  We use the LUQAS sample (Kim et al. 2004), a set of 27 high-resolution and
high signal-to-noise QSO absorption spectra at a median redshift of z=2.25, and
the data from Croft et al. (2002) at a median redshift of z=2.72, together with
a large suite of high-resolution large box-size hydro-dynamical simulations, to
estimate the linear dark matter power spectrum on scales 0.003 s/km < k <0.03
s/km. Our re-analysis of the Croft et al. data agrees well with their results
if we assume the same mean optical depth and gas temperature-density relation.
The inferred linear dark matter power spectrum at z=2.72 also agrees with that
inferred from LUQAS at lower redshift if we assume that the increase of the
amplitude is due to gravitational growth between these redshifts. We further
argue that the smaller mean optical depth measured from high-resolution spectra
is more accurate than the larger value obtained from low-resolution spectra by
Press et al. (1993) which Croft et al. used. For the smaller optical depth we
obtain a ~ 20% higher value for the rms fluctuation amplitude of the matter
density. By combining the amplitude of the matter power spectrum inferred from
the Lyman-alpha forest with the amplitude on large scales inferred from
measurements of the CMB we obtain constraints on the primordial spectral index
n and the normalisation sigma_8. For values of the mean optical depth favoured
by high-resolution spectra, the inferred linear power spectrum is consistent
with a LambdaCDM model with a scale-free (n=1) primordial power spectrum.
