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Fisher Forecast

Fisher Matrices can be used to forecast parameter uncertainties with few computational resources. They are encapsulated in the following struct

We implement two different approaches to evaluate the Fisher Matrix:

  • A field approach, where we consider the $a_{\ell m}$ to be the observable
  • An estimator approach, where we consider the $C_\ell$'s to be the observable

If correctly implemented, the two approaches give the same result (Hamimeche & Lewis 2008, Carron 2012).

Field approach

The Fisher Matrix in the Field approach is evaluated by the following method

CosmoCentral.ForecastFisherαβ — Method
ForecastFisherαβ(PathCentralCℓ::String, Path∂Cℓ::String,
InputList::Vector{Dict{String, Vector{Any}}}, CosmologicalGrid::CosmologicalGrid)

This function evaluate the Fisher Matrix according to the following formula:

\[F_{\alpha \beta}=\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \operatorname{Tr}\left \{[\boldsymbol{\Sigma}(\ell)]^{-1} \frac{\partial \mathbf{C}(\ell)}{\partial \alpha}[\boldsymbol{\Sigma}(\ell)]^{-1} \frac{\partial \mathbf{C}(\ell)}{\partial \beta}\right\},\]

where $\alpha$ and $\beta$ are parameters of the Fisher Matrix, $\ell$ are the multipoles, $\Sigma$ is the Covariance Matrix of the Field Approach and $\frac{\partial \mathbf{C}(\ell)}{\partial \alpha}$ is the Matrix of the derivatives of the $C_\ell$ wrt parameter $\alpha$.

source

Estimator approach

The Fisher Matrix in the Estimator approach is evaluated by the following method

CosmoCentral.ForecastFisherαβ — Method
ForecastFisherαβ(PathCentralCℓ::String, Path∂Cℓ::String,
InputList::Vector{Dict{String, Vector{Any}}}, CosmologicalGrid::CosmologicalGrid,
ciccio::String)

This function evaluate the Fisher Matrix according to the following formula:

\[F_{\alpha \beta}=\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \operatorname{vecp} \left(\frac{\partial \mathbf{C}(\ell)}{\partial \alpha}\right)^{T} \left(\boldsymbol{\Xi}(\ell)\right)^{-1} \operatorname{vecp}\left(\frac{\partial \mathbf{C}(\ell)} {\partial \beta}\right),\]

where $\alpha$ and $\beta$ are parameters of the Fisher Matrix, $\ell$ are the multipoles, $\Xi$ is the Covariance Matrix of the Field Approach CℓCovariance and $\frac{\partial \mathbf{C}(\ell)}{\partial \alpha}$ is the Matrix of the derivatives of the $C_\ell$ wrt parameter $\alpha$.

source

Plotting Fisher Matrix

Here we show plots for some Fisher Matrices (this code will probably soon released in a separate package).

canvas = FisherPlot.PrepareCanvas(BigLaTeXArray, central_values, limits, ticks, probes, colors,
PlotPars)
FisherPlot.PaintCorrMattrix!(canvas, central_values,
CosmoCentral.SelectCorrelationMatrix(FisherWL, pars_list), "deepskyblue3")
FisherPlot.PaintCorrMattrix!(canvas, central_values,
CosmoCentral.SelectCorrelationMatrix(FisherGCNew, pars_list), "darkorange1")
FisherPlot.PaintCorrMattrix!(canvas, central_values,
CosmoCentral.SelectCorrelationMatrix(FisherGCNewplusWL, pars_list), "green")
FisherPlot.PaintCorrMattrix!(canvas, central_values,
CosmoCentral.SelectCorrelationMatrix(FisherFinal, pars_list), "red")
canvas