Shape of the hot topological charge density spectral function

After motivating an interest in the shape of the topological charge density spectral function in hot Yang-Mills theories, we estimate it with the help of thermally averaged classical real-time simulations, for $N_{\mathrm{c}}=2,3$. After subtracting a perturbative contribution at large frequencies, we observe a non-trivial shape at small frequencies (a dip rather than a peak), interpolating smoothly towards the sphaleron rate at zero frequency. Possible frequency scales making an appearance in this shape are discussed. Implications for warm axion inflation and reheating, and for imaginary-time lattice measurements of the strong sphaleron rate, are recapitulated.