3.2. Application to WMAP+SDSS data

I will use the results from comparison of models to WMAP plus SDSS data given in Tegmark et al. (2003, henceforth T03). Much of the analysis in that paper focusses on a simple parameter set called the `vanilla' model or sometimes the `six parameter' model. Confusingly, it actually features seven parameters (they do not count the bias parameter, although it is an independent fit parameter). They are not quite the set given in Table 1; the radiation density parameter is omitted for reasons I explained above, while the spectral index n is included as an independent parameter. However n - 1 is not actually detected to be non-zero; its 1-sigma confidence range (table 4, column 6 of T03) is 0.952 < n < 1.016. In light of the above discussion, we might expect that the information criteria reject the inclusion of n - 1 as a useful parameter, and indeed that is the case.

The ² values quoted by T03 are derived using the WMAP likelihood code, and are defined as -2 ln (see Verde et al. (2003) and Spergel et al. (2003) for details). The total number of datapoints N (not corrected for the number of parameters in the fit) is N = 1367 (899 WMAP temperature spectrum, 449 WMAP polarization cross-correlation, 19 SDSS).

As seen in the upper two rows of Table 3, both information criteria prefer the base model, with n fixed at one, as opposed to letting n vary. As has been remarked before, there is presently no evidence that the parameter n - 1 is needed to fit present data. T03 draw the same conclusion on subjective grounds, and refer to the base model as `vanilla lite'.

A similar argument applies to other cosmological parameters. Unfortunately the other models analyzed by T03 include variation of n (their table 3) and so other parameters are not directly compared with the base model, but anyway the trend seen in Table 3 is clear - the more parameters included the higher the AIC and BIC as compared to the base model. The need for these additional parameters is strongly rejected by the information criteria, particularly the BIC which strongly penalizes additional parameters for a dataset of this size. ³ For example, the information criteria reject the need for _k as an independent parameter, instead identifying spatially-flat models as the preferred description of the data.

Table 3. AIC and BIC for the various models, with likelihood values taken from tables 3 and 4 of T03. The upper two rows compare the base model with the addition of n as an extra parameter. The lower entries show various other combinations of parameters. I drop the radiation density from the parameter list as it is not needed to fit these data.

Model	parameters	-2 ln	AIC	BIC
Base model	6	1447.9	1459.9	1491.2
Base + n	7	1447.2	1461.2	1497.7
Base + n, k	8	1445.4	1461.4	1503.2
Base + n,r	8	1446.9	1462.9	1504.7
Base + n,r, d n / d ln k, k	10	1444.4	1464.4	1516.6

³ It is interesting to note that recent applications of the Bayesian evidence to cosmological model selection have also found no significant evidence against the simplest model considered (Slosar et al. 2003; Saini et al. 2003; Niarchou et al. 2003). Back.