Cosmic Applications of Relative Energy between Quarks in Nucleons ()

F. C. Hoh^{}

Retired, Uppsala, Sweden.

**DOI: **10.4236/jmp.2019.1014108
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Retired, Uppsala, Sweden.

By taking into account the relative energy between the diquark and the quark in nucleons, the gravitational singularity in a black hole created from a collapsing neutron star can be removed; compatibility with quantum mechanics is restored. This black hole becomes a “black” neutron star. The negative relative energy identified as dark matter in the previous paper can account for the galaxy rotation curve. The positive relative energy identified as dark energy in the previous paper can explain the accelerating expansion of the universe. A possible scenario for cosmic ray generation is given.

Keywords

Gravitational Singularity, Relative Energy between Quarks, Scalar Strong Interaction Hadron Theory, “Weightless” Neutron, Black Neutron Star, Galaxy Rotation Curve, Accelerating Expansion of Universe, Cosmic Ray

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Hoh, F. (2019) Cosmic Applications of Relative Energy between Quarks in Nucleons. *Journal of Modern Physics*, **10**, 1645-1658. doi: 10.4236/jmp.2019.1014108.

1. Introduction

A neutron star with mass greater that the Tolman-Oppenheimer-Volkoff (TOV) limit *M _{TOV}*~3 solar mass

Dark matter needed to account for many observation, e. g., the galaxy rotation curve, and the dark energy required to drive the rapidly expanding universe remain hypothetical as they cannot be observed.

Recently, it was shown [3] that the “hidden”, unobservable relative energy between the diquark and quark in nucleons can interact with their ambient gravitational potential and possibly play the roles of dark matter and dark energy in the universe. The underlying theory is the scalar strong interaction hadron theory SSI [4] [5].

In this paper, it is shown that, as a neutron on the surface of a neutron star with mass ~*M _{TOV}* falls toward the star center, not only gravitational energy is released, but a negative relative energy also emerges simultaneously. These two energies can cancel each other and prevent the collapse of the star and formation of a mass singularity. The negative relative energy generated in an expanding galaxy can play the role of dark matter and account for the galaxy rotation curve. The positive relative energy generated in outer regions of the universe can play the role of dark energy, leads to accelerating expansion of the universe and may eventually give rise to cosmic rays.

The relevant parts of SSI are outlined in Sections 2-5. Creation of negative relative energy applied to neutron star collapse is considered in Sections 6-7. In Section 8, such negative relative energy reinforces the existing gravitational potential in an expanding galaxy to account for the galaxy rotation curve qualitatively. The accelerating expansion of the universe by means of the positive relative energy created in the outer parts of the universe is qualitatively outlined in Section 9. Some scenarios of the outer regions of the universe including cosmic ray generation are shown in Section 10. The so-obtained scenario of the universe is summarized in Section 11.

2. Outline of Construction of Baryon Wave Equations in SSI [4] [5]

The starting point is the Dirac equations for three quarks *A*, *B* and *C* with masses *m _{A}*,

${\partial}_{II}^{d\stackrel{\dot{}}{e}}{\chi}_{B\stackrel{\dot{}}{e}}\left({x}_{II}\right)-i\left({V}_{BC}\left({x}_{II}\right)+{V}_{BA}\left({x}_{II}\right)+{V}_{BG}\left({x}_{II}\right)\right){\psi}_{B}^{d}\left({x}_{II}\right)=i{m}_{B}{\psi}_{B}^{d}\left({x}_{II}\right)$ (2.1.a)

${\partial}_{II\text{\hspace{0.17em}}\stackrel{\dot{}}{e}f}{\psi}_{B}^{f}\left({x}_{II}\right)-i\left({V}_{BC}\left({x}_{II}\right)+{V}_{BA}\left({x}_{II}\right)+{V}_{BG}\left({x}_{II}\right)\right){\chi}_{B\stackrel{\dot{}}{e}}\left({x}_{II}\right)=i{m}_{B}{\chi}_{B\stackrel{\dot{}}{e}}\left({x}_{II}\right)$ (2.1.b)

where
${\partial}_{I}=\partial /\partial {x}_{I}$,
$\cdots $. The spinor indices run from 1 to 2 and the quark spinors *χ _{B}* and

To construct baryon wave functions, the three pairs of quark equations, one of them being (2.1), together with the cited strong potential equations, are multiplied together and the products of the quark spinors and those of the strong potentials are generalized to baryon wave functions and baryon potentials nonseparable in *x _{I}*,

In [ [4] [5] Sec. 9.3], each quark mass above has been generalized to a quark mass operator operatng on a flavour function in a fictitious, complex internal or flavour space *z ^{p}*, where

The baryon flavour function can now be removed leaving behind the ground state baryon wave equation [ [6] (2.9)], [ [4] [5] (9.3.16)]

${\partial}_{I}^{a\stackrel{\dot{}}{b}}{\partial}_{I}^{g\stackrel{\dot{}}{h}}{\partial}_{II\text{\hspace{0.17em}}\stackrel{\dot{}}{e}f}{\chi}_{\left\{\stackrel{\dot{}}{b}\stackrel{\dot{}}{h}\right\}}^{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f}\left({x}_{I},{x}_{II}\right)=-i\left({M}_{b}^{3}+{\Phi}_{b}\left({x}_{I},{x}_{II}\right)\right){\psi}_{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{\dot{}}{e}}^{\left\{ag\right\}}\left({x}_{I},{x}_{II}\right)$ (2.2a)

${\partial}_{I\text{\hspace{0.17em}}\stackrel{\dot{}}{b}c}{\partial}_{I\text{\hspace{0.17em}}\stackrel{\dot{}}{h}k}{\partial}_{II}^{d\stackrel{\dot{}}{e}}{\psi}_{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{\dot{}}{e}}^{\left\{ck\right\}}\left({x}_{I},{x}_{II}\right)=-i{\left({M}_{b}^{3}+{\Phi}_{b}\left({x}_{I},{x}_{II}\right)\right)}_{\left\{\stackrel{\dot{}}{b}\stackrel{\dot{}}{h}\right\}}^{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d}\left({x}_{I},{x}_{II}\right)$ (2.2b)

${M}_{b}=\left({m}_{A}+{m}_{B}+{m}_{C}\right)/2$ (2.2c)

The interbaryon strong potential Φ* _{b}* is a triple product of three strong interaction quark potentials of the type of

${\partial}_{I}^{a\stackrel{\dot{}}{b}}{\partial}_{II}^{f\stackrel{\dot{}}{e}}{\partial}_{I}^{e\stackrel{\dot{}}{f}}{\chi}_{0\stackrel{\dot{}}{b}}\left({x}_{I},{x}_{II}\right)=-i2\left({M}_{b}^{3}+{\Phi}_{b}\left({x}_{I},{x}_{II}\right)\right){\psi}_{0}^{a}\left({x}_{I},{x}_{II}\right)$ (2.3a)

${\partial}_{I\text{\hspace{0.17em}}\stackrel{\dot{}}{b}c}{\partial}_{II\text{\hspace{0.17em}}\stackrel{\dot{}}{e}h}{\partial}_{I\text{\hspace{0.17em}}\stackrel{\dot{}}{h}e}{\psi}_{0}^{c}\left({x}_{I},{x}_{II}\right)=-i2\left({M}_{b}^{3}+{\Phi}_{b}\left({x}_{I},{x}_{II}\right)\right){\chi}_{0\stackrel{\dot{}}{b}}\left({x}_{I},{x}_{II}\right)$ (2.3b)

where *χ*_{0} and *ψ*_{0} the wave functions of the doublet baryons.

3. Laboratory and Relative Spaces

Since quarks cannot be observed, their coordinate spaces are converted into an observable laboratory space *X ^{μ}* for the baryon and a relative space

${x}^{\mu}={x}_{II}^{\mu}-{x}_{I}^{\mu},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{X}^{\mu}=\left(1-{a}_{m}\right){x}_{I}^{\mu}+{a}_{m}{x}_{II}^{\mu}$ (3.1)

For observable particles, *a _{m}* is often determined by that

The baryon wave functions in (2.1) have been factorized into the form of [ [4] [5] (10.1.1)],

$\begin{array}{l}{\chi}_{0\stackrel{\dot{}}{b}}\left({x}_{I},{x}_{II}\right)={\chi}_{0\stackrel{\dot{}}{b}}\left(\underset{\_}{x}\right)\mathrm{exp}\left(-i{K}_{\mu}{X}^{\mu}+i{\omega}_{K}{x}^{0}\right)\\ {\psi}_{0}^{a}\left({x}_{I},{x}_{II}\right)={\psi}_{0}^{a}\left(\underset{\_}{x}\right)\mathrm{exp}\left(-i{K}_{\mu}{X}^{\mu}+i{\omega}_{K}{x}^{0}\right)\end{array}$ (3.2)

${K}_{\mu}=\left({E}_{K},-\underset{\_}{K}\right)$ [ [4] [5] (3.1.6)] = (3.3)

where *E _{K}* is the energy of the baryon and

In spherical coordinates,
$\underset{\_}{x}=\left(r,\theta ,\varphi \right)$ [ [4] [5] (3.1.7b)], the doublet wave functions in (3.2) with total angular momentum *j* = 1/2 and orbital angular momentum *l* = 0 read [ [4] [5] (10.2.3)]

${\psi}_{0}^{1}\left(\underset{\_}{x}\right)={g}_{0}\left(r\right){Y}_{00}\left(\theta ,\varphi \right)+i{f}_{0}\left(r\right)\sqrt{\frac{1}{3}}{Y}_{10}\left(\theta ,\varphi \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\psi}_{0}^{2}\left(\underset{\_}{x}\right)=i{f}_{0}\left(r\right)\sqrt{\frac{2}{3}}{Y}_{11}\left(\theta ,\varphi \right)$ (3.4)

where the *Y*’s are the usual spherical harmonics.
${\chi}_{0\stackrel{\dot{}}{a}}$ is found by changing the signs of *f*_{0}(*r*) in (3.4).

4. Radial Wave Equations in Relative Space, Solutions and Results

Consider baryons at rest, *K* = 0. *a _{m}* = 1/2 is set as in the meson case [ [4] (3.5.7)], [ [5] (5.7.2)]. Similarly, the “hidden” relative energy −

$\left[\frac{{E}_{0}^{3}}{8}+{M}_{b}^{3}+{\Phi}_{bd}\left(r\right)+\frac{{E}_{0}}{2}{\Delta}_{0}\right]{g}_{0}\left(r\right)+\left(\frac{{E}_{0}^{2}}{4}+{\Delta}_{0}\right)\left(\frac{\partial}{\partial r}+\frac{2}{r}\right){f}_{0}\left(r\right)=0\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}$ (4.1a)

$\left[\frac{{E}_{0}^{3}}{8}-{M}_{b}^{3}-{\Phi}_{bd}\left(r\right)+\frac{{E}_{0}}{2}{\Delta}_{1}\right]{f}_{0}\left(r\right)-\left(\frac{{E}_{0}^{2}}{4}+{\Delta}_{1}\right)\frac{\partial}{\partial r}{g}_{0}\left(r\right)=0$ (4.1b)

where the subscript *d* denotes doublet and [ [4] [5] (10.2.2a)] gives the diquark-quark strong interaction potential

${\Phi}_{bd}\left(r\right)=\frac{{d}_{b}}{r}+{d}_{b0}+{d}_{b1}r+{d}_{b2}{r}^{2}+{d}_{b4}{r}^{4}$ (4.2)

here, the nonlinear potential Φ* _{cd}*(

The two coupled third order Equations (4.1) have been converted into six first order equations [ [4] (10.7.5)], [ [5] (10.4.5)] which have six linearly independent solutions each associated with its own *λ*_{+} values in [ [4] [5] (10.2 8a)] found from the sixth order indicial equation [ [5] (10.4.4)]. These eventually led to two free parameters *w*_{(1)} and *w*_{(2)} in [ [4] [5] (11.1.1)]. It turned out that only *d _{b}*

Due to the large number of unknown constants, (4.1-2) could not be solved as a conventional eigenvalue problem. A less ambitious approach has been adopted. The known mass of the neutron is used as input for the eigenvalue *E*_{0} and the quark masses obtained from meson spectra given in [ [4] [5] Table 5.2] are used as input for *M _{b}* according to (2.2c) where

Here, the word “finite” on the first line of the second paragraph on [ [5] p239] turned out to be incorrect in subsequent numerical calculations and should be changed to “infinite”; *d _{b}*

These wave functions have led to the nearly correct predictions of the neutron life and the electron asymmetry parameter *A* or the neutrino asymmetry parameter *B* [ [5] Table 12.1] in its beta decay.

5. Relative Energy and Gravitation

If *a _{m}* = 1/2 and −

$\begin{array}{c}{\partial}_{I}^{a\stackrel{\dot{}}{b}}=\left(1-{a}_{m}\right){\partial}_{X}^{a\stackrel{\dot{}}{b}}-{\partial}^{a\stackrel{\dot{}}{b}}=\left(1-{a}_{m}\right)\left(-{\delta}^{a\stackrel{\dot{}}{b}}{\partial}_{X0}-{\underset{\_}{\sigma}}^{a\stackrel{\dot{}}{b}}{\partial}_{\underset{\_}{X}2}\right)+{\delta}^{a\stackrel{\dot{}}{b}}{\partial}_{0}+{\underset{\_}{\sigma}}^{a\stackrel{\dot{}}{b}}\underset{\_}{\partial}\\ ={\delta}^{a\stackrel{\dot{}}{b}}i\frac{1}{2}{E}_{0}+{\underset{\_}{\sigma}}^{a\stackrel{\dot{}}{b}}\underset{\_}{\partial}\to {\delta}^{a\stackrel{\dot{}}{b}}i\left(\left(1-{a}_{m}\right){E}_{0}+{\omega}_{0}\right)+{\underset{\_}{\sigma}}^{a\stackrel{\dot{}}{b}}\underset{\_}{\partial}\end{array}$ (5.1)

The expressions on both sides of the arrow will be equal if [ [7] (6.6) or [4] [5] (3.1.10a)]

${a}_{m}=\frac{1}{2}+\frac{{\omega}_{0}}{{E}_{0}}$ (5.2)

which keeps (4.1) invariant. The relative energy −*ω*_{0} is “hidden” with respect to strong and electromagnetic but not gravitational interactions [3]. This is indicated by *V _{BG}* in (2.1), which has been neglected in SSI because it is generally small next to the strong

Figure 1. *g*_{00}(*r*) and *f*_{00}(*r*) are *g*_{0}(*r*) and *f*_{0}(*r*) in (4.1) normalized according to [ [4] [5] (10.3.14)] for the *d _{b}*

In [3], −*ω*_{0} > 0 is identified as dark energy and −*ω*_{0} < 0 plays the role of dark matter. Both *E*_{0} and *ω*_{0} are observables with respect to gravitational forces. Lorentz invariance leads to that the associated baryon momentum *K* in (3.3) and the relative momentum represented by
$\underset{\_}{\partial}$ in (5.1) are also observables. Thus, the conjugate variables *X*, the coordinate of the baryon, and *x*, the relative coordinate in (3.1), are also observables, remembering that *x* is still a “hidden” variable with respect to strong forces. By (3.1), both the diquark coordinate *x _{I}* and quark coordinate

6. Neutron Star Collapse Scenario

Consider the following idealized scenario. A neutron star with a mass *M _{NS}* =

In Figure 2, an external neutron arrives on the surface of the neutron star at position *a* on the right part of the figure. The interquark distance is
${x}_{II,a}-{x}_{I,a}={r}_{a}~\text{4}$ fm according to Figure 1. *X _{a}* lies in the middle corresponding to

Figure 2. Illustration of the above idealized scenario. The horizontal line represents the radius *R* inside the neutron star. The center of the star is at *R* = 0. An external neutron arrives at *X _{a}* on the surface of this star at

shifted to the left to become *x _{I}*

${r}_{b}={x}_{II,b}-{x}_{I,b}\to {r}_{bG}={x}_{II,bG}-{x}_{I,b}={r}_{a}\alpha ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha ={R}_{b}/{R}_{a}<1$ (6.1)

As was mentioned at the end of Section 5, gravity interacts directly with the quark at *x _{II}* in (2.1), hence also with that in Figure 2. Phenomenologically, it also interacts with the diquark at

If *X _{b}* were taken to be the star radius

The radial wave Equations (4.1-2) hold for the arriving neutron in position *a*. At position *b*, (6.1) shows that the diquark-quark distance or radius *r* in (4.1) is reduced by a factor of *α*. Equations (4.1-2) remain invariant under the transformations
$r\to r\alpha $,
${E}_{0}\to {E}_{0}/\alpha $,
${M}_{b}\to {M}_{b}/\alpha $,
${d}_{b2}\to {d}_{b2}/{\alpha}^{5}$,
$\cdots $. The neutron mass *E*_{0} and the quark masses in *M _{b}* at position

7. Removal of Mass Singularity and “Black” Neutron Star

Application of (3.1), (5.2) and (6.1) to position *b* in Figure 1 yields

${a}_{m}=\frac{1}{2}+\frac{{\omega}_{0}}{{E}_{0}}=1+\frac{{X}_{b}-{x}_{II,bG}}{{x}_{II,bG}-{x}_{I,b}}=\frac{1}{2}\frac{{r}_{a}}{{r}_{bG}}=\frac{{R}_{a}}{2{R}_{b}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{\omega}_{0}}{{E}_{0}}=-\frac{{R}_{a}}{2}\left(\frac{1}{{R}_{b}}-\frac{1}{{R}_{a}}\right)$ (7.1)

where −*ω*_{0} is the relative energy gained when the external neutron arriving at position *a* falls to position *b*. In this fall, the gravitational energy *E _{G}* released is given by

$\frac{{E}_{G}}{{E}_{0}}=\frac{{R}_{Sa}}{2}\left(\frac{1}{{R}_{b}}-\frac{1}{{R}_{a}}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}_{Sa}=2G{M}_{NSa}$ (7.2)

where *G* is the gravitational constant and *R _{Sa}* is the Schwarzschild radius of this collapsing star with radius

${R}_{a}={R}_{Sa}$ (7.3)

*i*.*e*., when the neutron star in Figure 2 with the critical mass *M _{NSa}* has a radius

In this case, the fall of this external neutron actually gains no energy because *E _{G}* −

The radius *R _{a}* of the initial neutron star has been estimated by equating gravitational pressure to the pressure of the degenerate neutrons [ [1] p161]

${R}_{a}=0.0026{R}_{Earth}{\left(\frac{{M}_{SUN}}{{M}_{NSa}}\right)}^{1/3}$ (7.4)

which together with (7.2-3) yields *R _{a}* =

A possible scenario is as follows. As additional neutrons arrive at the star surface, a thin shell of weightless neutrons is added to the star at first. Let the mass of this shell be Δ*M _{NSa}*, the new Schwarzschild radius will be
${R}_{Sa\Delta}=\left(\text{1}+\Delta \right){R}_{Sa}$ and the new star radius will be
${R}_{a\Delta}={\left(1+\Delta \right)}^{1/3}{R}_{a}$,assuming that the neutron density in the shell is the same as that in the star.

Since *R _{Sa}*

In this scenario, the neutrons in such a black hole fill it up to its Schwarzschild radius. Such a black hole may more suitably be called a “black” neutron star. The absence of mass singularity here is consistent with the conjecture that an eventual future quantum gravity theory will not contain any singularity.

8. Galaxy Rotation Curve without Dark Matter

There is a large body of data that require the presence of dark matter. The first one is the galaxy rotation curve [ [2] Dark Matter §3.1]. However, nothing is known about this hypothetical dark matter. In [3], this matter has been identified as the negative relative energy −*ω*_{0} < 0 between the diquark and quark in nucleon. Here, the role of the dark matter in accounting for this rotation curve will be played by this −*ω*_{0}.

Consider again an idealized scenario as follows. A spiral galaxy with an appreciable part of its mass consisting of hydrogen gas was in its earlier stage of development. In that stage, this galaxy was smaller, denser and hotter according to the big bang model and the gravitational potential resisting such an expansion was insufficient. It therefore expanded. The situation is analogous to a violation of Jeans criterion for a gas cloud.

To illustrate the mechanism, let us turn off the gravity for a moment. Follow now the movement of a proton, denoted by *c*, in a hydrogen molecule. In this thermal expansion, this molecule will collide with other molecules, exchange energy and momentum with them and end up in a new position, labeled *d*, farther out from the galaxy center. In this journey, the forces involved are all Coulomb forces, between the proton and the orbiting electron and between the orbiting electrons in other molecules. The protons get dragged along; their quark structure is not involved in the expansion.

The situation is illustrated in Figure 3.

In Figure 3, the coordinate of proton *c* is *X _{c}* which lies in the middle between the diquark at

Turn now on gravity, the situation is reversed. Gravitational effects on electrons are negligible due to their small mass. Gravitational forces now act on this proton at *X _{d}* and try to pull it back together with other protons acted upon. However, they turn out to be too small to account for the galaxy rotation curve and prevent the escape of the outer stars. Here, such escapes are prevented by including the negative energies generated by differentiated gravitational pull on quarks and proton.

Just like the neutron star case mentioned below (6.1), gravity also acts directly on the quarks and tends to pull them towards the galaxy center, with the heavier diquark at *x*_{I}_{,d} closer to this center than does the lighter quark at *x _{I1,d}*,as is shown in Figure 3. Here, the diquark-quark distance
${x}_{II,c}-{x}_{I,c}={x}_{II,d}-{x}_{I,d}={r}_{a}\approx \text{4}$ fm as was mentioned below Figure 2. This distance is fixed by the strong interquark potential (4.2) and is unaffected by gravity here; the core collapse situation in position

Applying (3.1) the gravity pulled position *d* in Figure 3 yields the relative energy −*ω*_{0} given by

Figure 3. Illustration of the expanding galaxy scenario. The horizontal line represents the radius *R _{g}* inside the galaxy. The center of the galaxy is at

Figure 4. Illustration of the expanding universe scenario. The horizontal line represents the distance *R _{u}* from some unspecified inner region of the universe.

${a}_{m}=\frac{1}{2}+\frac{{\omega}_{0}}{{E}_{0}}=\frac{{X}_{d}-{x}_{I,d}}{{x}_{II,d}-{x}_{I,d}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{\omega}_{0}}{{E}_{0}}=\frac{1}{2}-\frac{{X}_{d}-{x}_{I,d}}{{r}_{a}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}_{a}={x}_{II,d}-{x}_{I,d}\approx 4\text{\hspace{0.17em}}\text{fm}$ (8.1)

This energy has the same negative sign as the gravitational potential energy produced by matter inside *X _{d} *and hence reinforces it to become large enough to keep the outer stars of the galaxy from escaping. It is due to the lag of the “hidden” quark coordinates

The required energy is -*w*_{0}*»* -5.5*E _{0}* per proton, when averaged over the whole universe, and is generated for “free” at “no cost”. This −

This scenario, characterized by Figure 3, is also expected to have dominated in the earlier stages of the universe when it was smaller, hotter, denser, and fast expanding.

9. Accelerating Expansion of the Universe without Dark Energy

The observed accelerating expansion of the universe is currently considered to be due to assumed dark energy in the outer regions of the universe. This hypothetical energy may here be replaced by the positive relative energy −*ω*_{0} > 0 corresponding to *a _{m}* < 1/2 in (8.1).

As in Sections 6 and 8, consider the following idealized scenario. The expansion mechanism in Section 8 applied to a galaxy can analogously be used in some later stage of the development of the universe. In the outer part of the universe, its expansion leads to that the hydrogen gas density decreases and the gas temperature drops there. The expansion nearly comes to a halt. In this region, the gas is tenuous, cold and experiences very weak gravitational force.

Consider a proton in a hydrogen molecule of this gas. The configuration of this proton in position *d* of Figure 3, with the galaxy replaced by the universe, has now largely returned to its original form in position *c *of Figure 3. This proton is also similar to the neutron in position *a* of Figure 2 with *a _{m}*

Let a second hydrogen molecule arrive at position *e* simultaneously. It contains a second proton with its diquark and quark at the same *x _{I}*

If its position is *X _{e}*

If its position is *X _{e}*

Current data show that dark energy exceeds the energy of ordinary matter by a factor of ≈14 when averaged over the universe. Identifying this energy with −*ω*_{0}, (8.1) leads to *a _{m}* = −13.5 ( [3] §6.1). This value and (8.1) give the “whole” size of this second proton at position

This scenario, characterized by Figure 4, also dominates in the later stages of the universe when it has become large.

10. Scenarios in the Outer Regions of the Uiniverse

10.1. Possible Plasma Creation

As the above “run way” situation continues, the “whole” size of this proton, *x*_{I}_{I}_{,f} − *X _{e}*

The relative energy corresponding to this case is by (8.1) −*ω*_{0} ≈ 1.29 × 10^{1}^{3} eV. In this scenario, hydrogen atoms in the expanding outer regions of the universe having this energy or greater turn into plasma.

10.2. Generation of Cosmic Rays

According to 10.1., the hydrogen gas in outer regions of the universe can expand until an average atom acquires an energy ~1.29 × 10^{13} eV, beyond which the atom becomes ionized. For atoms carrying higher energies, the gas becomes a tenuous plasma in intergalactic space. In this plasma, the protons continue to gain energy by the increasing −*ω*_{0} but the ejected electrons lag behind, inasmuch as gravity is absent in Dirac’s equation for an electron in a Coulomb field. This leads to a proton current flowing radially outwards which in its turn generates transverse magnetic field. As there are inhomogeneities in gas distribution in this region, the magnetic field will also vary in space.

As the “run away” expansion in Section 9 continues, some protons become very energetic. The magnetic field will cause them to move perpendicular to their path and to the direction of the magnetic field. Some of these high energy protons may be moved by magnetic fields such that they return to the inner part of the universe. Such protons can be a source of cosmic rays with energy > 10^{13} eV,

For a cosmic ray proton having a high energy of, say, 10^{20} eV, The “whole” size of this proton *x*_{I}_{I}_{,f} − *X _{e}*

10.3. Diquark-Quark “Flip” and Magnetic Curtailment

In Figures 2-4, the diquark is closer to the center of the star, galaxy or universe; *x _{I}* <

This may in principle go on forever. This pushing force is stronger on the heavier diquark than that on the quark and may eventually push the diquark past the quark farther out and ”flip” their positions from *x _{I,f}* <

But even before this scenario, another one may take place. The protons with energy > 10^{13} eV are part of a plasma with transverse magnetic field in §10.2. Some of them will eventually move in a direction perpendicular to the direction of *R _{u}* in Figure 4. In that motion, the acceleration mechanism in Section 9 is no longer active. In this scenario, infinite proton energy and infinite expansion of the universe’s outer regions may be curtailed by the magnetic fields.

11. Summary-Scenario of the Universe

The present results lead to the following scenario for the universe.

1) The universe contains neither dark matter nor dark energy.

2) A heavy neutron star with mass = *M _{TOV}* and radius = its Schwarzschild radius does not collapse into a mass singularity and may be called a “black” neutron star. If every neutron star gets heavier, becomes a black hole and passes through this stage in its development, there will be no mass singularity in the universe, in agreement with that quantum mechanics does not allow such a singularity.

3) The run away instability in Section 9 provides a mechanism for an accelerating expansion of the universe. The driving positive relative energy is “free” and “costs nothing”. The density and temperature of the universe will decrease with time. This scenario may in principle go on forever.

4) In outer parts of the universe, fast expanding hydrogen gas may turn into plasma and part of it may become cosmic rays. There may also exist a scenario in which the above free expansion can halt and eventually reverts to contraction.

The above results are derived phenomenologically by joining SSI to aspects of general relativity. A formal integration of these both theories is beyond reach; no quantum gravity theory exists presently.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

[1] | Ryan, S.G. and Norton, A.J. (2010) Stellar Evolution and Nucleosynthesis. Cambridge University Press, Cambridge. |

[2] |
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