Past exam of the astrophysics course of the University of Oxford / 2015 / 8 / e / Solution

For a small peculiar line-of-sight velocity $v_{\parallel }$ within the group, cosmological and Doppler factors multiply:At fixed group redshift this gives $\delta z=(1+z_g)v_{\parallel }/c$. ThereforeUsing $c{\sigma }_z$ directly would give a detector-frame redshift interval without removing the cosmological stretch. This is the one-dimensional line-of-sight velocity dispersion requested, not the three-dimensional RMS speed. Redshift measurement errors or a large unbound line-of-sight extent would require additional corrections; neither is supplied here.