Generalized Brans-Dicke theory: A dynamical systems analysis

The stability criteria for the generalized Brans-Dicke cosmology in a spatially flat, homogeneous and isotropic cosmological model is discussed in the presence of a perfect fluid. The generalization comes through the channel that the Brans-Dicke coupling parameter $\omega$ is allowed to be a function of the scalar field $\varphi$. This generalization can lead to a host of scalar-tensor theories of gravity for various choices of $\omega =\omega (\varphi )$. A very interesting general result has been found. Excepting for the case of a radiation distribution as the choice of the fluid, all other solutions find a natural habitat in the corresponding solutions in general relativity in an infinite $\omega$ limit. For the radiation distribution, the dependence of stability on $\omega$ is a bit obscure. If a scalar potential, function of the Brans-Dicke scalar field, is added to the action, the requirement of an infinite $\omega$ for stability is relaxed for a matter distribution with a nonzero trace whereas it becomes a possibility for a radiation distribution.

Figure 1

Phase plots of the class I system.

Figure 2

Projection of perturbations along $x$, $y$, $\lambda$ axis for Class II, without potential and $\gamma \ne \frac{4}{3}$.

Figure 3

Projection of perturbations along $x$, $y$, $\lambda$ axis for Class II, without potential and $\gamma =\frac{4}{3}$.

Figure 4

Evolution of the system from the fixed point (a) to the fixed point (b) for Class I, with potential $V(\varphi )=M^4(\frac{\varphi }{{\varphi }_0}{)}^{2n}$, $n=1$ and $\gamma \ne \frac{4}{3}$.

Figure 5

Evolution of the system from some arbitrary initial values to the fixed point (b) for Class I, with potential $V(\varphi )=M^4(\frac{\varphi }{{\varphi }_0}{)}^{2n}$, $n=1$ and $\gamma =\frac{4}{3}$.

Figure 6

Projection of perturbations along $x$, $y$, $z$, $\lambda$ axis for Class II, with potential $V(\varphi )=M^4(\frac{\varphi }{{\varphi }_0}{)}^{2n}$, $n=2$ and $\gamma \ne \frac{4}{3}$.

Figure 7

Evolution of the system from some arbitrary initial conditions to the fixed point(b) for Class II, with potential $V(\varphi )=M^4(\frac{\varphi }{{\varphi }_0}{)}^{2n}$, $n=1$ and $\gamma \ne \frac{4}{3}$.

Figure 8

Projection of perturbations along $x$, $y$, $z$, $\lambda$ axis for Class II, with potential $V(\varphi )=M^4(\frac{\varphi }{{\varphi }_0}{)}^{2n}$, $n=2$ and $\gamma =\frac{4}{3}$.