  Motivated by the lack of direct evidence with inelastic neutron scattering of
the well documented bound state of Heisenberg ferromagnets, we use the
time-dependent Thermal Density Matrix Renormalization Group algorithm to study
the temperature dependence of the dynamical spin structure factor of Heisenberg
ferromagnetic spin chains. For spin-1/2, we show that the bound state appears
as a well defined excitation with significant spectral weight in the
temperature range $J/12 \lesssim T \lesssim J/3$, pointing to the possibility
of detecting it with inelastic neutron scattering near $k=\pi$ provided the
temperature is neither too low nor too high - at low temperature, the spectral
weight only grows as $T^{3/2}$, and at high temperature the bound state peak
merges with the two-magnon continuum. For spin-1, the situation is more subtle
because the bound state with two neighboring spin flips competes with an
anti-bound state with two spin-flips on the same site. As a consequence, the
relative spectral weight of the bound state is smaller than for spin-1/2, and a
weak resonance due to the anti-bound state appears in the continuum. A clearer
signature of the bound state (resp. anti-bound state) can be obtained if a
negative (resp. positive) biquadratic interaction is present.
