Shear viscosity to entropy density ratio in higher derivative gravity with momentum dissipation

We investigate $\eta /s$ in the linear scalar field modified Gauss-Bonnet theory that breaks translation invariance. We first calculate $\eta /s$ both analytically and numerically and show its relationship with temperature in a log-log plot. Our results show that $\eta /s\sim T^2$ at low temperatures. The causality is also considered in this work. We then find that causality violation still happens in the presence of the linear scalar field and we suggest there is a Gauss-Bonnet coupling dependent lower limit for the effective mass of the graviton. If the effective mass of the graviton is big enough, then there will be no causality violation and hence no constraints for the Gauss-Bonnet coupling.

Figure 1

Log-log plot of $4\pi \eta /s$ as a function of $\alpha /T$, in a background with higher derivative Gauss-Bonnet gravity. The dotted red line represents the analytical results at small $\alpha /T$, and the green one shows that for large $\alpha /T$. Here we set $\lambda =0.1$ and $u_m=1/10$.

Figure 2

Log-log plot of the numerical results for Eq. (16). Here we set $\lambda =-0.1$, 0.08, and 0.15. The blue line represents the results for $\lambda =-0.1$, the red disks are data for $\lambda =0.08$, and the green ones are the numerical data obtained when $\lambda$ is set to be 0.15.

Figure 3

Diagram for the local speed of the graviton as a function of $u$. Here $\lambda =0.05$. The top blue line corresponds to ${\beta }^2=1/4$, the middle red one represents $c_g^2$ for ${\beta }^2=1$, and the bottom green line illustrates the case in which ${\beta }^2=4$.

Figure 4

The local speed of the graviton for $\lambda =0.2$. The blue, red, and green lines (from top to bottom) correspond to ${\beta }^2=1/4,1,4$, respectively.

Figure 5

Contour plot of $c_g^2$ as a function of ${\beta }^2$ and $\lambda$. From bottom to top, the lines correspond to $c_g^2=1.2,1.0,\dots ,0.4$, respectively. Here $u$ is chosen to be 0.5.