On model selection forecasting, Dark Energy and modified gravity

TL;DR

This work tackles predicting, in a Bayesian sense, how well future cosmological experiments can distinguish Dark Energy from modified gravity. It generalizes the Fisher-matrix approach from parameter estimation to model selection by employing a Laplace approximation to compute the expected Bayesian evidence, allowing the expected Bayes factor $\langle B\rangle$ to be derived from the Fisher matrices $F$ and $F'$ for nested models. The authors derive the central expression $\langle B\rangle = (2\pi)^{-p/2} \frac{\sqrt{\det F}}{\sqrt{\det F'}} \exp(-\tfrac{1}{2}\delta\theta_\alpha F_{\alpha\beta}\delta\theta_\beta) \prod_{q=1}^p \Delta\theta_{n'+q}$ and show how parameter offsets $\delta\theta'\!$ arise when the wrong model is assumed, with an accompanying Occam penalty from the prior-volume ratio. Applying the method to combinations of Planck, 3D weak lensing surveys, SN, and BAO, the study demonstrates that space-based wide-field lensing can decisively distinguish General Relativity from DGP-type modified gravity (with measurable growth-rate differences $\delta\gamma$), especially when distance probes are included. Overall, the approach provides a fast, principled design tool for assessing a survey's capacity to discriminate between competing cosmological theories, while acknowledging the Gaussian-likelihood assumption as a limitation.

Abstract

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.

Figures (2)

• Figure 1: Illustrating how assumption of a wrong parameter value can influence the best-fitting value of other model parameters. Ellipses represent iso-likelihood surfaces, and here in the simpler model, the parameter on the horizontal axis is assumed to take the value given by the vertical line. Filled circles show the true parameters in the more complicated model, and the best-fit parameters in the simpler model.
• Figure 2: The expected value of $|\ln B|$ from WL$_{NG}$ (solid), Pan-STARRS (dot-dashed) and DES (dashed), in combination with CMB constraints from Planck, as a function of the difference in the growth rate between the modified-gravity model and General Relativity. The crossover at small $\delta\gamma$ occurs because Occam's razor will favour the simpler (General Relativity) model unless the data demand otherwise. To the left of the cusps, GR would be likely to be preferred by the data. The dotted vertical line shows the offset of the growth factor for the DGP model. The horizontal lines mark the boundaries between 'inconclusive', 'significant', 'strong', and 'decisive' in Jeffreys' (1961) terminology.