  The temperature dependence of the topological susceptibility in QCD, chi_t,
essentially determines the abundance of the QCD axion in the Universe, and is
commonly estimated, based on the instanton picture, to be a certain negative
power of temperature. While lattice QCD should be able to check this behavior
in principle, the temperature range where lattice QCD works is rather limited
in practice, because the topological charge is apt to freezes at high
temperatures. In this work, two exploratory studies are presented. In the first
part, we try to specify the temperature range in the quenched approximation.
Since our purpose here is to estimate the range expected in unquenched QCD
through quenched simulations, hybrid Monte Carlo (HMC) algorithm is employed
instead of heatbath algorithm. We obtain an indication that unquenched
calculations of chi_t encounter the serious problem of autocorrelation already
at T~2Tc or even below with the plain HMC. In the second part, we revisit the
axion abundance. The absolute value and the temperature dependence of chi_t in
real QCD can be significantly different from that in the quenched
approximation, and is not well established above the critical temperature.
Motivated by this fact and precedent arguments which disagree with the
conventional instanton picture, we estimate the axion abundance in an extreme
case where chi_t decreases much faster than the conventional power-like
behavior. We find a significant enhancement of the axion abundance in such a
case.
