Question

Explain why the cosmological principle requires that ${\mathit{H}}_{\mathbf{0}}$ must have the same value everywhere in space, but does not require that it be constant in time.

Short answer

Because there is an equal speed of two different points at the same distance $r$obey the cosmological principle.

Step-by-step solution

Definition of speed

The term speed may be defined as the ratio of distance and time.

Determine the reason

According to Hubble’s law :

$v=H_{0}r$

Consider $H_0=H_0\left(r\right)$it means we have taken two points in which speed of recession is given as:

$v_1=H_0\left(r_1\right)r$, and $v_2=H_0\left(r_2\right)r$

Two speeds are equal and obey the cosmological principle at the same distance $r$.

$$\begin{gathered} H_0\left(r_1\right)r=H_0\left(r_2\right)r \\ {H}_0\left(r_1\right)=H_0\left(r_2\right) \end{gathered}$$

It means $H_0$does not depend on $r$,

According to the cosmological principle, the universe looks the same from any point at the same time so$H_0$ vary with time.

Hence, there is an equal speed of two different points at the same distance, $r$obeying the cosmological principle.