  Axions are one of the most attractive dark matter candidates. The evolution
of their number density in the early universe can be determined by calculating
the topological susceptibility $\chi(T)$ of QCD as a function of the
temperature. Lattice QCD provides an ab initio technique to carry out such a
calculation. A full result needs two ingredients: physical quark masses and a
controlled continuum extrapolation from non-vanishing to zero lattice spacings.
We determine $\chi(T)$ in the quenched framework (infinitely large quark
masses) and extrapolate its values to the continuum limit. The results are
compared with the prediction of the dilute instanton gas approximation (DIGA).
A nice agreement is found for the temperature dependence, whereas the overall
normalization of the DIGA result still differs from the non-perturbative
continuum extrapolated lattice results by a factor of order ten. We discuss the
consequences of our findings for the prediction of the amount of axion dark
matter.
