Simple EFT for Dark Energy

• Simple EFT for Dark Energy is a model-independent framework that captures low-energy modifications to gravity using time-dependent couplings and a minimal operator basis.
• It unifies various scalar-tensor theories such as quintessence, f(R), and Horndeski through a minimal EFT action defined on a flat FLRW background.
• The framework enables robust constraints on cosmic acceleration by linking background dynamics, perturbative stability, and UV consistency via observational data.

A simple Effective Field Theory (EFT) for dark energy is a model-independent framework that captures the most general low-energy modifications to gravity and dark sector dynamics compatible with the observed expansion history of the universe and the symmetries of cosmological spacetimes. It expresses cosmic acceleration in terms of time-dependent gravitational couplings, vacuum energy, and scalar kinetic terms, using a minimal basis of operators defined in a gauge where the dark energy degree of freedom is absorbed into the spacetime foliation. This concise, robust formalism enables the translation between a wide variety of scalar-tensor models—including quintessence, $f(R)$, Horndeski, and beyond-Horndeski theories—using a small set of free functions and facilitates the confrontation of theoretical scenarios with cosmological and local experimental data.

1. Minimal EFT Action and Operator Structure

The core of a simple EFT of dark energy is its action in unitary gauge, defined on a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) background. The scalar degree of freedom is fixed to uniform time hypersurfaces, so the action depends only on the metric and its geometric invariants. Up to quadratic order in perturbations and leading in derivatives, the general minimal action reads

$$S = \int d^4x \sqrt{-g} \left[ \frac{m_0^2}{2} \Omega(t) R - \Lambda(t) - c(t)\, \delta g^{00} \right] + S_{\mathrm{DE}}^{(2)}[\delta g^{00}, \delta K_{ij}, \ldots] + S_m[g_{\mu\nu}, \Psi],$$

where:

• $\Omega(t)$ is a time-dependent conformal coupling (Planck mass run rate),
• %%%%2%%%% generalizes the cosmological constant,
• $c(t)$ controls the kinetic normalization of the would-be Goldstone mode,
• $S_{\mathrm{DE}}^{(2)}$ collects quadratic and higher-order operators such as $(\delta g^{00})^2$, $\delta g^{00} \delta K^i_i$, $\delta K^2$, etc., encoding perturbative effects,
• $S_m$ is the minimally coupled matter action (Frusciante et al., 2013).

The three background functions $\Omega(t)$, $\Lambda(t)$, $c(t)$ fully determine the homogeneous expansion. The additional quadratic operators parameterize linear scalar, vector, and tensor perturbations and are central for computing structure formation and gravitational wave signatures (Gleyzes et al., 2013, Hu et al., 2013).

2. Background Dynamics, Parametrizations, and Physical Interpretation

Variation of the minimal action on a FLRW background yields two generalized Friedmann equations: $$\begin{aligned} 3m_0^2 \Omega H^2 + 3 m_0^2 H \dot{\Omega} &= \rho_m + \rho_r - \Lambda + 2c, \ m_0^2 (3\Omega H^2 + 2\Omega \dot{H} + \ddot{\Omega} + 2H\dot{\Omega}) &= - (p_m + p_r) - \Lambda. \end{aligned}$$ These equations, together with the matter/radiation continuity relations, can be recast as an effective dark energy continuity equation and solved for any viable expansion history by appropriate choice of the EFT functions.

A minimal ansatz ("zero-th order") often used is: $$\Omega(a) = \Omega_0\, a^{-\alpha_0}, \qquad c(a) - \Lambda(a) = (c-\Lambda)_0\, a^{-\lambda_0},$$ where $\alpha_0 \equiv -\dot{\Omega}/(H \Omega)$ and $\lambda_0 \equiv -\dot{(c-\Lambda)}/[H(c-\Lambda)]$ are constants closing the background system.

Physical roles of the functions:

• $\Omega(t)$ governs the running of the effective gravitational coupling,
• $\Lambda(t)$ generalizes the vacuum energy density,
• $c(t)$ acts as the kinetic coefficient for scalar fluctuations (positivity, $c>0$, is required to avoid ghosts) (Frusciante et al., 2013, Okamatsu et al., 2 Dec 2025).

Viable cosmologies at background level require:

• $c > 0$ (no ghost instability),
• $c_s^2 > 0$ (no gradient/Laplace instability for scalar perturbations),
• $w_{\mathrm{eff}} < -1/3$ at late times (to achieve acceleration),
• $\Omega(t) > 0$ (to preserve positive effective Newton constant),
• Positive matter and radiation densities (Frusciante et al., 2013, Okamatsu et al., 2 Dec 2025).

3. Mapping to Theoretical Models and EFT Function Reconstruction

Any single-scalar theory with a Jordan-frame action—including quintessence, $k$-essence, $f(R)$, Horndeski, and their higher-derivative generalizations—admits a mapping to this minimal EFT language through explicit identification of background and perturbation coefficients. As specific examples:

• Quintessence: $\Omega=1$, $c=\frac{1}{2} \dot{\phi}_0^2$, $\Lambda=V(\phi_0)$, higher perturbation operators vanish.
• $f(R)$ Gravity: $\Omega=f_R$ (scalaron), $c=0$, $\Lambda=(m_0^2/2)[f(R)-R f_R]$.

Model-independent reconstruction of EFT functions from observational data is enabled, for instance, by inverting the background Friedmann and acceleration equations with reconstructed $H(z)$ data (cosmic chronometers, Gaussian Process regression). For minimal coupling, the inversion yields: $$\Lambda(z) = -\frac{3}{2} M_{\mathrm{Pl}}^2 H_0^2 \Omega_{\mathrm{m0}} (1+z)^3 + M_{\mathrm{Pl}}^2 H[3H - (1+z) dH/dz],$$

$$c(z) = -\frac{3}{2} M_{\mathrm{Pl}}^2 H_0^2 \Omega_{\mathrm{m0}} (1+z)^3 + M_{\mathrm{Pl}}^2 H(1+z)\frac{dH}{dz},$$

where $H(z)$ is determined non-parametrically from data. This method constrains EFT backgrounds without model assumptions and can be extended to non-minimal coupling cases (Okamatsu et al., 2 Dec 2025).

4. Linear Perturbations, Stability, and Observational Signatures

In the EFT formalism, scalar, vector, and tensor perturbations are controlled by quadratic and higher-order operators in the action. For scalar perturbations, the EFTCAMB code implements the full set of linearized Einstein and π-field equations, evolving both background and perturbations without relying on the quasi-static approximation (Hu et al., 2013). The dynamical variables include generalized dimensionless α-functions (Bellini–Sawicki parametrization):

• $\alpha_M$: running of $M^2_*(t)$ (Planck mass),
• $\alpha_K$: kinetic energy parameter,
• $\alpha_B$: kinetic braiding (mixing between scalar and metric),
• $\alpha_T$: tensor speed excess,
• $\alpha_H$: beyond-Horndeski contributions (Gleyzes et al., 2013).

Ghost-freedom (no negative kinetic energy), positivity of sound speed ($c_s^2 \geq 0$), and avoidance of tachyonic or gradient instabilities are imposed for perturbative viability at each time step (Hu et al., 2013). The EFT approach thus unifies cosmological phenomenology, allowing efficient computation of $G_{\mathrm{eff}}(k,t)$ and slip $\eta(k,t)$, and anchoring model-independent tests of gravity and dark energy (Gleyzes et al., 2013).

5. Constraints from Solar-System Tests: PPN and Screening

Simple EFTs of dark energy must be consistent with local astrophysical and solar-system constraints. The parameterized post-Newtonian (PPN) formalism provides a dictionary mapping between EFT parameters (especially those entering higher-order operators in degenerate higher-order scalar-tensor, U-DHOST, or beyond-Horndeski theories) and the PPN parameters $\gamma$, $\beta$, $\alpha_1$, $\alpha_2$. Precision experiments bound deviations of these PPN parameters to $|\gamma-1|,\ |\beta-1|, |\alpha_1| < 10^{-5}$; $|\alpha_2| < 10^{-9}$.

A class of single-function degenerate higher-order EFTs constructed with a specific structure in the kinetic sector yields all PPN parameters to be exactly the same as in GR, while allowing for rich cosmological phenomenology. The Lagrangian takes the form

$$L = \frac12 M_{\mathrm{Pl}}^2 R + P(X) - M_{\mathrm{Pl}}^2\,\frac{\alpha(X)}{3X} \left[ \Box\phi + \frac{\phi^\mu \phi_{\mu\nu}\phi^\nu}{2X} \right]^2,$$

where only one free function $\alpha(X)$ is needed, and all solar-system constraints are automatically satisfied. On large scales, the beyond-Horndeski invariants can induce observable cosmological modifications without violating fifth force or monopole/dipole radiation bounds (Saito et al., 2024).

6. Positivity Bounds and UV Consistency

The EFT framework admits constraints from fundamental quantum field theory principles—most notably, positivity bounds derived from analyticity and unitarity of $2\to2$ scattering amplitudes in the ultraviolet (UV). For shift-symmetric Horndeski-type EFTs, the coefficients of higher-order operators in the action are related to the low-energy expansion of scattering amplitudes, which must satisfy inequalities such as $c_{20}>0$, $2M^2 c_{21} + 3c_{20}>0$. Expressed in phenomenological parameters ($c_T^2$, $\alpha_B$), these become: $$\alpha_B c_T^2 < 2 (c_T^2 -1), \qquad \alpha_B c_T^2 < 2 (c_T^2-1)(2c_T^2+1).$$ These "positivity bounds" carve out allowed wedges in parameter space, and analyses show that imposing such UV priors strongly restricts otherwise phenomenologically viable models. In particular, viable models typically require $c_T > 1$ and positive $\alpha_B$, leading to nontrivial implications for the properties of gravitational waves and structure growth (Melville, 31 Dec 2025).

Parameter	No UV prior (68%)	With UV prior (68%)
$\alpha_B/\Omega_{\rm DE}$	$0.71^{+0.90}_{-0.71}$	$0.26^{+0.46}_{-0.46}$
$(c_T^2-1)/\Omega_{\rm DE}$	$-1.00^{+1.25}_{-1.00}$	$0.46^{+0.64}_{-0.41}$
$\alpha_M/\Omega_{\rm DE}$	$-0.02^{+1.32}_{-0.89}$	$0.67^{+0.97}_{-0.58}$

The introduction of positivity priors sharpens the posterior distributions, reducing the allowed parameter space and enhancing the power to falsify unphysical scenarios (Melville, 31 Dec 2025).

7. Computational Implementations and Practical Usage

The EFT formalism has been implemented in public numerical frameworks such as EFTCAMB (Hu et al., 2013), which:

• Accepts arbitrary choices of background EFT functions or direct model mappings,
• Evolves linear perturbations without the quasi-static approximation,
• Dynamically checks for ghost, gradient, and tachyonic instabilities at every time step,
• Interfaces with cosmological datasets and supports modified gravity/dark energy parameter estimation via MCMC.

This approach offers a unified engine for exploring and constraining both model-dependent and model-independent theories of cosmic acceleration, yielding robust physical insights and enabling the systematic testing of gravity across cosmological and local scales.

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Simple EFTs for dark energy thus provide a compact and general structure for parameterizing and constraining cosmic acceleration, connecting theoretical model space to data and fundamental consistency bounds in a unified formalism (Frusciante et al., 2013, Okamatsu et al., 2 Dec 2025, Gleyzes et al., 2013, Saito et al., 2024, Melville, 31 Dec 2025, Hu et al., 2013).