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\section{The lattice-universe}  \subsection{The formation of the lattice-universe}  Whatever the mechanism of mass-space's accretion in the universe, the relations identifying any \emph{increasing} lattice-universe are given by  \begin{align}  R & = n \lambda\\  t & = n \tau \nonumber\\  H & = \frac{1}{t} \nonumber\\  N & = n^2 \nonumber  \end{align}  The increase of massive particles $m_i$ implies an increasing space and then an increase of the radius $R$ of the universe $U_i$ with spatial step given by $\lambda_i$.\\  The gravitational coupling constant $\alpha_i$takes the value:  \begin{equation}\label{gravitationalcoupling}  \alpha_i = G \frac{m_i^2}{\hbar c}  \end{equation}  We also noted that  \begin{equation}  \left ( \frac{m_i}{\lambda_i^3} \right )_{U_i} = m_i \frac{m_i^3 c^3}{\hbar^3} = \frac{m_i^2 c^4}{\hbar^2} \frac{m_i^4}{c \hbar} = \frac{1}{\tau^2} \frac{\alpha_i}{G}  \end{equation}  The first universe-lattice $U_i$ originated after the the birth of the spacetime could be built on the value of the Planck mass:  \begin{equation}  m_{Pl} = \sqrt{\frac{\hbar c^5}{G}}  \end{equation}  It will be denoted $U_{Pl}$, the Planck's Universe.\\  For another universe-lattice $U_i$, with $m_i < m_{Pl}$, we have:  \begin{equation}  \rho (n) = \frac{N m_i}{V} = \frac{3}{4 \pi} \frac{N m_i}{(n \lambda_i)^3} = \frac{3 \alpha_4}{4 \pi n G \tau^2}  \end{equation}  While the value of the critical density of each specific $U_i$ will be given by  \begin{equation}  \rho_C (n) = \frac{3}{8 \pi} \frac{H^2}{G} = \frac{3}{8 \pi n^2 G \tau^2}  \end{equation}  It should be noted that the universe $U_i$ reaches to the critical density when its the number of time steps $n_i$ is such that occurs  \begin{equation}\label{timesteps}  n_i = \frac{1}{\alpha_i} \equiv n_{\alpha_i}  \end{equation}  Thus it follows  \begin{equation}  \left ( \rho \left ( n_{\alpha_i} \right ) \right )_{U_i} = \left ( \rho_C \right )_{U_i}  \end{equation}  We point out that  \begin{equation}  \left ( \rho \left ( n < n_{\alpha_i} \right ) \right )_{U_i} < \left ( \rho_C \right )_{U_i}  \end{equation}  For $n < n_i$, $U_i$ is an open lattice-universe. Instead for $n > n_i$  \begin{equation}  \left ( \rho \left ( n > n_{\alpha_i} \right ) \right )_{U_i} > \left ( \rho_C \right )_{U_i}  \end{equation}  and $U_i$ is a closed lattice-universe.\\  So, by varying $n$ we have the following development of density:\\  (figure)\\  The time $\tau_C$, the \emph{critical age}, need to reach the critical density will be given by  \begin{equation}  \tau_C (U_i) = (n_i)_C \tau_i = n_{\alpha_i} \tau_i = \frac{\tau_i}{\alpha_i}  \end{equation}  We find that  \begin{equation}  \tau_C (U_i) = \frac{1}{2} \frac{\hbar}{m_i c^2} \frac{\hbar c}{G m_i^2} = \frac{\hbar^2}{2 G m_i^3 c}  \end{equation}  As examples, let us consider the lattice-universe of the electron, $U_e$, and the lattice-universe of the proton, $U_p$, finding the critical times  \begin{align}  \tau_c (U_e) & = \frac{\hbar^2}{2 G m_e^3 c} \approx 3.69 \cdot 10^{23} s \approx 1.2 \cdot 10^{16} Y\\  \tau_c (U_p) & = \frac{\hbar^2}{2 G m_p^3 c} \approx 5.9 \cdot 10^{23} s \approx 1.9 \cdot 10^{6} Y  \end{align}  It should be noted that the first time it is enormously larger than the likely present age of the universe $U$, while the second is much smaller.\\  We compare the two lattice-universes; in the critical phase it is  \begin{equation}  \frac{(\rho_C)_p}{(\rho_C)_e} = \frac{3 H_p^2}{8 \pi G} \frac{8 \pi G}{3 H_e^2} = \frac{H_p^2}{H_e^2}  \end{equation}  From (\ref{gravitationalcoupling}) and (\ref{timesteps}) we obtain  \begin{align}  \frac{\alpha_p}{\alpha_e} & = \frac{m_p^2}{m_e^2}\\  \frac{\alpha_p}{\alpha_e} & = \frac{n_{eC}}{n_{pC}}  \end{align}  and therefore  \begin{equation}  n_{pC} m_p^2 = n_{eC} m_e^2  \end{equation}  Expressing the mass through $\lambda_{p, e}$,  \begin{equation}  n_{pC} \left ( \frac{\hbar}{\lambda_p c} \right )^2 = n_{eC} \left ( \frac{\hbar}{\lambda_pe c} \right )^2  \end{equation}  it follows  \begin{equation}  n_{pC} \lambda_e^2 = n_{eC} \lambda_p^2  \end{equation}  Now we set  \begin{equation}\label{comptonsurface}  4 \pi \lambda_c^2 = S_c  \end{equation}  the \emph{Compton surface} of a particle. We will then  \begin{equation}  n_{pC} S_e = n_{eC} S_p  \end{equation}  In general we will have  \begin{equation}\label{unverse_surfaces}  n_i S_j = k n_j S_i  \end{equation}  Where we introduce a numerical constant of proportionality, $k$, related to the various physical cases that we can take into consideration. As well as (omitting for the moment the constant $k$)  \begin{equation}\label{criticaluniverses}  n_i \lambda_j^2 = n_j \lambda_i^2  \end{equation}  Taking the squareroot in (\ref{criticaluniverses}) we have:  \begin{equation}  \sqrt{n_i} \lambda_j = \sqrt{n_j} \lambda_i  \end{equation}  that we can rewrite as  \begin{equation}  n_i \lambda_j = \sqrt{n_i} \sqrt{n_j} \lambda_j  \end{equation}