Parameterizing Dark Energy at the density level: A two-parameter alternative to CPL

Abstract

We introduce a minimal two-parameter formulation of the dark energy (DE) density evolution normalized to its present-day value, $f_{\rm DE}(z) \equiv ρ_{\rm DE}(z)/ρ_{\rm DE,0}$, in terms of $f_p\equiv f_{\rm DE}(z_p)$ and the DE equation of state $w_p\equiv w(z_p)$, at a pivot redshift $z_p$. This provides an alternative framework for assessing the evidence for evolving DE, complementary to the established Chevallier-Polarski-Linder (CPL) parameterization. By parameterizing the DE density directly, the $(w_p,\,f_p)$ formulation avoids the approximate degeneracies intrinsic to the $(w_0,\,w_a)$ basis -- in particular the weak sensitivity of the expansion history to $w_a$ -- while reproducing the background evolution of representative quintessence models with equivalent accuracy. Confronting it with the latest baryon acoustic oscillation (BAO) measurements from DESI, a prior on early-universe parameters from Planck cosmic microwave background (CMB) observations, and Type Ia supernovae (SNe) data, we find that the $w_p$ and $f_p$ parameters are both tightly constrained and sensitive to distinct subsets of the data. Specifically, $w_p$ is measured to percent-level precision by BAO and CMB alone, while $f_p$ is pinned down by the independent matter density constraint that only SNe provide. Including the Pantheon+ SNe sample, we obtain $w_p = -1.04 \pm 0.04$ and $f_p = 1.07 \pm 0.04$, with similar results when using the DESY5 SNe sample. The preference for evolving DE over $Λ$CDM remains below $3σ$ across all dataset combinations, comparable to that obtained with CPL. Notably, the proximity of both $w_p$ and $f_p$ to their cosmological constant values of $(-1,1)$ -- precisely at the epoch where the data are most sensitive -- deepens the coincidence previously identified in the CPL framework, reinforcing the case for caution in interpreting the current evidence for dynamical DE.

Figures (13)

• Figure 1: Left panel: Contours of the maximum relative error $E_H$, Eq. \ref{['eq:EH']}, in the $(w_p,\,f_p)$ plane for a representative exponential quintessence scenario, Eq. \ref{['eq:exp']}, with $\lambda=0.7$. Contours correspond to $E_H = 0.5\%,\,1\%,\,5\%$ and $10\%$, colored as indicated in the legend, with the black diamond marking the best-fit point corresponding to $E_H\approx 0.1\%$. Dashed gray lines indicate the actual values of $w$ and $f_{\rm DE}$ at the pivot redshift $z_p$ for the underlying quintessence model. The inset on the lower left of the panel shows a zoomed-in view of the region enclosed by the 1% error contour. Right panel: Same as the Left panel but for the $(w_0,\,w_a)$ parameterization, with dashed gray lines now indicating the actual values of $w_0$ and $w_a$, defined as the zeroth- and first-order Taylor coefficients of $w(a)$ at $a=1$. The compact, non-degenerate contour structure in the $(w_p,\,f_p)$ plane, together with the near-coincidence of the model's physical properties at the pivot with the best fit, illustrates the fidelity and stability of the density-level parameterization relative to CPL.
• Figure 2: Left panel: Two-dimensional posteriors in $w_p$ and $f_p$ at the 68% and 95% confidence levels from the DESI + $Q_{\rm CMB}$ (black), DESI + $Q_{\rm CMB}$ + Pantheon+ (red), and DESI + $Q_{\rm CMB}$ + DESY5 (blue) dataset combinations. The pivot redshift is set to $z_p=0.5$, which sufficiently decorrelates the two parameters across all dataset combinations shown. The gray dashed lines mark the $\Lambda$CDM limit given by $(w_p,\,f_p) = (-1,\,1)$. Right panel: Same as the Left panel but for the $(w_0,\,w_a)$ parameterization, with dashed gray lines indicating again the $\Lambda$CDM limit, which here corresponds to $(w_0,\,w_a) =(-1,\,0)$. The inclusion of SNe dramatically tightens the constraints in both parameterizations, increasing the preference for evolving DE while simultaneously shifting all posteriors toward their $\Lambda$CDM values, with the $(w_p,\,f_p)$ formulation delivering percent-level measurements of both DE parameters individually.
• Figure 3: Normalized dark energy density, $f_{\rm DE}(z) \equiv \rho_{\rm DE}(z)/\rho_{\rm DE,0}$ ( Top panel), and corresponding equation of state parameter, $w(z)$ ( Bottom panel), as a function of redshift from the DESI + $Q_{\rm CMB}$ + Pantheon+ analysis. Results are shown for the $(w_p,\,f_p)$ (blue) and $(w_0,\,w_a)$ (gold) parameterizations, with solid lines indicating the posterior median and increasingly lighter shading denoting the 68% and 95% confidence level bands. The dashed gray line marks the $\Lambda$CDM limit, namely $f_{\rm DE} = 1$ and $w=-1$. Both parameterizations yield consistent reconstructions that remain close to the cosmological constant limit, with a DE density that rises mildly above unity at intermediate redshifts and an equation of state that remains marginally phantom over the range where the data have constraining power.
• Figure 4: Two-dimensional posteriors in $f_p$ and $H_0$ at the 68% and 95% confidence levels from the DESI + $Q_{\rm CMB}$ (black), DESI + $Q_{\rm CMB}$ + Pantheon+ (red), and DESI + $Q_{\rm CMB}$ + DESY5 (blue) dataset combinations, with the pivot redshift set to $z_p=0.5$ as in the main analysis presented in Sec. \ref{['sec:results']}. The dashed gray line marks the $\Lambda$CDM limit corresponding to $f_p=1$. All three dataset combinations yield constraints consistent with $\Lambda$CDM, with SNe tightening the posteriors significantly while shifting the central value mildly above unity.
• Figure 5: Same as Fig. \ref{['fig:1']} for a hilltop quintessence, Eq. \ref{['eq:hill']}, with $k=3/M_{\rm pl}$ and $\phi_i=10^{-2}\,M_{\rm pl}$. The same qualitative features are reproduced, with the $(w_p,\,f_p)$ plane again exhibiting compact contours that recover the physical DE properties at the pivot.