\label{our_universe} The time \(\tau_c\) need to reach the critical density of the flat-universe will be derived from the system of relations \[\left \lbrace
\begin{array}{cc}
R^*_{(p,e)} = & k n_{p, C} \lambda_e\\
R^*_{(p,e)} = & c \tau_C^{(p,e)}
\end{array} \right.\] it follows: \[\tau_C^{(p,e)} = \left ( \frac{k n_{p, C} \lambda_e}{c} \right )_{k=4}\] Remembering the relation between \((n_c)_i\) and \(\alpha_i\), we find \[\tau_C^{(p,e)} = \frac{4 n_{p, C} \lambda_e}{c} = \frac{2 \lambda}{c \alpha_p}\] Approximate values of the constants \cite{codata1987} in the fifth decimal place, we get the critical time of the universe \(U^*\): \[\tau_C^{(p,e)} = 2 \frac{3.86159 \cdot 10^{-13} \cdot 0.31688 \cdot 10^{-7}}{2.99792 \cdot 10^8 \cdot 5.90464 \cdot 10^{-39}} Y = 13.82539 \cdot 10^9 Y\] 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 \cite{planck2013}.
If our universe U is approximated by \(U^*_{p,e}\) then the critical age \(\tau_C^{(p,e)}\) would be very close (if not identical) to age \(\tau\) of the universe \(U\).
The observations \cite{planck2013} give an age of the universe equal to \[\tau_U \approx (13.82 \pm 0.20) \cdot 10^9 Y\] The data shows that the value of \(\tau_C^{(p,e)}\) coincides with age found by Planck. Therefore, by comparing the two times, we could assert that our universe may be in a critical phase with a flat metric, as is evidenced by the astronomical observations which really shown a universe with a flat metric. By the critical time value \(\tau_C^{(p,e)}\) (according to \ref{eq_tau}) a value of the Hubble’s constant \(H^*\) is given by: \[H_c^* = \frac{1}{\tau_C^{(p,e)}} = 70.78 \frac{km}{Mpc \cdot s}\] The \(H_c^*\) value is compared so to one experimental \(H_{exp} = (73 \pm 3) km/(Mpc \cdot s)\) derived by observations of distant galaxies by WAMP satellite \cite{wmap2007}, to which is given an universe old of \(\tau_U = (13.72 \pm 0.20) \cdot 10^9 Y\) and with critical density \(\rho_c \approx 9.58 \cdot 10^{-27} Kg/m^3\). So, from the experimental data, it follows that the \(H\) value derived from the distant galaxies (past time) is larger that the one derived from the nearest galaxises (present time).
Even if we take into account the experimental uncertainty is evident that \(H_c^*\) is closer to the value \(H_0 = (72 \pm 3 \pm 7) Km/(Mpc \cdot sec)\) derived from the analysis of Cepheids \cite{hst2001} and we conclude that the current universe is flat. This means that in the past the recession velocity would have been greater than that of the present time (nearby galaxies).
In a universe with variable metric \(U^*\) the transition from open metric to flat should obtained itself by the diagram red shift/galactic distance a greater gradient for remote galaxies (Fig. 16, curve 5) than the one (curve 3) provided for a universe in constant (flat) metric
However (as shown in figure \ref{fig_hubble_law_scp}) we see that the observations place galaxies more distant (curves 4-6) than those of a universe with an open metric (curves 5). This means that it would be an additional acceleration to that provided in a universe with an open metric. Here we talk of accelerating expansion \(a_{exp}\) of the Universe.
Now we calculate the critical density of the universe \(U^*\) \[\label{criticaldensity}
\rho_C^* = \frac{3}{8 \pi G \left ( \tau_C^{(p,e)} \right )^2} = 9.409 \cdot 10^{-27} \frac{kg}{m^3}\] that is compatible with the experimental critical density \cite{pdg2014} \(\rho_{\text{crit}} = 8.5 \cdot 10^{-27} kg / m^3\).
Now we can also calculate the radius \(R^*\): \[R^* = c \tau_C^{(p,e)} = 13.04 \cdot 10^{25} m\] By definition the current mass density of the universe depends on its amount of total mass (expressed in number of protons). We also known that \(\alpha_p\) provides us the number of temporal steps needed to reach the critical phase. Combining (\ref{timesteps}) with (\ref{eq_particles}) we can obtain the number of protons present in \(U^*\) when it will reach the critical phase: \[N_{pC} = \frac{1}{4\alpha_{pC}} = 7.17 \cdot 10^{75}\] And so the protonic density would be: \[\rho_{pC} = \frac{N_{pC} m_p}{V^*} = 1.611 \cdot 10^{-29} \frac{kg}{m^3}\]