The median density of the Universe

Despite the fact that the mean matter density of the universe has been measured to an accuracy of a few percent within the standard $\Lambda$CDM paradigm, its median density is not known even to order of magnitude. Typical points lie in low-density regions and are not part of a collapsed structure of any scale. Locally, the dark matter distribution is then simply a stretched version of that in the early universe. In this single-stream regime, the distribution of unsmoothed density is sensitive to the initial power spectrum on all scales, in particular on very small scales, and hence to the nature of the dark matter. It cannot be estimated reliably using conventional cosmological simulations because of the enormous dynamic range involved, but a suitable excursion set procedure can be used instead. For the Planck cosmological parameters, a 100 GeV WIMP, corresponding to a free-streaming mass $\sim 10^{-6}M_\odot$, results in a median density of $\sim 4\times 10^{-3}$ in units of the mean density, whereas a 10 $\mu$eV axion with free-streaming mass $\sim 10^{-12}M_\odot$ gives $\sim 3\times 10^{-3}$, and Warm Dark Matter with a (thermal relic) mass of 1 keV gives $\sim 8\times 10^{-2}$. In CDM (but not in WDM) universes, single-stream regions are predicted to be topologically isolated by the excursion set formalism. A test by direct N-Body simulations seems to confirm this prediction, although it is still subject to finite size and resolution effects. Unfortunately, it is unlikely that any of these properties is observable and so suitable for constraining the properties of dark matter.