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

We introduce a minimal two-parameter formulation of the dark energy (DE) density evolution normalized to its present-day value, $f_{\rm DE}(z) \equiv \rho_{\rm DE}(z)/\rho_{\rm DE,0}$, 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 $\Lambda$CDM remains below $3\sigma$ 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.