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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, wa} 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.
Highlights
Since the discovery of the acceleration of the expansion of the Universe in the late 1990s (Riess et al 1998; Perlmutter et al 1999), the Lambda Cold Dark Mattery (ΛCDM) model remains the most successful model in cosmology
The steps used in the numerical derivatives and integrals involved in the calculation of the Fisher matrix elements can be chosen in order to keep the precision of the Fisher matrix elements below this maximum amplitude
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
Summary
Since the discovery of the acceleration of the expansion of the Universe in the late 1990s (Riess et al 1998; Perlmutter et al 1999), the Lambda Cold Dark Mattery (ΛCDM) model remains the most successful model in cosmology It provides a simple and accurate description of the properties of the Universe, with a very limited number of parameters that are constrained at the level of a few percent by using measurements from the Planck CMB mission (Planck Collaboration VI 2020) and other experiments (see e.g., Percival et al 2001; Blake et al 2011; Alam et al 2017; Abbott et al 2018). In order to quantify the performance of a survey and, in particular, its ability to constrain the properties of dark energy, the Dark Energy Task Force (DETF) defined a metric known as the figure of merit (FoM, Albrecht et al 2006): a figure inversely proportional to the surface bounded by the confidence contours for the w0 and wa parameters from the Chevalier-Polarski-Linder
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