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Emerging gravitation in an accelerated expansion of the – Probable natures of inertia, and

Nima ZIAEI† † Physicist engineer at the ALTEN innovation center, Chaville, FRANCE

Abstract— Since the 2000s, many astrophysical observations measuring type Ia supernovae and their apparent have led to establishment of a standard model of , based magnitude [5]. These two discoveries have led to consider that on the existence of dark matter and dark energy to explain Universe is probably composed of dark matter and dark energy formation and the future of the Universe. Others theories like permitting respectively to explain formation of large-scale MOND (Modified Newtonian Dynamics) theory [1] or entropic structure of the Universe and the current positive measurement gravity theory [2] give different explanations on universal of acceleration of the expansion of the Universe despite the fact gravitation theory either in order to explain galaxy rotation curve 80 regardless existence of dark matter or to explain the origin of that Universe is composed of around 10 massive particles [6], gravitational field and of -time by the mass. In the which should normally decelerate its expansion. However, theory of relativity, the curvature of the space–time is imputed to nowadays, natures of dark matter and dark energy remain the presence of mass or energy but no explanation is given to link unknown and no dark matter particles as unknown particles presence of mass and curvature of the space-time. In other words, have been yet detected even with advanced sensor technology how, fundamentally, mass distorts space-time? How to explain, [7]. Initial performance of the modern COSINE-100 fundamentally, the equality between gravitational and inertial experiment reproducing the DAMA/LIBRA experiment masses? This paper proposes to establish some theories to explain questions its conclusions about detection of annual modulation origin of inertia and by consequent, explain how mass distorts space-time and creates gravitational field. For that, this article signal due to presence of dark matter particles [8]. Moreover, proposes first to establish a relation linking the gravitational we can ask ourselves why Milky Way’s dark matter halo which constant 푮 to the acceleration of the expansion of the Universe is supposed to be around 6.7 to 33 times more massive than called 횿. Moreover, this article proposes to model the evolution of radiant matter of the Milky Way [9][10] do not collapse to form the global Universe’s “scale factor” without taking account of dark matter black holes (even in form of a cloud) instead of . That permits to explain nature of dark energy having a spherical distribution? Even if that is not possible to and unifies Hubble constant 푯 to gravitational constant 푮 as well form a compact object of dark matter, it could however be as retrieving the literal value of the . possible theoretically to collect enough dark matter particles in Furthermore, explanation of origin of inertia needs to introduce a a volume included into a Schwarzschild radius to curb enough new form of gravitation field similar to magnetic field in the space-time to form massive black holes in the Universe. Maxwell electromagnetic theory and inspired by gravitoelectromagnetism theory. The new gravitational field, This affirmation is all the more relevant that dark matter does whose origin is linked to the movement of mass, permits to retrieve not interact electromagnetically and its compaction should be some general relativity’s results including polarization of much easier than ordinary matter () for which, gravitational waves predicted by Einstein as well as positions of electromagnetic interactions and electron degeneracy pressure or innermost stable circular orbit in the case of non- impeach gravitational collapse. rotary and electrically neutral central mass like a Schwarzschild Indeed, for a given density of matter 휌, Schwarzschild black black hole. It even permits to retrieve the general relativity‘s hole radius is given by: calculation of apsidal precession of an astronomical body’s orbit in case of weak field approximation. Finally, this article proposes 3푐2 푅∗ = √ (0) a model able to explain galaxy rotation curve as well as the 푆퐵퐻 8휋퐺휌 evolution of their characteristic size related to the evolution of scale factor of the Universe and regardless existence of dark With 퐺, the gravitational constant and 푐, the celerity of light in matter as unknown matter or regardless MOND theory. the vacuum. Even if we do not exactly know global density of dark matter in the Universe, an estimated density of dark matter −29 −3 Index Terms— Gravitational constant, Acceleration of the could be 휌퐷푀~3 × 10 g. 푐푚 [11], which is potentially expansion of the Universe, Energy density of the quantum higher than critical density. That means dark matter cloud could vacuum, Cosmological constant, Dark mater, Dark energy, form a black hole in our of nearly radius Hubble constant, Extraordinary gravitation field, Galaxy rotation of 8 billion light year, which is less than the radius of the curve observable Universe. Moreover, existence of this density of dark matter involves a curved shape of the Universe. Therefore, I. GENERAL INTRODUCTION imposing existence of dark matter, as a cloud unable to collapse Two major discoveries have permitted to advance the like ordinary matter, would lead to contradiction considering cosmology science during the last 70 years without counting the results from WMAP (Wilkinson Microwave Anisotropy Probe) advent of discoveries of and the cosmic microwave revealing that Universe is flat with 0.4% margin of error [11]. background. The first one is the discovery of the non-ordinary Our article does not try to deny existence of dark matter but mass distribution in galaxy M31 by Van de Hulst [3] and in questions about its nature. It proposes a debate about potential galaxy M33 by Louise Volders [4]. The second one is the nature of what dark matter could really be. This article proposes discovery of the acceleration of the expansion of the Universe also to establish the probable nature of dark energy and by two independent projects in 1998 (the Supernova highlights potential existence of a new gravitational field with Cosmology Project and the High-Z Supernova Search Team) in 2

푠 different physical properties, compare to the classical known Considering now the work, done by metric constraint 퐹 , 퐴 푚 gravitational field’s ones. applied on the cubic parallelepiped (푐푝). As metric constraint is distributed on each of the six faces of the cubic parallelepiped, II. THE MASS AS INERTIA OF THE EXPANSION OF SPACE- 푠 the total work, done by metric constraint 퐹 , becomes: TIME 퐴 푚 6푠 푑푊 = 푑푊 (5 푏푖푠) A. Physical concept using weak field approximation 푐푝 퐴 Let be an electrically neutral and non-rotary spherical mass 푚 If we consider that, energy density of quantum vacuum is scale with a radius 푅0. An infinitesimal volume in spherical invariant [12] thereby; we can suppose a proportional coordinate, if we supposed to be in a Euclidian space, has for mathematical relation between 푑푊푐푝 and Δ(푑푉) as: 2 푠 expression 푑푉 = 4휋푟 푑푟. In a Schwarzschild metric, Because 푑푊 = 휎 × Δ(푑푉) (8) of the central mass 푚 this infinitesimal volume is expanded of 푐푝 퐴 푟=푅0 −3 value Δ(푑푉) as: With 휎 a homogenous parameter as energy density in J. 푚 Δ(푑푉) = Δ(4휋푟2) × 푑푟 + 4휋푟2 × Δ(푑푟) (1) that supposed to be the average energy density of quantum 2퐺푚 vacuum. Equation (8) stipulates that expansion of the slice of Considering 푚 as a weak mass ( ≪ 푅 ≤ 푟), expression of 푐2 0 space with a thickness 푑푟 due to presence of mass must Δ(푑푉) becomes: accumulate energy because of the non-variant scale factor of 8휋퐺푚푟 푟 vacuum energy density. Δ(푑푉) ≅ 2 (2 × ln ( ) + 1) × 푑푟 (2) 푐 푅0 Thus (2), (5 푏푖푠) and (8) involve: Generating expansion of the volume 푑푉 has consequence to 8휋휎퐺푚 퐹 (푅 ) = (9) accumulate energy into expanded volume Δ(푑푉) due to the 푚 0 3푐2 work of a supposed constraint applied on slice thickness 푑푟 of We can note that 퐹푚(푅0) is independent of 푅0. the space. In Newton gravitation theory of weak field, gravitational force is in 푟−2, so by a reaction mechanism, we B. The theory can consider that for 푟 = 푅0 constraint applied on the volume 푑푉 surrounding mass 푚 has for expression: Our theory is based on the idea that the matter as inertia, 휂 interacts with space-time in its accelerated expansion and ( ) 퐹(푟 = 푅0) = 2 3 caused by the global acceleration of the expansion of the 푅0 With 휂, a homogenous parameter that we are not trying to Universe. Without presence of mass, space-time normally accelerate its expansion but, in presence of any mass, express. Metric constraint for 푟 = 푅0 applied to the entanglement between matter and space-time impeaches constraint 퐹, noted 퐹푚, is given by the following expression: acceleration of expansion of space-time inside the matter. The 푑퐹 2퐹(푟 = 푅0) 퐹 (푅 ) = − | = (4) space-time remains “trapped” into matter and is curved outside 푚 0 푑푟 푅 푟=푅0 0 of it because of its accelerated expansion. Mathematically, our Let be a cubic parallelepiped at 푟 = 푅0 with an area per face 3 theory stipulates that it exists a direct relation between 퐹푚(푅0) 2 푠 ≪ 퐴 = 4휋푅0 with a total volume of 푠2. Considering the and Ψ representing global acceleration of the expansion of the [20], total physical information of any Universe (in 푠−2) as: ratio of the expanded volume Δ(푑푉) is encoded on the surface 퐹푚(푅0) = 푚 × Ψ (10) surrounding it. Thus, because of the presence of the mass 푚, the Equations (9) and (10) permit to write: expansion of this cubic parallelepiped volume would be a ratio 3푐2 퐺 = Ψ (11) of Δ(푑푉) linked to value of its area 푠 such as at 푟 = 푅0 the 8휋휎 푠 In addition, we can note that acceleration of the Universe is expansion of its volume is worth Δ(푑푉)푟=푅0. Applied to the 퐴 given by: cubic parallelepiped at 푟 = 푅0, the metric constraint then 푠 퐻2휎 becomes 퐹푚(푅0). We can define an equivalent work of the 퐴 Ψ = 2 (11 푏푖푠) 휌푐푐 constraint 퐹(푟 = 푅0) on slice thickness 푑푟 of the space-time as: 1 With 퐻, the Hubble constant and 휌푐, the critical density of the 푑푊 = 퐹 (푅 )푅 푑푟 (5) matter. According to our theory, evolution of the value of Ψ 2 푚 0 0 We can note that in a non-relativist gravitation, this energy 푑푊 from epoch to present involves that the gravitational 2 constant 퐺 has undergone evolution during time [13]. Linking (proportional to 1/푅0) is equivalent in mathematical expression to gravitational energy contained in a volume 푑푉 with energy 퐺 to Ψ can explain, notably during inflation epoch to current density 푢 as: Universe that gravitational force has changed intensity from 푑푟 much higher values in the past to its current value. Our theory 푑푊 ≡ 푢 × 푑푉 ∝ (6) could thereby explain how large-scale structure in our current 푟2 With: Universe were formed regardless dark matter existence. Our 푔2 theory can also explain anisotropic measurement of temperature 푢 = (7) from cosmic microwave background due to quantum vacuum 8휋퐺 fluctuation of 휎 value and, thanks to (11), its implication to the With 푔, the local gravity. Of course, we can note that 푑푊 and possible spatial anisotropic value of 퐺 in the primordial 푢푑푉 are not of the same physical nature. Universe. With higher value of 퐺 in the past Universe, past 3

stellar evolution could be very different with a stellar lifetime We have also these following known equations implying the much shorter and with a number and emissivity of massive stars Hubble constant 퐻: higher than current astrophysical observations. It involves that 푅̇ 퐻 = (12) probable candidate for dark matter could be massive compact 푅 halo objects like black holes and primordial black holes. Our Ψ = 퐻2 + 퐻̇ (13) theory explains also how first galaxies could be formed so early From (12) and (13), we have: in the Universe. We can also remark that our theory involves 푅̈ that, in case of deceleration of the expansion of the Universe, Ψ = (14) 푅 gravitational force could be repulsive. If our theory permits to If we call by ℰ(푡), the total quantity of energy from thermic and know origin of gravitation in the Universe, it does not explain non-thermic in the Universe and 푃(푡), the total origin of Ψ itself whether it is positive or negative. quantity of energy per unit of time by radiation (Radiative power of emission) from stellar or no stellar objects, or more III. GLOBAL EXPANSION OF THE UNIVERSE commonly called luminosity, in the Universe, so we can write: 푑ℰ(푡) A. Physical concept and hypothesis = 푃(푡) − 퐻ℰ(푡) (15) In order to study and quantify theoretically expansion of the 푑푡 The gravitational potential energy of the Universe is worth: Universe, we are going to adopt five major hypothesis almost 4 all accepted by astrophysics science. The first one is the 퐸 = 휋2퐺휌2푅5 (16) 푝 15 cosmology principle, which considers that at any given epoch, 4 The gravitational field inside a global volume 푉 = 휋푅3 Universe is homogeneous and isotropic (at large scale), 3 4 electrically neutral, composed of energy and mass of formed or generated by mass 휋푅3휌 create a gravitational energy density not still formed compact objects (galaxies, clusters, quasars…). 3 given by (7). Integrating this all over the Universe give global In this case, compact objects are supposed to have no proper energy as: velocity due to movement of mass compared to an observer 8 (and so they have a null kinetic energy) except recessional 퐸 = 휋2퐺휌2푅5 (17) 푠 45 velocity due to expansion of the Universe. The second The displacement of mass because of expansion of the Universe hypothesis is that an observer on Earth could thought that he is create a current density of matter 푗⃗ at radius 푟 from the observer at the center of his own spherical Universe composed of as: massive matter with global density called 휌 with a scale factor 푗⃗(푟) = 휌퐻푟⃗ (18) noted 푅 representing the theoretical global radius of the whole At radius 푟 from the observer, the gravity field applied on Universe (and not only the observable Universe). The third matter is worth: hypothesis is that we consider Universe as a closed system that 4 is to say no transfer of mass or energy is possible inward or 푔⃗(푟) = − 휋퐺휌푟⃗ (19) 3 outward our Universe. The fourth hypothesis is that Universe is This global displacement of mass generate a work done by flat [11] and if we consider the perfect , gravitational interaction between matters and given by its the flatness of the global Universe has occurred in all the ages power as: of the Universe. It means that we can use Euclidian 2 to describe classical geometry evolution of the Universe 푃푊(푡) = ∫ 4휋푟 푗⃗(푟) × 푔⃗(푟) 푑푟 (20) through past and future time. Finally, we will consider that any Hence, according to (19) and (20): evolution of state of one part of the Universe included in a 16 sphere of volume 푉, with the observer at its center, can 푃 (푡) = − 휋2퐺휌2푅̇ 푅4 (21) 푊 15 influence simultaneously states of all the other parts of this Finally, it is necessary to introduce existence of energy from volume. This assertion permits to respect the first law of quantum vacuum and its global energy in the universe is worth: thermodynamics. This assertion puts forward the fact that 4 information could spread across Universe to compensate what 퐸 = 휋휎푅3 (22) 푣 3 we call horizon problem [14] at any epoch of the Universe and Expansion of the Universe must obey to conservation of the not only before . The holographic principle energy stated by the first law of thermodynamics. Breaking this could explain this [20], considering that a surface 푆 surrounding law should obey to another unknown law. We suppose that, any compact volume of space 푉 could contain all the necessary contrary to global thought, energy is a constant of time physical information of state of matter inside of the volume 푉. This quantity of Universe from Big Bang to present and it cannot be assertion permits to characterize state of a given matter (density otherwise. of matter, potential gravitational energy, gravitational energy According to the first law of thermodynamics applied to the 4 density…) inside a volume 푉 = 휋푟3 by decoding the Universe, we have for a 푑푡 flow of time between two instants 푡 3 necessary information on its surface 푆 = 4휋푟2, with 푟, the and 푡 + 푑푡: 2 distance from the observer without taking into account 푑퐸푝 + 푑퐸푠 + 푑퐸푣 + 푑ℰ(푡) + 푑푀푐 = 푃푊(푡)푑푡 (23) propagation time of information. Quantum entanglement With 푀, the mass of the Universe. phenomenon has showed that states of particles can be correlated and state’s affectation of one affects simultaneously the state of the other one independently of their distance. 4

B. Global equations and their involvements and, composed of around 2000 billion of galaxies [ퟏퟔ] According to (11), the paradigm that gravitation is linked to the themselves composed of around 100 billion stars emitting a 27 accelerated expansion of the Universe involves that 퐺 is variant power radiation of the order of magnitude of 10 푊, so we can 50 as function of time. Moreover, we can express temporal consider that 푃~10 푊. In this case, (30 푏푖푠) is true. evolution of the mass of the Universe as: 4 According to (11) and (32), we can also write 퐻 as: 푑푀 = 휋푅3푑휌 + 4휌휋푅2푑푅 (24) 1 푑퐺 3 퐻 ≈ − × (33) Thus, (12) and (15) to (24) permit to write global equation of 1.4퐺 푑푡 evolution of the scale factor of the Universe as: According to current value of 퐻 ≈ 2.2 × 10−18푠−1 −1 −1 37휋푐2휌2푅4 푑푅 (67,8Km. 푠 Mpc ), and regarding (33), we have current ((휎 + 휌푐2) × 4휋푅2 + Ψ) ̇ 30휎 푑푡 value of 퐺 as: 푑퐺 4 휋푐2휌푅5 푑휌 휋휌2푅5푐2 dΨ ≈ −2.06 × 10−28푚3. 푠−3. Kg−1 (34) + ( 휋푅3푐2 + Ψ) + × 푑푡 3 3휎 푑푡 6휎 dt From (32), we have a direct relation between Ψ and 푅 as: + 푃(푡) − 퐻 × ℰ(푡) = 0 (25) 퐴 If we consider in our Universe that main mass loss is due to Ψ = (35) 푅1.4 stellar or no stellar activities thanks to their radiative emission, With 퐴 a constant as: thus, we can write: 8휋휎퐺푅1.4 푑푀 퐴 = 0 ≈ 8.7 × 10−13푚1.4. 푠−2 (36) 푃(푡) + 푐2 ≈ 0 (26) 3푐2 푑푡 Numerical value of 퐴 in (36) is given for a value of 휎 ≈ From(24) and (26), we can conclude that: −29 −3 푑휌 −3푃(푡) 10 푔. 푐푚 and in considering that value of 푅0 ≈ 13.8 billion light-years, used to define the current observed ≈ 3 2 − 3휌퐻 (27) 푑푡 4휋푅 푐 radius of the observable Universe [15]. Thus, (27) represents Friedmann equation of evolution in time 푃(푡) Thus, according to (14) and (35), we can write: of density 휌, where quantity 3, called pressure 푝 in 4휋퐻푅 10 Friedmann-Lemaitre equations, represents the ratio between ̇ 0.6 0.6 ̇ 2 푅 = √ 퐴(푅 − 푅0 ) + 푅0 (37) total stellar power emission in the Universe (Luminosity of the 3 2 ̇ 0.6 2 Universe) and 4휋푅 푅 representing the volumetric flow rate of If we consider that 푅 ≫ 푅0 and 퐴푅 ≫ 푅̇0 so, we can write 푅 Universe expansion. So, we can write: as function of time 푡: 푃(푡) 10 7 푝 = 3 (27 푏푖푠) 4휋퐻푅 10 0.7 ( ) 푅(푡) ≈ (0.7√ 퐴 × (푡 − 푡0) + 푅0 ) (38) According to 26 and (27), we can rewrite (25) as: 3

7휋푐2휌2푅4 푑푅 휌푅2푃(푡) 휋휌2푅5푐2 dΨ Thus, future of the Universe is the scenario. (4휋휎푅2 + Ψ) − Ψ + × 30휎 푑푡 4휎 6휎 dt Hence, according to (12), 퐻 can be written as a function of − 퐻 × ℰ(푡) = 0 (28) time: 1 From (28), the Hubble constant is worth: 퐻(푡) ≈ (39) 2 2 5 2 1 휌푅 푃(푡) 휋휌 푅 푐 dΨ 0.7(푡 − 푡0) + Ψ − × 퐻0 퐻 = 4휎 6휎 dt (29) 7휋푐2휌2푅5 With 퐻0, the value of Hubble constant at 푡 = 푡0 as: 4휋휎푅3 + Ψ − ℰ(푡) 30휎 1 10 √ 퐻0 = 0.7 퐴 (39 푏푖푠) 1) Evolution of the current Universe 푅0 3 If we consider that 퐻 is in current time, independent of 휌 and 푅, According to (35) and (38), Ψ can be written as function of and also independent of global stellar power emission, so we time: have to consider the following approximation: 0.3 2 2 5 2 Ψ(푡) = (40) 휌푅 푃(푡) 휋휌 푅 푐 dΨ 1 2 | Ψ| ≪ | × | (30) (0.7(푡 − 푡0) + ) 4휎 6휎 dt 퐻0 In addition, we need also to suppose that: Thus, the current acceleration of the expansion of the Universe 7휋푐2휌2푅5 is linked to 퐻 as: |4휋휎푅3 − ℰ(푡)| ≪ | Ψ| (31) 2 30휎 Ψ = 0.3퐻 (40 푏푖푠) According to (29) and approximations (30) & (31): Thus, according to (11 푏푖푠) and (40 푏푖푠) we can estimate value 1 푑Ψ of 휎 from physical parameters: 퐻 ≈ − × (32) 휎 ≈ 0.3휌 푐2 (40 푡푒푟) 1.4Ψ 푑푡 푐 According to (32), approximation (30) becomes: With, as a reminder, 휌푐 the critical density that is a parameter 푃(푡) ≪ 퐻푀푐2 (30 푏푖푠) defining flatness of the spatial geometry of the Universe. With 푀, as a reminder, the mass of the Universe. Considering the Universe composed of 1080 equivalent proton particles [ퟔ] 5

2) Example of a specific past evolution of the Universe: The With: inflationary epoch 160휋휎ℰ 푅 퐾 = 0 0 (49) Considering a primitive age evolution of the Universe with a 21푀2푐2 time constant mass 푀 due to successive creation and Hence, we obtain from (48): disintegration of matter and without any presence of 10 physical process. The value of the primitive 푅̇ = √퐾(푅2 − 푅2) + 푅1.4(Ψ − 퐾)(푅0.6 − 푅0.6) + 푅̇ 2 (50) 0 3 0 0 0 0 Universe’s mass 푀 is much lower than the current one. The 2 energy of the primitive Universe was mainly in the form of Thus, if we consider that 퐾푅 is much higher than the rest of thermic photons. Globally, we can consider that (26) is true what is under the root square expression, evolution of 푅 as which means that average value of emitting power 푃 is null function of time is ascending exponential type: ∗ ∗ (there is as much radiative emission from massive matter’s 푅(푡) ≈ 푅0 exp[√퐾(푡 − 푡0)] (51) ∗ disintegration as photon absorption to create mass). Thus, we With 푅0 a specific scale factor respecting: are considering that 푃 ≈ 0 as mass is conserving during this 10 퐾푅∗2 ≫ 푅1.4(Ψ − 퐾)(푅∗0.6 − 푅0.6) + 푅̇ 2 − 퐾푅2 (52) Universe‘s specific primitive age. We consider that even in this 0 3 0 0 0 0 0 0 ∗ ∗ epoch of the Universe, space is flat considering that curvature With 푡0 the time of Universe after Big Bang to reach 푅0 scale of the Universe is undetectable at any epoch (known as flatness factor. 1 problem) that means that for any epoch of the Universe, we can We can observe that is the characteristic duration of the 푐 √퐾 consider that ≪ 1. According to (15), with 푃 = 0, energy of 퐻푅 inflationary epoch, proportional to the mass of the Universe at the free photons is worth as function of the scale factor 푅: inflationary epoch. ℰ0푅0 ℰ = (41) 푅 3) Potential nature of dark energy With ℰ0 and 푅0 the “initial” values of photon energy and scale According to (29), for the current Universe, Ψ is worth: factor of the Universe. We can admit that the value of ℰ is 2 0 − 휋휌2푅5푐2Ψ̇ − 4휎퐻(4휋휎푅3 − ℰ(푡)) almost equal to the value of the total energy of the past and Ψ = 3 (53) 14 present Universe if we consider that 푅0 has almost 휋푐2휌2푅5퐻 − 휌푅2푃(푡) length value and 푀푐2, energy of the primitive Universe’s mass, 15 As a reminder, if we consider that, our observable Universe is negligible compare to ℰ . 0 composed of around 2 × 1012 galaxies for 푧 < 8 [16] As mass remains constant, so density of matter 휌 evolves as composed of around 1011 stellar objects emitting function of 푅: 27 3푀 around 10 푊, so numerically, we can suppose that 푃 ≈ 휌 = (42) 1050푊. Moreover, if we consider that Universe is composed 4휋푅3 In addition: barely of tree hydrogen atoms per cubic meter of Universe 푑휌 9푀 (without counting dark matter), for 푅 > 13.8 billion light year, ̇ = − 4 푅 (43) we have: 푑푡 4휋푅 14 According to (25), (41) to (43), we have : 휋푐2휌2푅5퐻 15 2 > 21 (54) 2 2 2 2 휌푅 푃(푡) 21푀 푐 3푀 푐 푅̇ 14 (4휋휎푅2 + Ψ) 푅̇ + Ψ̇ − ℰ 푅 = 0 (44) Therefore, in neglecting 휌푅2푃(푡) compare to 휋푐2휌2푅5퐻, we 160휋휎푅2 32휋휎푅 푅2 0 0 15 can express acceleration of the expansion of the Universe from From (44), Hubble constant is worth for this Universe’s epoch: (53) as: 3푀2푐2 푑Ψ − × 5 Ψ̇ 30휎 4휋휎푅3 − ℰ(푡) 퐻 = 32휋휎푅 푑푡 (45) Ψ ≈ − × − × (55) 21푀2푐2 ℰ 푅 7 퐻 7 휋푐2휌2푅5 휎 × 4휋푅3 + Ψ − 0 0 160휋휎푅 푅 The equality between Ψ and the first term of (55) give Density of quantum vacuum is a constant of time, so for a equation (32). So, (32) is true if value of 푅 permits to neglect primitive Universe, we can consider that: the second term of (55). 2 2 21푀 푐 ℰ0푅0 Moreover, the NASA’s Fermi Gamma-ray Space telescope has |휎 × 4휋푅3| ≪ | 훹 − | (46) 160휋휎푅 푅 estimated that number of total photons in Universe is around Thus, from (45) and (46) we can write 퐻 as: 4 × 1084 [17] and if we consider that all of them are in average 3푀2푐2 푑Ψ wavelength of 0.5푛푚 (even it is much higher in average), and − × for 푅 > 13.8 billion light year, we have: 퐻 ≈ 32휋휎 푑푡 (47) 21푀2푐2 ℰ Ψ − ℰ 푅 < 5.7 × 10−2 (56) 160휋휎 0 0 4휋휎푅3 From (47), we can see that expansion of the Universe is According to (11), and (56), equation (55) becomes: impossible without presence of mass even if its value is small 40휋휎 푑퐺 120 휎2 compare to the current mass of the Universe. Ψ ≈ − × − × (57) 21퐻푐2 푑푡 7 푐2휌2푅2 From (14) and (47), we deduce that : 1.4 Thus, equation (57) shows that acceleration of the expansion 푅0 of the Universe is mainly linked to two terms including one that 푅̈ = 퐾푅 + (Ψ0 − 퐾) (48) 푅0.4 represents potential dark energy and linked to the evolution of 6

value of gravitational constant 퐺 as function of time. The other 120 term contributes to deceleration of the expansion of the 휎2휂 푅 = √ 7 (67) Universe. According to (57), considering a baryon density of 4 14 ( 휂 + ) 휋퐺푐2휌3 nearly tree hydrogen atom per cubic meter and 푅 > 13.8 billion 3 15 light year and if we consider that 휎 is worth exactly 1nJ per Thus, according to (66) (or even (54)) and (67), it is necessary cubic meter, therefore we have the following inequalities: to have a current Universe with an approximated scale factor 푑퐺 of: 2.1 × 10−28 < | | < 1.49 × 10−26(푚3. 푠−3. 퐾푔−1) (58) 푑푡 90휎2 Considering Friedmann equation: √ 푅 ≈ 2 3 (68) 4휋퐺 3푝 Λ푐2 7휋퐺푐 휌 Ψ = − (휌 + ) + (59) 3 푐2 3 If we still consider a density of matter of tree hydrogen atom With 푝 the pressure and Λ, the cosmological constant, we have per cubic meter, we must have a current Universe with a scale then, by comparison of (57) and (59) we can write: factor of around 246 billion light years. In this case, according 휌푐2 30휎2 to (63), the total radiative power emission must verify: 푝 = − + (60) 51 3 7휋퐺휌2푅2 푃 ≤ 3.4 × 10 푊 (69) Considering our current Universe with a density of tree Now, if we consider density of matter equivalent to around 17 hydrogen atoms per cubic meter and for 푅 = hydrogen atoms per cubic meter (presence of dark matter), the 13.8 billions light year, value of 푝 is: current scale of the Universe must be around 18 billion light- 푝 ≈ 4.76 × 10−8푃푎 (60 푏푖푠) years. In this case, according to (63), the total radiative power In addition, by comparison of (57) and (59) we can write: emission must verify: 49 40휋휎 푑퐺 푃 ≥ 2.67 × 10 푊 (70) Λ = − × (61) In these cases ((69) and (70)) relations (32) and (33) are true. 7퐻푐4 푑푡 Therefore, according to (61) and for the current Universe, Whether it is for (69) or (70) conditions, we can note according to (33), the cosmological constant can be written as: that theirs results are coherent considering that we estimated 8휋퐺휎 푃~1050푊 for our observable Universe with its scale factor of Λ = (62) 푐4 13.8 billion light years. Thus, Λ is consistent with the literature [17]. Moreover, by comparison of (53) and (59) if we suppose that Its current value is: condition (56) remains true, we have the quasi-exact Λ ≈ 2.07 × 10−52푚−2 (62 푏푖푠) expression of pressure 푝: Thus, the cosmological constant is linked to the density of the 휌푐2 4푐2휎2퐻푅 푝 = − + (71) vacuum energy and its value is a function of time. 14 3 퐺 × [ 휋푐2휌2푅3퐻 − 휌푃(푡)] According to (27 푏푖푠) and (60), the total electromagnetic 15 power emission of Universe is a determinist function as: Utilizing (27 푏푖푠) for our current Universe, total radiative 4 120 퐻휎2푅 power emission 푃(푡) of the Universe is solution of equation: 푃(푡) = − 휋휌푅3퐻푐2 + × (63) 4 16휋푐2휎2퐻2푅4 3 7 퐺휌2 푃(푡) = − 휋휌푅3퐻푐2 + (72) 14 Let be 휂 the following ratio: 3 [ 휋푐2휌2푅3퐻 − 휌푃(푡)] 퐺 14 15 휋푐2휌2푅5퐻 Still by comparison of (53) and (59), we can show that the 휂 = 15 (64) 휌푅2푃(푡) cosmological constant is worth: −16휋2휌푅3휎 1 푑퐺 From (64), we can note that 휂 is worth most commonly: Λ = × (73) 2 14 2 푀푐 휋푐2휌푅3퐻 − 3푃(푡) 푐 푑푡 휂 = 0.7 × 퐻 (65) 5 푃(푡) However, approximate expressions of 푝 (pressure), 푃(푡) With 푀, the current mass of the Universe. Therefore, we can (radiative power emission) and Λ given respectively in see that 휂 is a measurement of the ratio between the total current equations (60), (62) and (63) can give value close to reality energy in the form of mass and the total current energy in the accounting approximations done to reach their literal value. form of photons in the Universe. If we consider that, our Thus, according to (63) and (67), we can write: Universe is composed of average photon with a length wave 3 84 퐻휎 less than 500푛푚 and around 4 × 10 in number [18] and if we 푃(푡) = 3 7 퐾(휂) (74) consider that matter density is around 3 hydrogen atoms per √휋푐퐺2휌2 cubic meter, so 휂 must verify: With: 휂 > 1.84 × 103 (66) 3 120 2 120 With the presence of dark matter, minimum value of 휂 would 4 휂 120 휂 퐾(휂) = − ( 7 ) + √ 7 (75) be greater than the one given in (66). 3 4 14 7 4 14 We can note that (54) and (66) are not in accordance given that 휂 + 휂 + [ 3 15 3 15] the digital data for its quantification are not the same. Let be the parameter 푒 defined as: According to (63) and (64), we can write R as function of 휂: 7

120 휂 supposes that fluctuation of energy density of quantum vacuum 푒 = 7 (76) in the primordial Universe; whose 휎 is, as reminder, a 4 14 휂 + measurement of its average; had a direct impact on the spatial 3 15 anisotropy of the value of 퐺 and consequently on the spatial According to (65), (74), (75) and (76), 푒 is solution of anisotropy of the density of matter. It is because size of the equation: primordial Universe was at the same order of magnitude as the 14 3 7 푒 3 15 4 3 120 0.7푀푐 √휋퐺2휌2 spatial coherence of virtual particles composing quantum × [− 푒2 + √푒] = (77) 120 4 3 vacuum that the anisotropy of the value of 퐺 had large-scale − 푒 3 7 휎 7 3 effect for the primordial Universe and thereby, any proportion The approximate solution for 푒 is then: conserved, for the current Universe. It is resulting the current 휋푅2휌3푐2퐺 large-scale measurement of the spatial anisotropy of the density 푒 ≈ (78) 휎2 of matter in the cosmic microwave background map. A greater Hence, value of 퐺 permitted to form first galaxies without taking into 14 account existence of dark matter as extraordinary matter that 휋푅2휌3푐2퐺 휂 = 15 (79) currently, we supposed it is. It implies also that first stars were 120 4 휎2 − 휋푅2휌3푐2퐺 much more massive than the current ones [19] with a shorter 7 3 life cycle. Thus, with a greater value of 퐺, a larger number of We deduce that 휂 goes to zero as 휌 goes to zero with the generation of stars occurred. A significant quantity of ordinary expansion of Universe. In this case, for future Universe, it is matter remained in the form of red and brown dwarf but necessary to solve equation (72) to find the exact literal value especially in the form of black holes and primordial black holes. for 푃(푡). Therefore, we can deduce a new expression of 푃(푡) With the decline in the value of 퐺 as function of time and whatever the value of 휂: because of their primary rapid rotation curves, galaxies begun 289 16휋푐2휎2퐻2푅4 1 to grow larger and residual ordinary matter from past stellar 푃(푡) = √ 휋2휌2푅6퐻2푐4 − − 휋휌푅3퐻푐2 (80) 225 휌퐺 5 evolution has remained gravitationally trapped into them. We To ensure 푃 ≥ 0, it is necessary that density of matter verify the suppose that the current density of total matter is larger than the following inequality: supposed value of the current radiant baryon density such as 1 0.6휌푐 but its nature remains the same than the one of ordinary 90휎2 3 matter. If we imagine far in the future, the value of 푃, given 휌 ≥ ( ) = 휌 (81) 7휋푐2퐺푅2 푚푖푛 in (80), should decrease because of the expansion of the We can note that (81) ensures that 휂 remains positive. Universe. The current ordinary matter should therefore no Hence, for our current Universe, the scale factor 푅 must be longer irradiate. What future of Universe is for us today should be the same than today for the past of the Universe. That means greater than 108 billion light year if we consider that 휌 ≈ 휌푐 representing around 5.2 hydrogen atom per cubic meter. a portion of radiant matter in the first ages of the Universe is no Inequality (81) states that the mass of the Universe must be longer radiating today and compose what we call dark matter. greater than a minimum value dependent of the scale factor 푅 The current value of the critical density is around 5.2 hydrogen in order to generate radiation from massive matter as what atom per cubic meter. To explain formation of large-scale happen in the current core of the stars: structure in the Universe with the current value of 퐺 = 6.674 × 1 10−11Kg−1푚3푠−2, it is necessary to introduce presence of dark 2 2 7 640휎 휋 푅 3 matter with a density of around 5.2 time more than the density 푀 ≥ ( ) (82) 21푐2퐺 of ordinary matter [11]. That involves Universe could not be Hence, the current mass of the Universe must be greater than flat. However, the universe is flat according to the latest 3.8 × 1055Kg if we suppose that scale factor of the Universe is observations of satellite WMAP [11]. How to explain that? If greater than 108 billion light-year. Universe is flat, that means that matter density is less or equal From (73) and (80) we can write the quasi-exact expression of to critical density according to general relativity. If density of the cosmological constant without neglecting the quantity 푃(푡): the matter is less or equal to critical density, how could we 80휋휎 푑퐺 explain formation of large-scale structure of the Universe? − 17푐4퐻 푑푡 Establishing a proportional relationship between 퐺 and Ψ, Λ = (73 푏푖푠) assuming that a higher past value, than the current value, of Ψ 3 280 휌푚푖푛 occurred and it has decreased with time, permit to answer to the 1 − √1 − 3 289 휌 previous questions. We can note that we obtain the Λ literal value given in (62) if we consider 휌 = 휌푚푖푛 (equivalent to say that 푃 = 0) and IV. CONSEQUENCES OF THE ACCELERATION OF THE considering equation (33). Thus, Λ is a physical parameter EXPANSION OF THE UNIVERSE whose value is variable with time. A. The origins of inertia and equality between gravitational and inertial masses 4) Discussion around potential nature of dark matter and scenario of formation of galaxies The equivalence principle of Einstein states that an acceleration According to our theory, the value of 퐺 could be higher in the is equivalent to gravitation and any experiment, even that based past Universe. Contrary to what is admitted, our theory on gravitation itself, cannot permit to distinguish an accelerated 8

referential to a gravitational field. Therefore, an accelerated measured. Thus, the measured gravitational field noted 푔⃗푚 = person could not distinguish a gravitational field from an −grad⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗휙 can be expressed from (86) as: accelerated referential. Measurement of Ψ respects the 푟 2퐺 푟 2퐺푀 푟 푟 푔⃗ = 푔⃗ (푡 − ) − × 푀푎⃗ (푡 − ) + 푣 (푡 − ) × 푣⃗ (푡 − ) (87) equivalence principle of Einstein given the fact that, according 푚 푐 푟푐2 푐 푐2푟2 푟 푐 푐 푟 to our theory, its measurement needs to evaluate the perfect The first term 푔⃗ (푡 − ) is from classical gravitation attraction 푐 value of 퐺. Our theory states that any mass 푀 as inertia, plays due to, according to our theory, acceleration of the expansion a role of impeaching the space-time to expand in accelerating of the Universe and, in case of electrically neutral mass 푀, and and, the main consequence of it is space-time’s curvature is worth: around mass 푀. Thereby, as an inertia, if a mass is able to curb 퐺푀 space-time by creating gravitation field so, an accelerated mass 푔⃗ = − 푟⃗ (88) 푟3 can also distort space-time and generate a “felt” gravitational The second term is linked to the fact that the punctual mass has field proportional to its acceleration. Based on the idea of a specific acceleration compared to the referential of Dennis Sciama [21] and trying to demonstrate Mach’s measurement. As massive matter interacts with space-time, its principle, Woodward had developed an analogy between acceleration must also distort space-time, and generate an gravitation and electromagnetism [22]. Thus, if we call by 휙 the attraction or a repulsion depending on its vector’s orientation. gravitational potential energy per unit of mass, we can define Finally, the third term is linked to the fact that classical the gravitational field 푔⃗ as: Newtonian gravitation 푔⃗ changes in value because of radial 1 휕퐴⃗ velocity of the mass 푚 and has mathematical consequence to be 푔⃗ = −grad⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗휙 − 푔 (83) 푐 휕푡 always an attractive field. As said, equation (87) implies that a With 퐴⃗푔 the gravitation potential equivalent to the magnetic punctual mass 푀, undergoing a movement compare to an inertial frame of reference, must deform space-time of the same potential and grad⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗휙 is the mathematical gradient of potential 휙. nature that a classical gravitation field does it. Thus, we can By analogy with electromagnetism calculation of magnetic postulate that, in its movement, inertia is given to matter ⃗ potential, 퐴푔 can be calculated at a distance 푟 from mass source because of its interaction with space-time. According to (25) a as: Universe devoid of matter and energy has a static scale factor. 푟 2퐺 휇푣⃗ (푡 − ) According to our theory, this kind of Universe is therefore 퐴⃗ (푡, 푟) = − ∭ 푐 푑휏 (84) 푔 푐 푟 devoid of gravitation with 퐺 = 0 and, according to (87), 푟 preventing matter to distort space-time. Thus, in this kind of With 휇 as the local density of matter into volume 푑휏 and 푡 − 푐 Universe devoid of gravitation, interaction between matter and notation to symbolizing a retarded potential due to the fact that space-time is inexistent which means that interaction between propagation of gravitational potential is, as we will see in IV-C, matter and space-time exists only in an accelerated expansion celerity of light 푐. Moreover 푣⃗ is the vector velocity of the local of Universe. According to (53), acceleration expansion of the mass 휇푑휏. Universe occurs because of decreasing of value of Ψ (or 퐺) as According to Reissner-Nordström metric [23], 퐴⃗푔 becomes function of time and presence of density of matter. Thereby, with electrical charged mass: with the absence of matter in a specific Universe, any isolated 푟 mass 푚 does not distort space-time in its movement compare a 2퐺 휇푣⃗ (푡 − ) 퐴⃗ (푡, 푟) = − ∭ 푐 푑휏 frame referential. It is also important to note, thanks to 푔 푐 푟 푟 Reissner-Nordström metric, that a particle with a given mass 휅푣⃗ (푡 − ) 퐺 푐 and with an electrical charge generates less inertia than the same + 3 ∭ 2 푑휏 (84 푏푖푠) particle devoid of electrical charge [24] such as for 푄 ≠ 0: 4휋휀0푐 푟 2 푑푄 푚 1 푄2 With 휅 = , the squared charge density of matter into 푛푐 2 푑휏 푚푐 = − √푚푛푐 − (퐴) volume 푑휏 and 휀0, the vacuum permittivity. A generalized 2 2 4휋휀0퐺 expression of 퐴⃗푔 is given thereafter in equation (푂) according In equation (퐴), 푚푛푐 is the mass of an electrically neutral to Kerr-Newman metric. matter and 푚푐 is the apparent mass (its inertial mass) of the Gravitation is attractive and thereby is a centripetal field like same matter carrying electrical charge 푄). From (퐴), we can electric field generate by negative electric charge. As 휇 > 0, deduce that: thus it is necessary to include the sign “minus” in the expression 푚푐 ≤ 푚푛푐 (퐵) Inversely, we have: of 퐴⃗푔. At a distance 푟 from a punctual electrically neutral 푄2 ⃗ ( ) mass 푀, according to (84), 퐴푔 푡, 푟 is worth: 푚푛푐 = + 푚푐 (퐶) 푟 16휋휀0퐺푚푐 −2퐺푀푣⃗ (푡 − ) From (퐶), we can note that for 푄 = 0, 푚 = 푚 . and in the 퐴⃗ (푡, 푟) = 푐 (85) 푛푐 푐 푔 푟푐 case of the electron, if it becomes electrically neutral, its mass From (83) and (85), we can deduce that: would be around 9.5 × 1011Kg. For a proton, its electrically 2퐺 푟 2퐺푀 푟 푟 8 푔⃗ = −grad⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗휙 + × 푀푎⃗ (푡 − ) − 푣 (푡 − ) × 푣⃗ (푡 − ) (86) neutral mass would be around 5.2 × 10 Kg. In (퐴), the 푟푐2 푐 푐2푟2 푟 푐 푐 푑푟 following condition must be verified: With 푎⃗, the acceleration of mass 푀 and 푣 = the value of its 2 2 푟 푑푡 푄 ≤ 4휋휀0퐺푚푛푐 (퐷) radial velocity from the point where the gravitational field 푔⃗ is 9

With 퐺 tending to zero, 푄 is constraint to tend to zero which Thanks to Kerr metric, we obtain the same result of reduction implies that value of electrical charge must be dependent of 퐺. of inertia and interaction with space –time for a matter with a Thus, in a , no electrically charged particle can rotating axially symmetric. As for a charged particle, a matter exist. If we admit that 푚푛푐 is given to matter thanks to Higgs with a mass 푀 given by Higgs Boson could not have, if Boson and 푚푐 a constant of time mass for a given particle spinning, an angular momentum J exceeding a maximum value (electrical charged elementary particles are supposed to have of angular momentum J푚푎푥 such as: constant apparent masses independent of the Universe’s 퐺 J = 푀2 (퐽) epochs), so according to (퐶), the elementary electrical charge 푚푎푥 푐 is linked to 퐺 as: Thus, for example, measured 1Kg (equivalent to 푚푐) of pure 2 2 35 푄 = 4휋휀0퐺Γ (퐸) proton has a Higgs mass of 3.1 × 10 Kg (equivalent of 푚푛푐). 2 2 With Γ = 4(푚푛푐푚푐 − 푚푐 ) a constant of time depending of Hence, an object composed of pure proton with a measured the nature of the elementary particle. For electron and proton, mass of 1Kg could not have currently an angular momentum value of Γ is around 1.857 × 10−9Kg. Then, we will call Γ, the higher than 2.16 × 1053Kg. 푚2. 푠−1. We can note that in a static charged equivalent mass. Universe with 퐺 = 0, J푚푎푥 = 0 which means that angular We can suppose that for any known elementary particle: momentum of any matter is null even if a moment of force is Γ ≤ 푚푝 (퐹) applied on it. It implies that inertial forces are potentially inexistent and Mach’s principle is true. Besides electroweak With 푚푝 the Planck mass. Thus, maximum elementary charge and strong interactions, other forces have electrostatic and of any elementary particle cannot exceed the value 푄푚푎푥 such as: gravitational origins. In a static Universe, if gravitational interaction is inexistent, electrostatic interaction is, according 2 푄푚푎푥 = √4휋휀0퐺푚푝 = √2휀0ℎ푐 (퐺) to(퐸), also nonexistent. Thus, in a static Universe, moment of With ℎ, the Planck constant. Thus, value of 푄 is worth for force is also inexistent. 푚푎푥 We can note also that for an elementary particle, angular any epoch of the Universe (if we suppose 휀0, ℎ and 푐 as −18 momentum of spin is at the same order of magnitude as: constants of time) 푄푚푎푥 ≈ 1.875 × 10 퐶 representing the 퐺 2 Planck charge. Ĵ = 푚푝 (퐾) Existence of 푄 involves that 퐺 have a maximum value 퐺 푐 푚푎푥 푚푎푥 Thus, a matter, having an electrical charge or/and spinning such as: around an axis, has lower interaction between it and space-time 푄2 푚푎푥 and therefore, provide it with less inertia than if this same matter 퐺푚푎푥 = 2 (퐻) 4휋휀0Γ is electrically neutral without rotary movement. With Γ, the charged equivalent mass of proton or electron Thus, by postulating that interaction between matter and space- accounting the fact that there are no other elementary particle time generates less inertia, we postulate that origin of inertia in the standard model, with higher charged equivalent mass may be directly related to the fact that matter interacts with than proton or electron’s one. space-time through physical quantity 퐺푚푐 (and not only According to (퐸), (퐺) and (퐻), the ratio between the current quantity 푚푐) with 푚푐, for reminder, the apparent (or inertial) value of 퐺 and 퐺푚푎푥 is worth: mass of a given matter. Reciprocally, interaction between 2 퐺 푒 matter and space-time must generate curvature of space-time = = 훼 (퐼) 퐺푚푎푥 4휋휀0ℏ푐 whose equivalent Newtonian attraction between two apparent With 푒, the elementary charge; with ℏ, the reduced Planck (inertial) masses 푚1 and 푚2 should be proportionate to constant and 훼, the fine-structure constant. (퐺푚1) × (퐺푚2) accounting the fact that it is physical According to our theory, the physical quantity 퐺 varies as quantity "퐺푚푐", which is at the origin of space-time curvature. function of time. Therefore, according to (퐼), 훼 must also be a Thus, for gravitational interaction between apparent masses 푚1 time variant physical quantity. Current still non-detection of and 푚2 separated of distance 푟, the second law of Newton value’s variation of 훼 as function of time has the same origin applied to apparent mass 푚1 should fundamentally be written, than non-detection of value’s variation of Hubble constant as in a specific inertial frame of reference called ℛ, as: function of time whereas it is accepted that 퐻 must be a time 퐺푚1 × 퐺푚2 퐺푚 푎⃗ = 푟⃗ (퐿) variant physical quantity and demonstrated in this article. 1 1/ℛ 푟3 1→2 Indeed, measurement of 퐻 based on the study of different kind In the case of electrostatic interaction between charges 푞1 and of celestial objects lead to, without counting uncertainties of 푞2 separated of distance 푟, the second law of Newton applied to measurements, a quasi-constant value of 퐻 even when charge 푞1 with apparent mass 푚1 should fundamentally be measurement is done with the cosmic microwave background written, in a specific inertial frame of reference called ℛ, as: itself by WMAP [11]. This, lead us to ask the following 퐺푞1푞2 퐺푚1푎⃗1/ℛ = − 3 푟⃗1→2 (푀) question: “Do any measurements by any Universe’s 4휋휀0푟 observations are able to detect variability of any physical Introducing 퐺 in Coulomb’s law in (푀) is due to Reissner- quantity, including variation in the values of physical constants, Nordström metric [23], which combines the vacuum as function of time?” The answer of this question seems to be permittivity with the gravitational “constant”. “no”. Thus, no measurements made through Universe’s We can note that in replacing 푞1 and 푞2 by their charged observations may permit us to probe the past physical equivalent masses Γ1 and Γ2, previous equation (푀) becomes characteristics of our Universe. equivalent to gravitational interaction such as: 10

퐺Γ × 퐺Γ frame of reference of the mass 푀. Thus, it is as if, in the frame 퐺푚 푎⃗ = ± 1 2 푟⃗ (푁) 1 1/ℛ 푟3 1→2 of reference of mass 푀, mass 푚 undergoes gravitational In the practical case, simplifying equations (퐿) and (푀) lead to field 푔⃗푚. Here why equation (90) is true. admit that 푚1 plays directly as well role of inertia as role of Acceleration of the central mass 푀 in the referential of orbiting attraction (as well as 푚2) in the case of gravitational interaction. 푚 around 푀 is worth: However, virtues of the primary writings given thanks to (퐿) 휕푣⃗푀|푚 푎⃗ = (91) and (푀) permit to highlight nature of inertia and the type of 푀|푚 휕푡 physical evolution of the two kind of interactions in case of As we have: variation of the value of 퐺. Therefore, we can note that (84) or 푣⃗푀|푚 = −푣⃗푚|푀 (92) (84 푏푖푠) can be written more generally such as: We deduce from (91) and (92) that: 푟 2 푣⃗ (푡 − ) 푎⃗푀|푚 = −푎⃗푚|푀 (93) 퐴⃗ (푡, 푟) = − ∭ 푐 푑퐼 (푂) 푔 푐 푟 According to (90) and (93), we can conclude that: With 푑퐼, a term of repercussion of the interaction between 2퐺푀 푔⃗ + 3 2 (푣⃗푀|푚. 푟⃗)푣⃗푀|푚 matter and space-time at distance 푟 from the source, such as: 푎⃗ = 푟 푐 (94) 푚|푀 2퐺푀 휚푐2 휅 1 − 푟푐2 푑퐼 = 퐺 [휇 − − 2] 푑휏 (푃) 2퐺푟 8휋휀0푟푐 1) The photon sphere and the innermost stable circular orbit With 휚 the squared Kerr parameter (or length scale) density for a Schwarzschild black hole such as: The equation (94) permits to retrieve results of photon sphere 푑푎⃗2 and the innermost stable circular orbit for a massive particle 휚 = (푄) 푑휏 around a Schwarzschild black hole. Indeed, in case of circular 1 푑푢 With: orbit 푣⃗ . 푟⃗ = 0. Adopting Binet coordinates 푢 = , 푢′ = 푀|푚 푟 푑휃 1 푑⃗J 2 2 푎⃗ = × (푅) with 휃, the polar angle and 푎⃗푚|푀 = −퐿 푢 (푢" + 푢)푒⃗푟, with 푒⃗푟 휇푐 푑휏 the unitary vector of radial trajectory of mass 푚 compare to 푀, With ⃗J the angular momentum of the matter. we can write from (94): On the one hand, interaction between matter and space-time 퐺푀푢2 퐿2푢2(푢" + 푢) = (94.1) permits to a given mass, to undergo constraint of distorted 2퐺푀 1 − 푢 space-time as gravitational attraction. In the other hand, 푐2 interaction between matter and space-time generates inertia for With 퐿, the angular momentum per unit of mass. The set of the same given mass. It is because, gravitational attraction as circular trajectories is given, considering 푢" = 0, by: well as inertia are consequences of the same physical process 퐺푀 퐿2푢 = (94.2) (interaction of matter with space-time) that, gravitational and 2퐺푀 1 − 푢 inertial masses are both equals. 푐2 푐 In case of photon sphere, we can write 퐿 = . Thus, B. Examples of calculated theoretical results confirmed by 푢 the general relativity theory from (94.2), position of photon sphere is solution of equation: 2 푟 Considering a punctual central non-rotary electrically neutral = 푠 (94.3) mass 푀 (휚 = 0, 휅 = 0) and, a punctual non-rotary electrically 푢 1 − 푟푠푢 With 푟푠, the Schwarzschild radius. Thus, position of photon neutral mass 푚 (푚 ≪ 푀) orbiting with any trajectory around 3 sphere is at radius 푟 = 푟 . mass 푀. According to (87), in neglecting retarded effect, the 2 푠 gravitational field “felt” by the mass 푚 is: In a general context, solving (94.2) gives general solutions as: 2퐺푀 2퐺푀 푔⃗ = 푔⃗ − 푎⃗ + (푣⃗ . 푟⃗)푣⃗ (89) 1 1 푐2 푚 푟푐2 푀|푚 푟3푐2 푀|푚 푀|푚 √ 푢 = ± 2 − 2 (94.4) With 푎⃗푀|푚 and 푣⃗푀|푚 the acceleration and the velocity of the 2푟푠 4푟푠 2퐿 mass 푀 measured in the referential of the mass 푚. With 푟⃗ In the case of the innermost stable circular orbit, we admit that which represents the position of mass 푀 compare to 푚 (i.e. 푟⃗ = its angular momentum per unit of mass is equal to photon 3 "푚푀⃗⃗⃗⃗⃗⃗⃗⃗"). Finally, quantity (푣⃗ . 푟⃗) represents the dot product sphere’s one: 퐿 = 푟 푐. Thus, in taking this value for 퐿, (94.4) 푀|푚 2 푠 between 푣⃗푀|푚 and 푟⃗. If we consider that, the central mass 푀 is 3 has 2 solutions: 푟 = 푟푠 and 푟 = 3푟푠. We can deduce that an inertial frame of reference (Galilean reference frame), 2 position of the innermost stable circular orbit is at radius 푟 = according to the second law of Newton, we can write: 3푟 . The literal values of these solutions are consistent with the 푎 = 푚푔 (90) 푠 푚⃗⃗푚|푀 ⃗⃗푚 literature [25]. With 푎⃗푚|푀 the acceleration of mass 푚 measured in the referential of the mass 푀 supposed, as a reminder, to be an 2) The apsidal precession expression with a weak field inertial frame of reference. Indeed, even if 푔⃗푚 is the approximation gravitational field measured in the frame of reference of From (94), we can deduce that for a non-necessary circular mass 푚, it is curvature of the local space-time which orbit, 푎⃗푚|푀 cannot be radial. Thus, according to (92) and (94), generates . The same curvature is measured from inertial 푔⃗푚 the tangential component of 푎⃗푚|푀 is worth: 11

2 2 2퐺푀 퐺푀 4퐸푚 2퐺푀 4퐺 푀 3 2 (푣⃗푀|푚. 푟⃗) × (푣⃗푚|푀. 푒⃗휃) 푢" + 푢 ≈ (1 + ) × (1 + 푢) + 푢 (94. 푙) 푟 푐 2 2 2 2 2 (푎⃗푚|푀. 푒⃗휃)푒⃗휃 = − 푒⃗휃 (94. 푎) 퐿 푚푐 푐 푐 퐿 2퐺푀 4퐸푚 1 − In neglecting in front of 1, (94. 푙) can be written as: 푟푐2 푚푐2 With 푒⃗ , the unitary vector of tangential trajectory of mass 푚 퐺푀 6퐺2푀2 휃 푢" + 푢 ≈ + 푢 (94. 푚) compare to 푀. 퐿2 푐2퐿2 As 푟⃗ represents the position of mass 푀 compare to 푚 so by Thus general solutions in 푟(휃) of (94. 푚) are: noting by 푒⃗ , as a reminder, the unitary vector of radial 1 푟 푟(휃) = (94. 푛) 푟⃗ 퐺푀 trajectory of mass 푚 compare to 푀, we can write = −푒⃗푟. + 퐴 cos(휔휃 + 휙) 푟 퐿2휔2 ( ) Thereby, 94. 푎 becomes according to (92): 6퐺2푀2 퐺푀푒 2퐺푀 With 휔 = √1 − and 퐴 = , with 푒, the eccentricity of 퐿2푐2 퐿2휔2 2 2 (푣⃗푚|푀. 푒⃗푟) × (푣⃗푚|푀. 푒⃗휃) (푎⃗ . 푒⃗ )푒⃗ = − 푟 푐 푒⃗ (94. 푏) the orbit. 푚|푀 휃 휃 2퐺푀 휃 6퐺2푀2 1 − 2 Considering Taylor series to order 1, in , in case of weak 푟푐 퐿2푐2 From classical derivative, we can write: 퐿푐 ̈ ̇ mass 푀 in front of , the apsidal precession is given as: 푟휃 + 2푟̇휃 = (푎⃗푚|푀. 푒⃗휃) (94. 푐) 퐺 2휋 6휋퐺2푀2 From (94. 푏) and (94. 푐) we can write: 휀 = − 2휋 ≈ (94. 표) 4퐺푀 휔 퐿2푐2 ̇ 2 ̇ 2 푟̇휃̇ 2 푑(휃 ) 4푟̇휃 푟2푐2 In weak field approximation, we can show that: + = − (94. 푑) 2 2 2퐺푀 퐿 = 퐺푀푎(1 − 푒 ) (94. 푝) 푑푡 푟 1 − 푟푐2 With 푎, the elliptical semi major axis. We can note that 퐿푐 In removing dependence on time in (94. 푑), we can write: neglecting 푀 in front of means, for 푒 = 0, neglecting 4퐺푀 퐺 2 2 ̇ 2 푑(휃̇ ) 4휃̇ 2 2 휃 퐺푀 + = − 푟 푐 (94. 푒) quantity √ in front of 푐. As velocity of mass 푚 is considered 2퐺푀 푎 푑푟 푟 1 − 푟푐2 as non-relativistic, the approximation (94. 표) is good. By solving (94. 푒), we can note that angular momentum per Moreover, the third law of Kepler links the orbital period 푇 with unit of mass of mass 푚 is slightly different from its classical 푎 as: Newtonian value 퐿 and is worth 퐿′ such as: 푇2 4휋2 퐿 = (94. 푞) 퐿′ = 휃̇푟2 = (94. 푓) 푎3 퐺푀 2퐺푀 1 − From (94. 표), (94. 푝) and (94. 푞) we can deduce: 푟푐2 24휋3푎2 Thus, the real angular momentum per unit of mass is variant as 휀 = (94. 푟) 2 2( 2) function of 푟. 푇 푐 1 − 푒 The literal value of 휀 is consistent with the literature [26]. We can note that: We can note that even if calculations are done with the 퐿′ > 퐿 (94. 푔) Thus, the fact that angular momentum per unit of mass is hypothesis of weak field approximation, equation (87) is valid slightly higher than its classical Newtonian value implies in strong gravitational field due to the presence of existence of precession of the orbit (advance of the perihelion) Schwarzschild singularity in equation (94). With (87) of any celestial object. (or (94)), other results of general relativity can be obtained if By adopting Binet coordinates, the radial acceleration of 푎⃗ considering equation (푂), as the Lense-Thirring precession or 푚|푀 even existence of gravitational waves (see IV-B-3 Conclusion). is worth in case of weak field approximation (퐿′ ≈ 퐿): 2 2 2 2 Typically, a punctual non-rotary electrically neutral mass 푚 (푎⃗푚|푀. 푒⃗푟) = −퐿 푢 (푢" + 푢) + 표(퐿 푢 ) (94. ℎ) orbiting circularly around an observer at a distance 푟 from him From (92), (94), and (94. ℎ), we can write: and with velocity 푣 would distort a length ℓ of a quantity Δℓ, in 2퐺푀 2 accordance with (87) in taking value of radial velocity 푣 = 0, 퐺푀 + 2 (푣⃗푚|푀. 푒⃗푟) 푟 퐿2(푢" + 푢) = 푐 (94. 푖) such as: 2퐺푀 1 − 푢 2퐺푚푣2 푐2 Δℓ = 4 ℓ (94. 푠) The mechanical energy of mass 푚 noted 퐸푚, is a constant of 푐 푟 time. In weak field approximation, we can write 휙 = −퐺푚푀푢, For example, even if moon is not a punctual-non-rotary mass, thus: calculation from (94. 푠) shows that it distorts, at its apsis, a 2 2퐸푚 length ℓ = 1푚 situated at the center of Earth of value Δℓ ≈ (푣⃗ . 푒⃗ ) = − 퐿2푢2 + 2퐺푀푢 (94. 푗) −24 푚|푀 푟 푚 10 푚. In weak field approximation, with a Taylor series to order 1 in 퐺푀 푢, we can write from (94. 푖) and (94. 푗): 3) Conclusion 푐2 2퐺푀 2퐸 2퐺푀 By analogy to the electromagnetic field, it must exist another 퐿2(푢" + 푢) ≈ [퐺푀 + ( 푚 − 퐿2푢2 + 2퐺푀푢)] (1 + 푢) (94. 푘) 푐2 푚 푐2 field different from classical gravitational field 푔⃗ and equal In taking into account only terms in 푢 (Taylor series in order 1 to rot⃗⃗⃗⃗⃗⃗퐴⃗푔, with rot⃗⃗⃗⃗⃗⃗ the vector operator curl. Contrary to in 푢), we can write from (94. 푘): gravitoelectromagnetism theory, we postulate the principle that in terms of physics, this field has the same nature as 푔⃗ as a 12

gravitational field thanks to its dimensional analysis. However, With 푗⃗. 푔⃗, the dot product between 푗⃗ and 푔⃗ and with 푔⃗ × 휁⃗ the this new gravitational field is different from 푔⃗ in its physical cross product between 푔⃗ and⃗ ⃗휁⃗. behavior more analogous to a magnetic field. Thus, instead of In (103), 푢 represents the density of gravitational fields as: calling this new field “gravitomagnetic field”, we will prefer to 푔2 + 휁2 call this new gravitational field, “extraordinary gravitational 푢 = (104) field” (opposite to ordinary gravitational field 푔⃗), with the 16휋퐺 We can see that equation (104) is not consistent with (7), but ⃗ notation 휁. As space-time can only be distorted and, if we consider that energy density of field 휁⃗ in the Universe consequence of any distortion of space-time is emergence of contributes equally to that of 푔⃗ (as well as magnetic field has gravitation field, so nature of this new field cannot be different the same energy density as the electric field’s one in an from a gravitational field. Existence of this new gravitational electromagnetic field) we can then consider that equation (17) field explains also the existence of gravitational waves as well is correct. as their polarizations. From (103), we can deduce the gravitational Poynting vector C. The local equations of gravitational fields as: 푐 By analogy to the Maxwell electromagnetic local equations, Π⃗⃗⃗푔 = 푔⃗ × 휁⃗ (105) inspired by the gravitoelectromagnetism equations [27], and 8휋퐺 Like an accelerated charged particle, an accelerated mass lose according to previous equations, we admit equivalent energy in the form of gravitational waves. By analogy with gravitational local equations in case of electrically neutral and electromagnetic field, the gravitational Larmor formula [28] of non-rotary matter density, as: an accelerated mass losing energy, in the form of gravitational div푔⃗ = −4휋퐺휇 (95) wave, with a power emission of 푃 is: ⃗ 퐿 div휁 = 0 (96) 2퐺푚2푎2 1 휕휁⃗ 푃퐿 = 3 (106) rot⃗⃗⃗⃗⃗⃗푔⃗ = − (97) 3푐 푐 휕푡 With 푎, the proper acceleration of the mass 푚 in a given inertial 8휋퐺 1 휕푔⃗ frame of reference. This loss of energy contributes to a gain of rot⃗⃗⃗⃗⃗⃗휁⃗ = − 푗⃗ + (98) 푐 푐 휕푡 inertia for matter. With div, the divergence vector operator, 휇 and 푗⃗ are respectively the density of matter and the matter current density V. SPIRAL GALAXY ROTATION CURVE vector such as 푗⃗ = 휇푣⃗, with 푣⃗, the velocity of the matter’s current. A. Introduction and hypothesis From Kerr-Newman metric, for charged and rotary matter According to Hubble sequence, barred or regular spiral galaxies current density (휇, 휚, 휅), only equations (95) and (98) are are final formation of galaxies from elliptical galaxies modified as respectively (95.1) and (98.1) and we admit their through lenticulars [29][30][31][32][33]. In this chapter, we following expressions: want to model the curve of rotation of spiral galaxies given that 2휋휚푐2 퐺휅 spiral galaxies are the most quasi-static state in the evolution of div푔⃗ = −4휋퐺휇 + + 2 (95.1) galaxy formation. Any celestial object belonging to a spiral 푟 2휀0푟푐 8휋퐺 휚푐2 휅 1 휕푔⃗ galaxy undergoes gravitation field given thanks to ⃗⃗⃗⃗⃗⃗ ⃗ equation (87). We define the galaxy rotation curve as the set of rot휁 = − (1 − − 2) 푗⃗ + (98.1) 푐 2휇퐺푟 8휋휀0휇푟푐 푐 휕푡 tangential velocities of celestial objects composing the spiral With 푟 distance from the source. galaxy as function of distance at its center 푟. If we consider any For non-rotary electrically neutral matter, equations of object with a mass 푚 orbiting with a circular trajectory around propagation of gravitational fields are: a central mass 푀 ≫ 푚, supposed to be an inertial frame of 1 휕2푔⃗ 8휋퐺 휕푗⃗ Δ푔⃗ − = −4휋퐺grad⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗휇 − (99) reference, and with a tangential velocity 푣 at distance 푟, we can 푐2 휕푡2 푐2 휕푡 then write: 1 휕2휁⃗ 8휋퐺 푣2 Δ휁⃗ − = rot⃗⃗⃗⃗⃗⃗푗⃗ (100) 푎⃗ = − 푟⃗ (107) 푐2 휕푡2 푐 푚|푀 푟2 In introducing a gauge relation like: Thus, according to (89), (93) and (107), we can express the 1 휕휙 gravity field 푔⃗ applied to mass 푚 as: div퐴⃗ = − × (101) 푚 푔 푐 휕푡 2퐺푀 푣2 We can introduce the propagation equation of the gravitational 푔⃗푚 = 푔⃗ + × 푟⃗ (108) 푟3 푐2 potential vector as: As reminder, 푔⃗ is given by equation (88). 1 휕2퐴⃗ 8휋퐺 Thus, (108) can be written as: Δ퐴⃗ − 푔 = 푗⃗ (102) 푔 푐2 휕푡2 푐 2푣2 푔⃗ = 푔⃗ × (1 − ) (108 푏푖푠) From equations (97) and (98) we can demonstrate, thanks to 푚 푐2 Poynting’s theorem that the local conservation of the energy of As we suppose that celerity of celestial objects are not the gravitational fields is: relativistic, we will assume for the rest that: 휕푢 푐 = 푗⃗. 푔⃗ − div ( 푔⃗ × 휁⃗) (103) 푔⃗푚 ≈ 푔⃗ (109) 휕푡 8휋퐺 We can observe that (109) stays true if celestial objects have elliptical trajectories with a non-zero eccentricity (it is also the 13

case for value of the third term of (87) which can be considered Then, considering the great number of potential celestial objects as negligible compare to the value of 푔). situated at position 푟 from the center of the spiral galaxy, we Moreover, as said, we can calculate extraordinary gravitation can suppose that: field as: 〈푟̈〉 ≈ 0 (121)

휁⃗ = rot⃗⃗⃗⃗⃗⃗퐴⃗푔 (110) From (118), (120) and (121) we can write: 휕2〈푣2〉 휕〈푣2〉 As we consider a quasi-static evolution of spiral galaxy, we can 푟 = (1 − 훽〈푣2〉) × (122) 휕푔⃗⃗ 2 consider that ≡ 0⃗⃗. Then, we can write in quasi-static state for 휕푟 휕푟 휕푡 Considering the approximation lim〈푣2〉 = 0 [35], we deduce gravities 푔⃗ and 휁⃗ (at a distance 푟 from the center of the galaxy, 푟→0 from (122) that: ⃗ values of 푔⃗ and 휁 are considered as time-invariant) the 4 푟2 equivalent Biot and Savart law [34] for the extraordinary 〈푣2〉(푟) = × (123) 훽 푟2 + 푟2 gravitational field in case of 휅 = 0 and 휚 = 0: 0 2퐺 푗⃗ × 푟⃗ With 푟0 a characteristic value of radial distance. 휁⃗ = − ∭ 푑휏 (111) Thus, we can conclude that: 푐 푟3 4 Thus, we can write in order of magnitude that: lim 〈푣2〉 = (124) 푣 푟→+∞ 훽 ‖휁⃗‖ ≡ ‖푔⃗‖ (112) 푐 From (120), (121) and (123) we can deduce: 4 푟⃗ As celestial objects in spiral galaxy are not relativistic, (112) ( ) 푔⃗(푟) = − × 2 2 125 permits us to neglect the effects of extraordinary gravitation 훽 푟0 + 푟 field, compare to those of ordinary gravitation field. From (115) and (125), the gravitational potential energy 휙 can be expressed as: B. Spiral galaxies rotation curve modelling 2 2 2 푟0 + 푟 If we call by 휙 the gravitational potential energy per unit of 휙(푟) = ln ( ) (126) 훽 퐴 mass introduced in (83), we suppose that density of matter 휇 in any spiral galaxy is in accordance with the equivalent Maxwell- With 퐴, a constant of homogenization. Boltzmann statistics applied to matter; in the supposed inertial The gravitational potential energy 휙 is null for 푟 = 0. Indeed, frame of reference, which is the center of the galaxy; and has according to (113) and given the fact that maximum value for for expression: density 휇 is reached for 푟 = 0 [36] [37], and because 휙(푟) is a monotonically increasing function dependent of 푟, so we can 휇 = 휇0 exp(−훽휙) (113) With 훽 a homogeneous physical quantity in 푠2. 푚−2 and 휇 the conclude that 휙(0) = 0. Therefore, we have: 0 2 density of matter for 휙 = 0. 퐴 = 푟0 (127) From (113), (126) and (127) we can deduce that: According to (95) and (113), we can write: 휇 ( ) ( ) 0 div푔⃗ = −4휋퐺휇0 exp −훽휙 114 휇 = 2 (128) According to (109), we can write: 푟2 (1 + 2) 푔⃗ = −grad⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗휙 (115) 푟0 Thus, we can simply write from (114) and (115): From (95), (125) and (128) we can deduce that: 2 = 휋퐺휇 푟2훽 (129) grad⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ln(−div푔⃗) = 훽푔⃗ (116) 0 0 If we consider that a spiral galaxy has a constant thickness 퐸 With ln( ), the natural logarithm function to the base of independent of 푟, its mass called 푀푔 and is worth: the mathematical constant 푒 ≈ 2.71828 … +∞ As we suppose that quantity 푔⃗ is only dependent of radial 푀푔 = ∫ 휇(푟)2휋푟퐸푑푟 (130) distance 푟 from the center of the spiral galaxy, we can rewrite 0 (116) in cylindrical coordinate system as: According to (128) and (130), we can deduce that: 1 휕(푟푔) 푀 = 휋휇 퐸푟2 (131) 휕 ln ( ) 푔 0 0 푟 휕푟 푟⃗ From (129) and (131) we can show that: × = 훽푔⃗ (117) 휕푟 푟 2퐸 With 푔 = ‖푔⃗‖. Thus, from (117), we can write: 훽 = (132) 퐺푀푔 휕2(푟푔) ( ) Moreover, the angular momentum J of any spiral galaxy is quasi 휕푟2 1 = −훽푔 + (118) invariant in time if we consider no exchange of matter and no 휕(푟푔) 푟 ( ) major gravitation interactions between galaxies. 휕푟 Considering that, all the celestial objects, in a spiral galaxy, At distance 푟 from the center of the spiral galaxy, the tangential have quasi-circular orbit, the angular momentum of a spiral velocity of any object as function of local gravity 푔 can be galaxy is given as: written as: +∞ 푣2(푟) = 푟 × (푔 + 푟̈) (119) J = ∫ 휇(푟)2휋푟2퐸푣(푟)푑푟 (133) At a given distance 푟, the average value of 푣2(푟) on all the 0 celestial objects of the galaxy at distance 푟 from its center, noted According to (123), (128) and (133), we can write: 〈푣2〉(푟) is: 4 3 4 2 J = 휋휇 푟 퐸√ (134) 〈푣 〉(푟) = 푟 × (푔 + 〈푟̈〉) (120) 3 0 0 훽 14

According to (129) and (132), we can write that: Universe in the other side. According to (13) and (89) we can 푀 show that, in addition to the conventional gravitation force, it 휇 = 푔 (135) 0 2 exists a force density applied on any volume of any cosmic 휋퐸푟0 From (132), (134) and (135), we can express the spiral filament with density 휌 whose origin is linked to the presence galaxy‘s angular momentum as: of galaxy cluster of mass 푀 located at a filament node and worth 푓푣 such as: 4푀푔푟0 2퐺푀푔 2퐺휌푀 J = √ (136) 푓 = 퐻̇ (142) 3 퐸 푣 푐2 From (136), if we consider that values of 푀 and 퐸 are quasi- Thus, 푓푣 is independent of distance and is either attractive or 푔 ̇ invariant in time, even if it is not real the case, we can deduce repulsive according to the sign of 퐻. The current gravitational that value of 푟 increases in time as value of 퐺 decreases. Thus, density force is higher than current value of 푓푣 in our observable 0 ̇ galaxies have become bigger and bigger over time. Universe. However, past value of 퐻 implies that 푓푣 was higher Moreover, from (135) and (136), we can write: than gravitation density force and allowed matter to be torn out 4 from one galaxy cluster to another one. Nowadays, because of 32퐺푀푔 휇 = (137) weak value of 퐻̇ , flow of matter in cosmic filaments must be 0 9휋퐸2J2 faded or exist only due to attractive gravitational forces. We can consider that evolution of the mass of a galaxy per unit 푑푀 of time 푔 is due to algebraic agglomeration of mass per unit 푑푡 VI. SYNTHESIS AND CONCLUSION 휕푀 of time 푔 and its stellar radiative emission 푃 such as: This article was intended to give, modestly, solutions to some 휕푡 푔 푑푀 휕푀 푃 enigma of modern cosmology. This article proposes a new 푔 = 푔 − 푔 (138) 푑푡 휕푡 푐2 theory to explain origin of inertia and why matter, with a mass, Thus, in considering that J is a time-invariant quantity and if curves space-time. According to our theory, acceleration of the we still suppose that 퐸 is also a quasi-invariant of time quantity expansion of the Universe gives inertia to matter in permitting therefore, from (33), (136) and (138) we can deduce that it to interact with space –time. Hence, it is because any massive matter interacts with space-time that it tends to keep its velocity. evolution in time of 푟0 is given by the following equation: Consequently, because space-time is in constant accelerated 푑푟0 3 휕푀푔 3푃푔 7 = (− × + + 퐻) 푟 (139) expansion and interacts with massive matter as inertia of 푑푡 2푀 휕푡 2푀 푐2 10 0 푔 푔 expansion, that space-time is curved. Moreover, it is because We can consider that evolution of mass due to stellar radiation matter interacts with space-time that it undergoes its curvature is globally negligible compare to 퐻. Indeed, orders of effects in the form of gravitational interaction. The more matter magnitude are such as: interacts with space-time, the more its gravitational interaction 푃 퐿 푔 0 −21 −1 with other masses is strong. However, the more matter interacts 2 ~ 2 ≈ 10 푠 ≪ 퐻 (140) 푀푔푐 푀0푐 with space-time, the more its inertia is strong. Thus, it is With 퐿0 and 푀0, respectively the solar luminosity and mass. because gravitation interaction and inertia of a massive matter In addition, according to (39), from (139) and hypothesis are both from the same physical phenomenon that gravitational (140) we can write: mass equals to inertia mass. Concretely, in order to 3 conceptualize interaction between mass and space-time, this 푀 (푡 ) 2 푔 0 article proposes a theory, which unifies gravitational constant 퐺 푟0(푡) ≈ 푟0(푡0) × [1 + 0.7퐻0(푡 − 푡0)] × ( ) (141) 푀푔(푡) with the acceleration of the expansion of the Universe Ψ and Thus, considering (140) as true, therefore (141) permits to say density of vacuum energy 휎 in a single equation. This theory that characteristic size 푟0 of spiral galaxies have an evolution permits to consider that the gravitational constant must be a depending on how they accreted mass from surrounding time-variant physical quantity. Moreover, this article stipulates celestial objects during these last billion years. Their size’s that the total energy of the Universe must be a constant of time. evolution could be, over time, nearly linear and quasi- Thanks to this consideration, our article shows that the constant proportional to the expansion of the observable Universe‘s of Hubble is linked to gravitational constant as well as to its first scale factor. Their size could also have a time non-linear derivative as a function of time. It shows also that gravitation evolution in growing or shrinking due to mass gain or loss constant is a decreasing function of time and that dark energy is following galactic fusions and contribution of hydrogen matter just a consequence of the decline over time in the value of the due to cosmic filaments between galaxies clusters in different gravitational constant. Moreover, thanks again to this regions of the Universe. However, if galaxies ‘size has grown, consideration and thanks to Friedmann-Lemaitre equations, this residual primordial stars as well as their gas has migrated and article shows that radiative emission power (or luminosity) of are situated currently, at the edge of galaxies in the galactic the entire Universe is a determinist quantity depending on the halo. This is maybe why, for spiral galaxies, some physical knowledge of only few physical parameters like density of properties of galactic halo’s stars, as star’s metallicity and matter, Hubble constant or scale factor of the Universe. average age, are so different from the properties of less Furthermore, as gravitational constant is supposed to have peripheral stars [38]. According to (141), galaxy’s size can greater value in the past of the Universe, this article explains shrink or grow as function of time depending respectively on how current observed large-scale structure of the Universe was contribution of matter [39] in one side or expansion of the formed thanks to higher value of gravitational interaction 15

without taking into account necessarily of the presence of dark kind of expansion. Thus, interpretation of the given matter as a different nature matter from ordinary baryonic one. equation (11) above is the link between value of the celerity of The link between gravitational constant and acceleration of the any propagation wave based on the space-time structure, expansion of the Universe can also explain anisotropy in the including the electromagnetic wave as well as gravitational (black body temperature) mapping of the cosmic microwave wave, and the physical characteristic of space-time, which are background at a time when fluctuation of quantum vacuum its “rigidity” and the acceleration of its expansion. Thus, energy density generated spatial fluctuation in the value of equivalent to propagation of “classical” waves like acoustic gravitational constant 퐺 resulting in occurrence of primordial wave or a vibrating wire in a material medium, electromagnetic matter density’s anisotropy. Moreover, a past higher value of wave as well as gravitational wave propagate through space- gravitational constant has many consequences on past stellar time as immaterial medium of propagation with celerity 푐 given characteristics and evolutions. It is include the fact that past in the equation (11). It is because origin of inertia is interaction stars are more massive, much bigger and with a shorter life between space-time and matter that any mass cannot reach cycle than current ones. Consequently, this article proposes to celerity of gravitational waves. As photons are massless and so reconsider nature of what we call dark matter and it stipulates have no inertia that they can reach the same celerity of that the hidden mass of the Universe is in what remain of propagation of the gravitational waves. We consider in our previous generations of stars disappeared since the last 13 theory that variation of Ψ as function of time influences only billion years, mostly in the form of black holes or primordial the value of 퐺 and not the value of celerity 푐. Indeed, if black holes and are probably located in the halos of galaxies. acceleration of the expansion of the Universe permits matter to Consequence of this property is that visible stars in the galactic interact with space-time and give it inertia via the physical halo, including those of our milky way, ceased their formation quantity 퐺, therefore only value of 퐺 evolves with time via long ago, due to impoverishment of hydrogen gas, and tend to evolution of Ψ. Thus, in our theory, we consider that quantity 푐 be old and metal poor [38]. In an explanatory approach to the is a real constant of time independent of 퐺 and Ψ evolutions. nature of inertia, this article postulates the fact that inertial mass Of course, like all any others theories, those postulated in this of any elementary particle remains constant in time but it is article have to be verified, either by simulations or by necessary to consider that their elementary charge is linked to observations, before being confirmed or invalidated. gravitational constant and thus, evolves as function of time. Observations can be possible in the near future thanks to the This leads to consider that 퐺 has an upper bound and the ratio ESA and the NASA James Webb Space telescope replacement between current value of 퐺 and its maximum value is worth the of the Hubble telescope which will be able to study formation fine-structure constant. This article also proposes to take of first stars and galaxies [40] as well as to measure density of account of presence of two different kinds of gravitational field dark matter from gravitational lensing [41]. In parallel, the in the Universe. One is the classical ordinary gravitation field future Euclid Spacecraft by ESA has objective to understand created by presence of mass but, the second one, called in this nature of the dark energy by measuring acceleration of the article, extraordinary gravitation field, is, in its physical expansion of the Universe as well as measuring distribution of behavior, more analogous to the magnetic field and, is the dark matter and galaxies [42] in the Universe. generated by current mass density. This extraordinary gravitation field is different in nature from gravitomagnetism VII. 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