  We analyze the strong hexagonal warping of the Dirac cone of Bi$_2$Te$_3$ by
angle-resolved photoemission. Along $\overline{\Gamma}$$\overline{\rm M}$, the
dispersion deviates from a linear behavior meaning that the Dirac cone is
warped outwards and not inwards. We show that this introduces an anisotropy in
the lifetime broadening of the topological surface state which is larger along
$\overline{\Gamma}$$\overline{\rm K}$. The result is not consistent with
nesting. Based on the theoretically predicted behavior of the ground-state spin
texture of a strongly warped Dirac cone, we propose spin-dependent scattering
processes as explanation for the anisotropic scattering rates. These results
could help paving the way for optimizing future spintronic devices using
topological insulators and controlling surface-scattering processes via
external gate voltages.
