Probing Dark Energy Evolution Post-DESI 2024

The Early Data Release from the Dark Energy Spectroscopic Instrument (DESI, [1, 2, 3]) has motivated further exploration of dynamical dark energy (DE) in place of a cosmological constant in the standard cosmological model, $\Lambda$CDM [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Prominant $\Lambda$CDM alternatives include quintessence models [14, 15, 16, 17, 18, 19] and theories of modified gravity [20, 21, 22, 23, 24, 25, 26], many of which have been revised in light of the recent DESI data release [4]. The most common approach to look for potential deviations from the $\Lambda$CDM model is to represent the DE component using the equation of state parameter $w(a)$. This approach assumes there is an unknown component with energy density given by

This approach, in principle, admits a phantom behavior ($\omega<-1$), but does not allow the excursion to negative DE densities. However, the authors of [27, 28], demonstrate that the method is inherently constrained because the functions that connect observations with theory, such as the luminosity distance, rely on $\omega(z)$ through a complex integral relationship. This obscures detailed information about $\omega(z)$ and its temporal changes, reducing the precision with which $\omega(z)$ can be determined from existing data.

Rather than initially adopting a particular physical model for DE, given our lack of clear insight into its origin, in this work we opt for an exploratory approach, utilizing a probing function to represent the DE density. Our aim is to identify deviations from the $\Lambda$CDM model which assumes a constant DE component. This approach started in [29, 30, 31], where the authors used both a linear and a quadratic probe function for $X(z)=\rho_{DE}(z)/\rho_{DE}(0)$. For $\Lambda$CDM, $X(z)=1$, whilst if $X(z)\neq 1$ for any redshift $z$, this is an indication of DE evolution. In [32] it was found that low redshift observational data prefer that $X(z)$ decreases with $z$, even taking negative values for $z>1$. The same trend has been found using new data, such as the Pantheon sample [33, 34], and also using new methods, such as non-parametric and even biology inspired ones, discussed in [34, 35], with [35] making use of the Pantheon+ sample. This trend of including the possibility of a negative $X(z)$ is a challenge for DE modeling. Furthermore, the authors of [36, 13] verify that this trend holds true when datasets (Pantheon+ and DES 5YR, respectively) are confined to redshift bins.

The paper is organised as follows. The next section introduces the methodology and datasets used for the statistical analysis. The main results are described in section III. We make our concluding remarks in section IV.

Assuming an isotropic, homogeneous universe, characterised by a flat FLRW spacetime metric, the Hubble parameter is defined as follows,

where the energy densities parameters of matter, radiation, and dark energy are described by $\Omega_{\text{M}}$, $\Omega_{\text{R}}$, and $\Omega_{\Lambda}$, respectively. We restrict our analysis to the flat case, thus, $\Omega_{\Lambda}=1-(\Omega_{\text{M}}+\Omega_{\text{R}})$. Whilst $\Omega_{\text{M}}$ is a constant to be constrained directly, we make use of the relation,

for $\Omega_{\text{R}}$ in our analysis of BAO data [37]. $N_{\text{eff}}$ is effective number of neutrinos, $\Omega_{\gamma}$ is photon energy density, and $\Omega_{\nu}$ is neutrino energy density.

We adopt an agnostic approach in attempting to describe the behaviour of DE. Our analysis is not predicated on any physical assumptions about the nature of DE; instead, we choose to focus on dark energy as a purely geometric effect. As such, we devise a probe function with the capacity to detect dynamical dark energy if the data so prefer. Our probe function of choice is a second degree polynomial,

which is a natural non-linear deviation from unity, and advantageous over other parametrisations due to having few degrees of freedom. This parametrisation is a continuation of the work in [33, 32, 34]. Variables $x_{0},x_{1},$ and $x_{2}$ are constant values of $X(z)$ evaluated at $z_{0},z_{1}$, and $z_{2}$, where $z_{0}<z_{1}<z_{2}$. Using the same definitions as in [33], $z_{0}=0$, $z_{1}=z_{\text{m}}/2$, $z_{2}=z_{\text{m}}$, where $z_{\text{m}}=2.330$ is the maximum redshift value across the three datasets used in this analysis. The free parameters in $X(z)$ then become $x_{1}=X(z_{\text{m}}/2)$ and $x_{2}=X(z_{\text{m}})$. These definitions reduce (4) to

The first thing to note is the concordance between this cosmological model, which we shall refer to as $X(z)$CDM, and $\Lambda$CDM, in the case of no DE evolution. If the data prefer constant over dynamical DE density, then $x_{1}$ and $x_{2}$ will tend to unity, reducing (5) to the $\Lambda$CDM model. $X(z)$CDM, therefore, may be thought of as an extension of the mostly successful $\Lambda$CDM model, in pursuit of exploring the possibility of dynamical DE at low redshift.

The model is put under stress using four types of observational data: baryonic acoustic oscillation data from DESI [1], Type Ia supernovae from the Pantheon+ sample [38], redshift space distortions from a sample compiled by the authors of [39], and cosmic microwave background radiation data from the acoustic angular scale, also used in [1].

Tables 2(a), 2(b), and 2(c) summarise the results of MCMC sampling in the three different cases with varying constraints on $N_{\text{eff}}$, $\omega_{\text{b}}$, and $z_{\text{d}}$. We begin the analysis of our results by discussing the goodness of fit of both cosmological models. We proceed to analyse the effect of varying $N_{\text{eff}}$, $\omega_{\text{b}}$, and $z_{\text{d}}$ on the resulting cosmological parameters for both models. We compare $\Lambda$CDM and $X(z)$CDM values for $H_{0}$ and $\sigma_{8}$ with previous analyses. Finally, we reconstruct $X(z)$ and $q(z)$, the deceleration parameter, using best fit parameters, and explore the implications for DE evolution.

In our evaluation of goodness of fit we make use of two statistics: $\chi_{\text{red.}}^{2}$ and Durbin-Watson (DW). The combination allows for a more robust analysis, as the two statistics assess different qualities of a given model. The DW statistic tests for serial autocorrelations of the residuals, a characteristic that $\chi_{\text{red.}}^{2}$ cannot account for [81], with a desired value of 2. The more familiar statistic within the field of cosmology, $\chi_{\text{red.}}^{2}$, has an ideal value of 1, with lower values implying over-fitting [77].

The statistics in Table 2(a), when considered in aggregate, imply no significant preference for $X(z)$CDM over $\Lambda$CDM. For the combination of all three datasets with $\theta_{*}$, the $\chi_{\text{red.}}^{2}$ and DW values for $X(z)$CDM are 1.14 and 1.74, respectively. In the same case, for $\Lambda$CDM, $\chi_{\text{red.}}^{2}=0.902$ and DW $=1.70$. The proximity of the $\chi_{\text{red.}}^{2}$ and DW values to their desired values of 1 and 2, respectively, indicates successful data-fitting from both models. The case outlined above is the most constrained (fewest free parameters) and hence, shall be a focal point in our analysis. However, the success of $X(z)$CDM being comparable to that of $\Lambda$CDM extends to all cases. Furthermore, the DW statistic favours $X(z)$CDM universally. It should also be highlighted that all $\chi_{\text{red.}}^{2}$ and DW values across the three tables remain acceptably close to the desired values. In this way, both models are competitive in describing the observational data used, both fitting the data and obtaining a small autocorrelation. Hence, neither model should be outright rejected.

By a comparison of the results in Tables 2(a), 2(b), and 2(c) we observe – as expected – that increasing the number of free parameters increases the resulting best fit value for $H_{0}$ in the case of $\Lambda$CDM: DESI+Panth.+RSD+$\theta_{*}$. We find that $H_{0}$ for $N_{\text{eff}}$, $\omega_{\text{b}}$, and $z_{\text{d}}$ fixed lies $3.1\sigma$ lower than $H_{0}$ for $N_{\text{eff}}$, $\omega_{\text{b}}$, and $z_{\text{d}}$ all variable. We define $\sigma=\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}$, where $\sigma_{1}$ and $\sigma_{2}$ are the mean uncertainty values associated with the given results. This implies that increased uncertainty in early-time physics variables and, hence, fewer assumptions in our $\Lambda$CDM model, does impact best-fit cosmological parameters, specifically increasing $H_{0}$. We note that in both cases with $N_{\text{eff}}$ variable (Tables 2(c) and 2(b)), a consequence of this variability is $N_{\text{eff}}$ increasing to $3.4\pm 0.1$, a $1.5\sigma$ deviation from the prior distribution mean. This increase in cosmological parameters is not observed in the corresponding $X(z)$CDM results to a significant degree.

To comment briefly on $\sigma_{8}$, across Tables 2(a), 2(b), and 2(c), we observe lower values in the $X(z)$CDM analyses than those of $\Lambda$CDM both with and without $\theta_{*}$. Including $\theta_{*}$, the $X(z)$CDM value of $\sigma_{8}=0.731\begin{subarray}{c}+0.015\\ -0.017\end{subarray}$ in 2(a) differs from the 2018 reported Planck value of $0.811\pm 0.006$ at the 4.7$\sigma$ level.

To compare an $X(z)$CDM value of $H_{0}$ with those from other analyses, we confine the rest of our discussion to the results in Table 2(a), reflecting the analysis with the fewest free variables.

First, we consider the results of our analyses excluding $\theta_{*}$. For $X(z)$CDM, we report $H_{0}=73.38\begin{subarray}{c}+0.89\\ -0.85\end{subarray}$ km/s/Mpc, with $X(z)$ free parameters $x_{1}=0.37\begin{subarray}{c}+0.13\\ -0.15\end{subarray}$ and $x_{2}=-1.69\begin{subarray}{c}+0.48\\ -0.74\end{subarray}$. The corresponding $\Lambda$CDM results are $H_{0}=70.59\begin{subarray}{c}+0.75\\ -0.79\end{subarray}$ km/s/Mpc, a deviation of $2.4\sigma$ from the $X(z)$CDM case.

The addition of $\theta_{*}$ lowers $H_{0}$ universally, as expected, giving rise to a larger discrepancy of $3.6\sigma$ between the two models. In the $X(z)$CDM case, $H_{0}=72.33\begin{subarray}{c}+0.63\\ -0.80\end{subarray}$ km/s/Mpc, with $x_{1}=0.42\begin{subarray}{c}+0.14\\ -0.13\end{subarray}$ and $x_{2}=-1.74\begin{subarray}{c}+0.60\\ -0.59\end{subarray}$. Even with the addition of condensed CMB information, $H_{0}$ remains significantly closer to the Cepheid-SNe-Ia deduced value of $73.04\pm 1.04$ km/s/Mpc [50], differing by only $0.6\sigma$. When compared with the Planck value of $H_{0}=67.4\pm 0.5$ km/s/Mpc [40], a much larger discrepancy of $5.7\sigma$ arises, confirming that the $X(z)$CDM model fails to resolve the tension between Hubble constant values when constrained by these three datasets and the angular acoustic scale. An interesting extension of this work would be to incorporate additional CMB information beyond $\theta_{*}$ from Planck and the ACT to be used in combination with BAO, SNe-Ia, and RSD data.

The results of the $X(z)$CDM analysis, whilst consistent with model-independent, local $H_{0}$ determinations, remain in tension with those of Planck [40]. Hence, we conclude that $X(z)$CDM does not resolve the Hubble Tension, in so far as the model does not rectify the notorious discrepancy between $H_{0}$ values. Nonetheless, the model may still offer some insight into the nature of DE at low redshift. More specifically, in all $X(z)$CDM cases, the data prefer $0<x_{1}<1$ and $x_{2}<-1$. Both parameters showing significant deviation from unity is suggestive of dynamical DE at low redshift. Using the best fit parameters for $X(z)$ in Table 2(a), DE density is re-parametrised and plotted as a function of redshift ($0<z<2.33$), as shown in Fig. 2. As $z$ increases beyond $\sim 0.5$, DE density begins to exhibit dynamical behaviour and decays quadratically, assuming negative values beyond $z\sim 1.5$. DE density decreasing beyond zero is supported at the 2$\sigma$ above the best fit curve.

In our case, a negative dark energy (DE) density does not violate any physical principle, as we are using this parametrization strictly within a phenomenological framework. What remains entirely prohibited, however, is a negative total density. Actually, models with negative DE density have been considered before. In [82], the author explored a model inspired by unimodular gravity, which predicts fluctuations in the cosmological constant. These fluctuations are always on the order of the ambient density, making it unsurprising that the cosmological “constant” may sometimes take negative values. Additionally, in a similar context, using the back-reaction approach, the author of [83] found that the cosmological constant oscillates around half the total density on Hubble time scales. An interesting related study was conducted in [84], where the authors examined an interacting dark matter/dark energy model using a moderately general interaction term. One of their key conclusions was that, based on their examples, resolving the coincidence problem would require the DE density to have been negative in the past. This feature was first identified using this reconstruction method in [32], and similar results were examined in [85].

At low redshift, $z\lesssim 0.5$, $X(z)=1$, in agreement with $\Lambda$CDM, although this feature of the $X(z)$ reconstruction should be interpreted with caution. A quadratic parametrisation lacks the ability to detect fluctuations at low redshifts– it is constrained by the properties of a second degree polynomial. The authors of [4] assert that SNe-Ia data at very low redshift, $z<\mathcal{O}(0.1)$, is responsible for the preference of dynamical dark energy over a cosmological constant. Furthermore, by excluding this low redshift data, they show that no deviation from the $\Lambda$CDM model is necessary.

This prompts interesting questions about the nature of our local Universe, potentially challenging the cosmological principle (CP) for $z<\mathcal{O}(0.1)$. This is not a novel idea; the authors of [86] provide a comprehensive survey of observational data seemingly at odds with predictions of the CP. Perhaps there exists some local gravitational effect due to anisotropic matter distribution that we are yet to account for. Exploring this phenomenon is not within the scope of our analysis but its mentioning serves to contextualise our results at $z<\mathcal{O}(0.1)$. Hence, we emphasise that $X(z)$ tending to unity as $z$ tends to zero without any perturbations could purely be a consequence of a quadratic parametrisation.

We plot the deceleration parameter, $q(z)$, as a function of redshift in the range $0<z<2.33$, as shown in Fig. 3. Although the DE density clearly deviates from the $\Lambda$CDM model, even dipping into negative values, the reconstructed deceleration parameter remains consistent with the $\Lambda$CDM model. Notably, the redshift at the deceleration-to-acceleration transition, $z\simeq 0.8$, is identical in both models. Additionally, the current value of the deceleration parameter is slightly lower than the $\Lambda$CDM value, aligning closely with the standard value of $q_{0}=-0.55$.

In this paper we explore the implications of using a quadratic DE density parametrisation, $X(z)$, in place of a cosmological constant. With MCMC sampling, we analyse the effectiveness of the $X(z)$CDM model at describing the latest DESI BAO data in combination with SNe-Ia and RSD samples. Using reduced $\chi^{2}$ and the DW statistic to evaluate goodness of fit, we find no significant preference for $X(z)$CDM over $\Lambda$CDM. We note that both models perform very well according to these statistical measures. Further, we consider the effect on resulting cosmological parameters of constraining the parameter space with the angular acoustic scale. The addition of CMB information in this form lowers $H_{0}$ in all analyses.

We present a fundamental way of overcoming the $r_{\text{d}}$-$H_{0}$ degeneracy problem in the analysis of BAO data, using an integral equation for $r_{\text{d}}$. This method, nonetheless, is still dependent on external knowledge from early-time physics predictions and measurements. We consider the consequences of increasing uncertainty in this external information, finding that for $\Lambda$CDM, this has a notable impact on best fit parameters. Specifically, best fit values of $H_{0}$ increase with increased variability. We do not observe this trend in the $X(z)$CDM results.

Finally, our $X(z)$CDM analysis is indicative dark energy evolution at low redshift: in all cases, $x_{1}$ and $x_{2}$ deviate from unity. From our reconstruction of $X(z)$, we find that from $z\sim 0.5$ onwards, DE exhibits dynamical behaviour, and proceeds to assume negative values beyond redshift $\sim 1.5$.

LO would like to acknowledge Daniel Scolnic for his mentorship. VHC would like to thank CEFITEV-UV for partial support.