_{1}

^{*}

Dark matter is identified as negative relative energy between quarks in proton and is generated in cold hydrogen gas with pressure gradient in gravitational field. Positive relative energy PRE can be generated between quarks in protons in cold hydrogen gas in outskirts of the universe. The mechanisms for such creation of dark matter and PRE are reviewed and updated in greater detail and clearer manner. The so-generated dark matter in a galaxy can account for the galaxy’s rotation curve. Star formation in this galaxy uses up the hydrogen atoms and thereby reduces its dark matter content. Dark matter created in intergalactic hydrogen gas can form filaments. In a hypothetical model of the universe, a hydrogen atom with a small amount of negative relative energy or dark matter at the outskirts of this universe can via collisions with other atoms turn into one with a small positive relative energy PRE. Once such a sign change takes place, gravitational attraction switches to anti-gravity repulsion unopposed by any pressure gradient. This leads to a “run away” hydrogen atom moving away from the mass center of the universe and provides a basic mechanism for the accelerating expansion of the universe. This theoretical expansion and the measured redshift data are both compatible with the conception of an acceleratingly expanding universe and complement each other. But they cannot verify each other directly because the present model has been constructed for purposes different from those of the measurements. But it can be shown that both approaches do support each other qualitatively under certain circumstances for small velocities. Dark matter and PRE in the present model are not foreign objects like WIMPs and dark energy-cosmological constant but can only be created in cold hydrogen gas in gravitational field. To achieve this, infrequent collisions among the hydrogen atoms must take place. Dark matter was created first and can eventually later evolve into PRE in the outskirts of the universe and in the intergalactic void. Dark matter and PRE will disappear if the hydrogen atom carrying them becomes ionized as in stars.

The present standard model of big bang cosmology LCDM [

Further, general relativity is a classical theory in which mass, length and hence also mass density are continuous quantities that can be 0 and ¥. However, the smallest mass unit contributing to baryonic matter in the universe is the proton mass, a discrete quantity ≠ 0. In addition, the proton comprises of 3 point-like quarks which do not occupy the same spatial position and do have extension in space so that its mass density cannot be ¥. Therefore, general relativity breaks down at small distances and high mass densities when applied to the real universe and has to be “cut off” at suitable values from such 0 and ¥ where quark structure of matter enters.

In the parameter regions cut off from general relativity, an appropriate elementary particle theory is supposed to fill in. The obvious first choice is the current mainstream particle theory, the standard model SM [^{+} W^{−} bound state below). This model, including quantum-chromodynamics QCD, has turned out to be not useful; it cannot account for even the most basic meson spectra and does not describe the behavior of quarks in proton. Further, it cannot explain the existence of dark matter and dark energy.

Cosmologists have been attracted to SM by its Higgs mechanism which converts the energy created in the big bang to electron and quark masses. Such fermions obey Dirac’s equation and hence are observable, as does the electron. But the so-generated quarks cannot be observed and this contradicts the Higgs hypothesis. The situation reminds me of Einstein’s citation of a Bertrand Russell formulation: “Naive realism, if true, is false. Therefore, it is false”.

In its place, the scalar strong interaction hadron theory SSI has been proposed [^{+} W^{−} bound state [

Recently, it has been pointed out that such relative energy of a nucleon interacts with gravitational fields on equal footing as does the nucleon itself [

Negative relative energy or dark matter generated in a neutron on the Schwarzschild sphere of a neutron star falling towards its center can exactly cancel the gravitational energy gained in this fall. This neutron becomes weightless and the fall is halted. This mechanism can prevent the creation of gravitational singularity [

Positive relative energy can be generated in outskirts of the observable universe but not where dark matter is created. Conventionally, dark energy is assumed to permeate throughout the universe and has been associated with the cosmological constant, as in the ΛCDM model. Therefore, the earlier assignment of positive relative energy to dark energy in [

are adopted. The conventional meaning of dark energy remains unaltered. Like dark energy, PRE is also not observable but for different reasons.

The above results have been obtained using simple models to illustrate the mechanisms. The purpose of this paper is to review and update these mechanisms in greater detail and clearer manner. A simple model of a galaxy and a hypothetical model of the universe have been constructed for this purpose. Further, expansion of the universe via anti-gravity repulsion caused by PRE is treated semi-quantitatively.

In Section 2, the basic mechanism of the generation of dark matter in a galaxy model is reviewed and clarified in greater detail. The so-created dark matter phenomenologically accounts for the galaxy rotation curve and filaments. This also paves the way for the mechanism of creation of PRE in Section 3. By means of a hypothetical model of the universe, such PRE leads to anti-gravity repulsion which expands the universe in Section 4. The so-obtained nonlinear equations of motion for a test hydrogen atom participating in such expansion is solved on computer in Section 5. Relations between the so-obtained results and Hubble’s law and a partial comparison to the ΛCDM model are given in Section 6. The Appendix reproduces some earlier results for reference.

The mechanism of dark matter generation [

The coordinates of the diquark uu at x_{I} and the quark d at x_{II} in a proton cannot be observed. In SSI, they have been transformed into an observable laboratory coordinate X for the proton and an unobservable, “hidden” relative coordinate x (A2) between uu and d. The transformation constant a_{m} can in principle be any real number. In the plane wave expansion of the proton wave function (A3), the relative energy −ω is however connected to a_{m} via (A4), which insures that the proton mass and behaviour are unaffected by such a variable transformation [_{D} between the relative energy generated to the proton mass E_{0},

R D = − ω E 0 = − a m + 1 2 = − | X _ − x _ I | | x _ I I − x _ I | + 1 2 = − X p − x I r a + 1 2 (2.1)

where X_{p} is the proton coordinate. The time components have been left out. The distance between uu and d, | x _ I I − x _ I | , has been approximated by its average value r_{a} = 3.23 fm [_{p} in a “test” hydrogen atom in an expanding “test” galaxy.

This test hydrogen atom is acted upon by the ambient gravitational force and eventual centrifugal force due to its motion. These forces accelerate the proton and the electron in this atom equally and move the test atom as a single entity.

Phenomenologically, an average test atom is being pushed towards the right in _{p} in (2.1).

On the other hand, the gravitational pull from the galaxy center also acts directly on the quarks of the proton [_{I} in (2.1).

It is this left shift of the position of the uu-d aggregate relative to the position of the center of the proton and electron clouds corresponding to greater X_{p} – x_{I} values in (2.1) that generates the negative relative energy or dark matter between uu and d; –ω < 0. If this uu-d aggregate were shifted to the right, as is depicted on the right half of

The gravitational force exerts a greater pull on the heavier diquark uu than it does on the lighter quark d so that uu lies to the left of d and is closer to the galactic center in _{b} (A6).

The above dark matter generation mechanism is phenomenological. A formal treatment would require an integration of SSI and gravitation. This out of the scope of this paper. The mechanism in Section 2.1 holds for a single test hydrogen atom for some time before this test atom experiences a new collision and ends up in a new state in which the dark matter generated in Section 2.1 is altered or lost. Therefore, the gas containing this test hydrogen atom needs to be cold and tenuous so that collisions, necessary for such generation, are infrequent.

Under such circumstances, the direction of the pressure gradient in Section 2.1 will cause an average test proton to generate dark matter that, together with other similar hydrogen atoms, can lead to observable phenomena such as the galaxy rotation curve.

The amount of dark matter so-generated depends upon the value of the transformation constant a_{m} which can so far assume any real value. The interquark separation r_{a} in (2.1) depends upon the interquark strong potential (A6) in the relative space, decoupled from the laboratory space X, and is hence largely unaffected by an atomic collision. Thus, large a_{m} implies large X_{p} – x_{I} which in its turn depends upon the unknown collision parameters. However, X_{p} – x_{I} can be constrained in this test atom’s environment.

For example, consider a collision that kicks the electron to the right which in its turn drags the proton at X_{p} along in _{p} – x_{I} assumes some positive value and dark matter –ω < 0 is created according to (2.1). A harder collision leads to a greater X_{p} – x_{I} and yields a larger amount of dark matter. But if the collision is too hard, the electron can be knocked off its orbit around the proton so that the test atom becomes ionized. In this case, the mechanism in Section 2.1 no longer works; the generated dark matter vanishes. In this case, X_{p} – x_{I} is restricted by an unspecified nonionization limit. This restriction is practically satisfied because a temperature of > 50,000 ˚K is needed for ionization which far exceeds the temperature of the cold hydrogen gas environment of the test atom.

A milder form of restriction on X_{p} – x_{I} is the heuristic limit which requires that the uu-d aggregate belonging to a proton has to lie inside the proton cloud. How this can eventually be verified formally would require a formalism beyond the scope of this paper. This heuristic restriction limits X_{p} – x_{I} in _{0p} = 28.8 fm so that (2.1) becomes [

R D M = − ω E 0 = − a m + 1 2 ≥ − a o p r a + 1 2 = − 8.4 (2.2)

This magnitude is much smaller than that due to the above-mentioned nonionization limit.

The observed average over the entire visible universe is R D M → R D M E X P = – 5 . 6 [_{DM} (2.2) or to the unspecified nonionization R_{DM}. This is due to that these two R_{DM} values refer to a test atom in cold, tenuous hydrogen gas. Warm, rarified hydrogen gas and hydrogen molecule gas may eventually contribute to a less degree. On the other hand, R_{DMEXP} refers to all forms of ordinary matter in the observable universe including in addition ionized media, dust, stars, planets, etc in which no dark matter can be created by the mechanism of Section 2.1. Thus, R_{DMEXP} is created from the hydrogen gas part of the universe only. This part has therefore to produce |R_{DM}| > |R_{DMEXP}|. The actual |R_{DM}| may perhaps lie around 8.4 of the heuristic (2.2) but well below the much higher, unspecified nonionization value.

This situation appears to be qualitatively compatible with Milky Way data. Milky Way has 1% - 5% cold hydrogen gas in volume and hence also a small % in mass. Therefore, it is expected to have a fairly small |R_{DM}| value. The Milky Way dark matter density at the sun’s position is ~ 6 × 10^{4} times smaller than average mass density of the universe.

The heuristic limit |R_{DM}| is essentially the number of uu-d aggregates with size r_{a} = 3.23 fm that can be fitted into one side of the proton cloud with radius 28.8 fm in _{p} by its own size 3.23 fm, a new dark proton is created, up to 8.4 such in (2.2). The energy needed to move X_{p} 28.8 fm, negligible on atomic scale, is very small and is estimated to be well covered in the momentum exchange of the collision.

The so-produced negative relative energy or dark matter in a proton is on equal footing with the proton mass itself [

Such an effect is also expected to be prevalent in the early phases of the expansions of galaxies and the universe. Near the conters, the temperature and pressure and their gradients were high and the gravitational pull was strong. The former leads to greater X_{p} and the latter to smaller x_{I} in _{p} – x_{I} values yield greater amount of dark matter –ω < 0 via (2.1). No dark “energy” is created at this stage. These results are in agreement with our current conception.

Consider the following prototype scenario. A cold hydrogen gas cloud exists between two galaxy clusters. Consider a cylinder of this cloud between these clusters. The gas near the cylinder surface will fall towards the cylinder axis. This gas will according to the mechanism on the left side of _{DM}| < 8.4 in (2.2).

According to Section 2.1, dark matter can only be created in cold hydrogen gas in gravitational field. An exception can be some neutrons in neutron stars (see Section 1.3). However, stars are also being formed from the same gas. When a hydrogen atom in this gas is used to build a new star, it gets ionized and becomes part of the hot plasma in this star. The dark matter generated by this atom is lost and the so-produced free proton can, contrary to the proton in a hydrogen atom, not generate any dark matter via the mechanism of Section 2.1. Conversion of a hydrogen atom to a proton and an electron in a star implies a loss of dark matter in the gas.

Therefore, star formation in a galaxy reduces its dark matter content. The amount of dark matter that contributes to the galaxy rotation curve is diminished by star formation. This galaxy will appear to expand faster.

Similarly, if the universe runs out of hydrogen gas, all dark matter and PRE will also vanish, except possibly in neutron stars and some other exotic objects.

In helium, the simple two-body, uu and d, problem here turns into a many- body problem involving many quarks and relative spaces. This problem has not been investigated.

The mechanism of positive relative energy PRE generation described in [

As the test hydrogen atom in Section 2 moves outwards and eventually reaches the outskirts of the test galaxy, where the gas pressure gradient, eventual centrifugal force and gravitational pull acting on it become very small and nearly balance off each other so that the expansion of this galaxy nearly comes to a halt. In the absence of force acting on this hydrogen atom, its uu-d aggregate will move back to its normal positions centered at X_{p} (see

Although the net outward movement of this test hydrogen atom and its like may nearly vanish, they will still have some small random velocities corresponding at least to the cosmic microwave background average temperature in the universe ~ 2.8 ˚K. Over time, this test hydrogen atom will experience a collision with another hydrogen atom that happens to be moving inward towards the galaxy center in _{H} = X_{p} also to the left via Coulomb force. This atomic force is small compared to strong interaction forces in the relative space x. The uu-d aggregate centered in _{p}. This situation is equivalent to the right half in

In this switched configuration, from the left half of _{p} – x_{I} values in (2.1) in Section 2 turns negative so that the negative relative energy generated in Section 2 changes its sign in (2.1) and turns into positive relative energy. This energy has been assigned to dark energy earlier [

Applying the assumption in Section 2 that led to (2.2), (2.1) yields the heuristic upper limit of the ratio between the so-generated PRE to proton mass [

R P R E = − ω E 0 = − a m + 1 2 ≤ a o p r a − 1 2 = 8.4 (3.1)

which has the same magnitude as that in (2.2) due to the left-right symmetry of the proton orbit in

Again, such R_{PRE} limits refer to the by now cold hydrogen test atom in the outskirts of the test galaxy. Such R_{PRE} values can also not be directly compared to the measured ratio of dark energy to ordinary matter averaged over the universe R_{DEEXP} = 13.6 [_{DM}| given below (2.2). R_{DEEXP} refers to dark energies caused by entirely different mechanisms, i.e., cosmological constant throughout the universe in the ΛCDM model.

The above developments show that dark matter and PRE are not foreign objects in SSI, like those in the ΛCDM model, but are generated in hydrogen gas in gravitational field and can vary and transform into each other dependent upon the positions of the uu-d aggregates relative to the centers of the proton clouds.

The present scenario is compatible with a current view that dark matter appears first and dark energy later, about 6 × 10^{9} years ago.

The mechanism in Section 3 can be applied to the expansion of the universe. Anticipating greater velocities for the accelerated test hydrogen atom, a Lorentz boost is performed on the rest frame proton whereby the heuristic limit (3.1) is approximately modified to

R P R E = − ω γ E 0 ≤ a o p γ r a − 1 2 , γ = 1 / 1 − v 2 / c 2 (4.1)

for slow protons, where v is the proton velocity and c the light speed.

The following SSI model of the universe shown in

Note that this model is not realistic because there is no known center of the observable universe. It is employed mainly for a semi-quantitative treatment of the anti-gravity expansion mechanism.

From [_{US} = 4.4 × 10^{26} m. The mass of this universe is estimated from the average density 9.9 × 10^{−}^{30} g/cm^{3} of the universe and its volume and is M_{U} = 3.53 × 10^{54} kg. The test galaxy is taken to be a copy of the Milky Way in spherical shape. The radius is R_{GS} = 100 kly = 9.46 × 10^{20} m and the mass M_{G} = 2.4 × 10^{42} kg. This is about 3 times the mass of the Milky Way and reflects the modification of its disc form to sphere form here.

Let the position of the test hydrogen atom after the collision in Section 3 be R(t) from S in ^{0} and R(0) = 0. This test hydrogen atom will in time experience another collision. Before this happens, the proton in this atom obeys the equation of motion

d d t E 0 γ ( t ) v ( t ) = G [ M G ( R G S + R ( t ) ) 2 + M U ( R U S + R ( t ) ) 2 ] ( − ω − E 0 γ ( t ) ) , v ( t ) = d R ( t ) d t (4.2)

where G is the gravitational constant.

Inclusion of the relative energy −ω next to the proton energy E_{0}γ in (4.2) has been demonstrated in [^{0} and E_{0}X^{0} are on equal footing. Since the proton mass E_{0} is known to interact with gravitational field, the associated relative energy −ω has to do so also in view that x and X are both linear combinations of the quark coordinates x_{I} and x_{II} in (A2). Equation (4.2) is general and holds for all free hydrogen atoms.

Initially, t is small, R(t) can be neglected in (4.2) and γ ≅ 1 . Using the values given below

d v ( t ) / d t = − a c 0 ( 1 + ω / E 0 ) , a c 0 = 1.396 × 10 − 9 m/s 2 (4.3)

v ( t ) = v ( 0 ) − a c 0 ( 1 + ω / E 0 ) t , R ( t ) = v ( 0 ) t − a c 0 ( 1 + ω / E 0 ) t 2 / 2 (4.4)

where v(0) is the initial velocity of this test hydrogen atom. About 13% of the initial acceleration a_{c0} comes from the galaxy mass M_{G} and 87% from the mass M_{U} of the universe.

If (4.3 - 4) were applied to the test hydrogen atom in Section 2.1, a_{c}_{0} needs to be reduced by 87%. In the absence of relative energy, −ω = 0, (4.3) simply describes the “free fall” of this test atom towards the galaxy center. If dark matter is generated, −ω < 0, and this atom will fall faster. This inward movement is on the average largely balanced off by outward movement produced by the pressure gradient in the hydrogen gas in Section 2.1.

Picking up the “run away” test hydrogen atom mentioned above (3.1), which now lies at S in _{0}. Then the right side of (4.3) becomes positive and accelerates this test atom outwards; the above gravitational “free fall” now turns into anti-gravity “free rise”. A formal way to include such a collision in (4.2) is to put

− ω = ( 1 + Δ ) u ( t ) E 0 < ( 8.9 − γ ( t ) 2 ) E 0 (4.5)

Here, u(t) is a step function; u(t < 0) = 0 and u(t > 0) = 1 representing the effect of the collision which takes place at time t = 0.1 + Δ is the amplitude of this step function and has to be > 1 to overcome the gravitational pull on the test hydrogen atom; Δ > 0. Following the considerations on R_{PRE} below (3.1), the limit (4.1) is tentatively adopted on the right of (4.5). Inserting (4.5) into (4.2) yields

d d t v ( t ) = G [ M G ( R G S + R ( t ) ) 2 + M U ( R U S + R ( t ) ) 2 ] [ ( 1 + Δ ) γ ( t ) − 1 ] , Δ + 1 < 8.9 − γ ( t ) 2 (4.6)

For small −ω or negative relative energy −ω < 0, the right side of (4.2) is negative. The accompanying attraction force increases with time as R(t) < 0 increases the magnitude of this side. Acceleration of the gravitational “fall” increases with time. On the other hand, for PRE satisfying (4.5) with Δ > 0, the right side of (4.6) is positive but decreases with time because R(t) > 0 increases with time. This causes the acceleration of anti-gravity “rise” to slow down with time.

Here, the initial velocity v(0) of the test atom will be taken to be the mean thermal speed of hydrogen atoms corresponding to the average temperature of the universe of ~ 2.8 ˚K mentioned in Section 3,

v ( 0 ) = 263 m/s (5.1)

It is unknown how Δ in §4 can be evaluated. It will be regarded as a free parameter here tentatively limited by the heuristic 0 < Δ < 7.4 due to (3.1) and (4.5); the associated higher nonionization limit is ignored here.

The nonlinear (4.6) with the initial conditions R(t = 0) = 0, v(t = 0) = v(0) has been solved on computers at Uppsala University. Some results are given in

This table shows that the position R(t) of the test atom and its velocity v(t) increase with time compatible with an acceleratingly expanding universe [_{GS} and then R_{US}, (4.6) shows that the acceleration a_{c} slows down, mentioned at the end of Section 4.2. This is also reflected in

t(yr) | 0 | 10^{4} | 10^{5} | 10^{6} | 10^{7} | 10^{8} | 10^{9} | 10^{10} |
---|---|---|---|---|---|---|---|---|

Δ = 1.0 | ||||||||

R(ly) | 0 | 1.61 × 10^{−}^{2} | 0.821 | 74.2 | 7.32 × 10^{3} | 6.82 × 10^{5} | 6.43 × 10^{7} | 4.97 × 10^{9} |

v(m/s) | 263.3 | 703 | 4.66 × 10^{3} | 4.43 × 10^{4} | 4.38 × 10^{5} | 4 × 10^{6} | 3.83 × 10^{7} | 2.38 × 10^{8} |

a_{c}/a_{c}_{0} | 1.0 | 1.0 | 1.0 | 1.0 | 0.983 | 0.874 | 0.855 | 0.156* |

H/H_{0} | 1.0 | 2.67 | 17.7 | 167 | 1040 | 267 | 27.6 | 2.22* |

Δ = 4.6 | ||||||||

R(ly) | 0 | 4.25 × 10^{−}^{2} | 3.46 | 338 | 3.33 × 10^{4} | 3.05 × 10^{6} | 2.85 × 10^{8} | 8.85 × 10^{9} |

v(m/s) | 263.3 | 2290 | 2.05 × 10^{4} | 2.03 × 10^{5} | 1.98 × 10^{6} | 1.8 × 10^{7} | 1.64 × 10^{8} | 2.95 × 10^{8} |

a_{c}/a_{c}_{0} | 1.0 | 1.0 | 1.0 | 0.999 | 0.944 | 0.87 | 0.69 | 1.5 × 10^{−}^{11}* |

H/H_{0} | 1.0 | 8.69 | 77.9 | 740 | 2020 | 273 | 26.7 | 1.55* |

Δ = 7.4 | ||||||||

R(ly) | 0 | 6.31 × 10^{−}^{2} | 5.52 | 544 | 5.33 × 10^{4} | 4.87 × 10^{6} | 4.35 × 10^{8} | 9.32 × 10^{9} |

v(m/s) | 263.3 | 3250 | 3.28 × 10^{4} | 3.06 × 10^{5} | 3.16 × 10^{6} | 2.88 × 10^{7} | 2.38 × 10^{8} | 2.98 × 10^{8} |

a_{c}/a_{c}_{0} | 1.0 | 1.0 | 1.0 | 0.999 | 0.926 | 0.867 | 0.474* | 7.5 × 10^{−}^{12}* |

H/H_{0} | 1.0 | 13.4 | 125 | 1180 | 2030 | 274 | 25.4* | 1.48* |

For v(0) = 0, (4.3 - 5) show that R, v and a_{c} are all proportional to Δ. With (4.1), (4.6) is seen to behave similarly for small v. These three sets of values in _{DM}| = 5.6 for dark matter below (2.2), the test hydrogen atom will reach a distance of 3 × 10^{6} ly with a velocity of 1.8 ´ 10^{7} m/s 10^{8} years after its collision at S in

As was shown beneath (4.5), Δ > 0. Computer runs with Δ < 0 leads to negative v(t) and blueshift. The bulk of the random collisions will yield Δ < 0 and these do not contribute to expansion. But sooner or later, a subsequent collision will produce Δ > 0. Once this occurs, one of the colliding atoms will become of the “run away” type obeying (4.6) and starts to move outwards and leave the S region in

Δ is driven by (5.1). At the outskirts of a galaxy on an outskirt of the universe, the electron and proton clouds and the center of the uu-d pair all lie at the center in

It is this “lag” that produces PRE, positive relative energy, which is repulsed by the anti-gravity turned gravitational force from the mass inside the large circle in

Analogous to the considerations above Section 2.3, the momentum exchange in such a collision is sufficiently energetic to move the charged clouds up to a_{0p} = 28.8 fm (2.2) away from the uu-d pair in some of such collisions so as to produce all allowed Δ values up to 7.4.

If the above treatment for a single test hydrogen atom is to be applied to real expansion of the universe, the following ad-hoc assumptions need made:

1) All hydrogen atoms arriving at the outskirt of the observable universe indicated by the large circle in

2) All these atoms form hydrogen gas clouds in regions outside the large circle in

3) The protons in these stars cannot be accelerated according to (4.6) (see Section 2.4, end of

1) With the assumptions 1)-3) above, the velocity v(t) increases with time t and with the distance R(t) in

In this SSI model universe, hydrogen gas “leaks” out at its boundary (large circle in

2) The scenario in 1) is derived from (4.6) and holds only for R(t) ≪ R_{US} because the newly created energy in the shell with thickness R(t) outside the original universe (large circle in

In this qualitative manner, the hydrogen atoms “leaked” out from the universe inside the large circle in _{US} in _{US} + R(t). This process is repeated as this model universe expands. As was mentioned at the end of Section 4.2, this anti-gravity accelerating expansion itself slows down with increasing t and R(t) according to (4.6).

3) Another scenario concerns the assumed form of (4.5). The collision of the test hydrogen atom with another atom mentioned above (3.1) causes the relative energy −ω to change its sign. This implies that −ω, a constant in the relative space x, can depend upon the laboratory time coordinate X^{0} = t. There is no conflict here as such gravitationally induced time dependence is negligibly weak relative to those normally associated with strong inter-quark forces. Such a time dependence was included in (4.5) in form of the step function u(t) with a constant amplitude 1 + Δ. This may be regarded to be a first order t dependence. To second order, a linear dependence in form of βt, where β is another unknown constant, may be added to the above step function and modify (4.5) to

− ω = ( 1 + Δ ) u ( t ) E 0 + β t (5.2)

Inserting this into (4.2) renders the acceleration in (4.6) to increase with time instead and enhances the rate of acceleration. The heuristic limit in (4.5 - 6) may then need be modified. Furthermore, as the expansion velocities approach the light speed, relativistic effects become important so that (4.2) as well as the treatment in the Appendix no longer holds.

The mechanism of Section 5.2 1) for the expansion at the outskirts of the model universe can be taken over to apply to the outer edges of a galaxy, as was indicated above (3.1). The hypothetical center UC of the universe in

Evidence of the expansion of the universe comes from Hubble’s law

v H ( t ) = H ( t ) D ( t ) , H ( 0 ) = H 0 = 2.28 × 10 − 18 /s (6.1)

where v_{H} denotes redshift velocities of stars in distant galaxies measured from the earth, D the earth to stars distances, H(t) the Hubble parameter, and H_{0} the measured Hubble constant. Differentiation of (6.1) yields_{ }

d v H ( t ) d t = H ( t ) v H ( t ) + d H ( t ) d t D ( t ) , v H ( t ) = d D ( t ) d t d H ( t ) d t = − H 2 ( t ) + H ( t ) 1 v H ( t ) d v H ( t ) d t (6.2)

Let the earth be on a radius in _{ES}(t) and the light emitting stars be located near R(t), the position of the test hydrogen atom governed by (4.6). Then,

D ( t ) = R ( t ) + R E S ( t ) (6.3)

The Hubble constant H_{0} in (6.1) cannot be derived from the present model because R_{ES}(t) is unknown. Now R_{ES}(t) lies inside the observable universe and changes with t slowly compared to R(t) for a “run away” test atom accelerated via PRE near S in

v ( t ) = v H ( t ) (6.4)

where the unknown R_{ES}(t) drops out upon differentiation. Equations (6.4), (4.6) and the last of (6.2) leads to

d H ( t ) d t = − H 2 ( t ) + H ( t ) G v ( t ) [ M G ( R G S + R ( t ) ) 2 + M U ( R U S + R ( t ) ) 2 ] [ 1 + Δ γ ( t ) − 1 ] (6.5)

For small t, (4.6) can be replaced by (4.3 - 5) and (6.5) becomes

d H ( t ) d t = − H 2 ( t ) + H ( t ) Δ a c 0 v ( 0 ) + a c 0 Δ t (6.6)

For small Δ, the last term can be dropped and the solution is

H ( t ) = H 0 1 + H 0 t (6.7)

which reduces to the second of (6.1) at t = 0. This deceleration is extremely weak. A 1% reduction of H(t) will take 1.4 ´ 10^{8} years.

A linearized approximation, valid for small t, is achieved via H^{2}(t) H_{0}H(t) in (6.6) which now yields

H ( t ) = H 0 ( 1 + Δ a c 0 t v ( 0 ) ) exp ( − H 0 Δ t ) (6.8)

For Δ values in ^{2}(t) in (6.6) to 0. Equation (6.8) then becomes the same as the first of (4.4) using (4.5) if H(t) and H_{0} were replaced by v(t) and v(0), respectively. Thus, the Hubble parameter H(t) increases with time at the same rate as does the accelerating expansion v(t) in (4.4) for small t. As was mentioned above

This increase is due to that the last term in (6.6) arising from PRE acceleration is positive and large compared to the decelerating −H^{2}(t). This last term becomes large as H(t) become greater at larger times; (6.8) then no longer holds and the nonlinear (6.5) has to be solved on computer.

The computer results in ^{2}(t) term becomes large enough to cancel out the last, accelerating term in (6.5) so that H(t) reaches very large maxima there and then starts to decrease to smaller values at larger times, of the order of magnitude of 10^{8-9} years. At such large times, the test atom may have collided with other atoms so that the PRE gained via (6.5) was lost and the results in

These results are based upon the identification (6.4) which in its turn depends upon the gross, ad-hoc assumptions 1)-3) above Section 5.2. Therefore, the H(t) predictions in

The Hubble law (6.1) is empirical and valid in the part of the observable universe containing the earth and the galaxies visible from it. This scenario is more realistic in this respect but does not include dark matter. This law cannot be derived from any first principles’ theory without introducing additional concepts. The expansion mechanism is not clear; the mainstream candidate is the unidentified dark energy.

On the other hand, in the present SSI model in

Results from both of these approaches are compatible with the general conception of an acceleratingly expanding universe and complement each other. In the Hubble case, measurements can be made but no theory exists to account for them. In the SSI model, theory exists to explain the accelerating expansion but no measurement can be made to verify it. The theoretical results can thus not confirm data directly because they refer to different environments.

Nevertheless, with the aid of the assumptions 1)-3) above Section 5.2, the connection (6.4) could be set in a heuristic manner. This connection leads to qualitative compatability between data and present theory under such circumstances for small velocities.

Model | ΛCDM | SSI |
---|---|---|

Foundation | General relativity and additional basic postulates | First principles’ theory supported by hadron data. No additional basic postulate |

Dark Matter | Assumed CDM, cold dark matter, WIMPs not found | As negative relative energy generated among quarks in inhomogeneous, cold hydrogen gas in gravitational field |

Dark Energy | Difficulty with the cosmological constant problem | No dark energy. Only positive relative energy generated in hydrogen gas in outskirts of observable universe yielding anti-gravity expansion |

Distribution | Universe has no center, no boundary, supported by Hubble’s law | Universe has an unobserved center and is bounded by surrounding vacuum. into which it expands |

Another approximately mutually compatible scenario may be that evolution according to the SSI model took place long, perhaps 10^{9} - 10^{10} years, ago. Evolution of such a universe may eventually have ended up in the present state of the observable universe in which Hubble’s law (6.1) can be established.

As was mentioned in the caption of

Negative relative energy between the diquark uu and the quark d in a proton plays the role of dark matter. Positive relative energy PRE between these quarks can lead to anti-gravity expansion of the universe.

Inside the universe, the amount of negative relative energies or dark matter produced depends upon the gravitational force acting on the quarks in hydrogen gas opposed by the pressure gradients acting on the electrons in this gas. In a galaxy, such amount can prevent fast moving stars from escaping the galaxy and cause gravitational lensing. Star formation in this gas uses up the hydrogen atoms that create such dark matter and hence reduces the dark matter content in this galaxy and increases the galaxy’s apparent expansion rate. In intergalactic space, such dark matter can contribute to formation of filaments.

To account for the expansion of the universe, a hypothetical model of the universe is proposed. Near the outskirts of the observable universe, both the gas pressure gradient and gravitational force become very weak and the amount of dark matter generated nearly vanishes. Random collisions between hydrogen atoms in this region can flip such a small amount of negative relative energy or dark matter into a small positive relative energy or PRE in a hydrogen atom. Once such a sign change takes place, gravitational attraction switches into anti-gravity repulsion now unopposed by any pressure gradient. This leads to a “run away” hydrogen atom and provides the mechanism for an acceleratingly expanding universe.

This theoretical expansion and the measured Hubble data are both compatible with the conception of an expanding universe and complement each other. But they cannot verify each other directly because the present model has been constructed for purposes different from those of the measurements. On the other hand, the present accelerating expansion mechanism of the universe is based upon a first principles’ theory while Hubble’s law cannot be derived from any such theory. However, both approaches can under certain circumstances be shown to support each other qualitatively at small velocities.

Dark matter and PRE are not foreign objects like WIMPs and dark energy-cosmological constant but can only be created in cold hydrogen gas in gravitational field. To achieve this, collisions among the hydrogen atoms must take place. Dark matter was created first and can eventually later evolve into PRE in the outskirts of the universe. Dark matter and PRE will disappear if the hydrogen atom creating them becomes ionized as in stars.

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

Hoh, F.C. (2021) Dark Matter Creation and Anti-Gravity Acceleration of the Expanding Universe. Journal of Modern Physics, 12, 139-160. https://doi.org/10.4236/jmp.2021.123013

The equations of motion for baryons in SSI are [

∂ I a b ˙ ∂ I g h ˙ ∂ I I e ˙ f χ { b ˙ h ˙ } f ( x I , x I I ) = − i ( M b 3 + Φ b ( x I , x I I ) ) ψ e ˙ { a g } ( x I , x I I ) (A1a)

∂ I b ˙ c ∂ I h ˙ k ∂ I I d e ˙ ψ e ˙ { c k } ( x I , x I I ) = − i ( M b 3 + Φ b ( x I , x I I ) ) { b ˙ h ˙ } d ( x I , x I I ) (A1b)

M b = ( 2 m A + m B ) / 2 (A1c)

⌈ ⌉ ¯ _ I ⌈ ⌉ ¯ _ I ⌈ ⌉ ¯ _ I I Φ b ( x I , x I I ) = 1 4 g s 6 { χ { b ˙ h ˙ } f ( x I , x I I ) ψ f { b ˙ h ˙ } ( x I , x I I ) + c . c . } (A1d)

where the spinor indices run from 1 to 2, x_{I} is the coordinate of the diquark, x_{II} that of the quark, ∂ I = ∂ / ∂ x I , ∂ I I = ∂ / ∂ x I I , the m’s quark masses [_{b} the interquark potential dependent only upon the interquark distance | x _ I I − x _ I | for free baryons [_{ }

x μ = x I I μ − x I μ , X μ = ( 1 − a m ) x I μ + a m x I I μ , a m = ( X μ − x I μ ) / ( x I I μ − x I μ ) (A2)

χ { a ˙ c ˙ } e ( x I , x I I ) = χ { a ˙ c ˙ } e ( x _ ) exp ( i ω x 0 ) × exp ( − i K μ X μ ) ψ e ˙ { a c } ( x I , x I I ) = ψ e ˙ { a c } ( x _ ) exp ( i ω x 0 ) × exp ( − i K μ X μ ) , K μ = ( E K , − K _ ) (A3)

Here, E_{K} is the baryon energy, K _ its momentum and ω the relative energy between the diquark and the quark, and a_{m} a real constant. The baryon wave functions χ and ψ have 6 components each comprising of a spin 1/2 doublet part χ 0 a ˙ , ψ 0 a [

Consider the rest frame K _ = 0 doublet baryons and put [

a m = 1 / 2 + ω / E 0 (A4)

Substituting (A2-A4) into (A1) and put K _ = 0 , (A1) can be decomposed into a quartet part for the spin 3/2 baryons [

( i δ a b ˙ E 0 / 2 + σ _ a b ˙ ∂ _ ) ( E 0 2 / 4 + Δ ) χ 0 b ˙ ( x _ ) = i ( M b 3 + Φ b ( x _ ) ) ψ 0 a ( x _ ) ( i δ b ˙ c E 0 / 2 − σ _ b ˙ c ∂ _ ) ( E 0 2 / 4 + Δ ) ψ 0 c ( x _ ) = i ( M b 3 + Φ b ( x _ ) ) χ 0 b ˙ ( x _ ) , Δ = ∂ 2 / ∂ x _ 2 (A5)

For the plane wave solution in (A3), the normalized amplitude of the wave functions in (A3) with K _ = 0 vanishes so that the right side of (A1d) also drops out to yield [

Φ b ( x _ ) = d b r + d b 0 + d b 1 r + d b 2 r 2 + d b 4 r 4 , r = | x _ | (A6)

where the d_{b}’s are constants. The ansatz [

ψ 0 1 ( x _ ) = 1 4 π ( g 0 ( r ) + i f 0 ( r ) cos θ ) , χ 0 1 ˙ ( x _ ) = ( ψ 0 1 ( x _ ) ) ∗ ψ 0 2 ( x _ ) = 1 4 π i f 0 ( r ) sin θ exp ( i φ ) , χ 0 2 ˙ ( x _ ) = − ψ 0 2 ( x _ ) (A7)

where θ, f are angles in the “hidden” relative space x _ . Equations (A-6-7) have been inserted into (A5) which has been converted into a first order system [^{0} and X^{0} baryons. The associated d_{b} values needed to obtain confinement, g 0 ( r → ∞ ) → 0 and f 0 ( r → ∞ ) → 0 , as well as g_{0}(r) and f_{0}(r) themselves are given in [_{0}(r) and f_{0}(r) for the neutron that led to correct prediction of its life [

Due to the small differences in the mass E_{0} and quark masses (A1c) between the neutron and proton, the above neutron results can be taken over for proton here.