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We have that the proper distance to an object is given by

[tex]d_p (t_0 ) = c\int_{t_e }^{t_0 } {\frac{{{\rm{d}}t}}{{a(t)}}}[/tex]

and this goes for all possible universes described by the Robertson-Walker metric. Since we know that

[tex] 1 + z = \frac{1}{a(t_e)}[/tex]

does this mean that the proper distance at the time of

[tex]d_p (t_0 ) = c\int_{t_e }^{t_0 } {\frac{{{\rm{d}}t}}{{a(t)}}}[/tex]

and this goes for all possible universes described by the Robertson-Walker metric. Since we know that

[tex] 1 + z = \frac{1}{a(t_e)}[/tex]

does this mean that the proper distance at the time of

**emission**is always smaller than the proper distance at the time of observation by a factor of 1+z? And in this true for multi-component universes (e.g. matter + radiation) as well as single-component (e.g. only radiation, only matter ...) ?
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