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# S10. A Newtonian Homogeneous Expanding Universe - SOLUTIONS

Box $$\PageIndex{1}$$

Exercise 10.1.1: Derive Hubble's Law in this universe, $$v = H d$$, where $$H = \dot a/a$$

Solution $$\PageIndex{1.1}$$

\begin{equation*} \begin{aligned} d = a\ell \quad \Longrightarrow \quad \ell = \frac{d}{a} \end{aligned} \end{equation*}

Substituting for $$\ell$$ in $$v = \dot a \ell$$, we get

\begin{equation*} \begin{aligned} v = \frac{\dot a}{a}d \end{aligned} \end{equation*}

Now substituting in $$\dot a/a = H$$, we get

\begin{equation*} \begin{aligned} v = H d \end{aligned} \end{equation*}