  The heat transport in a system of $S=1/2$ large-$J$ Heisenberg spin chains,
describing closely Sr$_2$CuO$_3$ and SrCuO$_2$ cuprates, is studied
theoretically at $T\ll J$ by considering interactions of the bosonized spin
excitations with optical phonons and defects. Treating rigorously the
multi-boson processes, we derive a microscopic spin-phonon scattering rate that
adheres to an intuitive picture of phonons acting as thermally populated
defects for the fast spin excitations. The mean-free path of the latter
exhibits a distinctive $T$-dependence reflecting a critical nature of spin
chains and gives a close description of experiments. By the naturalness
criterion of realistically small spin-phonon interaction, our approach stands
out from previous considerations that require large coupling constants to
explain the data and thus imply a spin-Peierls transition, absent in real
materials.
