Machian Cosmological Solution in the Generalized Scalar-Tensor Theory of Gravitation

Abstract

The Machian cosmological solution satisfying $ϕ=O(ρ/ω)$ is discussed for the homogeneous and isotropic universe with a perfect fluid (with negative pressure) in the generalized scalar-tensor theory of gravitation. We propose $ω(ϕ)=η/(ξ-2)$ for the coupling function in the Machian point of view. The parameter $ξ$ varies in time very slowly from $ξ=0$ to $ξ=2$ because of the physical evolution of matter in the universe. When $ξ\to 2$, the coupling function diverges to $-\infty$ and the scalar field $ϕ$ converges to $G_{\infty}^{-1}$. The present mass density is precisely predicted if the present time of the universe is given. We obtain $ρ_{0}=1.6\times 10^{-29} g.cm^{-3}$ for $t_{0}=1.5\times 10^{10} yr$. The universe shows the slowly decelerating expansion for the time-varying coupling function.