How many cosmological parameters?

Constraints on cosmological parameters depend on the set of parameters chosen to define the model that is compared with observational data. I use the Akaike and Bayesian information criteria to carry out cosmological model selection, in order to determine the parameter set providing the preferred fit to the data. Applying the information criteria to the current cosmological data sets indicates, for example, that spatially flat models are statistically preferred to closed models, and that possible running of the spectral index has lower significance than inferred from its confidence limits. I also discuss some problems of statistical assessment arising from there being a large number of 'candidate' cosmological parameters that can be investigated for possible cosmological implications, and argue that 95 per cent confidence is too low a threshold to identify robustly the need for new parameters in model fitting. The best present description of cosmological data uses a scale-invariant (n = 1) spectrum of Gaussian adiabatic perturbations in a spatially flat Universe, with the cosmological model requiring only five fundamental parameters to specify it fully.