The lingering problem with high central densities in dark halos has arisen in the context of (L)CDM cosmologies with n = 1 scale-invariant initial power spectra. Although n = 1 is often justified by appealing to the inflation scenario, the choice is not generally justified. Specifically, inflation models with mild but important deviations from scale invariance (n ∼ 0.9) are not uncommon, and those with significant "running" of the spectral index are quite plausible. Even a mild deviation from scale invariance can be important because halo collapse times and densities depend on the relative amount of small-scale power. Here, we choose several popular, often well-motivated, models of inflation and work out the ramifications for galaxy central densities. For each model, we calculate its COBE-normalized primordial 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 ∼ 50 dark matter-dominated galaxies using a non-parametric measure of the density, ∆ V/2 , defined as the mean mass density, relative to the critical density, within the radius at which the rotation curve falls to half of its maximum value. 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, and even below, the range preferred by observations. We also show how the presence of massive (mν ∼ 0.5 eV) neutrinos can help to alleviate the central density problem, even with a scale invariant spectrum. 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 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 σ8, the rms overdensity on the scale 8h −1 Mpc. Our successful models tend to have values of σ8 ≈ 0.75, which is well within the range of recent determinations. Finally, models with n > 1 (or σ8 > ∼ 1) are highly disfavored.