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We attempt to find new symmetries in the space-time structure, leading to amodified gravitation at large length scales, which provides the foundations ofa quantum gravity at very low energies. This search begins by considering aunified model for electrodynamics and gravitation, so that the influence of thegravitational field on the electrodynamics at very large distances leads to areformulation of our understanding about space-time through the elimination ofthe classical idea of rest at quantum level. This leads us to a modification ofthe relativistic theory by introducing the idea of a universal minimum speedrelated to Planck minimum length. Such a speed, unattainable by the particles,represents a privileged inertial reference frame associated with a universalbackground field. The structure of space-time becomes extended due to such avacuum energy density, which leads to a cosmological anti-gravity, playing therole of the cosmological constant. The tiny values of the vacuum energy densityand the cosmological constant are successfully obtained, being in agreementwith current observational results. We estimate the very high value of vacuumenergy density at Planck length scale. After we find the critical radius of theuniverse, beyond which the accelerated expansion takes place. We show that sucha critical radius is $R_{uc}=r_g/2$, where $r_g=2GM/c^2$, being $r_g$ theShwarzschild radius of a sphere with a mass $M$ representing the totalattractive 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 acceleratedexpansion. For $R_u>R_{u0}$, the rate of acceleration decreases to zero at theinfinite, avoiding Big Rip.

Foundations of a quantum gravity at large scales of length and its consequences for the dynamics of cosmological expansion