CPL Dark Energy Parametrisation

• CPL parametrisation is a two-parameter model defining dark energy’s equation of state as w(a)=w₀ + wₐ(1-a), capturing late-time cosmic acceleration through a Taylor expansion about the present epoch.
• It is widely applied in BAO, SNIa, and CMB analyses to standardize performance metrics and forecast constraints despite limitations at high redshift.
• The model aids in comparing dynamical dark energy against ΛCDM but exhibits degeneracies and pathologies that require careful interpretation and further extensions.

The Chevallier–Polarski–Linder (CPL) parametrisation is an influential, widely adopted two-parameter model specifying the evolution of the dark energy equation of state (EoS) as a function of cosmological scale factor or redshift. Defined as a first-order Taylor expansion about the present epoch, CPL enables precision constraints on late-time cosmic acceleration, forms the basis of key survey analyses and Fisher forecasts, and serves as a standard benchmark for testing dynamical dark energy models against ΛCDM. However, CPL exhibits notable limitations in physical interpretation, flexibility at high redshift, and potential degeneracies, mandating careful critical assessment in the context of cosmological constraints and model selection.

1. Mathematical Definition and Parametric Form

The CPL parametrisation describes the dark energy EoS $w(a)$ as a linear function of the scale factor $a$ or, equivalently, redshift $z$: $w(a) = w_0 + w_a (1 - a)$

$w(z) = w_0 + w_a \frac{z}{1+z}$

where:

• $w_0 = w(a=1)$ is the present-day value of the EoS.
• $w_a = -\left.\frac{dw}{da}\right|_{a=1}$ quantifies the running or rate of departure from $w_0$ towards higher redshift (early times).

The functional form assures $w(a=1)=w_0$ and $w(a\rightarrow 0)=w_0 + w_a$, so the EoS at early times is $w_0 + w_a$. This structure directly informs the evolution of the dark energy density: $$\rho_{de}(a) = \rho_{de,0}\, a^{-3(1+w_0+w_a)} \exp[-3w_a(1-a)]$$

2. Physical Motivation and Theoretical Context

CPL is motivated as the lowest-order Taylor expansion of $w(a)$ about $a=1$, designed to efficiently capture mild time dependence in the EoS over the redshifts most relevant for late-time cosmology: $w(a) = \sum_{n=0}^\infty c_n (1-a)^n \Longrightarrow \text{CPL: retain only %%%%11%%%% terms}$ This minimality allows transparent physical interpretation: $w_0$ and $w_a$ single out the value and first derivative of $w(a)$ today. The linear-in-$(1-a)$ construction gives stability and simplicity on the fitting interval ($0 < a \le 1$), and is preferred in Fisher-matrix approaches, principal component analyses, and survey figure-of-merit calculations (Zhao et al., 2015).

Nevertheless, the CPL form imposes extra structure not required by data:

• All higher derivatives $d^p w/da^p$ for $p \ge 2$ are identically zero, an implicit prior that narrows the parameter space and can artificially strengthen exclusions or detections (Nesseris et al., 28 Mar 2025).
• Extrapolated to high redshift ($z \rightarrow \infty$), CPL yields $w(z) \rightarrow w_0 + w_a$, which may be inconsistent with more general scalar field or barotropic DE models, as well as physically nonviable if $w_0 + w_a < -1$ (phantom regime) (Artola et al., 5 Oct 2025, Notari et al., 12 Jun 2024).

3. Applications in Cosmological Data Analysis and Parameter Fitting

CPL has been extensively employed in large-scale survey analyses, likelihoods, and simulation pipelines:

• BAO, SNIa, CMB: Inclusion of $w_0$, $w_a$ in joint parameter spaces, with flat priors [0.1,0.5]$\times$[$-1.5,0.5$]$\times$[$-2,2$] or similar (Dong et al., 28 Oct 2025, Tsedrik et al., 19 Dec 2025, Giarè et al., 23 Jul 2024).
• Statistical inference: Likelihoods constructed as $\mathcal{L} \propto \exp(-\chi^2/2)$; parameter constraints obtained via MCMC or Fisher matrices; error ellipses and figures of merit (FoM) quantified (Zhao et al., 2015, Tsedrik et al., 19 Dec 2025).
• Distance measures and growth: The background expansion, observable distances (e.g., $D_A$, $D_V$, $E(z)$), and growth rates explicitly depend on CPL $w(a)$. Linear perturbations, clustering, and RSD are modeled by inputting $w(a)$ into modified Boltzmann solvers (Tsedrik et al., 19 Dec 2025, Sinha, 2021).
• Nontrivial extension to non-GR frameworks: CPL's $w(a)$ can be embedded in modified gravity scenarios or effective fluids derived from inhomogeneity backreaction, with appropriate mapping to model parameters (Yao et al., 30 May 2024).

4. Empirical Constraints and Observational Performance

Meta-analyses and recent data releases consistently yield tight confidence intervals on $(w_0, w_a)$:

• SDSS AP-only (z<0.7): $w_0 = -0.798^{+0.192}_{-0.102}$, $w_a = -0.165^{+0.610}_{-0.945}$ (Dong et al., 28 Oct 2025).
• SDSS AP + Pantheon+ SNIa + DESI BAO: $w_0 = -0.856^{+0.050}_{-0.044}$, $w_a = -0.124^{+0.334}_{-0.368}$ (Dong et al., 28 Oct 2025).
• DESI + Planck + Pantheon+: $w_0 = -0.84^{+0.11}_{-0.12}$, $w_a = -0.60^{+0.50}_{-0.40}$ (95% CL) (Barua et al., 22 Aug 2025).
• DESI DR1 + Y5 SNe: $w_0 = -0.726 \pm 0.069$, $w_a = -1.05^{+0.34}_{-0.28}$ (Giarè et al., 23 Jul 2024).
• DES Y3 + DESI DR2 + SN + CMB: $w_0 = -0.76 \pm 0.06$, $w_a = -0.77^{+0.23}_{-0.20}$ (Tsedrik et al., 19 Dec 2025).

All robust combinations remain consistent with a cosmological constant ($w_0 = -1$, $w_a = 0$) at better than $2\sigma$ once multiple late-universe probes are combined (Dong et al., 28 Oct 2025, Nesseris et al., 28 Mar 2025, Barua et al., 22 Aug 2025).

Addition of high-precision photometric data or larger redshift lever arms improves the DETF FoM and tightens constraints, but CPL's inherent degeneracy direction (e.g., banana-shaped error ellipses) persists, typically along approximately $w_a \approx -3.6(1 + w_0)$ (Tsedrik et al., 19 Dec 2025).

5. Physical Interpretation, Mapping to Fundamental Models, and Limitations

5.1 Scalar Field Realization (Quintessence)

For canonical quintessence, the CPL $w(a)$ induces a potential $V(\phi)$ through

$$\rho_{DE}(a) = \rho_{DE,0}\, a^{-3(1+w_0+w_a)}\,e^{3w_a(a-1)}, \quad \frac{d\phi}{da} = \sqrt{\frac{(1+w(a))\rho_{DE}(a)}{[a\,H(a)]^2}}$$

Mapping CPL onto quintessential $V(\phi)$ yields a narrow family of gently sloping, convex potentials, except along special lines (e.g., $w_a \simeq -1.5(1 + w_0)$ for thawing models), where a richer diversity of potentials is possible. Thus CPL does not generically span the range of physically motivated potentials (Sadri, 2017, Scherrer, 2015).

5.2 Barotropic Models

For barotropic fluids, CPL $w(a)$ implies $p(a) = w(a) \rho(a)$, but the relation $p(\rho)$ is encoded in a single combination $w_i = w_0 + w_a$. Thus, two CPL parameters reduce to a single degree of freedom, again highlighting degeneracy and potential parameter redundancy (Scherrer, 2015).

5.3 High-Redshift Limit and Pathologies

CPL enforces $w(z\to\infty) = w_0 + w_a$, which, for typical best fits with $w_a < 0$, can yield strongly phantom $w \ll -1$ behavior at early times. Empirical fits to DESI BAO alone drive $w_a$ to large negative values to compensate for discrepancies, pushing the model to unphysical regions characterized by rapid energy density decay or negative physical densities (Lee, 23 Jun 2025, Artola et al., 5 Oct 2025). Similarly, at $a \to \infty$, CPL can diverge, signaling a breakdown in the linear approximation well outside the observed epoch.

6. Extensions, Alternatives, and Model Selection

6.1 Higher-Order Expansions and Information Injection

Restricting $w(a)$ to CPL's first-order truncation imposes zero second and higher derivatives $(d^p w/da^p = 0)$, which is not required by data and acts as an unwarranted prior, artificially tightening constraints or even excluding $\Lambda$ ($w_0 = -1, w_a = 0$) spuriously. Allowing quadratic or cubic terms and marginalizing over them reliably restores consistency with $\Lambda$CDM (Nesseris et al., 28 Mar 2025, Notari et al., 12 Jun 2024, Pan et al., 2019).

6.2 Physically Consistent Realizations

Canonical quintessence cannot realize $w<-1$ (i.e., phantom behavior). CPL best-fit regions with $w_0 + w_a < -1$ violate this, implying that the linear parametrisation is not always physically realizable. Modifications such as the “ramp” model, where $w(a)$ matches CPL at late times but is frozen or capped at $w_i > -1$ at early times, can circumvent this and retain a physically consistent regime throughout cosmic history (Notari et al., 12 Jun 2024).

6.3 Smoothed and Alternative Parametrisations

Smoothed families (e.g., sigmoid, tanh) and other two-parameter models (exponential, Barboza–Alcaniz, Jassal–Bagla–Padmanabhan) have been proposed to mitigate or test CPL's high-$z$ artifacts. Empirical analyses reveal that, for $z\lesssim2$, these alternatives are observationally indistinguishable from CPL given current data (Artola et al., 5 Oct 2025, Giarè et al., 23 Jul 2024) but may offer greater flexibility and reduced pathologies.

7. Empirical Performance, Degeneracies, and Best Practices

Forecasts for Stage IV and V surveys (e.g., DESI, eBOSS, LSST) highlight CPL’s utility in survey design, providing standardized performance metrics (DETF FoM, eigenmode sensitivity) (Zhao et al., 2015). Nevertheless, degeneracies between $w_0, w_a$, and other cosmological parameters (e.g., $\Omega_m$, $H_0$, $\sum m_\nu$) complicate unique inference, and combined analyses (CMB+BAO+SNIa) are required to robustly break parameter degeneracies (Barua et al., 22 Aug 2025).

Best practices emerging from recent analyses:

• Do not claim exclusion of $w=-1$ unless higher-order terms are included and robustness to expansion order is demonstrated (Nesseris et al., 28 Mar 2025).
• Combine a diverse set of observables (BAO, SNIa, CMB, RSD) and deploy model-independent or higher-order parametrisations to cross-check results (Notari et al., 12 Jun 2024, Dong et al., 28 Oct 2025).
• When testing for dynamical dark energy, use principal component, binned, or non-parametric reconstructions alongside CPL for comprehensive coverage (Zhao et al., 2015).

Table: Representative Recent Constraints on CPL Parameters

Data Combination	$w_0$	$w_a$	Reference
SDSS AP-only (z<0.7)	$-0.80_{-0.10}^{+0.19}$	$-0.17_{-0.95}^{+0.61}$	(Dong et al., 28 Oct 2025)
AP+Pantheon+SNe+DESI BAO	$-0.86^{+0.05}_{-0.04}$	$-0.12^{+0.33}_{-0.37}$	(Dong et al., 28 Oct 2025)
DESI+Planck+Pantheon+	$-0.84^{+0.11}_{-0.12}$	$-0.60^{+0.50}_{-0.40}$	(Barua et al., 22 Aug 2025)
DES Y3+DESI DR2+SN+CMB	$-0.76\pm 0.06$	$-0.77^{+0.23}_{-0.20}$	(Tsedrik et al., 19 Dec 2025)
Planck+DESI+Pantheon-Plus	$-0.82 \pm 0.06$	$-0.77 \pm 0.28$	(Giarè et al., 23 Jul 2024)
Planck+DESI+DES Y5	$-0.73 \pm 0.07$	$-1.05^{+0.34}_{-0.28}$	(Giarè et al., 23 Jul 2024)

Summary and Outlook

The Chevallier–Polarski–Linder parametrisation remains a central tool for constraining the possible time evolution of the dark energy equation of state at late times, owing to its simplicity, analytic tractability, and direct relation to observable distances and growth. However, its built-in limitations—spurious rigidity, high-$z$ pathologies, degeneracy, and limited mapping onto fundamental model space—necessitate caution in physical interpretation, especially as Stage IV data become increasingly precise. Extending the parameter space (higher-order expansions, non-parametric forms), combining diverse observational probes, and explicitly testing the physical realizability of best-fit solutions are essential to robustly assessing dynamical dark energy beyond the CPL paradigm (Dong et al., 28 Oct 2025, Nesseris et al., 28 Mar 2025, Notari et al., 12 Jun 2024).