Abstract
The Fisher matrix formalism outlined in Sect. 5.3 gives a useful methodology for predicting the error ellipses around the fiducial model on the parameters of interest for future proposed astrophysical probes of cosmology. One of the most common Figures of Merit (FoM) used as a metric for comparing proposed probes is the inverse area of the error ellipse derived from the Fisher matrix formalism [1, 2] which gives a measure of the expected statistical power or ability of a probe to be able to constrain the parameters of interest. Alternative FoMs in higher dimesnions are given by [3]–Mortonson et al. [7], and a more general Bayeisan approach to FoMs is given in [8]. The purpose of the FoM is to evaluate in advance the expected statistical power of future probes. Survey parameters can be adjusted in order to maximise the statistical power of a particular probe, and proposed probes can be ranked by their FoM. This ranking can then assist in the decision making process of how to allocate limited resources to get the best science return.
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- 1.
Notice that as we average \(R\) along \(b\) we do not re-evaluate the Fisher matrix of the probe as a function of \(b\), but we simply translate the Fisher matrix found at the fiducial point (i.e., the true parameters values). The Fisher matrix typically depends only weakly on the fiducial model chosen, as long as we consider the models within the parameter confidence region. If the bias vector is not much larger than the statistical errors we can therefore approximate the Fisher matrix at the biased parameters values with the one evaluated at the fiducial point.
- 2.
An alternative test would be to check whether the \(N\)th probe is compatible with the previous \(N-1\) (assuming those are already available and they are free of systematics themselves). In this case the relevant quantity is
$$\begin{aligned} R_{\text{ N}}=\frac{p(d_{N}|d_{N-1}\dots d_{1})}{p(d_{N})p(d_{N-1}\dots d_{1})} \end{aligned}$$(8.37)which can be computed by appropriate substitutions in Eq. (8.22).
- 3.
For multiple parameters, there is ambiguity to define the worst-case error, since a sign of \(\delta m_{\mathrm{{ sys}}}(z_{\alpha })\) that makes excursion in \(w_{0}\) positive may actually make the \(w_{a}\) excursion negative or vice versa. (Footnote continued) We make a choice that the excursion in \(w_{0}\) is positive in a given redshift bin, which determines the sign of \(\delta m_{\mathrm{{ sys}}}(z_{\alpha })\); then the excursion in \(w_{a}\) in that bin is simply \(c_{\alpha }^{(w_{a})}\,\delta m_{\mathrm{{ sys}}}(z_{\alpha })\).
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March, M.C. (2013). Robustness to Systematic Error for Future Dark Energy Probes. In: Advanced Statistical Methods for Astrophysical Probes of Cosmology. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35060-3_8
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