Fisher Matrix Decomposition for Dark Energy Prediction

TL;DR

This paper examines how Fisher matrix eigendecomposition depends on basis set choice and expansion order in dark energy studies, highlighting its potential and limitations for constraining w(z).

Contribution

It demonstrates the sensitivity of Fisher matrix eigenfunctions to basis set and order, and discusses implications for dark energy parameter estimation.

Findings

Eigenfunctions are not unique and require very high order expansions for accuracy.

Common marginalised eigenfunction errors are sensitive to parameter priors.

Fisher eigendecomposition can predict experiment accuracy and redshift sensitivity.

Abstract

Within the context of constraining an expansion of the dark energy equation of state w(z) we show that the eigendecomposition of Fisher matrices is sensitive to both the maximum order of the expansion and the basis set choice. We investigate the Fisher matrix formalism in the case that a particular function is expanded in some basis set. As an example we show results for an all sky weak lensing tomographic experiment. We show that the set of eigenfunctions is not unique and that the best constrained functions are only reproduced accurately at very higher order N > 100, a tophat basis set requires an even higher order. We show that the common approach used for finding the marginalised eigenfunction errors is sensitive to the choice of non-w(z) parameters and priors. The eigendecomposition of Fisher matrices is a potentially useful tool that can be used to determine the predicted accuracy with which an experiment could constrain w(z). It also allows for the reconstruction of the redshift sensitivity of the experiment to changes in w(z). However the technique is sensitive to both the order and the basis set choice. Publicly available code is available as part of iCosmo at http://www.icosmo.org .