$2-5.$ Hubble's Law

The astronomer Edwin Hubble $(1889-1953)$ is famous for observing that each object in the universe is receding from the earth at a velocity that is proportional to its distance. Choosing earth as the origin. Hubble's law may be stated as $v=Hr,$ where $r$ is the position vector and $H$ is called Hubble's parameter.

$($a$)$ Show that Hubble's law would be the same for an observer in any galaxy. That is$,$ show that the relative velocity of any two points in the universe is described by the same equation.

$($b$)$ It has also been found that, on a large enough length scale $($on the order of megaparsecs, where $1$ parsec $=3{10}^{13}km),$ the density of the universe is approximately uniform. If the continuity equation is applied to these observations $($Hubble$\text{'}$s law and uniform density$)$ and if $H$ is assumed to be constant, what is the implied functional form of $(t)?$

$($c$)$ It is generally accepted that $H$ was much larger in the early universe than it is today. If it is assumed, for example, that $H=\frac{c}{t},$ where $c$ is a constant, how is $(t)$ affected?