We explore the dependence of the central logarithmic slope of dark matter halo density profiles α on the spectral index n of the linear matter power spectrum P(k) using cosmological N-body simulations of scale-free models i.e. P(k) ∝kn. These simulations are based on a set of clear, reproducible and physically motivated criteria that fix the appropriate starting and stopping times for runs, and allow one to compare haloes across models with different spectral indices and mass resolutions. For each of our simulations we identify samples of well-resolved haloes in dynamical equilibrium and we analyse their mass profiles. By parametrizing the mass profile using a ‘generalized’ Navarro, Frenk & White profile in which the central logarithmic slope α is allowed to vary while preserving the r−3 asymptotic form at large radii, we obtain preferred central slopes for haloes in each of our models. There is a strong correlation between α and n, such that α becomes shallower as n becomes steeper. However, if we normalize our mass profiles by r−2, the radius at which the logarithmic slope of the density profile is −2, we find that these differences are no longer present. This is apparent if we plot the maximum slope as a function of r/r−2– we find that the profiles are similar for haloes forming in different n models. This reflects the importance of concentration, and reveals that the concentrations of haloes forming in steep-n cosmologies tend to be smaller than those of haloes forming in shallow-n cosmologies. We conclude that there is no evidence for convergence to a unique central asymptotic slope, at least on the scales that we can resolve.