Exact cosmological solution of a scalar-tensor gravity theory compatible with theΛCDMmodel

Abstract

We consider the massive scalar-tensor theory in the Jordan frame $F(\ensuremath{\Phi})={K}^{2}{\ensuremath{\Phi}}^{2}$ and $U(\ensuremath{\Phi})=(1/2){m}^{2}{\ensuremath{\Phi}}^{2}$, where $F(\ensuremath{\Phi})$ corresponds to a constant Brans-Dicke parameter ${\ensuremath{\omega}}_{\mathrm{BD}}=1/4{K}^{2}$. 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 $\frac{\ensuremath{\Phi}(z)}{\ensuremath{\Phi}(0)}$ and $\frac{H(z)}{{H}_{0}}$ that depend only on two parameters, ${K}^{2}$ and ${\ensuremath{\Omega}}_{m,0}$. For $K\ensuremath{\le}1/400$, the expansion rate $H(z)$ can be practically superposed on the $\ensuremath{\Lambda}\mathrm{CDM}$ solution ${H}_{\ensuremath{\Lambda}}(z)$, up to high redshift $z$, but the equation of state ${w}_{\mathrm{DE}}(z)$ of the dark energy is not constant: it presents a very slight crossing of the phantom divide line $w=\ensuremath{-}1$ in the neighborhood of $z=0$ and becomes very slightly positive at high redshifts.