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We investigate the topological properties of $N_f = 2+1$ QCD with physicalquark masses, both at zero and finite temperature. We adopt stout improvedstaggered fermions and explore a range of lattice spacings $a \sim 0.05 - 0.12$fm. At zero temperature we estimate both finite size and finite cut-offeffects, comparing our continuum extrapolated results for the topologicalsusceptibility $\chi$ with predictions from chiral perturbation theory. Atfinite temperature, we explore a region going from $T_c$ up to around $4\,T_c$, where we provide continuum extrapolated results for the topologicalsusceptibility and for the fourth moment of the topological chargedistribution. While the latter converges to the dilute instanton gas predictionthe former differs strongly both in the size and in the temperature dependence.This results in a shift of the axion dark matter window of almost one order ofmagnitude with respect to the instanton computation.

Axion phenomenology and $\theta$-dependence from $N_f = 2+1$ lattice QCD