According to currently accepted ideas among astronomers, the universe came into existence about 15 billion years ago in an explosion called the Big Bang. Ever since that time the universe has been expanding in such a way that the velocity $v$ of a galaxy at distance $R$ from our galaxy (the Milky Way) is given by Hubble's law $v =HR$, where $H$ is Hubble's constant, about 16 km/s per million light-years (a light-year is about $9.47 x 10^12$ km). No one knows whether this expansion of the universe will continue indefinitely. If the universe contains enough matter, then the gravitational forces exerted by this matter on itself will ultimately slow down and stop the expansion. Then there will be a period of contraction ending in a complete gravitational collapse called the Big Crunch, in which the universe as we know it - space, time, matter, energy - will cease to exist.
(a) Show that the universe will continue to expand forever if the present density of matter $\delta$ (mass per unit volume) is less than the critical density $\delta_{c} = {3H^2\over 8\pi G}$.
Hint: What is the escape velocity at a distance R from our galaxy, due to the matter inside a sphere of radius R centered on us?
(b) Estimate the value of the critical density $\delta_{c}$. given that the gravitational constant G is approximately 6.67 x 10-20 km3/(kg·s2).
I am greatly puzzled about where to start from and what should be my approach.
My approach
$v = HR$
${dv\over dr} = H$
$a_{cceleration} = v{dv\over dr} = H^2R$
Given in hint which asks for escape velocity at a distance R from our galaxy, due to the matter inside a sphere of radius R centered on us.
$V_{escape}= \sqrt{2G M\over R}$
Given, density of universe is $\delta$
Mass inside a sphere of radius R
$M = {4\over 3}\pi R^3\delta $
$V_{escape}= \sqrt{2G {4\over 3}\pi R^3\delta \over R}$
$V_{escape}= \sqrt{ {8G\over 3}\pi R^2\delta }$
If we put $V_{escape}= HR$
Then we get $\delta = {3H^2\over 8\pi G}$
But that's not satisfactory as if the hint was not given how could we reach that point?
Any few step suggestions or worked out solution would be much much appreciated and I would be really grateful to you
The calculation suggested by the hint is based on the conceptual argument that our galaxy would not be able to expand if it is too dense to prevent the matter on the edge of galaxy from escaping. This requires that the Habble’s velocity $HR$ for the edge be greater than the escape velocity of the galaxy, i.e.
$$HR > \sqrt{2G M\over R}$$ which leads to $\delta <\delta_c= {3H^2\over 8\pi G}$. Given that the galaxy is currently expanding and its density is decreasing due to expansion, the expansion will continue indefinitely.