Dark energy constraints in light of theoretical priors

Problem Statement
Current efforts to constrain dark energy (DE) and modified gravity (MG) theories often rely on parametrizing the space of possible theories to derive model-independent observational bounds. A common approach involves two complementary strategies: (I) phenomenological parametrisations with minimal theoretical assumptions (e.g., modifying the Poisson equations via functions $\mu(a,k)$ and $\Sigma(a,k)$), and (II) theory-informed approaches based on Effective Field Theory of Dark Energy (EFTDE) or Horndeski scalar-tensor theories.

The central problem addressed in this paper is that the choice of parametrisation and the associated theoretical priors can significantly alter the resulting cosmological constraints. Without a clear understanding of how theoretical priors map to phenomenological observables, there is a risk of either missing well-motivated physical theories or falsely interpreting phenomenological deviations as evidence for physics that cannot arise from a consistent underlying theory. Specifically, the authors investigate how different theoretical priors—ranging from the functional form of time dependence to constraints derived from gravitational wave (GW) physics—affect the constraints on the phenomenology of dynamical dark energy, particularly the linear perturbations.

Methodology
The authors employ a Markov Chain Monte Carlo (MCMC) analysis using the MontePython code to constrain cosmological parameters against a comprehensive set of datasets:

• CMB: Planck 2018 likelihoods (including lensing and low-$\ell$ TT/EE/TE).
• Large Scale Structure (LSS): eBOSS DR16 joint Redshift Space Distortions (RSD) and Baryon Acoustic Oscillations (BAO) data.
• Supernovae: Pantheon+ likelihood.
• Integrated Sachs-Wolfe (ISW): Cross-correlations between CMB temperature and galaxy number counts.

The study compares two primary frameworks:

1. Phenomenological Parametrisations: Directly modeling the modifications to the Poisson equations using $\mu(a)$ and $\Sigma(a)$ (or the slip parameter $\gamma(a)$). The authors test two specific time-dependence ansätze: proportionality to the dark energy density fraction ($\propto \Omega_{DE}$) and proportionality to the scale factor ($\propto a$).
2. Theory-Informed (EFTDE) Parametrisations: Utilizing the EFTDE/Horndeski framework where linear perturbations are described by time-dependent functions $\alpha_B(a)$ (braiding) and $\alpha_M(a)$ (running of the Planck mass). The authors map these underlying parameters to the phenomenological $\mu$ and $\Sigma$ using the Quasi-Static Approximation (QSA) and scale-independent growth assumptions.

The analysis systematically varies theoretical priors, including:

• The functional time dependence of the underlying theory functions ($\alpha_i \propto \Omega_{DE}$ vs. $\alpha_i \propto a$).
• Theoretical constraints on the speed of gravitational waves ($\alpha_T = 0$ vs. free $\alpha_T$).
• Stability constraints in a GW background (requiring $|\alpha_B + \alpha_M| \lesssim 10^{-2}$).
• The interplay between background expansion history (fixed $\Lambda$CDM vs. free CPL $w(a)$) and perturbation dynamics.

Key Contributions and Results

• Mapping Priors to Phenomenology: The authors demonstrate that deriving $\mu$ and $\Sigma$ from an underlying EFTDE framework imposes a strong, non-trivial theoretical prior on the present-day values $\{\mu_{today}, \Sigma_{today}\}$. This prior restricts the allowed parameter space and introduces correlations not present in purely phenomenological fits. Notably, the region $\mu_{today} < 1, \Sigma_{today} > 1$ is entirely excluded by the gradient stability condition for models with luminal GW speeds, a restriction absent in unconstrained phenomenological models.
• Impact of Time Dependence: Comparing the $\propto \Omega_{DE}$ and $\propto a$ time dependences reveals qualitative differences. The $\propto a$ dependence affects a broader redshift range, leading to tighter constraints on $\mu_{today}$ due to the increased constraining power of high-redshift data. Furthermore, the $\propto a$ dependence results in a significant overlap between stable and unstable parameter spaces in the $\{\mu_{today}, \Sigma_{today}\}$ plane, unlike the well-separated spaces found in the $\propto \Omega_{DE}$ case.
• Posterior vs. Prior Volume: A counter-intuitive result is found in the EFTDE $\propto \Omega_{DE}$ model: the region $\mu_{today} > 1, \Sigma_{today} < 1$, which has the smallest prior volume (due to theoretical constraints), possesses the largest volume in the observational posterior. This indicates that current data (clustering and lensing) are sufficiently constraining to overcome prior volume effects and favor this specific quadrant.
• Distinguishability of Theories: For a specific class of shift-symmetric theories that satisfy the "no-slip" condition ($\mu = \Sigma$), the authors find that theoretically motivated time dependences cannot be distinguished from naive phenomenological parametrisations based on current constraints on $\mu_{today}$.
• Gravitational Wave Priors:
  • Allowing $\alpha_T$ to vary (relaxing the GW170817 constraint) broadens the posterior slightly and opens up the $\mu_{today} < 1, \Sigma_{today} > 1$ quadrant, which is otherwise forbidden.
  • Imposing stability in a GW background (effectively $\alpha_B = -\alpha_M$) reduces the parameter space to a single functional degree of freedom. This results in a one-dimensional posterior for $\{\mu_{today}, \Sigma_{today}\}$ and significantly tighter constraints.
• Background-Perturbation Interplay: While freeing the background expansion history ($w_0, w_a$) has a negligible effect on constraints for phenomenological models, the converse is not true for EFTDE. The choice of perturbation model (specifically the gradient stability condition) places a strong theoretical prior on the background expansion. Models with only one functional degree of freedom (e.g., shift-symmetric or GW-stability motivated) strongly disfavor expansion histories that deviate significantly from $\Lambda$CDM, even when data (like DESI DR2 BAO) hints at such deviations.

Significance
The paper argues that understanding the theoretical priors imposed by specific parametrisations is crucial for correctly interpreting observational constraints on dark energy. The authors demonstrate that "model-independent" phenomenological fits can yield misleading results if they do not account for the correlations and boundaries imposed by underlying physical theories (such as stability and GW propagation).

The work highlights that:

1. Theoretical priors can drastically shrink the allowed parameter space for modified gravity, excluding regions that phenomenological fits might otherwise allow.
2. The choice of time dependence ansatz is not merely a technical detail but qualitatively alters the relationship between stable and unstable regions and the resulting observational constraints.
3. There is a strong coupling between background expansion and perturbation dynamics in EFTDE models; constraints on perturbations can effectively rule out exotic background expansion histories preferred by data if those histories violate stability conditions.

The authors conclude that as Stage-IV Large Scale Structure surveys (like Euclid and DESI) provide tighter constraints, a rigorous understanding of these theoretical priors will be essential to distinguish between genuine new physics and artifacts of parametrisation choices.