We note that in any lattice at massive coupling (see Figure 3) is possible to define a dimension spatial and time. It is just this way which pushes us to affirm that a massive particle carries with itself the Space-Time, see [(D), (τ)]. Then we could admit that it is possible define the S-T in any frame reference built between only with massive objects, because these carry the S-T with themselves. It follows that only massive particles introduce in Universe the dimension “Space- Time”. Besides, if any massive particle is built using the transversal additional coupling on X-scalar field, we may state that in universe there is a (X_U) background field with structure scalar, in which an any massive particle is built by a particular structure of transversal couplings of quantum oscillators of X-field.

This presses us to assume that a massive particle is structure of articulated couplings between chains of oscillators of the X-scalar field of universe; we indi-

cate all this as “Structure Hypothesis” and we call (X_U) as the “Scalar Field of Universe”.

The propagation of a particle into Universe is so the propagation in (X_U)-field of a structure with “massive” coupling of quantum oscillators of X-scalar Field. Note that non massive (X_U)-field do is not coincident with “the ether” because this cannot constitute a reference frame (RF). On the contrary a massive particle (massive scalar field) with transversal coupling introduces a local RF (S₀) with a motion in the proper time (τ) and a spatial dimension (D₀). Then we can consider the universe as a set (S_U) of the local reference frames (S₀), associated with massive particles. The symmetry group of the universe based on the lattices is the Poincare group. Now, we can build a whole RF for every galaxy as well as for clusters of galaxies until whole universe. The reference frame (RF) of each galaxy of the universe coincides with the “comovent RF” [2] defined by actual cosmology, to which a proper cosmic time (τ_c) is associated equivalent to the age of the universe. We could associate an only and unique field of base to each “comovent RF” in any galaxy: this will be a massive scalar field of fund which will indicate with (X_G). So as we associated to single massive particle a reference frame at rest with a proper space and proper time, at same way we associated to whole galaxy an unique RF with proper ST (see the cosmic time).

Therefore, once more we will talk about Universe as “Universe Field” (X_U) ≡ (X_U, X_G), of which the fundamental nature is the Space-Time. But we have not believe that (X_U) is a sort of elastic ether pervading an absolute space, refuted by the theory of relativity, because the field (X_U) is itself the cosmic space-time (you see the RF comovent) which an any observer of the universe represents and uses. We could also conjecture that the dark matter is in relation with (X_U): [X_U ó (X_G) ó X_D], where (X_D) is the field of Dark Matter (you see the halos of dark matter around any galaxy). Because we asserted that a massive particle introduces in universe the dimensions of the space and time [(D_c), (τ)], we wonder if the expansion of universe is due to creation of massive particles which “add space” in universe: [m_i ó(D_i)]. To lattice of massive couplings that represents the massive particle (m_i) we can associate a field (X_i) seen as a system of chains of quantum oscillators which couple in way transversal (massive coupling), with step (D_i). We may so associate to the particle a field-lattice of chains of quantum oscillators. If a massive particle (m_i) is represented by massive lattice in (X_U)-scalar field, we can add to this lattice ulterior particles with same mass: we talk about “universe-lattice” of particles (U_i). If the number of massive particles (m_i) is increasing we may hypothesize that (U_i) is increasing also in space: (U_i) should be so increasing in “mass-space”. Following the Standard Model and the structure hypothesis we could suppose that only the basic fermions (electron, for leptons, and proton, viewed like quarks (u) and (d), for baryons) can determine some processes of space-mass creation in the universe, because they are basic structures of couplings of the Quantum Oscillators. We take in consideration the fermions because for each chain of field-lattice with transversal step (D_i), it is possible placing one only particle (one quantum).

All this allow us of conjecture that the creation of massive particle in U-Un- iverse could “creates” new space (see the D_c) and new “time” (see the cosmic time (τ)_c) or adds space-time into universe. By this asserts we can conjecture that the expansion of universe (X_U) could be the effect of a space increasing continually derived by Space creation in [(X_i) Ì (X_U)].

As we will show after, the creation of mass-space can determine lattice-fields with variable metric. However, in this paper we not talk about sources of matter in the various universe-lattices, showing solely what happens in a universe with lattice-fields increasing in mass-space and with variable metric. Only in the section “conclusions” we will mention to a possible origin of matter into Universe that builds it with variable metric without to contradict the principle that the “Universe is all that is”. Besides we introduce the idea that the universe crosses in the course of time different evolving phases. Then, we so consider a U*-universe constituted by different universe-phases [ U 1 ∗ , U 2 ∗ , ⋯ , U k ∗ ] each of them built on specific universe-lattice (U_i)*. The last universe-phase (that in which we are living) started in time of recombination: (U*)_rec.

The present theory of the “lattice shaped field” or “lattice-fields” is built on the hypothesis that the “mass” is the physical variable describing the additional “transversal” coupling between “chains” of quantum oscillators which represent a particle-field. In these systems (see par. 2.1) the “monotone mode” of oscillation is associated to the (ω₀) proper frequency (see the Equation (2.7)) which indicates the mass. So the nature of the mass is that of being an oscillatory characteristic; note: an oscillation frequency is always positive. Not only, recalling the relations of [frequency ó clock] and [mass ó energy at rest], we have establish (par. 2.1) a relation between the energy at rest and energy of “motion in the time”: so in the reference frame at rest (S°) of a massive particle we cannot give an any physical meaning to the reverse motion in time. In any other reference frame (S) the particle will move in the time (in forward) and also it will move in the space of this reference frame (S). In relativity, if we reverse the time in an any relativistic equation the energy will be negative: there will be particles with negative energy which in the reference frames at rest (S°) will have negative mass or negative energy a rest. For what has already been said, in our theory we cannot have particles with mass negative. If instead we insist to admit hypothetical particles with negative energy, the reading experimental will give always particles with positive energy going forward in time but in spatial direction opposite (e.g. see about a system of an atom source (A) of particle massive and a detector atom (B)): for negative energies we will see B as transmitter and A as receiver and if the emitted particle has an electric charge then from B to A we will see a particle with opposite charge). In this way particles with negative energy exist no experimentally: this happens both in the special relativity that in general. In fact, the gravity admits as sources only masses with positive values. Also when one takes in consideration a platform in proper rotation and its curvature is negative, the mass of the platform is always positive.

Also more dangerous is the association negative energy ó antimatter. In literature some authors (see Villata in arXiv:1302.3515v1 [physics. gen-ph] 2013) take in consideration this aspect which, in my opinion, produces some problems in physics. In fact, in Villata the reversal of time in CPT transformation is associated to antimatter but this determines a Lagrangian L_A opposite to that of matter L_M. The emerging contradiction is the one of not associate a negative mass to reversal of time, because in this case (with m < 0, t ® −t) the Lagrangian result invariant [L_A = L_M], aspect in contrast with the result of Villata where it revers only the time: [L_A = −L_M]. The negative Lagrangian determines:

• [H = −p₀ < 0]

• a gravitational field generated by antimatter which is indistinguishable by that generated by matter

• the possibility that a “local” CPT-transformation on the particles with negative Lagrangian and a local inversion of the time connected to the metric of the surrounding gravitational field determines repulsion with the surrounding matter

Following this road, Villata obtains that matter and antimatter are mutually repulsive and this should be the origin of the cosmic speed-up as well as the well-known universe expansion. The origin of this problematic aspect could be caused by two lacking considerations:

• The CPT-transformation operates in quantum systems and the charge electric is a variable correlated to wave function of particle but no correlated to the Space-Time of the reference frame from which an observer detects it.

• The mass and its gravitational field operate only in Space-Time “system” where we collocate the fields

We note that the antimatter in pair creation processes is always with positive energy: the coupling of two photons (positive energy) creates two particles with positive energy but with opposite charges. So in pair annihilation processes is an error to associate a negative value of mass to the positron, because matter and antimatter vanishing. Besides, two particles with positive energy have positive mass a rest and they determine a gravitational interaction to positive curvature: this result is in contrast with the Villata’s hypothesys that matter and antimatter repulse reciprocally.

As one knows, the positron has positive electric charge, see the conjugation operator (C), because a hypothetical particle whit negative energy (E < 0) going back in time (T) can be interpreted as a particle whit positive energy going in direction opposite (P) but with opposite electric charge (C). The conjugation of charge leads us to operate the CPT transformation when we speak of matteró antimatter. As it is know, the electric charge (±e) is correlated to the phase of the wave function of a particle because it is the generator of transformation U1 on the phase: [Q ó U1]. The gauge local invariance on the transformation of phase is guaranteed by photon which keeps the electric charge invariant in electromagnetic interaction. This shows that the electric charge is a variable relative to undulatory aspect of particle or to the oscillators of field (recall the field as a set of coupled quantum oscillators). In this way the CPT symmetry is a property typically of quantum systems or of systems with wave-particle duality. The antimatter is the transformed for CPT transformations of matter associated to solutions at negative energy of wave equation or to negative frequencies of the quantum oscillators of field. The CPT transformation as also the pair creation processes create particles always with positive energy (m > 0). All this leads us to admit that when we reverse the time the apparent negative energies are seen (thanks to CPT symmetry) as particles at positive energy but with electric charge opposite. The antimatter is the interpretation (the transformed) of the matter at apparent negative mass or negative frequencies. For overcome the problem of the presence of negative frequencies (or negative masses) it is need that the negative frequencies come “detected” without any reference to energy. Recall that in the oscillation theory we associate to the oscillation a rotation (see also the phase rotation). If the negative frequencies are associated to clockwise rotations of phase while that positives to anticlockwise rotations of phase (see Guido [3] ) and these rotations are associate to sign of electric charge, then only particles with negative electric charge or positive will exist but always with positive energy and mass. All this is possible only if a photon is able to detect the direction of phase rotation. In Guido ( [3] [4] ) this is showed. In this way we cannot associate negative value of mass to antimatter. Besides, the gravitational interaction do not perturbs the gauge invariance of electromagnetic interaction because it no causes variations to direction of phase rotation of wave function: the gravity do no transforms an electron in a positron. Nevertheless, the gravitational interaction is in turn a gauge invariant interaction: the graviton can cause some phase shifts (see interference phenomenon with neutrons in gravitational field) but, like photon, no changes the electric charge. Instead, the graviton acts on the tensorial fields which express the metric of S-T of reference frames (see the gravitational waves perturbing the S-T of “detectors”). In the gravitation the local gauge invariance on the metric is instead, the Equivalence Principle (EP) which not distinguish different values of mass (or so the signs ±) and so there is not any difference between positrons and electrons in free fall in earthling gravitational field. Any theory talking about repulsion between matter-antimatter should be in contrast to the EP. A cosmologic theory which explains the galactic acceleration through the matter-antimatter repulsion should be so in contrast with GR to the which any cosmologic theory is based. In this paper the mass express the additional coupling between chains of oscillators of a quantum field, building so a “lattice shaped field” or “massive lattice”. The (g_ik) tensor gives the metric of this lattice while the gravitational interaction between “lattices” is expressed by (R_ik) Riemann’s tensor. Any gauge transformation on the wave function of particle (massive lattice) not perturbs the symmetry of local gauge of gravity or this not distinguish positrons and electrons. In final, our theory being built on the lattices of field is, therefore, coherent to EP and can solely express a relativistic cosmology.

In literature the universe’s expansion is regarded as the expansion of a spherical surface. We see, like observers, that the galaxies are immersed in a 3-dim space, “increasing” in volume, however, the cosmological principle imposes galaxies placed above a surface. If we add the time axis at every point of the universe by then a spherical surface 4-dim. is given. To increase a surface portion of a plan is necessary to increase the (X, Y) axes of a (DX, DY). The “expansion” of the spherical surface portion corresponds to add elementary steps (D) along the X and Y axes, which on the surface “spherical” 2-dim. become two meridians: [(X + DX), (Y + DY)]. Remember, indeed the expansion of a flat geometric figure in a computer: the program that performs the graphics operation increases the number of components “points” in Figure 4:

Note that to extend in way isotropic the surface in Figure 4, the mouse of PC points at point (A) and moves along the OA diagonal. This makes increase the number of points (one adds other points of space) into circle which increases so in surface. In an isotropic universe any local observer O puts himself at the

center of a sphere 3-dim. with ray R_i and places on its surface the distant galaxies from O (Figure 5).

An observer sees galaxies move away radially in space (3-dim) but describes the isotropic expansion as dilatation of a 2-dim spherical surface, in which it placed itself into origin of the local spherical cap. Into an any U_i universe-lattice belonging to U k ∗ universe-phase [ U i ⊂ U k ∗ ] increasing in space, the ray R_i of any local spherical cap (3-dim) would increase as [ ( R n ) i = n ƛ i + ( R 0 ) i ] . Where it can be [ R ( n ≫ 1 ) ≫ R 0 ] , it follows [(R_n)_i ≈ nD_i]; it will be, omitting i-index:

{ S n = 4 π R n 2 R n = n ƛ ⇒ S n = 4 π n 2 ƛ 2 (4.1)

where S_n varies with increasing R_n; then the index (n) will be a function of time: n ® n(t); we calculate the following derivatives:

d S n d t = d d t ( 4 π R n 2 ) = ( 8 π R n ) ( d R n d t ) = ( 8 π R n ) V n (4.2)

where V_n is the velocity of recession of a galaxy placed on the spherical surface of ray R_n. For any observer an evolving universe (U_i) is described by means of an increasing time as [t = nτ], from which we obtain

n = ( t / τ )

We calculate the following derivative

d S n d t = d d t ( 4 π R n 2 ) = d d t ( 4 π n 2 ƛ 2 ) = ( 8 π n ƛ 2 ) d n d t = ( 8 π n ƛ 2 ) ( 1 τ ) (4.3)

and combining with the Equation (4.2) it follows

{ d S n d t = ( 8 π n ƛ 2 ) ( 1 τ ) d S n d t = ( 8 π R n V n ) (4.4)

We will have

( 8 π n ƛ 2 ) ( 1 τ ) = ( 8 π R n V n ) ⇒ ( n ƛ 2 ) ( 1 τ ) = ( R n V n )

we obtain

V n = ( 1 R n τ ) [ n ƛ 2 ] = ( 1 n ƛ τ ) [ n ƛ 2 ] = ƛ τ (4.5)

So we obtain a constant velocity. This is achieved because we did not consider the “aspect” of space increasing or in any time (τ) between two any points it is added ulterior space steps (D): this causes larger distances to increase at even greater times than the smaller ones (see thermal expansion of metals). Then we’ll have that a spherical surface increase more than those with a smaller radius. Then we may conjecture S = 4 π n R n 2 . This because in time (τ₀) other (n) steps (D) are added on surface S which pass from [ S = 4 π R n 2 ] to [ S = 4 π n R n 2 ] . It follows:

{ S n = 4 π n R n 2 R n = n 0 ƛ 0 ⇒ S n = 4 π n 3 ƛ 0 2 (4.6)

with a proportionality [D₀ = kD], [n = n₀] and [c = D₀/τ₀]; we calculate

d S n d t = d d t ( 4 π n R n 2 ) = d d t ( 4 π k n 3 ƛ 2 ) = ( 12 π k n 2 ƛ 2 ) d n d t = ( 12 π k n 2 ƛ 2 ) ( 1 τ 0 )

{ d S n d t = ( 12 π k n 2 ƛ 2 ) ( 1 τ 0 ) d S n d t = ( 8 π R n V n ) (4.7)

Computing

( 12 π k n 2 ƛ 2 ) ( 1 τ 0 ) = ( 8 π R n V n ) ⇒ ( k n 2 ƛ 2 ) ( 3 2 τ 0 ) = ( R n V n )

It follows

V n = ( 3 2 R n τ 0 ) [ k n 2 ƛ 2 ] = ( 3 k 2 R n τ 0 ) = 3 k R n 2 τ 0

If we assume [ 3 k / 2 τ 0 = H ] then it follows

V n = H R n (4.8)

finding again the Hubble’s law.

We avoid cosmological problems of difficult resolution if we restrict our attention to the individual Universes-lattices U_i.

We can admit that if the number of massive particles (m_i) is increasing in a universe-lattice U_i with spatial step (D_i), then there is more gravity in U_i; however, in U_i there will be even a space increasing with a consequent weakening of gravity, because to each particle with mass (m_i) which is created we have associate a spatial step (D_i): [m_i óD_i]. It derives that U_i is an increasing universe- Lattice or in expansion. Note that each U_i universe-lattice if increasing with mass-space could be at variable metric in relation of have more masses or more space: each U_i universe-lattice is at variable metric. We define the Universe as the set of all universes-lattices U_i.

We intuit, then, that during the accretion of the universe, any universe-phase U k ∗ it goes to cross, it is possible to have a time of its evolution where it could be there a just balancing between gravity and space. Remember a flat universe be consistent with the cosmological solution of general relativity that admits just a “equable balancing “ between kinetic energy (expansion) and potential energy (gravity). We note that even in the universe-lattice U_i with accretion of “space- mass” it’s possible to associate the kinetic energy (K_exp) to the creation of space because each galaxy moves away by others. Also, if the increase in mass is correlated with an increase of potential energy (more gravity) then the fair balance, for the general relativity, corresponds to a fair balance between mass and space in a flat universe.

There are distinct possibilities on balancing Space-Gravity:

1) The balancing Space-Gravity is an invariant of the universe, from its origins (see relativistic cosmology)

2) The balancing Space-Gravity is reached only at a particular time of cosmic evolution (see inflationary cosmology)

3) The balancing Space-Gravity is reached at a particular time (t_k) of each k- phase of cosmic evolution (universe at more evolutionary phases)

We may think that the universe crossed different phases in time (see U k ∗ ), each built on a particular universe-lattice U_k with variable metric. We remember that in general relativity the T mass-energy tensor of the universe (defined as everything that is physically observable) is constant, so determining a constant metric in time.

We avoid cosmological problems of the whole universe if we restrict our attention to the individual Universes-lattices U_i of any U k ∗ . We may admit in U k ∗ that the individual lattice U_i can be described by variable metric because there is a flow of mass-energy between different lattices U_j, with [ U j ⊂ U k ∗ ] . So we can suppose that into universe ( U k ∗ ), with increasing mass-energy, the mass-energy tensor T is variable. However, one would must explain the existence of something that “nurtures” the mass-space (or T) of each universe ( U k ∗ ). In a theory where “the mass is the source of both space both of gravity”, we highlight the following possibilities: the increase in the number of massive particles (and not) in the universe generates more gravity (increasing curvature ó metric variation), but also the space is increasing with a consequent weakening of gravity (decreasing curvature).

In case of variable metric it can believe, then, that during the increasing of the universe ( U k ∗ ), there is possible have a particular “moment” or “phase” in which the universe ( U k ∗ ) achieves a fair balance between gravity and space: “having so much gravity as much as space”.

A equal balancing between gravity (mass) and space will thus be expressed by the relation [ ( K esp = K escape ) = U gravitational ] .

By this condition is obtained the escape velocity:

v f 2 ( R ) = G 2 M R (4.9)

If we combine [2] with the Hubble law (note that with the speed of expansion of a sphere with ray R is equivalent to a speed of escape from a distant center R):

{ v R = H R R v e 2 ( R ) = G 2 M R R (4.10)

We obtain

( H R R ) 2 = G 2 M R R (4.11)

From which we get easily

ρ R = M R V o l = 3 M R 4 π R 3 = ( 3 8 π R 2 ) ( 2 M R R ) = ( 3 8 π R 2 ) ( H R 2 R 2 G ) = 3 H R 2 8 π G (4.12)

So we find the critical density, well-known in literature, of a flat universe ( U k ∗ ):

ρ C = 3 H 2 8 π G (4.13)

This relation is valid both in each universe-lattice U_i of any both in whole universe (in each of its evolving phases ( U k ∗ ) seen as overlap or intersection of

individual universes-lattices: [ U ∗ = U 1 ∩ U 2 ∩ ⋯ ∩ U n ] .

We admit so a universe that crosses different evolving phases where each phase is built on a particular universe-lattice U_k with variable metric, passing from an open metric to a closed through the flat phase.

The relations identifying any “lattice universe”, with [ U i ⊂ U k ∗ ] and U_i “increasing” in mass-space, are given by

{ [ R = n ƛ ] ,   [ τ = n τ ] ,   [ H = 1 / τ ] ,   [ N = n 2 ] } (5.1)

The increase of massive particles (m_i) in U_i implies an “increasing” space with ray R_i and spatial step given (D_i) on the increase. The gravitational coupling constant (α) takes the value:

( α i ) = ( G ℏ c ) ( m i 2 ) (5.2)

It’s also noted

( m i ƛ i 3 ) U i = ( m i ) ( m i 3 c 3 ℏ 3 ) = ( m i 2 c 4 ℏ 2 ) ( m i 2 ℏ c ) = ( 1 τ i 2 ) ( α i G ) (5.3)

The first lattice-universe U_i which has been originated in the first moments of the universe could be built on the value of the Planck mass

m p l = ℏ c 5 G (5.4)

We denote with (U_pl) the Planck’s lattice-universe. Instead for (m_i < m_pl) we have U_i universe-lattice in masse-space increasing (where is N = n²).

( ρ ( n ) ) U i = ( N m i V o l ) U i = ( 3 4 π ) ( N m i [ ( n ) ƛ i ] 3 ) U i = ( N n 3 ) ( 3 4 π ) ( m i ( ƛ i ) 3 ) U i = [ ( 3 4 π ) ( α i G τ i 2 ) ] ( 1 n ) U i (5.5)

The value of the specific critical density of each specific U_i (with (n) is the number of “time” steps) will be given by

( ρ C ( n ) ) U i = ( 3 8 π ) ( H 2 G ) U i = ( 3 8 π ) ( 1 G ( n τ ) 2 ) U i = [ ( 3 4 π ) ( 1 G τ i 2 ) ( 1 2 n ) ] [ ( 1 n ) ] (5.6)

The critical density, as already said, represents the balancing Space-Gravity.

It notes that this balance is achieved in each U_i lattice only at a certain moment of its evolution, where at passing time the Universe lattice U_i is increasing in mass (at increase of number N of particles) as well as in space. In fact (see Equation (5.5)) the U_i universe reaches his critical density when the (n) number of time steps is such that occurs:

n = 1 2 α i ≡ n _ α i (5.7)

Thus it follows

[ ρ ( n _ α i ) ] U i = ( ρ C ) U i (5.8)

We point out that

[ ρ ( n ) ] U i | n < n _ α i < ( ρ C ) U i (5.9)

For (n < n_i) U_i is an open lattice-universe. Instead for (n > n_i)

[ ρ ( n ) ] U i | n > n _ α i > ( ρ C ) U i (5.10)

U_i is a closed lattice-universe. By varying (n) we have the following development of density (you see Figure 6).

In this way U_i is a universe with a “variable metric” in the phase ( U k ∗ ), going from negative curvature (open universe for future reference) with accelerated expansion to one of positive curvature (closed universe) after to going cross an intermediate “critic” phase with a zero metric (flat universe). Now it is proved

that the time (τ_c) to reach the critical phase in ( U k ∗ ) is connected to the coupling gravitational constant (α_i) of each U_i. The time τ_c (critical age) to reach the critical density will be given by

τ c ( U i ) = ( n i ) c τ i = ( n _ α i ) τ i = ( τ i 2 α i ) (5.11)

We find

τ c ( U i ) = ( n _ i ) τ i = ( τ i 2 α i ) = ( 1 2 ) ( ℏ m i c 2 ) ( ℏ c G m i 2 ) = ( ℏ 2 2 G m i 3 c ) (5.12)

As examples, consider the universe-lattice of the electron (U_e) and that of the proton (U_p), finding the respective “critical” time

τ c ( U e ) = ( ℏ 2 2 G m e 3 c ) ≈ 3 .69 × 10 23 sec ≈ 1.2 × 10 16 y (5.13)

τ c ( U p ) = ( ℏ 2 2 G m p 3 c ) ≈ 5 .90 × 10 13 sec ≈ 1.9 × 10 6 y (5.14)

We compare the two lattice-universes; in the critical phase it is

( ( ρ c ) p ( ρ c ) e ) = ( 3 H p 2 8 π G ) ( 8 π G 3 H e 2 ) = ( H p 2 H e 2 ) (5.15)

As well as for the Equation (5.2) and Equation (5.7) we obtain

{ ( α p α e ) = ( m p 2 m e 2 ) ( α p α e ) = ( ( n e ) c ( n p ) c ) = ( ( n _ e ) ( n _ p ) ) (5.16)

And therefore:

[ ( n _ p ) m p 2 ] = [ ( n _ e ) m e 2 ] (5.17)

Expressing the mass through D

( ( n _ p ) ( ℏ ƛ p c ) 2 ) = ( ( n _ e ) ( ℏ ƛ e c ) 2 ) (5.18)

It follows

{ ( n _ p ƛ e 2 ) = ( n _ e ƛ p 2 ) } ( ρ = ρ c ) (5.19)

We place

4 π ƛ c 2 = S c (5.20)

where S_c is defined as “Compton’s Surface” of a particle. We will denote by [S_e º (S_c)_e, S_p º (S_c)_p]. We will be then

n _ p S e = n _ e S p (5.21)

We can have

( n _ i S j ) = k ( n _ j S i ) (5.22)

where we introduce a (k) numerical constant related to the physical various arguments that take into consideration. Remember that (n_i/n_j) is a rational number because (n_i, n_j) are whole numbers, while the ratio (D_i/D_j) cannot be irrational if (D_i) and (D_j) are not commensurable; therefore the numerical constant (k) can be useful for make commensurable the ratio (D_i/D_j). With constant (k); so it’s

( n _ i ƛ j 2 ) = k ( n _ j ƛ i 2 ) (5.23)

Taking the square root we have:

[ ( n _ i ) ƛ j ] = k [ ( n _ j ) ƛ i ] (5.24)

It follows even

n _ i [ ( n _ i ) ƛ j ] = n _ i [ ( k ) ( n _ j ) ƛ i ] (5.25)

And therefore:

k s ( ( n _ i ) ƛ j ) = [ ( n _ i ) ( n _ j ) ] ( ƛ i ) (5.26)

where k s = (1k)

We wonder what surface (S) can represent this equality [S = n_eS_p = n_pS_e]. The surface [(n_i)S_j] (see Equation (5.22)) can have physical meaning only in an universe-lattice U_(i,j) which “contains” the two distinct lattices U_i and U_j. This can be achieved only in a universe-lattice intersection of the two universe lattices U_i and U_j:

( U i ∩ U j ) = ( U i j ) (5.27)

We note that the real universe cannot be built on only type of particle (m_i ® U_i), you see the standard model. Then, one must conjecture a universe-lattice intersection of the universe lattices U_i of base. In an any universe-lattice the spatial step is given by (D_c), therefore to build the universe-lattice U_(i,j) as intersection of U_i and U_j, the respective spatial steps should penetrate an into the other. We distinguish two cases:

• if (D_i) and (D_j) are geometrically “commensurable” then the intersection will be represented as “superposition” of their direct lattices;

• if (D_i) and (D_j) are “incommensurable” then the intersection can be built by use of a “intermediary” field that constitutes a background field where the two respective lattices can be in “superposition”

Whatever be the case in consideration, however, we could assume that [(n_i) D_j] is the ray R* of U* º U_(i,j). The same will be for universe-lattice in the critical phase [(n_i) D_j] ó R c ∗ . Noting that if [n_eS_p = n_pS_e = S_U] for any U universe-lattice and if

[ S U = 4 π R U 2 ] → [ n _ e S p = 4 π R U 2 ]

By Equation (5.21) è [R_U = (n_e)^1/2D_p]; however we set in U* º U_(i,j):

[ R c ∗ = R U ( n _ e ) 1 / 2 = ( n _ e ) 1 / 2 ƛ p ( n _ e ) 1 / 2 = ( n _ e ) ƛ p ]

We will then (see the Equation (5.26)):

( R c ) U * = k s ( ( n _ i ) ƛ j ) U * = [ ( n _ i ) ( n _ j ) ] ( ƛ i ) (5.28)

where [k_s = (k)⁻¹]. Let us consider the intersection of the two universe-lattice U_p and U_e

( U ) ( p , e ) = ( U p ∩ U e ) (5.29)

We will geometrically (you see Figure 7).

Note that the actual universe U consists mainly of protons and electrons or Hydrogen, then we could assume that U-universe is “built” on an universe (U)_(p,e), intersection of the two lattices (U_p, U_e ); because into Universe there not is only Hydrogen lattice, we assume: [U É (U)_(p,e)]. Note that the U_H could be one of some evolving phases ( U k ∗ ) of Universe: we talk of the “Hydrogen phase”. The beginning of this Universe-Phase is coincident with recombination time. We wonder, therefore, if the hydrogen atom builds (through the intermediation of the photons, the quanta of the electromagnetic field) the intersection between the two lattices or [(U)_(p,e) = (U)_(H)]. The U_H will be said “Universe lattice at Hydrogen”. If we consider the diameter of a hydrogen atom, we can assert that along the diameter a bond between the proton and the “orbiting” electron is established.

( ( n _ p ) ( ℏ ƛ p c ) 2 ) = ( ( n _ e ) ( ℏ ƛ e c ) 2 )

The diameter is given by QuO-chain where the “oscillations” of three particles (proton, electron and photon) can be in “superposition”. In hydrogen atom the Compton wavelength of electron is (D_leg = D_e/α_em), where α_em is the constant of electromagnetic coupling or also known as the fine structure constant. It notes that D_leg is exactly the Bohr ray è [D_leg ≡D_H].

We will say then that the action of the electromagnetic field (photon) “adapts” the Compton wavelength of the electron to the one of proton to allow the intersection of the two lattices. Then the fine structure constant is a constant of “electromagnetic adaptation” between two electric charge with different structure. The adaptation happens along the diameter: for obtain the superposition of two lattices (U_p, U_e) the D_H must be projected along the chain-diameter because the wavelength l_p is projected along the chain diameter and an orbital electron is physically equivalent to an oscillating electron along diameter. Nevertheless, being electron and proton of different spatial “dimensions”, then it must make an adaptation in scale (a gauge transformation): we introduce a “gauge factor” derived by ratio of the two periods (τ_p,τ_e) or wave lengths, for adapting space dimensions between the representative quantum oscillators of particles (p,e). Remember that two oscillators in coupling phase need of adaptation: the coordination between two oscillators involves the reciprocal adaptions in periods and wave lengths. Into H-atom (intersection D) we will have to compare wave lengths of proton and bound electron; it is [(D_H/l_p) ≈ 10⁴]. We need to compare the orbital ray of Hydrogen with the Compton’s wave length of proton. Note that the electron moves long the circumference while the proton long the diameter. Then, after having made the “gauge wavelength” [n_eD_H(10⁻⁴)] or “scale reduction one must adapting the orbital movement of the electron with the proton oscillation along the diameter or adapting the circumference with diameter (you see the (2π) irrational value and remember èCirc/ray = 2π). Into intersection U p ∩ U e ) follows: [(2π)n_pD_p = n_eD_H(10⁻⁴)]. Where (n_p, n_e) are two any whole numbers. Point out the [(n_p/n_e]) = k_s]:

{ ( n p n e ) = ( ƛ H λ p ) ( 10 − 4 ) ( n p n e ) = k s (5.30)

thus it follows:

{ k s = ( ƛ H λ p ) ( 10 − 4 ) = ( 5 .2917733534 × 10 − 11 1 .32141002 × 10 − 15 ) × 10 − 4 = 4 .004 } (5.31)

(k_s) is so the ratio (D_H/l_p) scaled by (10⁴) factor. We define (k_s) as adapting ratio. The general relation is: [(k_s) = γ(l_x/l_y)(10^s)]. Where (s) is the exponential of the scale factor which makes (k_s) Î [1, 10]; (γ) is a coefficient relative at geometric aspect. In this case it is: γ = (2π)⁻¹. We will have in U_H (see Equation (5.28)) that:

( R c ) H = k s ( ( n _ p ) ƛ e ) = [ ( n _ p ) ( n _ e ) ] ( ƛ p ) (5.32)

This relation is valid in a universe being “critical”. Then we could develop the following system of relations:

{ ( R c ) H = k s ( n _ p ) ƛ e ( R c ) H = c ( τ c ) H (5.33)

These relations could allow us of calculating the τ_c critical age of the universe U_H. Since U_H is the intersection of two universe-lattices with variable metric, we can also assert that it evolves from an expansive phase with open metric to a phase of contraction with closed metric but passing through a phase a flat metric. By restoring (see Equation (5.22)), the multiplicative constant we can express this in U_H:

( n _ e ) S p = k ( n _ p ) S e (5.34)

here the numerical constant (k) will be in connection with numerical constant (k), because U_H is intersection of two lattices U_p and U_e. This leads (see Equation 5.34 and Equation 5.32) to following relation

R H = ( k ) − 1 ( ( n _ p ) ƛ e ) = k s ( ( n _ p ) ƛ e ) (5.35)

The Hydrogen phase is characterized by U_H lattice. The (τ_c)*º (τ_c)_H time to reach the critical density of the flat-universe will be derived from the system of relations (see Equation (5.33)):

{ R H = k s ( n p ) c ƛ e R H = c ( τ c ) * (5.36)

( τ c ) * = ( k s ( n p ) c ƛ e c ) (5.37)

Remembering that [k_s » 4] and the relation between (n_c)_i and α_i, it will be

( τ c ) * = ( k s ( n p ) c ƛ e c ) k = 4 = ( 4 ( n p ) c ƛ e c ) = ( 2 ƛ e c α p ) (5.38)

Approximate values of the constants in the fifth decimal place, we get

( τ c ) * = ( 2 ƛ e c α p ) = 2 ( 3 .86159 × 10 − 13 × 0 .31688 × 10 − 7 y 2 .99792 × 10 8 × 5.90464 × 10 − 39 ) = 13.825 × 10 9 y (5.39)

We have got the “critical” time of the universe U*. It should be noted that this value is very close to the estimated value of the age of our universe processing transmitted data from the Planck space telescope. If our universe U is approximated by U* then the “critical” age τ c ∗ would be very near (if not identical) to age τ of the universe U. The observations [5] give an age of the universe equal to

( τ U ) Planck ≈ ( 13 .82 ± 0 .20 ) × 10 9 y (5.40)

The data shows that the value of τ c ∗ “coincides” with age found by spatial telescope “Planck” [6] . Therefore, by comparing of the two times [ ( τ U ∗ ) ≈ ( τ c ∗ ) ] , we could assert that our universe may be in a phase “critical” to flat metric, as is evidenced by the astronomical observations which shown really an universe with flat metric. By the critical time value τ c ∗ (according to Equation (5.39)) a value of the Hubble’s constant H* is given:

H c * = ( 1 τ c * ) = ( 1 13.825 × 10 9 y ) = 70.78 ( km / Mpc ⋅ s ) (5.41)

The H c * value is compared so to one experimental H WAMP = ( 73 ± 3 ) [ km / ( Mpc ⋅ sec ) ] derived by observations [7] of remote galaxies through WAMP satellite, to which is given an universe old of τ U = ( 13.72 ± 0.20 ) × 10 9 y and with critical density ( ρ c ) U ≈ 9.58 × 10 − 27   kg . Note by data that the H c * value seems instead to be closer to the value [7] H Ceph = ( 70 ± 3 ) [ km / ( Mpc ⋅ sec ) ] derived from the analysis of Cepheids (for nearby galaxies, the source “HST and N. Wright.”). If we connect the value of H to a past time, because derived from observations of remote galaxies (H_pass º H_{(remote gal)} º H_WAMP), or H to present time (H_pres º H_{(nearby galaxies)} º H_Ceph), then it is clear that is: [ ( H pass ) > ( H pres ) ≈ H c * ] . By comparison of the two data (both the values of H and time τ_U) it is noted that the recession velocity in the past was higher. Having that [ H c * = H Ceph ] it follows that the current universe is flat. This means that in the past the recession velocity would have been greater than that of the “present” (nearby galaxies). It is noted then that the variable trend of the metric, from open to flat, be manifested into galactic “red shift” with gradient greater than that linear of the flat metric having origin in our present.

In a universe with variable metric ( U H * ) the transition from open metric to flat should obtained itself by diagram [5] “red Schift-galactic distance” a gradient greater for remote galaxies (Figure 8, curve 5) with respect to the linear (curve 3) provided for a universe in flat metric (next galaxies) (you see Figure 8).

However, (as shown in Figure 8) we see that the observations place more distant galaxies (curves (4,6) to those of an open universe metric (curves 5). This means that here it would be an additional acceleration to that provided in a universe with open metric (you see [5] ). Here we talk of accelerating expansion (a_exp) of the Universe. Nevertheless, in this cosmologic model an explanation can be given on this acceleration: the increasing space in space (or space addition inside space) could cause a “pressure” (because one introduces quanta, you see the equivalence massó space) with consequent galactic acceleration. This induce to think that the space pressure could represent the “dark” energy. Now we calculate the critical density (r_c)_U* for the Universe U* in (t = τ_c)

( ρ c ) U * = [ 3 8 π ⋅ 1 G ( τ U * 2 ) ] = 9 .401 × 10 − 27   kg / m 3 (5.42)

While the universe ray R* is:

R U * = c τ U * = 13.076 × 10 25 m (5.43)

It is evident that if τ c * ≈ τ U it follows that R* » R_U. By definition of the actual mass density of the universe derives the value of the total mass in number of protons. Nevertheless, we know the number of temporal steps needed to roach the critical phase in U_p,e. because these are coincident with number of protons (n_p) in U_p only that in U* the numbers of particles is in relation with spherical surface, therefore it will be

{ ( α p ) U p = ( 1 2 n p ) U p ( N p ) U * = ( n p 2 ) U * = ( n p ) U p 2 (5.44)

and

( N p ) U * = ( n p ) U * 2 = ( n p ) U p 2 = [ 1 4 ( α p 2 ) U p ] U * = ( 1 4 × 5 .90464 2 × 10 − 78 ) U * = 7 .172 × 10 75 (5.45)

If we want to consider the baryons thus we add a numeric ratio b in Equation (5.44) or N = b(n²).

And so the baryonic density would be:

ρ b = ( b N p m p V U ) = ( 3 b 4 π ) N p m p ( R U ) 3 = ( 3 4 π ) b × 7 .172 × 10 75 × 1 .6726231 × 10 − 27 kg 13.076 3 × 10 75 m 3 = b × 1 .281 × 10 − 30 kg / m 3 (5.46)

If we consider that the baryonic matter has been created in couple one obtains N_p ®2N_p:

ρ ( b , b ¯ ) = b × 2 .562 × 10 − 30 kg / m 3 (5.47)

The [ ρ ( b , b ¯ ) ] is, at less than factor of (10), next to experimental value of the density of visible matter: ρ v i s ≈ 5 × 10 − 29 kg / m 3

In U* we observe that:

Ω p = ( b ( ρ p ) c ( ρ c ) ) U * = b ( 1.281 × 10 − 30 9.401 × 10 − 27 ) = b × 1 .363 × 10 − 4 (5.48)

Note that:

( ρ c ) U * = ( 3 8π ) ( H 2 G ) = ( 3 8π ) ( 1 G τ U * 2 ) = ( 3 8π ) ( 1 G ) ( 2 c α p k ƛ e ) 2 = [ ( 3 8π ) ( 4 c 2 G k 2 ) ( α p 2 ƛ e 2 ) ] (5.49)

Besides this:

( ρ p ) U * = ( 3 4π ) ( M p R 3 ) = ( 3 4π ) ( N m p c 3 τ U 3 ) = ( 3 4π ) ( N m p c 3 ) ( 2 c α p k ƛ e ) 3 = [ ( 3 4π ) ( N m p 2 α p 3 k 3 ) ( 8 m p ƛ e 3 ) ] = [ ( 3 4π ) ( 1 4 α p 2 ) ( m p 2 α p 3 k 3 ) ( 8 m p ƛ e 3 ) ] = [ ( 3 8π ) ( m p 2 k 3 ) ( 4 α p m p ƛ e 3 ) ] = [ ( 3 8π ) ( α p ℏ c G k 3 ) ( 4 α p m p ƛ e 3 ) ] = [ ( 3 8π ) ( 4 c 2 G k 2 ) ( α p 2 ƛ e 2 ) ( ƛ p k ƛ e ) ] = ( ρ c ) [ ( 1 k ) ( ƛ p ƛ e ) ] (5.50)

From the Equation (5.31), we obtain that the protonic matter density is equivalent to critic density scaled by factor (10)⁴ around. In fact, if k = 4 we have (from Equation (5.50)):

Ω p = ( ( ρ p ) ( ρ c ) ) U * = [ ( ƛ p 4 ƛ e ) ] = [ 2 .103 × 10 − 16 4 × 3 .862 × 10 − 13 ] = 1 .361 × 10 − 4 (5.51)

Now we note that: the universe U contains all the universe lattices and all universe lattices reach their critical density at the same time. It thus follows that the universe U in “hydrogen phase” is flat together with all the universe lattices which compose it. However, remember that the U_p universe is contained in the lattice U_H (U_p Ì U_H) even if it (U_p) reaches its critical density at the same time to U_H (where U_H is an “expanded universe” because the time step is that of the proton while the space step is that of the electron). Nevertheless, the density is higher in U_H that in U_p (should be on the contrary less for the “expansion”), this can mean that there must be other particular baryonic mass or non-baryonic mass which compensates the defect of density. This value tells us that there must be more mass in the universe to reach the critical density. This aspect opens the question of the missing mass: the astronomical observations however detect not radiative mass in the massive halos enveloping each galaxy (dark mass). Even at the dark mass it is associated a universe lattice U_d.

Let us consider the intersection of the two universe-lattice U_α and U_π

( U ) ( α , π ) = ( U α ∩ U π ) ≡ U N (5.52)

where (α) indicates the α-particle and (π) the neutral pion. Note that the U_N could be a Universe-lattice representative of an evolving phases ( U k ∗ ) of Universe in which the first nucleus were formed: we talk of the “Nucleosynthesis phase” ( U N ∗ ). Note this phase is antecedent to that of recombination. The Nucleosynthesis phase is characterized by U_N lattice. The (τ_c)* º (τ_c)_N time to reach the critical density of the flat-universe derives from the system of relations (see Equation 5.33):

τ N = ( k N ƛ π ° 2 c η α α ) (5.53)

where (k_N) is adapting ratio: [(k_s) = γ(l_x/l_y)(10^s)] where if [l_x > l_y] ó[s < 1]. The (η) is a coefficient connected to probability of interaction with nucleons: [ £ 1]. In our study [4] on the quarks we found that the geometric relation between the l_p of pion and l_α particle (α) is given by (j) “aureus” ratio (golden ratio). Let’s go find the (k_N) adapting ratio:

{ k s = ( γ ) ( ƛ x λ y ) ( 10 − s ) } ⇔ { k s = ( φ ) ( ƛ π ° λ α ) ( 10 − s ) } (5.54)

We calculate (k_N) and (α_α), with (γ = j), where (j) is “aureus” ratio:

k N = ( φ ) ( ƛ π ° ƛ α ) ( 10 ) − s = ( φ ) ( 1.4643 × 10 − 15 m 5 .29399 × 10 − 17 m ) ( 10 − s ) = ( φ ) ( 0 .276597 ) × 10 2 × 10 − 2 = ( φ ) ( 0 .276597 ) (5.55)

and

α G ( α ) = G m α 2 ℏ c = 6 .67259 × 10 − 11 [ 44 .15146 × 10 − 54 1.05457266 × 10 − 34 × 2.99792458 × 10 8 ] = 9 .31842 × 10 − 38 (5.56)

Now we can calculate the critical time (τ_c)_N of universe-lattice U_N of Universe phase ( U N ∗ ):

( τ N ) * = ( k N ƛ π ° 2 c η α α ) = ( 0.276597 × 1.618 × 1 .4643 × 10 − 15 × 3 .169 × 10 − 8 y 2 η × 2 .99792 × 10 8 × 9 .31842 × 10 − 38 ) = ( η ) − 1 ( 3 .717 × 10 5 ) y (5.57)

We have got the “critical” time of the universe U N ∗ . The critical density (omitting h) is:

( ρ c ) U N = [ 3 8 π G ( τ N 2 ) ] = [ ( η ) − 1 ( 3 8 π × 6 .67259 × 10 − 11 × ( 3 .717 × 3 .156 × 10 12 ) 2 ) ] ( η = 1 ) kg / m 3 = 1 .3 × 10 − 17 kg / m 3 (5.58)

So one can see that in the antecedent phase to that of recombination (Nucleosynthesis phase) the universe reached a flat metric (you see the literature) a few moments prior to the recombination.