Why does the estimation of the distance of supernovae depends on the cosmological model?

Question

I was reading a paper from arxiv (https://arxiv.org/abs/1312.5798) when I stumbled upon this passage at pag 4

"However, even at low redshifts, there are limitations to how well the Type Ia supernova data can be interpreted with an empirical model such as $\Lambda$CDM, because the data cannot be determined independently of the assumed cosmology--- the supernova luminosities must be evaluated by optimizing at least 4 parameters simultaneously with those in the adopted model. This renders the data compliant to the underlying theory..."

but I know that supernovae distances are calculated by comparing the observed flux with the luminosity through the relation

$$ F = \frac{L}{4 \pi S_k (r)^2 (1+z)^2} $$

where $ 4 \pi S_k (r)^2 $ is the proper area and the $(1+z)^2$ comes from both the redshift of the photon wavelength and the time expansion, so i thought that the only thing that could influence the measure should be the value of the curvature parameter $ k$ , so what does the author mean with "optimizing at least 4 parameters simultaneously with those in the adopted model" ? thanks

Answer 1

The calculation of z for one supernova depends on the change in frequency of multiple measurements of specific atoms. The value of z does not depend on H_0. The values of z for multiple supernovae (a lot of them) lead to corresponding estimates of distances from Earth. Each combination of z and distance gives a supernova specific value for H_0. The average of these very many values for H_0, one for each supernova, is the calculated cosmological value for H_0. (This is a simplified explanation, more-or-less like is was about a century ago when Edwin Hubble first calculated what has become H_0.)