Exact cosmological solution of a Scalar-Tensor Gravity theory compatible with the $ΛCDM$ model

We consider the massive scalar-tensor theory in the Jordan frame $F(Φ) =K^{2}Φ^2$ and $U(Φ) =(1/2)m^{2}Φ^2$, where $F(Φ)$ corresponds to a constant Brans-Dicke parameter $ω_{BD}=1/4K^2$. The constraint of the Solar System experiments is $K^2<(1/400)^2$. 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 $\frac{Φ(z)}{Φ(0)}$ and $\frac{H(z)}{H_{0}}$ that depend only on two parameters, $K^2$ and $Ω_{m,0}$. For $K\leq1/400$ the expansion rate $H(z)$ can be practically superposed on the $Λ$CDM solution $H_Λ(z)$, 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.

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