L'Univers en expansion et probl\`emes d'\'energie

In this paper we first construct a mathematical model for the Universe expansion that started up with the original Big Bang. Next, we discuss the problematic of the mechanical and physical laws invariance regarding the spatial frame exchanges. We then prove the (theoretical) existence of a variable metric $g_t$, depending on time and satisfying to a simplified Einstein equation, so that all free ordinary trajectories are geodesics. This is done by considering the classical Galileo-Newtonian space and time relation, by using generalized Newtonian principles and adding the approved physical new ones (as covariance principle, Mach principle, the Einstein equivalence principle ...) in order to establish a new cosmological model of the dynamical Universe as being $(U(t))_{t\geq 0}=(B_e(O,t),g_t)_{t\geq 0}$, where $B_e(O,t)$ is the Euclidean ball of radius $t$ in $\mathbb R^3$. The cosmological metric $g_t$ is totally determined, at time $t$, by the mass energy distribution $E_t(X)$ on $B_e(O,t)$. We study also the black holes phenomenon and we prove that the total and global cosmological energy distribution $E_t(X)$ satisfies to the wave equation whose solutions are characterized by pseudo-frequencies depending on time and related to the spectrum of the Dirichlet problem on the unit ball $B_e(O,1)$ for the Laplace-Beltrami operator -$\Delta$. Our model is consistent in the sense that all Newtonian and classical physical laws are valid as particular cases in classical situations. Then, we show that all basic results of Modern Physics are still valid without using the second postulate of special relativity nor the uncertainty principle.