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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

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