Abstract

There are currently three important connected major unanswered questions in physics and astrophysics.

(1)  How can the theories of symmetry and the Higgs field be used to calculate the masses of the fundamental particles?

(2)  How can dark matter be explained and described?

(3)  How can dark energy be explained and described?

A self-consistent theory (called Model 1 in this paper) has been developed that appears to answer questions 2 and 3 quantitatively. In order to derive and justify Model 1, however, it became necessary to calculate the mass-energy of the proton and other fundamental particles that make it up, which answers question 1. This procedure then gave a path for calculating the mass of the Super Particle, which is the primary particle of Model 1. The super particle was found to be a Grand Unified Particle, which unifies the electromagnetic, weak and the strong forces. In investigating the properties of this super particle, it became obvious that it had the properties of dark matter, and when it breaks down, it generates dark energy. At this point, it was noticed that the Grand Unified Particle may be connected to the ultra high-energy cosmic rays that are observed with energy beyond the GZK cutoff (UHECR’s). In a sequence of related papers (Wald, Model 1-A; Wald, Model 1-B; Wald, Model 1-C; Wald, Model 1-D; and Wald, Model 1-F), Model 1 is detailed and expanded. In this paper, an explanation for the UHECR’s will be proposed that answers the question:

(4)  Where do the ultra high-energy cosmic rays with energy beyond the GZK cutoff come from?  

 

 

The Problem

The Unique Features of Model 1 

• There are two spaces in the universe, low-energy particle space and high-energy 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 particle space to vacuum space through black holes where they are converted into super particles (energy ~10¹⁷GeV).  They are then wrapped with a potential energy barrier shield (~10¹⁹GeV) to stabilize them. The barrier shield forms the boundary of high-energy vacuum space. These stabilized, shielded super particles are then able to escape from the black hole into particle space. The shielded super particles have a low interaction cross-section with ordinary particles except through gravity, and so are observed as dark matter.

• Dark matter particles interact with each other and form a slowly building bubble centered on a galaxy that stabilizes its outer edges. Corridors of dark matter are also generated which form a cosmic web between the galaxies. These corridors guide the development of new galaxies.

• The super particles can tunnel through the barrier into particle space. Upon reaching particle space, the super particles become unstable and break down into particles (cosmic ray protons) with ultra high kinetic energy (UHECR’s). In doing so, they give up potential energy from their barrier shields into particle space which becomes the dark energy that we observe as the cause of our accelerating, expanding universe.

 

The UHECR Observations

Blasi (Blasi, 1) describes the problem of a set of ultra high-energy cosmic rays (UHECRs), which are observed in an energy range where there should be no cosmic rays. The reason is as follows. A cosmic ray is usually a very high-energy proton. When it interacts with the earth’s upper atmosphere, it produces gammas that penetrate to earth and are detected there. It was found, however, that a proton with energy above a certain cutoff (the GZK cutoff ~ 4×10¹⁹ eV or ~ 4×10¹⁰ GeV), interacts with the microwave background in particle space to form other particles (pions). Since microwave background is everywhere in the universe, protons with extremely high energy (i.e. from super nova, black holes, etc.) should not last very long in particle space. This means that particle space should be dark above this cutoff energy unless the source is common within our galaxy and close to our solar system. Yet such sources (neutron stars, AGN’s, black holes, white dwarfs, radio lobes, etc) are not common enough in our neighborhood of the galaxy to account for the UHECR’s we observe.

The observations, as described in Blasi (Blasi, 2) are as follows. Cosmic rays have been observed from ~ 0.1 GeV up to 3 x 10¹¹ GeV. We note:

• A total of 59 events have been observed above 4 x 10¹⁰ GeV. The spectrum of these events from ~5 GeV to ~10⁶ GeV has a power law increase with slope of ~ 2.7. From  ~10⁶ GeV to ~10¹⁰ GeV, the slope increases to ~ 3.1. At energies larger than 10¹⁰ GeV, the spectrum begins to flatten out.
• The 59 events appear to be isotropic on the large scale..
• On the small scale, 5 possible doublets and 1 possible triplet were found.
• The primaries appear to be mostly protons. A recent study indicates less than 30% are possible photon primaries.
• The observed UHECR’s are omni directional, and the particles from neutron stars, AGN’s, white dwarfs and radio lobes, which must come from the direction of these phenomena, are not (Blasi, 2).
• There is an observed increase in the number of gamma rays in the center of our galaxy. This must come from a high-density gamma ray source centered on the galactic center.

 

The Accelerated Expansion of Space Observations

Two teams of astronomers determined the distance of super novas by measuring their intensities in the visible spectrum range. The red shift of each super nova was also measured, giving its relative velocity. It was found from these data, that the universe is undergoing an accelerated expansion of space (Smolin, 154)). This result means that contrary to expectations, the universe must contain dark energy in order to sustain this expansion.

 

The Solution

Model 1 Predictions for the UHECRs

Model 1 predicts high-energy cosmic rays with the following characteristics.

• Super particles will tunnel through the barrier between vacuum space and particle space with energies in a Gaussian distribution in the range ~10¹⁰ to10¹⁹ GeV. The portion of the super particle Gaussian less than 10¹¹ GeVis reduced because the vacuum space potential energy attenuates the super particles if they have kinetic energy less than this value (see Appendix 1). Note that the cosmic rays less than 10¹⁷ GeV are attenuated because they come from super particles in vacuum space with energy less than the potential energy of vacuum space (see Appendix 1). In particle space, they are unstable, so they break down (through spontaneous symmetry breaking) into extremely high-energy protons, giving up the extra potential energy to the particle space vacuum (becoming dark energy). The resulting protons in particle space will have energy in the range 10¹⁰ to 10¹⁹ GeV..
• The super particles in vacuum space have a Gaussian distribution with a mean, which depends on the age of the universe. Thus:

–        For a young universe, the mean will be low (energy range ~10¹⁰ to ~10¹¹ GeV), because the super particles coming in from the black hole (Wald, Model 1-C) meet a relatively large, empty vacuum space, so they expand into the volume and their temperature goes down.

–        For an early mature universe, the mean will be modest (energy range ~10¹¹ to ~10¹² GeV), because the addition of more energetic super particles will have increased the particle temperature and density in vacuum space, and the cosmic net is beginning to build, so conditions are right for making galaxies.

–        For a middle aged universe, the mean will be medium (energy range ~10¹² to 10¹⁸ GeV), because the super particle temperature and density will have increased enough to generate a mature cosmic net (Wald, Model 1-C), and the generation of galaxies along the net will be rapid.

–        For an old universe, the mean will be high (energy range ~10¹⁸ to 10¹⁹ GeV), because the super particle temperature and density will have increased enough to be near to the Planck energy preparing for a new big bang.

• The protons that result from super particles in particle space will be isotropic in velocity direction on the large scale, because they come from an isotropic distribution of super particles in thermodynamic equilibrium in vacuum space.
• On the small scale, doublets and triplets can form from reactions with resident particles in particle space.
• Ultra high-energy protons will result from the super particles, but collisions with particles in particle space can also cause ultra high-energy gammas.
• Because there is a bubble in the density of the super particles centered on the galactic center (Wald, Model 1-C), there will be a bubble in the density of the protons that have tunneled out of the super particle barrier shell centered on the galactic center. By colliding with resident dust and gas, these ultra high-energy protons will cause a bubble in the density of gammas centered on the galactic center. This bubble has been observed, as noted above. 

We note that the cosmic rays of Model 1 are omni-directional, and so match the UHECRs that are observed..

The spectrum of the UHECR’s matches the Model 1 spectrum for early in the “mature universe” stage. The (~10¹¹ to 10¹² GeV) slope shows the spectrum of ordinary cosmic rays plus the rising contribution of the cosmic rays caused by the high-energy tail of the super particle Gaussian. The rollover beyond ~10¹⁰ GeV is caused by the peak and then the drop-off of super particles as the Gaussian tails off. The isotropy, the doublets and triplets and the dominance of protons match the data satisfactorily. 

Note that a shift (if any) of the GZV cutoff to a higher value has no impact on this match, because it only impacts what happens to the particles after entering particle space. Such a shift makes other sources for UHECR’s possible in terms of the cutoff. However, the generation sources and the spectrum of the protons produced do not appear to match those observed. Nonetheless getting cosmic ray data for events up to 10¹⁹ GeV will clarify this issue by showing any new cutoffs.

 

Model 1 Predictions for the Accelerated Expansion of Space.

Peebles postulates the existence of a new real scalar field, and develops the equations for the field density ρ_φ, and the pressure p in space from general relativity.The expansion and contraction of space is controlled by the following field density (ρ_φ) and pressure (p) equations from general relativity (see Peebles, 396):

 

ρ_φ=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 average mass-energy density (ρ_m), pressure (p) and the cosmological constant (Λ) (see Peebles, 75):

 

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

      = acceleration of the cosmological expansion parameter

           

            Where:

            m = particle mass

            ρ_m = Σm/vol

            vol = volume of space containing the particles

 

Note from the field pressure equation (2), that if the potential energy density exceeds the field kinetic energy, the pressure is negative. The field kinetic energy term is slowly varying, however, because it depends primarily on the total mass in the general vicinity (see Wald, Model 1-B), so the potential energy controls the pressure. Then, if the potential energy increases enough, the negative pressure term in equation (3) can become large enough to exceed the mass density term in equation (3), and the acceleration of the cosmological expansion parameter (ä/a) turns positive, and space will eventually expand. If the potential energy V is small compared to the field kinetic energy term, however, the field pressure term is positive, and if Λ is small, the acceleration ä/a becomes negative, and space will eventually contract. Even if the (ä/a) term is negative, however, space will continue to expand for a while, thus maintaining its prior state, but eventually, expansion velocity will reduce below zero and space will contract. Thus, if ä/a is positive, space will expand faster and faster as time goes on. On the other hand, if ä/a is negative, space will expand slower and slower until the expansion velocity reverses sign, and then space will contract.

 

Testing Model 1 with Data

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

• The UHECR’s that cannot be accounted for by ordinary processes, but are still observed.
• The potential energy needed to explain the accelerated expanding universe we observe. .

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

 

Summary and Conclusions

Ultra high-energy cosmic rays (UHECR’s) and vacuum potential energy density have been studied to find their origin and characteristics. They are found to match the characteristics of Model 1 super particles (dark matter) that have tunneled through the barrier separating vacuum space from particle space in Model. We found that after tunneling, super particles brake down into a UHECR proton and potential energy. The UHECR proton is observed as a cosmic ray. The potential energy is observed as vacuum energy density in particle space. The observed spectrum of UHECR’s appears to match the expected spectrum of an “early mature universe”, which means that the mean energy of the super particle Gaussian distribution of dark matter is at ~10¹² GeV.

 

 

Appendix 1

The Barrier Equations

Consider the following non-relativistic three-dimensional time independent Schrödinger equation (see ref 7, AP4.7).

 

[-(h²/8p²m)Ѳ+V(x, y, z)]Y(x, y, z) = EY(x, y, z)

 

            Where:

            V(x, y, z) = barrier potential

            E = particle energy

 

If we convert to spherical coordinates, and let:

 

            Y(r, q, f) = R(r) Y(q, f)

            Where:

            Y(l, m) = Spherical Harmonics = (4p) ^-1/2  , if  l = 0 (spherical symmetry)

 

Now, let:

           

            R(r) = U(r)/r, then:

 

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

 

Now, consider:

 

[-(h²/2m d²/dr² + V(r)]U(r) = EU(r)

    

            Where

            E = the energy of the particle.

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

 

            Also:

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

            w = the potential zone width.

 

Note that any solution to the equation is unchanged if the step function is moved along the r axis to r_ob.  Then:

1. One can think of starting at 0 and moving to r_ov. Think of this as moving within the super particle itself, so the complex kinetic and potential energy distributions applicable within the particle exist here.
2. One then moves from r_ov to r_ob., so w = r_ob -r_ov .= v. This is vacuum space, so the vacuum potential is V(r) = V_ov[Q(r_ov) – Q( r_ov-v)] = the vacuum potential
3. One then moves from r_ob to r_ob + a, so w = a. this is the space within the barrier itself, i.e. between vacuum space and particle space, so the vacuum potential is V(r) = V_ob[Q(r_ov) – Q( r_ov-v)] = the vacuum potential.

 

Thus the equation governs passage from the super particle through vacuum space and then through the barrier and into particle space. For convenience, we will let r  = 0for solving the equation. Note that what we are describing is two concentric spherical shells with thicknesses v and a wrapped around a super particle. The shell with thickness v is vacuum space. The shell with thickness a is the barrier between vacuum space and particle space.

 

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 = w) along with their derivatives.

 

The solution can be used to generate the transmission through the barrier T , which is as follows:

 

            If E>V

            T = 1/(1+V₀²sin²(k₁w)/4E(E-V₀)

           

            If E<V,

            T = 1/(1+V_o² sinh²(k₁w)/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 is:

 

R = T h/2p k₁

 

The best fit for the vacuum space sphere and the barrier shell of the very rough data available is:

           

            r_ov = 10^-20 cm

            r_ob = 10^-11 cm

            a = 10^-31 cm

 

There are three important cases.

 

Case 1

            E > 10¹⁷ GeV

            V_ov = potential energy ~ 10¹⁷ GeV

            V_ov/cc = potential energy density ~ 10³² GeV/cc

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

 

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 gradually attenuates.

 

Case 2

            E ~ 10¹⁷ GeV

            V_ob = 10¹⁹ GeV

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

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

 

This is the case of leakage of super particles for the whole universe by tunneling through the barrier. When these particles reach particle space, they break down to ordinary particles and give up their phase change energy (10¹⁹ GeV/sp) 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.

 

References

1. L. H. Wald, Model 1-A “Mass and Function of the Standard Model Particles” www.Aquater2050.com/2017/01/
2. L. H. Wald, Model 1-B “The Origin of the Higgs Field for Model 1” www.Aquater2050.com/2017/01/
3. L. H. Wald, Model 1-C “Explaining the Dark Matter Halo” www.Aquater2050.com/2017/02/
4. L. H. Wald, Model 1-D “The Recycling Universe” www.Aquater2050.com/2017/02/
5. L. H. Wald, Model 1-F “How to Prove a Theory’s Correctness” www.Aquater2050.com/2017/03/
6. P. Blasi, “The Highest Energy Particles in the Universe: the Mystery and its Possible Solutions”, arxiv.org/pdf/astro-ph/0110401.pdf
7. L. Smolin, The Trouble with Physics, Boston, New York: Mariner Books, 2006
8. P. J. E. Peebles, Principles of Physical Cosmology, Princeton, New Jersey, Princeton University Press.