The Universe evolution is thoroughly studied in this article where the gravitational field is treated as a physical field of spins 2 and 0 and its source is the conserving in Minkowski space total energy-momentum tensor of matter and gravitational field. It is shown that the cosmological singularity is absent in the framework of this approach and that the spatial geometry of the Universe should be flat. By using the Causality Principle which is to be fulfilled by the evolution of the physical gravitational field we demonstrate that the expansion of the Universe cannot be infinite. The Universe develops cyclically. Therefore the theory is not compatible with the presence of constant cosmological term and with the phantom expansion. The presence of "quintessence" which density is falling as scale factor rises (slower than const/a²) is necessary to explain the observable accelerated expansion. Starting from data on anisotropy of the microwave relic radiation we obtain the upper limit for graviton mass, which is 5000 times stronger than existent estimations, and also the possible value of graviton mass in the experimental error corridor is given. The moments of time corresponding to the beginning and the end points of the present acceleration stage are determined from this value as well as the moment of maximum Universe expansion which later is replaced by contraction till some maximum density value r_max, this is another parameter of the theory along with the graviton mass.

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