We have to answer whether the universe will continue to expand forever. For attacking this problem, we make the reasonable assumption that the recessional speed of v of a galaxy a distance r from us is determined only by the matter that lies inside a sphere of radius r centred on us. If the total mass inside this sphere is M, the escape speed . (a) We have to show that the average density inside the sphere must at least equal to the value given by

to prevent unlimited expansion. (b) We have to evaluate this “critical density” numerically; and have to express our answer in terms of . Measurements of the actual density are difficult and complicated by the presence of the dark matter.

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We make the reasonable assumption that the recessional speed of v of a galaxy a distance r from us is determined only by the matter that lies inside a sphere of radius r centred on us. The amount of mass contained inside a sphere of radius r and uniform density is

If the total mass inside this sphere is M, the escape speed .

We thus have the following expression for the escape speed

According to the Hubble’s law, the recessional speed of a galaxy at a distance r from us is

, where H is the Hubble’s constant. If the universe is to be prevented from unlimited expansion

.

Therefore, the condition for the density for a closed universe is given by

(b)

We will evaluate this “critical density” numerically; and express our answer in terms of .