Second order viscous corrections to the harmonic spectrum in heavy ion collisions

We calculate the second order viscous correction to the kinetic distribution, $\delta f_{(2)}$, and use this result in a hydrodynamic simulation of heavy ion collisions to determine the complete second order correction to the harmonic spectrum, $v_n$. At leading order in a conformal fluid, the first viscous correction is determined by one scalar function, $\chi_{0p}$. One moment of this scalar function is constrained by the shear viscosity. At second order in a conformal fluid, we find that $\delta f(\p)$ can be characterized by two scalar functions of momentum, $\chi_{1p}$ and $\chi_{2p}$. The momentum dependence of these functions is largely determined by the kinematics of the streaming operator. Again, one moment of these functions is constrained by the parameters of second order hydrodynamics, $\tau_\pi$ and $\lambda_1$. The effect of $\delta f_{(2)}$ on the integrated flow is small (up to $v_4$), but is quite important for the higher harmonics at modestly-large $p_T$. Generally, $\delta f_{(2)}$ increases the value of $v_n$ at a given $p_T$, and is most important in small systems.