Exact cosmological solution of a Scalar-Tensor Gravity theory compatible with the $\mathrm{\Lambda }CDM$ model

We consider the massive scalar-tensor theory in the Jordan frame $F(\Phi) =K^{2}\Phi^2$and$U(\Phi) =(1/2)m^{2}\Phi^2$,where$F(\Phi)$ corresponds to a constant Brans-Dicke parameter ${\omega }_{BD}=1\mathrm{/}4K^2$. The constraint of the Solar System experiments is $K^2<(1\mathrm{/}400{)}^2$. For dustlike matter in a spatially flat homogeneous isotropic universe, we reduce the equations of motion to a system of two differential equations of first order which can be exactly solved. We obtain simple and explicit expressions for $\frac{\mathrm{\Phi }(z)}{\mathrm{\Phi }(0)}$ and $\frac{H(z)}{H_0}$ that depend only on two parameters, $K^2$ and ${\mathrm{\Omega }}_{m,0}$. For $K\le 1\mathrm{/}400$ the expansion rate $H(z)$ can be practically superposed on the $\mathrm{\Lambda }$CDM solution $H_{\mathrm{\Lambda }}(z)$, up to high redshift $z$, but the equation of state $w_{DE}(z)$ of the dark energy is not constant: it presents a very slight crossing of the phantom divide line $w=-1$ in the neighborhood of $z=0$ and becomes very slightly positive at high redshifts.