Testing Dispersion Relations of Quantum $κ$-Poincaré Algebra on Cosmological Ground

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 derived from the quantum $\kappa$-Poincar\'e algebra. We find that depending on the choice of initial conditions of the perturbations, the spectrum is either $n^3 P \sim \frac1n$ for initial conditions minimizing energy density, or the flat one $n^3 P \sim n^0$ for instantaneous Minkowski vacuum. This latter spectrum leads to the observed scale-invariant Harrison-Zel'dovich spectrum in the Friedmann epoch.