Why cosmological data analysis of dark energy may inadvertently favor phantom energy models

1 Introduction

Despite the great success of Λ cold dark matter (ΛCDM), the still unknown nature of cold dark matter (CDM) and the nature of the cosmological constant (Λ), which according to measurements each contribute roughly 25% and 70%, respectively, to the present-day energy density of the universe, remain open questions. Many cosmological observation campaigns analyze their data not only in light of testing ΛCDM but also to explore possible extensions to ΛCDM, which replace Λ with various alternative models of dark energy (DE) that use a time-dependent equation of state (EoS) parameter.

Prominent examples of campaigns that test a dynamical model of dark energy (DDE) component with a time-dependent EoS parameter include cosmic microwave background (CMB) measurements by the Planck Collaboration (2020), or the Dark Energy Survey (DES) (see, for example, the Year 3 results (DES-Y3) in Abbott et al. (2023) and the DESI Collaboration et al. (2025); both of the latter derive measurement data from baryonic acoustic oscillations (BAOs).

All these studies find a DDE component whose EoS parameter increases from the early universe to the present day but interpret their results differently. The Planck Collaboration (2020) and DES-Y3 (Abbott et al., 2022, Abbott et al., 2023) interpret their results as confirmation of the cosmological constant Λ, while the DESI Collaboration et al. (2025) interpret their results as confirmation of a DDE model and as evidence of the new physics considered necessary to explain DE.

In this study, we discuss these results in the light of a degeneracy in the customary computation procedure for the expansion history of cosmological models, as we identified by Foidl and Rindler-Daller (2024) (henceforth “Paper I”), which affects the computation of cosmological models that replace Λ in the ΛCDM concordance model with a DDE component with an increasing EoS parameter. The degeneracy is in the sense that these models perfectly reproduce ΛCDM results, although the DDE component is clearly different from a cosmological constant Λ. The DESI Collaboration et al. (2025) report a similar degeneracy but attributes it to their data and not to the computational procedure.

We discuss in detail the cause of the degeneracy, as well as the conditions that must be met for it to distort a Markov-chain Monte Carlo (MCMC) data analysis of cosmological measurement data.

Furthermore, we re-evaluate the interpretation of an oscillating EoS parameter of the DDE component, reported by DESI (see also Keeley et al., 2025), which they interpret as evidence for the new physics that they believe is necessary to explain DE.

First, we briefly introduce the equations used in the following discussion. Friedmann (1922); Friedmann (1924) derived solutions to Einstein’s field equations for universes with constant curvature, without restriction—apart from the cosmological principle—to specific physical or astronomical assumptions. He considered matter as the only energy content of the universe and found a solution Friedmann (1922); Friedmann (1924) known as the “First Friedmann Equation (Equation 1)which describes the expansion history of model universes. H is the Hubble parameter (we use the term “expansion rate” interchangeably) defined as H ( t ) = a ̇ / a, where a is the scale factor and the dot refers to the derivative with respect to cosmic time t. Its present-day value, the Hubble constant, is denoted as H 0.

H 2 t = 8 π G 3 c 2 ρ − k c 2 a 2 t , (1)

In order to study a variety of cosmological models, it has become customary to add “curvature” as well as the cosmological constant “Λ” to the energy content of the universe by defining operationally “effective” energy densities for them: ρ k = − 3 k c 2 / ( 8 π G a 2 ) for curvature (k) and ρ Λ = Λ c 2 / ( 8 π G ) for the cosmological constant. The Friedmann equation in modern cosmology reads aswith the time-dependent background energy densities for radiation ( ρ r ), baryons ( ρ b ), CDM ( ρ CDM ), and curvature ( ρ k ), as well as ( ρ Λ ), where ρ k = 0 in ΛCDM.

H 2 t = 8 π G 3 c 2 ρ r t + ρ b t + ρ CDM t + ρ k t + ρ Λ t , (2)

Friedmann Equations 1, 2 have, in general, more than one variable—respectively, the scale factor a ( t ) and the background energy densities ρ i ( t ). The Friedmann equations alone are therefore not sufficient to determine the expansion history of the universe. We need additional equations that include these quantities—one for each cosmic component—in order to solve the Friedmann equations. Therefore, the first law of thermodynamics as applied to the adiabatic expansion of a comoving volume in the universe yields the energy conservation equation separately for each component i = CDM, b, r:where ρ i and p i stand for the respective background energy densities and pressures. Like the Friedmann equation, they only depend upon time t or scale factor a but not on spatial coordinates. The energy densities and pressures are each related by their respective EoS,where w i is often called the “EoS parameter,” which can also generally change with time. However, in ΛCDM, w i is assumed to be a constant for every component i = CDM, b, r, Λ and k. Assuming a constant EoS parameter w i and using (4) in (3), it follows that ρ i ̇ / ρ i = − 3 ( 1 + w ) a ̇ / a, which is readily integrated to yield the well-known relationThis describes the evolution of the background energy densities as a function of the scale factor a for constant w i (the dependence on t is indirect via a = a ( t )), and Ω i , 0 and ρ crit , 0 refer to their present-day values, derived from observations (Planck-Collaboration, 2020).

∂ ρ i ∂ t + 3 H ρ i + p i = 0 , (3)

p i t = w i t ρ i t , (4)

ρ i a = Ω i , 0 ρ crit , 0 a − 3 1 + w i . (5)

The background evolution of the standard cosmic components is thus given by with (Equation 6a) for radiation (its EoS parameter in (Equation 4) is w r = 1 / 3), (Equation 6b) for baryonic matter and for CDM ( w m = 0 ), (Equation 6c) for curvature ( w k = − 1 / 3 ), and (Equation 6d) for the cosmological constant Λ ( w Λ = − 1 ).

ρ r ( a ) = Ω r , 0 ρ crit , 0 / a 4 , (6a)

ρ m ( a ) = Ω m , 0 ρ crit , 0 / a 3 (6b)

ρ k ( a ) = Ω k , 0 ρ crit , 0 / a 2 (6c)

ρ Λ = Ω Λ , 0 ρ crit , 0 (6d)

A commonly used approach to test models of DE as possible answers to the Hubble tension or explain DE is the parametrization ansatz, such as the so-called “CPL parametrization” by Chevallier and Polarski (2001) and Linder (2003) for the EoS parameterwhich is defined for a ∈ [ 0,1 ], where a is the scale factor and has two parameters: w 0 is the value of the EoS parameter in the present and w a is the difference to this value in the early universe (where scale factor a is close to zero). A negative w a defines an increasing EoS parameter w ( a ), and a positive w a defines a decreasing EoS parameter w ( a ). The cosmological constant is recovered for w 0 = − 1 and w a = 0.

w a = w 0 + 1 − a w a , (7)

The CPL parametrization (Equation 7) is simple and empirically defined, with no physical motivation for the evolution of w ( a ) being linear in a. The CPL parametrization, however, also provides an analytical solution to the integration of energy conservation Equation 3, which reads as

ρ ∝ a − 3 1 + w 0 + w a e − 3 w a 1 − a . (8)

Therefore, Equation 8 offers a very straightforward way to incorporate CPL-based models of DE into existing cosmological codes by replacing Equation 5.

This study is organized as follows. Section 2 explains the reasons for degeneracy in the customary computation procedure for the expansion history of cosmological models that replace Λ with a DDE component. In Section 3, we discuss the reported CPL-based DDE models of the major cosmological observation campaigns and reassess the presumed evidence for the new physics reported by the DESI project. Furthermore, we re-evaluate their physical plausibility by comparing them with important observations in cosmology. Section 4 explains the conditions under which an MCMC analysis of cosmological measurement data is likely to be trapped in providing a phantom-energy-type DE component. In Section 5, we summarize and discuss the results.

All calculations of cosmological models presented in the main text were performed using the public version of the open-source cosmological Boltzmann code CLASS ¹ .

2 Degeneracy in the computation procedure

The degeneracy, pointed out in Paper I, in the customary computation procedure for the expansion history of cosmological models, which is applied in the major cosmological codes (e.g., CLASS and CAMB ² ), is in the sense that cosmological models that include a DDE component with an increasing EoS parameter w, replacing Λ in the ΛCDM concordance model, perfectly reproduce the ΛCDM results. The DDE model starts in the early universe as phantom energy with an EoS parameter well below − 1.

Figure 1 shows the computation of the temperature and the matter power spectrum of a cosmological model with the DDE component described by the CPL parametrization (solid magenta lines), with mean values w 0 = − 0.6 and w a = − 1.8 derived by DES-Y3 in an MCMC data analysis, where their measurement data was not combined with external data, as depicted by the light blue region in the left-hand panel in Figure 4 of Abbott et al. (2023). We compare the results with ΛCDM shown by the solid blue lines. We can observe a perfect agreement with ΛCDM, although this DDE model is significantly different from a cosmological constant.

The key to understanding this degeneracy lies in apprehending the CPL parametrization within the framework of concordance cosmology. Significant contributions of w a are found only in the early universe, close to a = 0. However, DE in the early universe was very subdominant by many orders of magnitude. Hence, w a has practically no influence on the overall evolution of the early universe. Close to a = 1, on the other hand, w 0 dominates over w a. The energy density of Λ or DE, only contributes significantly to the total sum of all the energy components when the scale factor a ≥ 0.75 (Figure 2).

The customary procedure for computing the expansion history starts in the present with the critical density determined by H 0 of the ΛCDM concordance model and evolves the energy densities of the individual components of ΛCDM back in time according to Equation 5 or Equations 6a–6d. To explain the cause of the degeneracy, we show in Figure 3 the evolution of the densities in a cosmological model with the DDE component of Figure 1, corresponding to the original measurements of DES-Y3, computed using Equation 8 instead of Equation 6d.

Taking into account the characteristic of the CPL parametrization, the evolution of the energy densities in ΛCDM, and the customary back-in-time integration of the energy densities, starting with their concordance values at the present, the degeneracy is explained as follows. The CPL-parameter w a has no significant impact onto the evolution of DE’s EoS parameter w in the late stages of the evolution of the universe, which is dominated by w 0. The energy density of DE only becomes dominant in the late phases of the universe’s evolution. Therefore, the w 0 dominance explains why DE is nearly constant (slightly concave) at a ≥ 0.8 and therefore very close to the cosmological constant. Hence, the energy densities in the final stages of the universe’s evolution are very close to that of ΛCDM. The negative value of w a causes the energy density of the DE component to drop far below that of Λ toward the early universe, while its EoS parameter becomes a value less than − 1 that defines phantom energy.

To reiterate, this result is due to the concurrence of three conditions: DE’s increasing EoS parameter w, the cosmic inventory of the ΛCDM concordance model, and the back-in-time integration of the energy conservation (Equation 3), using a value of H 0 close to the concordance value.

The slightly concave evolution of the DDE component’s energy density (Figure 3) has an important impact on the evolution of the expansions rate H, as shown in Figure 4, which depicts the evolution of the expansion rate H of the cosmological model with the CPL-based DDE component shown in Figure 1 (magenta solid line).

The bottom inset depicts the evolution of the expansion rate H of this model relative to its evolution in the ΛCDM model (gray line). The DDE component has a linearly (in scale factor a) increasing EoS parameter (i.e., the EoS does not oscillate). However, the evolution of the expansion rate H shows an oscillation relative to the ΛCDM model. We will return to this characteristic of CPL models with a negative w a (i.e., an increasing EoS parameter) in the Section 3.

3 Discussion of DDE models

Based on an MCMC analysis of BAO measurement data, DES-Y3 results (Abbott et al., 2023) claimed to have confirmed the cosmological constant, as the CPL parameters for Λ, w 0 = − 1, and w a = 0 were within the 1σ region of the resulting probability distribution of the CPL parameters w 0 = − 0.95 and w a = − 0.4, depicted as the purple region in their Figure 4 (left-hand panel) and Figure 5, respectively. This was only true due to the combination of their original measurement results (light blue region) with other data (see their Figure 5). Their originally derived CPL parameters were something like w 0 ∼ − 0.6 and w a ∼ − 1.8. The Planck Collaboration (2018), based on CMB measurements, reported similar results and also claimed to confirm the cosmological constant. Most recently, the DESI Collaboration et al. (2025) favored a DDE model, based on the MCMC analysis of BAO measurement data, resulting in a CPL DDE model with negative w a—an increasing EoS parameter—as well. Thus, very similar CPL models, all with a negative w a of the DE component, provide contradicting interpretations.

These DDE models appear in the early universe as phantom energy with w well below the phantom line − 1. At redshift z ∼ 0.45, they cross the phantom line, yet before DE becomes the dominant energy density in the universe. The EoS parameter w continues to increase, linear in scale factor a, until the present, as shown, for example, in Figure 2 of DESI Collaboration et al. (2025). Here, the DESI Collaboration et al. (2025) reported that the crossing of the phantom line ( w ( z ) = − 1 ) coincides with the maximum in the energy density of their DDE model. They interpret this as indicating that DE appears to be much more complex than traditionally assumed and as evidence for a new physics necessary to explain DE.

We cannot follow this argument, since this coincidence is exactly what we expect when computing the expansion history of a model with a DDE component with increasing w. It is well-known (Section 1) that in order to solve Friedmann Equation 2, each energy component of the cosmological model is coupled to the energy conservation Equation 3. Inspecting the exponent − 3 ( 1 + w i ) in Equation 5, we can easily recognize the slope of the evolution of the energy density at any point in time. Therefore, it is not surprising that the slope at phantom crossing ( w = − 1 ) is zero as this is consistent with the CPL parameters of a cosmological constant (w 0 = − 1 and w a = 0). Thus, prior to phantom crossing ( w < − 1 ), the slope is increasing, and subsequent to the phantom crossing ( w > − 1 ), the slope is decreasing.

To support our argument, we show in Figure 5 the evolution of the energy density of the DESI CPL-based DDE component with w 0 = − 0.85 and w a = − 0.6 relative to the density at z = 0, computed using Equation 8, which corresponds to the curve in the bottom panel in Figure 2 of DESI Collaboration et al. (2025).

Both curves agree well, from which we can conclude that there is not a “…more complex dark sector…” in the universe than traditionally assumed (DESI Collaboration et al., 2025). It is simply a consequence of the energy conservation (Equation 3), coupled to with Friedmann Equation 2—no exceptional indication whatsoever of new physics.

Additionally, the DESI project claims to have found oscillations in the DE’s EoS, interpreted as even more convincing evidence for a new physics that is considered necessary to explain DE.

DESI derives from their BAO measurements the evolution of the expansion rate H, showing oscillations relative to the expansion history of ΛCDM. Keeley et al. (2025) apply a more sophisticated parametrization than CPL, which allows rapid changes of w to directly relate the oscillations of the expansion rate H to the DE component’s EoS parameter w.

As already discussed in the description of Figure 4, an oscillation in the expansion rate is not necessarily connected to an oscillating EoS of the DDE component, especially if the DDE model was derived by using a CPL parametrization, as shown in Figure 2 of DESI Collaboration et al. (2025).

Relating the oscillations of H directly to w oversimplifies the process of transitioning from the matter-dominated universe to the DE-dominated universe, as shown in Figure 6, showing the evolution of the density parameters of a model which replaces Λ (red dotted line) by the DESI CPL-based DDE component (solid red line). We observe that Ω d e overshoots Ω Λ of ΛCDM due to the slightly concave shape of the evolution of its energy density (Figure 3), but that toward a = 1 the two curves come closer again. This is due to the decreasing energy density of the DDE component, as shown in Figure 5 and the bottom panel of Figure 2 in DESI Collaboration et al. (2025).

The evolution of the sum total of the energy densities yields the expansion history of this model (Figure 7). The gray solid line at the bottom displays the oscillations relative to ΛCDM.

During the transition from the matter-dominated universe to the DE-dominated universe, the interplay between the decreasing relative energy densities of the matter components and the increasing relative energy density of the DDE component cause the oscillations of H relative to ΛCDM, although the evolution of the CPL model’s EoS parameter w is linear in scale factor a, exhibiting no oscillations at all, which we also discussed in Figure 4.

The final drop to the configured value of H 0 is due to the customary backward integration of the energy conservation (Equation 3), which starts close to the concordance value. Hence, directly relating H’s oscillation to w leads to a biased interpretation of an oscillating EoS parameter of DE, since the CPL-based DDE component’s EoS parameter w increases linearly in scale factor a.

We now focus on the physics of Friedmann Equation 2 in the late stages of the evolution, where Λ or DE become dominant. A decreasing energy density of the DE component (Figure 5) corresponds to a decreasing expansion rate H (Figure 7). Nobel-Prize-winning observations (Perlmutter et al., 1999; Schmidt et al., 1998; Riess et al., 1998) have shown the accelerated expansion of the universe by directly measuring the decreasing deceleration of the expansion of the universe. These direct measurements were insensitive to the peculiarities of a cosmological model. However, a decreasing expansion rate H corresponds to increasing deceleration, which is inconsistent with the observed late-time accelerated expansion. We can therefore conclude that these models are incompatible with the observed accelerated expansion of the universe in the late phase and are therefore physically improbable.

4 Trapped in phantom energy

In Section 2, we discussed the cause of the degeneracy in the customary computation procedure of the expansion history for cosmological models, which we first reported in Paper I. This degeneracy causes cosmological models with a phantom energy DDE component with increasing w to perfectly reproduce the ΛCDM results for a great range of parameter w a of the CPL parametrization with respect to the matter power spectrum and the CMB temperature spectrum (Figure 1). In Section 3, we discussed the plausibility of those models from a physical perspective and concluded that they are physically very improbable and do not agree with the observation of the accelerated expansion of the universe.

The question remains why the large cosmological campaigns’ MCMC analyses (e.g., Planck, DES, and DESI) of their measurement data yield precisely these models as results, which we discuss in the course of this section.

We start by briefly summarizing the MCMC method. Based on the definition of the parameter space for the model, described by the priors to be fitted to measurement data, the sampler usually generates very many parameter sets that comprise all possible combinations (or reasonable combinations for minimizing the necessary number of computations of models) of the possible parameter values for the cosmological model. For each sampled parameter set, the computation of the model is carried out, followed by the computation of the likelihood, such as by least-mean-squares to measurement data (e.g., Planck fitted the computed CMB temperature spectrum to their measurement data, whereas DES and DESI fitted the computed matter power spectrum to their measurement data). In the final step, a standardized Bayesian statistics method uses the calculated likelihoods and computes the probability distribution for all the parameters and provides the best-fitting parameter set, the posteriors, for the sampled models—for example, the well-known density parameters of the ΛCDM model (Planck Collaboration, 2020).

Thus, the computation of every single sampled model appears as the elementary building block of the data analysis. Every issue that may impact the result of a model computation is very likely to bias the result of the data analysis, which may lead to a misinterpretation of the measurement data.

In Figure 1, we observe that cosmological models with a phantom-energy-like DDE component replacing Λ in the concordance model perfectly reproduce the ΛCDM’s matter power spectrum and CMB temperature spectrum. This explains the high probability of these models obtained in an MCMC data analysis. This corresponds, for example, to the results shown in Figure 4 (left-hand panel) of Abbott et al. (2023). In Section 2, we discussed that one of the necessary conditions for the occurrence of degeneracy is the back-in-time integration of the energy conservation (Equation 3) and the use of a value of H 0 close to the concordance value.

However, the introduction of DDE models extends the necessary parameter space of H 0 so that values obtained through measurements in the local universe (i.e., which could provide a solution to the Hubble tension, as described in Paper I) are also covered. Applying a chain, which does not account for this situation, could unintentionally and unwillingly restrict the prior of H 0 to only a small range around the concordance value, thus trapping the analysis in phantom energy models.

For example, the Planck Collaboration (2018) and DES-Y3 used a two-step approach, applying Markov chains when carrying out their data analysis to evaluate cosmological models beyond ΛCDM. They performed an MCMC data analysis, fitting sampled models to the measured CMB temperature spectrum or the measured matter power spectrum to derive in a first step the base cosmology—the density parameters of the ΛCDM model. In order to derive a CPL-based DDE model replacing Λ, they perform a second MCMC data analysis using the parameter space of the CPL parametrization chained to the results of the base cosmology, which limits the range of H 0’s prior distribution to be close to the value of the base cosmology. This explains the results of their original measurements with clearly negative average values of w a, as shown by the light blue and red regions in Figure 4 (left-hand panel) of Abbott et al. (2023).

Figure 5 of Abbott et al. (2023) depicts their final result, where they combined their original measurement data with external low-redshift data, depicted by the green dot-dashed region, which pulls the value of w a up to a “less negative” value (purple region). Inspired by this low-redshift data, corresponding to a CPL-based DDE component with w a ∼ + 0.1 and w 0 ∼ − 0.9, thus a decreasing EoS parameter, we computed the corresponding cosmological model. In Figure 8, we show the expansion history of this model by the magenta solid line, where we used the value H 0 = 73.04 from Riess et al. (2022) derived from measurements in the local universe.

This is justified by the low-redshift origin of the data. With respect to the expansion history, this model can be regarded a promising result as it could provide a solution to the Hubble tension.

Nevertheless, this model suffers from low probability compared to the phantom-energy-type CPL models, which can be observed in Figure 4 of Abbott et al. (2023). This is a systematic problem and is explained by Figure 9, where we use the temperature spectrum of the CMB to exemplify to which extent the model agrees with measurement data.

We observe a deviation from the measured CMB temperature spectrum (labeled “ΛCDM”) by approximately 15% due to a shift of the spectrum caused by the increase of the expansion rate H in the late stages of the expansion history.

Therefore, we can conclude that due to the degeneracy explained in Section 2, an MCMC analysis of cosmological measurement data, using a CPL-based DDE component replacing Λ, always favors an increasing EoS parameter w (i.e., w a < 0). Even if the range of values of the prior of H 0 is sufficiently large to cover DDE models with decreasing w (i.e., w a > 0), these will not be favored in the result since they do not agree well with the measurement data, as shown in Figure 9. These are precisely the results found in the Planck Collaboration (2020) and DES-Y3 (Abbott et al., 2023), shown in Figure 4 of both.

5 Summary and discussion

In Section 2 we discussed the cause of the degeneracy in the customary computational procedure for the expansion history of cosmological models, which evolves the individual energy densities of the components of a cosmological model back-in-time, based on the critical density at the present. The degeneracy is in the sense that cosmological models, which replace Λ in the ΛCDM concordance model by a DDE model with an increasing EoS parameter, perfectly reproduce ΛCDM results, although the DE component significantly differs from a cosmological constant. We also found that this is connected to using a value of H 0 close to the concordance value.

The major cosmological campaigns, like DES-Y3 (Abbott et al., 2023), the Planck Collaboration (2018), and recently the DESI Collaboration et al. (2025), report similar results for their analysis of DE. All of them found DDE models with an increasing EoS parameter and used the CPL parametrization. However, their interpretations of the results differed: Planck and DES claimed to have confirmed Λ, whereas DESI favored a DDE component replacing Λ.

In Section 3, we discussed DDE components with an increasing EoS parameter replacing Λ in the concordance ΛCDM model. We found that there is no evidence to suggest that new physical principles are necessary to explain DE. The coincidence of crossing the phantom line in the evolution of the EoS and the maximum in the energy density of the DE component, pointed out by DESI Collaboration et al. (2025), is a simple consequence of the energy conservation (Equation 3) coupled with Friedmann Equation 2 for the DDE component. In addition, we could not identify evidence for an oscillating EoS of the DE component, as reported by Keeley et al. (2025). Most importantly, we recognized that these models do not agree with the observations of the accelerated expansion of the universe (Perlmutter et al., 1999; Schmidt et al., 1998; Riess et al., 1998). We concluded that these models are very improbable from a physical point of view.

In Section 4, we explored why MCMC data analyses of cosmological measurement data applying CPL-based DDE components favor models with a clearly negative w a parameter. We found that this is caused by the degeneracy in the customary computation procedure for the expansion history discussed in Section 2. Although models with a decreasing EoS parameter ( w a > 0 ), computed with a value of H 0 compatible with local measurements (Riess et al., 2022) could provide a solution to the Hubble tension, they are disfavored in the MCMC analyses as they do not agree well with measurement data (CMB temperature spectrum or BAOs, respectively). This is connected with the increase of the expansion rate H in the late stages of the expansion history dominated by DE, depicted by the solid gray line in the bottom inset of Figure 8. This suggests that there is a systematic problem in the parameterization of the cosmological model and not just the DDE model.

It is well-known that the effects of spatial curvature and DE are degenerate (Hu and Dodelson, 2002, Plates 4a and 4b). Both shift the CMB spectrum, which is also observed in Figure 9. Therefore, we conclude that adding the parameter Ω k , 0 instead of using Ω k , 0 = 0, as in the concordance model, might solve the problem. In Foidl and Rindler-Daller (2026) and Foidl and Rindler-Daller (2025), we provided a physical explanation by taking into account that our peculiar motion against the microwave background, which can be observed in the CMB-dipole, deflects us from the position of perfectly comoving observers due to the spatial curvature in our local cosmic environment (see Tully et al., 2014). The assumption of Ω k , 0 = 0 is justified only for perfectly comoving observers as they move on geodesics. Consequently, we concluded that we must consider spatial curvature in the cosmological model.

Since we focus on the reasoning of a possible solution to MCMC analyses being trapped in phantom energy models of DDE, we refer to the simple proof-of-concept procedure of Foidl and Rindler-Daller (2026). Otherwise, a full MCMC analysis including all parameters of the model would be preferred. We fitted the parameters Ω k , 0 and Ω de , 0 (the other parameters of the model shown in Figure 9 remained unchanged) to the CMB spectrum of ΛCDM and to the value H 0 = 73.04 km/s/Mpc derived by Riess et al. (2022). As is customary, we call the resulting model the “owCDM model” (Figure 10). We found Ω k , 0 = − 0.0197, which is in agreement with the final Planck data release (Tristram et al., 2024). The CMB temperature spectrum then perfectly agrees with the ΛCDM spectrum. We can conclude that this model agrees well with measurement data, which leads to a high probability in the result of an MCMC analysis and possibly solves the problem of MCMC analyses being trapped in phantom energy.

In conclusion, we summarize our main results as follows:

• DDE models with an increasing EoS parameter seem to be not compatible with the observed late-time accelerated expansion of the universe and could therefore be regarded as physically improbable.
• DDE models with a decreasing EoS parameter can possibly provide a solution to the Hubble tension.
• In the MCMC data analysis, the prior distribution of H 0 must cover a sufficiently large range to allow values of H 0 that agree with measurements in the local universe to be sampled. A one-step full MCMC analysis including all priors of the cosmological model and the DDE parametrization is strongly recommended.
• Spatial curvature and DE lead to degeneracy and therefore must be considered in the parameters of the cosmological model (and the priors of the MCMC data analysis) to avoid disfavoring DDE models with decreasing EoS parameter in an MCMC data analysis due to a higher expansion rate H than ΛCDM in the late stages of the expansion history.
• The customary backward integration procedure applied in an MCMC data analysis can then provide results compatible with ΛCDM’s CMB temperature spectrum, BAO measurement data, and an expansion history consistent with measurements of H 0 in the local universe. In particular, it will provide physically probable parameters of a DDE model.
• The reported models of DDE do not agree with the observations of accelerated expansion of the universe. Furthermore, we find no compelling evidence for an oscillating EoS, which is sometimes interpreted as the need for new physics that is necessary to explain the results (e.g., Kevork N. Abazajian in Zack Savitsky (2025, 03 19) Shocking Dark Energy Findings Challenge the Standard Model of the Universe. Scientific American, [https://www.scientificamerican.com/article/frozencosmic- sound-bubbles-suggest-dark-energy-isshockingly-changeable/](https://www.scientificamerican.com/article/frozencosmic- sound-bubbles-suggest-dark-energy-isshockingly-changeable/)).

In conclusion, we comment on our findings of Section 3, where we concluded that DDE components with an increasing EoS parameter replacing Λ in the concordance ΛCDM model do not agree with the observation of the accelerated expansion of the universe and we regarded these models as physically improbable. In principle, these conclusions can be found by performing in-depth cross-checks of the results yielded by an MCMC data analysis.

However, the extended MCMC procedure we introduced in Paper I can help avoid MCMC data analyses to yield cosmological models that are physically improbable as it incorporates consistency checks of the sampled cosmological models based on a complementary computation procedure, which breaks the discussed degeneracy. We provide a brief summary of this optional procedure in Supplementary Appendix A.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fspas.2026.1818402/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Hong-sheng Zhao, University of St Andrews, United Kingdom

Reviewed by: Dario Bettoni, University of León, Spain

Ghulam Mustafa, Zhejiang Normal University, China

Footnotes

The code CLASS is publicly available at https://github.com/lesgourg/class_public.

The code CAMB is publicly available at https://github.com/cmbant/CAMB.