The relativistic theory of gravitation (RTG) [ 1,2,3 ] disagrees with the Einstein's general relativity (GR) in the crucial point: it denies the total geometrization and considers the gravitation on the basis of the classical Faraday-Maxwell's field approach. This means that there is a topologically simple background spacetime of Minkowski type, which can be restored in any situation. As a result, we can detach the physical content from an arbitrary geometrical game with co-ordinates. This converts the gravitation from the tensor-geometrical concept to the tensor-field one and puts it on the unified level with another fields. [26]
Formally, the RTG can be considered as the bi-metric theory of gravitation [ 4, 5 ]. However, in the RTG the effective Riemannian spacetime produced by the gravitational field is essentially separated from the Minkowski background because the latter is presented in the field equations. Naturally, this transforms the solutions of the field equations and has the pronounced physical consequences. For example, the singularity disappears and the graviton acquires the nonzero mass. Nevertheless, the basic observational consequences of the RTG coincide with those in the GR (for instance, Mercury perihelion motion, time delay and spectral shift in the gravitational field, see [2]).
The application of the RTG for the cosmology produces some astonishing results, viz., in virtue of the field equations the Friedmann-Robertson-Walker cosmology admits only the flat global efficient Riemannian spacetime without initial singularity and with oscillating time behavior [ 2, 3 ]. The initial cosmological expansion is stimulated by the antigravitation, which is caused by the massive gravitons in the strong gravitational fields. The initial temperature is defined by the graviton mass and can be too low to create the undesirable relics (e.g. monopoles). [19] So, the problems of the cosmological spacetime flatness, the source of the initial expansion, the cosmological singularity and the absence of the relics find in the RTG a natural solution. However, in this theory there are some disagreements with the modern observational data. As it is known, the latter suggests the accelerated expansion of the universe at present (see, for example, [ 6, 7 ]). But in the RTG the accelerated expansion is possible only during a very short stage of the initial evolution and the subsequent expansion has a definitely decelerated character.
As it is well known, the accelerated cosmological expansion in the framework of the GR can be obtained "by hand" due to an insertion of the so-called cosmological constant in the field equations (for a review see [ 8 ]). This constant can be considered as a part of the geometrical structure of the GR because it is a natural consequence of the variational principle [ 9 ]. Alternatively, it is possible to treat the cosmological constant as the vacuum zero-point energy. But in the both cases its value is too small and can not be attributed to any known physical scale. The situation in the RTG is more complicated by virtue of the vacuum stability principle: the absence of the material fields reduces the effective Riemannian spacetime to the Minkowski one. Hence, the cosmological constant can not be introduced by hand and is to have the gravitational nature concerned with the nonzero graviton mass. As a result, the cosmological constant-like action of the massive graviton in the RTG produces the deceleration of the cosmological expansion. Nevertheless there exists an approach, which considers the accelerated expansion of the universe as a manifestation of some matter possessing an unusual state equation p = wp (where p is the pressure and p is the density). This matter usually is called as X-matter or quintessence. If its state parameter w lies between the limits of the strong and week energy conditions the domination of such matter produces a repulsion causing the accelerated expansion of the universe [ 10 ]. The best candidate here is a certain scalar field whose potential energy dominates at present (the survey can be found in [ 11 ], for example). In this article we shall consider the implementation of this idea in the RTG framework. As a result, some restrictions on the key parameter of the theory, i.e. the graviton mass, will be obtained.