Measurements of distance from the close supernovæ

From the parameters of the lightcurves, we want to build an estimator of distance. The magnitude out of B in the reference frame of the supernovæ is the estimator most usually used (perlmutter1999, riess1998).

We saw before ( 6 ) that several measurements made it possible to reduce the dispersion of the luminosity to the maximum (0.5 magnitude) in a significant way and in particular measurement of the factor of stretching and the color to the maximum $(B-v)$.

In the continuation, we will use these two parameters to build our estimators of distance. By using relation 2.20 , we can write:

$$m(z) - M = 5\log_{10}({\cal D}(z,\Omega_{\rm M}, \Omega_\Lambda )) = \mu(z,\Omega_{\rm M},\Omega_\Lambda )$$ (10.1)

where $M = { \cal M }+25 - 5 \log_{10}(H_0)$ is a term of harmful effect which takes into account the constant of Hubble and the intrinsic luminosity of the supernovæ.

If we apply the relations of standardization, by using the factor of stretching and the color to the maximum, we can reexpress our module of distance like:

$$m(z) - M = \mu(z,\Omega_{\rm M},\Omega_\Lambda ) + \alpha(1-s) + \beta({\it col})$$ (10.2)

The close supernovæ will be used to us as standard for measurement of $\Omega_{\rm M } = 0, \Omega_\Lambda = 0$) for several cosmological models according to the redshift. We see that the degeneration between the various models is gradually raised starting from redshifts around 0.2. In this diagram, the close supernovæ will fix the base line, the remote supernovæ will make it possible to differentiate the various models.

Figure 10.5: Variation with the model of universe empties for several models of universe in magnitude in the filter B for supernovæ of the Ia type according to the redshift. The line continuous is the model `` standard '' with a nonnull cosmological constant, in dotted line, the model Einstein-With Sitter ( $\Omega_\Lambda = 0$), in indent-dotted line, a model Einstein-With Sitter with gray dust.