On model selection forecasting, Dark Energy and modified gravity

The Fisher matrix approach (Fisher 1935) allows one to calculate in advance how well a given experiment will be able to estimate model parameters, and has been an invaluable tool in experimental design. In the same spirit, we present here a method to predict how well a given experiment can distinguish between different models, regardless of their parameters. From a Bayesian viewpoint, this involves computation of the Bayesian evidence. In this paper, we generalise the Fisher matrix approach from the context of parameter fitting to that of model testing, and show how the expected evidence can be computed under the same simplifying assumption of a gaussian likelihood as the Fisher matrix approach for parameter estimation. With this `Laplace approximation' all that is needed to compute the expected evidence is the Fisher matrix itself. We illustrate the method with a study of how well upcoming and planned experiments should perform at distinguishing between Dark Energy models and modified gravity theories. In particular we consider the combination of 3D weak lensing, for which planned and proposed wide-field multi-band imaging surveys will provide suitable data, and probes of the expansion history of the Universe, such as proposed supernova and baryonic acoustic oscillations surveys. We find that proposed large-scale weak lensing surveys from space should be able readily to distinguish General Relativity from modified gravity models.