Dark energy constraints in light of theoretical priors

TL;DR

This study investigates how theoretical priors on dark energy parametrisations influence cosmological constraints, focusing on linear perturbations described by μ and Σ or by EFTDE-inspired α_i functions within Horndeski theories. By mapping EFTDE parameters to phenomenological modifications under the quasi-static, scale-independent regime, the authors quantify how priors shape present-day limits on μ_{ ext{today}} and Σ_{ ext{today}}, and they compare two time-dependence choices, Ω_DE and a, as well as gravitational-wave (GW) priors (e.g., α_T=0) and GW-background stability. Key findings include: (i) EFTDE priors tightly constrain μ_{ ext{today}} and Σ_{ ext{today}} and forbid μ_{ ext{today}}<1, Σ_{ ext{today}}>1 in luminal-GW scenarios, (ii) the two time dependences yield qualitatively different shapes for the stable/unstable regions in μ_{ ext{today}}–Σ_{ ext{today}} space, (iii) shift-symmetric, no-slip theories with μ=Σ>0 impose strong but no-signature constraints on w_0,w_a, while GW-stability priors can reduce the perturbation-parameter space to effectively one dimension, and (iv) in EFTDE, background-perturbation coupling via gradient stability means the choice of background model can strongly affect constraints on the expansion history. These insights guide interpretation of Stage IV data and inform theoretical priors for future gravity tests.

Abstract

In order to derive model-independent observational bounds on dark energy/modified gravity theories, a typical approach is to constrain parametrised models intended to capture the space of dark energy theories. Here we investigate in detail the effect that the nature of these parametrisations can have, finding significant effects on the resulting cosmological dark energy constraints. In order to observationally distinguish well-motivated and physical parametrisations from unphysical ones, it is crucial to understand the theoretical priors that physical parametrisations place on the phenomenology of dark energy. To this end we discuss a range of theoretical priors that can be imposed on general dark energy parametrisations, and their effect on the constraints on the phenomenology of dynamical dark energy. More specifically, we investigate both the phenomenological $\{μ,Σ\}$ parametrisation as well as effective field theory (EFT) inspired approaches to model dark energy interactions. We compare the constraints obtained in both approaches for different phenomenological and theory-informed time-dependences for the underlying functional degrees of freedom, discuss the effects of priors derived from gravitational wave physics, and investigate the interplay between constraints on parameters constraining only the background evolution vs. parameters controlling linear perturbations.

Figures (26)

• Figure 1: The allowed parameter space for $\{\alpha_{B,0},\alpha_{M,0}\}$ for the EFTDE parametrisation $\alpha_B, \alpha_M \propto \Omega_{DE}$ (explicitly, \ref{['eqn:baseline_alphas']}). The coloured (white) regions correspond to $\{\alpha_{B,0},\alpha_{M,0}\}$ values where the gradient stability condition $c_s^2 > 0$ is obeyed (violated) today. The bounds at $\alpha_{M,0}=10$ and $\alpha_{B,0}=-7$ are imposed by hand to limit the size of the region, while the lower bound on $\alpha_{M,0}$ and upper bound on $\alpha_{B,0}$ are physical boundaries imposed by the gradient stability condition. The colours indicate the quadrant of $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ space that a particular $\{\alpha_{B,0},\alpha_{M,0}\}$ maps to from evaluating \ref{['mu_qsa', 'Sigma_qsa']} at $a=1$. Note the different prior volumes of the quadrants: in particular, the $\mu_{\rm{today}} > 1, \Sigma_{\rm{today}} < 1$ quadrant has the smallest prior volume, in accordance with the conjecture of Pogosian:2016ji. The image of this mapping in the $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ space is shown in \ref{['mapping_omegaDE']}.
• Figure 2: The image of the theoretical parameter space in \ref{['musigma_signs_omegaDE']} in the $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ space under the mapping \ref{['mu_qsa', 'Sigma_qsa']}. The points shown here are images of a uniform sampling of the $\{\alpha_{B,0},\alpha_{M,0}\}$ parameter space enclosed in \ref{['musigma_signs_omegaDE']}. The axis ranges are limited for clarity. Blue (grey) points are derived from $\alpha_{B,0}, \alpha_{M,0}$ values where the gradient stability condition $c_s^2 > 0$ is obeyed (violated) today, which corresponds to the coloured (white) regions in \ref{['musigma_signs_omegaDE']}. Note the clear separation between the stable and unstable $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ parameter spaces, and the fact that even the combination of stable and unstable parameter spaces does not cover the entire $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ plane. Both these features are due to the mathematical nature of the nonlinear mapping from the $\{\alpha_{B,0},\alpha_{M,0}\}$ space to the $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ space, \ref{['mu_qsa', 'Sigma_qsa']}.
• Figure 3: Left panel: Comparison of observational constraints on $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ from phenomenological and EFTDE parametrisations. The grey contour shows the allowed parameter space of the EFTDE prior, with its boundaries arising from the mapping \ref{['mu_qsa', 'Sigma_qsa']}. We only show the prior boundaries and not the density of points in the prior, as the relative density of different regions in the prior depends on the prior on the EFTDE parameters as explained in \ref{['sec-mapping']}. Working with a theory-informed EFTDE parameter basis clearly results in significantly tighter bounds, which is the combined consequence of the EFTDE prior shown here, and the different time evolution of $\{\mu, \Sigma\}$ shown in \ref{['musigma_evolution_omegaDE']}. Right panel: Comparison of observational constraints with the theoretically allowed parameter space for the $\alpha_B, \alpha_M \propto \Omega_{DE}$ model. An interesting feature of this posterior is that the $\mu_{\rm{today}} > 1, \Sigma_{\rm{today}} < 1$ region which has the smallest relative volume in the prior space has the largest relative volume in the posterior.
• Figure 4: The time evolution of $\mu$ and $\Sigma$ for EFTDE and phenomenological parametrisations. The solid lines are the time evolution in the EFTDE parametrisations, while the faint dashed lines are the time evolution in the phenomenological parametrisations with the same $\mu_{\rm{today}}, \Sigma_{\rm{today}}$ values. We show the time evolutions for one point in each of the three observationally allowed quadrants in the $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ space for EFTDE parametrisations (see \ref{['sec-mapping']}). The phenomenological parametrisation is not able to reproduce features seen in the EFTDE parametrisation such as an increase in $\mu$ at higher redshifts, or a sign change of $\mu - 1$ or $\Sigma - 1$.
• Figure 5: The allowed parameter space for $\{\alpha_{B,0},\alpha_{M,0}\}$ for the EFTDE parametrisation $\alpha_B, \alpha_M \propto a$. The coloured (white) regions correspond to $\{\alpha_{B,0},\alpha_{M,0}\}$ values where the gradient stability condition $c_s^2 > 0$ is obeyed (violated) today. The bounds at $\alpha_{M,0}=10$ and $\alpha_{B,0}=-7$ are imposed by hand to limit the size of the region, while the lower bound on $\alpha_{M,0}$ and upper bound on $\alpha_{B,0}$ are physical boundaries imposed by the gradient stability condition. The colours indicate the quadrant of $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ space that a particular $\{\alpha_{B,0},\alpha_{M,0}\}$ maps to from evaluating \ref{['mu_qsa', 'Sigma_qsa']} at $a=1$. Note the different prior volumes of the quadrants: in particular, the $\mu_{\rm{today}} > 1, \Sigma_{\rm{today}} < 1$ quadrant has the smallest prior volume, in accordance with the conjecture of Pogosian:2016ji. The image of this mapping in the $\{\mu_{\rm{today}},\Sigma_{\rm{today}}\}$ space is shown in \ref{['mapping_scale']}.