Inflation, cold dark matter, and the central density problem

The lingering problem with high central densities in dark halos has arisen in the context of (L)CDM cosmologies with $n=1\mathrm{}$ scale-invariant initial power spectra. Although $n=1\mathrm{}$ 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 $(n\sim 0.9)$ 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 semianalytic method calibrated against N-body simulations. We compare our predictions to a sample of $\sim 50$ dark matter-dominated galaxies using a nonparametric measure of the density, ${\Delta }_{\mathrm{V}/2\mathrm{}},$ 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 $n=1\mathrm{}$ LCDM halos are overdense by a factor of $\sim 6,$ 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 ${(m}_{\nu }\sim 0.5\hspace{0.5em}\mathrm{eV})$ 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 ${\sigma }_8,$ the rms overdensity on the scale ${8h}^{-1}\mathrm{Mpc}.$ Our successful models tend to have values of ${\sigma }_8\approx 0.75,$ which is well within the range of recent determinations. Finally, models with $n>\mathrm{}1\mathrm{}$ (or ${\sigma }_8\gtrsim 1)$ are highly disfavored.