Reconstruction of dark energy and expansion dynamics using Gaussian processes

TL;DR

This work tackles the challenge of inferring the dark energy equation of state $w(z)$ from observations without relying on restrictive parametrizations. It introduces a non-parametric, fully Bayesian Gaussian Process approach to reconstruct the distance measure $D(z)$ and its derivatives, from which $H(z)$, $q(z)$, and $w(z)$ are derived. Applying the method to DES-like mock data and real Union2.1 SN data, the authors demonstrate that DES can recover a slowly evolving $w(z)$ with tight low-redshift errors and increasing uncertainties at higher redshift, while $H(z)$ and $q(z)$ are inferred with smaller relative errors. The GP framework provides a robust, flexible tool for model-independent probes of expansion dynamics and dark energy, with publicly available code to compute functions and derivatives from data.

Abstract

An important issue in cosmology is reconstructing the effective dark energy equation of state directly from observations. With few physically motivated models, future dark energy studies cannot only be based on constraining a dark energy parameter space, as the errors found depend strongly on the parameterisation considered. We present a new non-parametric approach to reconstructing the history of the expansion rate and dark energy using Gaussian Processes, which is a fully Bayesian approach for smoothing data. We present a pedagogical introduction to Gaussian Processes, and discuss how it can be used to robustly differentiate data in a suitable way. Using this method we show that the Dark Energy Survey - Supernova Survey (DES) can accurately recover a slowly evolving equation of state to sigma_w = +-0.04 (95% CL) at z=0 and +-0.2 at z=0.7, with a minimum error of +-0.015 at the sweet-spot at z~0.14, provided the other parameters of the model are known. Errors on the expansion history are an order of magnitude smaller, yet make no assumptions about dark energy whatsoever. A code for calculating functions and their first three derivatives using Gaussian processes has been developed and is available for download at http://www.acgc.uct.ac.za/~seikel/GAPP/index.html .

Figures (9)

• Figure 1: Left: Dark energy reconstruction using Eq. \ref{['w']} for $\Omega_k=0$, when $D(z)$ and its derivatives are exactly known ($\Lambda$CDM with $\Omega_m=0.275$). The uncertainty in the reconstructed $w$ (the blue shaded regions show the 68% and 95% CL) only comes from the prior on the matter density, $\Omega_m=0.275\pm 0.016$ (WMAP7 Komatsu). Right: The same plot for fixed matter density, $\Omega_m=0.275$, and a prior on the curvature of $\Omega_k=0.00 \pm 0.01$.
• Figure 2: Left: Prior of the Gaussian process as given by Eq. \ref{['prior']}. Note that the hyperparameters have not been trained yet. Right: Posterior of the Gaussian process as given by Eq. \ref{['post']}.
• Figure 3: $f^{(10)}(x)$ from equation \ref{['sin-waves']} and its first and second derivative.
• Figure 4: The function $f^{(N)}(x)$ from Eq. \ref{['sin-waves']} and its first and second derivatives are reconstructed using a GP, for different numbers of superpositions of sine waves $N$. We show the fractions of the range interval where the true function (top left) and its derivatives (first, top right; second, bottom left) a lie between the expected 1- and 2-$\sigma$ limits (1-$\sigma$: red points, 2-$\sigma$: blue points; the red and blue lines are the respective expectation values). Each point is the result of averaging over 200 mock data sets. Data sets with a smaller number of data are slightly shifted to the left, those with larger numbers to the right.
• Figure 5: Reconstruction of $D(z)$, $D'(z)$ and $D"(z)$ obtained from a mock data set following DES specifications and assuming a $\Lambda$CDM model with $\Omega_m=0.3$ (red line). The shaded blue regions are the 68% and 95% CL of the reconstruction.