  Following the procedure proposed recently by Martin and Brandenberger we
investigate the spectrum of the cosmological perturbations in the case when the
``trans-Plackian'' dispersion relations are motivated by the quantum
$\kappa$-Poincar\'e algebra. We find that depending on the choice of initial
conditions of the perturbations, the spectrum either differs from the flat one
for for instantaneous Minkowski vacuum or, in the case of initial conditions
minimizing energy density, leads to the observed scale-invariant
Harrison-Zel'dovich spectrum in the Friedmann epoch.
