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Astron. Astrophys. 331, 815-820 (1998)

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3. Results

Using tabulated data (Hamuy, 1996) from the 29 Cal [FORMULA] n/Tololo supernovae and keeping q0 fixed at 0.385, we search the space of b and R for the value of H0 yielding the lowest [FORMULA]. This best fit results in an unusually small reduced [FORMULA] of 0.51 to be compared to 0.97 expected for 29 data points and 3 parameters (b, R, and H0), leading to an overly high confidence level (CL) of 0.98. For this minimum in [FORMULA], b = 0.52, R = 2.09 and H0 = 60.1. Fig. 2 displays H0 vs. [FORMULA], [FORMULA], and z for the above fit. All show that H0 = 60.1 describes the data very well over the full ranges of these quantities. Fig. 2c, shown for completeness, is the analogue of the conventional Hubble diagram but with a clearer display of the total error for each supernova.

[FIGURE] Fig. 2. a  Plot of H0 vs. the [FORMULA] color, b  H0 vs. [FORMULA], and c  H0 vs. z for the best fit of 29 Cal [FORMULA] n/Tololo SNe for a [FORMULA] slope parameter b = 0.52 and a [FORMULA] color parameter R = 2.09. A best-fit value of H0 = 60.1 passes within 1.5 [FORMULA] of all 29 supernovae, indicating that the fit is much better than can be expected with these errors. Quantitatively, the reduced [FORMULA] = 0.51, leading to a confidence level CL = 0.98.

Thus, by introducing one additional parameter R, we have achieved a very substantial reduction in [FORMULA] compared to our previous analysis. There, without a color correction, the best fit obtained when the smaller sample of 26 Cal [FORMULA] n/Tololo SNe that passed the color cut [FORMULA] plus the six cosmological SNe yielded a large reduced [FORMULA] = 1.47 with a low confidence level of CL = 0.05. Since these two studies use somewhat different sets of data, a closer comparison would be to the best fit of 29 Cal [FORMULA] n/Tololo supernovae alone which yields for R set to zero, b = 1.09 and CL = [FORMULA] while a fit to 26 Cal [FORMULA] n/Tololo supernovae passing the color cut, yields b = 0.80 and CL = 0.027 for R = 0.

It could be argued that the low [FORMULA] merely reflects the introduction of yet another uncertainty [FORMULA]) multiplied by R in Eq. (3). However a convincing argument that this is not the case can be made by arbitrarily setting both the measurement uncertainties [FORMULA]) and [FORMULA] to zero. When this is done and the [FORMULA] minimized, the reduced [FORMULA] becomes 0.94 (CL = 0.55), i.e. a highly probable representation of the data using for measurement uncertainties only the apparent magnitude uncertainties [FORMULA] and the assumed peculiar velocity [FORMULA], which is generally considered a minimal estimate of peculiar velocity. Thus the introduction of a second parameter R has indeed improved the fit and is fully justified. As an alternative (and more realistic) way to obtain a more likely confidence level of about 0.5, all measurement errors of the 29 Cal [FORMULA] n/Tololo SNe could be scaled by a factor of 0.55.

The above three fits with different treatment of errors, as well as others with b or R or both set to zero, are tabulated in Table 1. It is interesting to note that setting b = 0 yields an acceptable confidence level for all 29 SNe, while setting R = 0 does not. Thus the [FORMULA] color correction appears to be more effective in standardizing SNe Ia than does the presently popular shape correction [FORMULA], although both appear to be necessary as can be seen in Fig. 3. Fig. 3a,b shows the impact of setting both b and R to zero. A best-fit value of H0 = 53.9 agrees with neither plot; each displays a strong dependence of H0 on [FORMULA] or [FORMULA]. Fig. 3c,d and Fig. 3e,f show the effect of setting b = 0 or R = 0 respectively. Here the b = 0 solution despite a satisfactory confidence level, nonetheless shows a downward trend in H0 vs. [FORMULA]. Even more evident is the consequence of setting R = 0 which produces a visibly non-zero slope of H0 vs. [FORMULA]. As can be seen from Fig. 3e, this problem persists even for the color-selected sample of 26 Cal [FORMULA] n/Tololo SNe previously investigated by Hamuy et al. (1996) and Tripp (1997). The best-fit values found when these 26 supernovae are fitted by themselves under the various conditions are also listed in Table 1.


Table 1. Quantities associated with the best fits to all 29 and to a color-selected sample of 26 SNe when the parameters b or R or both are set to zero or are free parameters. In the latter case, results of three treatments of the measured uncertainties are listed.

[FIGURE] Fig. 3. Examples of unacceptable fits that occur without b and/or R corrections. Plot of H0 vs. the [FORMULA] color and H0 vs. [FORMULA] for the best-fit value of H0 for the 29 Cal [FORMULA] n/Tololo SNe: a,b  without corrections (b = 0 and R = 0) where H0 = 53.9, c,d  with no [FORMULA] correction (b = 0, R = 2.8) where H0 = 57.1, e,f  with no color correction (R = 0, b = 1.09) where H0 = 60.9.

Fig. 4 is a plot of confidence level (with measurement uncertainties scaled by 0.55) as a function of b and R, showing that there is some negative correlation between these parameters. Consequently by setting one parameter to zero as can be noted from Table 1, there is for the best-fit solution a compensatory increase in the other parameter. Further consequences of these omissions, as mentioned above and seen in Fig. 3, are less-than-satisfactory fits of H0 vs. the neglected quantity.

[FIGURE] Fig. 4. Confidence level for 29 supernovae as a function of b and R using the measurement uncertainties reduced by a factor of 0.55. The uncertainties for b and R, including correlations, are [FORMULA].

Table 1 also lists under the various conditions the dispersion of the data points expressed in magnitudes. This is obtained from the standard deviation in the calculated values of H0 divided by H0 ln(10)/5 in order to express it in magnitudes. From the table it can be seen that the uncorrected data (b = R = 0) has for 29 SNe a dispersion of 0.35 mag and for the color-selected sample of 26 SNe a dispersion of 0.25 mag. This is to be compared with typically 0.15 mag of dispersion for various b and R corrected good and overly good fits. Since the dispersion itself takes no account of measurements uncertainties and changes by only a modest amount in going from completely unacceptable fits to overly good fits, it is not a useful indication of the quality of the fit. A much more sensitive measure of whether a parametrization fits the data is obtained from the confidence level or, alternatively, reduced [FORMULA]. However, even then, as noted above and seen in Fig. 3d, one should also inspect the fit to be reassured that an adequate [FORMULA] does not hide a trend in the data.

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Online publication: March 3, 1998