An alternative to dark matter ? A cosmological model of galaxy rotation using cosmological gravity force

............................................................................................... 2 Table of contents ..................................................................................... 5 1Introduction: Formulation of the model, initial concept ........................... 7 2Equation of state for the temperature, pressure, volume .......................... 8 3State equation, evolution of photon gas, temperature, volume and pressure .. 10 4Increase in the number of photons ...................................................... 12 5Energy gain .................................................................................. 15 6A possible solution to the horizon problem? .......................................... 20 7Early baryogenesis (protons, neutrons) and leptons (electrons, neutrinos) .... 24 8Electrons ..................................................................................... 28 9Cosmic neutrinos from SN1987A ....................................................... 30 10Temperature variations in the CMB ................................................... 35 11Expanding 3d-sphere of matter .......................................................... 37 12Pressure in the CMB and the Casimir effect: A possible age of the universe .. 39 13A possible baryonic matter-free zone caused by proton and electron time lags 52 14Cosmological constant Λ estimated values ............................................. 57 15The energy form of the Friedmann equation ........................................ 71 16Some comparison with data from the ΛCDM model ............................... 73 17Cosmological gravity force, FΛ .......................................................... 75 18Attractive cosmological gravity, FΛ, and galaxy rotation (simplified model) .. 83 19Mass rotation equation and tangential velocity ...................................... 88 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 May 2019 doi:10.20944/preprints201905.0335.v1


Abstract
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 , cosmological constant Λ, curvature of space k, energy density , age of the universe Ω etc. Some assumptions were put forth in order to solve these equations. Cosmology fascinates. Sky-watching has forever been an integral part of human experience.
Unfortunately, we do not have all the data we need to fully understand the distant past, what we call the beginning of all things, until today, or even until the so-called end. Nevertheless, we do have numerous findings that allow us to reconstruct, to a greater or lesser extent, the sequence of events from the very beginning, if at all possible, using the laws of physics. The model herein is based on the following key premises, some of which are tested, while others are purely speculative.
The following are the key premises of the model: -The macroscopic laws of physics applied after the Planck era; -At the beginning (1 ), all of the energy in the universe was electromagnetic ( -The law of conservation of energy applies to universe-size scales; -The cosmological principle is not necessarily adhered to; -The Hubble constant of the Hubble-Lemaître law is used to solve the Friedmann equations and find values for Λ(t) and k(t).

Equation of state for the temperature, pressure, volume
The photon gas equation that applies when photon numbers are high enough to be considered a gas (N>>1) is written as: kb N T = f(t) where f(t) represents a function of cosmic time. Observations show that the universe is expanding with time r(t). Expansion of the universe is isotropic (̇ isotropic) and in accordance with the Hubble-Lemaître law. The volume V of space (photon propagation) thus generated is isotropic (large-scale isotropic, ̇) . The mechanism behind the evolution pattern for V is unknown but, as we will see later, it is represented by the evolution of energy associated with curvature k. It starts with the initial Planck time tp, and time evolves freely as t+tp. At every step, tp, V, T and P evolve, but the triggering mechanism for this evolution is unknown. V, T and P evolve in some sort of sequence, which is probably as follows: t+tp, V+dV, N+dN, T-dT, P-dP, E-dE. The expanding volume (spacetime) is a sphere whose radius evolves in line with cosmic time. The Hubble-Lemaître law takes the following simple form: In this version, H varies according to cosmic time. We can observe H at t0, written as ̅ 0 (~70[km −1 −1 ]) (Guo, R.Y., JF Zhang, X Zhang, 2019). This yields r = ct + r as the mean evolution of r over time. The radius can undergo local, spontaneous variations that are different than ct, but the average is still equal to ct.
Let us write the equation of state for photon gas in the form of the variation, freely choosing the negative form of the variations, which allows to denote the possible existence of a singularity at the beginning of the evolution of the universe. Moreover, CMB observations reveal a decay of T: Developing the right-hand side yields: The final term on the right is retained as it contains the potential existence of a singularity at the beginning of the evolution of the universe.
Let us develop V, dV, P and dP: The equation for temperature variations in line with the Hubble constant yields different scenarios of evolution for T(t). First, integration creates a problem since dt appears in both the numerator and denominator. The presence of ̃ in the denominator is caused by the term dVdP/VP. If this term is left aside, we get a conventional form of -H dt. Integration can be done by considering the process as a summation along cosmic time t for the numerator dt, with H/(-1+4H̃). Then, the term 4H̃ can be processed in various ways. Moreover, the value of H can vary according to different expansion scenarios. In this version of the model, we assume that the Hubble constant decreases monotonically with time. Let us assume that this term remains constant for the main integration of dt, therefore: where H = 1/t, or /̈ = 2 + ̇ = 0, or still q=0 (for the boundary of the universe).
Note that the acceleration factor q of the boundary of the universe is zero, but we will see later that it is not zero for the mass of the universe.
The equation for T in relation to cosmic time yields interesting characteristics. First, two constants, or unknowns, a4 and ̃ , are required to determine the evolution process of T. Second, ̃ is normally positive, because time is positive and so is ̃. Third, ̃ can be considered a time limit in the flow of time t, which is causal. The smallest ̃ time limit could be a unit of Planck time, tp.

State equation, evolution of photon gas, temperature, volume and pressure
No data is available on the evolution of temperature in the universe due to the limited time since the beginning of T measurements. CMB temperature has been measured, as well as spatial variation ∆ . We also know Planck temperature, Tp, which is normally considered the maximum temperature of any element. If we take T(0) = Tp, (Lima and Trodden, 1996), which denotes the maximum energy in the universe at positive temperature, we get: If we assume that the temperature must remain positive at the beginning and all along the cosmic timeline, then the constant a4 is also positive. This choice of positive temperature is debatable, and a negative temperature at the beginning of the universe leads to a positive temperature after a time delay of 4 ̃. However, the use of a negative temperature requires the support of an extra element, which is not included in this model.
Let us define the age of the universe as t, and CMB temperature as TΩ, or that of the universe as we see it today. Therefore: The value of ̃ for this condition is: To develop an equation for T, we can start with: Finally, we can assume (T −Tp) ~ −Tp, then the final expression for T is:  (Guth and Steinhardt, 1984)).
The most important point to note about this timespan or delay, expressed as = − Ω −1 , is the fact that it allows to slow the decrease in temperature down to a characteristic value of ~10 -14 [s].
We will see that during that delay, the number of photons increases at a quasi-constant temperature and pressure, which allows finding a possible explanation for the event horizon problem.
Photon gas pressure is expressed as ( With r = (Planck length).

Increase in the number of photons
If the expressions lp and at t=0 are used, the number of photons at the beginning of the universe, The cosmic time expression can be used as a progression of n Planck time units, which then yields the following expression of the number of photons in relation to the number of Planck time units: The above expression of the number of photons relative to time is unusual. Indeed, we find that the number of photons increases according to a geometrical progression of ~n 3 over a characteristic time of ~10 -9 [s] for an age of 76.1 [Gy], up to a maximum where it remains constant. However, the energy necessary to expand the number of photons is not known or at least it is not in the electromagnetic form (photonic). This energy of expanding the number of photons could be identified as the one often mentioned void energy. An important point to emphasize, the model is based on the idea of an original big bang but to the difference that the total energy of the universe is created during this characteristic time of 10 -9 [s]. In summary, the creation of the universe begins with 1 photon originally at t=0 and subsequently the following photons are created during this period. One can call this period, the inflation of photons and the original photons the Alphaton.
We will see how this progression in the number of Alphatons relative to time will make it possible to solve the complex horizon problem. [ ] (Fig 3).

Energy gain
Energy at the beginning of the universe is expressed as the energy of a single photon, the value of which is slightly lower than Planck energy, Ep. For N=1: (0) = 0,9NkbTp = 0,9kbTp = 0,9Ep = 0,9c 2 √ ℎ 2 = 1,76 x 10 9 [J] From a macroscopic standpoint, we assume that the universe does not undergo energy transfers with other universes. Also, conventional energy is preserved in relation to time.
-1,0E+89 period of photons inflation Photon gas energy in relation to time can be expressed in several equivalent ways for N >> 1: (3) N kb T ~ 2,7 N kb T With the expression for N(t) obtained earlier: It can be written as: Or still as: Where n is the whole number of Planck time units, tp, and k' is a constant of universe (CMB temperature is considered constant, as well as the age of the universe, 76,1 [Gy]), therefore: Mass has not yet been created at this time because the temperature is in the order of 3,5x10 31 [K].
To get an idea of the sheer magnitude of energy, assuming that the entire mass created is in the order of 10 52 [kg], with relativistic energy-mass equivalence (=0,9), this corresponds to 2x10 69 [J]; still an infinitissimal fraction of the energy in the universe. unexplained, but this is due to the expansion of the universe volume, V and the availability of an unknown energy. Such colossal energy comes from an existing potential which enchances the proliferation of photons, since nothing other than the above equations can predict energy levels.
The number of photons increases in a geometrical progression of nearly n 3 , where n is the number of Planck time units, tp.  Let us express the photon volume quotient to the volume of the universe relative to the number of Planck time units, n, and the number of photons N.
After manipulation, the expression can be written as: In the above expression, the only variables that evolve are the number of Planck time units, n, and number of photons, N. The value of the quotient found for the entire age of the universe is: What does this result mean? We have found that the volume occupied by photons, which increases in geometric progression, is always slightly higher than the volume of the universe, and its boundary is moving at the speed of light. Obviously, the value 2 is not accurate because the photons are contained within the volume of the universe.  6). Hence, even during the photon inflation period, the information exchange between photons cannot be entirely causal, that information is not necessary from a thermodynamic standpoint because the states of T and P remain more or less constant (Fig. 7). This mechanism makes it possible to solve the horizon problem for the photon inflation period, or energy creation period, if the high-energy exchange principle is accepted. Photon-photon exchange are a fact that has been confirmed at CERN (Kłusek-Gawenda, Lebiedowicz and Szczurek, 2016). Photon exchange energy, gg for that experiment was an estimated ~15-20 [GeV], while the energy of photons at the beginning was ~0.9 Ep, or ~10 19 [GeV]. Of course, this goes beyond the purpose of this paper since exchange will require much more study. However, the process makes it possible to solve the event horizon problem, as the photon energy volume is always in phase with that of the volume of the universe. In brief, these periods are: The CMB is at z~1100, or well after the start of the causality recovery period. We will see that the last scattering surface of the model is ~69 [My] after the beginning. This leaves ~10 58 Planck time units to restore causality. It can be reasonably assumed that at recombination time the universe had enough time to recover all of the causality, and that is why we can observe isotropy in the CMB (McCoy, 2014).

Early baryogenesis (protons, neutrons) and leptons (electrons, neutrinos)
Interactions between photons and matter are complex and beyond the scope of this paper.
Moreover, relativistic effects have to be considered as particle speeds approach the speed of light upon creation. In this paper, we describe a creation mechanism for the main particles (p, n, e and ) to demonstrate the coherence of the model. During early baryogenesis, at very high temperature (mc 2 <<kT), the Maxwell-Juttner M-J (relativist) statistical law is used to predict particle properties (fermions and letpons). Moreover, the presence of antiparticles must be considered, along with the creation-annihilation process. In this paper, we want to estimate the total barionic mass produced at the end of baryogenesis. We are able to estimate the full potential of mass creation in the universe using the mass-energy equivalence, since we are estimating total energy. The following expression is used to find the mass creation potential. Note that here, we assume that the energy in the universe is conventional: We can see that the mass creation potential is relative to the cube of the age of the universe. For comparison purposes, for a universe aged 13,8 [Gy] (=0), the maximum total mass that can be produced is 4,81x10 48 [kg], which is ~10 4 smaller than the approximate estimated mass of the universe (10 52 à 53 [kg]) (Carvalho, 1995). This clearly shows that to maintain this estimated mass, the existence of a source of non-conventional energy, or dark energy, has to be considered. Another possibility is to extend the age of the universe. Evidently, the precise mass of the universe is unknown. Supposing an estimated mass variation factor of 10 2 and conventional energy, we have to assume, based on the above equation, that the universe is much older than 13, and 2 ( ) the modified Bessel function of the second kind. In this distribution, the stop temperature for the definitive creation of protons and neutrons must be specified, as well as the relativistic speed of created fermions. The value of poses a problem, in fact, a lower value allows to create more mass and conversely also. We will see further from the energetic form of the Friedmann equation that an average value of can be estimated at ~0,866.
However, the global energy equation imposes a maximum value for beta to ~0,998 in order to maintain the positive energy balance at the scale of the universe (for the entire cosmic time): The temperature can be estimated based on the total energy of a proton or neutron at  This mean photon energy appears at proton and neutron temperature and time, or tpr,ne , after the beginning of expansion: Therefore: tpr,ne = b -  neutrons, the respective creation and annihilation of antiparticles must be considered. To do so, we assume that baryonic asymmetry prevails according to a normally accepted proportion of one stable baryon created for every 10 9 ̅ and ̅ annihilations (Dolgov, 1998 Also, an equation can be found for the baryon-photon ratio, . Initially assuming that the baryonphoton ratio can be expressed as the proton and neutron creation potential after annihilation and disintegration, expressed in a number of protons (at ) only, after manipulation, we get the following equation and a maximum value for the ratio: The above constant ratio solely depends on associated with protons during (relativistic) creation, and the modified Bessel function of the second kind, 2 ( ) (Maxwell-Juttner distribution), as well as the numbers, e, and Riemann constant, (3). The value 10 −9 is the oft-used matter-antimatter annihilation factor, ̅ . The maximum value is for = 0, or = 1 and 2 (1) = 1,62. Therefore: = 10 −9 (1− 2 ) 2 ( ) 6,53 = 10 −9 1,62 6,53 = 2,48x10 -10 The resulting value of 2,48x10 -10 is lower than the results of the estimates yielded by the ΛCDM model (Kirilova and Panayotova, 2015), based on Planck measurements. Indeed, the estimated quotient is not a direct measurement, but rather an estimate that is partly based on ΛCDM model assumptions and observations, or: = = 6,108 ± 0,038 x10 -10 However, a small change in the oft-stated ~10 -9 particle-antiparticle annihilation factor and can proportionally change the result.

Electrons
The Maxwell-Juttner statistical distribution for electrons:  To estimate the final number of electrons, the respective antiparticle creation and annihilation must be considered. To do so, let us assume that lepton asymmetry prevails according to a proportion of one stable electron created for every 10 9 ̅ annihilations.
For Ω =76,1 [Gy]: Finally, the following total mass for the creation of electrons, protons and neutrons is achieved: The ratio of positive (p) to negative (e) charges is strictly equal to one, since the beta disintegration of a neutron produces one proton and one electron. Therefore, the Maxwell-Juttner relativistic distribution predicts an electrically neutral universe in terms of protons, neutrons and electrons.
Based on this relativistic distribution and for a specific cosmological model, the following dynamic temperature-time relation must be met during the proton-electron production process. Indeed, the exact mass ratio is known:

Cosmic neutrinos from SN1987A
Cosmic neutrino mass can be estimated using the above relation. Indeed, cosmic neutrino mass can be expressed according to proton or electron mass, as: The above equation can be developed with the electron temperature equation along with electron creation time. After some manipulations, we get the following expression for cosmic neutrino mass: The only undetermined variable in the above equation is the mean  of cosmic neutrinos during their creation. The use of  is not an easy choice since this particle is still relatively unknown and has three known states (oscillations). Using 1987 , estimated from Stodolsky's observations of SN1987A in (Stodolsky, 1988), (≤0,999999998), the maximum neutrino mass can be expressed as: While this is too high a mass for electron neutrinos (<2,5 [eV −2 ]), it fits well for muon neutrinos In addition, this found value is within the estimated limit of Benes et al, (2005) for the sterile neutrino mass of SN1987A (10-100 [keV −2 ]). Also, Bezrukov (2018), from a detailed analysis of the possibilities for the mass of the sterile neutrino, find a value ~ 3,3 [keV −2 ] that it identifies as a possibility that dark matter is made of sterile neutrinos. However, we will see that the amount of neutrino generated cannot explain the abundance of dark matter predicted by the ΛCDM model This maximum mass is situated between that of the electron neutrino and muon neutrino, or: The resulting mass for cosmic neutrinos is ~ 10 times lower than that of electrons, and their speed is practically the speed of light c. Of course, cosmic neutrinos can be found to have different masses depending on the assumptions made for  The goal here is not to derive precise neutrino mass, which is beyond the scope of this paper. Using the neutrino mass obtained above, the time, temperature, quantity, and total mass of cosmic neutrinos can be achieved using the Maxwell-Juttner distribution: Therefore: A conclusion can be made here, neutrino mass (without annihilation) represents a maximum ~4,2 % of proton mass. Based on the model, cosmic neutrino mass cannot explain the origin of the missing mass. Furthermore, based on the Maxwell-Juttner distribution, cosmic neutrinos appeared before electrons, but after baryons. Another way to proceed involves using the known neutrino mass and look at the creation period and predicted mass, but we still get a predicted neutrino mass that is much smaller than that of baryons.
Let us revisit the total predicted mass of ~7 x10 50 , which is relatively lower (17 to 350 times) than the oft-mentioned total mass of the universe (1,25x10 52 to 2,5x10 53 ). However, total mass is relative to the age of the universe. Hence, baryon mass could be increased by increasing the age of the universe or by reducing the particle-antiparticle annihilation factor. However, we will see that the so-called missing mass is not that essential to explain galaxy rotation. The mass can be increased, but we will see that the data from the Planck probe give us the mass vs. energy ratio, which allows us to calculate an approximate age of the universe that partly meets the proportions.
We will come back to this argument later. With the energy-mass equivalence, when the ratio of total created mass-energy to total universe energy at the time of electron production (around the end of the main leptogenesis) is obtained, we get =0.001, or a low non-relativistic speed of the baryonic mass, but still within the range of velocity for the MW: This energy ratio confirms that the universe, during early leptogenesis, or at the end of the creation of the particles that make up most of the mass, was vastly influenced by radiation (radiation universe) and that the effects associated with mass, such as gravity, were negligible compared to the electromagnetic impact of photon gas. Such density is much lower than the approximate density of a proton (~6,7x10 17 [kg −3 ]), showing that the universe could have contained that amount of mass at that time.
We have not yet considered the electrostatic energy associated with protons and electrons. Let us assume that the Coulomb charge was attributed to protons and electrons at the time of baryogenesis and leptogenesis. Indeed, the electrostatic energy of protons and electrons contained in the sphere with a radius of rpr and rel at the time of protons and electrons is quite significant or, respectively: However, because the quantity of protons, npr, and electrons, nel, created is identical, we get (including neutron disintegration): npr = nel = 3,9x10 77 Therefore, the total charge becomes neutral, and the potential energy disappears in the aftermath of electron production. However, the electrostatic potential remains active for ~666 days, which corresponds to the time difference from the appearance of protons and electrons. We will see that the time difference or delay is the cause of a major so-called baryon-free (empty) zone, except for cosmic neutrinos and others neutral particules.
Thus, the actual baryon-photon ratio for the entire universe (~0,001) can be estimated: This baryon-photon ratio is ~1000 times smaller that the Bernreuther estimate (2002). This is due to the calculated baryon mass, which is 500 to 1000 times smaller, ~10 50 [kg], than the oftsuggested ~10 53 [kg].

Temperature variations in the CMB
A possible way to address partially the temperature variations in the CMB is found in variations in the energy of the universe during baryogenesis and leptogenesis. Indeed, when protons, neutrons, and electrons were created, a considerable amount of energy was drawn from the photons for the creation of the particles. That one-time energy shift in the early expansion of the universe When that energy is put in relation with that of the blackbody, the energy ratio can be expressed in terms of temperature as: Following measurements made by Planck, the analysis and explanation of temperature variations in the CMB became priorities. Ever since the initial analyses and Fixsen's synthesis (Fixsen, 2009 This shows that baryogenesis and leptogenesis, or variation of energy for the creation of protons, electrons and neutrinos, is in the order of magnitude of the overall temperature variations in the CMB (energy disruption or negative energy jump of the photons during the creation of matter).
Could those temperature variations in the CMB be partially caused by successive energy jumps during particle creation, in addition to the vibrational mode of baryons (Eisenstein, Zehavi, Hogg et al., 2005) ? Moreover, analyses of the variations do not seem to show any anisotropy, except for great empty zones. This supports the notion of isotropic energy variations for the entire volume that is compatible with the creation of a uniform mass in the volume. Finally, because protons, neutrons and electrons, and the particle fusion cycles, occurred at different times and different energy levels for the photons in the photon gas, notable variations (Δ / ) could be found in the variations of energy spectrum of the CMB in line with the energy levels successively implicated in beryogenesis and leptogenesis, and at successive times for the protons-neutrons, electrons, deuterium, etc.

Expanding 3d-sphere of matter
An order of magnitude for the avarage speed of baryonic matter can be calculated with a theoretical mean mass density of the universe, the Hubble-Lemaître expansion law, the cosmic time and the assumption that the boundary of the universe is moving constantly at the speed of light.
Let us suppose that this sphere of matter was at state 1 at the time of early creation of great structures like galaxies (<2 [Gy]), whose boundaries were expanding at the speed of light towards state 2, or the current age of the universe, written as tΩ. Let us also suppose a material point in the sphere in state 1 (e.g. the original bulbe of matter at the center of the MW), which undergoes expansion until today. That point is not located at the mathematical centre of the sphere, but at a given location written as r1 at state 1. The material point evolves towards a material position 2 in state 2, moving at a mean speed ̅ (non-relativist). Moreover, considering expansion and displacement at the mean speed in the direction of expansion, the following equation yields the position of the material point at state 1 at time t0 in the sphere of matter at the time of state 2 (universe age tΩ): The first term is the expansion of the material point in the expanding volume during the time period, and the second term is the effect of the speed modulated by the inverse of expansion. The equation has four mathematically independent variables that must be compatible from a physics standpoint.
Indeed, for each quartet ( 1 , 1 , Ω , ̅ ), the value of 0 must be lower than or equal to Ω , which limits possibilities, or still, forces a restriction on variable ̅ . In this paper, we only consider the mean value of β ̅ for a sphere of matter undergoing Hubble-Lemaître expansion, the boundary of which is moving at β=1. The cosmological principle states, at least, that there are no preferred positions. However, expansion of the universe occurs in a precise order of events, each appearing at its own cosmic time, which leads to the idea that for a much larger universe than what we can observe today, one can imagine relative positions within that chronological universe. Moving forward with that idea, one can estimate an approximate position for the MW in the sphere universe.
Indeed, we will see in the next section, dealing with a mass rotation model for a few galaxies with the combined action of gravitational force and cosmological gravity, that initial formation of the That number must be seen as sufficient to create the required energy for the universe to generate a baryonic mass that is close to the mass estimated from observations of the cosmos, while providing a possible explanation for the formation periods and rotations of the galaxies being studied.

Pressure in the CMB and the Casimir effect: A possible age of the universe
The Casimir Effect is often used to explain what authors call vacuum energy or vacuum force.
There is a model we can use to further analyze this effect and see if it can be partially explained and provide useful information.
Readers can refer to numerous works on the Casimir Effect and its electromagnetic origin (Kawka, 2010). If the Casimir force is expressed as shown in works where parallel plates are used, we get the following equation: Where represents the distance between the parallel conductive plates, and S is the surface of the plates. The constant is obtained from the integration of potential photon vibration modes between the plates (the space between the plates act as a resonant cavity for the photons). This normally attractive force can be expressed as radiation pressure: The quantities of energy in the universe on a per-era basis are known, which can be expressed in the form of mean density of energy in the volume, as: From the photon gas energy expression, an expression of Casimir force, from a standpoint of properties at time t, is written as: Where N is the constant number of photons after the photon inflation period, or about 10 -13 [s] (N~6,4x10 89 ). Moreover, if we postulate that Casimir pressure is generated by CMB photons at our position t0, then: The above Casimir Effect equation makes it possible to calculate pressure at time t0 (at our position in the universe) when the mean wavelength of photons in the CMB is known. As with CMB temperature, Casimir pressure is an observable property of the universe. That wavelength is well known and derived from the Wien's law, as: In a manner of speaking, that pressure is the same as theoretical pressure in a vacuum (CMB radiation pressure), considering the fact the energy of the universe decreased when the particles were created. To determine that pressure, we could estimate the position of the observer, t1, in the universe. To do so, we know the expression for photon gas pressure at the same time, t1, and we get the following expression to determine a possible position in the universe or cosmic time: The wavelength of the CMB, as perceived by an observer at point t1, is not modified by the scale factor: Then with the temperature equation: Or with the expression using the Lambert function: This is a surprising result, as it implies that the following ratio of physics constants is relative to position in the universe, or: Of course, if that equation holds true, its cosmological implications are important. The equation can be rewritten assuming that Wien's law is universal and that the speed of light for photons is always the product of wavelength times frequency, or: The ratio of -origin photon frequency to temperature T is strictly constant (1,034x10 11 [s -1 K -1 ]) from the initial Planck time tp up to 76,1 [Gy]. Finally, we get: ℎ = ( Ω ) (function of position in the universe or cosmic time) The implications of that equation are beyond the scope of this paper. The previous section, Expanding 3d-sphere of matter, we arrived at the following expression, which we equate to the result we obtained for 0 : This constant ratio is surprising! It implies that mass speed increases with time as the universe ages, in order to conserve a quasi constant quotient for a given structure (or a given position, t1).
In other words, using the MW as an example, its speed would appear to increase with the increase in the age of the universe. Therefore, for a sphere of matter beginning at 1 [Gy], we use the following to determine the speed of the MW at t0 (13,8 [Gy] and r1/R1 assumed to be 0,181314 in the 1 [Gy] sphere to derive the speed of the MW today): The following three figures (8, 9 & 10) show the form of that evolving speed, or = , acceleration, , and the intrinsic deceleration factor, q, of the MW relative to the age of the universe for a sphere of matter starting at 1 [Gy] and expanding. The MW is at position ~0,181314 [Gy] in that sphere (start of bulbe formation). We use 1 [Gy] sphere because the MW started to expand after its creation, or an initial sphere larger than 181 [My]. Note that the speed of the MW today is an estimated ~ 600 [ −1 ]. That value for the current speed of the MW corresponds relatively well with the estimates made by Kraan-Korteweg et al. (1998).
As for acceleration, we find a very reliable number, which is nevertheless not zero: In brief, the MW was moving slowly in the direction of the beginning (closed universe) after principal formation up to ~ 2 [Gy]. Then, expansion of the mass began, and the MW started to accelerate towards the boundary (open universe). Also, the variation of acceleration, ̇, is slightly showing that the mass accelerates in the direction of expansion.
Finally, for an intrinsic deceleration factor, we get the following expression, which is based on the conventional definition. Moreover, it should be noted that in this version of the model, the deceleration factor, q, of the boundary of the universe is zero, as it moves at constant speed c.
However, mass in the volume of the universe is moving with a negative deceleration factor (acceleration). This is an important difference because the observation of motion in supernovas does not automatically guarantee that such motion applies without distinction at the boundary of the universe. For the deceleration factor of a given mass (intrinsic) we get (based on the definition of ): It is apparent here that the deceleration factor tends towards -1 as the age of the universe increases.
This means that expansion is constantly accelerating and the universe is open. Here, t1 is understood to be the starting value (sphere) of the expansion factor computation, or after the initial formation of the great structures (1-2 [Gy]). The deceleration factor, (z), can be obtained either according to the relative distance to the MW, or to z, the relative cosmological redshift to the MW: By substituting the expression for z in q, the following equation for the deceleration factor is achieved: be seen that at the beginning of expansion, the universe, or the mass, decelerated to >5,9 (t~2 [Gy]). Then, the mass accelerated. Measurements by Reiss et al. (1998) and Kiselev (2003) are shown on the curves. Therefore, the model seems to perform rather well in terms of deriving values of q for the low values of z. However, the model predicts a deceleration-acceleration transition earlier than most other predictive models for q(z). For comparison purposes, is closer to 0,7 according to Giostri et al (2012), who used a calibrated parametrical model with a prescribed constant of ( ) = 1/2 for → 0. That prescribed value is in fact being questioned by researchers.
Based on the model, the deceleration of mass in the universe is quite substantial. Then, after ~ 2 [Gy], expansion starts to increase, and the mass accelerates in small steps.
In the above equation, if the age of the universe is assumed to be 76,1 [Gy], then q= −0,986. However, it should be noted that validation of the Hubble-Lemaître law principally comes from the observation of galaxies, a period of the existing universe after their formation, around 0,1 to 2 [Gy], or the expansion of a sphere at time 1 towards another sphere at time 2 , and not from a dimensionless starting point towards a sphere. This is an important detail because it puts into perspective the fact that the Hubble-Lemaître law is experimental, resulting from the observation of great structures over a period of time which logically begins when those structures have already been formed.
Let us return to Casimir pressure which, relative to , is: Based on this approach, such minimum or zero Casimir energy pressure, 0 , would be lower than what can be obtained from our position in the universe, and only corresponds to the pressure found with the original photons and no matter. This may correspond to the volumic energy state from point zero to our position. Today, pressures as low as ~10 -10 , or extreme vacuum, have been measured at (Conseil Européen pour la Recherche Nucléaire, 2018). Expressing that pressure in terms of amplified pressure between two parallel reflecting plates at distance from each other (cavity), the maximum distance required to arrive at that minimum pressure is in the order of 0,1 [mm], or: To see if that minimum pressure corresponds closely with experimental results designed to determine whether the theoretical value obtained for that pressure is in the order of magnitude of the estimated pressure. Decca et al. (2007) tested the Casimir effect using a torsion oscillator between two gold-coated parallel plates. The smallest pressure mentioned is in the order of 3 [mPa], or one billion times greater than the minimum pressure obtained, 0 . They reported the following measurements (table 1): By using the Casimir Effect, we get amplification of that pressure by photon resonance in the CMB in the different experimental setups and, in particular, in the cavity between the reflecting plates.
That amplification can be expressed as: , because at any greater distance the pressure would be below the minimum value of 0 at our position in the universe. Figure 12 shows the Casimir zero pressure and the photon gas pressure relative to the age of the universe. In brief, with this model we note that photon pressure in the CMB (~1,291x10 -11 [Pa]) at our position, t0, provides a possible explanation for the Casimir effect, as the photons produce an amplified pressure of that value. This leads to the following question: If the Casimir effect is generated by photons in the CMB, how is it that in laboratory experiments, in the total absence of CMB photons, when they are not physically in the presence of experimental setups, their effects are nevertheless measured by the instruments? A first part of the answer could be that the universe has stored the presence of the original photons in 'memory'. This helps us to partially understand how this effect is found in many types of experiments and phenomena (Klimchitskaya, Mohideen and Mostepanenko, 2009) : It is a fundamental characteristic of our universe, where the effects of CMB photons are stored as some sort of property of spacetime in the form of energy which we put into action and measure in diverse experimental setups with more or less pronounced amplification effects.

A possible baryonic matter-free zone caused by proton and electron time lags
This model shows that, assuming that recombination ends when the temperature drops below This is a surprising result, as it matches the sequence between the temperature drop to the recombination level, around 3000 [K], and the time period associated with recombination with the estimated age of the universe. Moreover, the redshift is calculated according to the scale factor for the universe, and not that of the MW; therefore, it applies to the entire universe rather than a onetime object within the universe. Indeed, during recombination, free photons end up on this last scattering surface, travelling in all directions, including that of expansion at the same speed as the physical boundary of the universe, c, (we chose H=1/t). That is why CMB photons appear as an omnipresent gas in all directions and close to us. Finally, such a late recombination time allows solving the horizon problem paradox from a standpoint of last scattering surface dimension. Indeed, the diameter of the universe at recombination was ~138 [My], making it possible to estimate the dimension of the last scattering surface with the equation for the angular dimension of a structure relative to redshift, z, and Sitter's apparent angular dimension ∆ . For an apparent angular dimension of this last scattering surface, which covers the entire celestial half-sphere (∆ = ), we can solve for d or t: Then, a smaller value than the diameter of the universe at recombination, or: We can see that the last scattering surface is included in the universe at that time, which suggests that the inflation mechanisms may no longer be in play, at least from the standpoint of the physical dimensions of the original CMB. Finally, for ( ), which represents the average position of proton motion towards the perimeter during electron production, we get the following differential equation:   (Petrosian, 1974)). This value varies greatly throughout the age of the universe. Moreover, the constant is not a true constant; indeed, it varies with the age of the universe, that is to say the effects of expansion and the production of mass, or the decrease of non-massive energy in the universe.
At the beginning, during the primitive formation of large structures like galaxies over a time period of about 0,2 to 2 [Gy], the energy is mostly in the form of radiation (over 90% of the energy is radiation), and for this period of a few [Gy], the second term, which depends on total mass, MT, is far less important. Figure 14 shows the mass/rad ratio. If different Planck quantities are used, the following expression can be used for the constant: Also, this expression is for the beginning when t→tp:  universe. This closely corresponds with the value found for deceleration transition, q, around 2 [Gy] (Fig. 8). That these two values are relatively close is promising in terms of model constancy.
As concerns energy density, we find two distinct contributions: one associated with radiation and the other, with mass (for b~0, valid for t > 10 -  In short, as concerns energy density variation in the universe, we find a ratio to the power of four between temperature variation and Planck temperature variation, with a multiplication factor. We can see that the volumic mass associated with the cosmological constant, Λ, is equivalent to that of photon gas minus the baryonic mass. Therefore, the cosmological constant reveals the existence of radiation energy. As concerns space curvature, we get a value that can turn negative according to the value of the curve (closed universe). This is important data because it is the only term that can become negative and act in opposition to gravity and mass-energy equivalence. If we express volumic masses based on the critical value corresponding to Λ=k=0, or a flat universe whose only energy comes from mass, we get:   Figure 20 shows the equivalent densities. Here, the contribution of curvature is negative for an age below 2,9 [Gy], a closed universe, as already discussed with the q curve (deceleration). Then, that value of curvature increases rapidly to about 4 [Gy]. Thereafter, all values decrease in monotonic fashion and at different rates. Note that the total value is very close to the critical value, but always smaller. Figure 21a shows the values of associated contributions as they relate to critical density. We can see that curvature, k, is the key factor that can explain sustained expansion of the universe. We know that the contribution of mass, along with the cosmological constant, are based on conventional energy (mass-energy, radiation). In the case of space curvature, k, that form of energy cannot be so easily explained.

The energy form of the Friedmann equation
To determine the type of energy behind the expansion of the universe, the Friedmann equation can be expressed in terms of energy. Indeed, if all the terms of the equation are multiplied by 5 −1 −3 , we get: In short, with the Friedmann equation and the assumptions of this model, we find that energy of unknown origin is acting on the expansion of the universe through an enormous power that is equal to Planck power multiplied by cosmic time. That expansion energy is not directly expressed in a model variable. Moreover, it is positive via Planck power, which represents conventional energy acting in opposition to gravity (or ) and cosmological gravity force Λ (or ).The expansion power is not associated to mass (baryonic) or radiation (photonic via ). This unknown energy of expansion is possibly contained in a potential form available in the volume and at the frontier of the universe that acts by an expansion effect of space in the manner of a stretching of space. This Planck power can be expressed by the Planck force multiplied by c. In this model, we consider that the frontier of the universe moves at speed c. It is seen that the idea of an internal and external force (multiverse) of the magnitude of Planck force acts at the boundary to stretch the space at speed c.
One can determine the expression of the volume expansion force of the universe ⃗ ⃗ using the theorem of divergence in spherical coordinates knowing the expansion force at the border ⃗ ⃗ .

⃗ ⃗ =
if one accepts that spherical symmetry applies to the scale of the universe: The solution found with the divergence theorem is: The result found is remarkable. Indeed, we find that a constant Planck force acts at all points of space, radial direction outwards to realize the expansion of the universe. Of course, the result found brings more questions than answers. At first glance, however, the result seems logical and presupposes an energy associated with space itself. Finally, for comparison, this total expansion energy can be estimated with the spherical symetry.

Some comparison with some data from the ΛCDM model
The 1,5E+70 0,1 0,9 1,7 2,5 3,3 4,1 4,9 5,7 6,5 7,3 8,1 8,9 9,7 10,5 11,3 12,1 12,9 13,7 energies sources (J) Finally, at the beginning, the energy associated with space curvature, k, is relatively small compared to radiation. That energy is of unknown origin and possibly acting at the boundary. As concerns the curvature of space, k, the energy source is not identified. However, in this version of the model, that energy form does not behave like mass-energy equivalence, as is the case with the cosmological constant.

Cosmological gravity force, FΛ
For the time period when radiation was dominant, a central force associated with can be determined using mass-energy equivalence. Indeed, we know the value for via the evolution of energy in the universe. Let us assume an element with mass m in rotation according to a Kepler model in a central gravity field of mass M. Another attractive force is a work around mass m, this time associated with the non-baryonic energy density, which acts through mass-energy equivalence of the interior sphere whose boundary is determined by the rotation radius, r, of mass m. That central force has been suggested by several authors, including Martin (2012). However, after mathematical elaboration, they note that the force is repulsive, and not attractive. This can be explained through mathematical calculations using the cosmological constant, which predicts a repulsive rather than attractive effect when placed on the left side of the general relativity equation.
In this model, we consider that the force is attractive simply through mass-energy equivalence, which can also be achieved with the General Relativity Theory (see below), meaning that a positive energy mass is associated with a positive energy, such as the energy of photons associated with constant Λ, and that energy mass exerts an attractive force on surrounding masses the same way the inertial mass (baryonic) does. What's more, the notion of mass-energy (or electromagnetic) was addressed initially by Langevin (1913), a contemporary of Einstein.
We can see that the mass-energy associated with the cosmological constant (photon gas) depends on a zone demarcated by the assumed radius, r. The full action of this force is unknown, but it is gravitational, meaning that this cosmological gravity force acts together with conventional gravity and that other such couplings are possible. This can partially explain the issues with the cosmological constant, Λ. In fact, that gravity force can be put into action in the general relativity equation through the existence of the cosmological constant, as put forth by Einstein but for a different reason than the static universe he proposed. Indeed, the cosmological constant was later added by Einstein as an opposing force to gravity. Therefore, when the term Λ is moved to the right-hand side, the side of the energy-momentum tensor, we get a repulsive effect associated with Λ: With the signature of the metric tensor (+,-,-,-), the energy-momentum tensor can be expressd as:

= −
In this case, the resulting force is repulsive, as Einstein wanted. However, it is also possible to make the effects of that energy appear directly in the energy-momentum tensor as a source of additional mass-energy through the mass-energy principle, as: Hence, the energy density component of the tensor, 00 , is entirely positive: The solution for the spherical geometry is found in the Newton equation for low velocities: The potential being: A potential in r 2 is said harmonic and the equation of the trajectory of a mass ′ in harmonic potential is a closed curve like that Newtonian in r -1 (Bertrand's problem). The acceleration of a mass ′ in this field is expressed as the gradient of potential : We can see that, at this time, solving the equation predicts an attractive force associated with constant Λ and of the same type as the baryonic mass. The r term can be related to the Hooke ellipse. Moreover, it is surprising to note here that at the beginning of the formation of the structures of the universe the two forces in k r -2 and k r acted simultaneously which, certainly would be likely to reconcile, if it were possible Newton and Hooke. It would make sense to call the potential found NcH for Newton-cosmological-Hooke. Finally, in a detailed form, the NcH potential is expressed as: The force can be expressed in relation to the age of the universe: This attractive force can be attributed to the cosmological constant, which translates conventional energy density that is not in the form of conventional baryonic mass. Moreover, the force of gravity, which varies in r, is active everywhere on the same basis as baryonic mass gravity. Note that such a force has never been detected around us because the cosmological constant is extremely small today (~10 -54 ). However, at the time of primitive galaxy formation, the cosmological constant was much greater (Λ~10 -48 at t~0,5 [ ]). Also, when we include the great galaxy or cluster radii, we will see that the cosmological gravity played a large part in galaxy rotation. For comparison purposes, let us calculate the ratio between the cosmological gravity and Newton's force for the solar system: Note that the value for gΛ is much too small to be detectable by current instruments. However, over the first billion years, let us calculate the ratio of the cosmological gravity to the force of gravity for the universe with a critical volumic mass of 3H 2 /8G: Note that the attractive effect of cosmological gravity is huge and greatly surpasses that of gravity alone during the formation of great structures like galaxies. At 500 [My], the ratio was ~34. Figures   22 and 23 show the mean ratio FΛ/FG for the time period starting at proton time tpr. Note that the cosmological gravity makes it possible for the great structures like galaxies to form much faster than simply under gravity. This notion of additional force to gravity could provide a possible explanation for the production of primitive black holes at the very beginning of the universe (6 <z< 30) (Lupi, Colpi, Devecchi et al., 2014). Indeed, the ratio FΛ/FG is ~54 aound 400 [My], which may accelerates the accumulation of mass beyond the Eddington limit.
Today, those effects are potentially limited to the great structures, such as galaxy clusters or superclusters, as it increases with an increase in radius. The time period when cosmological gravity was greater than gravity alone can be determined with: Where  is the volumic mass of matter in the zone concerned. For the entire universe at critical density, we get: With the expression derived for the cosmological constant, we get the following expression, which yields the cosmic time at which cosmological gravity was greater than gravity force alone: With the values for  and  already obtained, cosmic time is found to be: This expression of ( ) was proposed repeatedly by many authors as part of a family of models called: Dark matter, dark energy dynamical scalar field (quintessence) (Amendola et al, 2018).
The general form of the equation proposed is: The value mentioned for compatible with the CMB is (0 < < 0,06). We find this value of for the MW. According to the author, while that force is negligible today on our scale, it was central to the formation of our universe and the great structures within it. 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, North (2011). A relatively complete model has been put forth by Martig et al. (2018), 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 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 massenergy equivalence: Note that cosmological attractive force associated with Λ is supplemental to conventional gravity (baryonic). Moreover, that force cannot be attributed to negative masses or so-called "dark" unobservable forces. In fact, the denominator 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, FΛ, 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 FΛ. 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 probabal 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 FΛ on galaxy rotation, let us find an expression of rotation speed relative to time: the time at which the galaxy started to form, ti, the total formation time of the galaxy, tT, with variable force, FΛ, acting during that formation time, tT -ti. For the masses of the bulbe and disc, we get a simplified expression: In the above equation for vt, 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 FΛ due to the cosmological constant through the residual mass-energy equivalence of the universe at the beginning of formation, ti, of the galaxy acting throughout formation time, tT-ti. This equation contains the essential elements for predicting the rotation curve of the luminous mass of galaxies.
Force F 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 formation of that galaxy.
This means that the type of rotation curve (concave ͡ or convex ͝ ) lets us know, in part, whether the galaxy was formed in the early days of the universe, or later (concave = older; convex = younger). So, a weaker cosmological gravity should lead to Keplerian rotation, or a convex curve.
Finally, we have assumed a very simple galaxy radius growth rate that is linear over time. Other, more realistic models can be introduced in the equation to better illustrate the generic growth of an isolated galaxy. Of course, the impacts of galactic collisions and agglomerations are not considered here.

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: The actual mass distribution and radial velocity of galaxies is 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. (luminous mass of 6x10 10 ʘ ), and several spiral arms. Also, the MW may have collided with Andromeda in the past, but this is an unverified assumption. In other words, the velocity profile of the MW may have been disrupted by past events that the velocity prediction model cannot consider due to a lack of relevant data. ). 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 [ ]) could be made possible by a direct collapse mechanism (Natarajan, Pacucci, Ferrara et al., 2017). Recently, a team discovered a candidate galaxy,

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 (Corbelli, 2003). If the dark matter is removed, the following luminous masses remain:  To avoid making too many speculative simulations regarding center and disc masses, we chose the following values as constants: ~ 9x10 9 ʘ ~5,0x10 8 ʘ Figure 27 shows five rotation curves for M33, derived from the three remaining parameters of the equation: ti, tT and tb, along with the rotation curve for the luminous mass only (Kepler baryonic, FΛ=0). Note that the rotation of this galaxy is not as well represented for the greater radii. The beginning of formation is assumed to be ti=0,16 to 0,2 [Gy]. Note also that a start time closer to the beginning increases the concave nature of the velocity profile, due to the stronger effect of the cosmological constant and the longer formation time, which flattens the velocity profile, as the cosmological constant decreases more sharply in remote areas with large radii. Further, the luminous mass is not sufficient to explain the rotation of the outer radius, as the speeds decrease sharply beyond 9 [kpc]. This confirms the presence of dark matter (baryonic non-luminous) in this galaxy because cosmological gravity alone is not enough to accurately simulate 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 (5,59x10 10 ʘ ). Note the strong correspondence between the estimated and measured speeds, clearly showing the existence of non-uminous matter in M33 and similar galaxies.  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: = 0,69~ 0,69 (4,6x10 11 ʘ )~ 3,18x10 11 ʘ = 0,31~ 0,31 (4,6x10 11 ʘ )~ 1,42x10 11 ʘ Figure  . 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 (Gentile, Józsa, Serra et al., 2013).  , 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.

UGC2885, Sc D
One of the largest of spiral galaxies observed to date has been the object of many studies. Figure   30 shows two velocity profile curves. The first for the estimated observable mass, 2x10 12 Mʘ and the three remaining parameters of the equation: ti, tT and tb, along with the cosmological gravity, FΛ, 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, 4x10 10 Mʘ, is in the same order of magnitude as the 10 10 Mʘ estimated by Gentile (2013

NGC253, Sculptor
The rotation curve of this southern sky galaxy was measured by Pence (1981), with over 3,700 measurements made (Fabry-Perot) along the great axis of the galaxy. Figure 31 shows an approximation of the low and high values (range) derived from the means on groups of ten values.
The profile becomes nearly flat beyond 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 1,08x10 11 and 1,54x10 11 Mʘ, if the estimated mass of the galaxy is   (20,5 arc-minutes). The total mass is an estimated 1,5x10 11 Mʘ, or twice that of the measurement zone (observed mass of 5-7 x 10 10 Mʘ for r ≤ 7,9 [kpc]). Center mass is an estimated 1,8x10 10 Mʘ. 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

Irregular dwarf galaxy DDO161
Dwarf galaxy DDO161 was chosen to show the later effects of cosmological gravity. Côté et al. (2000) 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 mdark/mlumi ~8 to 9 to explain the observed rotation speeds. The observed luminous mass (stars and gases) is: during formation, which lasted between 1 and 5 [Gy], 3-15 times longer than the MW. Figure 33 shows the rotation speed relative to formation time, with a start time around 0,177 [ ], around the same time as the MW. 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 nonluminous 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 Zwicki (1937) and thereafter those of Mayall (1960), Van Albada (1961), Omer et al. (1965), Pebbles (1970), Rood et al. (1972), Kent et al. (1982), Merrit (1987), White et al. (1993) and recently Gavazzi (2009). 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 ( ʘ / ʘ ).
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 estimate the luminous mass M, the time of the start of formation of the cluster ti, the formation time of the cluster tT and the size of the cluster r. As an example, White et al. (1993), estimates the mass of stars and hot gas in the cluster from the observations of Godwin et al. (1977) To: = 1 ± 0,2 10 13 ℎ 70  We see that the observed and estimated characteristics are relatively variable between the authors.
However, one observation is constant, i.e. the estimated observed (luminous) mass is less than 2 to 10 times compared to that estimated necessary to explain the observed rotational velocities of 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: ]+ 2 2 2 6( + ) 4 We know that for a spherical geometry, mean quadratic radial velocity (line of sight) can be expressed from the quadratic velocity v, i.e.: Three unknowns are to be determined, the time of the start of the formation of the cluster ti, the duration of the formation of the cluster tT and the variable speed of progression of the formation of the cluster with exponant b. Several combinations are possible but we possess the measured values of the radial velocity of the galaxies according to the radius < ̅ > of the cluster as well as the standard deviations of the velocities R. The following 3 figures 34, 35, 36 present the profile of the standard deviation R for different plausible combinations of ti, tT and (b) and the measures taken by Kent (1982), Rood (1972) and Chincarini (1975). Several observations can be made: -The standard deviations R of < ̅ > are variable for small radii, calibration is more difficult for this area.    Kent et al, 1982measured, Rood et al, 1972measured, Chincarini et al, 1976 Figure 36: Coma cluster radial velocity dispersion profil 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 [ ]. In addition, the growth rate of the cluster appears to be higher at the beginning (b<1). If we choose the following preferred values (ti= 0,7[ ], tT= 2,2 [ ] and b=0,5), we obtain the following velocity curves for the Coma cluster ( Figure 37). Coma cluster radius ([kpc]) case 9, ti=0,72 Ga, tt=2,0 Ga, b=0,3 case 10, ti=0,72 Ga, tt=2,0 Ga, b=0,4 case 1, ti=0,72 Ga, tt=2,0 Ga, b=0,5 case 11, ti=0,72 Ga, tt=2,0 Ga, b=0,6 case 12, ti=0,72 Ga, tt=2,0 Ga, b=0,7 measured, Kent et al, 1982measured, Rood et al, 1972measured, Chicarini et al, 1976 Figure 37: Coma cluster radial velocity dispersion profil We can draw the following conclusions about the Coma cluster and the cosmological force FΛ.
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.  (White, 1993) of the Coma cluster is ~ 500 times greater which allows to maintain the cosmological gravitational force in the case of a cluster of galaxies. In Figure 38 are realistic because the maximum observable speeds of < ̅ > are precisely those of this tangential velocity when this one is practically aligned with the line of sight.

3-
However, without the cosmological gravitational force (Kepler only), it is not possible to obtain such large values of < ̅ > unless you increase the mass of the cluster by a 300x factor. [Gy] when joined to the cluster which is a lower value than the formation time of dragonfly 44 estimated at ~5 [Gy]. 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.

5-
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, Tao

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]. Recently, Wang et al (2019) have observed the existence of 39 massive and mature galaxies only 2 [ ] after the beginnings of the universe. However, the mass accumulation model (radius growth) is very basic, and a fullcapacity accumulation model based on existing forces would be more realistic and would surely yield more accurate galactic growth rates. The model uses cosmological gravity, a major force during the initial billion years when the first galaxies were formed. Also, variation of the constant G (as G(r,t)), used to adjust gravity forces for large radii, is not used in this mass rotation speed simulation model (Brownstein and Moffat, 2006).
Observation of early galaxies is difficult due to their low brightness. Bouwens et al. (2006) reported that very few galaxies were formed before 700 [My]. However, some very old stars have been detected in the MW, which tends to confirm primitive formation of the galaxy before 700 wide-open question and delaying the start times of galaxy formation can be done by changing the accumulation rates or increasing the age of the universe, because cosmological gravity depends on the cosmological constant, which is cosmic time dependent, tΩ. With cosmological gravity in play, there is less of a need to turn to lacking or unobserved mass (like dark matter -existing but nonluminous baryonic matter) to explain the rotation patterns of many galaxies. In brief, from the standpoint of observable and unobservable masses of these eight galaxies, we had recourse to dark matter (non-luminous) for two of the galaxies (M33 and NGC3198). As for the other galaxies, only the observable luminous mass and cosmological gravity were used.
What is interesting with this model is the fact that it was derived from the analysis of a model of the universe that estimates the evolution of energy to calculate new dynamic parameters, such as the cosmological constant and cosmological gravity. Moreover, the mass rotation model for galaxies provides estimated formation times from early formation, ti, as well as formation time, tt, by adjusting these two parameters with the observed rotation curves and observed masses. In our opinion, no other model uses these dynamic parameters for the formation of galaxies with the cosmological constant. Finally, the mass accumulation model described herein is a very simple one; nevertheless, the results of rotation speed simulations are promising. Of course, a much greater number of galaxies should be studied with this rotation model to further improve and develop its potential. However, we can generate a graph (Fig. 38) showing the relation between total observed mass and formation times of these galaxies, with the exception of UDG44, which does not fit the model due to a significant amount of time of formation. The graph shows that formation time increases with total mass, which is quite plausible.   , , , which leaves a fairly large margin of error. 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 Zhou et al. (2017). Indeed, from the study of observed acceleration variations, gobs, they determined two precise but diametrically opposed galactic directions (l,b) and (l+180º,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 a0 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 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 a0 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 Mʘ and r=40 [kpc]). For smaller structures, r→0, 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 0 through time can be obtained, knowing that the cosmological constant will vary. Randriamampandry et al. (Randriamampandry and Carignan, 2014) use the MOND theory for the study of the rotation of 15 galaxies and they mention the need to vary the constant a0 in order to adjust the rotation curves (a0~ 0.34 to 2 x10 -10 ). The figure 39 shows the values of 0 for three typical masses, M (10 9 , 10 10 and 10 11 Mʘ), and radii, r (20 and 40 [kpc]). MOND, a 0 typical value, 1,2x10 -10 a 0 ,Lake, 1989, 2,5x10 -11 a 0 ,Schubert, 2006, 1,2x10 -8 a 0 , Randriamampandry, 2014 concordance of photon volume with universe volume, the causality recovery period after z=10 26 and last scattering surface z=1098. The model questions certain elements of the cosmological principle, that is the idea that there is no preferred position. The model assumes that the MW occupies a precise location (cosmic time 13,8 [Gy]), and not a central one in this universe of possible ~76 [Gy] cosmic age. Moreover, we do not have sufficient data from cosmological observations to claim with assurance that the universe is the same in all directions and, more specifically, to the high values of z, excluding the CMB, which appears in the early universe before the formation of structure that we observe, which in turn is subject to a different chronology. Indeed, the observed percentage of this universe is extremely low, especially as concerns galaxies. If the number of galaxies is an estimated ~2x10 12 , less than ~10 -6 percent have been indexed (90,000 galaxies) (Vipers, 2016). The model can partially describe the rotation of certain galaxies without recourse to dark matter (halo), but rather uses the cosmological gravity effect, which has a heavy impact during the early formation period. The galaxies studied herein appear very early in the after the beginning (Hoag, Bradač, Brammer et al., 2018); (Hashimoto, Laporte et al., 2018).
Another galaxy detected at z = 6,027 already has a population of stars aged 800 [ ] (z ~ 18) or ~ 200 [ ] after the beginning, Richard et al. (2011). 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). The baryonic mass of the universe could be as much as ~500 times smaller than formerly estimated and accepted. However, it should be noted that no one has been able to estimate that mass without the use of predictive models, meaning that such mass could be in fact lower than normally accepted values. Finally, in the context of this model, which uses the cosmological constant, the value of constant a0 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 a0 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.

Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this paper.

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

Acknowledgement
The author would like to thank the members of his family, especially his spouse (Danielle) who with patience to bear this work as well his childrens (Pierre-Luc, Vincent, Claudia), for their encouragement to persevere despite the more difficult periods. Also, a big thank you to Mrs. Nadia Villeneuve of UQAC who has prepared the article and references in an acceptable version. Finally, thanks to the University of Quebec at Chicoutimi and to the colleagues of the Department of Applied Sciences for their supports in the realization of this work.