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

TL;DR

This paper derives exact solutions for a specific scalar-tensor gravity model that closely mimics the standard $ ext{ extLambda}$CDM cosmology, revealing subtle deviations in dark energy behavior at different redshifts.

Contribution

It provides explicit analytical solutions for a scalar-tensor theory compatible with $ ext{ extLambda}$CDM, including the evolution of the scalar field and Hubble parameter.

Findings

The model's expansion rate closely matches $ ext{ extLambda}$CDM up to high redshift.

Dark energy equation of state slightly crosses the phantom divide near $z=0$.

The solutions depend only on two parameters, $K^2$ and $oxed{ extOmega_{m,0}}$.

Abstract

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 $\omega_{BD}=1/4K^2$. 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 $\frac{\Phi(z)}{\Phi(0)}$ and $\frac{H(z)}{H_{0}}$ that depend only on two parameters, $K^2$ and $\Omega_{m,0}$. For $K\leq1/400$ the expansion rate $H(z)$ can be practically superposed on the $\Lambda$CDM solution $H_{\Lambda}(z)$, 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.