We use LUQAS, a sample of 27 high resolution high signal-to-noise UVES quasar (QSO) spectra (Kim et al. 2004), and the Croft et al. (2002) sample together with a set of high resolution large box-size hydrodynamical simulations run with the code GADGET-II to recover the linear dark matter power spectrum at $z>2$ and at scales of $1-40\,h^{-1}\,$ Mpc. These scales cannot be probed by any other observable. We address some of the uncertainties in the theoretical modelling of the Lyman-$\alpha$ forest structures such as the difference between full hydro-dynamical simulations and a more simplified scheme based on the HPM (Hydro-Particle-Mesh) technique. We combine these data sets with the Cosmic Microwave Background (CMB) data of WMAP in order to get tighter constraints on cosmological parameters. We focus on the recovered values of the power spectrum amplitude, the primordial spectral index and the running of the primordial spectral index. By considering models of slow-roll inflation we give constraints on inflationary parameters. We explore the implications for the mass of a warm dark matter particle, including light gravitinos and sterile neutrinos, and from the perspective of constraining the neutrino mass.