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How would you modify this relationship to use it in cosmology, where distances are given in units of $h^{-1}$Mpc?

\begin{equation} \label{eq1} \begin{split} m - M & = 5\log_{10}\Big(\frac{d}{10pc}\Big) \end{split} \end{equation}

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The distance modulus is used in cosmology exactly as you quote it. The $h^{-1}$ Mpc units are not used universally in cosmology either. It's better to use appropriate units for whatever the situation is.

The parameter $h^{-1}$ is simply the inverse of the dimensionless Hubble parameter and is hence just a number, typically quoted as around $h = 0.7$. You have quoted the distance modulus in units of $10$pc; all you need to do is convert that to Megaparsecs.

For the distance, $d$, in parsecs, the distance modulus is $m-M = 5 \log_{10} \frac{d}{\rm pc} - 5$.

One Megaparsec is $10^6$ parsecs, i.e. $1$ pc = $10^{-6}$ Mpc.

$$ \begin{align} m -M &= 5 \log_{10} \frac{d}{10^{-6}\rm Mpc} - 5, \\ &= 5 \left( \log_{10} \frac{d}{\rm Mpc} + \log_{10} 10^6 \right) - 5, \\ &= 5 \log_{10} \frac{d}{\rm Mpc} + 5\cdot 6 -5, \\ &= 5 \log_{10}\frac{d}{\rm Mpc} + 25. \end{align} $$

For the distance, $d$, in Megaparsecs, the distance modulus is therefore $m-M = 5 \log_{10} d + 25$.

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