An Alternative to the Dark Matter? Part 2: A Close Universe (10−9 s to 3 Gy), Galaxies and Structures Formation

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 the energy of the universe U, cosmological constant Λ, the curvature of space k, energy density e Λ ρ (part 1). The age of the universe in cosmic time that is in line with positive energy conservation (in terms of conventional thermodynamics) and the creation of proton, neutron, electron, and neutrino masses, is ~76 [Gy] (observed 1 1 0 ~ 70 km s Mpc H − −   ⋅ ⋅   ). In this model, what is usually referred to as dark energy actually corresponds to the energy of the universe that has not been converted to mass, and which acts on the mass created by the energy-mass equivalence principle and the cosmological gravity field, FΛ , associated with the cosmological constant, which is high during the primordial formation of the galaxies (<1 [Gy]). A look at the Casimir effect makes it possible to estimate a minimum Casimir pressure 0 c P and thus determine our possible relative position in the universe at cosmic time 0.1813 ( [ ] [ ] 0 13.8 Gy 76.1 Gy t tΩ = ). Therefore, from the observed age of 13.8 [Gy], we can derive a possible cosmic age of ~76.1 [Gy]. 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.

). In this model, what is usually referred to as dark energy actually corresponds to the energy of the universe that has not been converted to mass, and which acts on the mass created by the energy-mass equivalence principle and the cosmological gravity field, F Λ , associated with the cosmological constant, which is high during the primordial formation of the galaxies (<1 [Gy]). A look at the Casimir effect makes it possible to estimate a minimum Casimir ). Therefore, from the ob-

Expanding 3d-Sphere of Matter
An order of magnitude for the average 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 the 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 t Ω . 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 r 1 at state 1. The material point evolves towards a material position 2 in state 2, moving at a mean speed β (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 t 0 in the sphere of matter at the time of state 2 (universe age t Ω ): 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 , , , r t t Ω β , the value of t 0 must be lower than or equal to t Ω , which limits possibilities, or still, forces a restriction on variable β . In this paper, we only consider the mean value of β for a sphere of matter undergoing Hub- 1 1 0.15 -0. 19 Gy Gy r R = . Moreover, by selecting β according to an equation developed in the next section ( 3 2 10 − × β ), and t 0 , the age of the universe calculated by Planck (13.8 [Gy]) at our observation position, we get an approximate range of ages for the universe today: [ ] 73 to 92 Gy t Ω 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 [1]. If the Casimir force is expressed as shown in works where parallel plates are used, we get the following equation: where l 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 acts 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: 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, t 1 , in the universe. To do so, we know the expression for photon gas pressure at the same time, t 1 , 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 t 1 , is not modified by the scale factor: Then with the temperature equation: Or with the expression w σ using the Lambert function: The ratio of ν-origin photon frequency to temperature T is strictly constant (1.034 × 10 11 [s −1 ·K −1 ]) from the initial Planck time t p up to 76.1 [Gy]. Finally, we (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 t 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, t 1 ). In other words, using the MW as an example, its speed would appear to increase with the increase in the age of the  Figures 1-3 show the form of that evolving speed, or v c = β , acceleration, a, 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 [km·s −1 ]. That value for the current speed of the MW corresponds relatively well with the estimates was made by [2] Kraan-Korteweg et al.
As for acceleration, we find a very reliable number, which is nevertheless not zero: ( )        showing that the MW 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 q): 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, t 1 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, ( ) m q 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:  Measurements by [3] Riess et al. and [4] Kiselev 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, z t is closer to 0.7 according to [5] Giostri et al., who used a calibrated parametrical model with a prescribed constant of ( ) 1 2 q z = for 0 t → . 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. Let us return to Casimir pressure which, relative to z, is: 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. [7] 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 c P . They reported the following measurements (Table 1):    Figure 5 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 (~1.291 ×  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? The 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 [9]: 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 ~3000 [K], recombination occurred much later than the previously assumed, or ~69.  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 one-time 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 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 the 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 ∆θ. For an apparent angular dimension of this last scattering surface, which covers the entire celestial half-sphere ( ∆ = π θ ), 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.
A possible zone of empty matter due to the time lag during photon and electron and the electrostatic force acting before recombination, around 69.  Indeed, protons and neutrons appear approximately 666 days before electrons.
At that time, the electrostatic repulsive force of protons is dominant and much greater than gravity (10 42 times greater). This repulsive action of protons, which pushes them towards the physical boundary of the universe, can be estimated.
Indeed, assuming that the minimum energy principle applies at this time period of the universe, which is much greater than Planck time (t p = 10 31 ), the electrostatic energy difference between an evenly distributed proton configuration in the volume at the time of electrons vs. evenly distributed protons around the perimeter, is: where: e cte = ρ ρ : the volumic density of proton charge in the R-radius sphere σ : the surface density of proton charge at r radius (at electron time) * V : the electric potential Q: the total charge of protons, Note that the minimum energy is for the proton configuration around the perimeter of the volume at electron time. The mean speed of proton motion towards the perimeter, discounting the effects of gravity force, which is much smaller than the Coulomb force, can be estimated using the proton motion equation with energy conservation and work done: With the last two expressions and derivation, we get: Finally, for ( ) p r t , which represents the average position of proton motion towards the perimeter during electron production, we get the following differential equation:

Cosmological Constant Λ Estimated Values
The Friedmann equation (FLRW metric) for an isotropic universe made up of matter in the presence of energy associated with the cosmological constant can Journal of High Energy Physics, Gravitation and Cosmology be written in relation with the terms that contribute to the expansion or contraction of the universe, H, with gravity, G, the existence of energy other than baryonic through Λ and the space curvature, k, or: where the scale factor is a [-], k is the space curvature, [m −2 ] and ρ, the density of conventional mass [kg·m −3 ]. In this form, the equation represents the expansion of the universe expressed with the Hubble constant. In this model, we consider and assess the evolution of conventional energy (photon gas and mass-energy equivalence). An expression for the cosmological contant, Λ, can be obtained using the Friedmann equation. Indeed, assuming the existence of mass-energy equivalence (non-baryonic), represented by constant Λ, along with zero acceleration (H = 0) of that mass-energy equivalence, that equation, which represents the non-baryonic residual volumic mass-energy equivalence of the universe, is written as: The effects of each term of the equation are clearly seen. The first term is the closing effect caused by gravity, G, via mass density, ρ; the second is the closing effect caused by the residual mass-energy equivalence (non-baryonic) via cosmological constant Λ; and the last term is the opening effect, or expansion, caused by an unknown element, but represented by the Hubble constant. Figure   6 shows that the space curvature, k (equation k (H)), in relation to the other variables: ρ, Λ and H = 1/t. The value of k today, time t 0 , is very close to zero, but slightly negative (open).       [12]. 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.
In 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, M T , is far less important. Figure 7 shows the Λ mass /Λ rad ratio.
Therefore, the Λ mass /Λ radiation ratio at our time, t 0 , is equal to ~ 0.163. It is interesting to note that the ratio obtained is in the same order of magnitude as this mentioned for baryonic matter to that of dark matter     After manipulation, another expression for Λ rad is found: The above equation contains a scale factor that varies inversely with the radius of the universe, 2 u r , modulated by a power ratio, or the quotient of output power of the universe, P u , taken as a blackbody at T, time t, and Planck power P p . This clearly shows that the cosmological constant diminishes relative to the squared radius and dissipated energy of the universe, leading to the great variation of the two factors combined, scale and energy. These two variations of magnitude (squared scale factor and dissipated energy) lead to the great variation of the constant. Indeed, the only variation of the energy factor (P u /P p ) leads to a variation of ~10 4 , and that of the squared radius, to a variation of ~10 126 . In brief, it is principally the expansion of the universe that leads to the reduction of the con- Also, this expression is for the beginning when where pr t t ≥ Journal of High Energy Physics, Gravitation and Cosmology    The expression for energy density at t p can be written as: ( ) 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 0 k Λ = = , or a flat universe whose only energy comes from mass, we get:  Figure 9 and Figure 10 show the values for i ρ and i Ω calculated according to the age of the universe. Figure 13 shows the equivalent densities. Here, the contribution of curvature is negative for age below 2.9 [Gy], a closed universe, as already discussed with Journal of High Energy Physics, Gravitation and Cosmology 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 14(a) 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 (massenergy, 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    [Gy]). We see the impact of the mass on the transition.
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 P P multiplied by cosmic time. That expansion energy E curv is not directly expressed in a model variable. Moreover, it is positive via Planck power, which represents conventional energy acting in opposition to gravity F G (or E mass ) and cosmological gravity force F Λ (E radiation or E Λ ). The expansion power is not associated to mass (baryonic) or radiation (photonic via Λ). 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 stretching of space. This Planck power P P can be expressed by the Planck force F P 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. 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 energy associated with space itself. A summary calculation, based on the work PdV done by this Planck force to create space, shows that for every m 3 of space in our position (MW) the energy used to create space is worth ~1.8 × 10 −9 [J·m −3 ]. However, at the beginning of the Planck era, this space creation energy was worth ~1.
We find the same result for an empty universe (without total mass and radiation energy E mass and E radiation ). Figure 14

Age of the Universe from the Friedmann Equation
The values obtained from the model for our position (a 0 = 1) are (see Figure   14 in the usual case. We find the following expression for the integral:

Some Comparison with Some Data from the ΛCDM Model
The Table 2 below shows some of the major differences between this model and the ΛCDM model [13]. The numbers are averages over a time period ranging from z = 0 to ~z re (~7.70), or ~1.  The cosmic neutrino is estimated with the muonic neutrino with β SN1987A

Cosmological Gravity Force, FΛ
For the time period when radiation was dominant, a central force associated with Λ rad can be determined using mass-energy equivalence. Indeed, we know the value for Λ rad 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 [15] Martin. 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 massenergy 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 [  Note that the value for g Λ 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 3 H 2 /8πG: 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. Figure 15 and Figure 16 show the mean ratio F Λ /F G for the time period starting at proton time t pr . Note that the cosmological Journal of High Energy Physics, Gravitation and Cosmology 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 30 z < < ) (Lupi, Colpi, Devecchi et al., 2014). Indeed, the ratio F Λ /F G is ~54 aound 400 [My], which may accelerate 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:    the development of more sensitive sensors will determine whether or not the GR theory will be a definitive, or not, theory of gravity as it has been formulated in 1916 [20].
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.

Conclusion
The cosmological observations to claim with the 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 the 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 ~2 × 10 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 (part 3). 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.