Topology in the $SU(N_f)$ chiral symmetry restored phase of unquenched QCD and axion cosmology

We investigate the topological properties of unquenched QCD on the basis of numerical results of simulations at fixed topological charge, recently reported by Borsanyi et al. We demonstrate that their results for the mean value of the chiral condensate at fixed topological charge are inconsistent with the analytical prediction of the large-volume expansion around the saddle point, and argue that the most plausible explanation for the failure of the saddle-point expansion is a vacuum energy density that is $\theta$-independent at high temperatures, but surprisingly not too high $(T\sim 2T_c)$, a result which would imply a vanishing topological susceptibility and the absence of all physical effects of the $U(1)$ axial anomaly at these temperatures. We also show that under a general assumption concerning the high-temperature phase of QCD, where the $SU(N_f{)}_A$ symmetry is restored, the analytical prediction for the chiral condensate at fixed topological charge is in very good agreement with the numerical results of Borsanyi et al., all effects of the axial anomaly should disappear, the topological susceptibility and all the $\theta$ derivatives of the vacuum energy density vanish, and the theory becomes $\theta$ independent at any $T>T_c$ in the infinite-volume limit.