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Various conclusions can be drawn from the information criterion approach. Most importantly, they provide a simple objective criterion for the inclusion of new parameters into the standard cosmological model. For example, it is sometimes said that the WMAP analysis actually mildly favours a closed cosmological model, as their best-fit value is Omegak = 1.02 ± 0.02 (at 1-sigma). However, the information criteria lead to the opposite conclusion: they say that the most appropriate conclusion to draw is that the spatial curvature is not needed as a parameter, and hence it is more likely that the observations were generated in a spatially-flat Universe. That's not to say that future observations might not show that the Universe is closed, but a much higher significance level than 1-sigma is needed before it becomes the best description of the data in hand. Similar arguments can be applied also to parameters such as running of the spectral index; even in the absence of controversy over the use of lyman-alpha forest data, it seems likely that the information criteria would reject the running as a useful parameter (I can't test it, as the WMAP team don't feel the data are reliable enough even to quote a maximum likelihood). In general, a 95% `detection' of a particular new parameter cannot be taken to imply that the base model, without that parameter, is ruled out at anything like that significance.

According to the information criteria, the best current cosmological model features only five fundamental parameters and two phenomenological ones, as listed in Table 1. While there is an elegant simplicity to this model which is satisfying, such simplicity does come at a cost, because the cosmological parameters are what tells about the physical processes relevant to the evolution of the Universe. That there are so few parameters is telling us that there is very little physics that we are currently able to probe observationally. Accordingly, we should be hoping that new observational data is powerful enough to promote parameters from the candidate list to the base list; for example, we won't be able to say anything quantitative about how cosmological inflation might have taken place unless n - 1, and ideally r as well, make their way into the standard cosmological model.

The information criteria appear well suited to providing an objective criterion for the incorporation of new parameters, and have had considerable testing across many scientific disciplines. The BIC appears to be preferred to the AIC for cosmological applications. For the size of the current dataset the BIC penalizes extra parameters very strongly, indicating that a very high-significance detection is needed to justify adoption of a new parameter.


This research was supported in part by PPARC. I thank Charles Goldie for directing me to the literature on the information criteria, and Sarah Bridle, Martin Kunz, Sam Leach and Max Tegmark for helpful discussions.

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