Five-dimensional metric $f(R)$ gravity and the accelerated universe

The metric $f(R)$ theories of gravity are generalized to five-dimensional spacetimes. By assuming a hypersurface-orthogonal Killing vector field representing the compact fifth dimension, the five-dimensional theories are reduced to their four-dimensional formalism. Then we study the cosmology of a special class of $f(R)=\alpha R^m$ models in a spatially flat Friedmann-Robertson-Walker spacetime. It is shown that the parameter $m$ can be constrained to a certain range by the current observed deceleration parameter, and its lower bound corresponds to the Kaluza-Klein theory. It turns out that both expansion and contraction of the extra dimension may prescribe the smooth transition from the deceleration era to the acceleration era in the recent past as well as an accelerated scenario for the present Universe. Hence, five-dimensional $f(R)$ gravity can naturally account for the present accelerated expansion of the Universe. Moreover, the models predict a transition from acceleration to deceleration in the future, followed by a cosmic recollapse within finite time. This differs from the prediction of the five-dimensional Brans-Dicke theory but is inconsistent with a recent prediction based on loop quantum cosmology.

Figure 1

Allowed range for the values of $m$ and $n$ with $q_0\in (-0.47,-0.67)$ [32]. The horizontal axis stands for different value of $n$, while the vertical axis represents $m$.

Figure 2

The relation between $n$ and the possible range for $\alpha$. The range is spanned by the uncertainty of the deceleration parameter $q_0$.

Figure 3

The evolution of $a(t)$ (solid line) and $q(t)$ (dashed line) with $m=1.5$, $n=5.4$. Here, $t=0$ corresponds to today, and the present value of the deceleration parameter is $q_0=-0.6$.

Figure 4

The evolution of $\lambda (t)$ with $m=1.5$, $n=5.4$. Note that $\lambda (t)$ is decreasing at $t=0$.

Figure 5

The evolution of $\varphi (t)$ with $m=1.5$, $n=5.4$.

Figure 6

The evolution of $\tilde{G}(t)$ with $m=1.5$, $n=5.4$.

Figure 7

Numerical simulation for a longer period of cosmic evolution. The deceleration parameter $q(t)$ becomes divergent at $t=(13.5+47.6)\mathrm{Gyr}$ (dashed line), as $a(t)$ (solid line in the subfigure) approaches a constant value.

Figure 8

The evolution rate of $\frac{\dot{a}}{a}$ (solid line), $\frac{\dot{\lambda }}{\lambda }$ (dashed line), and $\frac{\dot{\varphi }}{\varphi }$ (dash-dotted line) in the five-dimensional BD theory, with parameters $\omega =1.2$, $n=3$ in 16.

Figure 9

The evolution of $\frac{\dot{a}}{a}$ (solid line), $\frac{\dot{\lambda }}{\lambda }$ (dashed line), and $\frac{\dot{\varphi }}{\varphi }$ (dash-dotted line) in five-dimensional $f(R)$ gravity with parameters $m=1.5$, $n=5.4$.