Birkhoff's Theorem and the Void Matter Density
  According to the discussion on Birkhoff's theorem by Peebles (1993), a void with a negative perturbation \delta may evolve as a separate homogeneous universe with a local expansion parameter H_V\simeq H_0(1-\delta/3). This slightly low density ``universe'' will diverge from the mean, producing a void of ever lower density. As a result, the contents of voids will ``fall'' outward at a velocity equal to the difference between the local, and the mean expansion parameters times the radius of the void. Observational constraints on the outfall velocity can be placed as a result of the fortuitous event that void, and non-void hydrogen LyA absorbers have distinct characteristics -- both in their equivalent width distributions and their Doppler parameter distributions -- which are clearly distinguished when cloud environments are measured in terms of the sum of the tidal fields of surrounding galaxies acting on the cloud (Manning 2002). These constraints dictate that the radial outfall velocity from an average void of radius ~ 15 h_{75}^{-1} Mpc must be less than \~100 km/s. The implications of this are probed with a ``negative'' top-hat simulation, using a 1-D Lagrangian code, to determine the relationship between the mass deficit in voids and the local expansion parameter in a flat lambda cosmology. The outfall velocity constraint shows that the void matter density must be greater than 75% of the mean matter density. I argue that this implies that \Omega_m >~ 0.86$. Thus the total amount of matter in voids is of order twice the total mass in the filamentary structures. 
C. V. Manning