12  Fisher Matrices in Cosmology

The discussion here parallels Chap. 10 in the Huterer textbook. I will take a Bayesian approach here, since that is most prevalent in cosmology.

12.1 Terminology

In cosmology, we are interested in inferring cosmological parameters given observational data. More formally, we are interested in the posterior probability distribution \(P(\text{params}|\text{data})\). By Bayes theorem, we can write this as \[ P(\text{params}|\text{data}) \propto P(\text{data}|\text{params}) P(\text{params}) \] where the first factor on the right is referred to as the likelihood while the second is the prior. The prior is meant to quantify our prior beliefs about the value of the data.

Let us write the data as a vector \(\mathbf{d}\) and the model similarly as \(\mathbf{\mu}(\{\theta \})\), where the parameters are \(\{\theta\}\). A very useful form of the likelihood \(\mathcal{L}\) is the Gaussian form \[ \mathcal{L}(\{\theta\}) = \frac{1}{(2\pi)^{N/2}|\mathbf{C}|^{1/2}} \exp\left(-\frac{1}{2}(\mathbf{d} - {\mu}(\{\theta\}))^\top \mathbf{C}^{-1} (\mathbf{d} - {\mu}(\{\theta\}))\right) \] where \(\mathbf{C}\) is the covariance matrix.

12.2 Fisher Matrices

Suppose you were designing an experiment and wanted to understand how the errors on your measurements translated into cosmological parameters. One approach would be to compute the Fisher matrix \[ F_{ij} = \bigg\langle - \frac{\partial^{2} \ln \mathcal{L}}{\partial \theta_{i} \partial \theta_{j}} \bigg\rangle \] The inverse of the Fisher matrix then gives you the best possible errors you can find (the Cramer-Rao bound) \[ \mathbf{C}_{\theta} = \mathbf{F}^{-1} \] For our Gaussian likelihood, you can show that \[ F_{ij} = \mu_{,i}^\top \mathbf{C}^{-1} \mu_{,j} + \frac{1}{2} \text{Tr}[\mathbf{C}^{-1} \mathbf{C}_{,i} \mathbf{C}^{-1} \mathbf{C}_{,j}] \] where we allow for the fact that our covariance depends on the parameters. If not, then we only have the first term.

Fisher forecasts are ubiquitous in cosmology. Below are two examples from the Frieman et al, 2008 review of dark energy.