Dark energy in Horndeski theories after GW170817: A review

Dark energy in Horndeski theories after GW170817: A review

Ryotaro Kase Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku, Tokyo 162-8601, Japan    Shinji Tsujikawa Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku, Tokyo 162-8601, Japan
July 27, 2019
Abstract

The gravitational-wave event GW170817 from a binary neutron star merger together with the electromagnetic counterpart showed that the speed of gravitational waves is very close to that of light for the redshift . This places tight constraints on dark energy models constructed in the framework of modified gravitational theories. We review models of the late-time cosmic acceleration in scalar-tensor theories with second-order equations of motion (dubbed Horndeski theories) by paying particular attention to the evolution of dark energy equation of state and observables relevant to the cosmic growth history. We provide a gauge-ready formulation of scalar perturbations in full Horndeski theories and estimate observables associated with the evolution of large-scale structures, cosmic microwave background, and weak lensing by employing a so-called quasi-static approximation for the modes deep inside the sound horizon.

In light of the recent observational bound of , we also classify surviving dark energy models into four classes depending on different structure-formation patterns and discuss how they can be observationally distinguished from each other. In particular, the nonminimally coupled theories in which the scalar field has a coupling with the Ricci scalar of the form , including gravity, can be tightly constrained not only from the cosmic expansion and growth histories but also from the variation of screened gravitational couplings. The cross correlation of integrated Sachs-Wolfe signal with galaxy distributions can be a key observable for placing bounds on the relative ratio of cubic Galileon density to total dark energy density. The dawn of gravitational-wave astronomy will open up a new window to constrain nonminimally coupled theories further by the modified luminosity distance of tensor perturbations.

Contents

I Introduction

The late-time cosmic acceleration was first discovered in 1998 from the observations of distant supernovae type Ia (SN Ia) SN1 (); SN2 (). The source for this phenomenon, which was dubbed dark energy Huterer (), occupies about 70 % of today’s energy density of the Universe. The existence of dark energy has been independently confirmed from the observation data of Cosmic Microwave Background (CMB) WMAP (); WMAP9 (); Planck (); Planck2015 (); Planck2018 () and baryon acoustic oscillations (BAO) BAO (). Despite the tremendous observational and theoretical progress over the past two decades dreview1 (); dreview2 (); dreview2a (); dreview3 (); dreview4 (); dreview5 (); Tsuji10 (); dreview6 (); dreview7 (); deRham12 (); dreview8 (); Joyce16 (); dreview9 (); dreview10 (); dreview11 (); dreview12 (), the origin of dark energy has not been identified yet.

A key quantity describing the property of dark energy is the equation of state (EOS) , where and are the dark energy density and pressure respectively. If we use the EOS parametrization of the form CP (); Linder (), where are constants and is a scale factor (with today’s value ), the joint data analysis of Planck 2018 combined with the SN Ia and BAO data placed the bounds and at 68 % confidence level (CL) Planck2018 (). This shows the overall consistency with the EOS , but the time variation of around is also allowed from the data. We also note that the parametrization is not necessarily versatile to accommodate models with a fast varying dark energy EOS Bruce1 (); Cora03 (); Bruce2 (); LH05 () or models in which has an extremum Savvas12 (); Lasenby ().

In the theoretical side, the simplest candidate for dark energy is the cosmological constant characterized by the EOS . If the cosmological constant originates from the vacuum energy in particle physics, however, the theoretically predicted value is enormously larger than the observed dark energy scale Weinberg (); MartinLa (); Tony (). There is a local theory of the vacuum energy sequestering in which quantum vacuum energy is cancelled by an auxiliary four-form field Kaloper2 () (see also Ref. Kaloper1 () for a nonlocal version). In this formulation, what is left after the vacuum energy sequestering is a radiatively stable residual cosmological constant . The value of is not uniquely fixed by the underlying theory, but it should be measured to match with observations. To explain today’s cosmic acceleration, it is fixed as eV.

If the cosmological constant problem is solved in such a way that the residual vacuum energy completely vanishes or it is much smaller than today’s energy density of the Universe, we need to find an alternative mechanism for explaining the origin of dark energy. There are dynamical models of dark energy in which changes in time. The representative example is a minimally coupled scalar field with a potential energy , which is dubbed quintessence quin1 (); quin2 (); Ratra (); quin3 (); quin4 (); quin4b (); quin5 (); Joyce (); quin6 (); quin7 (). The quintessence EOS varies in the region . While the “freezing” quintessence models Caldwell () in which the deviation of from has been large by today are in strong tension with observations, the “thawing” models in which starts to deviate from at late times have been consistent with the data for today’s EOS CDT ().

We also have other minimally coupled scalar-field theories dubbed k-essence in which the Lagrangian is a general function of and its derivative kinf (); kes1 (); kes2 (); kes3 (). One of the examples of k-essence is the dilatonic ghost condensate model gcon1 (); gcon2 () ( and are constants), in which case the typical evolution of is similar to that in thawing quintessence. It is also possible to construct unified models of dark energy and dark matter in terms of a purely kinetic Lagrangian unifiedkes (); Ber1 ().

There are also theories in which the scalar field is nonminimally coupled to the Ricci scalar in the form Fujiibook (). One of the representative examples is Brans-Dicke (BD) theory Brans () with a scalar potential . This theory is given by the Lagrangian , where is a constant called the BD parameter. The gravity whose equations of motion are derived under the variation of metric tensor Bergmann (); Ruz (); Staro () corresponds to the special case of BD theory with Ohanlon (); Chiba03 (). The application of gravity to the late-time cosmic acceleration has been extensively performed in the literature fRearly1 (); fRearly1b (); fRearly2 (); fRearly3 (); fR1 (); fR2 (); fR3 (); fR4 (); fR5 (). In this case, it is possible to realize the dark energy EOS smaller than without having ghosts fR1 (); fR4 (); Moto (). Moreover, the growth rate of matter perturbations is larger than that in General Relativity (GR) fR2 (); fR4 (); Tsuji07 (). Hence it is possible to observationally distinguish gravity models from quintessence and k-essence. The construction of dark energy models in BD theories with the scalar potential is also possible BD1 (); BD2 (), in which case observational signatures are different depending on the BD parameter. In these models, the propagation of fifth forces can be suppressed in local regions of the Universe Bunn (); Navarro (); fR1 (); Capo07 (); Tamaki (); Brax08 () under the chameleon mechanism chame1 (); chame2 ().

There are other modified gravity theories dubbed Galileons Nicolis (); Galileons (); Galileons1 () containing scalar derivative self-interactions and nonminimal couplings to gravity. In the limit of Minkowski space, the equations of motion following from the Lagrangian of covariant Galileons are invariant under the Galilean shift . The cubic Galileon of the form arises in the Dvali-Gabadadze-Porrati braneworld model due to the mixture between longitudinal and transverse gravitons DGP () and also in the Dirac-Born-Infeld decoupling theory with bulk Lovelock invariants DTolley (). This derivative self-interaction can suppress the propagation of fifth forces in local regions of the Universe Cede (); Luty (); Ratta (); Babichev (); Babichev1 (); Burrage (); Tana (); Brax (); DKT12 (); KKY12 (); deRham (); Hira (); Koyama13 (); Andrews (); Babi13 (); Kase13 () under the Vainshtein mechanism Vain (), while modifying the gravitational interaction at cosmological distances Kase10 (). For covariant Galileons including quartic and quintic Lagrangians, there exist self-accelerating de Sitter attractors responsible for the late-time cosmic acceleration Sami (); DT10 (); DT10b (). The self-accelerating solution of full covariant Galileons preceded by a tracker solution with is in tension with observational data of SN Ia, CMB, and BAO NDT10 (); AppleLin (); Neveu (); Barreira1 (); Barreira2 (); Renk (); Peirone2 (). However, this is not necessarily the case for covariant Galileons with a linear potential KTD15 () or the model in which an additional term is present to the Galileon Lagrangian Kase18 ().

The aforementioned theories belong to a subclass of more general scalar-tensor theories–dubbed Horndeski theories Horndeski (). In 2011, Deffayet et al. Def11 () derived the action of most general scalar-tensor theories with second-order equations of motion after the generalizations of covariant Galileons. Kobayashi et al. KYY () showed that the corresponding action is equivalent to that derived by Horndeski in 1974 Horndeski () (see also Ref. Char11 ()). The application of Horndeski theories to the late-time cosmic acceleration was extensively performed in the literature DKT11 (); DT12 (); Char12 (); Zhao (); Cope12 (); Amen12 (); Zuma13 (); Sil13 (); Raveri14 (); Hu13 (); OTT13 (); HKi14 (); Bellini (); Zuma17 (); Ken (); Peirone (); Ken (). Since different modified gravity theories predict different background expansion and cosmic growth histories, it is possible to distinguish between dark energy models in Horndeski theories from the observations of SN Ia, CMB, BAO, large-scale structures, and weak lensing.

The recent detection of gravitational waves (GWs) by GW170817 GW170817 () from a binary neutron star merger together with the gamma-ray burst GRB 170817A Goldstein () constrained the propagation speed of GWs, as Abbott ()

(1)

for the redshift . If we strictly demand that and do not allow any tuning among functions in Horndeski theories, the Lagrangian needs to be of the form Lon15 (); GWcon1 (); GWcon2 (); GWcon3 (); GWcon4 (); GWcon5 (); GWcon6 (). This includes the theories such as quintessence, k-essence, gravity, BD theories, and cubic Galileons, but the theories with quartic-order derivative and quintic-order couplings do not belong to this class. Now, we are entering the era in which dark energy models in Horndeski theories can be tightly constrained from observations.

In this article, we review the application of Horndeski theories to cosmology and discuss observational signatures of surviving dark energy models. After reviewing Horndeski theories in Sec. II, we devote Sec. III for deriving the background equations of motion and the second-order action of tensor perturbations on the flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime in the presence of matter. In Sec. IV, we obtain the second-order action of scalar perturbations and their equations of motion in full Horndeski theories without fixing any gauge conditions. We note that this gauge-ready formulation was carried out in a subclass of Horndeski theories Hwang () and in scalar-vector-tensor theories HKT18 (). The formalism developed in Ref. HKT18 () encompasses both Horndeski gravity and generalized Proca theories Heisenberg14 (); Tasinato (); Allys (); Jimenez () as specific cases.

In Sec. V, we derive conditions for the absence of ghost and Laplacian instabilities of scalar perturbations in the small-scale limit by choosing three different gauges and show that these conditions are independent of the gauge choices. In Sec. VI, we also apply the results in Sec. IV to the computation of observables relevant to the linear growth of matter density perturbations and the evolution of gravitational potentials in the gauge-invariant way. We also classify surviving dark energy models into four classes depending on their observational signatures. In Secs. VII, VIII, IX, and X, we review the behavior of dark energy observables in each class of theories, i.e., (A) quintessence and k-essence, (B) gravity and BD theories, (C) kinetic braidings, and (D) nonminimally coupled scalar with cubic derivative interactions. This is convenient to distinguish between surviving dark energy models in current and future observational data. We conclude in Sec. XI.

Throughout the review, we adopt the metric signature . We also use the natural unit in which the speed of light , the reduced Planck constant , and the Boltzmann constant are equivalent to 1. The reduced Planck mass is related to the Newton gravitational constant , as .

Ii Horndeski theories

The theories containing a scalar field coupled to gravity are generally called scalar-tensor theories Fujiibook (). This reflects the fact that the gravity sector has two tensor polarized degrees of freedom. Horndeski theories Horndeski () are the most general scalar-tensor theories with second-order equations of motion. Due to the second-order property, there is no Ostrogradski instability Ostro (); Woodard () associated with the Hamiltonian unbounded from below.

Horndeski theories are given by the action KYY ()

(2)

where is a determinant of the metric tensor , and

(3)

Here, the symbol stands for the covariant derivative operator with , is the Ricci scalar, is the Einstein tensor, and

(4)

The functions depend on and , with and . Originally, Horndeski derived the Lagrangian of scalar-tensor theories with second-order equations of motion in a form different from Eq. (3) Horndeski (), but their equivalence was explicitly shown in Ref. KYY (). Depending on the papers KYY (); DT12 (); Bellini (); Gleyzes14 (), different signs and notations were used for the quantities and , so we summarize them in Appendix A to avoid confusion.

Below, we list the theories within the framework of the action (2).

  • (1) Quintessence and k-essence

    K-essence kinf (); kes1 (); kes2 (); kes3 () is given by the functions

    (5)

    Quintessence quin1 (); quin2 (); Ratra (); quin3 (); quin4 (); quin4b (); quin5 (); Joyce (); quin6 (); quin7 () corresponds to the particular choice:

    (6)

    where is the potential of .

  • (2) BD theory

    In BD theory Brans () with the scalar potential , we have

    (7)

    In the limit that , we recover GR with a quintessence scalar field. We note that there are more general nonminimally coupled theories given by the couplings , , , Gas92 (); Damour1 (); Damour2 (); Amen99 (); Uzan (); Chiba99 (); Bartolo99 (); Perrotta (); Boi00 (); Gille (). Since the basic structure of such theories is similar to that in BD theories, we do not discuss them in this review.

  • (3) gravity

    The action of gravity Bergmann (); Ruz (); Staro () is given by

    (8)

    where is an arbitrary function of . The metric gravity, which corresponds to the variation of (8) with respect to , can be accommodated by the Lagrangian (3) for the choice

    (9)

    where . In this case, the scalar degree of freedom arises from the gravity sector. Comparing Eq. (7) with Eq. (9), it follows that metric gravity is equivalent to BD theory with and the scalar potential Ohanlon (); Chiba03 ().

  • (4) Covariant Galileons

    In original Galileons Nicolis (), the field equations of motion are invariant under the shift in Minkowski spacetime Nicolis (). In curved spacetime, the Lagrangian of covariant Galileons Galileons () is constructed to keep the equations of motion up to second order, while recovering the Galilean shift symmetry in the Minkowski limit. Covariant Galileons are characterized by the functions

    (10)

    where and are constants. In absence of the linear potential , there exists a self-accelerating de Sitter solution satisfying DT10 (); DT10b (); Ginf1 (); Ginf2 ().

  • (5) Derivative couplings

    There is also a derivative coupling theory in which the scalar field couples to the Einstein tensor of the form Amen93 (); Germani (). This corresponds to the choice

    (11)

    where is a scalar potential and is a constant. Integrating the term by parts, it is equivalent to up to a boundary term.

  • (6) Gauss-Bonnet couplings

    One can consider a coupling of the form Lovelock (), where is a function of and is the Gauss-Bonnet curvature invariant defined by

    (12)

    The theories given by the action NOS (); Mota (); Mota2 (); TS06 ()

    (13)

    can be accommodated in the framework of Horndeski theories for the choice KYY ()

    (14)

    where .

  • (7) gravity

    There is also modified gravitational theories given by the action NO05 (); DT08 ()

    (15)

    where is a function of . Since the Lagrangian is equivalent to , the action (15) belongs to a subclass of Horndeski theories with the couplings

    (16)
  • (8) Kinetic braidings and its extensions

    There are theories given by the Lagrangian

    (17)

    As we will show later, the Lagrangian (17) corresponds to most general Horndeski theories with the tensor propagation speed equivalent to 1. The kinetic braiding scenario braiding1 (); braiding2 () corresponds to the minimally coupled case, i.e., . The cubic Galileon given with belongs to a subclass of kinetic braidings. The dark energy scenario given by Kase18 () is also in the framework of kinetic braidings. In presence of the nonminimal coupling , it is known that a self-accelerating solution characterized by exists for the model with Silva (); Koba1 (); Koba2 (); DTge (). There is also a nonminimally coupled model given by Babi11 (); KTD15 (), which recovers the minimally coupled cubic Galileon in the limit .

Thus, Horndeski theories can accommodate a wide variety of scalar-tensor theories with second-order equations of motion. Except for quintessence and k-essence, the scalar field has derivative self-interactions and nonminimal/derivative couplings to gravity. In such cases, we need to confirm whether there are neither ghost nor Laplacian instabilities. In Secs. III.2 and V, we will address this issue after deriving the second-order actions of tensor and scalar perturbations on the flat FLRW background in full Horndeski theories. The same analysis also allows one to identify the speed of gravitational waves on the isotropic cosmological background. Under the observational bound (1), this result can be used to pin down viable Horndeski theories relevant to the late-time cosmic acceleration. In Sec. VI.4, we classify surviving dark energy models into four classes depending on their observational signatures.

To discuss the dynamics of cosmic acceleration preceded by the matter-dominated epoch, we take into account a matter perfect fluid minimally coupled to gravity. The vector perturbations are nondynamical in Horndeski theories, so we only need to consider the evolution of scalar and tensor perturbations. The perfect fluid without the vector degree of freedom can be described by the Schutz-Sorkin action Sorkin (); DGS ():

(18)

where is a fluid density, is a scalar quantity, and is a four vector associated with the fluid number density , as

(19)

Varying the action (18) with respect to , we obtain

(20)

where corresponds to a normalized four velocity, and .

In this review, we focus on the theories given by the sum of (2) and (18), i.e.,

(21)

After deriving the background and perturbation equations of motion, they can be applied to subclasses of Horndeski theories listed above.

Iii FLRW background and tensor perturbations

iii.1 Background equations of motion

Let us consider the flat FLRW spacetime given by the line element

(22)

where is a lapse and is a scale factor. The lapse is introduced here for deriving the Friedmann equation, but we finally set after the variation of . The scalar field depends on the cosmic time alone on the background (22). For the matter sector, the temporal component of in Eq. (19) is given by

(23)

where is the background value of . Then, the Schutz-Sorkin action (18) reduces to

(24)

Varying this matter action with respect to , it follows that

(25)

where corresponds to the total conserved fluid number.

Now, we compute the action (21) on the spacetime metric (22) and vary it with respect to , and . Setting in the end, we obtain the background equations of motion:

(26)
(27)
(28)

where is the Hubble expansion rate, and is the matter pressure defined by . Substituting into Eq. (20), the temporal component of four velocity is equivalent to and hence . Thus, the matter pressure can be written as

(29)

From Eq. (25), the fluid number density obeys the differential equation . On using the property and the relation (29), the conservation of total fluid number translates to

(30)

We note that this continuity equation also follows from Eqs. (26)-(28).

The quantity in Eq. (27) is defined by

(31)

which is associated with the no-ghost condition of tensor perturbations discussed later in Sec. III.2. The definitions of coefficients appearing in Eqs. (27) and (28) are presented in Appendix B. As we will show in Sec. IV, they also appear as coefficients in the second-order action of scalar perturbations. We note that Eq. (27) has been derived by varying the action with respect to and then eliminating the term by using Eq. (26).

Solving Eqs. (27)-(28) for and , it follows that

(32)
(33)

where

(34)

To avoid the divergences on the right hand sides of Eqs. (32) and (33), we require that

(35)

As we will show later, the determinant is related to the no-ghost condition of scalar perturbations. The no-ghost condition corresponds to , in which case and can remain finite.

In the presence of matter, the condition for the cosmic acceleration is given by

(36)

We can express Eqs. (26) and (27) in the forms

(37)
(38)

where the density and pressure of the “dark” component are given, respectively, by

(39)
(40)

We define the dark energy EOS, as

(41)

which is different from the effective EOS (36) due to the presence of additional matter (dark matter, baryons, radiation). The necessary condition for the late-time cosmic acceleration is given by , but we caution that this is not a sufficient condition. In quintessence and k-essence we have , so the time variation of leads to the deviation of from . In other theories listed in Sec. II, the quantity generally differs from , so the term in Eq. (41) also contributes to the additional deviation of from . Since the evolution of is different depending on dark energy models, it is possible to distinguish between them from the observations of SN Ia, CMB, and BAO.

iii.2 Tensor perturbations

The speed of gravitational waves on the flat FLRW background can be derived by expanding the action (21) up to second order in tensor perturbations . As a by-product, we can identify conditions for the absence of ghost and Laplacian instabilities in the tensor sector. The perturbed line element, which contains traceless and divergence-free tensor perturbations obeying the conditions and , is given by

(42)

Without loss of generality, we can choose nonvanishing components of in the forms

(43)

where and characterize the two polarization states.

Expanding the Horndeski action (2) up to second order in perturbations and integrating it by parts, the quadratic action contains the time derivative , the spatial derivative , and the mass term (where ). The second-order action of tensor perturbations arising from the matter action (18) can be written in the form

(44)

where and with . Then, Eq. (44) reduces to

(45)

where is the matter pressure given by Eq. (29). Using the background Eqs. (26) and (27), the terms proportional to identically vanish from the total second-order action . Finally, we can express in a compact form:

(46)

where was already introduced in Eq. (31), and is the tensor propagation speed squared given by

(47)

To avoid the ghost and Laplacian instabilities, we require that

(48)
(49)

Varying the action (46) with respect to , we obtain the tensor perturbation equation of motion in Fourier space, as

(50)

where is a coming wavenumber. The time variation of and the deviation of from 1 lead to the modified evolution of compared to that in GR.

The observational bound (1) of places tight constraints on surviving dark energy models. To realize the exact value , we require the following condition

(51)

If we do not allow any tuning among functions, the dependence of on and on is forbidden. Then, the Horndeski Lagrangian is restricted to be of the form Lon15 (); GWcon1 (); GWcon2 (); GWcon3 (); GWcon4 (); GWcon5 (); GWcon6 ()

(52)

Among the theories listed in Sec. II, the theories (5), (6), (7) lead to different from 1. In particular, as long as the derivative and Gauss-Bonnet couplings contribute to the late-time cosmological dynamics, the deviation of from 1 is large and hence such couplings are excluded from the bound (1) as a source for dark energy Gong ()111Besides this problem, matter perturbations in gravity are subject to strong instabilities attributed to the negative sound speed squared FMT10 (), so cosmological models are not cosmologically viable..

Quintessence and k-essence, BD theory, and gravity give rise to the exact value , so they automatically satisfy the bound (1). The quartic and quintic Galileon couplings and lead to the deviation of from 1, but the Galileon Lagrangian up to the cubic interaction is allowed (which includes the model with the additional term to the cubic Galileon Lagrangian Kase18 ()). The kinetic braidings braiding1 (); braiding2 () and its extensions Silva (); Koba1 (); Koba2 (); DTge (); Babi11 (); KTD15 () are also consistent with the bound (1).

Even if is constrained to be close to 1, there is another modification to Eq. (50) arising from the time variation of . This leads to the modified luminosity distance of GWs relative to that of electromagnetic signals Maggiore1 (); Amen17 (); Maggiore2 (). Since for the Lagrangian (52), it is possible to distinguish between nonminimally and minimally coupled theories from the luminosity distance measurements of GWs together with the electromagnetic counterpart. After the accumulation of GW events in future, the luminosity distance can be a key observable to probe the existence of nonminimal couplings.

For the Lagrangian (52), we have and , so the dark energy EOS (41) reduces to

(53)

In Secs. VII-X, we will study the evolution of for concrete models of the late-time cosmic acceleration in the framework of Horndeski theories.

Iv Gauge-ready formulation of scalar perturbations

In this section, we derive the second-order action of scalar perturbations in full Horndeski theories without fixing gauge conditions. The resulting linear perturbation equations of motion are written in a gauge-ready form Hwang (); HKT18 (), so that one can choose convenient gauges depending on the problems at hand. For generality, we do not restrict the Horndeski Lagrangian to the form (52).

We begin with the linearly perturbed line-element given by Bardeen (); Kodama (); MFB92 (); BTW05 (); Malik ()

(54)

where are scalar metric perturbations, and the symbol stands for the partial derivative . We do not take into account vector perturbations, as they are nondynamical in scalar-tensor theories. The scalar field is decomposed into the form

(55)

where and are the background and perturbed values, respectively. In the following, we omit the bar from the background quantities.

iv.1 Second-order matter action

We first expand the Schutz-Sorkin action (18) up to second order in perturbations. We decompose the temporal and spatial components of into the background and perturbed parts, as GPcosmo (); GPGeff ()

(56)

where is the conserved background fluid number defined by Eq. (25), and are scalar perturbations. The spatial component of four velocity can be expressed as , where is the velocity potential. On using Eq. (20), the scalar quantity contains the perturbation . Then, can be decomposed as

(57)

where the first contribution to the right hand side corresponds the background quantity. Defining the matter density perturbation

(58)

where (the same latin subscripts are summed over), the perturbation of fluid number density , up to second order, is expressed as

(59)

Since reduces to at linear order, this confirms the consistency of the definition (58).

Now, we are ready for expanding the Schutz-Sorkin action (18) up to quadratic order in scalar perturbations. This manipulation gives the second-order matter action HKT18 ():

(60)

where is the matter sound speed squared given by

(61)

Variation of the action (60) with respect to leads to

(62)

Substituting the relation (62) into Eq. (60), we obtain

(63)

where we used Eq. (29).

iv.2 Full second-order scalar action and linear perturbation equations of motion

We also expand the Horndeski action (2) up to second order in scalar perturbations and take the sum with Eq. (63). Using the background Eq. (26), the term in Eq. (63) is cancelled by a part of contributions proportional to arising from the Horndeski action. After the integration by parts, we can write the full second-order action of in the gauge-ready form

(64)

where

(65)