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It is shown that using the relativistic field theory of gravity (RTG) andmeasured value of $\Omega_{tot}$ one can obtain the upper limit on the gravitonmass with 95%C.L.: $m\leq 1.6\cdot 10^{-66}$ [g]; within the $(1\sigma)$ rangeits probable value is $m_{g}= 1.3\cdot 10^{-66}$ [g]. It is pointed out thataccording to RTG the presence of the quintessence is necessary to explain theUniverse accelerated expansion. Experimental data on the Universe age and darkmatter density allow one to determine the range of possible values of the $\nu$parameter in the equation of quintessence state and indicate characteristictime, which corresponds to the beginning and cessation of the acceleratedexpansion epoch, as well as the time period of the maximal expansion, whichcorresponds to the half-period of the oscillatory evolution of the Universe.Comment: 17 pages, 4 jpeg figure

Graviton mass, quintessence, and oscillatory character of Universe evolution