Observational signatures of the theories beyond Horndeski

# Observational signatures of the theories beyond Horndeski

Antonio De Felice Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan    Kazuya Koyama Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK    Shinji Tsujikawa Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku, Tokyo 162-8601, Japan
September 22, 2019
###### Abstract

In the approach of the effective field theory of modified gravity, we derive the equations of motion for linear perturbations in the presence of a barotropic perfect fluid on the flat isotropic cosmological background. In a simple version of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, which is the minimum extension of Horndeski theories, we show that a slight deviation of the tensor propagation speed squared from 1 generally leads to the large modification to the propagation speed squared of a scalar degree of freedom . This problem persists whenever the kinetic energy of the field is much smaller than the background energy density , which is the case for most of dark energy models in the asymptotic past. Since the scaling solution characterized by the constant ratio is one way out for avoiding such a problem, we study the evolution of perturbations for a scaling dark energy model in the framework of GLPV theories in the Jordan frame. Provided the oscillating mode of scalar perturbations is fine-tuned so that it is initially suppressed, the anisotropic parameter between the two gravitational potentials and significantly deviates from 1 for away from 1. For other general initial conditions, the deviation of from 1 gives rise to the large oscillation of with the frequency related to . In both cases, the model can leave distinct imprints for the observations of CMB and weak lensing.

## I Introduction

The constantly accumulating observational evidence for the late-time acceleration of the Universe SNIa ; WMAP ; BAO ; Planck implies that there may be at least one additional degree of freedom to the system of the Einstein-Hilbert action with non-relativistic matter and radiation. One simple example is a canonical scalar field with a sufficiently flat potential – dubbed quintessence quin . The -Cold-Dark-Matter (CDM) model corresponds to the non-propagating limit of quintessence (i.e., the vanishing kinetic energy) with the constant potential .

The scalar degree of freedom also arises in modified gravitational theories as a result of the breaking of gauge symmetries present in General Relativity (GR) moreview . In gravity, for example, the presence of non-linear terms in the 4-dimensional Ricci scalar gives rise to the propagation of an extra gravitational scalar degree of freedom dubbed scalarons Staro ; fRearly ; fRreview . Provided that the functional form of is well designed fRviable , it is possible to realize the late-time cosmic acceleration, while suppressing the propagation of the fifth force in regions of the high density through the chameleon mechanism chameleon .

Another well-known example of single-scalar modified gravitational theories is the covariant Galileon Galileons , in which the field derivatives have couplings with the Ricci scalar and the Einstein tensor (see Ref. Nicolis for the original Minkowski Galileon). In this case the field kinetic terms drive the cosmic acceleration Sami ; DT10 , while recovering the General Relativistic behavior in local regions through the Vainshtein mechanism Vain .

Many dark energy models proposed in the literature (including quintessence, gravity, and covariant Galileons) can be accommodated in Horndeski theories Horndeski –most general second-order scalar-tensor theories with a single scalar field (see also Refs. Deffayet ). Using the linear perturbation equations of motion on the flat Friedmann-Lemaître-Robertson-Walker (FLRW) background DKT , the dark energy models in the framework of Horndeski theories can be confronted with the observations of large-scale structure, CMB, and weak lensing Hoobser . In such general theories the screening mechanisms of the fifth force in local regions were also studied in Refs. screening .

There is another approach to the unified description of modified gravity– the effective field theory (EFT) of cosmological perturbations Cheung -Gao1 . By means of the Arnowitt-Deser-Misner (ADM) formalism with the 3+1 decomposition of space-time ADM , one can construct several geometric scalar quantities from the extrinsic curvature and the 3-dimensional intrinsic curvature , e.g., , , . The EFT of modified gravity is based on the expansion of a general Lagrangian in unitary gauge that depends on these geometric scalars, the lapse , and the time . In fact, Horndeski theories can be encompassed in such a general framework with several conditions imposed for the elimination of spatial derivatives higher than second order Piazza .

Recently, Gleyzes et al. GLPV proposed a generalized version of Horndeski theories by extending the Horndeski Lagrangian in the ADM form such that two additional constraints are not imposed. For example the Horndeski Lagrangian involves the term , where the functions and , which depend on and , have a particular relation . In GR the ADM decomposition of the Einstein-Hilbert term , where is the reduced Planck mass, leads to the Lagrangian with . The theories with belong to a class of GLPV theories.

On the flat FLRW background, the Hamiltonian analysis based on linear cosmological perturbations shows that GLPV theories have only one scalar propagating degree of freedom GLPV ; Lin ; GlHa ; Gao2 . One distinguished feature of GLPV theories is that the scalar and matter sound speeds are coupled to each other Gergely ; GLPV . For example, in the covariantized version of the Minkowski Galileon where partial derivatives in the Lagrangian are replaced by covariant derivatives, the scalar propagation speed squared becomes negative in the matter-dominated epoch due to a non-trivial kinetic-type coupling Kasecs . Unlike the covariant Galileon with positive during the matter era, the covariantized Galileon mentioned above (a class of GLPV theories) is practically excluded as a viable dark energy scenario.

In this paper, we develop the analysis of cosmological perturbations further to confront dark energy models in GLPV theories with observations. First, the linear perturbation equations of motion are derived in the presence of a barotropic perfect fluid for a general Lagrangian encompassing GLPV theories. We provide a convenient analytic formula for and show that even a slight deviation from Horndeski theories generally gives rise to a non-negligible modification to the scalar sound speed.

We also apply our general formalism to a simple dark energy model with a canonical scalar field in which the function differs from . In this case the tensor propagation speed squared is different from 1. We show that this deviation leads to a significant modification to whenever the field kinetic energy is suppressed relative to the background energy density . The scaling solution characterized by the constant is a possible way out to avoid having large values of in the early cosmological epoch.

For the scaling dark energy model described by the potential ( and ), we study the evolution of cosmological perturbations and resulting observational consequences. For the initial conditions where the contribution of the oscillating mode to the velocity potential is suppressed, the evolution of perturbations is analytically known during the scaling matter era. In particular, the anisotropic parameter between the two gravitational potentials and exhibits a large deviation from 1 for away from 1. If the oscillating mode gives a non-negligible contribution to initially, the rapid oscillations with frequencies related to arise for the perturbations like and . Thus the model in the framework of GLPV theories can be clearly distinguished from that in Horndeski theories.

This paper is organized as follows. In Sec. II the extension of Horndeski theories to GLPV theories is briefly reviewed. In Sec. III the perturbation equations of motion are derived in the presence of a barotropic perfect fluid according to the EFT approach encompassing GLPV theories. In Sec. IV we obtain convenient formulae for the matter and scalar propagation speeds in the small-scale limit. In Sec. V we present observables associated with the measurements of large-scale structure, CMB, weak lensing, and discuss the quasi-static approximation on sub-horizon scales. In Sec. VI we propose a simple dark energy model in the framework of GLPV theories and study its observational signatures by carefully paying attention to the oscillating mode induced by . Sec. VII is devoted to conclusions.

## Ii The theories beyond Horndeski

The EFT of cosmological perturbations is a powerful framework to deal with low-energy degrees of freedom in a systematic and unified way for a wide variety of modified gravity theories. It is based upon the 3+1 ADM decomposition of space-time described by the line element ADM

 ds2=gμνdxμdxν=−N2dt2+hij(dxi+Nidt)(dxj+Njdt), (1)

where is the lapse function, is the shift vector, and is the three-dimensional spatial metric. The extrinsic curvature is defined by , where is a unit vector orthogonal to the constant hyper-surfaces and a semicolon represents a covariant derivative. We also introduce the three-dimensional Ricci tensor on . Then we can construct a number of geometric scalar quantities:

 K≡Kμμ,S≡KμνKμν,R≡Rμμ,Z≡RμνRμν,U≡RμνKμν. (2)

Horndeski theories Horndeski are the most general scalar-tensor theories with second-order equations of motion in generic space-time. The action of Horndeski theories is given by with the Lagrangian Deffayet

 L = G2(ϕ,X)+G3(ϕ,X)□ϕ+G4(ϕ,X)R−2G4,X(ϕ,X)[(□ϕ)2−ϕ;μνϕ;μν] (3) +G5(ϕ,X)Gμνϕ;μν+13G5,X(ϕ,X)[(□ϕ)3−3(□ϕ)ϕ;μνϕ;μν+2ϕ;μνϕ;μσϕ;ν;σ],

where , and the four functions () depend on a scalar field and its kinetic energy , with . Choosing the unitary gauge on the flat FLRW background described by the line element , the Lagrangian (3) can be expressed in terms of the geometric scalars introduced above, as Piazza

 L=A2(N,t)+A3(N,t)K+A4(N,t)(K2−S)+B4(N,t)R+A5(N,t)K3+B5(N,t)(U−KR/2), (4)

where . Up to quadratic order in the perturbations we have , where is the Hubble parameter (a dot represents a derivative with respect to ).

Horndeski theories satisfy the following two conditions Piazza

 A4=2XB4,X−B4,A5=−13XB5,X. (5)

More concretely, the coefficients in Eq. (3) and in Eq. (4) are related with each other, as

 A2=G2−XF3,ϕ,A3=2(−X)3/2F3,X−2√−XG4,ϕ, A4=−G4+2XG4,X+XG5,ϕ/2,B4=G4+X(G5,ϕ−F5,ϕ)/2, A5=−(−X)3/2G5,X/3,B5=−√−XF5, (6)

where and are auxiliary functions obeying the relations and . Since in unitary gauge, the functional dependence of and can translate to that of and .

It is possible to go beyond the Horndeski domain without imposing the two conditions (5) GLPV . This generally gives rise to derivatives higher than second order, but it does not necessarily mean that an extra propagating degree of freedom is present. In fact, the Hamiltonian analysis on the flat FLRW background shows that the theories described by the Lagrangian (4), dubbed GLPV theories, do not possess an extra scalar mode of the propagation GLPV ; Lin ; GlHa ; Gao2 .

## Iii Perturbation equations of motion

The Lagrangian (4) depends on , , but not on . The dependence on appears in the theories with spatial derivatives higher than second order Gao1 ; KaseIJMPD , e.g., in Hořava-Lifshitz gravity Horava . In the following we shall focus on the theories described by the action

 S=∫d4x√−gL(N,K,S,R,U;t)+∫d4x√−gLm(gμν,Ψm), (7)

where is the Lagrangian of the matter field . We consider a metric frame in which the scalar field is not directly coupled to matter (dubbed the Jordan frame). For the matter component we consider a scalar field characterized by

 Lm=P(Y),Y=gμν∂μχ∂νχ, (8)

whose description is the same as that of the barotropic perfect fluid Hu ; Arroja ; Mukoh . The perfect fluids of radiation and non-relativistic matter can be modeled by and with , respectively, where are constants Scherrer ; Kasecs .

The linearly perturbed line element on the flat FLRW background with four metric perturbations and tensor perturbations is given by Bardeen

 ds2=−(1+2A)dt2+2∂iψdtdxi+a2(t)[(1+2ζ)δij+2∂i∂jE+γij]dxidxj. (9)

Then the shift vector is related to the perturbation , as

 Ni=∂iψ. (10)

In the following we choose the gauge conditions

 δϕ=0,E=0, (11)

under which temporal and spatial components of the gauge-transformation vector are fixed. The background values (denoted by an overbar) of the ADM geometric quantities are

 ¯Kμν=H¯hμν,¯K=3H,¯S=3H2,¯Rμν=0,¯R=¯U=0. (12)

Around the FLRW background we consider the scalar perturbations , , , and , where and are the first-order and second-order perturbations respectively. The scalar , which is a perturbed quantity itself, obeys the relation up to a boundary term, where is an arbitrary function with respect to .

Decomposing the scalar field as and omitting the overbar in the following discussion, the kinetic term , expanded up to second order, can be expressed in the form , where

 δ1Y = 2˙χ2δN−2˙χ˙δχ, (13) δ2Y = −˙δχ2−3˙χ2δN2+4˙χ˙δχδN+2˙χa2δij∂iψ∂jδχ+1a2(∂δχ)2, (14)

and . The energy-momentum tensor of the field is given by . Defining the linear perturbations of energy density, momentum, and pressure, respectively, as , , and , it follows that

 δρ=(P,Y+2YP,YY)δ1Y,δq=2P,Y˙χδχ,δP=P,Yδ1Y. (15)

These quantities appear in the perturbation equations of motion presented later.

### iii.1 Background equations of motion

Expanding the action (7) up to linear order in scalar perturbations, we obtain the first-order action with Piazza ; Gergely

 L1=a3(¯L+L,N−3HF−ρ)δN+3(¯L−˙F−3HF+P)a2δa−2a3P,Y˙χ˙δχ+a3Eδ1R, (16)

where

 F ≡ L,K+2HL,S, (17) E ≡ L,R+12˙L,U+32HL,U, (18) ρ ≡ 2YP,Y−P. (19)

The last term of Eq. (16) is a total derivative irrelevant to the background dynamics. Varying Eq. (16) with respect to , , and , respectively, we obtain the background equations of motion

 ¯L+L,N−3HF=ρ, (20) ¯L−˙F−3HF=−P, (21) ddt(a3P,Y˙χ)=0. (22)

These correspond to the Hamiltonian constraint, the momentum constraint, and the equation of motion for , respectively. Equation (22) is equivalent to the continuity equation by using the definition (19) of the field energy density.

### iii.2 Perturbation equations of motion

We expand the action (7) up to second order in scalar perturbations to derive the linear perturbation equations of motion. In doing so, we use the following properties

 δKij = (˙ζ−HδN)δij−12a2δik(∂kNj+∂jNk), (23) δRij = −(δij∂2ζ+∂i∂jζ), (24)

where . Expansion of the action (7) gives rise to the terms in the forms , , and , which generate the spatial derivatives higher than second order. The absence of these higher-order terms requires that Piazza ; Gergely

 L,KK+4HL,SK+4H2L,SS+2L,S=0, (25) L,KR+2HL,SR+12L,U+HL,KU+2H2L,SU=0, (26) L,RR+2HL,RU+H2L,UU=0. (27)

The GLPV Lagrangian (4) obeys all these conditions, so we assume the conditions (25)-(27) in the following discussion.

The second-order Lagrangian density of reads

 L(2)m=−ρδNδ√h−2P,Y˙χ˙δχδ√h+a3(P,Yδ2Y+12P,YYδ1Y2+P,YδNδ1Y), (28)

where . The Lagrangian density contains the second-order contribution . On using the background Eq. (20), this cancels the first term of Eq. (28). Then, expansion of the total action (7) leads to the second-order action with

 L(2)T = a3[{L,N+12L,NN−3H(W−2L,SH)}δN2+{W(3˙ζ−∂2ψa2)−4(D+E)∂2ζa2}δN+4L,S˙ζ∂2ψa2 (29) −6L,S˙ζ2+2E(∂ζ)2a2−(P,Y+2YP,YY)(˙χδN−˙δχ)2−6P,Y˙χ˙δχζ−2P,Y˙χδχ∂2ψa2+P,Y(∂δχ)2a2],

where

 W ≡ L,KN+2HL,SN+4L,SH, (30) D ≡ L,NR−12˙L,U+HL,NU. (31)

Varying Eq. (29) with respect to , , , and respectively, and employing Eqs. (15) and (22), we obtain the following linear perturbation equations of motion

 (2L,N+L,NN−6HW+12L,SH2)δN+(3˙ζ−∂2ψa2)W−4(D+E)∂2ζa2=δρ, (32) WδN−4L,S˙ζ=−δq, (33) 1a3ddt(a3Y)+4(D+E)∂2δNa2+4Ea2∂2ζ+6P,Y˙χ2δN=3δP, (34) 1a3ddt[a3(P,Y+2YP,YY)(˙δχ−˙χδN)]+3P,Y˙χ˙ζ−˙χP,Y∂2ψa2−P,Y∂2δχa2=0, (35)

where

 Y≡4L,S∂2ψa2−3δq. (36)

On using Eq. (22), it is easy to show that the momentum perturbation obeys

 ˙δq+3Hδq=−(ρ+P)δN−δP. (37)

Similarly, Eq. (35) can be expressed in the following form

 ˙δρ+3H(δρ+δP)=−(ρ+P)(3˙ζ−∂2ψa2)−∂2δqa2. (38)

We note that Eqs. (37) and (38) also follow from the continuity equations and , respectively. Substituting Eq. (36) into Eq. (34) and using Eq. (37), it follows that

 (˙L,S+HL,S)ψ+L,S˙ψ+(D+E)δN+Eζ=0, (39)

where the integration constant is set to 0. The dynamics of scalar perturbations is known by solving Eqs. (32), (33), (39) together with Eqs. (37) and (38).

For the tensor perturbation the second-order action reads , where Piazza ; Tsuji15 ; DT15

 L(2)h=a3L,S4δikδjl(˙γij˙γkl−c2ta2∂γij∂γkl). (40)

Here, the propagation speed squared is

 c2t=EL,S. (41)

Then the equation of motion for gravitational waves is given by

 ¨γij+(3H+˙L,SL,S)˙γij−c2t∂2γija2=0. (42)

Provided that and , the ghost and Laplacian instabilities are absent for the tensor mode.

## Iv Propagation speeds of scalar perturbations

We derive the propagation speeds of the gravitational scalar and the matter field in the small-scale limit. In doing so, we first express and in terms of , and their derivatives by using Eqs. (32) and (33). Substituting these relations into Eq. (29), the Lagrangian density can be expressed in the form Gergely ; Kasecs

 L2=a3(˙→XtK˙→X−∂j→XtG∂j→X−→XtB˙→X−→XtM→X), (43)

where , , , are matrices and . The components of the two matrices and are given, respectively, by

 K11=Qs+16L2,SM2plW2˙χ2K22,K22=(2˙χ2P,YY−P,Y)M2pl,K12=K21=−4L,S˙χMplWK22, G11=2(˙M+HM−E),G22=−P,YM2pl,G12=G21=−M˙χL,SMplG22, (44)

where

 Qs ≡ 6L,S+8L2,SW2(2L,N+L,NN−6HW+12H2L,S), (45) M ≡ 4L,S(D+E)W. (46)

The scalar ghosts are absent as long as the determinants of principal sub-matrices of are positive, which translates to the two conditions and . These conditions are satisfied for and .

The dispersion relation following from Eq. (43) in the limit of the large wave number is given by . The scalar propagation speed , which is related to the frequency as obeys

 (c2sK11−G11)(c2sK22−G22)−(c2sK12−G12)2=0. (47)

In Horndeski theories there is the specific relation and hence . In this case the two solutions to are given by

 c2m = G22K22=P,YP,Y−2˙χ2P,YY=δPδρ, (48) c2H = 1Qs[G11−(K11−Qs)G22K22]=2Qs⎛⎝˙M+HM−E+8L2,S˙χ2P,YW2⎞⎠. (49)

In GLPV theories the relation no longer holds, so we define the parameter GlHa

 αH≡D+EL,S−1=c2t−1+DL,S, (50)

which characterizes the deviation from Horndeski theories. We also introduce the following quantity

 βH≡2c2m(K11Qs−1)αH=16L2,S(ρ+P)W2QsαH. (51)

If , then the parameter is simply related to the deviation of the tensor propagation speed squared from 1, as . This is the case for the theories with and constant . In Sec. VI we shall discuss the dynamics of cosmological perturbations for a simple model satisfying the condition .

Eliminating the terms , , , and in Eq. (47) with the help of Eqs. (48), (49), (51) and the relation , the two solutions to Eq. (47) can be expressed as

 ~c2m = 12[c2m+c2H−βH+√(c2m−c2H+βH)2+2c2mαHβH], (52) c2s = 12[c2m+c2H−βH−√(c2m−c2H+βH)2+2c2mαHβH]. (53)

For non-relativistic matter with , Eqs. (52) and (53) reduce to and respectively.

For the general perfect fluid with , we consider the case in which the deviation from Horndeski theories is small, i.e., . Then the propagation speeds (52) and (53) are approximately given, respectively, by

 ~c2m ≃ c2m−c2m2(c2H−c2m−βH)αHβH, (54) c2s ≃ c2H−βH+c2m2(c2H−c2m−βH)αHβH. (55)

The condition, , does not necessarily mean that is also much smaller than 1. In the covariantized Galileon model GLPV where the partial derivatives of the original Minkowski Galileon Nicolis are replaced by the covariant derivatives, we have and during the matter era for late-time tracking solutions (in which regime is much smaller than 1) Kasecs . The reason why is not as small as comes from the fact that the variable in Eq. (51) is much smaller than 1 in the early cosmological epoch. Since in this case, the covariantized Galileon is plagued by the Laplacian instability on small scales. On the other hand, for the covariant Galileon Deffayet , we have during the matter era DT10 .

Generally, the sound speed squared is subject to the modification arising from the deviation from Horndeski theories, such that . Meanwhile, provided that , the correction to the matter sound speed squared , i.e., the second term on the r.h.s. of Eq. (54), is suppressed to be small. This shows that the effect beyond Horndeski theories arises for the scalar sound speed rather than the matter sound speed . In Sec. VI we shall apply the results in this section to a concrete model that belongs to a class of GLPV theories.

## V Confrontations with observations

In this section we discuss several physical quantities associated with the measurements of large-scale structures, CMB, and weak lensing in order to confront GLPV theories with observations. We then proceed to the discussion of the quasi-static approximation for the perturbations deep inside the sound horizon.

### v.1 Observables

We first introduce the gauge-invariant combinations of the matter density contrast and the velocity perturbation , as

 δm≡δ−3Hv,vm≡v+(1+w)δϕ˙ϕ, (56)

where

 δ≡δρρ,v≡δqρ,w≡Pρ. (57)

Since we choose the unitary gauge (), the perturbation is equivalent to itself. In Fourier space we can rewrite Eqs. (37) and (38), respectively, as

 ˙vm+3H(c2m−w)vm=−(1+w)δN−c2mδm, (58) ˙δm+3(Hvm)⋅+3H(c2m−w)(δm+3Hvm)=−(1+w)(3˙ζ+k2a2ψ)+k2a2vm, (59)

where is defined by Eq. (48).

Since we are interested in the growth of structures after the onset of the matter era, we shall focus on the case of non-relativistic matter characterized by and . Taking the time derivative of Eq. (59) and using Eq. (58), we obtain

 ¨δm+2H˙δm+k2a2Ψ=−3¨B−6H˙B, (60)

where , and is the gauge-invariant gravitational potential defined by Bardeen

 Ψ≡δN+˙ψ. (61)

If is not exactly 0 and the term on the r.h.s. of Eq. (58) is not neglected, this gives rise to the term on the l.h.s. of Eq. (60). This works as a pressure that prevents the gravitational growth induced by the source term . The matter propagation speed squared in GLPV theories is not equivalent to , but, in the limit , they are identical to each other. On the other hand, the scalar sound speed squared is generally subject to a non-negligible change even by the slight deviation from Horndeski theories.

In order to know the evolution of the matter density contrast , we need to relate the gravitational potential in (60) with . Usually, this relation is expressed in the following form

 k2a2Ψ=−4πGeffρδm, (62)

where is the effective gravitational coupling. In GR, is equivalent to the Newton gravitational constant . In modified gravitational theories, generally differs from . In Horndeski theories, for example, the quasi-static approximation provides the analytic expression of for the perturbations deep inside the sound horizon DKT . Provided that the terms on the r.h.s. of Eq. (60) are negligible compared to those on the l.h.s., the evolution of is known by integrating Eq. (60). The growth rate of is related to the peculiar velocity of galaxies Kaiser . For the observations of redshift-space distortions, the quantity is usually introduced Tegmark , where

 f≡˙δmHδm, (63)

and is the amplitude of over-density at the comoving Mpc scale ( is the normalized Hubble constant km sec Mpc).

In order to confront modified gravity models with the observations of CMB and weak lensing, we also introduce the following gauge-invariant gravitational potential

 Φ≡ζ+Hψ. (64)

The effective gravitational potential associated with the deviation of light rays is given by Sapone

 Φeff≡12(Ψ−Φ). (65)

Introducing the anisotropic parameter

 η≡−ΦΨ, (66)

Eq. (65) can be expressed as . In GR we have and hence . We caution that the definition (66) is valid only for . If the gravitational potential crosses 0 with oscillations, we should compute from Eq. (65) rather than using . As we will see in Sec. VI, the crossing of can actually occur in GLPV theories if the oscillating mode initially dominates the perturbation .

### v.2 The quasi-static approximation on sub-horizon scales

For the observations of large-scale structure and weak lensing, we are primarily interested in the evolution of perturbations for the modes deep inside the sound horizon (). In the presence of a propagating scalar degree of freedom, there is an oscillating mode of the field perturbation in addition to the mode induced by matter perturbations. Provided that the oscillating mode of perturbations is suppressed relative to the matter-induced mode, the time derivatives of metric perturbations (like and ) can be neglected relative to the terms involving their spatial derivatives quasi . In Horndeski theories, this quasi-static approximation was first employed in Ref. DKT to derive the analytic expression of , , and .

In GLPV theories, let us discuss what kind of difference from Horndeski theories arises. First of all, Eq. (39) can be written in the form