On the Analysis Dependence of DESI Dynamical Dark Energy

Abstract

We continue scientific scrutiny of the DESI dynamical dark energy (DE) claim by explicitly demonstrating that the result depends on the analysis pipeline. Concretely, we define a likelihood that converts the $w_0{w}_a$CDM model back into the (flat) $\Lambda$CDM model, which we fit to DESI constraints on the $\Lambda$CDM model from DR1 Full-Shape (FS) modeling and BAO. We further incorporate CMB constraints. Throughout, we find that $w_0$ and $w_a$ are within $1\sigma$ of the $\Lambda$CDM model. Our work makes it explicit that, in contrast to DR1 and DR2 BAO, there is no dynamical DE signal in FS modeling, even when combined with BAO and CMB. Moreover, one confirms late-time accelerated expansion today $(q_0<0)$ at ≳3.4$\sigma$ in FS modeling + BAO. On the contrary, DR1 and DR2 BAO fail to confirm $q_0<0$ under similar assumptions. Our analysis highlights the fact that trustable scientific results should be independent of the analysis pipeline.

1. Introduction

The DESI collaboration has presented a suite of dynamical DE claims [1,2,3]. These claims begin with baryon acoustic oscillation (BAO) data but become more statistically significant in the presence of external datasets. Overlooked in this discussion is the observation from [4] that DESI Full-Shape (FS) modeling of galaxy clustering exhibits no hint of a dynamical DE signal. Nevertheless, both DESI BAO and FS modeling prefer a lower value of the (flat) $\Lambda$CDM parameter ${\Omega }_m$ relative to the Planck [5]. In combination with external datasets, this still suffices to produce a statistically significant dynamical DE signal.

DESI recently released the DR2 BAO results [3], showing that the combination DR2 BAO+CMB prefers dynamical DE at $3.1\sigma$. As remarked in [1,6,7,8,9,10,11,12], there are noticeable fluctuations in both DR1 and DR2 BAO, especially in luminous red galaxies (LRGs), which primarily drive the dynamical DE signal in DESI data alone. When combined with CMB and Type I supernovae (SNe) datasets, the overall discrepancy in ${\Omega }_m$ between the datasets increases the significance [3].

Given that there is explicitly a dynamical DE signal in BAO [1,2,3] ($w_0>-1$) but implicitly no dynamical DE signal in FS modeling [4], there is a degree of confusion. What compounds the confusion is the omission of explicit FS modeling + BAO entries for the Chevallier–Polarski–Linder (CPL) $w_0{w}_a$CDM model [13,14] in Table 2 of the DESI DR1 FS modeling paper [2]: only FS + BAO in combination with both the CMB and SNe datasets are considered. FS + BAO in combination with CMB alone is not. The reason for the omission is the documented projection effects in Bayesian posteriors in the absence of SNe data [2] (see [15] for frequentist analysis).

On the other hand, Table 3 of the DR1 BAO paper [1] shows the analogous constraints for BAO. Thus, the point of our letter is to make it explicit that fits of the CPL model to DESI FS modeling and BAO constraints [4], both with and without CMB, lead to results consistent with $\Lambda$CDM within $1\sigma$. This addresses a gap in the current literature. The upshot of this outcome is that one can confirm late-time accelerated expansion today, $q_0<0$, in DESI FS modeling + BAO alone confronted with the CPL model at ≳3.4$\sigma$. As highlighted initially in [16], the same cannot currently be said for DESI DR1 and DR2 BAO.

2. Analysis

We employ a technique we have used in previous papers [6,17], allowing one to map the CPL model back into the $\Lambda$CDM model at a given redshift $z_i$:

$${\displaystyle \frac{D_M\left(z_i\right)}{D_H\left(z_i\right)}}=E\left(z_i\right){\int }_0^{z_i}{\displaystyle \frac{z}{E\left(z\right)}}.$$

(1)

In the above, the left-hand side is computed for the CPL model,

$$\begin{array}{cc}\hfill D_M\left(z\right)=c{\int }_0^z{\displaystyle \frac{z^{\prime }}{H\left(z^{\prime }\right)}},& D_H\left(z\right)={\displaystyle \frac{c}{H\left(z\right)}}\hfill \\ \hfill H^2\left(z\right)=H_0^2[{\Omega }_m{(1+z)}^3+(1-& {\Omega }_m{)(1+z)}^{3(1+{\omega }_0+{\omega }_a)}{e}^{-{\displaystyle \frac{3{\omega }_{a}z}{1+z}}}],\hfill \end{array}$$

(2)

while the right-hand side of (1) is computed for the $\Lambda$CDM model, for which

$$E^2\left(z\right)=1-{\tilde{\Omega }}_m+{\tilde{\Omega }}_m{(1+z)}^3.$$

(3)

Note that the matter density of the CPL model ${\Omega }_m$ is distinct from that of the $\Lambda$CDM model ${\tilde{\Omega }}_m$. Henceforth, we denote the $\Lambda$CDM matter density parameter with a tilde to avoid confusion. Observe also that $H_0$ drops out from the left-hand side, so we simply fix $H_0=70$ km/s/Mpc. As a result, one is fitting only the $({\Omega }_m,w_0,w_a)$ parameters from the CPL model. We employ (1) only at low redshift where any contribution from radiation is negligible. We will discuss how one incorporates CMB in due course.

The mapping in (1) can be applied beyond CPL more generally to any FLRW model on the left-hand side to map it into the $\Lambda$CDM parameter ${\tilde{\Omega }}_m$. If there are no deviations from $\Lambda$CDM behaviour, one recovers a constant ${\tilde{\Omega }}_m$ at all redshifts probed. There is overlap with the $Om\left(z\right)$ diagnostic [18], but $Om\left(z\right)$ is usually continuous, necessitating a reconstruction of $E\left(z\right)≔H\left(z\right)/H_0$, whereas (1) begins from the cosmological distances at discrete redshifts typically constrained more directly by observations, e.g. BAO. In addition, $Om\left(z\right)$ typically blows up at lower z (one can try propagating redshift errors to ameliorate this), whereas (1) propagates errors in both the numerator and denominator and cannot blow up at lower z. One could simply equate $H{\left(z\right)}^{\mathrm{FLRW}}=H{\left(z\right)}^{\Lambda \mathrm{CDM}}$, but then one would need to assume that $H_0^{\mathrm{FLRW}}=H_0^{\Lambda \mathrm{CDM}}$ to obtain a constraint on ${\tilde{\Omega }}_m$. Here, we do not need to assume $H_0^{\mathrm{FLRW}}=H_0^{\Lambda \mathrm{CDM}}$ since $H_0$ drops out in the ratio.

In our analysis we use of the constraints on ${\tilde{\Omega }}_m$ provided by the DESI collaboration from FS modeling + BAO [4] in

Table 1. Constraints on ${\tilde{\Omega }}_m$ from FS modeling + BAO from different tracers at different effective redshifts. Redshifts and constraints are reproduced from Table 1 and Table 10 of [4].

Tracer	${\mathit{z}}_{\mathbf{eff}}$	${\tilde{\mathbf{\Omega }}}_{\mathit{m}}$
BGS	$0.295$	$0.284\pm 0.024$
LRG1	$0.510$	${0.307}_{-0.020}^{+0.018}$
LRG2	$0.706$	$0.287\pm 0.020$
LRG3	$0.919$	$0.304\pm 0.023$
ELG2	$1.317$	${0.310}_{-0.034}^{+0.027}$
QSO	$1.491$	${0.314}_{-0.039}^{+0.029}$

. One could use the constraints from FS modeling alone, but it will not change the conclusions since BAO does not have a strong bearing on FS modeling (see Table 10 of [4]). We define a log-likelihood $\mathcal{L}({\Omega }_m,w_0,w_a)$ with input parameters $({\Omega }_m,w_0,w_a)$ from the CPL model. For each $z_i\in z_{\mathrm{eff}}$ in Table 1, we solve Equation (1) to identify the corresponding ${\tilde{\Omega }}_m\left(z_i\right)$ value from the $\Lambda$CDM model. From there, we define the log-likelihood

$$log\mathcal{L}({\Omega }_m,w_0,w_a)=-{\displaystyle \frac{1}{2}}{\chi }^2=-{\displaystyle \frac{1}{2}}\sum _i{\displaystyle \frac{{({\tilde{\Omega }}_m\left(z_i\right)-{\tilde{\Omega }}_m^i)}^2}{{\sigma }_{{\tilde{\Omega }}_m^i}^2}},$$

(4)

where ${\tilde{\Omega }}_m^i$ denotes the central value of the ${\tilde{\Omega }}_m$ values, and ${\sigma }_{{\tilde{\Omega }}_m^i}$ denotes the errors in Table 1. Given the antisymmetric errors in Table 1, if ${\tilde{\Omega }}_m\left(z_i\right)>{\tilde{\Omega }}_m^i$ we use the upper error, and vice versa. This ensures that our log-likelihood properly takes account of the differences in antisymmetric errors. Note, we are unaware of any work in the literature fitting model A to model B constraints by rewriting it as model B. This may be a novel workaround for model comparison.

Once the log-likelihood is defined, one marginalizes over the CPL parameters $({\Omega }_m,w_0,w_a)$ with Markov Chain Monte Carlo (MCMC) to identify the parameters that best fit the DESI FS + BAO constraints. We employ emcee [19] with the DESI priors, $w_0\in [-3,1],w_a\in [-3,2]$ and $w_0+w_a<0$ [1]. The result of this exercise is presented in Figure 1 and

Table 2. Constraints on the CPL parameters and reconstructed deceleration parameter $q_0$ from FS modeling + BAO with and without CMB.

Data	${\mathbf{\Omega }}_{\mathit{m}}$	${\mathit{w}}_0$	${\mathit{w}}_{\mathit{a}}$	${\mathit{q}}_0$
FS + BAO	${0.307}_{-0.051}^{+0.036}$	$-{1.02}_{-0.11}^{+0.12}$	$-{0.03}_{-1.1}^{+0.76}$	$-{0.58}_{-0.13}^{+0.17}$
FS + BAO + CMB	${0.3025}_{-0.0069}^{+0.0070}$	$-{1.042}_{-0.097}^{+0.099}$	${0.03}_{-0.36}^{+0.34}$	$-0.59\pm 0.11$

, where we use getdist [20] to plot the posteriors. The ($w_0,w_a$) values are within $1\sigma$ of $\Lambda$CDM, $(w_0,w_a)=(-1,0)$, so there is no trace of dynamical DE as expected. There is a concern that our methodology, namely fitting the CPL model through the $\Lambda$CDM model to $\Lambda$CDM constraints, risks washing out a dynamical DE signal even if one is present. In the Appendix A, we perform a consistency check to show that we can recover the dynamical DE signal in DESI DR1 BAO using both Bayesian and frequentist methods. Our MCMC posteriors are impacted by priors and projection/volume effects, but in a frequentist analysis it is clear that $w_0>-1$.

At this stage, there is a further consistency check one can perform. It is evident from the ${\tilde{\Omega }}_m$ values in Table 1 and the blue constraints in Figure 2 that there is a mild increasing trend in the ${\tilde{\Omega }}_m$ central values. We expect to see this in the best-fit CPL model when it is mapped back to $\Lambda$CDM. Given the MCMC chain, one can use (1) and redshifts in the range $z\in [0.25,1.5]$ separated by a uniform $\Delta z=0.025$ to reconstruct a distribution of ${\tilde{\Omega }}_m$ values at each redshift. One then isolates the 16th and 84th percentiles of ${\tilde{\Omega }}_m$ at each redshift as the limits of the 68% confidence intervals and the median as the central value. What one expects to find through the consistency check is that the increasing trend evident in the blue constraints is mirrored in the resulting confidence interval band. As can be seen from Figure 2, one sees this feature in the green band.

The next step is to incorporate CMB to see if this makes a difference to the conclusions. Here, we follow the DESI collaboration and introduce the Gaussian priors on $({\theta }_{\ast },{\omega }_b,{\omega }_m)$ from appendix A of [3], where we define ${\theta }_{\ast }=r_{\ast }/D_M\left(z_{\ast }\right)$, ${\omega }_b={\Omega }_b{h}^2$ and ${\omega }_m={\Omega }_m{h}^2$. ${\Omega }_b$ is the baryonic matter density parameter, $r_{\ast }$ denotes the comoving sound horizon at last scattering and $h≔H_0/\left(100\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\right)$. This reintroduces $H_0$, which drops out of the log-likelihood in (4) but is relevant for the CMB constraints. Furthermore, ${\Omega }_b$ appears as a second additional parameter. Finally, we fix $z_{\ast }=1090$ and introduce a radiation sector in the CPL model (2) with the standard $a^{-4}$ scaling with fixed coefficient ${\Omega }_r=4.18\times {10}^{-5}/h^2$. We add the CMB log-likelihood dependent on $v=({\theta }_{\ast },{\omega }_b,{\omega }_m)$ to the log-likelihood for FS modeling + BAO, which results in a log-likelihood that depends on $(H_0,{\Omega }_m,{\Omega }_b,w_0,w_a)$:

$$\begin{array}{cc}\hfill log\mathcal{L}(H_0,{\Omega }_m,{\Omega }_b,w_0,w_a)=& -{\displaystyle \frac{1}{2}}\sum _i{\displaystyle \frac{{({\tilde{\Omega }}_m\left(z_i\right)-{\tilde{\Omega }}_m^i)}^2}{{\sigma }_{{\tilde{\Omega }}_m^i}^2}}\hfill \\ & -{\displaystyle \frac{1}{2}}\Delta v·C^{-1}·\Delta v,\hfill \end{array}$$

(5)

where we define $\Delta v=v-v_{\mathrm{theory}}$. Expressions for v and C can be found in appendix A of [3] and one calculates $v_{\mathrm{theory}}=({\theta }_{\ast },{\omega }_b,{\omega }_m)$ from the log-likelihood input parameters. Marginalizing over the parameters through MCMC, while isolating $({\Omega }_m,w_0,w_a)$ for comparison, one obtains the result in Figure 1, where the corresponding $68\%$ confidence intervals can be found in Table 2. The consistency check of reconstructing ${\tilde{\Omega }}_m$ appears in Figure 2, where we see that the red (brownish) confidence interval band shows the expected increasing redshift trend.

A number of comments are in order. First, comparing FS + BAO with and without CMB from Table 2, one can see the difference that CMB makes. CMB greatly increases the ${\Omega }_m$ precision, and decreases the $w_a$ errors by a factor of 2–3, but there is no great increase in $w_0$ precision. This is also reflected in the reconstructed ${\tilde{\Omega }}_m$ in Figure 2. What one sees is that the green and red (brownish) confidence intervals show little difference at lower redshifts, where the DE sector parameterized by $(w_0,w_a)$ is most relevant, whereas at higher redshifts in the matter dominated regime, the confidence intervals contract appreciably. One also notes that the CMB is forcing the reconstruction to a relatively lower canonical ${\tilde{\Omega }}_m\sim 0.3$ value at higher redshifts. Secondly, whether one employs CMB constraints or not, we find $(w_0,w_a)$ values that are within $1\sigma$ of $\Lambda$CDM.

Finally, we come to a key remark. In [16] it was noted that DESI DR1 [1] and DR2 BAO [3], when confronted with the CPL model, fail to confirm late-time accelerated expansion today, which is characterized by a negative deceleration parameter:1

$$q_0={\displaystyle \frac{1}{2}}\left[1+3w_0(1-{\Omega }_m)\right]<0.$$

(6)

In contrast, from Table 2 and Figure 3, we see that FS + BAO confirms late-time accelerated expansion at $3.4\sigma$ without CMB and at $5.4\sigma$ with CMB. This highlights a key difference in the physical implications of BAO alone compared to FS + BAO.

3. Outlook

Where will future DESI results take us? What should be clear is that the DESI BAO and DESI FS modeling results need to converge. It is evident from the analysis in [16] that DESI BAO shows good consistency between DR1 and DR2 at higher redshifts but that fluctuations are still present in LRG and potentially the lowest redshift emission line galaxy (ELG) bin. More concretely, while LRG1 was primarily responsible for the dynamical DE signal in DR1 BAO data alone [6], in DR2 BAO, LRG2 is the main driver of the dynamical DE signal [16]. When data points move about to this extent, the risk is that fluctuations are present.

The point of this letter is to make the implicit explicit and provide results that were omitted in Table 2 of [2], namely constraints on the CPL model from DESI DR1 FS + BAO and FS + BAO + CMB. Projection effects were used to justify the omission of the results on the grounds that the mean and mode of posteriors did not agree [2]. From a mathematical perspective, this is puzzling since if the $\Lambda$CDM model is well constrained in redshift bins, this has implications for the CPL dark energy model; (1) is a mapping between any late-universe FLRW cosmology and $\Lambda$CDM. With additional nuisance parameters beyond cosmological parameters, projection effects can easily arise due to degeneracies between the parameters.

We have demonstrated that there is no hint of a dynamical DE signal in FS modeling, even when combined with CMB. To this end, we have employed a mapping between the CPL model and $\Lambda$CDM. This strategy of mapping model A to model B to fit model A to model B constraints may be a novel workaround; we are unaware of any examples. We test our method in Appendix A. However, given that one can run a horizontal constant ${\tilde{\Omega }}_m$ line through the blue constraints in Figure 2, this is the expected and obvious result. Finally, while DESI DR1 and DR2 BAO fail to confirm late-time accelerated expansion today [16], we see that any combination with FS modeling does this in excess of $3\sigma$.

The interesting question now is, assuming BAO converges to FS modeling, what will happen when the blue constraints in Figure 2 shrink as FS modeling + BAO results are upgraded from DR1 to DR2? If the central values do not shift, this will leave an increasing ${\tilde{\Omega }}_m$ trend with redshift. We remind the reader that in [6] it was conjectured that an increasing ${\tilde{\Omega }}_m$ signal would emerge from DESI data, in particular BAO. The flip side of an increasing matter density parameter with redshift in the $\Lambda$CDM model is a decreasing Hubble constant $H_0$ with redshift in the $\Lambda$CDM model2, thereby corroborating Hubble tension, a discrepancy in $H_0$ between the early (high redshift) and late (low redshift) universe [25]. More generally, see [26] for a review of earlier observations of redshift-dependent $\Lambda$CDM parameters—a hallmark of model breakdown—in different observables. See also [27] for a recent relevant observation that redshift-dependent $\Lambda$CDM fitting parameters3 can be found in SDSS data.

Finally, great care is required when combining BAO and SNe datasets. One needs to check that BAO and SNe agree on cosmological distances in overlapping redshift ranges, e.g., [17]. Two groups [28,29] have recently reported a breakdown in the distance duality relation when DESI BAO is combined with SNe. The physical implications of such a breakdown are so profound (giving up conservation of photon propagation in a metric theory of gravity) that systematics must be present.