  We explore the dependence of the central logarithmic slope of dark matter
halo density profiles $\alpha$ on the spectral index $n$ of the linear matter
power spectrum $P(k)$ using cosmological $N$-body simulations of scale-free
models (i.e. $P(k) \propto k^n$). For each of our simulations we identify
samples of well resolved haloes in dynamical equilibrium and we analyse their
mass profiles. By parameterising the mass profile using a ``generalised''
Navarro, Frenk & White profile in which the central logarithmic slope $\alpha$
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 $\alpha$ and $n$, such that $\alpha$
becomes shallower as $n$ becomes steeper. However, if we normalise 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. We
conclude that there is no evidence for convergence to a unique central
asymptotic slope, at least on the scales that we can resolve.
