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. 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 5, 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 working out this model, it became necessary to establish the characteristics of light speed in vacuum space and barrier space as well as particle space because the speeds appear to be different. In this paper, these characteristics will be analyzed, and connected to Model 1.

 

The Problem

In Reference 1, AP4.7F, a problem was discovered with respect to travel near the speed of light. In order to gather the particles from the entire observable universe to a Principal Black Hole for a big bang, travel of high-energy particles at a speed faster than the speed of light in particle space was needed. Such a speed has been proposed for extremely high-energy particles (Magueijo, 31 and Amelino-Camelia, 1), and found to be theoretically sound, but the theory has not been connected with Model 1. In this paper, this connection will be made.

 

Solution

In order to make relativity and a constant speed of light compatible with a constant Planck length, Magueijo and Amelino-Camelia, note (Magueijo, 31 and Amelino-Camelia, 1) that a modified or deformed speed of light for photons is needed: They worked out this deformed speed of light, but it is not immediately applicable to Model 1 because of the different spaces in Model 1. Model 1 requires that the modified speed of light be defined in each of three spaces:

• Particle space
• Vacuum space
• Barrier space

Here we start by noting that the speed of light c is no longer assumed to be a universal constant in this paper. We need a new maximum speed to use in the equations of relativity for the Lorenz factor instead of c, where the Lorenz factor is.

                        γ = 1/(1- v²/c²)^½  

In this paper, we will call this new maximum speed v_o.

In order to accomplish this task, we start in particle space with the Planck constants. Specifically, Planck length, time and energy are considered to be the operating parameters of a Planck granule:

                        l_p= 1.62 x 10^-33 cm

                        t_p= 5.38 x 10^-44 sec

                        E_pl = 1.22 x 10¹⁹ GeV

Here we assume space is quantized or granular on a Planck scale (see Appendix 3). Then a particle (light or massive particle) travels along a series of Planck granules of size l_p separated by spaces or gaps of size l_p. This size cannot be reduced.The granules have potential energy E_pl, and the gaps have zero potential energy. Now a quantum of light travels a Planck length (which is a minimum and unchanging length) in no less than a Planck time (which is the minimum time possible). Also, there is a time needed for a particle to cross the gap from one granule to another. In order to calculate this time, we must define creation and destruction operators.

We construct a field normalized to a single quantum of energy ω and momentum k. The super particle (and a super photon) can be thought of as being created and destroyed by creation and destruction operators from the field as follows (Kane, 21):

            Φ = 1/(2ω) [ae^{ί(k·r – ωt)} + a^†e^{-ί(k·r – ωt)} ]

            Where:

            a^† creates quanta.

            a destroys quanta.

            k = k₁= (8p²m(V₀-E)/h²) ^½ = momentum of the super particle in vacuum space, or momentum of the particle in particle space depending on E and V₀.

            V₀ = potential energy.

            E = p²/2m+ E_mo

            E_ke = kinetic energy = p²/2m

            E_mo = rest mass energy = m_oc²

Note that relativistic foreshortening cannot reduce the Planck length because it is the minimum possible, so a modified speed of light is required. It is important to note that the potential energy V₀ that is shown above is the potential energy of the particle we are observing. It is low in particle space (<1 ev), high in vacuum space (~10¹⁷ GeV), and super high in barrier space (~10¹⁹ GeV). At the same time, the potential energy of the Planck granule is 1.22 x 10¹⁹ GeV– the highest possible.

Now each time a particle (or photon) crosses a granule, it takes a minimum time

                       ∆t = t_p,

to move a distance:

                        ∆l = l_p

and this movement is lossless because losses only occur at the edges of the granule where energy changes.

Each time a granule boundary is crossed, the rest mass must cross out of the potential energy zone of the Planck granule into the gap and then back into the potential energy zone of next Planck granule. This is destruction of a particle with potential energy, and creation of a particle without potential energy, and then destruction of a particle without potential energy and creation of a particle with potential energy again. The distance moved in the gap is l_p again because it is the minimum possible. According to the uncertainty principle, this process requires a time:

                        ∆t_c = gap crossing time = ћ/∆E_c

Then, if n granules of uniform space are crossed, the maximum velocity that the particle travels is:

                        v_o = 2n∆l /[n∆t + n∆t_c]                     

                              = 2∆l /∆t[(1 + ∆t_c /∆t

            Where:

                        ∆t = granule crossing time

                        ∆t_c = gap crossing time

Here, we have noted that in crossing a Planck granule and a gap, we must move a distance 2∆l. Then, noting the Planck values and the uncertainty principal, we get:

                        l_p/ t_p = 2.99 x 10¹⁰ cm/sec = c  

                        ∆t_c /∆t = (ћ/∆E_c) / (ћ/∆E) = ∆E / ∆E_c  

We also note that the total energy change is:

                        ∆E= (V₀-E), and E = E_ke + E_mo

So for a particle to go from a Planck granule down into the gap where V₀ is zero, the energy change is,

                        ∆E= -(-E_ke – E_mo) 

Then, to get up into the next granule, a particle moves its energy from the gap ground level to E_p where V₀ is non-zero. The energy change required is,.

                        ∆E_c = +(V₀ – E_ke – E_mo )

It is important to note that a particle or wave packet with energy near the Planck energy has a size near the Planck size, so each particle or packet will be absorbed by sequential granules with high probability. For particles or wave packets of lower energy, the wave front is very much wider than a Planck granule, so it can pass by one Planck granule of the vacuum with high probability unless the wave front meets the edge of a large number of granules in a uniform layer all at once, so the probability of absorption is increased. Remember, however, that the particle or photon must be absorbed by only one granule according to the rules of quantum mechanics. The substance (energy content) of the particle or wave is subject to the energy-time uncertainty principle for the layer, so that substance must respond appropriately. The uniformity of the layer needed to command a response is dependent on the energy of the particle or photon. The higher the energy, the more precise the wave front, so the more uniform the layer must be. This requirement is the basis for the equations use above. In deep space, regularity in arrangement would be expected, that is the arrangement of granules in space will be uniform and regular across a broad area and thus present a potential barrier made up of many Planck granules across a complete wave front. Thus the above ∆E_c and ∆Eequations will be applicable (see Appendix 2 for details). Irregularities can happen at some boundaries in space, however, such as at the edge of a galaxy, of a solar system, a cosmic web edge or the edge of the observable universe, where shock zones exist. There, a new equation applies that accounts for a different potential and kinetic energy changes in those zones. This point will become important in case 5 below.

Several cases are important for Model 1.

1. For photons in particle space, the rest mass (E_mo) is zero, so:

            v_o = 2∆l /∆t [(1 + ∆E/∆E_c] = l_p/ t_p ½[(1 – E_ke/(V₀-E_ke)]

                  = 2.99 x 10¹⁰ / ½[(1 + E_ke/(E_ke-V₀] cm/sec  

We note that for normal temperatures, E_ke is greater than V_0. In fact, V₀<E_ke<10¹⁷ GeV, since for E_ke  >10¹⁷.the photon enters vacuum space. For these moderate temperature conditions, speed is constant at 2.99 x 10¹⁰ cm/sec. If the temperature is extremely low (cryogenic temperatures), E_ke nears V_0, and v_o can be very small. In fact, it can even go to zero. Thus light speed is constant except in extreme regimes as relativity requires, and ∆l = l_p = constant, as quantum mechanics requires

2. For particles in particle space with non-zero rest mass, E_mo >> E_ke, then:       

            v_o = 2∆l /∆t [(1 + ∆E/∆E_c] =  l_p/ t_p ½[1 – E_mo/(V₀-E_mo)]

                  = 2.99 x 10¹⁰ / ½[1 + E_mo/( E_mo-V₀)]

Note that V₀ is small compared to the particle rest mass. We also note that V₀<E_mo<10¹⁷ GeV, because a particle with energy E_mo >10¹⁷.GeV will become a super particle and enter vacuum space. Note also that particles with non-zero rest mass will always have a speed less than the maximum for photons, as relativity requires.

3. For super photons in vacuum space, the rest mass (E_smo) is zero, and E_ske > V_so so:

            v_so = 2∆l /∆t[1 + ∆E/∆E_c] =  l_p/ t_p ½[1 – E_ske/(V_so-E_ske)]

                   = 2.99 x 10¹⁰ / ½[1 + E_ske/(E_ske-V_so)] cm/sec            

The potential energy V_so ~10¹⁷ GeV, so the energy occupation zone is 10¹⁷< E_ske< 1.22 x 10¹⁹ GeV. Thus photon velocity becomes constant, as E_ske gets larger than V_so. It gets small as E_ske nears V_so.

4. For super particles in vacuum space with non-zero rest mass, E_smo+E_ske > V_so so:         

            v_so = 2∆l /∆t[(1 + ∆E/∆E_c] =  l_p/ t_p ½[(1 – (E_smo+ E_ske)/( V_so-E_smo-E_ske)

                    = 2.99 x 10¹⁰ / ½[(1 + (E_smo+ E_ske)/(E_smo+E_ske– V_so)]

Again the potential energy V_so ~10¹⁷ GeV, so the energy occupation zone is 10¹⁷<E_smo+E_ske< 1.22 x 10¹⁹ GeV. Thus photon velocity becomes constant as (E_smo+E_ske) gets larger than V_so.

5. For super particles in barrier space the potential energy V_sbo 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. Thus 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:

            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

 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 = 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 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.

Thus we are able to conclude that according to Model 1, the super particle (and photon) velocity is large enough to allow for the gathering of the high-energy particles of the distribution to the big bang from the entire observable universe when the super particle energy mean approaches the Planck energy and starts to flow out as a big bang. This velocity is not infinite, however. This extreme velocity cannot be maintained in vacuum space at lower energies, so lower energy super particles will not be completely gathered to the big bang, and the entropy of the new universe will be slightly increased with each big bang cycle.
 

Testing Model 1 with Data

Now the idea that light speed is variable is known to be controversial, and so it will be of interest to know what happens to the predictions of Model 1 if it were not true.

• If Model 1 has a constant light speed in barrier space instead of the varying speed, the black holes that are spilling super particles rapidly over the top of the barrier will be operating independently, and the large, universal “big bang” will not occur. What will happen is a group of little bangs instead, and that means that the origin of each new universe will be more spread out in space and time than is currently thought to be true. Current data support a small spatial and temporal distribution for the big bang.
• The little bangs would impact the initial conditions of the following universe significantly through the residual cosmic web for the new universe. The result would be a very fine grained distribution of galaxy groupings rather than the coarse grained distribution we observe (see ref 1, AP4.7F).
• The little bangs would also change the expected spatial structure of the cosmic microwave background radiation. Instead of the large peak in the distribution we observe, we would expect a set of smaller peaks and a flatter distribution (see ref 1, AP4.7F).

 

Summary and Conclusions

In Reference 1, AP4.7F, a problem was discovered with respect to travel of particles with energies near the Planck energy. In order to gather the particles from the entire observable universe to a Principal Black Hole so that a big bang can occur, travel of high-energy particles at a speed faster than the speed of light in particle space is needed. Also, a variable light speed is needed to insure the constancy of the Planck length for energies near the Planck energy. In this paper, we established that super particles that near Planck energy should indeed be able to travel faster than the speed of light in a vacuum, and thus fulfill the conditions for a single, universal big bang as well as constant light speed in particle space and a constant Planck length.

 

Appendix 1

We consider the equation for the barrier to vacuum space (see ref 4, AP4.7 for more details)

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

            Where:

            E = the energy of the particle.

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

            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

Note that what we will calculate is the transmission probability density (T = t²  = r) or probability of transmission. The solution to the equation is a combination of left and right moving wave functions that are continuous at the boundaries of the barrier (r = 0 and r = a) along with their derivatives.

The solution can be used to generate the transmission through the barrier T (see Ref 8), which is as follows:

            If E>V

            T = 1/(1+V₀²sin²(k₁a)/4E(E-V₀)          

            If E<V,

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

            Where k₁= (8p²m(V₀-E)/h²)) ^1/2

Here we have set up the equations for 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

            E = 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.

Appendix 2

Here we consider what happens as a particle or photon passes between two vacuum granules. First we consider the wave front problem. Consider the two-slit experiment. This experiment shows interference fringes for particles or photons impinging on the two slits. The spacing of the fringes w at a distance z from the slits is as follows:

            w = zλ/d

            Where:

            d = spacing of the slits.

            λ = wavelength of the particle or photon.

If we think of the slit as the gap between vacuum granules, then d = l_p. Now for almost all particles and photons, λ/d >>1 and w is very wide, which means that wave front recovers its planar form even at a distance z = l_p from the two slits. This is equivalent to saying that the wave front of a particle or photon recovers its shape within one Planck length after passing through a layer of Planck granules as long as the wavelength is not near the Planck length. Thus the wave front of a particle or photon passes undisturbed through the vacuum as long as the vacuum granules are regularly spaced and arranged.

Next, we consider the particle or photon length problem. The wave front meets the edge of large number of vacuum granules at the same time. This edge and the edge of the next layer of granules can be seen as a slit formed between two layers at very high potential, and the particle or photon passes between these layers as it moves on its way. A particle or photon delays long enough after passing the first edge and before passing the second edge of the slit to accomplish the creation and destruction of the particle (or photon) during the passage. This process creates energy while it happens, and the energy is without size and shape between the layers. The particle or photon is then reformed at the edge of the next granule a bit (a Planck time) at a time, and the particle (or photon) does not have to be all in one granule while it happens. Thus even though the photon or particle length is much longer than a Planck granule, the substance (energy content) of the photon or particle must respond to the high potential energy of the Planck layer edges according to the energy-time uncertainty principle as described in the main text above.

Note, of course, that any disturbance in the regularity or potential energy content of the Planck granules will make the layers uneven, so the response will be irregular. Then the response to creation and destruction will have an energy change different from that normally seen between granules, and the number of granule distances traveled with the same energy change n will not be the same as the number of gaps crossed. This mismatch is important for case 5 above. 

Appendix 3

In 1955, John Wheeler proposed that a concept called Quantum Foam be used to describe the foundation of the fabric of the universe (ref 9). At scales of the order of a Planck length, the Heisenberg uncertainty principle allows energy to decay into virtual particles and anti particles and then annihilate back into energy without violating physical conservation laws. At the Planck size, the energy density involved is extremely high, and so according to General Relativity, it would curve space-time tightly and cause a significant departure from the smooth space-time observed at larger scales. Note that the potential energy of the tightly curled grain would be stored in the curvature of space. Thus at these tiny scales, space-time would have a “foamy” or “grainy” character. The resulting grains would pop in and out of existence with Planck energy and Planck size for a Planck time. According to this proposal, the Planck size is the smallest possible size, and the Planck energy is the largest possible energy, and the separation of these grains must be at least a Planck distance. Note that the time of existence as determined by the uncertainty principle is extremely small, so the flickering of the energy to virtual particles and back within the grains would not be noticeable to the real particles or photons passing through them, but they would notice the grainy structure due to the high potential energy. The real particles and photons operate on a much longer time scale unless the energy is near the Planck energy. Thus we have defined a vacuum composed of granules with high potential energy that flicker in and out of existence so fast that a particle or photon passing through will only see the high potential energy grains. Here the potential energy is stored in the curvature of space within the grain.   

Without a basic quantum gravity theory, it is not possible to describe the graininess of space, and its interaction with particles or photons in more detail. Quantum Loop Gravity (Smolin, 250) has made some initial steps in the basic theory, but it has not yet been used to describe the Quantum Foam, so more work is needed. For the purposes of this paper, this describes the “grains” of space mentioned in the main text above. The model that results from this description appears to predict the nature of the interactions between vacuum and particles or photons as seen in the main text above. The details of Quantum Gravity may provide more on the details of the interaction, but it is not expected to change its nature because it currently agrees with the experimental data.

 

References

1. L. H. Wald, “AP4.7F GATHERING DARK MATTER FOR THE BIG BANG AND ITS IMPACT ON MBR” www.Aquater2050.com/2015/12/
2. J. Magueijo, New Varying Speed of Light Theories, arxiv.org/pdf/astro-ph/0305457.pdf
3. I. G. Amelino-Camelia, “Testable Scenario for Relativity with Minimum-Length” hep-th/0012238. 
4. L. H. Wald, “AP4.7 DARK MATTER AND ENERGY-FUNDAMENTAL PROBLEMS IN ASTROPHYSICS” www.Aquater2050.com/2015/11/
5. L. H. Wald, “AP4.7D HOW TO PROVE A THEORYS CORRECTNESS” www.Aquater2050.com/2015/12/