  This paper deals with the long time behaviour of solutions to the spatially
homogeneous Landau equation with hard potentials . We prove an exponential in
time convergence towards the equilibrium with the optimal rate given by the
spectral gap of the associated linearized operator. This result improves the
polynomial in time convergence obtained by Desvillettes and Villani
\cite{DesVi2}. Our approach is based on new decay estimates for the semigroup
generated by the linearized Landau operator in weighted (polynomial or
stretched exponential) $L^p$-spaces, using a method develloped by Gualdani,
Mischler and Mouhot \cite{GMM}.
