Effective field theory of cosmological perturbations

Anytime we search for stable, universal and model-independent statements in physics, symmetry is a good direction to look to. In particular, the Nambu–Goldstone phenomenon associated with the spontaneous breaking of a continuous global symmetry allows to grasp the low energy spectrum and dynamics of a theory in a completely general way, i.e. without knowing the details at higher energy of the theory itself.

Consider a multiplet of N fields Φ_i(x) whose theory (i.e. the action) is invariant under 'rotations' in some matrix group G,

$$\begin{equation} \Phi _i(x) \ \rightarrow \ \gamma _{ij} \Phi _j(x)\, , \qquad \gamma _{ij} \in G . \end{equation} \tag{ 6 }$$

If the fields acquire vacuum expectation values 〈Φ_i〉₀ ≠ 0—which we assume constant in space and time—they will in general 'spontaneously break' the symmetry G, in the sense that there will be some elements γ_ij of the group G such that γ_ij〈Φ_j〉₀ ≠ 0. Quite intuitively, field configurations that differ from the vacuum by a symmetry transformation will be distinct 'equivalent vacua', all at the same energy. In order to explore such configurations we can act on 〈Φ_i〉₀ with a generic group element that breaks the symmetry. Such a group element can be written as an exponential of a combination of the broken generators τ_a of the symmetry group, $\gamma _{ij} \ \sim \ ({\rm e}^{{\rm i} \pi _a \tau _a})_{ij}$. We now promote the parameters at the exponent to fields, π_a → π_a(x),

$$\begin{equation} \gamma (x) \ = \ {\rm e}^{{\rm i} \pi _a(x) \tau _a} . \end{equation} \tag{ 7 }$$

Note that the π_a parameterize a subset of all possible field configurations and that they must correspond to massless excitations, because in the limit of zero gradients they just interpolate between different but energetically equivalent vacua. As opposed to the original fields Φ_i that transform linearly (6) under the entire symmetry group G, the Goldstones π_a transform linearly only under the unbroken subgroup of G that leaves the vacuum invariant, and are in general nonlinear realizations of G itself [52]. Once the field space Φ_i is parameterized in terms of the Goldstone fields π_a, other ('radial') directions will generally be heavy and thus decouple from the low-energy theory. There is also the possibility of other 'accidentally' flat directions in the field space. The corresponding fields ('moduli'), however, are not protected by any symmetry and will generally acquire a mass by the effect of quantum corrections. We conclude that the entire low-energy dynamics is encoded in the Goldstone fields π_a(x). Up to a limited number of free coefficients, such a dynamics is entirely fixed by the symmetry breaking pattern.

In order to promote a global symmetry to a local (or gauge) symmetry, we need to introduce covariant derivatives, ∂_μ → ∇_μ = ∂_μ + igA_μ. We impose that the gauge fields A_μ, that can be seen as matrices acting on the original multiplet Φ_i, transform in such a way to counterbalance the effect that such spacetime dependent transformations have on the derivative terms:

$$\begin{equation} A_\mu (x) \rightarrow \gamma (x) \left(A_\mu (x) - \frac{{\rm i}}{g} \partial _\mu \right) \gamma (x)^\dagger . \end{equation} \tag{ 8 }$$

It is known that such a prescription produces couplings of the type $\sim g^2 (\Phi _i^* \Phi _i) \, A_\mu A^\mu$. When the symmetry is broken, these are effectively mass terms for the gauge fields.

As in the case of global symmetries, we can still parameterize the Φ_i sector with the Goldstones π_a plus other heavy 'radial' fields. Remarkably, the gauge fields A_μ(x) and the Goldstones π_a(x) are redundant, in the sense that different configurations of A_μ and π_a related by a gauge transformation correspond to the same physical situation. Among the possible gauge fixing conditions to get rid of such a redundancy,

• (I)  
  the unitary gauge is defined by simply setting all Goldstone fields to zero.

In the well-known example of the electro-weak theory, the gauge group is SU(2) × U(1) spontaneously broken to a diagonal U(1). In this case, the complex Higgs doublet is reduced, by the unitary gauge prescription, to a single real 'radial' component:

$$\begin{equation} \Phi _j(x) \ =\ \left({\begin{array}{@{}c@{}}\Phi _1(x) \\ \Phi _2(x) \\ \end{array}} \right) \ \ \rightarrow \ \ \left({\begin{array}{@{}c@{}}v + h (x) \\ 0 \\ \end{array}} \right) . \end{equation} \tag{ 9 }$$

In other words, the unitary gauge prescription picks up, among all equivalent configurations, the particular representative of the Φ_i sector that does not contain any fluctuation along the symmetry direction. In Weinberg's words [4], this choice 'makes manifest the menu of physical particles in the theory': all the fields have a straightforward particle interpretation, they directly represent physical states with well-defined (positive) probabilities, from which the name 'unitarity' or 'unitary'. Also interactions are particularly transparent in the unitary gauge, most physical processes appearing already at tree-level as interaction terms in the Lagrangian.

Quite intuitively, since a precise gauge choice has been made,

• (II)  
  a Lagrangian written in the unitary gauge is no longer invariant under the broken symmetries, while it is still invariant under the unbroken symmetries.

The above can serve as a guidance to parameterize our ignorance about higher energy physics. With the massive vectors $A_\mu ^a$ and the various matter fields (leptons and quarks), we can assemble effective Lagrangians to parameterize the physics beyond the standard model. Such a Lagrangian can be organized in a series of terms of increasing inverse powers of a 'high energy' scale Λ, typically that of new massive degrees of freedom, or of new physics. Low energy observables (amplitudes, decay rates etc) can then be systematically calculated as a power series in E/Λ, E being the energy relevant for the process in question.

For other uses—such as understanding the behavior of the theory at high energy—other gauge choices are more convenient.

• (III)  
  Starting from a Lagrangian written in the unitary gauge, gauge invariance can be restored by the 'Stückelberg mechanism', i.e., by forcing on the fields a gauge transformation that reintroduces the Goldstones.

In a way, this is like 'undoing' the gauge fixing (9). The Goldstones fields π_a will reappear by forcing on the vector bosons A_μ the transformation (8) with γ defined in (7). For example, consider a Lagrangian for massive vector bosons in the unitary gauge,

$$\begin{equation} \mathcal{L} = - \case{1}{4} {\rm Tr}\, F_{\mu \nu } F^{\mu \nu } - \case{1}{2} m^2 {\rm Tr}\, A_\mu A^\mu , \end{equation} \tag{ 10 }$$

By applying (8) and expanding at quadratic order in the Goldstones π, we obtain

$$\begin{equation} \mathcal{L} = - \case{1}{4} {\rm Tr}\, F_{\mu \nu } F^{\mu \nu } - \case{1}{2} (\partial _\mu \pi _c )^2- \case{1}{2} m^2 {\rm Tr}\, A_\mu A^\mu + {\rm i} m \partial _\mu \pi _c A^\mu , \end{equation} \tag{ 11 }$$

where we have defined the canonically normalized Goldstone fields π_c ≡ (m/g)π. By introducing redundant degrees of freedom, one can just make gauge invariant a theory of massive vector fields A_μ.

Having emphasized that the gauge redundancy is not really a symmetry, one might then wonder what it really means, after all, to spontaneously break it. The issue is subtle and beyond the scope of the present review. Naively, one can always keep as a reference the 'global part' of the symmetry and say that

• (IV)  
  the gauge symmetry is spontaneously broken if its 'global part' is.

In the above example, having a vacuum expectation value in the Φ sector different from zero, 〈Φ_i〉₀ ≠ 0, is a gauge invariant statement. Whatever field configuration we choose to represent it, such configuration will break, at least formally, the global SU(2) × U(1). More pragmatically, one can also look at the action in the unitary gauge and say that

• (V)  
  the gauge symmetry is 'spontaneously broken' if, by applying the Stückelberg 'trick' (8) to the action written in the unitary gauge, interacting Goldstone particles π_a are produced.

Finally, note that in the limit m → 0 and g → 0 keeping m/g constant, the Goldstone bosons decouple from the gauge fields A_μ. In other words, at high energies E ≫ m it is convenient to use π to describe the scattering of massive vector fields. In writing (10) as (11) we have neglected cubic and higher order terms in π_c suppressed by m²/g², suggesting that the Goldstone boson self-interactions become strongly coupled at energies E ≫ 4πm/g. The decoupling limit is thus well defined in the regime m ≪ E ≪ 4πm/g.

General Relativity is a gauge theory because of its invariance under coordinate changes, x^μ → x'^μ = x'^μ(x^ν)—the metric field g_μν playing the role of the gauge fields sector. In a cosmological context it is useful, in particular, to look at time reparameterizations, t → t' = t'(x^ν). How can we say whether such a symmetry is spontaneously broken? According to point (IV) above, we should start by looking at its global version. A global symmetry of Minkowski space is time translations. More generally, any time-like killing vector defines a global symmetry. Therefore, in the sense above specified, we can say that time translations are broken by any solution—any spacetime—that does not have a time-like killing vector field. Since we are interested in the local dynamics, it is not too important whether this time-like killing vector is defined globally or not. For instance, the de Sitter space is obviously not a static solution. However, among its many killing vectors, we can choose one that is time-like in a finite patch. In the usual 'cosmological' coordinates

$$\begin{equation} {\rm d}s^2 = -{\rm d}t^2 + {\rm e}^{2 H t} {\rm d} \vec{x}^2 \end{equation} \tag{ 12 }$$

consider the dilation isometry

$$\begin{equation} t \ \rightarrow \ t + \Delta t\, , \qquad \vec{x} \ \rightarrow \ {\rm e}^{- H \Delta t} \, \vec{x} . \end{equation} \tag{ 13 }$$

The corresponding killing vector is time-like in an entire finite patch around the origin of the coordinates, until we hit the horizon, $|\vec{x}| {\rm e}^{H t} < H^{-1}$. Therefore, for all practical purposes, the de Sitter space is a state of gravity with unbroken time translations.

The example of de Sitter is relevant because it represents the limiting case of most inflationary models. Crucially, the expansion during inflation is quasi-de Sitter but not quite so, because of the empirical evidence of a red tilt in the primordial power spectrum [53] and because we need to exit the accelerating phase at some point, and it is quite natural to think this transition to happen smoothly. Since inflation is not completely de Sitter, we deduce that it must be accompanied with a Goldstone excitation. This is because we have argued above in point (V) that any spontaneous breaking of time translations—or of any gauge symmetry for that matter—is associated with a Goldstone excitation π(x) upon application of the Stückelberg trick. This field must transform linearly under the unbroken space translations and rotations, which simply means, in this case, that it must be a three-dimensional scalar. It is not difficult to show (e.g. [54], section 3.3) that π(x) can always be 'completed' into a proper four-dimensional scalar field. Therefore, rather than postulating a scalar field ab initio, the EFT of cosmological perturbations shows that the presence of a scalar is just the inevitable consequence of broken time translations. This is a powerful point of view to address one's possible unease about postulated 'fundamental' scalars in inflationary theories.

To build the EFT of cosmological perturbations we start working in the unitary gauge. The Goldstone field is absent by definition (I) and, in the case of inflation, matter fields are not there either. We deduce that the minimal model of inflation is described in the unitary gauge by an action for the metric field alone. By point (II), the unitary gauge action must be invariant under the unbroken symmetries of the problem. As we have argued, a FRW background that is neither Minkowski nor de Sitter only breaks time translations and boosts, but leaves spatial diffeomorphisms unbroken.

The EFT of cosmological perturbations can thus contain [8]

• (1)  
  four-dimensional diff-invariant scalars (e.g. any curvature invariant such as the Ricci curvature R) in general multiplied by functions of the time t.
• (2)  
  Four-dimensional covariant tensors with free upper 0 indices such as g⁰⁰, R⁰⁰ etc. All spatial indices must be contracted.
• (3)  
  Three-dimensional objects belonging to the t = const. surfa\textbackslashce, such as the extrinsic curvature K^ij and its trace K, the three-dimensional curvatures ⁽³⁾R, ⁽³⁾R_ij etc.

The last two points deserve a bit more of explanation. By breaking the invariance under time-reparameterization we are allowed to write functions of the metric g_μν that contain information about the specific choice of the chosen time coordinate. We can thus contract covariant tensors with the unitary vector orthogonal to the t = const. surfaces,

$$\begin{equation} n_\mu \ = \ - \frac{\delta _\mu ^0}{\sqrt{-g^{00}}} , \end{equation} \tag{ 14 }$$

thereby producing free upper 0 indices. But we can also use geometric quantities describing such a surface. By defining the induced metric h_μν = g_μν + n_μn_ν, we can use the extrinsic curvature

$$\begin{equation} K_{\mu \nu } \equiv h^{\ \sigma }_\mu \, \nabla _\sigma n_\nu , \end{equation} \tag{ 15 }$$

as well as the three-dimensional Ricci tensor ⁽³⁾R_μν[h_ρσ].

2.2.1. Inflation

Before starting to write an action with all possible combinations of the above ingredients, it is worth trying to address, at the same time, another important point. Now that we are left with the metric field as the only dynamical variable, it is straightforward to write such an action also already expanded in number of perturbations, which is one of the main desiderata expressed in the introduction. At zeroth order in perturbations, the metric is just that of a spatially flat FRW solution,

$$\begin{equation} {\rm d}s^2 = -{\rm d}t^2 + a^2(t) \, {\rm d} \vec{x}^{\, 2} , \end{equation} \tag{ 16 }$$

and some of the basic ingredients listed above read (in order of appearance)

$$\begin{eqnarray} &&\fl R_{(0)} = 12 H^2 + 6 \dot{H},\qquad g^{00}_{\ \ (0)} = -1,\qquad K^{i j}_{(0)} = H \delta ^{ij}, \qquad K_{(0)} = 3 H, \qquad {\rm etc.} \end{eqnarray} \tag{ 17 }$$

We can thus start writing a completely general EFT action for inflation, naively of the form

$$\begin{eqnarray} &&\fl S = \int {\rm d}t \, {\rm d}^3x\, \sqrt{h} [\Lambda _0(t) + c_1(t) ( g^{00} + 1) + c_2(t) (K - 3H(t)) + c_3(t) (R - R_{(0)}(t)) + \cdots ], \nonumber\\ \end{eqnarray} \tag{ 18 }$$

where $\sqrt{h}$ is the three-dimensional volume element that takes care of the invariance under 3D diffeomorphisms, Λ₀ is the zeroth-order term in the perturbations, and the other terms start at the first order in the perturbations.

However, in practice, it is very convenient to rearrange the terms above in a slightly different way. First, it is convenient to use directly the 4D volume element $\sqrt{-g} = \sqrt{h}/\sqrt{-g^{00}}$ instead of $\sqrt{h}$. This is useful whenever we need to integrate by parts 4D covariant derivatives or just derive Einstein equations by variations of the action with respect to g_μν. Moreover, ${\rm d}^4 x \sqrt{-g}$ is the invariant volume element and, as we will see, is left unaffected by the Stückelberg trick. Related to this last point, it is also useful to have the combination $\sqrt{-g} R$ sticking out, and merging R₍₀₎(t) together with the zeroth-order piece. Moreover, in the absence of matter fields, it is always possible to absorb the time-dependent coefficient c₃(t) by redefining an Einstein metric through a conformal transformation. Finally, by using (15), the term linear in the extrinsic curvature K can be integrated by parts giving a function of g⁰⁰,

$$\begin{equation} \fl\int {\rm d}^4x \sqrt{-g} \mathcal{F}(t) K=-\int {\rm d}^4x \sqrt{-g} n^\mu \nabla _\mu \mathcal{F}(t) =-\int {\rm d}^4x \sqrt{-g} \sqrt{-g^{00}} \dot{\mathcal{F}}(t) . \end{equation} \tag{ 19 }$$

We are thus lead to the following action

$$\begin{equation} S = \int {\rm d}^4x \sqrt{-g} \left[\frac{M_{\rm Pl}^2}{2} R - \Lambda (t) - c(t) g^{00}\right] + S^{(2)}. \end{equation} \tag{ 20 }$$

In the above, S⁽²⁾ is made by terms that start already at quadratic order in the number of perturbations. The fact that only three operators (and only two tunable functions of the time, c and Λ) determine the background evolution is a non-trivial consequence of the symmetries of FRW. A rigorous derivation of such a result is contained in appendices A and B of [8].

2.2.2. Dark energy

As already stressed, one of the insights of the EFT of inflation is the inevitability of a propagating scalar degree of freedom on a general FRW background that is not de Sitter or Minkowski—for which time translations are unbroken. Indeed, such inevitable scalar fluctuations are nothing else than the adiabatic perturbations, as will be clearer from the discussion of the Stückelberg mechanism in section 3. When extending this formalism to late-time cosmology, one has to decide how to involve matter fields (baryons, dark matter, radiation etc) in the game. In fact, it is convenient to apply this formalism (and thus to write the most general Lagrangian for the metric g_μν in the unitary gauge etc) to the dark energy-gravitational sector only of the theory. For the matter sector we will assume that the WEP is valid, so that matter fields ψ_m couple to the metric g_μν universally and through a covariant action S_m[g_μν, ψ_m]. In other words, we assume the existence of a 'Jordan metric' g_μν and we will work with that. It would be more complicated but technically straightforward to consider different matter sectors coupled to different metrics.

This marks the main difference with the case of inflation: if we want to stick with the Jordan frame metric that minimally couples to matter we now need to allow a general free function of time f(t) in front of the Ricci scalar in equation (20),

$$\begin{equation} S \ = \ S_m[g_{\mu \nu }, \Psi _i] \ + \int {\rm d}^4x \sqrt{-g} \left[\frac{M_{*}^2}{2} f(t) R - \Lambda (t) - c(t) g^{00}\right] + S_{DE}^{(2)}. \end{equation} \tag{ 21 }$$

The part of the action that contributes at quadratic and higher order in (20) and (21) can be read off from the second, third and fourth line of (1). It contains terms such as δg⁰⁰ = g⁰⁰ + 1, quantities of first order in the perturbations. In this respect, it is useful to define the perturbation of the extrinsic curvature as

$$\begin{equation} \delta K_{\mu \nu } = K_{\mu \nu } - H h_{\mu \nu }, \end{equation} \tag{ 22 }$$

where h_μν = g_μν + n_μn_ν is the (perturbed) three-dimensional metric of the t = const. surface.

Apart from the Einstein–Hilbert term in the background part of the action (first line), instead of four-dimensional Riemann, Ricci tensors and their contractions we find it convenient to deal with three-dimensional quantities belonging to the t = const. surface (⁽³⁾R_αβγδ, ⁽³⁾R_μν and K_μν) because they do not explicitly contain higher time derivatives. In order to relate four-dimensional with three-dimensional one can make use of the Gauss–Codazzi equation [55, 56],

$$\begin{equation} {}^{(3)}\!R_{\alpha \beta \gamma \delta } = h^\mu _\alpha h^\nu _\beta h^\rho _\gamma h^\sigma _\delta R_{\mu \nu \rho \sigma } - K_{\alpha \gamma } K_{\beta \delta } + K_{\beta \gamma } K_{\alpha \delta }, \end{equation} \tag{ 23 }$$

and its contracting forms.

In a general (perturbed) FRW universe, $\phi (t, \vec{x}) = \phi _0(t) + \delta \phi (t, \vec{x})$. By choosing the coordinate t to be a function of ϕ, t = t(ϕ), we thus simply have δϕ = 0. Therefore, the action written in this gauge only displays metric degrees of freedom. For instance, a canonical kinetic scalar term (∂ϕ)² is written in the unitary gauge as

$$\begin{equation} -\case{1}{2} (\partial \phi )^2 \ \equiv \ - \case{1}{2} g^{\mu \nu } \partial _\mu \phi \partial _\nu \phi \ \rightarrow \ - c_0(t) g^{00} . \end{equation} \tag{ 38 }$$

Note however that c₀ is only one of the potentially many contributions to the term c(t) in the actions (20) and (21). For example, the covariant operator (∂ϕ)²R that represents a higher derivative coupling between the metric and the scalar field can be expanded in perturbations as

$$\begin{equation} (\partial \phi )^2 R\ \ = \ \dot{\phi }_0^2 [-R + R^{(0)}(t) + R^{(0)}(t) g^{00} + \delta g^{00} \delta R], \end{equation} \tag{ 39 }$$

with R⁽⁰⁾ the background value of the Ricci scalar. The first three terms in brackets contribute to the EFT terms displayed in (21), while the forth is already explicitly second order in the perturbations.

By generalizing (38), it is immediate to see how action (1) includes also k −inflation and k-essence models [59, 60]. There, the Lagrangian has at most one derivative acting on each field ϕ, $\mathcal{L} = P(\phi , X)$, where X = ∂_μϕ∂^μϕ (note that X is sometimes defined with a −1/2 factor). In the unitary gauge this is of the form $P( \phi _0(t) , \dot{\phi }_0^2 g^{00})$, which can be expanded in powers of $\dot{\phi }_0^2 \delta g^{00}$. By redefining the field in such a way that ϕ₀ = t, it is straightforward to see the various contributions to action (1),

$$\begin{eqnarray} &&\fl \Lambda (t) = c (t) -P(t, -1), \;\,\quad c(t) = - \left. \frac{\partial P}{\partial X} \right|_{X=-1} , \;\,\quad M_n^4 (t) = \left. \frac{\partial ^n P}{\partial X^n} \right|_{X=-1} \quad (n\ge 2) . \end{eqnarray} \tag{ 40 }$$

In a way, Brans–Dicke [61] and F(R) theories [62, 63] are even easier to include in this formalism because, at least in their basic versions, they do not need any higher order operator and are completely described by the operators explicitly displayed in (21).

An detailed dictionary for writing covariant operators of increasing complexity in the unitary gauge can be found in section 3 of [33]. In the following subsection, we summarize the results by considering the full Horndenski Lagrangian.

In four dimensions, the most general scalar–tensor theory having field equations of second order in derivatives is a combination of the generalized Galileon Lagrangians [46, 49, 64],

$$\begin{equation} L = L_2 + L_3 + L_4+ L_5 , \end{equation} \tag{ 41 }$$

where

$$\begin{equation} L_2 = G_2{}, \end{equation} \tag{ 42 }$$

$$\begin{equation} L_3 = G_3{}\Box \phi , \end{equation} \tag{ 43 }$$

$$\begin{equation} L_4 = G_4{}R - 2 G_4{}_{X} (\Box \phi ^2 - \phi ^{; \mu \nu } \phi _{; \mu \nu }), \end{equation} \tag{ 44 }$$

$$\begin{equation} L_5 = G_5{}G_{\mu \nu } \phi ^{;\mu \nu } +\case{1}{3} G_5{}_{X} \big(\Box \phi ^3 - 3 \Box \phi \phi _{;\mu \nu }\phi ^{;\mu \nu } + 2 \phi _{;\mu \nu } \phi ^{;\mu \sigma } \phi ^{; \nu }_{\ ; \sigma }\big), \end{equation} \tag{ 45 }$$

and G₂, G₃, G₄ and G₅ are functions of ϕ and X.

It is possible to translate this theory in the EFT language by first rewriting the above Lagrangian in terms of 3D geometrical objects induced on uniform ϕ hypersurfaces. In particular, we can first define the future directed unitary vector orthogonal to these hypersurfaces. Up to a factor γ, it is proportional to the gradient of ϕ,

$$\begin{equation} n_\mu \equiv - \gamma \phi _{; \mu }, \qquad \gamma \equiv {1}/{\sqrt{-X}} . \end{equation} \tag{ 46 }$$

The metric induced on the ϕ = const. hypersurface is h_μν ≡ n_μn_ν + g_μν. Finally, we can define the extrinsic curvature as $K_{\mu \nu } \equiv h_\mu ^\sigma \, n_{\nu ;\sigma }$ and the 3-Ricci tensor computed from the induced metric h_μν as ⁽³⁾R_μν. The key ingredient is then to decompose the covariant derivative of n_ν as $n_{\nu ; \mu } = K_{\mu \nu } - n_\mu {\dot{n}}_\nu$, where the acceleration vector ${\dot{n}}_\mu$ is defined as ${\dot{n}}_\mu = n^\nu \, n_{\mu ; \nu }$. By means of the quantities just defined, we can finally decompose the second derivative of the scalar field as

$$\begin{equation} \phi _{; \mu \nu } =- \gamma ^{-1}(K_{\mu \nu } - n_\mu {\dot{n}}_\nu - n_\nu {\dot{n}}_\mu )+ \frac{\gamma ^2}{2} \phi ^{; \lambda } X_{; \lambda } n_\mu n_\nu . \end{equation} \tag{ 47 }$$

Making use of this decomposition and of the Gauss–Codacci relation (23) and its contractions, after several manipulations it is very lengthy but straightforward to show that the above Lagrangian can be rewritten, up to boundary terms, as [33]

$$\begin{eqnarray} &&\fl L = -\case{1}{3} (-X)^{3/2} G_5{}_X (K^3 - 3 K K_{\mu \nu }K^{\mu \nu } + 2 K_{\mu \nu } K^{\mu \sigma } K^\nu _{\ \sigma }) - \sqrt{-X} F_5{}\left( K^{\mu \nu } {}^{(3)}\!R_{\mu \nu } - \case{1}{2} K {}^{(3)}\!R\right) \nonumber\\ &&+\, ( 2 X \tilde{G_4{}}_X - \tilde{G_4{}})(K^2 - K_{\mu \nu }K^{\mu \nu }) + \tilde{G_4{}}{}^{(3)}\!R\nonumber\\ &&-\, \sqrt{-X} (2G_4{}_\phi + 2X F_3{}_X) K - X F_3{}_\phi +G_2{}. \end{eqnarray} \tag{ 48 }$$

The auxiliary functions F₅ and F₃ are defined by

$$\begin{equation} G_3{}\equiv F_3{}+ 2 X F_3{}_{X}, \qquad G_5{}_{X} \equiv F_5{}_{X} + {F_5{}}/({2X}), \end{equation} \tag{ 49 }$$

and the function $\tilde{G_4{}}\equiv G_4{}+ X(G_5{}_\phi - F_5{}_\phi ) /2$ has been introduced to simplify the notation.

In the unitary gauge $\phi (t,\vec{x})= \phi _0(t)$, the functions G_i and F_i on (ϕ, X) become dependent on $(\phi _0(t), \dot{\phi }_0^2(t)g^{00})$. These functions can be thus expanded in powers of δg⁰⁰ with time-dependent coefficients. It is now straightforward to write the Lagrangian above in the unitary gauge in the EFT language by integrating by parts the term linear in K and expanding K and K_μν in the other terms around their background values. One obtains

$$\begin{eqnarray} &&\fl S =S_0 +\! \int \! {\rm d}^4x \sqrt{-g} \bigg \lbrace \frac{M_2^4 (t)}{2} (\delta g^{00})^2 - \frac{m_3^3(t)}{2} \delta K \delta g^{00} - m_4^2(t)\! \left(\delta K^2 - \delta K^\mu _{ \ \nu } \delta K^\nu _{ \ \mu } - \frac{1}{2} {}^{(3)}\!R\delta g^{00} \right) \nonumber\\ &&+\, \frac{m_5(t)}{3} \bigg[ \delta K^3 - 3 \delta K \delta K_{\mu \nu } \delta K^{\mu \nu } + 2 \delta K_{\mu \nu } \delta K^{\mu \sigma } \delta K^\nu _{\ \sigma }\nonumber\\ &&-\, \delta g^{00} \left( K^{\mu \nu } {}^{(3)}\!R_{\mu \nu } - \frac{1}{2} K {}^{(3)}\!R\right) \bigg] + \cdots \bigg \rbrace , \end{eqnarray} \tag{ 50 }$$

where the dots ... stand for cubic or higher order terms containing the same four operators explicitly written in the action times higher powers of δg⁰⁰; for instance, (δg⁰⁰)³, δK(δg⁰⁰)², etc. The explicit relations between the six time-dependent coefficients f, Λ, c, $M_2^4$, $m_3^3$, $m_4^2$ is given in [33]. Here we just note that the three coefficients Λ, c and $M_2^4$ are affected by all the four Galilean Lagrangians L_i; $m_3^3$ is not affected by L₂ while f and $m_4^2$ are only affected by L₄ and L₅. Finally, m₅ is only affected by L₅. Indeed, in the unitary gauge $\delta (\sqrt{-X} F_5{}_X ) = \sqrt{-X} G_5{}_X \delta X$, which can be derived from equation (49). Using this relation in equation (48) and comparing it with the Lagrangian in equation (50), one finds

$$\begin{equation} m_5 (t) = - \dot{\phi }_0^3(t) G_5{}_X \big(\phi _0(t), \dot{\phi }_0^2(t)g^{00}\big) . \end{equation} \tag{ 51 }$$

Since L₄ and L₅ start differing only by the operator proportional to m₅ which is cubic, at a quadratic order in the action L₄ and L₅ carry the same dynamics. The first line—i.e. the action up to the second order—is equivalent to the first two lines of action (1), which for $m_4^2= \tilde{m}_4^2$ contain the set of quadratic operators that are known not to generate higher derivatives in the linear equations of motion [33, 34]. This implies, remarkably, that the dynamics of linear perturbations can be more general than that of Horndeski while remaining second order.

Let us first consider the universal part of action (1),

$$\begin{equation} S_0 = \int {\rm d}^4x \sqrt{-g} \left[\frac{M_{*}^2}{2} f(t) R - \Lambda (t) - c(t) g^{00} \right], \end{equation} \tag{ 52 }$$

that contains the only operators which are also zeroth and first order in the perturbations.

We will use the ADM formalism to study this action. The ADM metric is

$$\begin{equation} {\rm d}s^2=-N^2 {\rm d}t^2 +{h}_{ij}({\rm d}x^i + N^i {\rm d}t)({\rm d}x^j + N^j {\rm d}t) , \end{equation} \tag{ 53 }$$

where h_ij is the induced spatial metric on constant time hypersurfaces and N and Nⁱ are respectively the lapse and the shift. We decompose R in (52) using the contracting form of the Guass–Codazzi relation (23),

$$\begin{equation} R = {}^{(3)}\!R+ (K_{\mu \nu } K^{\mu \nu } - K^2) + 2 \nabla _\nu (n^\nu \nabla _\mu n^\mu -n^\mu \nabla _\mu n^\nu ), \end{equation} \tag{ 54 }$$

and employ the ADM expression for the extrinsic curvature,

$$\begin{equation} K_{ij} = \frac{1}{N} E_{ij}, \qquad E_{ij} \equiv \frac{1}{2} (\dot{h}_{ij} - \nabla _iN_j-\nabla _jN_i), \end{equation} \tag{ 55 }$$

where the covariant derivative ∇_i are taken with respect to the 3D spatial metric h_ij (note that K^0μ = 0), and for the upper time–time component of the metric, g⁰⁰ = −N⁻². Integrating by parts the last term on the RHS of (54), the action becomes

$$\begin{equation} \fl S_0 = \int {\rm d}^4x \sqrt{h} \bigg \lbrace \frac{M_{*}^2 f }{2} [ N {}^{(3)}\!R + N^{-1}(E_{ij} E^{ij} - E^2) - 2 ({\dot{f}}/{f})N^{-1} E ] - N \Lambda + N^{-1}c \bigg \rbrace . \end{equation} \tag{ 56 }$$

The background equations can be obtained by varying the homogenous action with respect to N and a (using $\sqrt{h} = a^3)$. This yields

$$\begin{equation} 3 M_{*}^2 (H^2 f +H \dot{f} ) = \Lambda + c , \end{equation} \tag{ 57 }$$

$$\begin{equation} M_{*}^2 (2 f \dot{H} - H {\dot{f}} + {\ddot{f}}) = - 2 c . \end{equation} \tag{ 58 }$$

By varying action (56) with respect to Nⁱ and N we find the momentum and Hamiltonian constraint, respectively

$$\begin{equation} \fl 0 =\mathcal{P}_{0i} \equiv \nabla _k\big [-M_{*}^2fN^{-1}\big(E_i^k-E\delta _i^k\big) + M_{*}^2\dot{f}N^{-1}\delta _i^k\big ], \end{equation} \tag{ 59 }$$

$$\begin{equation} \fl 0 = \mathcal{H}_{0} \equiv M_{*}^2 f \big [ {}^{(3)}\!R- N^{-2} (E_{ij} E^{ij} - E^2 ) + 2 ({\dot{f}}/{f}) N^{-2} E \big ] - 2 \Lambda - 2 N^{-2} c . \end{equation} \tag{ 60 }$$

We only need the linear solution of these equations—second order terms in N or Nⁱ will multiply the constraints and will thus vanish [65]. We expand N ≡ 1 + δN and decompose the shift into a scalar and a transverse part, $N^i \equiv \partial _i \psi + N_T^i$, with $\partial _i N^i_T=0$. Since here we are only concerned with scalar perturbations we pose (see [33] for a derivation of the quadratic action of tensor modes)

$$\begin{equation} {h}_{ij}\ =\ a^2(t) {\,\rm e}^{2\zeta }\, \delta _{ij}. \end{equation} \tag{ 61 }$$

The following expressions, which are exact in the unitary gauge, will be also useful,

$$\begin{equation} {}^{(3)}\!R_{ij} = - \partial _i \partial _j \zeta + \partial _i \zeta \partial _j \zeta - \delta _{ij} [ \partial ^2 \zeta + (\partial \zeta )^2 ], \end{equation} \tag{ 62 }$$

$$\begin{equation} E^i_j = (H + \dot{\zeta }- \partial \zeta \partial \psi ) \delta ^i_j - \partial _i \partial _j \psi - \case{1}{2} \big(\nabla _{i} N^{j}_T + \nabla ^{i} N_{j}^T\big). \end{equation} \tag{ 63 }$$

Solving the momentum constraint at the first order gives

$$\begin{equation} \delta N = \frac{\dot{\zeta }}{A_0} , \qquad A_0 \equiv H+\frac{\dot{f}}{2f}, \end{equation} \tag{ 64 }$$

$$\begin{equation} N_T^i=0, \end{equation} \tag{ 65 }$$

Using this equation and the background equation (57), the Hamiltonian constraint yields

$$\begin{equation} \partial ^2 \psi = \frac{1}{A_0} \left [ \left ( \frac{3}{4} \frac{\dot{f}^2}{f^2} + \frac{c}{f^2 M_{*}^2} \right ) \frac{\dot{\zeta }}{A_0} - \frac{\partial ^2 \zeta }{a^2} \right ] . \end{equation} \tag{ 66 }$$

One can expand the action (56) up to the second order and replace δN using equation (64). We do not need to use the solution of the Hamiltonian constraint, equation (66). Indeed, the shift Nⁱ only appears either as a linear term proportional to ∇_iNⁱ or in the combination ∇_iN_j∇^jNⁱ − (∇_iNⁱ)². Because of equation (65), both these terms can be integrated out of the action. Thus, we find

$$\begin{eqnarray} &&\fl S_0 = \ \int {\rm d}^4x a^3 {\rm e}^{3\zeta } \bigg \lbrace \left (1 + \frac{\dot{\zeta }}{A_0} \right ) \big[ - {M_{*}^2 f} ( 2 \partial ^2 \zeta + (\partial \zeta )^2 ) a^{-2} {\rm e}^{-2\zeta } - \Lambda \big] \nonumber\\ &&+ \ \frac{1}{\left(1 + \frac{\dot{\zeta }}{A_0} \right)} \big [ -3M_{*}^2f (H+\dot{\zeta })^2 -3M_{*}^2 \dot{f} (H + \dot{\zeta }) +c \big ] \bigg \rbrace , \end{eqnarray} \tag{ 67 }$$

where we have used $\sqrt{h} = a^3{\rm e}^{3\zeta }$. Collecting all the terms in powers of ${\dot{\zeta }}/{A_0}$, one can use the background equation (57) to simplify terms proportional to $(\dot{\zeta }/A_0)^2$ and to show that those proportional to ${\dot{\zeta }}/{A_0}$ vanish. Terms proportional to $(\dot{\zeta }/A_0)^0$ also vanish, as one can check using the background equations (57) and (58) and an integration by parts. Thus, using again the background equations and the definition of A₀ the final action reads

$$\begin{equation} \fl S_0 = \int {\rm d}^4x a^3\left[ \alpha _0 \dot{\zeta }^2 - \beta _0 \frac{1}{a^{2}}(\partial \zeta )^2 \right], \qquad \alpha _0 \equiv \beta _0 \equiv \frac{1}{A_0^2}\left(c +\frac{3}{4} \frac{\dot{f}}{f}^2\right) . \end{equation} \tag{ 68 }$$

As expected, this corresponds to a propagating d.o.f. with unity sound speed, $c_s^2 \equiv \beta _0/\alpha _0=1$.

We can now add all quadratic operators that are known not to generate higher derivatives [33] in the linear equations of motion,

$$\begin{eqnarray} &&\fl S =S_0 + \int \! {\rm d}^4x \sqrt{-g} \bigg[ \frac{M_2^4}{2} (\delta g^{00})^2 - \frac{m_3^3}{2} \delta K \delta g^{00} - m_4^2\big(\delta K^2 - \delta K^\mu _{ \ \nu } \delta K^\nu _{ \ \mu } \big) + \frac{\tilde{m}_4^2}{2} {}^{(3)}\!R\delta g^{00} \big] . \nonumber\\ \end{eqnarray} \tag{ 69 }$$

Another operator that does not generate higher derivatives in the equations of motion is

$$\begin{equation} {}^{(3)}\!R_{\mu \nu } \delta K^{\mu \nu } - \case{1}{2} {}^{(3)}\!R\delta K . \end{equation} \tag{ 70 }$$

However, we did not explicitly include it in equation (69) because at quadratic order it can be re-expressed as the operator $\tilde{m}_4^2$ using the relation

$$\begin{equation} \lambda (t) \left({}^{(3)}\!R_{\mu \nu } \delta K^{\mu \nu }- \frac{1}{2} {}^{(3)}\!R\; \delta K \right) =\frac{\dot{\lambda }(t)}{4} {}^{(3)}\!R\delta g^{00}, \end{equation} \tag{ 71 }$$

which is valid up to boundary terms (see appendix A of [33]).

Using δK_ij = −δNHh_ij + δE_ij and δg⁰⁰ = 2δN, variation of the full action with respect to Nⁱ and N yields the momentum and Hamiltonian constraints,

$$\begin{equation} 0 = \mathcal{P}_{0i} + \nabla _k \big [ 2 m_4^2 \big(\delta E_i^k - \delta E \delta _i^k\big) + \big( m_3^3 - 4 H m_4^2\big)\delta N \delta _i^k \big ], \end{equation} \tag{ 72 }$$

$$\begin{equation} 0 = \mathcal{H}_0 + 2 \big ( 2 M_2^4 + 9 H m_3^3 - 6 m_4^2 H^2 \big ) \delta N - \big ( m_3^3 + 4 H m_4^2 \big ) \delta E + \tilde{m}_4^2 {}^{(3)}\!R, \end{equation} \tag{ 73 }$$

with $\mathcal{P}_{0i}$ and $\mathcal{H}_{0}$ defined in equations (59) and (60). Their solutions are

$$\begin{equation} \delta N = \frac{\dot{\zeta }}{A}, \qquad A \equiv H + \frac{M_{*}^2 \dot{f} - m_3^3}{2 \big(f M_{*}^2+ 2 m_4^2\big)} , \end{equation} \tag{ 74 }$$

$$\begin{equation} \partial ^2 \psi = \frac{1}{A} \left [ \left ( \frac{3}{2} (A-H)^2 + \frac{c + 4 M_2^4}{f^2 M_{*}^2+2 m_4^2 } \right) \frac{\dot{\zeta }}{A} - \left ( \frac{M_{*}^2 f + 2 \tilde{m}_4^2}{M_{*}^2 f + 2 m_4^2} \right ) \frac{\partial ^2 \zeta }{a^2} \right ], \end{equation} \tag{ 75 }$$

and equation (65).

Action (69) can be expanded up to the second order and one can replace δN using equation (74). As in the previous subsection, we do not need to use the solution of the Hamiltonian constraint as the shift Nⁱ only contributes to boundary terms. As before, by using the background equations one can check that mass terms cancel, as expected; moreover, terms of the type $\dot{\zeta }\partial ^2 \zeta$ can also be reduced to the form (∂ζ)² after integrations by parts. We finally obtain

$$\begin{equation} S = \int {\rm d}^4x a^3\left[ \alpha \dot{\zeta }^2 - \beta \frac{1}{a^2}(\partial \zeta )^2 \right], \end{equation} \tag{ 76 }$$

with

$$\begin{equation} \alpha \equiv \frac{1}{A^2} \left[c + 2 M_2^4 + \frac{3}{4} \frac{\big(M_{*}^2 \dot{f} - m_3^3\big)^2}{ M_{*}^2 f+ 2 m_4^2} \right], \end{equation} \tag{ 77 }$$

$$\begin{equation} \beta \equiv - M_{*}^2 f + \frac{1}{2 a} \frac{{\rm d}}{{\rm d}t} \left[\frac{2 \big(M_{*}^2 f + 2 \tilde{m}_4^2 \big)a}{A} \right]. \end{equation} \tag{ 78 }$$

Stability (absence of ghosts) is ensured by the positivity of α, equation (77), i.e. the coefficient in front of the time kinetic term. The speed of sound squared is given by $c_s^2 = \beta /\alpha$ and its expression simplifies by use of the background equation of motion (58) when $m_4^2 = 0 = \tilde{m}_4^2$, in which case [31]

$$\begin{equation} c_s^2 = \frac{c + \frac{3}{4} M_{*}^2 {\dot{f}^2}/{f} - \frac{1}{2} m_3^3 {\dot{f}}/{f} - \frac{1}{4} { m_3^6}/\big( {M_{*}^2} f\big) + \frac{1}{2}\big( \dot{ m}^3_3 + H m_3^3\big) }{c + 2 M_2^4 + \frac{3}{4} M_{*}^2 {\dot{f}^2}/{f} - \frac{3}{2} m_3^3 {\dot{f}}/{f} + \frac{3}{4} { m_3^6}/ \big( {M_{*}^2} f\big) }. \end{equation} \tag{ 79 }$$

As mentioned earlier, the quadratic operators appearing in action (69) do not yield higher derivatives in the linear dispersion relation. In particular, it is straightforward to verify using equations (55) and (63) that δK² contains a higher spatial derivative term, (∂²ψ)², while $\delta K^\mu _{\ \nu } \delta K_\mu ^{\ \nu }$ contains (∂_i∂_jψ)². However, taken in the combination as in equation (69), these higher derivative terms combine and give an irrelevant boundary term.

Independent operators that generate higher spatial—but not time—derivatives in the linear equations of motion are

$$\begin{equation} S_{\rm h.s.d.} = \int \! {\rm d}^4x \sqrt{-g} \left[ - \bar{m}_4^2(t) \delta K^2 + \frac{\bar{m}_5(t)}{2} {}^{(3)}\!R\delta K + \frac{\bar{\lambda }(t)}{2} {}^{(3)}\!R^2 \right] . \end{equation} \tag{ 80 }$$

We have already mentioned δK². The operator ⁽³⁾R δK contains ∂²ψ∂²ζ and, finally, ⁽³⁾R² = 16(∂²ζ)²/a⁴ so that they are both higher derivative terms. Note that ⁽³⁾R_μν⁽³⁾R^μν = [5(∂²ζ)² + (∂_i∂_jζ)²]/a⁴. Thus, to quadratic order it can be rewritten as ⁽³⁾R² up to a total derivative. Finally, one could take quadratic combinations of the 3D Riemann tensor such as ⁽³⁾R_μνρσ⁽³⁾R^μνρσ. However, in three dimensions, the Riemann tensor can be expressed in terms of the Ricci scalar and tensor¹⁰. Thus, at quadratic order in the perturbations, actions (69) and (80) seem to exhaust all the possible independent operators.

When one of these operators is present in the action the dispersion relation of the propagating mode receives corrections ∝k⁴ at large momenta, so that the dispersion relation becomes $\omega ^2 = c_s^2 k^2 + k^4/M^2$, where M is a mass scale. These corrections may become important in the limit of vanishing sound speed, such as in the model of the ghost condensate [5] or for deformations of this particular limit [6, 30].

No obvious distinction between modifying gravity and simple quintessence can be made at the level of the 'unifying' action (1). Having decided to write everything in the unitary gauge, action (1) is just the most general option: a generic functional of the metric in the presence of broken time translations and compatible with the residual unbroken three-dimensional space diffeomorphisms. Whether or not the operators in the action (1) display departures from General Relativity is ultimately encoded in the behavior of the probes—the matter fields—under the influence of the metric g_μν. A more direct way of studying departures from General Relativity is that of making explicit the scalar degree of freedom of the theory as we did in the last section for inflation, and see what type of coupling it has with the metric field. If this coupling is at the level of the kinetic terms, this is a smoking gun for genuine modifications of gravity.

As we did for inflation in the last section, in order to make the scalar degree of freedom explicit we apply the Stückelberg trick, i.e. we force a diffeomorphism t → t + π(x) upon the unitary gauge action (1), as outlined in section 3. The simplest way to generate a dynamical π field is to consider a non-vanishing coefficient c(t). In this case, the Stückelberg trick generates π with a relativistic kinetic Lagrangian $\dot{\pi }^2 - (\vec{\nabla }\pi )^2$. A more involved example is constituted by the operator $M_2^4$. In order to fix the ideas, once we have moved out of unitary gauge through Stückelberg, let us consider scalar linear perturbations in the Newtonian gauge, which is frequently used for late-time cosmology,

$$\begin{equation} {\rm d}s^2 = -(1+2\Phi ){\,\rm d}t^2 + a^2(t) (1-2 \Psi ) \delta _{i j} {\rm d}x^i {\rm d}x^j . \end{equation} \tag{ 92 }$$

By making use of equation (33) and of the expression for g⁰⁰, one finds $\delta g^{00} \rightarrow 2(\Phi -\dot{\pi }) + 4 \Phi \dot{\pi }- \dot{\pi }^2 + a^{-2}(\vec{\nabla }\pi )^2$. Thus, the Lagrangian of π reads

$$\begin{equation} \fl {-}c \; \delta g^{00} + \frac{M_2^4}{2} (\delta g^{00})^2 \ = \ \big(c + 2 M_2^4\big) \dot{\pi }^2 - c ( \vec{\nabla }\pi )^2 - 4\big(c+ M_2^4\big) \dot{\pi }\Phi + \cdots . \end{equation} \tag{ 93 }$$

We see that $M_2^4$ does not mix π with gravity at the highest energies. The first coupling appears at the level of terms that are quadratic in the fields but with only one derivative in total. Therefore, at high energy the last term can be neglected, π decouples from gravity and propagates with a speed of sound $c_s^2 = c/(c + 2 M_2^4)$. As previously discussed for inflation, this is the so-called decoupling limit [8], which takes place at an energy higher than $E_{\rm mix} \sim (c + M_2^4)/[(c + 2 M_2^4)^{1/2} M_{*}]$.

For other operators, decoupling is not necessarily at work and π and gravity may be mixed already at the kinetic level. This can be verified by inspection, considering the explicit (linear) expressions of the curvatures in Newtonian gauge,

$$\begin{eqnarray} &&K_{ij} = {\rm e}^{-\Phi } (H-\dot{\Psi }) h_{ij}, \nonumber\\ &&{}^{(3)}\!R_{ij} = \partial _i \partial _j \Psi + \delta _{ij} \partial ^2 \Psi . \end{eqnarray} \tag{ 94 }$$

So, for instance, the operator $m_3^3$ after the Stückelberg trick (33)–(34) gives

$$\begin{equation} {-} \frac{m^3_3}{2} \delta K \delta g^{00} \ \rightarrow \ - m^3_3\, (3 \dot{\Psi }\dot{\pi }+ a^{-2} \vec{\nabla }\pi \vec{\nabla }\Phi + a^{-2} \dot{\pi }\nabla ^2 \pi + \dots \, ), \end{equation} \tag{ 95 }$$

where the ellipsis stand for terms of lower order in derivatives. The presence of kinetic mixing between π and the gravitational perturbation changes the structure of the theory already at the level... of the propagator. The specific type of modification of gravity that the operator $m_3^3$ is responsible for has been named 'kinetic gravity braiding' [95], although it was previously studied in [6, 8, 30].

A more standard kinetic mixing is the one provided by a non-constant f(t),

$$\begin{equation} \fl f (t) R \ \rightarrow \ 2 f [ - 3 \dot{\Psi }^2 - 2 \vec{\nabla }\Phi \vec{\nabla }\Psi + (\vec{\nabla }\Psi )^2 + 3 ({\dot{f}}/{f}) \dot{\Psi }\dot{\pi }- ({\dot{f}}/{f}) \pi (\nabla ^2 \Phi - 2 \nabla ^2 \Psi )] . \end{equation} \tag{ 96 }$$

This is nothing else than a modification of gravity of the Brans–Dicke type [61].

Above are just two examples of the universality and generality of the EFT approach. Just in terms of few operators one has all the relevant effects that have been studied at length by specific explicit models. All versions and types of modifications of gravity are distilled in a finite number of terms. One can then wonder what are, more in detail, the cosmological consequences of the various operators. By briefly reviewing the more general results of [33] (see, e.g. also [32, 88, 96]), here we limit ourselves to the operators of the second line of (1): those that do not give higher derivatives in the equations of motion.

An ambitious target of the future galaxy surveys such as EUCLID [1, 2] and BigBoss [3] is that of constraining the linear growth factor that determines the growth rate of the LSS. On these scales, for models with $c_s \sim \mathcal{O} (1)$ we can take the quasi-static approximation, i.e. neglect anisotropic stresses and the time derivatives in the equations of motion. In this case, the evolution of perturbations is described by

$$\begin{equation} \mathcal{M}_{ab} \, V_b = \delta _{a3} \, \bar{\rho }_m \, \Delta _m , \end{equation} \tag{ 97 }$$

where V^a = (Φ, Ψ, π), $\bar{\rho }_m$ and Δ_m are respectively the unperturbed density and the density contrast in comoving gauge of non-relativistic matter and $\mathcal{M}$ is a matrix given in terms of the time-dependent coefficients in front of the quadratic operators (see [33] for details). The effects of modification of gravity on the linear growth factor are encoded in a Poisson equation with a modified Newton constant,

$$\begin{equation} -\frac{k^2}{a^2} \Phi \equiv 4 \pi G_{\rm eff}(t,k) \bar{\rho }_m \Delta _m, \end{equation} \tag{ 98 }$$

where, using equation (97), G_eff(t, k) can be written once and for all in terms of the quadratic operators, $4 \pi G_{\rm eff} = - [ \mathcal{M}^{-1}]_{13}$. If no higher derivative terms are considered, it has the following structure,

$$\begin{equation} G_{\rm eff}(t,k) \ =\ G_{\rm eff}^{(0)}(t) \ + \ G_{\rm eff}^{(-2)}(t) \left(\frac{k}{a}\right)^{-2} + \, \cdots , \end{equation} \tag{ 99 }$$

for $k \gg a \sqrt{G_{\rm eff}^{(0)}/G_{\rm eff}^{(-2)}}$. Already the renormalization of the Newton constant at high momenta—i.e. $G_{\rm eff}^{(0)}(t)$—signals that we are in the presence of a modification of gravity. Another quantity often used to parameterize deviations from General Relativity is the ratio between the gravitational potentials γ ≡ Ψ/Φ which, using again equation (97), is given by $\gamma = {[{\rm com}( \mathcal{M})]_{32}}/{[{\rm com} (\mathcal{M})]_{31}}$, where ${\rm com} ( \mathcal{M})$ denotes the comatrix of $\mathcal{M}$. Schematically, at the lowest order in derivatives we have

$$\begin{equation} \gamma \ =\ \gamma ^{(0)} \ + \ \gamma ^{(-2)} \left(\frac{k}{a}\right)^{-2} + \, \cdots , \end{equation} \tag{ 100 }$$

for $k \gg a \sqrt{\gamma ^{(0)}/\gamma ^{(-2)}}$. The actual coefficients in terms of the various operators are rather complicated and it is not worth reproducing them here. They can be found in [33] where indeed also the contribution of higher derivative terms were considered.