Curves of light and measurement of distances

There are several methods to estimate the distances with supernovæ of the Ia type:

• Method MLCS rests on the construction of owner of lightcurve owner using a batch of drive, the lightcurves are then adjusted by varying two parameters of forme(riess1998 & tonry2003).

• Correction of $\Delta m_{15}$ (phillips1993, phillips1999, hamuy199ã, riess1998 & tonry2003).

• Correction of factor of stretching (perlmutter1999, goldhaber2001 & knop2003).

These methods rest all on the use of the luminosity to the maximum of the lightcurve in the filter B standard of the reference frame of the supernova.

The estimator of distances is:

$$m_{B}^{Ref} = {\cal M}_B^{max} + 25 - 5\log_{10}H_0 + 5\log_{10}{{\cal D}(z,\Omega_{\rm M}, \Omega_\Lambda )}$$ (6.4)

By using the factor of stretching $s_b$, this one becomes:

$$m_{B}^{Ref} = {\cal M}_B^{max} + 25 - 5\log_{10}H_0 + 5\log_{10}{{\cal D}(z,\Omega_{\rm M}, \Omega_\Lambda )} -\alpha ( 1-s_b)$$ (6.5)

where $\Delta m_{15}$, is written:

$$m_{B}^{Ref} = {\cal M}_B^{max} + 25 - 5\log_{10}H_0 + 5\log_{10}{{\cal D}(z,\Omega_{\rm M}, \Omega_\Lambda )} -\gamma ( \Delta m_{15} - 1.1)$$ (6.6)

There too, the parameter $\gamma$ is given at the time of the adjustment of cosmology.

Lastly, we saw that the supernovæ can undergo an absorption either in their galaxy host, or in the intergalactic medium. A measurement of the color will indicate if the supernovæ have undergoes or not a reddening, witness of this absorption and will allow a correction. This correction can be made in an explicit way by using a galactic extinction law as we saw before (knop2003).

This method presents several disadvantages however:

• It rests on laws of extinction determined in our Galaxy. Moreover, as we saw in chapter 2 , this law is not strictly universal. In particular, the evolution of the metalicity within the galaxy host could have an influence on these laws.

• The laws of extinction take into account only absorption in the galaxy host. An intergalactic extinction of different nature could have an influence.

• These laws rest on knowledge a priori intrinsic color of the supernova to the maximum. This thus excludes very taken into account from the intrinsic dispersion of color.

An alternative method rests on a statistical estimate of the extinction law according to the color of the supernovæ (regnault2000). In the same way that for the factor of stretching, one seeks to estimate the parameter $\beta$ such as:

$$m_{B}^{Ref} = {\cal M}_B^{max} + 25 - 5\log_{10}H_0 + 5\log_{10}{{\cal D}(z,_om, \Omega_\Lambda )} -\alpha ( 1-s_b) - \beta(m_B^{Ref} - m_V^{Ref})$$ (6.7)

This technique has the advantage of not making an assumption on the extinction law. Moreover, one observes residuals and thus the weakest dispersion for this method.

It is this method which we will use in the continuation to take our measurements of distances and cosmology.

We have from now on tools to make measurements of precise distance from the supernovæ of the type Ia. Explorons now the whole of the effects which could bias measurements.