The essence of quintessence and the cost of compression

TL;DR

The paper demonstrates that common two-parameter compressions of the dark energy equation of state $w_{DE}(z)$ inadequately capture dynamic behavior, including rapid evolution that can fit SN-Ia data well. It compares Taylor-type expansions (in redshift, scale factor, and logarithmic bases) up to higher orders with a physically-motivated four-parameter Kink model, using SN-Ia constraints, information criteria, and Bayesian evidence. The results show that two-parameter models can falsely exclude rapid dynamics and misestimate the acceleration redshift $z_{acc}$, while higher-order, decorrelated parametrisations reveal degeneracies and caution against over-interpretation; nonetheless, $\,\Lambda$CDM remains favored by model selection metrics. The study emphasizes the need for at least three-parameter, decorrelated compressions to provide robust, high-precision inferences for future dark energy cosmology and survey design.

Abstract

Standard two-parameter compressions of the infinite dimensional dark energy model space show crippling limitations even with current SN-Ia data. Firstly they cannot cope with rapid evolution - the best-fit to the latest SN-Ia data shows late and very rapid evolution to w_0 = -2.85. However all of the standard parametrisations (incorrectly) claim that this best-fit is ruled out at more than 2-sigma, primarily because they track it well only at very low redshifts, z < 0.2. Further they incorrectly rule out the observationally acceptable region w << -1 for z > 1. Secondly the parametrisations give wildly different estimates for the redshift of acceleration, which vary from z_{acc}=0.14 to z_{acc}=0.59. Although these failings are largely cured by including higher-order terms (3 or 4 parameters) this results in new degeneracies which open up large regions of previously ruled-out parameter space. Finally we test the parametrisations against a suite of theoretical quintessence models. The widely used linear expansion in z is generally the worst, with errors of up to 10% at z=1 and 20% at z > 2. All of this casts serious doubt on the usefulness of the standard two-parameter compressions in the coming era of high-precision dark energy cosmology and emphasises the need for decorrelated compressions with at least three parameters.

Figures (4)

• Figure 1: Expansion order is more important than parametrisation. 1-d marginalised likelihoods for the different classes of parametrisations. The narrow curves are the linear likelihoods (constant $w_{DE}$, $n=0$, is dotted, first-order expansions, $n\leq 1$, are dashed). The solid lines show the likelihoods at second-order ($n\leq 2$) and are much wider due to degeneracies and the absence of a lower-bound on $w$ for $z\geq 1$. The curves correspond to expansions in (a) scale-factor, (b) log and (c) redshift respectively. Finally the 1-d likelihood for $w_m$ of the Kink is shown (dot-dashed) and exhibits strongly non-Gaussian wings that extend to very large values of $|w_m|$. For definition of the order of the expansion see eq. (\ref{['taylor']}).
• Figure 2: Parametrisations struggle with rapid evolution. Maximised limits on $w_{DE}(z)$ for the redshift (red dashed line), scale-factor (green dash-dotted), logarithmic (blue dotted) and Kink (solid black line) parametrisations. The best-fit kink solution passes well outside the limits of all the parametrisations (except for the kink) both at $z\sim 0$ and at $z\sim 0.2$, showing their inability to capture rapid dynamics which leads to their incorrectly ruling it out.
• Figure 3: When did acceleration begin? The marginalised 1d likelihood for $z_{acc}$ as estimated from the same SN-Ia data for each of the various dark energy parametrisations. The wide variance casts doubt on their usefulness. All of them peak at a lower $z_{acc}$ than in the standard $\Lambda$CDM, perhaps giving novel evidence for dark energy dynamics. Only the scale-factor expansion, eq. (\ref{['linder']}) shares significant overlap with $\Lambda$CDM, but that is expected since the transition is typically slow in that parametrisation.
• Figure 4: Plot showing the mean quadratic error of the best-fit, eq. (\ref{['res']}), to the different quintessence models as function of $z_{max}$ for each parametrisation for $n\leq 1$. At $z\sim 1$ the expansion to first order in $z$ ('redshift') and $\log(1+z)$ ('logarithmic') typically show errors around 10% while the kink and scale-factor parametrisations have errors typically at the 1% level at $z \sim 1$.