S.S. Gershtein, A.A. Logunov, and M.A. Mestvirishvili UPPER LIMIT ON THE GRAVITON MASS

The problem of the existence of nonzero invariant mass of the graviton can have the fundamental significance. The estimate for the upper bound on the graviton mass (m g < 2 · 10 −62) has been derived in Ref. [1], where the authors used the data on the existence of the gravitational coupling between the galaxy clusters, which is not cut off by the Yukawa potential at least up to distances ∼500 Kpc. In this paper we will present the estimates for upper limit on the graviton mass basing on the observable parameters of the Universe expansion. The fact that in this case typical distances are 3-4 orders of magnitude larger than those between gravitationally bound galaxy clusters allows one to strengthen the estimates on the upper limit on the graviton mass by few orders of magnitude, respectively. It should be noted that introduction of the nonzero invariant mass of the graviton requires to go beyond the General Theory of Relativity (GTR). This can be done naturally by using the notions of the gravitational field in the Minkowsky space [2,3]. In Ref. [3] the complete energy-momentum tensor t µν (including the gravitational field), which is conserved in the Minkowsky space, is considered as a source of the gravitational field described by symmetric tensor Φ µν. In arbitrary fixed (not necessarily inertial) frame of the Minkowsky space with metric tensor γ µν the equations for the density of the gravitational field˜Φ µν can be written analogously to the Maxwell equations and Lorentz condition for the electromagnetic field as follows: (γ αβ D α D β + m 2 g) ˜ Φ µν = 16π˜t µν , (1) D ν ˜ Φ µν = 0 , (2) where D α is the covariant derivative in the Minkowsky space, m g is the graviton mass (¯ h = c = G = 1), and˜Φ µν , ˜ t µν are the densities of tensors: ˜ Φ µν = √ −γΦ µν , ˜ t µν = √ −γt µν , γ = det(γ µν) = det(˜ γ µν). (3) Condition (2) singles out polarization states with spin values of 2 and 0 and provides the conservation of the density of the energy-momentum tensor 1