4.1. Type Ia supernova Hubble diagram

The luminosity distance receives a cosmology dependent correction as z increases; in a way pulls down d_L and pushes it up. (In the first order of z the correction enters in the combination of q₀ = /2 - , so this is often referred to as a q₀ test.) The discovery of two groups that distant supernovae are fainter than are expected from the local sample, even fainter than are expected for q₀ = 0, points to the presence of > 0 (Riess et al. 1998; Schmidt et al. 1998; Perlmutter et al. 1999).

The general difficulty with such a Hubble diagram analysis is that one has to differentiate among a few interesting cosmologies with small differences of brightness. For instance, at z = 0.4 where many supernovae are observed, the difference is m = 0.12 mag between (, ) = (0.3,0.7) and (0, 0), and m = 0.22 from (0, 0) to (1.0,0). Therefore, an accuracy of ( 5%) must be attained including systematics to conclude the presence of . On the other hand, there are a number of potential sources of errors:

(i) K corrections evaluated by integrating spectrophotometric data that are dominated by many strong features;

(ii) relative fluxes at the zero point (zero mag) across the colour bands;

(iii) dust obscuration in a host galaxy;

(iv) subtraction of light from host galaxies;

(v) identification of the maximum brightness epoch, and estimates of the maximum brightness including a m₁₅ correction;

(vi) selection effects (for high z SNe);

(vii) evolution effects.

Except for (vii), for which we cannot guess much ⁽⁵⁾, the most important seems to be combined effects of (i), (ii) and (iii). It is not easy a task to reproduce a broad band flux by integrating over spectrophotometric data convoluted with filter response functions, especially when spectrum contains strong features. (Even for the spectrophotometric standard stars, the synthetic magnitude contains an error of 0.02-0.05 mag, especially when the colour band involves the Balmer or Paschen regions.) Whereas Perlmutter et al. assigns 0.02 mag to the error of (i) [and (ii)], a comparison of the two values of estimated maximum brightness in their 1997 paper (Perlmutter et al. 1997, where they claimed evidence for a high universe) and the 1999 paper (TABLE 8) shows a general difficulty in the evaluation of the K correction (the difference dominantly comes from different K corrections). Schmidt et al. claim that their K correction errors are 0.03% mag. Dust obscuration (iii) is also an important source of errors, since the error of (i)+(ii) propagates to E(B - V) and is then amplified with the R factor. So a 0.02 mag error in colour results in a 0.06 mag error in A_V.

We note that each SN datum contains ± 0.2 mag (20%) error. The issue is whether this error is almost purely of random nature and systemtics are controlled to a level of 0.05.