\label{universe_formation} The relations identifying any increasing lattice-universe are given by
\[\begin{aligned} R & = n \lambda\\ t & = n \tau \label{eq_tau}\\ H & = \frac{1}{t}\\ N & = n^2 \label{eq_particles}\end{aligned}\]
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: \[\label{gravitationalcoupling}
\alpha_i = G \frac{m_i^2}{\hbar c}\] We also noted that \[\left ( \frac{m_i}{\lambda_i^3} \right )_{U_i} = m_i \frac{m_i^3 c^3}{\hbar^3} = \frac{m_i^2 c^4}{\hbar^2} \frac{m_i^4}{c \hbar} = \frac{1}{\tau^2} \frac{\alpha_i}{G}\] The first universe-lattice \(U_i\) originated after the the birth of the spacetime could be built on the value of the Planck mass: \[m_{Pl} = \sqrt{\frac{\hbar c^5}{G}}\] It will be denoted \(U_{Pl}\), the Planck’s Universe.
For another universe-lattice \(U_i\), with \(m_i < m_{Pl}\), we have: \[\label{eq-density}
\rho (n) = \frac{N m_i}{V} = \frac{3}{4 \pi} \frac{N m_i}{(n \lambda_i)^3} = \frac{3 \alpha_i}{4 \pi n G \tau^2}\] While the value of the critical density of each specific \(U_i\) will be given by \[\label{eq-critical_density}
\rho_C (n) = \frac{3}{8 \pi} \frac{H^2}{G} = \frac{3}{8 \pi n^2 G \tau^2}\] 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 \[\label{timesteps}
n_i = \frac{1}{\alpha_i} \equiv n_{\alpha_i}\] Thus it follows \[\rho^{U_i} \left ( n_{\alpha_i} \right ) = \rho^{U_i}_C\] We point out that \[\rho^{U_i} \left ( n < n_{\alpha_i} \right ) < \rho^{U_i}_C\] For \(n < n_i\), \(U_i\) is an open lattice-universe. Instead for \(n > n_i\) \[\rho^{U_i} \left ( n > n_{\alpha_i} \right ) > \rho^{U_i}_C\] and \(U_i\) is a closed lattice-universe.
So, by varying \(n\), we have the development of density prictured by the figure \ref{fig_density}.
The time \(\tau_C\), the critical age, need to reach the critical density will be given by \[\tau^{U_i}_C = n_{i, C} \, \tau_i = n_{\alpha_i} \tau_i = \frac{\tau_i}{\alpha_i}\] We find that \[\tau^{U_i}_C = \frac{1}{2} \frac{\hbar}{m_i c^2} \frac{\hbar c}{G m_i^2} = \frac{\hbar^2}{2 G m_i^3 c}\] 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
\[\begin{aligned} \tau_c (U_e) & = \frac{\hbar^2}{2 G m_e^3 c} \approx 3.69 \cdot 10^{23} s \approx 1.2 \cdot 10^{16} Y\\ \tau_c (U_p) & = \frac{\hbar^2}{2 G m_p^3 c} \approx 5.9 \cdot 10^{23} s \approx 1.9 \cdot 10^{6} Y\end{aligned}\]
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 \[\frac{\rho^p_C}{\rho^e_C} = \frac{3 H_p^2}{8 \pi G} \frac{8 \pi G}{3 H_e^2} = \frac{H_p^2}{H_e^2}\] From (\ref{gravitationalcoupling}) and (\ref{timesteps}) we obtain \[\begin{aligned}
\frac{\alpha_p}{\alpha_e} & = \frac{m_p^2}{m_e^2}\\
\frac{\alpha_p}{\alpha_e} & = \frac{n_{eC}}{n_{pC}}\end{aligned}\] and therefore \[n_{p, C} m_p^2 = n_{e, C} m_e^2\] Expressing the mass through \(\lambda_{p, e}\), \[n_{p, C} \left ( \frac{\hbar}{\lambda_p c} \right )^2 = n_{e, C} \left ( \frac{\hbar}{\lambda_e c} \right )^2\] it follows \[n_{p, C} \lambda_e^2 = n_{e, C} \lambda_p^2\] Now we set \[\label{comptonsurface}
4 \pi \lambda_c^2 = S_c\] the Compton surface of a particle. We will then \[n_{p, C} S_e = n_{e, C} S_p\] In general we will have \[\label{unverse_surfaces}
n_i S_j = k n_j S_i\] 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 \(k\)) \[\label{criticaluniverses}
n_i \lambda_j^2 = n_j \lambda_i^2\] Taking the squareroot in (\ref{criticaluniverses}) we have: \[\sqrt{n_i} \lambda_j = \sqrt{n_j} \lambda_i\] that we can rewrite as \[n_i \lambda_j = \sqrt{n_i} \sqrt{n_j} \lambda_j\]