# Energy Conditions in a Generalized Second-Order Scalar-Tensor Gravity

Authors: M. Sharif; Saira Waheed
Journal: Advances in High Energy Physics
 (2013)

---

## Abstract

The study of energy conditions has many significant applications in general relativistic and cosmological contexts. This paper explores the energy conditions in the framework of the most general scalar-tensor theory with field equations involving second-order derivatives. For this purpose, we use flat FRW universe model with perfect fluid matter contents. By taking power law ansatz for scalar field, we discuss the strong, weak, null, and dominant energy conditions in terms of deceleration, jerk, and snap parameters. Some particular cases of this theory likek-essence model, modified gravity theories and so forth. are analyzed with the help of the derived energy conditions, and the possible constraints on the free parameters of the presented models are determined.


---

## Body

## 1. Introduction

“The increasing rate of cosmic expansion in current phase” is one of the primal facts in modern cosmology that is supported by some sorts of energy with negative pressure as well as hidden characteristics refereed to dark energy (DE) (its existence is affirmed by the recent data of many astronomical observations) [1–4]. The investigation of this hidden unusual nature of DE has been carried out in two ways: one approach utilizes the modified matter sources, that is, different models like Chaplygin gas [5], quintessence [6–9], k-essence [10–12], cosmological constant [13, 14], and so forth which are introduced in the usual matter contents within the gravitational framework of general relativity (GR), while in second approach, GR framework is modified by the inclusion of some extra degrees of freedom [15, 16]. Examples of some well-known modified gravity theories include f(R) gravity [17, 18], scalar-tensor theories like Brans-Dicke (BD) gravity [19, 20], Gauss-Bonnet gravity [21, 22], f(T) theory [23, 24], and f(R,T) gravity [25]. Modified matter sources are rather interesting but each faces some difficulties and hence could not prove to be very promising. Modified gravitational theories being large-distance modifications of gravity have brought a fresh insight in modern cosmology. Among these, scalar-tensor theories are considered to be admirable efforts for the investigation of DE characteristics, which are obtained by adding an extra scalar degree of freedom in Einstein-Hilbert action.Scalar field provides a basis for many standard inflationary models, leading to an effective candidate of DE. In literature [26–28], many inflationary models have been constructed like chaotic inflation, small/large inflation, k-inflation, Dirac-Born-Infeld (DBI) inflation, single field k-inflation, and so forth. All of these are some peculiar extensions of the k-essence models. Although the k-essence scalar models are considered to be the general scalar field theories described by the Lagrangian in terms of first-order scalar field derivatives, that is, L=L(ϕ,∇ϕ). However, Lagrangian with higher-order scalar field derivatives (L=L(ϕ,∇ϕ,∇∇ϕ) can be taken into account which fixes the equations of motion (obtained by metric and scalar field variations of the Lagrangian density) to second-order [29, 30]. Horndeski [29] was the pioneer to discuss the concept of most general Lagrangian with single scalar field. Recently, this action is discussed by introducing a covariant Galilean field with second-order equations of motion [30]. Kobayashi et al. [31] developed a correspondence between these Lagrangians. This theory has fascinated many researchers and much work has been done in this context, for example, [31–35].The energy conditions have many significant theoretical applications like Hawking Penrose singularity conjecture that is based on the strong energy condition [36], while the dominant energy condition is useful in the proof of positive mass theorem [37]. Furthermore, null energy condition is a basic ingredient in the derivation of second law of black hole thermodynamics [38]. On the cosmological grounds, Visser [39] discussed various cosmological terms like distance modulus, look back time, deceleration, and statefinder parameters in terms of red shift using energy condition constraints. These conditions are originally formulated in the context of GR and then extended to modified theories of gravity. Many authors have explored these energy conditions in the framework of modified gravity and found interesting results [40–43].Basically, modified gravity theories contain some extrafunctions like higher-order derivatives of curvature term or some functions of Einstein tensor or scalar field, and so forth. Thus it is a point of debate that how one can constrain the added extra degrees of freedom consistently with the recent observations. The energy conditions can be used to put some constraints on these functions that could be consistent with those already found in the cosmological arena. Recently, these energy conditions have been discussed inf(T) [44] and f(R,T) [45] theories.In this paper, we study the energy condition bounds in a most general scalar-tensor theory. The paper is designed in the following layout. Next section defines the energy conditions in GR as well as in a general modified gravitational framework. Section3 provides basic formulation of the most general scalar-tensor theory. In the same section, we formulate the energy conditions in terms of some cosmological parameters within such modified framework. In Section 4, we provide some specific cases of this theory and discuss the corresponding constraints. Finally, we summarize and present some general remarks.


## 2. Energy Conditions

In this section, we discuss the energy conditions in GR framework and then express the respective conditions in a general modified gravity. In GR, the energy conditions come from a well-known purely geometric relationship known as Raychaudhuri equation [38, 46] together with the lineament of gravitational attractiveness. In a spacetime manifold with vector fields uμ and kμ as tangent vectors to timelike and null-like geodesics of the congruence, the temporal variation of expansion for the respective curves is described by the Raychaudhuri equation as
(1)dθdτ=-13θ2-σμνσμν+ωμνωμν-Rμνuμuν,dθdτ=-12θ2-σμνσμν+ωμνωμν-Rμνkμkν.
Here Rμν, θ, σμν, and ωμν represent the Ricci tensor, expansion, shear, and rotation, respectively, related with the congruence of timelike or null-like geodesics.The characteristic of the gravity that is attractive leads to the conditiondθ/dτ<0. For infinitesimal distortions and vanishing shear tensor ωμν=0, that is, zero rotation (for any hypersurface of orthogonal congruence), we ignore the second-order terms in Raychaudhuri equation, and consequently integration leads to θ=-τRμνuμuν=-τRμνkμkν. It further implies that
(2)Rμνuμuν≥0,Rμνkμkν≥0.
Since GR and its modifications lead to a relationship of the matter contents, that is energy-momentum tensor in terms of Ricci tensor through the field equations, therefore the respective physical conditions on the energy-momentum tensor can be determined as follows:
(3)Rμνuμuν=(Tμν-T2gμν)uμuν≥0,Rμνkμkν=(Tμν-T2gμν)kμkν≥0,
where Tμν and T are the energy-momentum tensor and its trace, respectively. For perfect fluid with density ρ and pressure p defined by
(4)Tμν=(ρ+p)uμuν-pgμν,
the strong and null energy conditions, respectively, are defined by the inequalities ρ+3p≥0 and ρ+p≥0, while the weak and dominant energy conditions are defined, respectively, by ρ≥0 and ρ±p≥0.Raychaudhuri equation being a geometrical statement works for all gravitational theories. Therefore, its interesting features like focussing of geodesic congruences as well as the attractiveness of gravity can be used to derive the energy constraints in the context of modified gravity. In case of modified gravity, we assume that the total matter contents of the universe act like perfect fluid, and consequently these conditions can be defined in terms of effective energy density and pressure (matter sources get modified, and we replaceTμν and T in (3) by Tμνeff and Teff, resp.). These conditions can be regarded as an extension of the respective conditions in GR given by [43]
(5)NEC:ρeff+peff≥0,SEC:ρeff+peff≥0,ρeff+3peff≥0,WEC:ρeff≥0,ρeff+peff≥0,DEC:ρeff≥0,ρeff±peff≥0.
For a detailed discussion, we suggest the readers to study a recent paper [47].The DE requires negative EoS parameterω≤-1/3, for the explanation of cosmic expansion. Indeed, for cosmological purposes, we are curious for a source with ρ≥0; in that case, all of the energy conditions require ω≥-1 [48]. The role of possible DE candidates with ω<-1 was pointed out by Caldwell, who referred to null DEC violating sources as phantom components. It is argued that DE models with w≥-1 such as the cosmological constant and the quintessence satisfy the NEC, but the models with ω<-1 (predicted for instance by the phantom theory), where the kinetic term of the scalar field has a wrong (negative) sign, does not satisfy. However, quintom models can also satisfy NEC as they yield the phantom era for a very short period of time [49]. Usually, the discussions on energy conditions for cosmological constant are available in literature by introducing it in some other type of matter like electromagnetic field [50]. The cosmological constant will trivially satisfy all these energy conditions except SEC.


## 3. Energy Conditions in the Most General Scalar-Tensor Gravity

The most general scalar-tensor theory in 4 dimensions is given by the action [31–35]
(6)S=∫‍d4x-g×[(∇β∇μϕ)}K(ϕ,X)-G3(ϕ,X)□ϕ+G4(ϕ,X)R+G4X{(□ϕ)2-(∇μ∇νϕ)(∇μ∇νϕ)}+G5(ϕ,X)Gμν(∇μ∇νϕ)-16G5X×{(□ϕ)3-3(□ϕ)(∇μ∇νϕ)(∇μ∇νϕ)+2(∇μ∇αϕ)(∇α∇βϕ)(∇β∇μϕ)}+Lm],
where ϕ is the scalar field, g is the determinant of the metric tensor, Lm denotes the matter part of the Lagrangian, and X represents the kinetic energy term defined by X=-(1/2)∂μϕ∂μϕ. Moreover, Gμν is the Einstein tensor, R is the Ricci scalar, ∇μ is the covariant derivative operator, and □=∇μ∇μ is the de’Alembertian operator. The functions K(ϕ,X) and Gi(ϕ,X);i=3,4,5 are all arbitrary functions and GiX=∂Gi/∂X. In this action, the term G3(ϕ,X)□ϕ is the Galilean term, G4(ϕ,X)R can yield the Einstein-Hilbert term and G5(ϕ,X) leads to the interaction with Gauss-Bonnet term. This indicates that it covers not only several DE proposals like k-essence, f(R) gravity, BD theory and Galilean gravity models, but it also contains 4-dimensional Dvali, Gabadadze, and Poratti (DGP) model (modified), the field coupling with Gauss-Bonnet term, and the field derivative coupling with Einstein tensor as its particular cases.By varying the action (6) with respect to the metric tensor, the gravitational field equation can be written as
(7)Gμν=1G4Θμνeff=1G4[Tμνm+Tμνϕ],
where Θμνeff is the modified energy-momentum tensor, Tμνm is the source of usual matter field that can be described by the perfect fluid, while Tμνϕ provides the matter source due to scalar field and hence yields the source of DE, defined in the Appendix. The scalar wave equation for such modified gravity has been described in the literature [32–35].By inverting (7), the Ricci tensor can be expressed in terms of effective energy-momentum tensor and its trace as follows:
(8)Rμν=Tμνeff-12gμνTeff,
where the effective energy-momentum tensor Tμνeff and its trace Teff are
(9)Tμνeff=Tμν+12KX∇μϕ∇νϕ-12G3X□ϕ∇νϕ∇μϕ-∇(μG3∇ν)ϕ+12G4X×R∇μϕ∇νϕ+12G4XX×[(□ϕ)2-(∇α∇βϕ)2]∇μϕ∇νϕ+G4X□ϕ×∇μ∇νϕ-G4X∇λ∇μϕ∇λ∇νϕ-2∇λG4X∇λ∇(μϕ∇ν)ϕ+∇λG4X×∇λϕ∇μ∇νϕ-2[G4XRλ(μ∇ν)ϕ∇λϕ-∇(μG4X∇ν)ϕ□ϕ]-G4X×Rμανβ∇αϕ∇βϕ+G4ϕ∇μ∇νϕ+G4ϕϕ∇μϕ∇νϕ-2G4Xϕ∇λϕ×∇λ∇(μϕ∇ν)ϕ+G4XX∇αϕ∇α∇μϕ∇βϕ∇β∇νϕ-G5XRαβ∇αϕ×∇β∇(μϕ∇ν)ϕ+G5XRα(μ∇ν)ϕ∇αϕ□ϕ+12G5XRαβ∇αϕ∇βϕ×∇μ∇νϕ+12G5XRμανβ∇αϕ∇βϕ□ϕ-G5XRαλβ(μ∇ν)ϕ∇λϕ∇α∇βϕ-G5XRαλβ(μ∇ν)∇λϕ∇αϕ∇βϕ+12∇(μ[G5X∇αϕ]∇α∇ν)ϕ□ϕ-12×∇(μ[G5ϕ∇ν)ϕ]□ϕ+∇λ[G5ϕ∇(μϕ]∇ν)∇λϕ-12[∇λ(G5ϕ∇λϕ)-∇α(G5X∇βϕ)∇α∇βϕ]×∇μ∇νϕ-∇αG5∇βϕRα(μν)β+∇(μG5Rν)λ×∇λϕ-12∇(μG5X∇ν)ϕ×[(□ϕ)2-(∇α∇βϕ)2]+∇λG5Rλ(μ∇ν)ϕ-∇α[G5X∇βϕ]∇α∇(μϕ∇β∇ν)ϕ+∇βG5X×[□ϕ∇β∇(μϕ-∇α∇βϕ∇α×∇(μϕ]∇ν)ϕ-12∇αϕ∇αG5X[□ϕ∇μ∇νϕ-∇β∇μϕ∇β∇νϕ]+12G5X×Gαβ∇α∇βϕ∇μϕ∇νϕ+12G5X□ϕ×∇α∇μϕ∇α∇νϕ-12G5X(□ϕ)2×∇μ∇νϕ-112G5XX[(□ϕ)3-3(□ϕ)(∇α∇βϕ)2+2(∇α∇βϕ)3]×∇μϕ×∇νϕ-12∇λG5Rμν∇λϕ,(10)Teff=K+∇λG3∇λϕ-2(G4ϕ□ϕ-2XG4ϕϕ)-2{[12(ϕ)2-(∇α∇βϕ)2]-2G4Xϕ∇α∇βϕ∇αϕ∇βϕ+G4XX∇α∇λϕ×∇β∇λϕ∇αϕ∇βϕ+12G4X[(□ϕ)2-(∇α∇βϕ)2]}+2[G4X×Rαβ∇αϕ∇βϕ-∇λG4X∇λϕ□ϕ]-12R∇λG5∇λϕ-2{-16G5X[(□ϕ)3-3□ϕ(∇α∇βϕ)2+2(∇α∇βϕ)3]+∇αG5Rαβ∇βϕ-12∇αG5ϕ∇αϕ□ϕ+12∇αG5ϕ×∇βϕ∇α∇βϕ-12∇αG5X∇αX□ϕ+12∇αG5X×∇βX∇α∇βϕ-14∇λG5X∇λϕ[(□ϕ)2-(∇α∇βϕ)2]+12G5XRαβ∇αϕ∇βϕ□ϕ-12G5X×Rαλβρ∇αϕ×∇βϕ∇λϕ∇ρϕ-16}+T+12KX(∇μϕ)2-12G3X□ϕ(∇μϕ)2-∇μG3∇μϕ+12G4XR(∇μϕ)2+12G4XX[(□ϕ)2-(∇α∇βϕ)2](∇μϕ)2+G4X□ϕ∇μ∇μϕ-G4X∇λ∇μϕ∇λ∇μϕ-2∇λG4X∇λ(∇ϕ)2+∇λG4X×∇λϕ∇μ∇μϕ-2[G4XRλμ∇μϕ∇λϕ-∇μG4X∇μϕ□ϕ]-G4XRμαμβ∇αϕ∇βϕ+G4X∇μ∇μϕ+G4ϕϕ(∇μϕ)2-2G4Xϕ∇λϕ×∇λ(∇μϕ)2+G4XX∇αϕ∇α×∇μϕ∇βϕ∇β∇μϕ-G5XRαβ∇αϕ∇β(∇μϕ)2+G5XRα(μ∇μ)ϕ∇αϕ□ϕ+12G5XRαβ∇αϕ∇β×ϕ∇μ∇μϕ+12G5X×Rμαμβ∇αϕ∇βϕ□ϕ-G5XRαλβμ∇μϕ∇λϕ∇α∇βϕ-G5XRαλβμ×∇μ∇λϕ∇αϕ∇βϕ+12∇(μ[G5X∇αϕ]∇α∇μ)ϕ□ϕ-12∇(μ[G5ϕ∇μ)ϕ]□ϕ+∇λ[G5ϕ∇(μϕ]∇μ)∇λϕ-12[∇λ(G5ϕ∇λϕ)-∇α(G5X∇βϕ)∇α∇βϕ]×∇μ∇μϕ-∇αG5∇βϕRαμμβ+∇μG5Rμλ∇λϕ-12∇μG5X×∇μϕ[(□ϕ)2-(∇α∇βϕ)2]+∇λG5Rλμ∇μϕ-∇α[G5X∇βϕ]∇α∇μϕ×∇β∇μϕ+∇βG5X[□ϕ∇β∇(μϕ-∇α∇βϕ∇α∇(μϕ]∇μ)ϕ-12∇αϕ×∇αG5X[□ϕ∇μ∇μϕ-∇β∇μϕ∇β∇μϕ]+12G5XGαβ∇α∇βϕ(∇μϕ)2+12G5X□ϕ∇α∇μϕ∇α∇μϕ-12G5X(□ϕ)2∇μ∇μϕ-112G5XX×[(∇α∇βϕ)3(□ϕ)3-3(□ϕ)(∇α∇βϕ)2+2(∇α∇βϕ)3](∇μϕ)2-12∇λG5R∇λϕ+12∇αG5X∇βX∇α∇βϕ-14∇αG5X∇λϕ[(□ϕ)2-(∇α∇βϕ)2]+12G5XRαβ∇αϕ∇βϕ□ϕ-12G5XRαλβρ.
Evaluating the temporal and spatial components of the effective energy-momentum tensor and its trace defined previously and using these values in (3), we can find the energy conditions for any spacetime.Let us consider the spatially homogeneous, isotropic, and flat FRW universe model witha(t) as a scale factor described by the metric
(11)ds2=-dt2+a2(t)(dx2+dy2+dz2).
The background fluid is taken as perfect fluid given by (4) with uμ=(1,0,0,0), and the null like vector is taken as kμ=(1,a,0,0). Furthermore, we assume that the scalar field is a function of time only. The Friedmann equations for the generalized scalar-tensor theory in terms of effective energy density and pressure are given by [32–35]
(12)3H2=ρeffG4,-(3H2+2H˙)=peffG4,
where
(13)ρeff=12[ρm+2XKX-K+6Xϕ˙HG3X-2XG3ϕ+24H2X×(G4X+XG4XX)-12HXϕ˙G4Xϕ-6Hϕ˙G4ϕ+2H3Xϕ˙(5G5X+2XG5XX)-6H2X(3G5ϕ+2XG5Xϕ)],(14)peff=12[pm+K-2X(G3ϕ+ϕ¨G3X)-12H2XG4X-4HX˙G4X-8H˙XG4X-8HXX˙G4XX+2(ϕ¨+2Hϕ˙)G4ϕ+4XG4ϕϕ+4X(ϕ¨-2Hϕ˙)×G4Xϕ-2X(2H3ϕ˙+2HH˙ϕ˙+3H2ϕ¨)G5X-4H2X2ϕ¨G5XX+4HX(X˙-HX)G5Xϕ+2[2(HX).+3H2X]G5ϕ+4HXϕ˙G5ϕϕ].
Here G4, being an arbitrary function of ϕ and X, acts as a dynamical gravitational constant, and it should be positive for any gravitational theory. Furthermore, ρm and pm are density and pressure, respectively, for ordinary matter. We shall discuss its different forms in the next section. Using these values in (5), it can be checked that the NEC, WEC, SEC, and DEC require the following conditions to be satisfied:
(15)NEC:ρeff+peff≥0⇒12G4[(ρm+pm)+2XKX+6Xϕ˙HG3X-4XG3ϕ+12H2XG4X+24H2X2G4XX-20XHϕ˙G4Xϕ-2Hϕ˙G4ϕ+6H3Xϕ˙G5X+4H3X2ϕ˙×G5XX-16H2X2G5Xϕ-12H2XG5ϕ-2Xϕ¨G3X-4HX˙G4X-8H˙X×G4X-8HX˙XG4XX+2ϕ¨G4ϕ+4XGϕϕ+4Xϕ¨G4Xϕ-2X(2HH˙ϕ˙+3H2ϕ¨)G5X-4H2X2ϕ¨G5XX+4HXX˙G5Xϕ+4(HX).G5ϕ+4HXϕ˙G5ϕϕ]≥0,WEC:ρeff+peff≥0,ρeff≥0⇒12G4[ρm+2XKX-K+6XHϕ˙G3X-2XG3ϕ+24H2X×(G4X+XG4XX)-12XHϕ˙-6Hϕ˙G4ϕ+2H3Xϕ˙(5G5X+2XG5XX)-6H2X(3G5ϕ+2XG5Xϕ)]≥0,SEC:ρeff+peff≥0,ρeff+3peff≥0⇒12G4[(ρm+3pm)+2XKX+2K+6XHϕ˙G3X-8XG3ϕ-6Xϕ¨G3X-12H2XG4X+24H2X2×G4XX-36XHϕ˙G4Xϕ+6Hϕ˙G4ϕ+6ϕ¨G4ϕ-2H3Xϕ˙G5X+4H3X2ϕ˙G5XX-24H2X2G5Xϕ+12HXX˙G5Xϕ+12(HX).G5ϕ+12HXϕ˙Gϕϕ-12H2X2ϕ¨G5XX-12XHH˙ϕ˙G5X-18XH2ϕ¨G5X+12Xϕ¨G4Xϕ-24HXϕ˙G4Xϕ+12XG4ϕϕ-24HXX˙×G4XX-24H˙XG4X-12HX˙G4X-36H2XG4X]≥0,DEC:ρeff≥0,ρeff+peff≥0,ρeff-peff≥0⇒12G4[(ρm-pm)+2XKX-2K+6XHϕ˙G3X+2Xϕ¨G3X+36H2XG4X+24H2XG4XX-4XHϕ˙G4Xϕ-10Hϕ˙G4ϕ+14H3Xϕ˙G5X+4H3X2ϕ˙×G5XX-24H2XG5ϕ-4(HX).G5ϕ-4HXϕ˙Gϕϕ-8H2X2G5Xϕ+4HX˙G4X+8H˙XG4X+8HXX˙G4XX-2ϕ¨G4ϕ-4XG4ϕϕ-4Xϕ¨G4Xϕ+2X(2HH˙ϕ˙+3H2ϕ¨)G5X+4H2X2G5XXϕ¨-4HXX˙G5Xϕ]≥0.In a mechanical framework, the terms velocity, acceleration, jerk, and snap parameters are based on the first four time derivatives of position. In cosmology, the Hubble, deceleration, jerk, and snap parameters are, respectively, defined as(16)H=a˙a,q=-1H2a¨a,j=1H3a⃛a,s=1H4a⃜a.
In order to have a more precise picture of these conditions (15), we use the relations of time derivatives of Hubble parameter in terms of cosmological quantities like deceleration, snap, and jerk parameters as
(17)H˙=-H2(1+q),H¨=H3(j+3q+2),H⃛=H4(s-2j-5q-3).
Moreover, we assume that the scalar field evolves as a power of scale factor, that is, ϕ(t)~a(t)β [51, 52], which leads to
(18)ϕ˙~βHaβ,ϕ¨~βaβ(H˙+βH2)=βaβH2(β-1-q),X~β2H2a2β2,X˙~β2H3a2β(β-1-q).
Here β is a nonzero parameter (β=0 yields constant scalar field, β>0 yields expanding scalar field, and β<0 corresponds to contracting scalar field). Clearly, ϕ remains as a positive quantity.Introducing these quantities in the energy conditions given by (15), it follows that
(19)1G4[KX-2G3,ϕ+6H2G4X-6H2G5ϕ+2G4ϕϕ]×β2H2a2β+[3HG3X-10HG4Xϕ+3H3G5X+2HG5ϕϕ]×β3H3a3β+(6H2G4XX-4H2G5Xϕ)×β4H4a4β-2H2βaβG4ϕ+H8β5a5βG5XX-β3H4a3β(β-1-q)G3X-4H4a2ββ2(β-1-q)×G4X+4H4a2ββ2(1+q)G4X-4H6β4a4β×(β-1-q)G4XX+2H2aβ(β-1-q)βG4ϕ+2β3H4a3β(β-1-q)G4Xϕ+2H6(1+q)β3a3βG5X-3H6β3a3β(β-1-q)G5X-H8β5a5β×(β-1-q)G5XX+2H6β4a4β×(β-1-q)G5Xϕ+4H4β2a2β(β-1-q)G5ϕ-2H4(1+q)β2a2βG5ϕ+(ρm+pm)≥0,1G4[ρm+3pm+2K+3β3a3βH4G3X-4β2H2×a2βG3ϕ-3β3a3βH4(β-1-q)G3X-6H4β2a2βG4X+3H6β4a4βG4XX-18H4β3a3βGXϕ+6H2β×aβG4ϕ+6H2βaβ×(β-1-q)G4ϕ-H6β3a3βG5X+2H8β5a5βG5XX-3H6β4a4βG5Xϕ+6H6β4a4β×(β-1-q)G5Xϕ-6H4β2a2β(1+q)G5ϕ+12H4β2a2β(β-1-q)G5ϕ+6H4β3a3βG5ϕϕ-3H8β5a5β(β-1-q)×G5XX+H6β3a3β(1+q)G5X-9H6β3a3β(β-1-q)G5X+6H4β3a3β×(β-1-q)G4Xϕ-12H4β3a3βG4Xϕ+6H2β2a2βG4ϕϕ-12H6β4a4β×(β-1-q)G4XX+12H4β2a2β×(1+q)G4X-12H4β2a2β(β-1-q)G4X-18H4β2a2βG4X+β2a2βH2KX]≥0,1G4[H2β2a2βKX-K+3H4β3a3βG3X-H2β2a2βG3ϕ+12H4β2a2βG4X+6H6β4a4βG4XX-6H4β3a3βGXϕ-6H2βaβG4ϕ+5H6β3a3βG5X+H8β5a5βG5XX-9H4×β2a2βG5ϕ-3H6β4a4βG4Xϕ+ρm]≥0,1G4[ρm-pm+β2H2a2βKX-2K+3H4β3a3βG3X+β2H3a2β(β-q-1)×G3X+18β2H4a2βG4X+12H4β2a2βG4XX-2H4β3a3βG4Xϕ-10H2×βaβG4ϕ+7H6β3a3βG5X+H8β5a5βG5XX-12H4β2a2βG5ϕ-4H4β2×a2βG5ϕ+2H4β2a2β(1+q)G5ϕ-2H4β3a3βG5ϕϕ-4H4β2a2βGXϕ+4×H4β2×a2β(β-q-1)G4X-4H4β2a2β(1+q)G4X+4H6β4a4β×(β-1-q)G4XX-2H2aββ(β-1-q)G4ϕ-2H2β2a2βG4ϕϕ-2H4β3a3β(β-q-1)G4Xϕ-2H6β3a3β(1+q)G5X+3H6β3a3β(β-1-q)G5X+H8β5×a5β(β-1-q)G5XX-2H6β4a4β(β-1-q)G5Xϕ]≥0.
These are the most general energy conditions that can yield the energy conditions for various DE models like k-essence and modified theories in certain limits. In order to satisfy these conditions, it must be guaranteed that the function G4 is a positive quantity. However, we have discussed earlier that G4, being a gravitational constant, would be positive in all cases (if it is not so, then we impose this condition and restrict the free parameters). Clearly, these conditions are only dependent on the Hubble, deceleration parameters and arbitrary functions, namely, K, G3, G4, and G5. Once these arbitrary functions are specified, the energy bounds on the selected models can be determined by using these conditions.In order to have a better understanding of these constraints, we can use either the power law ansataz for the scale factor, for example, [45] or we can use the estimation of present values of the respective parameters available in literature. In this study, we consider the present value of the Hubble parameter H0=0.718, the scale factor a0=1, and the deceleration parameter q=-0.64 as suggested by Capozziello et al. [53]. Since it is well known that the energy constraints are satisfied for usual matter contents like perfect fluid, therefore we shall focus on validity of the energy constraints for the scalar field terms only (either we take vacuum case or assume that the energy conditions for ordinary matter hold). It is interesting to mention here that the respective energy conditions in GR can be recovered by taking the arbitrary functions K,G3, and G5 zero with G4 as constant.


## 4. Energy Conditions in Some Particular Cases

Now we discuss application of the derived conditions to some particular cases of this theory. The violation of energy conditions leads to various interesting results. In particular, for a canonical scalar field, violation of these conditions yields instabilities and ghost pathologies. It is important to discuss the violation of these energy conditions in order to check the existence of instabilities in Horndeski theory. The procedure for FRW universe model in most general scalar-tensor theory based on tensor and scalar perturbations is available in literature [31]. By introducing perturbed metric, it has been shown that for the avoidance of ghost and gradient instabilities, the tensor perturbations suggest
(20)ℱT=2[G4-X(ϕ¨G5X+G5ϕ)]>0,𝒢T=2[G4-2XG4X-X(Hϕ˙G5X-G5ϕ)]>0,
while scalar perturbations impose
(21)ℱS=1addt(aΘ𝒢T2)-ℱT>0,𝒢S=ΣΘ2𝒢T2+3𝒢T>0,
where the quantities Σ and Θ are defined in [31]. We simply plug the values in these conditions for the following cases and show that violation of energy conditions leads to the existence of ghost instabilities.


### 4.1.k-Essence Models in General Relativity

Thek-essence dynamical models of DE play a dominant role in the solution of various problems in cosmological context [54]. The action (6) can be reduced to the action for k-essence model in GR framework defined by S=∫-g[K(ϕ,X)+(Mpl2/2)R+Lm]d4x with the following choice of the functions:
(22)K=K(ϕ,X),G4=Mpl22,G3=G5=0,
where Mpl2 is Planck mass. The k-essence models can be classified into three forms:  (i)
K(ϕ,X)=K1(X) (Kinetic case),(ii)
K(ϕ,X)=K1(X)B(ϕ),(iii)
K(ϕ,X)=K1(X)+B(ϕ).For the choice of arbitrary functions given by (22), the energy conditions (15) take the following forms:
(23)NEC:1Mpl2[2XKX+ρm+pm]≥0,WEC:ρeff+peff≥0,1Mpl2[2XKX-K+ρm]≥0,SEC:ρeff+peff≥0,1Mpl2[2XKX+2K+ρm+3pm]≥0,DEC:ρeff+peff≥0,ρeff≥0,1Mpl2[2XKX-2K+ρm-pm]≥0.
Here K(ϕ,X) is arbitrary.In order to see how the functionK(ϕ,X) can be constrained by using the previous energy conditions, we choose a particular model of k-essence [55] as follows:
(24)K(ϕ,X)=12[C2-C1-2Bϕ2]+12(C1+C2)X+12M04(X-1)2,
where C1, C2, B, and M0 are arbitrary constants. In this case, WEC requires the following conditions:
(25)1Mpl2[12X(C1+C2)+2XM04(X-1)-12M04(X-1)2-12(C2-C1-2Bϕ2)+ρm]≥0,1Mpl2[X(C1+C2)+2M04X(X-1)+ρm+pm]≥0,
where Mpl2>0. For the interpretation of the previous inequalities, we consider the power law ansatz for the scalar field ϕ~aβ; β≠0, which further yields X~(β2a2βH2/2). Consequently, the WEC (25) turns out to be
(26)[12β2a2βH22(C1+C2)+2β2a2βH22M04(β2a2βH22-1)-12M04(β2a2βH22-1)2-12(C2-C1-2Bϕ2)+ρm]≥0,(27)[β2a2βH22(C1+C2)+2M04β2a2βH22×(β2a2βH22-1)+ρm+pm]≥0.It is difficult to find the admissible ranges of all constantsC1, C2, B, M0, and β from the previous conditions. In order to find the constraints on these parameters, we consider that these conditions are satisfied for ordinary matter, that is, ρm>0 and ρm+pm>0. Moreover, we take the present value of Hubble parameter and choose some particular values of the constants C1 and C2 to find the ranges of B, M0, and β, consistent with the WEC. It turns out from the graphs that we can take the parameter β as follows β>1.4, 0<β<1.4 and β<0, while B and can be positive or negative. For the consistency of condition (27), we restrict the parameter M0 as 0<M0<1. From the condition (26), it can be observed that energy conditions are satisfied only when we take β>1.4, with arbitrary B and β<1.4 with B>10 only. Other choices of these parameters lead to violation of WEC. Figures 1(a) and 1(b) show that WEC is satisfied with these fixed input parameters by taking β>1.4, 0<M0<1, and 0<B<1.Plot (a) showsρeff+peff≥0 versus β and M0. Plot (b) represents ρeff≥0 versus parameters β and B with M0=0.2. In both cases, we take the present values of cosmological parameters with C1=1, C2=2.
(a)
ρ
eff
+
p
eff
≥
0
(b)
ρ
eff
≥
0


### 4.2. Brans-Dicke Theory

Brans-Dicke gravity with action∫-g[(MplXω/ϕ)-V(ϕ)+(1/2)MplRϕ+Lm]d4x can be defined by the following choice of functions:
(28)K=MplXωϕ-V(ϕ),G3=G5=0,G4=12Mplϕ.
Here ω is the BD parameter and V is the field potential. The action for general scalar-tensor gravity can be obtained by taking F(ϕ) instead of ϕ in G4. In this case, the energy conditions take the form
(29)NEC:1Mpl[ϕ˙2ϕ2Mplω-Hϕ˙ϕMpl+ϕ¨ϕMpl+ρm+pmϕ]≥0,WEC:ρeff+peff≥0,1Mpl[ωMplϕ˙22ϕ2+V(ϕ)ϕ-3Hϕ˙ϕMpl+ρmϕ]≥0,SEC:ρeff+peff≥0,1Mpl[2ϕ˙2ϕ2Mplω-2V(ϕ)ϕ+3Hϕ˙ϕMpl+3ϕ¨ϕMpl+ρm+3pmϕ]≥0,DEC:ρeff+peff≥0,ρeff≥0,1Mpl[2V(ϕ)ϕ-5Hϕ˙ϕMpl-ϕ¨ϕMpl+ρm-pmϕ]≥0.
A suitable choice of the field potential has always been a matter of debate in BD gravity. We use power laws for the scalar field and potential as ϕ=ϕ0aβ and V=V0ϕm, respectively, where m and β are nonzero parameters. Consequently, the WEC restricts the parameters as
(30)V0≥3H02β-ω2β2H02,β2H02ω-H02β+H02aββ2-aββH02(1+q0)≥0.For accelerated expanding universe, the observed range of BD parameter is-2<ω<-3/2 [56]. Clearly, both of these conditions are independent of the parameter m which shows that these conditions are valid for both the positive and inverse power law potentials; however, these conditions depend on the present value of the field potential. Figure 2(a) shows that the condition ρeff≥0 leads to β<0, which further yields positive present value of the BD field potential V0. Moreover, the second condition will be satisfied if we take β<0 with arbitrary ω and β>0 with ω>0 only. Figure 2(b) indicates that ρeff+peff≥0 is satisfied for a particular choice of β<0 and -2<ω<-1.5. Clearly, the energy condition ρeff+peff≥0 is violated for β>0 with ω<0. It is easy to check that this choice of parameters (β=2,ω=-1.8) also leads to the violation of condition imposed by scalar perturbations given by the expression
(31)2Hϕ2+4ϕϕ˙2Hϕ+ϕ˙2-2ϕ2(2Hϕ˙+2H˙ϕ+ϕ¨)(2Hϕ+ϕ˙2)2>aϕ,4ϕ2(Xω/ϕ-3H2ϕ-Hϕ˙/2(2Hϕ+ϕ˙2)2)+3ϕ>0
and consequently yields the ghost instabilities for the model. However, the conditions imposed by tensor perturbation are trivially satisfied.Plots (a) and (b) show WEC versus parametersω and β for H0=0.718 and q0=-0.64.
(a)
ρ
eff
+
p
eff
≥
0
(b)
ρ
eff
≥
0 (V0≥0)


### 4.3.f(R) Gravity

The action forf(R) gravity described by S=∫-g[(Mpl2/2)f(R)+Lm]d4x can be obtained from the action (6) for
(32)K=-Mpl22(RfR-f),G3=G5=0,G4=Mpl2ϕ,ϕ=Mplf,R.
Here f(R) is an arbitrary function of the Rici scalar. Using these values in energy conditions (15), we obtain
(33)NEC:(R¨-R˙H)f′′(R)+R˙2f′′′(R)+ρm+pm≥0,WEC:ρeff+peff≥0,ρm-12(f(R)-Rf′(R))-3HR˙f′′(R)≥0,SEC:ρeff+peff≥0,(f(R)-Rf′(R))+3(R¨+R˙H)f′′(R)+3R˙2f′′′(R)+ρm+3pm≥0,DEC:ρeff+peff≥0,ρeff≥0-(5HR˙+R¨)f′′(R)-R˙2f′′′(R)-(f(R)-Rf′(R))+ρm-pm≥0,
where prime denotes the derivative with respect to R. Let us consider the example of logarithmic model in f(R) gravity defined by [57]
(34)f(R)=R[log(αR)]q-R;α<0,q>0.
In this case, the WEC (ρeff≥0) leads to
(35)-6H02(1-q0)[log(-6αH02(1-q0))]q+6H02(1-q0)+6H02(1-q0)×[(log(-6αH02(1-q0)))q-1(log(-6αH02(1-q0)))q+q(log(-6αH02(1-q0)))q-1-1]-36H04×(j0-q0-2)×[q6H02(1-q0)(log(-6αH02(1-q0))q)+q(q-1)6H02(1-q0)×(log(-6αH02(1-q0)))q-2]≥0,
where we have used R˙=-6H2(1-q) and the present values of respective parameters. Notice that we have taken only the condition ρeff≥0 as the other condition involves the present value of snap parameter s which is not correctly estimated in the literature yet. Figure 3 shows that the WEC is satisfied for a suitable range of both the parameters q and α.Figure 3
Plot shows WEC versus free parametersq and α. Here we take q0=-0.64, H0=0.718, and j0=1.41.


### 4.4. Kinetic Gravity Braiding Model

Kinetic gravity braiding is defined by the action [58]
(36)S=∫‍d4x-g(K(ϕ,X)-G3(ϕ,X)□ϕ+Mpl22R+Mpl22×[(□ϕ)2-(∇μ∇νϕ)(∇μ∇νϕ)]+LmMpl22),
where the functions K and G3 are arbitrary (while other functions are taken to be zero in action (6)). A particular choice of these functions proposed by Dvali and Turner [59] is given by K=-X and G3=cXn, where c and n are constants. For this choice of model, the respective energy conditions turn out to be
(37)NEC:1Mpl2[-2X+6ncHϕ˙Xn-2cnXnϕ¨+ρm+pm]≥0,WEC:ρeff+peff≥0,1Mpl2[-X+6ncHϕ˙Xn+ρm]≥0,SEC:ρeff+peff≥0,1Mpl2[-4X+6ncHϕ˙Xn-6cnXnϕ¨+ρm+3pm]≥0,DEC:ρeff+peff≥0,ρeff≥0,1Mpl2[6ncHϕ˙Xn+2cnXnϕ¨+ρm-pm]≥0.
By taking the power law evolution of the scalar field, WEC yields
(38)[-β2a2βH22+6ncHϕ˙(β2a2βH22)n]≥0,[-2β2a2βH22+6ncHϕ˙(β2a2βH22)n-2cn(β2a2βH22)nϕ¨]≥0.
Here we have taken the present values of the ordinary density and pressure to be zero. Using the present values of the Hubble parameter and the scale factor, WEC can be satisfied only when both the parameters n and β remain positive as indicated in Figure 4, where we have taken 0<n<1 and 1<β<2. In this case, scalar perturbations lead to the following constraints:
(39)H2(2+q)-cnHXnϕ˙a+cn2Xn-1X˙ϕ˙+cnXnϕ¨>(H-cnXnϕ˙)2,(-X+12cnHXnϕ˙+6nc(n-1)HXnϕ˙-3H2Mpl2(HMpl2-cnXnϕ˙)2)×Mpl2>-3.
It is easy to check that the energy conditions are violated for 0.6<n<1 and negative range of β (e.g., β=-10). For this choice of parameters, the previous constraints are also violated, and hence the ghost instabilities occur. However, constraints imposed by tensor perturbations are trivially satisfied as Mpl2ϕ>0.Plots (a) and (b) show WEC, namely,ρeff≥0 and ρeff+peff≥0 versus parameters n and β.
(a)
ρ
eff
≥
0
(b)
ρ
eff
+
p
eff
≥
0


### 4.5. Covariant Galilean Model

In the absence of potential, the covariant Galilean model [59] is defined by the following choice of parameters in action (6):
(40)K=-c2X,G3=c3M3X,G4=Mpl22-c4M6X2,G5=3c5M9X2,
where c2, c3, c4, and c5 are dimensionless constants, while M is constant with dimensions of mass. Using these values in (15), it follows that
(41)NEC:12((Mpl2/2)-(c4/M6)X2)×[-2Xc2+6XHϕ˙c3M3-72H2X2c4M6+60H3X2c5M9-2Xϕ¨c3M3+24HXX˙c4M6+16H˙X2c4M6-12X2×(2HH˙ϕ˙+3H2ϕ¨)c5M9-24H2X2ϕ¨c5M9+ρm+pm]≥0,WEC:ρeff+peff≥0,12((Mpl2/2)-(c4/M6)X2)×[-Xc2+6XHϕ˙c3M3-96H2X2c4M6+84H3X2ϕ˙c5M9+ρm]≥0,SEC:ρeff+peff≥0,12((Mpl2/2)-(c4/M6)X2)×[-4Xc2+6XHϕ˙c3M3-6Xϕ¨c3M3+24XH2c4M6-48H2X2c4M6+12H3X2ϕ˙c5M9-180H2X2ϕ¨c5M9-72X2HH˙ϕ˙×c5M9+48XX˙Hc4M6+48X2H˙c4M6+24HXX˙×c4M6+72H2X2×c4M6+ρm+3pm]≥0,DEC:ρeff+peff≥0,ρeff≥0,12((Mpl2/2)-(c4/M6)X2)×[6XHϕ˙c3M3+2Xϕ¨c3M3-72H2X2c4M6-48XH2c4M6+108H3X2ϕ˙c5M6-24HXX˙c4M6-16H˙X2c4M6+12X2(2HH˙ϕ˙+2H2ϕ¨)c5M9+24H2X2ϕ¨c5M9+ρm-pm]≥0.By taking the power law ansataz for scalar field and consequently, for kinetic termX, the WEC in terms of present values of the involved parameter require the following inequalities:
(42)1(1-(c4β4H04/2M6))×[-c2H02β2+3H04c3β3M3-18H06β4c4M6+15H07βc5M9-β3H04(β-1-q0)c3M3+24H06β4(β-1-q0)c4M6-16H06β4×(1+q)c4M6-3H04β4(3H04β(β-1-q0)-2H04β(1+q))c5M9-6H08β5(β-1-q0)c5M9]≥0,1(1-(c4β4H04/2M6))×[-H02β2c22+3c3H04β3M3-24H06β4c4M6+ρ0m]≥0.
Clearly, these conditions are satisfied when both G4 and terms inside the brackets are positive. Since G4>0 requires β>(7.5M6/c4)1/4, therefore a suitable choice of all these parameters yield the consistency with WEC if parameter β remains small and positive (β<7.8), while c2 remains negative as shown in Figure 5. Here we have taken -5<c2<-1 or -50<c2<-10, and 5.5<β<8. In this case, tensor perturbations suggest the following conditions for the avoidance of ghost instabilities:
(43)ℱT=Mpl2-2X2c4M6-12X2ϕ¨c5M9>0,𝒢T=Mpl2+6X2c4M6-12X2ϕ˙Hc5M9>0.
Clearly, the energy conditions are violated for β>8 with -5<c2<-1. For this choice of parameters, the previous constraints imposed by tensor perturbations are also violated, and consequently the ghost instabilities exist. However, scalar perturbations lead to very complicated expressions, so we consider only the conditions imposed by tensor perturbations.Plots (a) and (b) show WEC versus parametersc2 and β with c3=2, c4=4, c5=3, and M=2.
(a)
ρeff+peff≥0
(b)
ρ
eff
≥
0


## 4.1.k-Essence Models in General Relativity

Thek-essence dynamical models of DE play a dominant role in the solution of various problems in cosmological context [54]. The action (6) can be reduced to the action for k-essence model in GR framework defined by S=∫-g[K(ϕ,X)+(Mpl2/2)R+Lm]d4x with the following choice of the functions:
(22)K=K(ϕ,X),G4=Mpl22,G3=G5=0,
where Mpl2 is Planck mass. The k-essence models can be classified into three forms:  (i)
K(ϕ,X)=K1(X) (Kinetic case),(ii)
K(ϕ,X)=K1(X)B(ϕ),(iii)
K(ϕ,X)=K1(X)+B(ϕ).For the choice of arbitrary functions given by (22), the energy conditions (15) take the following forms:
(23)NEC:1Mpl2[2XKX+ρm+pm]≥0,WEC:ρeff+peff≥0,1Mpl2[2XKX-K+ρm]≥0,SEC:ρeff+peff≥0,1Mpl2[2XKX+2K+ρm+3pm]≥0,DEC:ρeff+peff≥0,ρeff≥0,1Mpl2[2XKX-2K+ρm-pm]≥0.
Here K(ϕ,X) is arbitrary.In order to see how the functionK(ϕ,X) can be constrained by using the previous energy conditions, we choose a particular model of k-essence [55] as follows:
(24)K(ϕ,X)=12[C2-C1-2Bϕ2]+12(C1+C2)X+12M04(X-1)2,
where C1, C2, B, and M0 are arbitrary constants. In this case, WEC requires the following conditions:
(25)1Mpl2[12X(C1+C2)+2XM04(X-1)-12M04(X-1)2-12(C2-C1-2Bϕ2)+ρm]≥0,1Mpl2[X(C1+C2)+2M04X(X-1)+ρm+pm]≥0,
where Mpl2>0. For the interpretation of the previous inequalities, we consider the power law ansatz for the scalar field ϕ~aβ; β≠0, which further yields X~(β2a2βH2/2). Consequently, the WEC (25) turns out to be
(26)[12β2a2βH22(C1+C2)+2β2a2βH22M04(β2a2βH22-1)-12M04(β2a2βH22-1)2-12(C2-C1-2Bϕ2)+ρm]≥0,(27)[β2a2βH22(C1+C2)+2M04β2a2βH22×(β2a2βH22-1)+ρm+pm]≥0.It is difficult to find the admissible ranges of all constantsC1, C2, B, M0, and β from the previous conditions. In order to find the constraints on these parameters, we consider that these conditions are satisfied for ordinary matter, that is, ρm>0 and ρm+pm>0. Moreover, we take the present value of Hubble parameter and choose some particular values of the constants C1 and C2 to find the ranges of B, M0, and β, consistent with the WEC. It turns out from the graphs that we can take the parameter β as follows β>1.4, 0<β<1.4 and β<0, while B and can be positive or negative. For the consistency of condition (27), we restrict the parameter M0 as 0<M0<1. From the condition (26), it can be observed that energy conditions are satisfied only when we take β>1.4, with arbitrary B and β<1.4 with B>10 only. Other choices of these parameters lead to violation of WEC. Figures 1(a) and 1(b) show that WEC is satisfied with these fixed input parameters by taking β>1.4, 0<M0<1, and 0<B<1.Plot (a) showsρeff+peff≥0 versus β and M0. Plot (b) represents ρeff≥0 versus parameters β and B with M0=0.2. In both cases, we take the present values of cosmological parameters with C1=1, C2=2.
(a)
ρ
eff
+
p
eff
≥
0
(b)
ρ
eff
≥
0


## 4.2. Brans-Dicke Theory

Brans-Dicke gravity with action∫-g[(MplXω/ϕ)-V(ϕ)+(1/2)MplRϕ+Lm]d4x can be defined by the following choice of functions:
(28)K=MplXωϕ-V(ϕ),G3=G5=0,G4=12Mplϕ.
Here ω is the BD parameter and V is the field potential. The action for general scalar-tensor gravity can be obtained by taking F(ϕ) instead of ϕ in G4. In this case, the energy conditions take the form
(29)NEC:1Mpl[ϕ˙2ϕ2Mplω-Hϕ˙ϕMpl+ϕ¨ϕMpl+ρm+pmϕ]≥0,WEC:ρeff+peff≥0,1Mpl[ωMplϕ˙22ϕ2+V(ϕ)ϕ-3Hϕ˙ϕMpl+ρmϕ]≥0,SEC:ρeff+peff≥0,1Mpl[2ϕ˙2ϕ2Mplω-2V(ϕ)ϕ+3Hϕ˙ϕMpl+3ϕ¨ϕMpl+ρm+3pmϕ]≥0,DEC:ρeff+peff≥0,ρeff≥0,1Mpl[2V(ϕ)ϕ-5Hϕ˙ϕMpl-ϕ¨ϕMpl+ρm-pmϕ]≥0.
A suitable choice of the field potential has always been a matter of debate in BD gravity. We use power laws for the scalar field and potential as ϕ=ϕ0aβ and V=V0ϕm, respectively, where m and β are nonzero parameters. Consequently, the WEC restricts the parameters as
(30)V0≥3H02β-ω2β2H02,β2H02ω-H02β+H02aββ2-aββH02(1+q0)≥0.For accelerated expanding universe, the observed range of BD parameter is-2<ω<-3/2 [56]. Clearly, both of these conditions are independent of the parameter m which shows that these conditions are valid for both the positive and inverse power law potentials; however, these conditions depend on the present value of the field potential. Figure 2(a) shows that the condition ρeff≥0 leads to β<0, which further yields positive present value of the BD field potential V0. Moreover, the second condition will be satisfied if we take β<0 with arbitrary ω and β>0 with ω>0 only. Figure 2(b) indicates that ρeff+peff≥0 is satisfied for a particular choice of β<0 and -2<ω<-1.5. Clearly, the energy condition ρeff+peff≥0 is violated for β>0 with ω<0. It is easy to check that this choice of parameters (β=2,ω=-1.8) also leads to the violation of condition imposed by scalar perturbations given by the expression
(31)2Hϕ2+4ϕϕ˙2Hϕ+ϕ˙2-2ϕ2(2Hϕ˙+2H˙ϕ+ϕ¨)(2Hϕ+ϕ˙2)2>aϕ,4ϕ2(Xω/ϕ-3H2ϕ-Hϕ˙/2(2Hϕ+ϕ˙2)2)+3ϕ>0
and consequently yields the ghost instabilities for the model. However, the conditions imposed by tensor perturbation are trivially satisfied.Plots (a) and (b) show WEC versus parametersω and β for H0=0.718 and q0=-0.64.
(a)
ρ
eff
+
p
eff
≥
0
(b)
ρ
eff
≥
0 (V0≥0)


## 4.3.f(R) Gravity

The action forf(R) gravity described by S=∫-g[(Mpl2/2)f(R)+Lm]d4x can be obtained from the action (6) for
(32)K=-Mpl22(RfR-f),G3=G5=0,G4=Mpl2ϕ,ϕ=Mplf,R.
Here f(R) is an arbitrary function of the Rici scalar. Using these values in energy conditions (15), we obtain
(33)NEC:(R¨-R˙H)f′′(R)+R˙2f′′′(R)+ρm+pm≥0,WEC:ρeff+peff≥0,ρm-12(f(R)-Rf′(R))-3HR˙f′′(R)≥0,SEC:ρeff+peff≥0,(f(R)-Rf′(R))+3(R¨+R˙H)f′′(R)+3R˙2f′′′(R)+ρm+3pm≥0,DEC:ρeff+peff≥0,ρeff≥0-(5HR˙+R¨)f′′(R)-R˙2f′′′(R)-(f(R)-Rf′(R))+ρm-pm≥0,
where prime denotes the derivative with respect to R. Let us consider the example of logarithmic model in f(R) gravity defined by [57]
(34)f(R)=R[log(αR)]q-R;α<0,q>0.
In this case, the WEC (ρeff≥0) leads to
(35)-6H02(1-q0)[log(-6αH02(1-q0))]q+6H02(1-q0)+6H02(1-q0)×[(log(-6αH02(1-q0)))q-1(log(-6αH02(1-q0)))q+q(log(-6αH02(1-q0)))q-1-1]-36H04×(j0-q0-2)×[q6H02(1-q0)(log(-6αH02(1-q0))q)+q(q-1)6H02(1-q0)×(log(-6αH02(1-q0)))q-2]≥0,
where we have used R˙=-6H2(1-q) and the present values of respective parameters. Notice that we have taken only the condition ρeff≥0 as the other condition involves the present value of snap parameter s which is not correctly estimated in the literature yet. Figure 3 shows that the WEC is satisfied for a suitable range of both the parameters q and α.Figure 3
Plot shows WEC versus free parametersq and α. Here we take q0=-0.64, H0=0.718, and j0=1.41.


## 4.4. Kinetic Gravity Braiding Model

Kinetic gravity braiding is defined by the action [58]
(36)S=∫‍d4x-g(K(ϕ,X)-G3(ϕ,X)□ϕ+Mpl22R+Mpl22×[(□ϕ)2-(∇μ∇νϕ)(∇μ∇νϕ)]+LmMpl22),
where the functions K and G3 are arbitrary (while other functions are taken to be zero in action (6)). A particular choice of these functions proposed by Dvali and Turner [59] is given by K=-X and G3=cXn, where c and n are constants. For this choice of model, the respective energy conditions turn out to be
(37)NEC:1Mpl2[-2X+6ncHϕ˙Xn-2cnXnϕ¨+ρm+pm]≥0,WEC:ρeff+peff≥0,1Mpl2[-X+6ncHϕ˙Xn+ρm]≥0,SEC:ρeff+peff≥0,1Mpl2[-4X+6ncHϕ˙Xn-6cnXnϕ¨+ρm+3pm]≥0,DEC:ρeff+peff≥0,ρeff≥0,1Mpl2[6ncHϕ˙Xn+2cnXnϕ¨+ρm-pm]≥0.
By taking the power law evolution of the scalar field, WEC yields
(38)[-β2a2βH22+6ncHϕ˙(β2a2βH22)n]≥0,[-2β2a2βH22+6ncHϕ˙(β2a2βH22)n-2cn(β2a2βH22)nϕ¨]≥0.
Here we have taken the present values of the ordinary density and pressure to be zero. Using the present values of the Hubble parameter and the scale factor, WEC can be satisfied only when both the parameters n and β remain positive as indicated in Figure 4, where we have taken 0<n<1 and 1<β<2. In this case, scalar perturbations lead to the following constraints:
(39)H2(2+q)-cnHXnϕ˙a+cn2Xn-1X˙ϕ˙+cnXnϕ¨>(H-cnXnϕ˙)2,(-X+12cnHXnϕ˙+6nc(n-1)HXnϕ˙-3H2Mpl2(HMpl2-cnXnϕ˙)2)×Mpl2>-3.
It is easy to check that the energy conditions are violated for 0.6<n<1 and negative range of β (e.g., β=-10). For this choice of parameters, the previous constraints are also violated, and hence the ghost instabilities occur. However, constraints imposed by tensor perturbations are trivially satisfied as Mpl2ϕ>0.Plots (a) and (b) show WEC, namely,ρeff≥0 and ρeff+peff≥0 versus parameters n and β.
(a)
ρ
eff
≥
0
(b)
ρ
eff
+
p
eff
≥
0


## 4.5. Covariant Galilean Model

In the absence of potential, the covariant Galilean model [59] is defined by the following choice of parameters in action (6):
(40)K=-c2X,G3=c3M3X,G4=Mpl22-c4M6X2,G5=3c5M9X2,
where c2, c3, c4, and c5 are dimensionless constants, while M is constant with dimensions of mass. Using these values in (15), it follows that
(41)NEC:12((Mpl2/2)-(c4/M6)X2)×[-2Xc2+6XHϕ˙c3M3-72H2X2c4M6+60H3X2c5M9-2Xϕ¨c3M3+24HXX˙c4M6+16H˙X2c4M6-12X2×(2HH˙ϕ˙+3H2ϕ¨)c5M9-24H2X2ϕ¨c5M9+ρm+pm]≥0,WEC:ρeff+peff≥0,12((Mpl2/2)-(c4/M6)X2)×[-Xc2+6XHϕ˙c3M3-96H2X2c4M6+84H3X2ϕ˙c5M9+ρm]≥0,SEC:ρeff+peff≥0,12((Mpl2/2)-(c4/M6)X2)×[-4Xc2+6XHϕ˙c3M3-6Xϕ¨c3M3+24XH2c4M6-48H2X2c4M6+12H3X2ϕ˙c5M9-180H2X2ϕ¨c5M9-72X2HH˙ϕ˙×c5M9+48XX˙Hc4M6+48X2H˙c4M6+24HXX˙×c4M6+72H2X2×c4M6+ρm+3pm]≥0,DEC:ρeff+peff≥0,ρeff≥0,12((Mpl2/2)-(c4/M6)X2)×[6XHϕ˙c3M3+2Xϕ¨c3M3-72H2X2c4M6-48XH2c4M6+108H3X2ϕ˙c5M6-24HXX˙c4M6-16H˙X2c4M6+12X2(2HH˙ϕ˙+2H2ϕ¨)c5M9+24H2X2ϕ¨c5M9+ρm-pm]≥0.By taking the power law ansataz for scalar field and consequently, for kinetic termX, the WEC in terms of present values of the involved parameter require the following inequalities:
(42)1(1-(c4β4H04/2M6))×[-c2H02β2+3H04c3β3M3-18H06β4c4M6+15H07βc5M9-β3H04(β-1-q0)c3M3+24H06β4(β-1-q0)c4M6-16H06β4×(1+q)c4M6-3H04β4(3H04β(β-1-q0)-2H04β(1+q))c5M9-6H08β5(β-1-q0)c5M9]≥0,1(1-(c4β4H04/2M6))×[-H02β2c22+3c3H04β3M3-24H06β4c4M6+ρ0m]≥0.
Clearly, these conditions are satisfied when both G4 and terms inside the brackets are positive. Since G4>0 requires β>(7.5M6/c4)1/4, therefore a suitable choice of all these parameters yield the consistency with WEC if parameter β remains small and positive (β<7.8), while c2 remains negative as shown in Figure 5. Here we have taken -5<c2<-1 or -50<c2<-10, and 5.5<β<8. In this case, tensor perturbations suggest the following conditions for the avoidance of ghost instabilities:
(43)ℱT=Mpl2-2X2c4M6-12X2ϕ¨c5M9>0,𝒢T=Mpl2+6X2c4M6-12X2ϕ˙Hc5M9>0.
Clearly, the energy conditions are violated for β>8 with -5<c2<-1. For this choice of parameters, the previous constraints imposed by tensor perturbations are also violated, and consequently the ghost instabilities exist. However, scalar perturbations lead to very complicated expressions, so we consider only the conditions imposed by tensor perturbations.Plots (a) and (b) show WEC versus parametersc2 and β with c3=2, c4=4, c5=3, and M=2.
(a)
ρeff+peff≥0
(b)
ρ
eff
≥
0


## 5. Summary

The most general scalar-tensor theory being a combination of various DE proposals provides a vast gravitational framework for the discussion of accelerated expansion of the universe. The modified theories involve some extra degrees of freedom that are described by the models with some unknown parameters. It would be interesting to restrict these parameters on physical grounds. In cosmology, this can be done by making compatibility with local gravity tests. In a gravitational theory, energy conditions can be used as an approach to restrict these parameters. In the present paper, we consider the most general scalar-tensor gravity with the field equations involving second-order derivatives. Firstly, we have explored the effective energy-momentum tensor and its trace by inverting the generalized field equations which can be used to find the energy condition bounds for any spacetime manifold.In order to describe these conditions for specific cases, we consider flat FRW universe model with perfect fluid. By defining the effective energy density and pressure, we have expressed the strong, weak, null, and dominant energy conditions. For the sake of convenience, we have assumed that the scalar field evolves as a power of scale factor. Also, the derivative terms are removed by expressing these conditions in terms of cosmological quantities like deceleration, snap and jerk parameters. An estimation to the present values of these parameters is available in literature [53] that can be used to find the constraints on free parameters of the model. The derived energy conditions are the most general in nature involving many arbitrary functions K, G3, G4, and G5 that correspond to different DE proposals.For the application of these energy conditions, we have taken different choices of the functionsK,G3,G4, and G5 and have deduced the energy conditions for k-essence model, BD gravity, f(R) theory, kinetic gravity braiding, and covariant Galilean model. In each case, there are many free parameters. It is not possible to find their range that would be consistent with energy conditions as well. For this reason, we have specified some of these parameters and restricted the others. The results can be summarized in the following.  (i)
The WEC ink-essence model is satisfied only if the free parameters satisfy 0<M0<1, β>1.4 with arbitrary B and β<1.4 with B>10.(ii)
In the literature [40–42], using the equivalence between BD and f(R) gravity (ω=0), it has been shown that V0 should be negative. In our case (ω≠0), the WEC restricts the parameter β to be negative (the scalar field should be of contractive nature) for the positive present value of BD field potential, that is, V0>0 which is physically correct. However, the condition ρeff+peff≥0 is satisfied only when we take β<0 with arbitrary ω and β>0 with ω>0. Thus it can be concluded that expanding scalar field with negative range of BD parameter allowed for cosmic expansion is inconsistent with the WEC.(iii)
Inf(R) gravity, many authors have used the energy conditions to find the constraints on the models like f(R)=αRn or R+αRn [40–42]. In the present case, we have found the constraints on the logarithmic f(R) model. It is seen that the restrictions on free parameters α<0 and q>0 are consistent with WEC.(iv)
In kinetic gravity braiding, the WEC is satisfied for the presented model only whenβ>0 (that shows expanding scalar field) and n>0.(v)
In covariant Galilean model of DE, the WEC is satisfied when free parameters of the model satisfyβ>(7.5M6/c4)1/4 and c2<0.All these results are also shown through graphs. Further, we have determined the conditions for the avoidance of ghost instabilities using the constraints based on the scalar and tensor perturbations proposed by Kobayashi et al. [31]. It is concluded that the violation of these energy conditions leads to the occurrence of ghost and gradient instabilities in the above-mentioned cases of the most general second-order scalar-tensor theory. It would be interesting to investigate the constraints on other DE models like exponential model of f(R) gravity, other forms of potentials for BD gravity, and so forth by making them consistent with the energy conditions.



---

*Source: 253985-2013-02-26.xml*