We attempt to find new symmetries in the space-time structure, leading to a modified gravitation at large length scales, which provides the foundations of a quantum gravity at very low energies. This search begins by considering a unified model for electrodynamics and gravitation, so that the influence of the gravitational field on the electrodynamics at very large distances leads to a reformulation of our understanding about space-time through the elimination of the classical idea of rest at quantum level. This leads us to a modification of the relativistic theory by introducing the idea of a universal minimum speed related to Planck minimum length. Such a speed, unattainable by the particles, represents a privileged inertial reference frame associated with a universal background field. The structure of space-time becomes extended due to such a vacuum energy density, which leads to a cosmological anti-gravity, playing the role of the cosmological constant. The tiny values of the vacuum energy density and the cosmological constant are successfully obtained, being in agreement with current observational results. We estimate the very high value of vacuum energy density at Planck length scale. After we find the critical radius of the universe, beyond which the accelerated expansion takes place. We show that such a critical radius is $R_{uc}=r_g/2$, where $r_g=2GM/c^2$, being $r_g$ the Shwarzschild radius of a sphere with a mass $M$ representing the total attractive mass contained in our universe. And finally we obtain the radius $R_{u0}=3r_g/4(>R_{uc})$ where we find the maximum rate of accelerated expansion. For $R_u>R_{u0}$, the rate of acceleration decreases to zero at the infinite, avoiding Big Rip.
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