Will the universe continue to expand forever? To attack this question, make the (reasonable?) assumption that the recessional speed v of a galaxy a distance r from us is determined only by the matter that lies inside a sphere of radius r centered on us. If the total mass inside this sphere is M, the escape speed v e from the sphere is v e = √2GM/r, (Equation 13-28).

(a) Show that to prevent unlimited expansion, the average density p inside the sphere must be at least equal to p = 3H2 / 8πG

(b) Evaluate this "critical density" numerically; express your answer in terms of hydrogen atoms per cubic meter. Measurements of the actual density are difficult and are complicated by the presence of dark matter.