Validating the Fisher approach for stage IV spectroscopic surveys

¹ Université de Toulouse, UPS-OMP, IRAP and CNRS, IRAP, 14, avenue Edouard Belin, 31400 Toulouse, France
e-mail: alain.blanchard@irap.omp.eu
² Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy
³ INFN – Istututo Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
⁴ INAF – Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Torino, Strada Osservatorio 20, 10025 Pino Torinese, Italy
⁵ AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, 91191 Gif-sur-Yvette, France
⁶ Université PSL, Observatoire de Paris, Sorbonne Université, CNRS, LERMA, 75014 Paris, France
⁷ CEICO, Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, Praha 8, Czech Republic
⁸ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr, Pasadena, CA 91109, USA
⁹ School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK
¹⁰ Université St Joseph; UR EGFEM, Faculty of Sciences, Beirut, Lebanon
¹¹ Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
¹² Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain
¹³ Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

Received: 13 December 2019
Accepted: 28 July 2020

Abstract

In recent years, forecasting activities have become an important tool in designing and optimising large-scale structure surveys. To predict the performance of such surveys, the Fisher matrix formalism is frequently used as a fast and easy way to compute constraints on cosmological parameters. Among them lies the study of the properties of dark energy which is one of the main goals in modern cosmology. As so, a metric for the power of a survey to constrain dark energy is provided by the figure of merit (FoM). This is defined as the inverse of the surface contour given by the joint variance of the dark energy equation of state parameters {w₀, w_a} in the Chevallier-Polarski-Linder parameterization, which can be evaluated from the covariance matrix of the parameters. This covariance matrix is obtained as the inverse of the Fisher matrix. The inversion of an ill-conditioned matrix can result in large errors on the covariance coefficients if the elements of the Fisher matrix are estimated with insufficient precision. The conditioning number is a metric providing a mathematical lower limit to the required precision for a reliable inversion, but it is often too stringent in practice for Fisher matrices with sizes greater than 2 × 2. In this paper, we propose a general numerical method to guarantee a certain precision on the inferred constraints, such as the FoM. It consists of randomly vibrating (perturbing) the Fisher matrix elements with Gaussian perturbations of a given amplitude and then evaluating the maximum amplitude that keeps the FoM within the chosen precision. The steps used in the numerical derivatives and integrals involved in the calculation of the Fisher matrix elements can then be chosen accordingly in order to keep the precision of the Fisher matrix elements below this maximum amplitude. We illustrate our approach by forecasting stage IV spectroscopic surveys cosmological constraints from the galaxy power spectrum. We infer the range of steps for which the Fisher matrix approach is numerically reliable. We explicitly check that using steps that are larger by a factor of two produce an inaccurate estimation of the constraints. We further validate our approach by comparing the Fisher matrix contours to those obtained with a Monte Carlo Markov chain (MCMC) approach – in the case where the MCMC posterior distribution is close to a Gaussian – and finding excellent agreement between the two approaches.