A hierarchical universe can have an average density of zero, while containing infinite mass.
Hannes Alfvén (1908–1995), who pioneered the field of magnetohydrodynamics, against initial skepticism, to give us a universe permeated by what are now called "Alfvén waves," never relinquished his own skepticism concerning the Big Bang. "They fight against popular creationism, but at the same time they fight fanatically for their own creationism," he argued in 1984, advocating, instead, for a hierarchical cosmology, whose mathematical characterization he credited to Edmund Edward Fournier d'Albe (1868—1933) and Carl Vilhelm Ludvig Charlier (1861–1932). Hierarchical does not mean isotropic, and observed anisotropy does not rule it out.
Gottfried Wilhelm Leibniz (1646-1716), a lawyer as well as a scientist, believed that our universe was selected, out of an infinity of possible universes, to produce maximum diversity from a minimal set of natural laws. It is hard to imagine a more beautiful set of boundary conditions than zero density and infinite mass. But this same principle of maximum diversity warns us that it may take all the time in the universe to work the details out.