\label{increasing_pressure} In a universe with variable metric the \(a_{exp}\) could be related with additional pressure resulting from Space-Mass increasing.
In fact, if we admit that to put into a massive chain some IQuO (spatial cells) increases both the mass of the universe as well as space, we can think that this introduction will cause an additional pressure on the already existing mass.
This is because the inclusion of a certain amount of space-mass between contiguous cells of space determines a pressure on the latter.
By analogy we can think that in a square crowded with people suddenly a person it materialize: it is clear that this will push the people around him, and the pressure at the borders will be more little than close to the source (see for example the figure \ref{fig_pressure}).
Now we are going to calculate the pressure (pressure of space-mass increasing).
Since in the model of expanding universe the galaxies are placed on a 4-dim surface, which continuously expands in all directions, we consider a portion of that surface.
The emergence of a IQuO (with a spatial step \(\lambda\)) at a point of this surface, produce a pressure that will be distributed all around the IQuO-chains of surface; it is determined so a cap with area \(S = 4 \pi R_i^2\), where \(R_i\) is the radius of the surface corresponding to the Compton’s length associated to the IQuO’s cell, spatial step of the massive lattice defined on the surface. It follows that \(S = 4 \pi \lambda^2\), and the boost will be given by \(F = \frac{\Delta p}{\Delta t}\), where \(p\) is the quantum of motion: \[F = \frac{p}{\tau} = \frac{\hbar c}{\lambda^2}\] The additional pressure or pressure of space-mass increasing \(P_{incr}\) will be: \[P_{incr} = \frac{F}{S} = \frac{\hbar c}{\lambda^2} \frac{1}{4 \pi \lambda^2} = \frac{\hbar c}{4 \pi \lambda^4}\] The pressure \(P_{incr}\) of the mass on the increase is similar to a gas pressure, with a consequent acceleration of the galactic gas; from (\ref{eqgravity}) it follows that \[\label{pincr}
P_{incr} = g_{incr} \frac{c^2}{4 \pi R G}\] Here we use \(R\) instead of \(\lambda\) becouse the pressure to the borders of the surface.
We specify that the \(P_{incr}\), although it is a local pressure, will be present in the entire universe. This is a tipical behaviour of an hydrostatic pressure.
One important observation: if the universe were closed the accretion pressure would be nothing. It follows that the accretion pressure is only present in a universe with open metric or flat.
At the beginning of the birth of the lattice massive universe had to have: \[\label{pincr_l}
P_{incr} = g_{incr} \frac{c^2}{4 \pi \lambda G}\] The accretion pressure can be interpreted as a negative pressure that hinders the gravitational.
If we set \(P_{incr} = P_{gas}\), then we are able to find the expression of \(g_{incr}\), which we can see as an acceleration of anti-gravity, \(g_{anti-g}\). In this way we have that \(g_{exp} = g_{incr} = g_{anti-g}\), and so it follows: \[\label{gincr}
g_{incr} = g_{anti-g} = \frac{F_{gas}}{m} = \frac{\hbar c}{\lambda^2} \frac{1}{m} = \frac{\hbar c}{\lambda^2} \frac{\lambda c}{\hbar} = \frac{c^2}{\lambda}\] Substituting (\ref{gincr}) in (\ref{pincr}) we get: \[\label{pchi}
P_{incr} = g_{anti-g} \frac{c^2}{4 \pi \lambda G} = \frac{c^2}{\lambda} \frac{c^2}{4 \pi \lambda G} = \frac{c^4}{4 \pi \lambda^2 G} = \frac{2}{\lambda^2 \chi}\] Where \(P_{incr}\) is the pressure on the IQuO’s unit cell, \(\chi = \frac{8 \pi G}{c^4}\) is the constant setting in the equation of general relativity.
In this way, close to the border, we have: \[\label{eq_antig}
g_{incr} = g_{anti-g} = \frac{c^2}{R}\] The acceleration of anti-gravity could intend as a centrifugal acceleration \(a_f\) given by \(v_{exp}^2 / R\), because the force of the pressure of space-mass pushing out. In this way \(g_{exp} = a_f = v_f^2 / R\) and, putting \(v_f = c\), it follows \[g_{exp} = g_{incr} = g_{anti-g} = \frac{c}{R^2}\] In this way \(g_{exp}\), like \(g_{gas}\), will be given by: \[\label{eq_gexp}
g_{exp} = \frac{4 \pi G R P_{exp}}{c^2}\] The pressure accretion can well describe the negative pressure that is associated with the cosmological constant \(\Lambda\). \[\label{eq_plambda}
P_\Lambda = \frac{\Lambda c^4}{8 \pi G}\] So \(P_\Lambda = P_{exp}\). We now confront the cosmological pressure \(P_\Lambda\) (\ref{eq_plambda}) with (\ref{eq_gexp}) \[\label{eq_gexpbis}
g_{exp} = \frac{R \Lambda c^2}{2}\] Now, confronting \(g_{exp}\) (\ref{eq_gexpbis}) with \(g_{incr}\) (\ref{eq_antig}) \[\Lambda = \frac{2}{R^2}\] We recall that the present universe is constructed on the lattice \(U_{p,e}\) whose step is given by \(\lambda_e\). Now, in the early time of \(U_{p,e}\), we have: \[\Lambda = \frac{2}{\lambda_e^2} = 1.34121 \cdot 10^{25} m^{-2}\] So we can calculate the pressure \[P = \frac{\Lambda c^4}{8 \pi G} = 2.39764 \cdot 10^{66} Pa\] Instead, in the present universe: \[\Lambda = \frac{2}{R_U^2} = 1.17 \cdot 10^{-52} m^{-2}\] where \(R_U = 13.1 \cdot 10^{25}m\).
This result is very close to the experimentally calculated.
Thus the cosmological constant seems a time depending parameter while it is really constant throughout the space for a given instant of the cosmological time of each reference system.
In addition, we can say the dark energy is identified in the flow of half-quanta from \(\Theta\) to the universe \(U\).