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We study the finite temperature, low energy, long wave-length spectrum of thedynamic structure factor of the spin-$1/2$ antiferromagnetic Heisenberg chainin the presence of exchange anisotropy and external magnetic fields. Usingimaginary-time quantum Monte-Carlo we extract parameters, relevant tocharacterize a {\it renormalized} Luttinger liquid. For small momentum ourresults are consistent with a change from propagating spinon density waves tospin diffusion, described by a finite-frequency spin-current relaxation rate.Results for this relaxation rate as well as other Luttinger liquid parametersare presented versus temperature, momentum, magnetic field, and anisotropy,including finite-size analysis, and checks for anomalous diffusion. Our resultsare consistent with exact diagonalization and Bethe Ansatz, where available,and with corroborate findings of other previous studies using bosonization,transfer matrix renormalization group, and quantum Monte-Carlo.

Hydrodynamic spin fluctuations in the antiferromagnetic Heisenberg chain