Parameterization of Dark-Energy Properties: a Principal-Component Approach

TL;DR

It is argued that, in the absence of a compelling model for dark energy, the parametrizations of functions about which the authors have no prior knowledge, such as w(z), should be determined by the data rather than by their ingrained beliefs or familiar series expansions.

Abstract

Considerable work has been devoted to the question of how to best parameterize the properties of dark energy, in particular its equation of state w. We argue that, in the absence of a compelling model for dark energy, the parameterizations of functions about which we have no prior knowledge, such as w(z), should be determined by the data rather than by our ingrained beliefs or familiar series expansions. We find the complete basis of orthonormal eigenfunctions in which the principal components (weights of w(z)) that are determined most accurately are separated from those determined most poorly. Furthermore, we show that keeping a few of the best-measured modes can be an effective way of obtaining information about w(z).

Figures (2)

• Figure 1: The principal components of $w(z)$ for $\Omega_M$ perfectly known. The four best-determined and two worst-determined eigenvectors are shown and labeled for clarity. We have marginalized over the magnitude offset $\mathcal{M}$.
• Figure 2: Reconstruction of $w(z)$ by keeping only a fraction of eigenvectors so as to minimize risk. Top panel: illustration of the minimization of risk. Bottom left: optimal reconstruction (68% C.L. error bars shown) of fiducial $w(z)$ (solid line) that goes to zero at high-redshift end. Bottom right: optimal reconstruction of $w(z)$ that does not go to zero at high-redshift end. Also shown on this panel is optimal reconstruction of $1+w(z)$ for the same $w(z)$ model.