An Alternative to Dark Matter? Part 3: An Open Universe (3 Gy to 76 Gy) Galaxies and Structures Rotation

A cosmological model was developed using the equation of state of photon gas, as well as cosmic time. The primary objective of this model is to see if determining the observed rotation speed of galactic matter is possible, without using dark matter (halo) as a parameter. To do so, a numerical application of the evolution of variables in accordance with cosmic time and a new state equation was developed to determine precise, realistic values for a number of cosmological parameters, such as energy of the universe U, cosmological constant Λ, curvature of space k, energy density e ρΛ , age of the universe tΩ (part 1). That energy of the universe, when taken into consideration during the formation of the first galaxies (<1 [Gy]), provides a relatively adequate explanation of the non-Keplerian rotation of galactic masses (part 2). Indeed, such residual, non-baryonic energy, when considered in Newton’s gravity equation, adds the term ( ) F r Λ , which can partially explain, without recourse to dark matter, the rotations of some galaxies, such as M33, UGC12591, UGC2885, NGC3198, NGC253, DDO161, UDG44, the MW and the Coma cluster. Today, in the MW, that cosmological gravity force is in the order of 10 times smaller than the conventional gravity force. The model predicts an acceleration of the mass in the universe (q~−0.986); the energy associated with curvature Ek is the driving force behind the expansion of the universe, rather than the energy associated with the cosmological constant EΛ. An equation to determine expansion is obtained using the energy form of the Friedmann equation relative to Planck power PP and cosmic time or Planck force FP acting at the frontier of the universe moving at c. This constant Planck force, from unknown sources, acts everywhere to the expansion of the universe as a stretching effect on the volume. Finally, the model partly explains the value a0 of the MOND theory. Indeed, a0 is not a true constant, but depends on the cosmological constant at the time the great structures were How to cite this paper: Perron, J. (2021) An Alternative to Dark Matter? Part 3: An Open Universe (3 Gy to 76 Gy) Galaxies and Structures Rotation. Journal of High Energy Physics, Gravitation and Cosmology, 7, 844-872. https://doi.org/10.4236/jhepgc.2021.73048 Received: March 17, 2021 Accepted: June 22, 2021 Published: June 25, 2021 Copyright © 2021 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access


Attractive Cosmological Gravity, FΛ, and Galaxy Rotation (Simplified Model)
The formation and evolution of galaxies is a very complex field of study, and the associated mechanisms have not yet been fully interpreted. Indeed, the number of phenomena in play during galactogenesis, such as supplemental forces to gravity, the birth of stars and internal structures, energy dissipation effects, and the quantity and type of neighbouring matter being absorbed are only some of the factors involved in galaxy formation [1]. A relatively complete model has been put forth by [2] Martig et al., which assumes the presence or existence of dark matter that is as much subject to gravity (Kepler) as baryonic matter. In this article, as aforementioned, we do not consider the existence of dark matter, but rather energy at time t (non-massive) and the mass-energy equivalence acting through the cosmological constant. This has already been discussed by [3] Gessner, where the cosmological constant did not vary during the formation process of the structures. This may be due to the lack of a predictive model for Λ(t), which hinders the simulation of velocity profiles for structures with either small or great radii. With such a predictive model, the impact of this attractive force on galaxy rotation can be seen. We do know the values of the cosmological constant, Λ, at the time of primitive galaxy formation (1)(2)). We can calculate that attractive force and see its effects on the rotation of some galaxies. Put simply, for a given circular rotation orbit, the tangential rotation speed of a mass is expressed through the balance of the main forces considered in the model: gravity and cosmological gravity via mass-energy equivalence: tor of the second term is not the inverse of the radius, which confirms that the force is not due to the effects of mass as such, but to a mass-energy equivalence associated with Λ. Finally, because that force is relative to Λ, which is relative to the age of the universe, the rotation profile of masses like galaxies is in turn relative to time from the standpoint of forces in play. In other words, the rotation profile should take into consideration the evolution of Λ as the galaxy absorbs matter over time. The actual process behind the action of this cosmological gravity, F Λ , on the rotation dynamics of galaxies is complex, as it is relative to both time and the radius of any given galaxy: Solving this equation is beyond the scope of this paper because we would need to know the density profile of matter in the galaxy relative to time, t, meaning the formation mechanismes of that galaxy from a dynamic standpoint (mass accumulation process and rate). A simulator like Millenium could derive that term associated with F Λ . However, in this paper, we want to demonstrate that assuming the existence of dark matter is not necessary at first to describe the galaxy formation process and rotation curves as we see them today. To do so, the galaxy formation process can be simplified by assuming that mass accumulates according to a simple function of time, and that Λ(t) also evolves according to time (bottom-up model). The simplified equation of galaxy rotation has three terms, the effects associated with the bulbe, or denser central area, and with the disc around the central area, and the effects of Λ(t) at formation time t and radius r(t). At first, we will not consider dark matter, also called halo mass, although such dark baryonic mass (non-radiating) surely must exist within galaxies. We will see that for some galaxies, such as M33, the observable mass (luminous) is not sufficient to explain the observed rotations, meaning that we have to assume the probable existence of baryonic dark masses.
The time at which a galaxy started to form is important because it influences the effective value of Λ. Then, the formation time of the galaxy is just as important (acceleration rate of the mass), since this yields the total variation of Λ on the rotation process. To initially demonstrate the effects of force F Λ on galaxy rotation, let us find an expression of rotation speed relative to time: the time at which the galaxy started to form, t i , the total formation time of the galaxy, t T , with variable force, F Λ , acting during that formation time, t T -t i . For the masses of the bulbe and disc, we get a simplified expression: r r t r ≤ ≤ , for the disc: r b : bulbe radius determined at the end of galaxy formation; r T : disc and bulbe radii determined at the end of galaxy formation; M b : bulbe mass determined at the end of galaxy formation; M d : disc mass determined at the end of galaxy formation.
A simple law can be used to calculate mass accumulation at a constant rate: α : galaxy radius growth rate (accumulation).
For the mass, we get: For rotation speed, we get: In the above equation for v t , the first term is for the attraction of the bulbe on the rotating mass, the second, for the attraction of the disc, and the third, for the attraction of force F Λ due to the cosmological constant through the residual mass-energy equivalence of the universe at the beginning of formation, t i , of the galaxy acting throughout formation time, t T -t i . This equation contains the essential elements for predicting the rotation curve of the luminous mass of galaxies. Force F Λ decreases over time, or the age of the universe, but one must consider that the prevailing conditions of galaxy formation are still present in the space-time continuum of that galaxy. In other words, we will see that, in simulations of the rotation of some galaxies, the time at which mass started to accumulate is crucial for the development of the type of rotation because cosmological gravity varies like t 4 , or inversely with the age of the universe during the forma-

Mass Rotation Equation and Tangential Velocity
The rotation equation involves five parameters to determine, at first glance, the rotation of a galaxy assuming that it has not undergone severe transformations, such as collisions with other massive bodies. In this study, we propose a bottom-up approach with the following parameters: -Disc mass accumulated during disc formation time.
The actual mass distribution and radial velocity of galaxies are complex, other parameters have to be considered, such as the presence of gases, and small neighbouring structures or more massive structures nearby (like other galaxies), etc. However, we will see that the equation requires careful consideration to significantly reduce the need to consider the dark matter halo (invisible) to explain rotation speeds. Dark matter is not considered in this model, but we do consider non-luminous baryonic matter.

MW, S(B)bc I-II
Many studies have been conducted to try and determine the velocity profile and mass of the MW. Several variations of the luminous mass have been reported, and many authors include an estimation of the dark matter halo to validate certain observations or conclusions. Indeed, according to a number of studies, the total mass of the MW can vary by as much as a factor of seven   ). The galaxy's mass is considered constant after its main formation. Realistically, however, accumulation is a continuous process.
In this model, we will see that the main formation of galaxies seems to have occurred around the beginning of the universe (<1.5 [Gy]), and that accumulation progressively decreases thereafter, even though the intrinsic motion of galaxies continues over time and events (collisions, restructurings, amalgamations). In fact, the early formation of structures like massive black holes and galaxies (<500 [My]) could be made possible by a direct collapse mechanism [6] [7]. Recently, a team discovered a candidate galaxy, SPT0615-JD, at z ~10 -11, that may have existed around 400 [My] after the beginning [8]. Also, another team reported the lens-effect observation of a star dating back to earlier than 250 [My] in galaxy MACS1149-JD1 [9]. Lately, the ALPINE-ALMA project to confirm the existence of 118 galaxies matures much earlier than was thought possible [10].

J. Perron Journal of High Energy Physics, Gravitation and Cosmology
The simulation process is as follows: The primitive formation of the galaxy is determined by trial and error using the aforementioned five-paremeter equation.
Then, the galaxy undergoes expansion of the universe on a scale factor until today. The simulation can be extended into the MW's future. At the beginning, the MW had a radius of about r = 0. 34 [kpc]. This is smaller than stated by Martig et al. [2], a disc dimension derived from dynamic galaxy simulations (r core ~1.79 [kpc], case G10). However, taking into consideration the appropriate scale factor for the simulations by Martig (z = 2) (4.6 [Gy]/13.8 [Gy]), the starting dimension of the galaxy disc for case G10 is reduced to ~0.59. Rotation speed of the center was quite high, around 1100 [k·ms −1 ]. Then, the MW increased in size by a scale factor and speeds dropped. Around 6 [Gy], its diameter was around 1.6 [kpc]. Figure 3 shows three velocity profiles [6] with observed masses and the three remaining parameters of the equation: t i , t T and t b , along with cosmological gravity, F Λ , calculated at the time the MW was formed.

M33 (SA(s)cd) (of the Triangle)
Studying M33 to explain the radial velocity equation is an arbitrary choice, but we need a galaxy that has apparently not collided with another galaxy in the past, and which contains a large amount of dark matter. In fact, this galaxy is reportedly 85% dark matter [12]. If the dark matter is removed, the following luminous masses remain: ( ) 10 9 0.15~0.15 6 10~9 10 To avoid making too many speculative simulations regarding center and disc masses, we chose the following values as constants:  the rotation. To determine the effects of non-luminous matter, the last curve represents a mass total that is six times greater than the estimated luminous mass ( 10 5.59 10 M ×  ). Note the strong correspondence between the estimated and measured speeds, clearly showing the existence of non-uminous matter in M33 and similar galaxies.

UGC12591, S0/Sa (Pegasus)
Galaxy UGC12591 was chosen to demonstrate the significant effects of cosmological gravity, F Λ , on the formation and faster rotation speeds of early galaxies. Careful studies by [13] Giovanelli et al. and [14] Xinyu Dai show very large amounts of dark matter (84%). Removing the dark matter from the reported total mass ( 12 2.7 10 M ×  ) yields the following luminous mass: ( ) 12 11 0.17~0.17 2.7 10~4. 6 10 Center mass is not specified as such. The rotation curve shows that the center mass should be greater than the disc mass to be able to closely simulate the observed rotation speeds: . Note here that this formation period is called primitive as this is when most of the mass is accumulated. Of course, the evolution of galaxies is dynamic and continuous. Finally, the rotation curve shows that this galaxy's luminous mass is sufficient to generate the observed rotation speeds. The luminous mass of this galaxy is ~4.4 times greater than that of the MW, and its center mass alone is ~21 times greater, which partly explains the great rotation speeds starting in the first 5 [kpc] of the radius.

NGC3198, Sc C
This spiral galaxy has been the object of many studies to determine its velocity profile and the mass of hydrogen gas outside its planar disc [15]. Figure 6 shows three velocity profile curves. The first shows the estimated luminous mass, 10 1.08 10 M ×  and the three remaining parameters of the equa-  My]), if the amount of lacking mass is considered. Note that non-luminous mass must be considered here, which tends to confirm that non-luminous mass can make up significant proportions of galaxies, even when cosmological gravity is in full force. Journal of High Energy Physics, Gravitation and Cosmology

UGC2885, Sc D
One of the largest spiral galaxies observed to date has been the object of many studies. Figure 7 shows two velocity profile curves. The first for the estimated observable mass, 12 2 10 M ×  and the three remaining parameters of the equation: t i , t T and t b , along with the cosmological gravity, F Λ , calculated at the formation time of the galaxy. Note that the mass here is sufficient to generate the rotation speeds. The peak rotation velocity near the center is accurately predicted, but the measured peak is more spread out. The value used for the center mass, 10 4 10 M ×  , is in the same order of magnitude as the 10 10 M  estimated by Gentile [15]. Velocities at the outer radius are greater than those measured and quasi-constant at 298 km·s −1 . However, the mass accumulation model we use is a very simple one, meaning that the galaxy's mass accumulation rate at the outer radius could be smaller, thus reducing the rate of speed increase. The last curve is the Kepler speed curve only.
The galaxy may have started forming around 180 [My] and the main accumulation probably lasted nearly 1.2 [Gy], far longer than any of the other galaxies described herein. For example, the calculated formation time for the MW is 320 [My], or 3.7 times shorter than UGC2885. The luminous mass of this galaxy is sufficient to generate its rotation speeds with the presence of the cosmological gravity.

NGC253, Sculptor
The rotation curve of this southern sky galaxy was measured by [16] Pence, with over 3700 measurements made (Fabry-Perot) along the great axis of the galaxy.   an estimated 2.25 [kpc] distance from the center. In fact, Pence suggests a mean rotation speed of 205 km·s −1 for the measurement zone. He studied several rotation models as well as several estimated masses derived from these models (six estimated masses), varying between 1.08 × 10 11 and [kpc]). Center mass is an estimated 10 1.8 10 M ×  . The velocity profile seems to correspond fairly well with measured values, with a tendency to increase. Adjustment of the velocity profile shows that this galaxy started to form before the MW, but that the center formation process took much longer to complete, or 42 [My], compared to 13 [My] for the MW. The mass of the MW center is a bit lower than NGC253, which is of larger estimated dimension, however (2.2 [kpc] compared to 1 [kpc]), which could partly explain the longer formation time of its center. The luminous mass of this galaxy is sufficient to generate the observed rotation speeds.

Irregular Dwarf Galaxy DDO161
Dwarf galaxy DDO161 was chosen to show the later effects of cosmological gravity. [17] Côté et al. studied eight irregular dwarf galaxies with the Australian telescope, and reported large amounts of dark matter. In the case of DDO161, they predicted a large ratio of dark matter vs. luminous matter m dark /m lumi ~8 to 9 to explain the observed rotation speeds. The observed luminous mass (stars and gases) is:

UDG44, Dragonfly
To demonstrate the powerful effects of the cosmological constant, UDG44, a diffuse galaxy in the Coma cluster, was studied by [18] Van Dokkum et al., who concluded that dark matter makes up 98% of the galaxy's total mass ( ). This galaxy is highly diffuse, although very massive according to researchers, and does not behave like the MW, being considered a "failed MW". Authors report that the galaxy's velocity profile is not structured, and that the mean speed is around 9 [km·s −1 ] with large dispersion (unstructured), or σ~47 [km·s −1 ]. It is obvious that a Kepler rotation model would not apply here. However, to perceive the effects of cosmological gravity, the model can be used to see its effects during formation, which lasted between 1 and 5 [Gy], 3 -15 times longer than the MW. Figure 10 shows the rotation speed relative to formation time, with a start time around 0.177 [Gy], around the same time as the MW.  Note that the longer formation takes, the greater the decrease in cosmological gravity, meaning that rotation becomes almost Kepler-type, except for speeds in the smaller radii, which do not fit well with the Kepler model (r < 5 [kpc]). Note also the need to use cosmological gravity for the early formation of this galaxyas a reminder, the observations made by Van Dokkum et al. showed no established rotation. Nevertheless, we know that when a galaxy takes a very long time to form, the effects of the cosmological gravity diminish and the velocity profile points to the Kepler model; but the Kepler model is not adequate for simulating rotation speeds for smaller radii, while cosmological gravity does so relatively Journal of High Energy Physics, Gravitation and Cosmology well. In short, based on this model, for some unknown reason, but likely due to a lack of neighbouring matter, this galaxy seems to have taken a very long time to form. Knowing this, with the cosmological gravity getting smaller and smaller after ~1 [Gy], the galaxy never had the impetus to generate conventional rotation, making it a diffuse galaxy or, according to this model, a late-developing galaxy from the standpoint of mass accumulation. In fact, there are very many diffuse galaxies of this type in the Coma cluster (>40), which tends to confirm the idea that the matter content in this area of space is rather poor, leading to the formation of diffuse galaxies.

Galaxies Cluster of Coma
We have seen that the prediction model of the luminous mass rotation of a few galaxies, using the cosmological force of gravity, predicts fairly correctly the observed velocities. Of course, the non-luminous baryonic material exists but the quantities necessary to explain the rotational velocities are greatly diminished. Now it would be interesting to check whether on a larger scale (500 to 1000 times), this cosmological force of gravity can explain other mechanisms of rotation of matter. To do this, we apply the model of mass rotation at the scale of a galaxy cluster like that of the Coma cluster. Indeed, this cluster has been studied extensively since the 1930s with among others the studies of [19] Zwicky and thereafter those of [20] Mayall, [21] Van Albada, [22] Omer et al., [23] Pebbles, [24] Rood et al., [25] Kent et al., [26] Merrit, [27] White et al. and recently [28] Gavazzi. In summary, the various studies have all shown, to varying degrees, that the observed velocities of the ~1000 galaxies of the cluster can not be explained again by the presence of the luminous mass only estimated from the brightness-mass of galaxies ratio ( M L   ). It was also with the study of this cluster that the concept of dark matter was proposed (Zwicki). The application of the rotation model is more complicated in the case of a galaxy cluster for mainly four reasons are: -The clusters are much larger than the galaxies, with a rather spherical shape as well as a difficult boundary to determine precisely considering the surrounding objects. -The clusters are remote and the Hubble-Lemaître expansion effect is considerable (Hubble-Lemaître flow). -Most galaxies and other objects in the cluster are not bright and more difficult to characterize. -The velocities of the galaxies are observed from a line of sight that crosses the cluster in the direction of sight which causes a large variation of the observed velocities.
For the Coma cluster, for predicting the velocity of galaxies, we need to esti-  The following graphs show the results obtained for the Coma cluster. In order to allow for longer development and variable accretion in time of mass of a cluster compared to a galaxy, the expression of the growth of the radius r (and mass) is modified slightly like this: For a circular rotation model, the tangential velocity is expressed as: We know that for spherical geometry, mean quadratic radial velocity (line of sight) can be expressed from the quadratic velocity v, i.e.: are variable for small radii, calibration is more difficult for this area. In summary, we observe that the beginning of the formation of the cluster is posterior to ~0. 6 [Gy] and the duration of the formation is less than ~2.5 [Gy]. In addition, the growth rate of the cluster appears to be higher at the beginning (b < 1). If we choose the following preferred values (t i = 0.7 [Gy], t T = 2.2 [Gy] and b = 0.5), we obtain the following velocity curves for the Coma cluster ( Figure 14).
We can draw the following conclusions about the Coma cluster and the cosmological force F Λ .
1) Excluding this cosmological force, it is not possible to reproduce the observed radial velocities using only the luminous mass and gravitational force only. Indeed, the rotational velocities are too low. If the mass of the cluster is increased by 40 times, the observed elevated velocities can be obtained without the cosmological gravitational force. Journal of High Energy Physics, Gravitation and Cosmology    2) The cosmological constant Λ is to much decrease for a time of formation initial t i of 0.7 [Gy] compared to that of the MW of 0.18 [Gy] (228 times smaller) but, the r dimension of the Coma cluster is ~500 times greater which allows to maintain the cosmological gravitational force in the case of a cluster of galaxies.  . This seems to indicate that the formation of galaxies is parallel to that of galaxy clusters, that is, the UDG galaxies as dragonfly 44 are probably not fully mature or formed when joined to the formation of a galaxy cluster. In summary, with respect to the formation of a cluster of galaxies using the cosmological gravitational force, we observe that as in the case of galaxies, the formation appears faster than most estimates. However, lately, [33] Wang et al. observed a cluster called (protocluster) consisting of 17 massive galaxies and x-ray emission observations, suggests that the cluster formed rapidly from a dense nucleus of 80 [kpc] like a galaxy or even a black hole and the universe had only 2.5 [Gy]. This suggests that the rapid formation of the clusters does indeed exist. Besides other clusters or protocluster seems to have been discovered according to the same authors.

Summary of the Galaxy Rotation Model
We have seen that the five-parameter model performs relatively well for the simulation of mass velocity profiles for the seven galaxies and Coma cluster described above. The model shows mainly early formation of galaxies, which is not usually considered, although recent observations have shown that organized strucutures did exist as early as 400 [My]. [34] Wang et al. have observed the existence of 39 massive and mature galaxies only 2 [Gy] after the beginnings of the universe. Lately, the ALPINE-ALMA project, Le Fèvre et al. [10], reports the existence of 118 galaxies already formed from 1 to 1.5 [Gy] after the origin. However, the mass accumulation model (radius growth) is very basic, and a full-capacity accumulation model based on existing forces would be more realistic and would surely yield more accurate galactic growth rates. The model uses Journal of High Energy Physics, Gravitation and Cosmology galactic center was completed around 176 [My] after the beginning (see Table 2 below However, if the notion of this model for an approximate time of formation of the bulbe of these galaxies is accepted and we also accept that there is a characteristic or preferable time to the formation of galaxies, this could open a way to determine a preferred direction towards the beginning. This idea of a definite direction was addressed by [37] Zhou et al. Indeed, from the study of observed acceleration variations, g obs , they determined two precise but diametrically opposed galactic directions (l, b) and (l + 180˚, −b), where the accelerations of 147 galaxies show systematic differences that lead to two most likely directions. They used the MOND theory to derive these directions, along with values for a 0 which, as we will see later, are fundamentally related to the cosmological constant, which depends on cosmic time, and to the formation time of the structure.
Therefore, a more methodical study of the rotation of many galaxies around the galactic sphere would help to determine, with rotation curves and estimated masses, if a formation trend before or after the MW could yield a specific direction, and thus confirm or reject the notion of a possible direction towards the beginning.

MOND Theory and Cosmological Constant
The cosmological constant can be used to find a possible fundamental explanation for the MOND theory. Indeed, by equalizing the expression of rotation speed for the mass of a great structure, as predicted with the MOND theory, to that obtained using conventional and cosmological gravity, we get the following equation of equality:  First, constant a 0 is not independent of time. Indeed, it varies with the age of the universe via the cosmological constant, radius, and mass of the structure.
Hence, when the value for a 0 is adjusted, or selected, those three parameters are fixed. However, we know that the value of Λ is time dependent, so that the choice of r and M, in particular, fix the value of Λ, or the mean formation time of the structure. Selecting a typical mass and typical radius for a galaxy is easy (e.g. 10 10 M  and r = 40 [kpc]). For smaller structures, 0 r → , the last two terms tend towards zero, which brings us back to Newton's theory: By selecting a typical mass and radius, a specific value for a 0 through time can be obtained, knowing that the cosmological constant will vary [38]. Randriamampandry et al. use the MOND theory for the study of the rotation of 15 galaxies and they mention the need to vary the constant a 0 in order to adjust the rotation curves (a 0 ~0.34 to 2 × 10 −10 [m·s −2 ]). Figure 16 shows the values of a 0 for three typical masses, M (10 9 , 10 10 and 11 [kpc]).
Note that the selected value for a 0 (~1.2 × 10 −10 [m·s −2 ]) corresponds to the formation periods of structures from about 0.32 [Gy] to 1 [Gy] (Figure 17), showing that the MOND theory, in the context of this model, assumes that the galaxies were formed during that ~600 [My] period, so a short period of time also, but later than advanced in this model, in which formation starts around 200 [My] (MW).    [44]. Cosmological gravity is behind such early formation, prior to the accepted normal period of a few billion years. Of course, this does not exclude the relative activity of galaxies thereafter (accumulations, collisions, amalgamations, breakups). Finally, in the context of this model, which uses the cosmological constant, the value of constant a 0 of the MOND theory is more fundamentally explained, allowing to highlight the fact that the theory is an explicit form of cosmological gravity acting on the formation of galaxies. Constant a 0 is not fundamentally a constant, and it does not question Newton's law of gravity for great structures. Finally, the model described herein seems interesting for several reasons, but further development is required before its foundations can be validated (complete particle generation, atoms, fusion, etc.). The model is still one among many, fine tuning and improvements are to be expected.

Funding Statement
Funding for this article was supported by the University of Quebec at Chicoutimi.