  We emphasize that the estimation of cosmological parameters from cosmic
microwave background (CMB) anisotropy data, such as the recent high resolution
maps from BOOMERanG and MAXIMA-1, requires assumptions about the primordial
spectra. The latter are predicted from inflation. The physically best-motivated
scenario is that of slow-roll inflation. However, very often, the unphysical
power-law inflation scenario is (implicitly) assumed in the CMB data analysis.
We show that the predicted multipole moments differ significantly in both
cases. We identify several misconceptions present in the literature (and in the
way inflationary relations are often combined in popular numerical codes). For
example, we do not believe that, generically, inflation predicts the relation
n_S - 1 = n_T for the spectral indices of scalar and tensor perturbations or
that gravitational waves are negligible. We calculate the CMB multipole moments
for various values of the slow-roll parameters and demonstrate that an
important part of the space of parameters (n_S, n_T) has been overlooked in the
CMB data analysis so far.
