Revealing evolution of Dark Energy density from observations

Abstract

We present a model--independent reconstruction of the normalized dark energy density function, $X(z) \equiv \rho_{\mathrm{de}}(z)/\rho_{\mathrm{de}}(0)$, derived directly from the DES-SN5YR Type~Ia supernova sample. The analysis employs an inversion formalism that relates the derivative of the distance modulus, $\mu^{\prime}(z)$, 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 $\mu^{\prime}(z)$ 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 $\Lambda$CDM, CPL, and the quadratic $X^2(z)$ parameterization -- where CPL and $X^2(z)$ each introduce two additional free parameters relative to $\Lambda$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.