Thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics

TL;DR

It is conjecture that divergences in zeta/s for black hole horizons are related to extrema of the entropy density as a function of temperature.

Abstract

We consider classes of translationally invariant black hole solutions whose equations of state closely resemble that of QCD at zero chemical potential. We use these backgrounds to compute the ratio zeta/s of bulk viscosity to entropy density. For a class of black holes that exhibits a first order transition, we observe a sharp rise in zeta/s near T_c. For constructions that exhibit a smooth cross-over, like QCD does, the rise in zeta/s is more modest. We conjecture that divergences in zeta/s for black hole horizons are related to extrema of the entropy density as a function of temperature.

Figures (1)

• Figure 1: LEFT: A comparison of different $c_s^2(T)$ curves. The solid red curve corresponds to the potential \ref{['FirstPotential']} (Type I black holes). The dashed magenta curve corresponds to the potential \ref{['TunedV']} (Type II black holes). The dot-dashed orange curve corresponds to the potential in \ref{['ConfinementPotential']} (Type III black holes) with $a= 1$ and $b$ adjusted so that the dimension of the operator dual to $\phi$ is $\Delta \approx 3.93$. We also show lattice results for pure glue (solid black curve) and $2+1$-flavor QCD (open black squares), as well as $c_s^2(T)$ for a $2+1$-flavor quasiparticle model (QPM). The pure glue curve is based on Boyd:1996bx and private communications from F. Karsch. The $2+1$-flavor lattice QCD points are based on Cheng:2007jq; for these points, we take $T_c=187 \, {\rm MeV}$, as estimated from the halfway point of the initial rise in an $(\epsilon-3p)/T^4$ curve from Cheng:2007jq. This differs from the value $T_c = 196(3) \, {\rm MeV}$ computed by a different method in Cheng:2007jq. The QPM points are based on Bluhm:2007nu. RIGHT: A comparison of $\zeta/s$ results for the black holes described above (see legend) and the results of Meyer:2007dyKharzeev:2007wb. Lattice results for pure glue from Meyer:2007dy are shown in blue. The solid curves correspond to the sum rule result for QCD with $2+1$ flavors from Kharzeev:2007wb, using three representative values of the frequency parameter $\omega_0$.