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\subsection{The Hubble
law} law}\label{hubble_law}
We will show now that a increasing universe in space through a specific mode of increase is an expanding universe described by Hubble's
law. \\ law.\\
There are two IQuO ($O$-IQuO and $A$-IQuO ) placed at a distance $\lambda$, represented by the
$OA$-IQuO connecting.\\ link between $O$ and $A$ (the connecting $OA$-IQuO; see figure \ref{fig_iquo_expansion01}).\\
The three IquO (you see fig.3) could became the constituents of an emerging space line of the basic field of the universe
$\underline{\Xi}^\circ$, $\Xi^0$, built using IQuO described by the parameters $m$, $\tau$, $\lambda$ (see
eq.\ref{eq_iquolattice}).\\
(figure 3)\\ \ref{eq_iquolattice}).\\
Given the time
$tau$, $\tau$, we suppose that into the system $[O\text{-IQuO}, \, OA\text{-IQuO}, \, A\text{-IQuO}]$ a splitting of $OA$-IQuO in $OB$-IQuO and $BA$-IQuO is originated with the simultaneous creation of another IQuO, $B$-IQuO.\\
So the splitting will produce two connecting IQuO ($OB$-IQuO and $BA$-IQuO) and the distance between $O$ and $A$ will become the double than in the past: $OA = OB + BA = 2 \lambda$.\\
After the first splitting, in the next time interval $\tau$ it is possible a further doubling of IQuO connecting (see figure \ref{fig_iquo_expansion01}).\\
Object $A$ goes away from $O$ with a recession velocity given by:
\begin{displaymath}
v_{OA} = \frac{\Delta_{OA} (\tau)}{\tau}
\end{displaymath}
We observe that $v_{OA} = 2 v_{OB}$, because $OA = 2 OB$.\\
The doubling process of any IQuO goes on in time, determining an IQuO-chain in continuous growth (see figure \ref{fig_iquo_expansion01}), where velocity $v_{OA}$ doubles to the doubling of distance.\\
% \label{fig_iquo_expansion01}
This will involve in an expansion phenomenon (elongation of the line $OA$) described in figure \ref{fig_iquo_expansion02}, analogous to the elongation of the line $OA$ described mathematically by the expansion parameter $a(t)$ present in the Friedman's equations \cite{friedmann1922,friedmann1924}.\\
% \label{fig_iquo_expansion02}
This mechanism of IQuO creation and duplication, with resulting the creation of space and mass, depends to the presence of a \emph{fund} of energy composed by
not coupled uncoupled sub-oscillators. This fund it coincident with the no-field $\Theta$.\\
Therefore this $\Theta$-background provide the energy for the construction and the expansion of
$\underline{\Xi}^\circ$, $\Xi^0_i$, and for the creation of the particles, that are geometrical structures in
$\Xi$ $\Xi^0$ and
$\underline{\Xi}$.\\
The creation of an IQuO chain $\underline{\Xi}^\circ$, or better an $\underline{\Xi}^\circ$-lattice, could be caused by the coupling between two or more empty sub-oscillators of the $\Theta$-background, such that they create an $\underline{\Xi}^\circ$-IQuO that, subsequently, could couples with one or more empty sub-oscillators\footnote{articolo arXiv} of the $\Theta$-background.\\
If any quantum field $\Phi$ consists by coupling oscillators (excited or in vacuum), $\Theta$ could be considered like a no-field, because is formed by uncoupled sub-oscillators with not-null energy: $\Theta$, in this way, becomes an energy reservoir to create and feed the scalar fields $\Xi$, $\underline{\Xi}$ of the universe and so indirectly the other field of the Universe, included the Higgs field\footnote{Dovevamo inserire una nota o un riferimento?}.\\ $\Xi$.\\
In conclusion, whatever it could be the mechanism of creation of IQuOs with massive coupling, the expansion of the space is caused by mutual coupling of the sub-oscillators of $\Theta$ or between IQuOs belonging to any chain of the basic field
$\underline{\Xi}$ $\Xi$ and the $\Theta$-background.\\
For now, we do not specify the mechanism of creation of an IQuO and about the doubling of the connecting IQuOs.\\
In the next time interval $\tau$ it is possible a further doubling of IQuO connecting caused by the same mechanism mentioned above.\\
We denote by \emph{cell space} the connecting IQuO. Now the number of added cells ($OC$, $CB$, $BD$, $DA$) will become double than the previous one. This will involve a expansion phenomenon (elongation of the line $OA$) described in Figure 3, analogous to the elongation of the line $OA$ described mathematically by the expansion parameter $a(t)$ present in the Friedman's equations.\\ The homogeneity of the elongation will be guaranteed if for each space \emph{cell}, an additional cell is originated for doubling, as mentioned above and a shown in figure 3-b, where we show an increasing massive
chain $\underline{\Xi}^\circ$; chain; it should be noted immediately that this is only possible if $\Theta$-background is isotropic and
homogeneous.\\
The recession velocity is given by:
\begin{displaymath}
v_{OA} = \frac{\Delta_{OA} (\tau)}{\tau}
\end{displaymath}
We observe homogeneous, like an expanding gas.\\
Even we can admit that
$v_{OA} = 2 v_{OB}$, because $OA = 2 OB$.\\
We also note between two whatever connecting IQuOs, a $B$-IQuO is generated and powered by the background $\Theta$. So the $B$-IQuO becomes source of half-quanta flowing in $\Xi^0$ which can induce the already described doubling process or even become source for the enhancement of the elastic tension during the massive coupling which generates the \emph{curvature} into chains, or in other words the \emph{gravitation}.\\
% \label{fig_iquo_expansion03}
So we can affirm that the B-IQuO can be considered as the physical object through which the energy of the $\Theta$-background flows forming chains of
$\underline{\Xi}^\circ$ $\Xi$ increasing in
space-mass. space and mass (gravitation). A useful analogy could be a marine volcano that erupting lava it gradually builds itself and, on the sea's surface, builds the island.\\
Similarly, the universe has built itself by taking energy from the
$Theta$-background $\Theta$-background to build a lattice based on
$\underline{\Xi}^\circ$, $\Xi_U$, particle-fields, and galaxies (like islands). From the analogy cited before, we deduce that supermassive blacks holes could be the source objects that build the universe, because they are at the center of almost all
galaxies\footnote{cercare riferimenti}. It remains to determine how the blacks holes feed the $\underline{\Xi}^\circ$-lattice. galaxies \cite{merritt2013}. We remind you that recently someone begins to argue that blacks holes are not so
black.\\
So, we should admit the possibility of having the IQuO-sources, along already existing lines (B-IQuO, C-IQuO, ...).\\
(figure)\\ black \cite{hawking2014}.\\
If we repelace
to O-, B-, A-IQuO we replace $O$-, $B$-, $A$-IQuO with galaxies then we gets an expanding universe where the recession velocity from
O $O$ increases with the distance between them.\\
From the figure we observe that the increase in velocity follows the sequence $v$, $2v$, $4v$, $8v$, $16v$, $\cdots$, corresponding to the distances $d$, $2d$, $4d$, $8d$, $\cdots$.\\
Mathematically, the recession velocity follows the following law:
\begin{equation}
v_{n,0} = \frac{2^n \lambda}{n \tau}
...
v = H(t) \cdot d
\end{equation}
with $H$ a physical quantity that represents the rate of space increasing.\\
It is immediately clear that the scale factor describing the expansion in cosmology is $d(t) = 2^\frac{1}{\tau} \lambda$.\\
However it should be noted that this expansion is \emph{exponential}, but it proves to be the only one that maintains the symmetry in every IQuO-point.\\
It is evident that only a uniform distribution of IQuO-sources and the doublings of IQuO produces the Hubble law.\\
A different mode of formation of the IQuO-sources and their distribution, can determine different forms of the Hubble function, such as:\\
(figure)\\
It follows then that only a doubling in every point ensures homogeneous and isotropic expansion.\\
The creation of space-cells in a space segment between any two objects is equivalent to a mutual spacing of the extremes for which it will be that velocity of expansion, $v_{\text{exp}}$, is equal to velocity of recession, $v_{\text{rec}}$, as it placed in the standard cosmology.\\
We also must observe that the Doppler effect of the light propagating along an IQuO-chain is dued by the recession between the extrems of the chain, but also by the time delay dued by the doubling of the number of sub-oscillators that the photon is going through.
