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

The Machian cosmological solution satisfying $\phi =O(\rho /\omega)$ 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 $\omega (\phi)=\eta /(\xi -2)$ for the coupling function in the Machian point of view. The parameter $\xi$ varies in time very slowly from $\xi =0$ to $\xi =2$ because of the physical evolution of matter in the universe. When $\xi \to 2$, the coupling function diverges to $-\infty$ and the scalar field $\phi$ converges to $G_{\infty}^{-1}$. The present mass density is precisely predicted if the present time of the universe is given. We obtain $\rho_{0}=1.6\times 10^{-29} g.cm^{-3}$ for $t_{0}=1.5\times 10^{10} yr$. The universe shows the slowly decelerating expansion for the time-varying coupling function.