each term becomes:

            (Electromagnetic factor U(1)) = (1)

            (Isospin factor SU(2)) = (2)

            (Color factor SU(3)) = (3)

            (Higgs factor SU(4)) = (4)

So the (symmetry factor) = (1x2x3x4) = (n!) where only the quantum numbers active in the particle appear in the factor.

 

 The Charge factor specifies which charge is active in a particle:

            (Electromagnetic charge) = q_e

            (Isospin charge) = q_i

                (Color charge) = q_c

                (Mass charge) = m_p = the charge that generates mass-energy (ref 2, Model 1B).

So the (charge factor) = (q_e q_i q_c m_p) = q_t where only the charges active in the particle appear in the factor.

 

The field factor is more complicated. Two Higgs fields are important in mass generation. One Higgs field is from the distributed mass in the universe, which is relatively uniform on a large scale. We call this the ambient field. It can be described by the equation:

            φ_h1² = K_ho Σ m_p /r² = K_ho Σ m _η²/r² ~ K_ho n

Here it is assumed the density of particles is small, so on average a test particle is not close to another massive particle. 

The second Higgs field is due to self-interaction, and so is controlled by the mass of Higgs particles m _η that are close. This field can then be described by the equation:

            φ_h2² =  K_ho λ_t m _η²/r², where λ_t is the self interaction constant.

Recalling that m _η² is small unless r is such that the distance traveled in a pattern orbit is an integral multiple of the particle wavelength (n), we find:

            φ_h2² =  λ_t (1/n), Where λ_t is the self interaction constant.

If m nearby Higgs particles are active with charge, and recalling that the self-interaction term is negative (see 2), then the total is:

            φ_h² =  φ_h1² + φ_h2² =  n² (1- λ_t (m/n))             (3)

 

But as Kane noted (Kane, 105), as part of the Higgs mechanism, the Higgs field must be assigned an SU(2) doublet     m’, where:

            m’ = either of the two quantities:

            m = Higgs charge operating with ambient field

            (m+(m-1)) = Higgs charge operating with ambient and internal fields

Then:

            n² becomes n^2m’

Comparison with the field equation derived by Kane (2) allows us to replace it with (3) to get the quantized field factor:

            (Field factor) = n^2m’(1- λ_t(m/n_m)),              (4)

           Where:

            n = symmetry number for the particle of interest

            n_m = 4 = quantum number for mass.

            m’ = m (up), or

                  = (m+(m-1)) (down)

 

Then the total potential energy term is:

            V_n= m _η²f ² = (symmetry factor)(charge factor)(field factor)                

                 = n! q_t hcK_o n^2m’ (1- λ_tm/4)          (5)

                 = mass energy of the primary particles (see equation 2)      

            Where:

            q_t = total charge

            hcK_o = Higgs mass-energy constant (analogous to R_∞(q_e) for the electron case)

            λ_t = self interaction factor

            m = Higgs field number

Now we have the power to calculate the mass of the primary particles, but we still must establish how to calculate the confinement mass of composite particles such as the proton and the neutron. For composite particles, a procedure similar to the above one yields:

           V_nc = (Particle 1 factor)(Particle 2 factor)(Particle 3 factor)

                     (Symmetry factor)(Charge factor)(Field factor)

                  = (V_n1)(V_n2)(V_n3)(n! q_t hcK_o n^2m’ (1- λ_tm/4))

We note here that if many particles become active together, we get modifications in equation (5). There are three cases.

(1)  If three bosons are acting together on fermions (as in two cases of the Higgs mechanism with SU(2) symmetry), the self interaction term becomes:                    

            (1-λ_tm/4) = (1-λ_t(1/3×2/4)) if one particle of three is acting on m at a time                

                               = (1-λ_t(2/3×2/4)) if two particles of three are acting on m at a time

(2)  If three bosons and two fermions are acting together at a time (as in the second case of the Higgs mechanism with SU(2) symmetry), then:          

            (1-λ_tm/4)  = (1-λ_t2/3(2/4(2/4+3/4)) = (1–λ_t 2/3×5/8)

(3)  If two fermions are acting together (as in proton formation from quarks), the self interaction term becomes:

            (1-λ_tm/4) = (1-λ_t(2/4(2/4+3/4)) = (1-λ_t5/8)

 

Finally, we must specify the units of the mass-energy for the particles. There are two cases.

(1)  If hcK_o = 1, and λ_t = 1, the energy for each Fermion is given in MeV

(2) If hcK_o = 1, and λ_t = 1, the energy for each Boson is given in GeV

Kane notes that three forces appear to achieve the same value at ~10¹⁷ GeV. It also appears that four forces (including gravity) may unify at a somewhat higher value (10¹⁹ GeV?) (see Kane, 281), and this fact fuels the speculation that the forces unify at high energy. We note the following combinations:

• U(1)xSU(2)xSU(3) = SU(6) = The electromagnetic, weak and strong forces are all combined (Grand Unification)
• U(1)xSU(2)xSU(3)xSU(4) = SU(10) = All forces are combined. (Complete Unification)

In contrast, we note the following:

• SU(5) = SU(2)xSU(3) = unification of only weak and strong forces.
• SU(7), SU(8) and SU(9) have the same problem. They can unify only part of the forces.

The symmetry term shown above, demands that we must account for the symmetry of all of the lower symmetry forces in order to calculate the mass of a particle. Thus all combinations are forbidden except SU(6) and SU(10). As a result, the proton decay lifetime experiment, with short lifetime calculations based on SU(5), is expected to fail, as it appears to be doing (Kane, 289). But the super proton SU(6) and the barrier shell SU(10) are expected to be formed along with the standard model particles if the energy conditions are correct.

For the quark (3,3) case above, the (1-1/4) value for internal states was expected, which would give a mass value of 3.28GeV, but this value is a poor fit to data. However, the (1-0) value, which was used, gave the best fit to data. The (3,(3+2)) case would give the same value for either case. But the (1-0) value goes with the (4,6) case, which is the Grand Unification value. Note that the (3,2) value is too high as well, and it is also a down quark. It appears that Grand Unification case dominates the particle formation process, and it even distorts the (3, 2) value for d. It appears that we have accidentally stumbled on the components of a Grand Unification proton in cases (3,3) and (3,(3+2)). When we combine these quarks into a super proton, as shown above, we get a mass energy value of ttb-5.8×10¹⁶ GeV, a value very close to the expected ~10¹⁷ GeV. Also, the τ particle (1,777MeV), case (2,(3+2)) appears to be the electron equivalent for this super particle, and will be called the super electron.

The super proton is expected to be unstable in low energy particle space, and break down into an ultra high kinetic energy proton and give up potential energy. In order to maintain a high potential energy environment where it can recombine into a super particle and so maintain its lifetime, a potential energy barrier shell is required. It must have a potential energy greater than the super proton energy (5.8×10¹⁶ GeV) to contain it, but less than the Planck energy (1.22×10¹⁹ GeV), say ~10¹⁹ GeV. The (4,10) Higgs field appears to fit at potential energy 4.00×10¹⁸ GeV. Note also that as a Higgs field, it acts on mass, so electromagnetic force and photons penetrate the barrier, but mass does not, except under special circumstances.

 

Appendix 2

Here, we explore the characteristics of super particles to see if they can be observed in particle space. We ask are they dark? In Appendix 5, we ask are they massive? First, super particles do not show charges associated with the electromagnetic, weak, and strong forces. They are combined into one super charge and hidden behind the barrier potential. The super particle spin, if any, would not show beyond the barrier as well. They have only the super charge associated with the unified force. Thus they will not interact with the detectors we normally use. Particles in particle space will scatter off the potential barrier surrounding the super particle, however, so it is necessary to calculate this scattering cross section. This scattering cross section is like the scattering of a proton off a neutron, but with different energies. This scattering cross-section has been calculated (Halliday, 47), and is as follows:

 

            σ = 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:   σ  = ~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.

 

Appendix 3

Kane shows (Kane, 112) that 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. 

 

Appendix 4

A perhaps 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. 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 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 5

As particles move toward the black hole center, they encounter a zone with two layers separated by a barrier. In these two layers, the time rate of change of field f’ due to the motion of particles across layers varies according to the layer:

1. Contraction layer— As particles move past the event horizon toward the black hole center, they become more and more concentrated, and the mass density ρ_m increases. This increase causes a corresponding increase in field f according to equation 4. So particles with high average velocity toward the black hole center will pass through this layer with high φ’. Potential energy V increases as radius to the center decreases (Misner, 911), but φ’ ²/2 increases faster, so φ’ ²/2 exceeds V, and this is a space contraction zone.
2. Expansion layer—As the particle concentration increases deeper in the contraction layer, diffusion moves particles away from the deep, concentrated zone toward the shallow lower particle density zone, and particle density gradient diminishes (see equation 5). The result for a sinking particle is a reduction in φ’ ²/2. This movement to lower density is faster for higher kinetic energy particles. Thus φ’ reduces, but V is increasing, so φ’ ²/2 eventually becomes smaller than V . and this zone becomes an expansion zone.
3. Barrier—When particles move across a boundary between low and high potential energy, the boundary acts like a barrier, and if the kinetic energy is lower than the potential energy, most of the particles are reflected. This reflection accentuates the action of the diffusion, and reduces φ’ ²/2 even more.

Thus there will be a layer where space will contract, and just inside it there will be a shell where space will expand. The boundary between will reflect particles. So a particle will descend through the contraction zone toward the black hole center, and then most will be reflected back by the expansion zone inside it. The higher the kinetic energy of the reflected particle, the closer it will get to the event horizon. If it is high enough, the kinetic energy will near the Planck energy (1.22×10¹⁹ GeV), and the speed of light will increase above 2.99×10¹⁰ cm/sec in order for the Planck length to remain constant under relativistic foreshortening. When the speed increases above 2.99 x 10¹⁰ cm/sec, it is above the escape velocity of the event horizon formed at lower energies, so the shielded super particle will escape the black hole. These escaping particles may be the radiation from the black hole proposed by Hawking. If the kinetic energy is too low, the shielded super particle will fall back to the barrier and gain kinetic energy in the fall. There it will be reflected again and thus recycle until the particle gains enough kinetic energy to escape the event horizon. Note that there is no mass singularity at the center of the black hole. The diffusion will tend to drive particles from the high density interior, and the potential barrier will slow their return, so the growth of the central mass of the black hole is controlled. 

 

Appendix 6

In the diffusion zone, the average kinetic energy is 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 expected to be valid (Cobine, 51). 

 

            V⁺ = -D⁺/n⁺ dn⁺/dx + K⁺E

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

 

            Where:

            D = Diffusion coefficient

            K = Ion mobility under the influence of electric field 

            V = Ion velocity

            n = Ion concentration

            E = Electric field

           

We set:

            V+ = V-  = V ;  n⁺ = n ^– = n ;  dn⁺/dx = dn^–/dx = dn/dx

 

We solve these equations, and get:

 

            n = (N_o(r)/4pDt)^3/2 exp(–r²/4Dt)         (5)

 

            Where:

            D = (D⁺ K^– + D^– K⁺ ) / ( K⁺+ K^– – 2 K⁺ K^– ) = 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 can represent the time span that the black hole is 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 resulting bubble of particles to flatten out) can be long for intense matter intakes. 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 formation rate of super particles in the diffusion zone of the black hole.

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.

 

The mean free path of the super particles is small enough to be called diffusion in the diffusion zone of the black hole. These super particles within their barrier shell are called dark matter (n = ρ_d(r)), because they have a low interaction cross-section with protons (~10^-45 sq cm) and so are difficult to detect with visible matter detectors (see Wald Model 1-A). 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. Thus the super particle (dark matter) density (ρ_m (r) will vary as:

 

            (v_m (r) ~ K / r².

 

When the super particles reach intergalactic space, the density will gradually 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 the central black hole of the galaxy.

 

References

1. L. H. Wald, “Model 1-A” www.Aquater2050.com/2017/01/
2. P. J. E. Peebles, Principles of Physical Cosmology, Princeton, New Jersey, Princeton University Press.
3. B. A. Schumm, Deep Down Things, Baltimore, Johns Hopkins University Press, 2004.
4. 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)
5. D. Halliday, Introductory Nuclear Physics, John Wiley and Sons, New York, 1955.
6. G. Kane, Modern Elementary Particle Physics, Ann Arbor, Michigan, Perseus Publishing.
7. Misner, Thorne and Wheeler, Gravitation, New York, Freeman and Co., 1973.
8. J. D. Cobine, Gaseous Conductors, Dover publications, Inc., New York, 1958