Generalized Brans-Dicke theories

TL;DR

The paper addresses realizing late-time cosmic acceleration in scalar-tensor gravity without a potential, while preserving General Relativity in the early Universe. It introduces generalized Brans-Dicke theories with nonlinear self-interactions and power-law couplings $F(phi)$, $B(phi)$, and $xi(phi)$, and analyzes de Sitter fixed points via an autonomous system in variables $(x,y,Omega_r)$ to determine stability and viability. The analysis yields a stable de Sitter attractor for $2 leq n leq 3$ and omega < -n(n-3)^2, with no-ghost condition $Q_s>0$ and scalar speed $c_s^2>0$, and shows radiation and matter epochs precede the de Sitter phase with subluminal propagation during those eras. Numerical evolution confirms a viable background history with observed late-time densities and highlights model-specific signatures in structure growth and ISW effects, motivating perturbation studies to test against data.

Abstract

In Brans-Dicke theory a non-linear self interaction of a scalar field allows a possibility of realizing the late-time cosmic acceleration, while recovering the General Relativistic behavior at early cosmological epochs. We extend this to more general modified gravitational theories in which a de Sitter solution for dark energy exists without using a field potential. We derive a condition for the stability of the de Sitter point and study the background cosmological dynamics of such theories. We also restrict the allowed region of model parameters from the demand for the avoidance of ghosts and instabilities. A peculiar evolution of the field propagation speed allows us to distinguish those theories from the LCDM model.

Figures (4)

• Figure 1: The propagation speed squared at the dS point versus $x_{\rm dS}$ for several different values of $n$. One has $0 \le c_s^2<1$ for $2 \le n \le 3$.
• Figure 2: The allowed and excluded regions in the $(n, \omega)$ plane. The vertical axis is plotted in terms of $\log_{10} (-\omega)$ with $\omega<0$. The allowed region corresponds to $2 \le n \le 3$ and $\omega<-n(n-3)^2$. The black line shows $\omega=-n(n-3)^2$, whereas the dotted line correspond to the border at which the propagation speed squared temporally reaches $c_s^2=1$ during the course of the cosmological evolution.
• Figure 3: Evolution of $\Omega_{\rm DE}$, $\Omega_m$, $\Omega_r$, and $w_{\rm eff}$ versus the redshift $z=1/a-1$ for $n=2.5$ and $\omega=-10$ (left panel). The initial conditions are chosen to be $x=10^{-18}$ and $y=(3-n)\Omega_m/8$ with $\Omega_m=1.28 \times 10^{-5}$. We identify the present epoch ($z=0$) as $\Omega_{\rm DE}=0.72$, $\Omega_m=0.28$, and $\Omega_r=8 \times 10^{-5}$. The right panel shows the evolution of the variables $x$, $y$ and $\Omega_r$ with a logarithmic scale for the same model parameters and initial conditions as those in the left panel.
• Figure 4: Evolution of the propagation speed squared $c_s^2$ versus the redshift $z$ for the cases (a) $n=2$, $\omega=-188$, (b) $n=2.5$, $\omega=-10$, and (c) $n=3$, $\omega=-1$. The present epoch ($z=0$) is identified as $\Omega_{\rm DE}=0.72$, $\Omega_m=0.28$, and $\Omega_r=8 \times 10^{-5}$.