Shear Viscosity to Entropy Density Ratio in Higher Derivative Gravity with Momentum Dissipation

We investigate $\eta \mathrm{/}s$ in linear scalar fields modified Gauss-Bonnet theory that breaks translation invariance. We first calculate $\eta \mathrm{/}s$ both analytically and numerically and show its relationship with temperature in log-log plot. Our results show that $\eta \mathrm{/}s\sim T^2$ 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.