# The effective field theory of inflation/dark energy and the Horndeski theory

###### Abstract

The effective field theory (EFT) of cosmological perturbations is a useful framework to deal with the low-energy degrees of freedom present for inflation and dark energy. We review the EFT for modified gravitational theories by starting from the most general action in unitary gauge that involves the lapse function and the three-dimensional geometric scalar quantities appearing in the Arnowitt-Deser-Misner (ADM) formalism. Expanding the action up to quadratic order in the perturbations and imposing conditions for the elimination of spatial derivatives higher than second order, we obtain the Lagrangian of curvature perturbations and gravitational waves with a single scalar degree of freedom. The resulting second-order Lagrangian is exploited for computing the scalar and tensor power spectra generated during inflation. We also show that the most general scalar-tensor theory with second-order equations of motion–Horndeski theory–belongs to the action of our general EFT framework and that the background equations of motion in Horndeski theory can be conveniently expressed in terms of three EFT parameters. Finally we study the equations of matter density perturbations and the effective gravitational coupling for dark energy models based on Horndeski theory, to confront the models with the observations of large-scale structures and weak lensing.

## 1 Introduction

The inflationary paradigm, which was originally proposed to solve a number of cosmological problems in the standard Big Bang cosmology Sta80 ; oldinf , is now widely accepted as a viable phenomenological framework describing the accelerated expansion in the early Universe. In particular, the Cosmic Microwave Background (CMB) temperature anisotropies measured by COBE COBE , WMAP WMAP1 , and Planck Planck satellites support the slow-roll inflationary scenario driven by a single scalar degree of freedom. Inflation generally predicts the nearly scale-invariant primordial power spectrum of curvature perturbations oldper , whose property is consistent with the observed CMB anisotropies. In spite of its great success, we do not yet know the origin of the scalar field responsible for inflation (dubbed “inflaton”).

The observations of the type Ia Supernovae (SN Ia) Riess ; Perlmutter showed that the Universe entered the phase of another accelerated expansion after the matter-dominated epoch. This has been also supported by other independent observations such as CMB WMAP1 and Baryon Acoustic Oscillations (BAO) BAO . The origin of the late-time cosmic acceleration (dubbed “dark energy”) is not identified yet. The simplest candidate for dark energy is the cosmological constant , but if it originates from the vacuum energy appearing in particle physics, the theoretical value is enormously larger than the observed dark energy scale Weinberg ; CST . There is a possibility that some scalar degree of freedom (like inflaton) is responsible for dark energy quinpapers .

Although many models of inflation and dark energy have been constructed in the framework of General Relativity (GR), the modification of gravity from GR can also give rise to the epoch of cosmic acceleration. For example, the Starobinsky model characterized by the Lagrangian Sta80 , where is a Ricci scalar and is a constant, leads to the quasi de Sitter expansion of the Universe. The recent observational constraints on the dark energy equation of state (where and is the pressure and the energy density of dark energy respectively) imply that the region is favored from the joint data analysis of SN Ia, CMB, and BAO WMAP9 ; CDT ; Planck . If we modify gravity from GR, it is possible to realize without having a problematic ghost state (see Refs. moreview for reviews).

Given that the origins of inflation and dark energy have not been identified yet, it is convenient to construct a general framework dealing with gravitational degrees of freedom beyond GR. In fact, the EFT of inflation and dark energy provides a systematic parametrization that accommodates possible low-energy degrees of freedom by employing cosmological perturbations as small expansion parameters about the Friedmann-Lemaître-Robertson-Walker (FLRW) background Cremi ; Cheung ; Quin . This EFT approach allows one to facilitate the confrontation of models with the cosmological data.

Originally, the EFT of inflation was developed to quantify high-energy corrections to the standard slow-roll inflationary scenario Weinberg2 . Expanding the action up to third order in the cosmological perturbations, it is also possible to estimate higher-order correlation functions associated with primordial non-Gaussianities ng1 . The EFT formalism was applied to dark energy in connection to the large-distance modification of gravity Park -Gergely . The advantage of this approach is that practically all the single-field models of inflation and dark energy can be accommodated in a unified way Bloom .

Starting from the most general action that depends on the lapse function and other geometric three-dimensional scalar quantities present in the ADM formalism, Gleyzes et al. Piazza expanded the action up to quadratic order in cosmological perturbations of the ADM variables. In doing so, the perturbation of a scalar field can be generally present, but the choice of unitary gauge () allows one to absorb the field perturbation in the gravitational sector. Once we fix the gauge in this way, introducing another scalar-field perturbation implies that the system possesses at least two-scalar degrees of freedom. In fact, such a multi-field scenario was studied in Ref. Gergely to describe both dark energy and dark matter.

By construction, the EFT formalism developed in Refs. Cremi ; Cheung ; Bloom2 ; Piazza keeps the time derivatives under control, while the spatial derivatives higher than second order are generally present. Imposing conditions to eliminate these higher-order spatial derivatives for the general theory mentioned above, Gleyzes et al. Piazza derived the quadratic Lagrangian of cosmological perturbations with one scalar degree of freedom. If the scalar degree of freedom is responsible for inflation, for example, the resulting power spectrum of curvature perturbations can be computed on the quasi de Sitter background (along the same lines in Refs. Muka ; KYY ; Tsujinon ; XGao ; Tsujinon2 ). In this review, we evaluate the inflationary power spectra of both scalar and tensor perturbations expressed in terms of the ADM variables.

In 1973, Horndeski derived the action of the most general scalar-tensor theories with second-order equations of motion Horndeski . This theory recently received much attention as an extension of (covariant) Galileons Nicolis ; cova ; Char . One can show that the four-dimensional action of “generalized Galileons” derived by Deffayet et al. Deffayet is equivalent to the Horndeski action after a suitable field redefinition KYY . Gleyzes et al. Piazza expressed the Horndeski Lagrangian in terms of the ADM variables appearing in the EFT formalism (see also Ref. Bloom2 ). This allows one to have a connection between the Horndeski theory and the EFF of inflation/dark energy. In fact, it was shown that Horndeski theory belongs to a sub-class of the general EFT action Piazza .

For the background cosmology, the EFT of inflation/dark energy is characterized by three time-dependent parameters , , and Cremi ; Cheung ; Quin . This property is useful to perform general analysis for the dynamics of dark energy Silve . In the EFT of dark energy, Gleyzes et al. Piazza obtained the equations of linear cosmological perturbations in the presence of non-relativistic matter (dark matter, baryons). This result reproduces the perturbation equations in Horndeski theory previously derived in Ref. Koba . We note that the perturbation equations in the presence of another scalar field (playing the role of dark matter) were also derived in Ref. Gergely . These results are useful to confront modified gravitational models of dark energy with the observations of large-scale structures, weak lensing, and CMB.

In this lecture note, we review the EFT of inflation/dark energy following the recent works of Refs. Piazza ; Gergely .

In Sec. 2 we start from a general gravitational action in unitary gauge and derive the background equations of motion on the flat FLRW background.

In Sec. 3 we obtain the linear perturbation equations of motion and discuss conditions for avoiding ghosts and Laplacian instabilities of scalar and tensor perturbations.

In Sec. 4 the inflationary power spectra of scalar and tensor perturbations are derived for general single-field theories with second-order linear perturbation equations of motion.

In Sec. 5 we introduce the action of Horndeski theory and express it in terms of the ADM variables appearing in the EFT formalism.

In Sec. 6 we discuss how the second-order EFT action accommodates Horndeski theory as specific cases and provide the correspondence between them.

In Sec. 7 we apply the EFT formalism to dark energy and obtain the background equations of motion in a generic way. In Horndeski theory, the equations of matter density perturbations and the effective gravitational coupling are derived in the presence of non-relativistic matter.

Sec. 8 is devoted to conclusions.

Throughout the paper we use units such that , where is the speed of light and is reduced Planck constant. The gravitational constant is related to the reduced Planck mass GeV via . The Greek and Latin indices represent components in space-time and in a three-dimensional space-adapted basis, respectively. For the covariant derivative of some physical quantity , we use the notation or . We adopt the metric signature .

## 2 The general gravitational action in unitary gauge and the background equations of motion

The EFT of cosmological perturbations allows one to deal with the low-energy degree of freedom appearing for inflation and dark energy. In particular, we are interested in the minimal extension of GR to modified gravitational theories with a single scalar degree of freedom . The EFT approach is based on the choice of unitary gauge in which the constant time hypersurface coincides with the constant hypersurface. In other words, this corresponds to the gauge choice

(1) |

where is the field perturbation. In this gauge the dynamics of is “eaten” by the metric, so the Lagrangian does not have explicit dependence about the flat FLRW background.

The EFT of cosmological perturbations is based on the decomposition of the ADM formalism ADM . In particular, the splitting in unitary gauge allows one to keep the number of time derivatives under control, while higher spatial derivatives can be generally present. As we will see later, this property is especially useful for constructing theories with second-order time and spatial derivatives. The ADM line element is given by

(2) |

where is the lapse function, is the shift vector, and is the three-dimensional metric. Then, the four-dimensional metric can be expressed as , , and . A unit vector orthogonal to the constant hypersurface is given by , and hence with . The induced metric on can be expressed as , so that it satisfies the orthogonal relation .

The extrinsic curvature is defined by

(3) |

where is the acceleration (curvature) of the normal congruence . Since there is the relation , the extrinsic curvature is the quantity on . The internal geometry of can be quantified by the three-dimensional Ricci tensor associated with the metric . The three-dimensional Ricci scalar is related to the four-dimensional Ricci scalar , as

(4) |

where is the trace of the extrinsic curvature.

In the following we study general gravitational theories that depend on scalar quantities appearing in the ADM formalism Bloom ; Bloom2 ; Piazza . In addition to the lapse , we have the following scalars

(5) |

The Lagrangian of general gravitational theories depends on these scalars, so that the action is given by

(6) |

We do not include the dependence of the scalar quantity
coming from the shift vector,
since such a term does not appear even in the most general
scalar-tensor theories with second-order equations of motion
(see Sec. 5).
In the action (6), the time dependence is also explicitly
included because in unitary gauge its dependence is directly
related to the scalar degree of freedom, such that .
The field kinetic term^{1}^{1}1We caution that the notation of
the field kinetic energy is the same as that used in
Refs. Piazza ; Gergely , but the notation of
used in Refs. KYY ; Tsujinon ; XGao ; Tsujinon2 ; Koba is times different.

(7) |

depends on the lapse and the time . The field enters the equations of motion through the partial derivatives and .

Let us consider four scalar metric perturbations , , , and about the flat FLRW background with the scale factor . The general perturbed metric is given by

(8) |

where represents a covariant derivative with respect to , and . Under the transformation and , the perturbations and transform as

(9) |

where a dot represents a derivative with respect to . Choosing the unitary gauge (1), the time slicing is fixed. The spatial threading can be fixed with the gauge choice

(10) |

On the flat FLRW background with the line element , the three-dimensional geometric quantities are given by

(11) |

where a bar represents background values and is the Hubble parameter. We define the following perturbed quantities

(12) |

where the last equation arises from the first equation and the definition of . Since and vanish on the background, they appear only as perturbations. Up to quadratic order in perturbations, they can be expressed as

(13) |

where and are first-order and second-order perturbations in , respectively. The perturbation is higher than first order. The first equality (12) implies

(14) |

where the last term is a second-order quantity.

In order to derive the background and perturbation equations of motion, we expand the action (6) up to quadratic order in perturbations, as

(15) | |||||

where a lower index of the Lagrangian denotes the partial derivatives with respect to the scalar quantities represented in the index. From the second and third relations of Eq. (12), the expansion of the term up to second order reads

(16) | |||||

where

(17) |

In the second line of Eq. (16), the term has been integrated by using the relation , as

(18) |

where the boundary term is dropped. Note that we have also expanded the term up to second order in Eq. (16).

The term satisfies the relation

(19) |

where is an arbitrary function of . Using this relation and the fact that is a perturbed quantity, it follows that

(20) | |||||

where the first term on the r.h.s. is the first-order quantity, whereas the rest is second-order.

Summing up the terms discussed above, the zeroth-order and first-order Lagrangians of (15) are given, respectively, by

(21) | |||||

(22) |

where

(23) |

Defining the Lagrangian density as , where is the determinant of the three-dimensional metric , the zeroth-order and first-order terms read

(24) | |||||

(25) |

The last term is a total derivative, so it can be dropped. Varying the first-order Lagrangian (25) with respect to and , we can derive the following equations of motion respectively:

(26) | |||

(27) |

On using Eq. (27), the zero-th order Lagrangian density (24) vanishes. Subtracting Eq. (26) from Eq. (27), we obtain

(28) |

Two of Eqs. (26)-(28) determine the cosmological dynamics on the flat FLRW background.

As an example, let us consider the non-canonical scalar-field model given by kinf ; kes

(29) |

where is an arbitrary function with respect to and . Using Eq. (4) and dropping the total divergence term, it follows that

(30) |

where . Since , , and on the flat FLRW background, Eqs. (26) and (28) read

(31) | |||||

(32) |

which match with those derived in Ref. kinf . Taking the time derivative of Eq. (31) and using Eq. (32), we obtain the field equation of motion

(33) |

which is equivalent to , this reduces to the well-known equation . . For a canonical field characterized by the Lagrangian

## 3 Second-order action for cosmological perturbations

In order to derive the equations of motion for linear cosmological perturbations, we need to expand the action (6) up to quadratic order. The Lagrangian (15) reads

(34) | |||||

where

(35) | |||||

(36) | |||||

(37) | |||||

(38) | |||||

(39) |

Then, we obtain the second-order Lagrangian density, as

(40) | |||||

For the gauge choice (10), the three-dimensional metric following from the metric (8) is . Then, several perturbed quantities appearing in Eq. (40) can be expressed as

(41) |

where and .

From Eq. (3) the extrinsic curvature can be expressed in the form

(42) |

For the perturbed metric (8), the first-order extrinsic curvature reads

(43) |

where we have used the fact that the Christoffel symbols are the first-order perturbations for non-zero values of . Since the shift is related to the metric perturbation via , the trace of can be expressed as

(44) |

On using the relations (41), (43), and (44), the second-order Lagrangian density (40), up to boundary terms, reduces to

(45) | |||||

where we have used the background equation (28) to eliminate the term . Variations of the second-order action with respect to and lead to the following Hamiltonian and momentum constraints, respectively:

(46) | |||

(47) |

where

(48) |

From Eqs. (46) and (47) one can express and in terms of and . The last three terms in Eq. (45) give rise to the equations of motion containing spatial derivatives higher than second order. If we impose the three conditions

(49) | |||

(50) | |||

(51) |

then such higher-order spatial derivatives are absent. Under the conditions (49)-(51), we obtain the following relations from Eqs. (46) and (47):

(52) | |||||

(53) |

where . Substituting these relations into Eq. (45), we find that the second-order Lagrangian density can be written in the form , where are time-dependent coefficients. After integration by parts, the term reduces to up to a boundary term. Then, the second-order Lagrangian density reads Piazza ; Gergely

(54) |

where

(55) | |||||

(56) |

and

(57) |

Varying the action with respect to the curvature perturbation , we obtain the equation of motion for :

(58) |

This is the second-order equation of motion with a single scalar degree of freedom. Provided that the conditions (49)-(51) are satisfied, the gravitational theory described by the action (6) does not involve derivatives higher than quadratic order at the level of linear cosmological perturbations. As we will see in Sec. 5, Horndeski theory satisfies the conditions (49)-(51).

While we have focused on scalar perturbations so far, we can also perform a similar expansion for tensor perturbations. The three-dimensional metric including tensor modes can expressed as

(59) |

where is traceless and divergence-free such that . We have introduced the second-order term for the simplification of calculations Maldacena . On using the property that tensor modes decouple from scalar modes, we substitute Eq. (59) into the action (6) and then set scalar perturbations 0. We note that tensor perturbations satisfy the relations , , , and . The second-order action for tensor perturbations reads

(60) | |||||

One can express in terms of two polarization modes, as . In Fourier space, the transverse and traceless tensors and satisfy the normalization condition for each polarization ( is a comoving wavenumber), whereas . The second-order Lagrangian (60) can be written as the sum of two polarizations, as

(61) |

where

(62) | |||||

(63) |

Each mode () obeys the second-order equation of motion

(64) |

In order to avoid the appearance of ghosts, the coefficient in front of the term needs to be positive and hence . The small-scale instability associated with the Laplacian term is absent for . Then, the conditions for avoidance of the ghost and the Laplacian instability associated with tensor perturbations are given, respectively, by Piazza ; Gergely

(65) | |||

(66) |

Similarly, the ghost and the Laplacian instability of scalar perturbations can be avoided for and , respectively, i.e.,

(67) | |||

(68) |

where we have used the condition (65). The four conditions (65)-(68) need to be satisfied for theoretical consistency.

## 4 Inflationary power spectra

The scalar degree of freedom discussed in the previous section can give rise to inflation in the early Universe. Moreover, the curvature perturbation generated during inflation can be responsible for the origin of observed CMB temperature anisotropies oldper . The tensor perturbation not only contributes to the CMB power spectrum but also leaves an imprint for the B-mode polarization of photons.

In this section we derive the inflationary power spectra of scalar and tensor perturbations for the general action (6). We focus on the theory satisfying the conditions (49)-(51). In this case, the equations of linear cosmological perturbations do not involve time and spatial derivatives higher than second order. Since the Hubble parameter is nearly constant during inflation, the terms that do not contain the scale factor slowly vary in time. Let us then assume that variations of the terms , , , and are small, such that the quantities

(69) |

are much smaller than unity.

We first study the evolution of the curvature perturbation during inflation. In doing so, we express in Fourier space, as

(70) |

where

(71) |

Here, is the conformal time, is the comoving wavenumber, and are the annihilation and creation operators, respectively, satisfying the commutation relations

(72) |

On the de Sitter background where is constant, we have and hence . Here, we have set the integration constant 0, such that the asymptotic past corresponds to .

Using the equation of motion (58) for , we find that each Fourier mode obeys

(73) |

For large , the second term on the l.h.s. of Eq. (73) is negligible relative to the third one, so that the field oscillates according to the approximate equation . After the onset of inflation, the term starts to decrease quickly. Since the second term on the l.h.s. of Eq. (73) is of the order of , the third term becomes negligible relative to the other terms for . In the large-scale limit (), the solution to Eq. (73) is given by

(74) |

where and are integration constants. As long as the variable changes slowly in time, approaches a constant value . The field starts to be frozen once the perturbations with the wavenumber cross oldper ; kinfper ; BTW .

We recall that the second-order Lagrangian for the curvature perturbation is given by Eq. (54). Introducing a rescaled field with , the kinetic term in the second-order action can be rewritten as , where a prime represents a derivative with respect to . This means that is a canonical field that should be quantized Muka ; Tsujinon . Equation (73) can be written as

(75) |

On the de Sitter background with a slow variation of the quantity , we can approximate . In the asymptotic past (), we choose the Bunch-Davies vacuum characterized by the mode function . Then the solution to Eq. (75) is given by kinfper ; XGao ; Tsujinon ; Tsujinon2

(76) |

The deviation from the exact de Sitter background gives rise to a small modification to the solution (76), but this difference appears as a next-order slow-roll correction to the power spectrum Chen ; Hornshape .

In the regime , the two-point correlation function of is given by the vacuum expectation value at . We define the scalar power spectrum , as

(77) |

Using the solution (76) in the limit, it follows that

(78) |

Since the curvature perturbation soon approaches a constant for , it is a good approximation to evaluate the power spectrum (78) at during inflation. From the Planck data, the scalar amplitude is constrained as at the pivot wavenumber Mpc Planck .

The spectral index of is defined by

(79) |

where and are given by Eq. (69), and

(80) |

The slow-roll parameter is much smaller than 1 on the quasi de Sitter background. Given that the variations of and are small during inflation, we can approximate the variation of at , as . Since we are considering the situation with and , the power spectrum is close to scale-invariant ().

We also define the running of the spectral index, as

(81) |

which is of the oder of from Eq. (79). With the prior , the scalar spectral index is constrained as