# Should luminosity distance be 0 at z=0?

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• johnio09
johnio09
I am working on coding up the luminosity function for blazars but I have ran into a problem. In equations 1-3 of this paper https://arxiv.org/pdf/1912.01622 they state that the flux can be broken down into two components: one where z=0 and one part that is the evolutionary factor. The problem I have noticed is that in their equation for z=0 (eq. 3), there is 𝐿_𝛾 in the denominator, which is a function of the luminosity distance (eq. 2) https://ned.ipac.caltech.edu/level5/Hogg/Hogg4.html, the comoving distance would be 0 when z=0, resulting in luminosity distance being 0 when z=0 (according to 𝑑_𝐿=(1+𝑧)d_c). Hence this results in a divide by 0 error when trying to use their formula. Could somebody please help me understand what's going on? Thank you.

At z=0 the comoving distance is not zero, but equal to the luminosity distance. I.e. it's just the same then as the distance one would infer from the inverse square law when looking at a source of known intrinsic luminosity in a non-expanding space (eq. 19 in Hogg).
With growing z the two distances diverge (by the 1+z factor).

The only case when the comoving distance is zero, is when you're at the source (as with any other distance).

Bandersnatch said:
At z=0 the comoving distance is not zero, but equal to the luminosity distance. I.e. it's just the same then as the distance one would infer from the inverse square law when looking at a source of known intrinsic luminosity in a non-expanding space (eq. 19 in Hogg).
With growing z the two distances diverge (by the 1+z factor).

The only case when the comoving distance is zero, is when you're at the source (as with any other distance).
eq 19 in Hogg, however, is a function of luminosity and flux. In eq 2 of the other paper they state that luminosity distance is purely a function of redshift, if I am not mistaken. How would I be able to relate Hogg's equation to make it so that it is only a function of redshift?

I find the paper a bit too hard to follow, but I'd wager they just mean the d_L=(1+z)d_c relation.

Looking over it I came to similar conclusion. What I find interesting is that they aren't showing the influence of the matter/radiation density evolution at different redshifts.
Hoggs if I recall addresses this in his comologicsl distance measures article.

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