Effective Field Theory of Cosmic Acceleration

The effective field theory (EFT) of cosmic acceleration organizes linear cosmology when a time-dependent background selects preferred constant-time hypersurfaces. It does not assert a particular microscopic origin for acceleration. Instead it lists the operators allowed by spatial diffeomorphisms, separates those fixed by the background from those visible only in perturbations, and attaches every prediction to a cutoff and a matter-coupling prescription.

Helpful background. Einstein Equations supplies the background constraint equations; Gravitational Actions and Boundary Terms supplies variation of curvature and extrinsic-curvature operators; Formulations of General Relativity supplies the lapse, shift, and hypersurface decomposition; Gravity Beyond General Relativity supplies EFT power counting and extra-degree-of-freedom logic; Causality and Global Structure supplies the characteristic-cone checks; and Gravitational Radiation and Waveforms supplies the tensor normalization against which propagation operators are compared.

Unitary gauge and its operator basis

Suppose a scalar “clock” has a timelike background gradient. Choose time slices on which its perturbation vanishes. Time reparameterizations are then hidden, while arbitrary spatial diffeomorphisms remain manifest. The building blocks include $g^{00}$, the extrinsic curvature $K^i{}_j$ of the slices, their intrinsic three-curvature, and covariant spatial derivatives. Define

$$\delta g^{00}=g^{00}+1, \qquad \delta K^i{}_j=K^i{}_j-H\delta^i{}_j, \qquad \delta K=\delta K^i{}_i.$$

A representative action through quadratic order is

$$\begin{aligned} S=\int d^4x\sqrt{-g}\Bigg[& \frac{M^2(t)}{2}R-\Lambda(t)-c(t)g^{00} +\frac{M_2^4(t)}{2}(\delta g^{00})^2 \\ &-\frac{\bar M_1^3(t)}{2}\delta g^{00}\delta K -\frac{\bar M_2^2(t)}{2} \left(\delta K^2-\delta K^i{}_j\delta K^j{}_i\right) +\cdots\Bigg]+S_{\mathrm m}[g,\psi]. \end{aligned}$$

Here the future-directed unit normal has $K_{ij}=h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu$, so $K^i{}_j=H\delta^i{}_j$ on the background. The displayed basis is illustrative, not exhaustive. Operators involving the intrinsic three-curvature, acceleration of the normal, more spatial derivatives, or parity violation may be needed for a chosen theory class. Integration by parts and background identities also move coefficients between bases, so coefficient names have meaning only with the operator definitions attached.

On exact FLRW, every explicitly perturbed operator vanishes. Only $M^2$, $\Lambda$, and $c$ affect the homogeneous equations at this order. By contrast, $M_2^4$, $\bar M_1^3$, and $\bar M_2^2$ alter the kinetic or gradient structure of perturbations without changing the selected background. This is why matching $H(t)$ never fixes linear growth uniquely. The construction and its mapping from covariant scalar–tensor actions are given in Gubitosi, Piazza, and Vernizzi 2013, §§ 2–3, Eqs. (2), (6), and (27)–(30).

Restoring the clock fluctuation

To expose the scalar mode, restore broken time diffeomorphisms with a Stückelberg field $\pi$ by replacing every time-dependent coefficient according to

$$f(t)\longrightarrow f(t+\pi) =f(t)+\dot f(t)\pi+\frac12\ddot f(t)\pi^2+\cdots,$$

and transforming $g^{00}$ and $K^i{}_j$ consistently. In the high-frequency decoupling regime, $\pi$ displays the scalar kinetic term and its mixing with the metric. The sign convention for $\pi$ is conventional; observable stability conditions are unchanged if the entire transformation is changed consistently.

For example, freeze the background and coefficients over a short local interval and neglect metric mixing. Then $g^{\mu\nu}\partial_\mu(t+\pi)\partial_\nu(t+\pi)=-(1+\dot\pi)^2+(\boldsymbol\nabla\pi)^2/a^2$. The two operators $-c g^{00}$ and $M_2^4(\delta g^{00})^2/2$ contribute

$$\mathcal L_{\pi}^{(2)}=(c+2M_2^4)\dot\pi^2 -c\frac{(\boldsymbol\nabla\pi)^2}{a^2}, \qquad c_s^2=\frac{c}{c+2M_2^4}.$$

The local decoupled mode therefore requires $c+2M_2^4\gt0$ and $c\gt0$ for positive two-derivative kinetic and gradient terms. These simple conditions change when metric mixing and the remaining operators matter. They illustrate why an operator invisible in the background can control the sound speed.

Restoration also makes power counting transparent. An operator retained at quadratic order can generate cubic interactions whose strong-coupling scale lies below the nominal gravitational cutoff. Therefore the conditions “no ghost” and “positive sound speed squared” must be supplemented by $E\ll\Lambda_{\mathrm{EFT}}$ for every frequency and physical wavenumber used. A fitted time function outside that range is not an extrapolation licensed by EFT.

A compact description of linear Horndeski cosmology

For theories with one scalar degree of freedom and second-order field equations, Bellini and Sawicki reorganize the time-dependent coefficients into an effective Planck mass $M_*^2$ and four dimensionless functions:

• $\alpha_K$, the scalar kineticity;
• $\alpha_B$, kinetic “braiding” between the scalar and metric;
• $\alpha_M=d\ln M_*^2/d\ln a$, the Planck-mass running;
• $\alpha_T=c_T^2-1$, the tensor-speed excess.

These functions are a coordinate system on a restricted linear theory space, not four observables. Stability depends on combinations of them and on the background. Some beyond-Horndeski or degenerate higher-order theories require additional functions; theories with extra vectors, multiple scalars, masses, nonlocal prescriptions, or Lorentz breaking need a larger basis. The precise Horndeski reduction and quadratic stability coefficients appear in Bellini and Sawicki 2014, §§ 3.1 and 3.3, Eqs. (3.12)–(3.14), and Appendix A, Eqs. (A.6)–(A.10).

A mapping from a covariant action to EFT functions is one-way unless extra assumptions are imposed. Arbitrary smooth EFT functions need not possess a local covariant completion with the desired symmetries, nor remain stable on the entire cosmological history. Conversely, distinct covariant actions can give identical linear EFT functions over a finite redshift range and differ only nonlinearly or above the cutoff.

Checks before using the parameterization

First verify that the background equations are satisfied; otherwise linear terms in the perturbation action have been incorrectly retained. Next eliminate lapse and shift constraints and test the eigenvalues of the reduced kinetic and gradient matrices, not isolated operator coefficients. Then check that $M_*^2$ stays positive, that denominators used in the reduction do not vanish, and that the physical scales remain below the cutoff. Finally state the frame in which matter is minimally coupled, because rods, clocks, redshifts, and conservation equations refer to that metric.

This workflow preserves the central advantage of EFT: many covariant models can be compared in common variables while their limitations remain explicit. It also prevents the common mistake of treating a phenomenological function as though it were already a complete, nonlinear theory.

Exercises

Take constant $M$ and retain only $-\Lambda(t)-c(t)g^{00}$, with minimally coupled matter of density $\rho$ and pressure $p$. Show how $c$ and $\Lambda$ are fixed by a chosen $H(t)$.

Solution

On the FLRW background the clock sector has energy density $\rho_{\mathrm{DE}}=c+\Lambda$ and pressure $p_{\mathrm{DE}}=c-\Lambda$. The two Einstein equations are

$$3M^2H^2=\rho+c+\Lambda, \qquad -2M^2\dot H=\rho+p+2c.$$

Solving gives

$$c=-M^2\dot H-\frac12(\rho+p),$$ $$\Lambda=M^2(3H^2+\dot H)-\frac12(\rho-p).$$

Thus the background fixes these two functions once the matter history is specified. It leaves coefficients such as $M_2^4$ and $\bar M_1^3$ undetermined because their operators vanish on exact FLRW. Those coefficients can therefore change perturbations while preserving the same expansion history.

References

• Bellini, Emilio, and Ignacy Sawicki. “Maximal Freedom at Minimum Cost: Linear Large-Scale Structure in General Modifications of Gravity.” Journal of Cosmology and Astroparticle Physics 2014 (07), 050 (2014). DOI. Open PDF
• Gubitosi, Giulia, Federico Piazza, and Filippo Vernizzi. “The Effective Field Theory of Dark Energy.” Journal of Cosmology and Astroparticle Physics 2013 (02), 032 (2013). DOI. Open PDF