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

Abstract

The cosmological constant $(1/2){\lambda }_1{\varphi }_{,\mu }{\varphi }^{,\mu }/{\varphi }^2$ is introduced to the generalized scalar-tensor theory of gravitation with the coupling function $\omega (\varphi )=\eta /(\xi -2)$ and the Machian cosmological solution satisfying $\varphi =O(\rho /\omega )$ 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 for $0\leqq \xi <2$ leads to $\eta >3$. If ${\lambda }_1<0$ and $0\leqq -\eta /{\lambda }_1<2$, the universe shows the slowly accelerating expansion. The coupling function diverges to $-\infty$ and the scalar field $\varphi$ converges to $G_{\infty }^{-1}$ when $\xi \to 2$ ($t\to +\infty$). The cosmological constant decays in proportion to $t^{-2}$. Thus the Machian cosmological model approaches to the Friedmann universe in general relativity with $\ddot{a}=0$, $\lambda =0$, and $p=-\rho /3$ as $t\to +\infty$. General relativity is locally valid enough at present.

10 pages, LaTeX2e