Solved: Will the universe continue to expand forever To attack this

Will the universe continue to expand forever? To attack this question, assume that the theory of dark energy is in error and that the recessional speed v of a galaxy a distance r from us is determined only by the gravitational interaction of 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 ve from the sphere is ve = (2GM/r (Eq. 13-28).
(a) Show that to prevent unlimited expansion, the average density r inside the sphere must be at least equal to
(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.
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Solved: Will the universe continue to expand forever To attack this