  A problem with high central densities in dark halos has arisen in the context
of LCDM cosmologies with scale-invariant initial power spectra. Although n=1 is
often justified by appealing to the inflation scenario, inflationary models
with mild deviations from scale-invariance are not uncommon and models with
significant running of the spectral index are plausible. Even mild deviations
from scale-invariance can be important because halo collapse times and
densities depend on the relative amount of small-scale power. We choose several
popular models of inflation and work out the ramifications for galaxy central
densities. For each model, we calculate its COBE-normalized power spectrum and
deduce the implied halo densities using a semi-analytic method calibrated
against N-body simulations. We compare our predictions to a sample of dark
matter-dominated galaxies using a non-parametric measure of the density. While
standard n=1, LCDM halos are overdense by a factor of 6, several of our example
inflation+CDM models predict halo densities well within the range preferred by
observations. We also show how the presence of massive (0.5 eV) neutrinos may
help to alleviate the central density problem even with n=1. We conclude that
galaxy central densities may not be as problematic for the CDM paradigm as is
sometimes assumed: rather than telling us something about the nature of the
dark matter, galaxy rotation curves may be telling us something about inflation
and/or neutrinos. An important test of this idea will be an eventual consensus
on the value of sigma_8, the rms overdensity on the scale 8 h^-1 Mpc. Our
successful models have values of sigma_8 approximately 0.75, which is within
the range of recent determinations. Finally, models with n>1 (or sigma_8 > 1)
are highly disfavored.
