Reconstruction of dark energy and late-time cosmic expansion using the Weighted Function Regression method

TL;DR

This work tackles the potential bias introduced by truncating the dark energy equation-of-state expansion, by applying Weighted Function Regression (WFR) to combine information from multiple truncation orders via Bayesian weights. The method reconstructs key background functions, including $w_{\rm DE}(z)$ and $\rho_{\rm DE}(z)$, as well as cosmographical quantities $H(z)$, $q(z)$, and $j(z)$, in a model-agnostic way using two bases: a $w$-basis and a $\rho$-basis. Across Planck, DESI, and SNIa data (PantheonPlus or DES-Y5), the WFR reconstructions consistently favor dynamical dark energy with a phantom-crossing transition around $z_{\rm cross} \sim 0.4$, though the statistical significance weakens somewhat with PantheonPlus. The Hubble parameter remains consistent with Planck/$\Lambda$CDM values, and no compelling evidence for negative DE density is found within $z\lesssim 3$, implying that late-time DE dynamics are detectable but not yet decisively extreme given current data. The results demonstrate the robustness of WFR to basis choice and data sets, highlighting the role of model-averaging in avoiding biases introduced by specific parametrizations and informing the interpretation of DE dynamics in the era of precision cosmology.

Abstract

Recent data from multiple supernova catalogs and DESI, when combined with CMB, suggest a non-trivial evolution of dark energy (DE) at the $2.5-4σ$ CL. This evidence is typically quantified using the CPL parametrization of the DE equation-of-state parameter which corresponds to a first-order Taylor expansion around $a = 1$. However, this truncation is to some extent arbitrary and may bias our interpretation of the data, potentially leading us to mistake spurious features of the best-fit CPL model for genuine physical properties of DE. In this work, we apply the Weighted Function Regression (WFR) method to eliminate the subjectivity associated with the choice of truncation order. We assign Bayesian weights to the various orders and compute weighted posterior distributions of the quantities of interest. Using this model-agnostic approach, we reconstruct some of the most relevant background quantities, examining the robustness of our results against variations in the CMB and SNIa likelihoods. Furthermore, we extend our analysis by allowing for negative DE. Our results corroborate previous indications of dynamical DE, now confirmed for the first time using the WFR method. The combined analysis of CMB, BAO, and SNIa data favors a DE component that transitions from phantom to quintessence at redshift $z_{\rm cross}\sim 0.4$. The probability of phantom crossing lies between 96.21% and 99.97%, depending on the SNIa data set used, and hence a simple monotonic evolution of the DE density is excluded at the $\sim 2-4σ$ CL. Moreover, we find no significant evidence for a negative dark energy density below $z\sim 2.5-3$. Our reconstructions do not address the Hubble tension, yielding values of $H_0$ consistent with the Planck/$Λ$CDM range. If SH0ES measurements are not affected by systematic biases, the evidence for dynamical dark energy may need to be reassessed. [abridged]

Figures (10)

• Figure 1: First row: Representative curves of $w_{\rm DE}(z)$ and $f_{\rm DE}(z)$ for different truncation orders in the $w$-basis. The rightmost plot shows the asymptotic behavior of $f_{\rm DE}$; Second row: Corresponding curves in the $\rho$-basis. The examples are shown mainly to illustrate the flexibility of the functions in the redshift range of interest, despite their differing asymptotic behavior in the two bases.
• Figure 2: First row: Reconstructed DE equation of state $w_{\rm DE}(z)$ (left plot) and DE density normalized to its current value $f_{\rm DE}(z)$ (right plot) obtained with Planck18+DES-Y5+DESI for each individual model in the $w$-basis. Solid lines indicate the most probable values, and the shaded areas represent the 68% confidence intervals; Second row: The same, but for the analysis with PR4+PantheonPlus+DESI; Third row: Results obtained with PR4+PantheonPlus+DESI for the various models of the $\rho$-basis.
• Figure 3: Left plot: Scatter plot in the $w_0-w_a$ plane obtained with with Planck18+DES-Y5+DESI for ${\rm CPL}^+$. We also show the corresponding values of the parameter $w_b$ (cf. Table \ref{['tab:tab1']}). The intersection of grey dashed lines represents the $\Lambda$CDM point, i.e., $(w_0,w_a)=(-1,0)$. Right plot: Same as in the left plot, but with ${\rm CPL}^{++}$. In this case, the color pallette is for the parameter $w_c$.
• Figure 4: Reconstructed equation of state, energy density normalized to the current value, deceleration parameter and jerk with Planck18+DES-Y5+DESI using the $w$-basis (left-most column), and with PR4+PantheonPlus+DESI using the $w$-basis (central column) and the $\rho$-basis (right-most column). We also plot the reconstructed Hubble rate and energy density normalized to the Planck PR4 values for $\Lambda$CDM, for which $\Omega_{\rm m}^0= 0.315$ and $H_0=67.26 \, {\rm km/s/Mpc}$Rosenberg:2022sdy. The solid colored lines represent the most-probable value and the shaded regions show the 68% and 95% confidence intervals around it. The grey dashed lines correspond to $\Lambda$CDM values. In the plots of $q(z)$ we also show in black dash-dotted line the border between deceleration ($q>0$) and acceleration ($q<0$) regimes, i.e., $q=0$. Notice that the vertical axis of the last two rows are not aligned for visualization purposes.
• Figure 5: EoS diagram. The pink and orange areas are the standard quintessence ($-1\leq w\leq-1/3$) and phantom ($w\leq-1$) DE regions, respectively. In the white area the strong energy condition is fulfilled, i.e., $\rho+3p\geq 0$ and $\rho+p\geq 0$. In particular, non-relativistic matter ($w=0$) and radiation ($w=1/3$) live there, satisfying $p,\rho\geq 0$. Phantom matter ($w\leq -1$, with $\rho<0$ and $p>0$) occupies the remaining part of the white region, where the strong energy condition holds Grande:2006nnMavromatos:2021urxGomez-Valent:2024tdGomez-Valent:2024ejh. The contribution of curvature to the Friedmann equations can be thought of as an effective fluid with $w=-1/3$ and $\rho>0$ (open universe) or $\rho<0$ (closed universe). The blue region is the one for species with negative energy density, with decreasing absolute value. Inside that region, the triangle between the "closed universe" line and the border of the "phantom matter" area can be of phenomenological interest. We coin that form of energy as negative quintessence, since it has the same EoS as ordinary quintessence ($-1/3 > w> -1$), but has negative energy. See the main text for further details.