Stable Palatini $f(\mathcal{R})$ braneworld

We consider the static domain wall braneworld scenario constructed from the Palatini formalism $f(\mathcal{R})$ theory. We check the self-consistency under scalar perturbations. By using the scalar-tensor formalism we avoid dealing with the higher-order equations. We develop the techniques to deal with the coupled system. We show that under some conditions, the scalar perturbation simply oscillates with time, which guarantees the stability. We also discuss the localization condition of the scalar mode by analyzing the effective potential and the fifth-dimensional profile of the scalar mode. We apply these results to an explicit example, and show that only some of the solutions allow for stable scalar perturbations. These stable solutions also give nonlocalizable massless mode. This is important for reproducing a viable four-dimensional gravity.

Figure 1

The plot of the coordinate transformation $r(y)$ determined by the transformation function $\zeta$, with $n=1/7$ for the left and $n=5/3$ for the right. For $0<n<1/6$ the coordinate $r$ has a finite range, while for $n>1$ the coordinate $r$ has an infinite range.

Figure 2

The effective potential $V_{\mathrm{eff}}(r)$ and the corresponding ${\eta }_0/\sqrt{\zeta }$ in $y$ coordinate, with $n=1/7$ (up) $n=5/3$ (bottom) respectively. For $n=1/7$, $V_{\mathrm{eff}}(r)$ is an infinitely high potential well, and it has boundary since $r$ has finite range. The plot of ${\eta }_0/\sqrt{\zeta }$ reveals that the massless mode ${\eta }_0$ has nodes, so it is not the lowest state, which implies the existence of tachyon. For $n=5/3$, the potential is positive definite. The mode ${\eta }_0/\sqrt{\zeta }$ cannot be normalized.