Varying Cosmological Constant and the Machian Solution in the Generalized Scalar-Tensor Theory

Abstract

The cosmological constant $(1/2)λ_{1}ϕ_{, μ}ϕ^{, μ}/ϕ^{2}$ is introduced to the generalized scalar-tensor theory of gravitation with the coupling function $ω(ϕ)=η/(ξ-2)$ and the Machian cosmological solution satisfying $ϕ=O(ρ/ω)$ is discussed for the homogeneous and isotropic universe with a perfect fluid (with negative pressure). We require the closed model and the negative coupling function for the attractive gravitational force. The constraint $% ω(ϕ)<-3/2$ for $0\leqq ξ<2$ leads to $η>3$. If $λ_{1}<0$ and $0\leqq -η/λ_{1}<2$, the universe shows the slowly accelerating expansion. The coupling function diverges to $-\infty $ and the scalar field $ϕ$ converges to $G_{\infty}^{-1}$ when $ξ\to 2$ ($t\to +\infty $). The cosmological constant decays in proportion to $t^{-2}$. Thus the Machian cosmological model approaches to the Friedmann universe in general relativity with $\ddot{a}=0$, $λ=0$, and $p=-ρ/3$ as $t\to +\infty $. General relativity is locally valid enough at present.