Revealing evolution of Dark Energy density from observations

TL;DR

This work develops a model-independent inversion framework to reconstruct the normalized dark energy density $X(z)$ directly from the luminosity-distance data of the DES-SN5YR supernova sample by exploiting the derivative of the distance modulus, $\mu'(z)$. It contrasts this nonparametric approach with parametric forms such as the CPL $w(z)$ and a quadratic $X^2(z)$ parametrization, applying careful binning strategies (17 and 34 bins) and Monte Carlo error propagation to control noise. The reconstructed $X(z)$ is statistically consistent with a constant density ($X(z) \approx 1$) across the redshift range, with only a mild curvature at intermediate to high redshift that the quadratic form can accommodate; CPL provides the best fit among the tested models, albeit modestly. Overall, the method offers a transparent, data-driven diagnostic of dark energy evolution and is well poised to leverage future high-precision data from DESI, LSST, Euclid, and the Roman Space Telescope to reveal or constrain subtle time variation in the dark energy density.

Abstract

We present a model--independent reconstruction of the normalized dark energy density function, $X(z) \equiv ρ_{\mathrm{de}}(z)/ρ_{\mathrm{de}}(0)$, derived directly from the DES-SN5YR Type~Ia supernova sample. The analysis employs an inversion formalism that relates the derivative of the distance modulus, $μ^{\prime}(z)$, to the expansion history, allowing the data to determine the shape of $X(z)$ without assuming a specific equation--of--state or dark energy density parameterization. A statistically optimized binning of the supernova sample (using 17 intervals following the Freedman--Diaconis criterion and 34 following Scott's rule) ensures a stable estimation of $μ^{\prime}(z)$ and a controlled propagation of uncertainties throughout the inversion process. The resulting $X(z)$ remains statistically consistent with a constant value within one standard deviation across the entire redshift range, showing no significant evidence for an evolving dark energy component at present. In a direct comparison among $Λ$CDM, CPL, and the quadratic $X^2(z)$ parameterization -- where CPL and $X^2(z)$ each introduce two additional free parameters relative to $Λ$CDM -- the CPL model attains the best statistical agreement with the data, albeit only marginally and strictly within this restricted model set. These outcomes indicate that current observations are compatible with an almost constant dark energy density ($w \simeq -1$), while the inversion framework remains sensitive to subtle departures that forthcoming high--precision surveys could resolve.

Figures (2)

• Figure 1: Derivative of the distance modulus $\mu'(z)$ reconstructed from the DES-SN5YR supernova sample. The smooth black curve represents the spline interpolation, while the shaded band indicates the $1\sigma$ uncertainty derived from Monte Carlo realizations.
• Figure 2: Comparison of the reconstructed dark energy density $X(z)$ using two statistically motivated binning schemes. Blue points with light–blue error bars show the non–parametric inversion; the black dashed line represents $\Lambda$CDM, and the red and green curves correspond to the CPL and quadratic $X^2(z)$ parameterizations, respectively.