Research Article Energy Conditions in a Generalized Second-Order Scalar-Tensor Gravity

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1 Advances in High Energy Physics Volume 03, Article ID 53985, 5 pages Research Article Energy Conditions in a Generalized Second-Order Scalar-Tensor Gravity M. Sharif and Saira Waheed Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan Correspondence should be addressed to M. Sharif; msharif.math@pu.edu.pk Received 7 November 0; Accepted 7 January 03 Academic Editor: Piero Nicolini Copyright 03 M. Sharif and S. Waheed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 secondorder 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 like k-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.. 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) [ 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 [0 ], cosmological constant [3, 4], and soforthwhichareintroducedintheusualmattercontents within the gravitational framework of general relativity (GR), while in second approach, GR framework is modified by the inclusion of some extradegrees of freedom [5, 6]. Examples of some well-known modified gravity theories include f(r) gravity [7, 8], scalar-tensor theories like Brans-Dicke (BD) gravity [9, 0], Gauss-Bonnet gravity [, ], f(t) theory [3, 4], and f(r, T) gravity [5]. Modified matter sources are rather interesting but each faces some difficulties and hence couldnotprovetobeverypromising.modifiedgravitational 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 [6 8], many inflationary models have been constructed like chaotic inflation, small/large inflation, kinflation, Dirac-Born-Infeld (DBI) inflation, single field kinflation, 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(, ). However, Lagrangian with higher-order scalar field derivatives (L = L(,, ) 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 [9, 30]. Horndeski [9] 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. [3] developed a correspondence between these Lagrangians. This theory has fascinated many researchers and much work has been done in this context, for example, [3 35].

2 Advances in High Energy Physics 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. Thusitisapointofdebatethathowonecanconstrainthe 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 in f(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. Section 3 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, weprovidesomespecificcasesofthistheoryanddiscuss the corresponding constraints. Finally, we summarize and present some general remarks.. 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 u μ and k μ 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 dθ dτ = 3 θ σ μν σ μν +ω μν ω μν R μν u μ u ν, dθ dτ = θ σ μν σ μν +ω μν ω μν R μν k μ k ν. Here R μν, θ, σ μν, and ω μν 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 condition dθ/dτ < 0. For infinitesimal distortions and () vanishing shear tensor ω μν = 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= τr μν u μ u ν = τr μν k μ k ν. It further implies that R μν u μ u ν 0, R μν k μ k ν 0. () SinceGRanditsmodificationsleadtoarelationshipof 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: R μν u μ u ν =(T μν T g μν)u μ u ν 0, R μν k μ k ν =(T μν T g μν)k μ k ν 0, where T μν and T are the energy-momentum tensor and its trace, respectively. For perfect fluid with density ρ and pressure p defined by (3) T μν =(ρ+p)u μ u ν pg μν, (4) the strong and null energy conditions, respectively, are defined by the inequalities ρ+3p 0and ρ+p 0, while the weak and dominant energy conditions are defined, respectively, by ρ 0and 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 modifiedgravity,weassumethatthetotalmattercontents oftheuniverseactlikeperfectfluid,andconsequentlythese conditions can be defined in terms of effective energy density and pressure (matter sources get modified, and we replace T μν and T in (3) byt eff μν and Teff, resp.). These conditions can be regarded as an extension of the respective conditions in GR given by [43] NEC :ρ eff +p eff 0, SEC :ρ eff +p eff 0, ρ eff +3p eff 0, WEC :ρ eff 0, ρ eff +p eff 0, DEC :ρ eff 0, ρ eff ±p eff 0. For a detailed discussion, we suggest the readers to study a recent paper [47]. The DE requires negative EoS parameter ω /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 [48]. The role of possible DE candidates with ω < was pointed out by Caldwell, who referred to null DEC violating sources as phantom components. It is argued that DE models with w such as the cosmological constant (5)

3 Advances in High Energy Physics 3 and the quintessence satisfy the NEC, but the models with ω < (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 veryshortperiodoftime[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 [3 35] S= d 4 x g [K(,X) G 3 (, X) + G 4 (, X) R By inverting (7), the Ricci tensor can be expressed in terms of effective energy-momentum tensor and its trace as follows: R μν =T eff μν g μνt eff, (8) where the effective energy-momentum tensor T eff μν and its trace T eff are T eff μν =T μν + K X μ ν G 3X ν μ (μ G 3 ν) + G 4X R μ ν + G 4XX [( ) ( α β ) ] μ ν +G 4X μ ν G 4X λ μ λ ν λ G 4X λ (μ ν) + λ G 4X λ μ ν [G 4X R λ(μ ν) λ (μ G 4X ν) ] +G 4X {( ) ( μ ν ) ( μ ν )} +G 5 (, X) G μν ( μ ν ) 6 G 5X {( ) 3 3( )( μ ν ) ( μ ν ) + ( μ α ) ( α β ) ( β μ )} + L m ], (6) G 4X R μανβ α β +G 4 μ ν +G 4 μ ν G 4X λ λ (μ ν) +G 4XX α α μ β β ν G 5X R αβ α β (μ ν) where is the scalar field, g is the determinant of the metric tensor, L m denotes the matter part of the Lagrangian, and X represents the kinetic energy term defined by X = (/) μ μ.moreover,g μν 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(, X) and G i (, X); i = 3, 4, 5 are all arbitrary functions and G ix = G i / X. Inthisaction,thetermG 3 (, X) is the Galilean term, G 4 (, X)R can yield the Einstein-Hilbert term and G 5 (, 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 G μν = Θ eff μν G = [T m μν 4 G +T μν ], (7) 4 where Θ eff μν is the modified energy-momentum tensor, Tm μν is the source of usual matter field that can be described by the perfect fluid, while T μν providesthemattersourcedueto scalar field and hence yields the source of DE, defined in the Appendix.Thescalarwaveequationforsuchmodifiedgravity has been described in the literature [3 35]. +G 5X R α(μ ν) α + G 5XR αβ α β μ ν + G 5XR μανβ α β G 5X R αλβ(μ ν) λ α β G 5X R αλβ(μ ν) λ α β + (μ [G 5X α ] α ν) (μ [G 5 ν) ] + λ [G 5 (μ ] ν) λ [ λ (G 5 λ ) α (G 5X β ) α β ] μ ν α G 5 β R α(μν)β + (μ G 5 R ν)λ λ (μg 5X ν) [( ) ( α β ) ]+ λ G 5 R λ(μ ν) α [G 5X β ] α (μ β ν) + β G 5X [ β (μ α β α (μ ] ν) α α G 5X [ μ ν β μ β ν ]

4 4 Advances in High Energy Physics + G 5X G αβ α β μ ν + G 5X μ μ + G 5X R μαμβ α β α μ α ν G 5X( ) μ ν G 5XX [( ) 3 3( )( α β ) +( α β ) 3 ] μ ν λg 5 R μν λ, T eff =K+ λ G 3 λ (G 4 XG 4 ) { G 4X α β α β +G 4XX α λ β λ α β + G 4X [( ) ( α β ) ]} +[G 4X R αβ α β λ G 4X λ ] R λg 5 λ { 6 G 5X [( ) 3 3 ( α β ) +( α β ) 3 ] + α G 5 R αβ β αg 5 α + αg 5 β α β αg 5X α X + αg 5X (9) β X α β 4 λ G 5X λ [( ) ( α β ) ] + G 5XR αβ α β G 5X R αλβρ α β λ ρ } + T + K X( μ ) G 3X ( μ ) μ G 3 μ + G 4XR( μ ) + G 4XX [( ) ( α β ) ]( μ ) +G 4X μ μ G 4X λ μ λ μ λ G 4X λ ( ) + λ G 4X λ μ μ [G 4X R λμ μ λ μ G 4X μ ] G 4X R μαμβ α β +G 4X μ μ +G 4 ( μ ) G 4X λ λ ( μ ) +G 4XX α α μ β β μ G 5X R αβ α β ( μ ) G 5X R αλβμ μ λ α β G 5X R αλβμ μ λ α β + (μ [G 5X α ] α μ) (μ [G 5 μ) ] + λ [G 5 (μ ] μ) λ [ λ (G 5 λ ) α (G 5X β ) α β ] μ μ α G 5 β R αμμβ + μ G 5 R μλ λ μg 5X μ [( ) ( α β ) ] + λ G 5 R λμ μ α [G 5X β ] α μ β μ + β G 5X [ β (μ α β α (μ ] μ) α α G 5X [ μ μ β μ β μ ] + G 5XG αβ α β ( μ ) + G 5X α μ α μ G 5X( ) μ μ G 5XX [( ) 3 3( )( α β ) +( α β ) 3 ]( μ ) λg 5 R λ + αg 5X β X α β 4 α G 5X λ [( ) ( α β ) ] + G 5XR αβ α β G 5XR αλβρ. (0) 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 with a(t) as a scale factor described by the metric ds = dt +a (t) (dx +dy +dz ). () The background fluid is taken as perfect fluid given by (4) with u μ = (, 0, 0, 0), andthenulllikevectoristakenas k μ = (, 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 [3 35] +G 5X R α(μ μ) α + G 5XR αβ α β 3H = ρeff G 4, (3H + H) = peff G 4, ()

5 Advances in High Energy Physics 5 where WEC :ρ eff +p eff 0, ρ eff 0 ρ eff = [ρm +XK X K+6X HG 3X XG 3 + 4H X (G 4X +XG 4XX ) HX G 4X 6H G 4 +H 3 X (5G 5X +XG 5XX ) G 4 [ρ m +XK X K+6XH G 3X XG 3 + 4H X (G 4X +XG 4XX ) XH 6H G 4 +H 3 X (5G 5X +XG 5XX ) 6H X (3G 5 +XG 5X )] 0, 6H X(3G 5 +XG 5X )], SEC :ρ eff +p eff 0, ρ eff +3p eff 0 p eff = [pm +K X(G 3 + G 3X ) H XG 4X 4H XG 4X 8 HXG 4X 8HX XG 4XX +( +H ) G 4 +4XG 4 +4X( H ) G 4X X(H 3 +HH +3H ) G 5X 4H X G 5XX +4HX( X HX)G 5X + [ (HX). +3H X] G 5 +4HX G 5 ]. (3) (4) Here G 4, being an arbitrary function of and X, actsasa dynamical gravitational constant, and it should be positive for any gravitational theory. Furthermore, ρ m and p m are density and pressure, respectively, for ordinary matter. We shall discuss its different forms in the next section. Using these values in (5),itcanbecheckedthattheNEC,WEC,SEC, and DEC require the following conditions to be satisfied: NEC :ρ eff +p eff 0 G 4 [(ρ m +p m )+XK X +6X HG 3X 4XG 3 + H XG 4X + 4H X G 4XX 0XH G 4X HG 4 +6H 3 XG 5X +4H 3 X G 5XX 6H X G 5X H XG 5 X G 3X G 4 [(ρ m +3p m )+XK X +K+6XH G 3X 8XG 3 6X G 3X H XG 4X + 4H X G 4XX 36XH G 4X +6H G 4 +6 G 4 H 3 XG 5X +4H 3 X G 5XX 4H X G 5X + HX XG 5X +(HX). G 5 + HXG H X G 5XX XHH 8XH G 5X + XG 4X 4HX + XG 4 4HX X G 4XX G 4X G 5X 4 HXG 4X H XG 4X 36H XG 4X ] 0, DEC :ρ eff 0, ρ eff +p eff 0, ρ eff p eff 0 G 4 [(ρ m p m )+XK X K+6XH G 3X +X G 3X + 36H XG 4X + 4H XG 4XX 4XH G 4X 0HG 4 + 4H 3 XG 5X +4H 3 X 4H XG 5 4(HX). G 5 4HX G 8H X G 5X +4H XG 4X G 5XX +8 HXG 4X +8HX XG 4XX G 4 4XG 4 4XG 4X +X(HH +3H ) G 5X 4H XG 4X 8 HX G 4X 8H XXG 4XX + G 4 +4XG +4XG 4X X (HH +3H ) G 5X +4H X G 5XX 4HXXG 5X ] 0. (5) 4H X G 5XX +4HX XG 5X +4 (HX). G 5 +4HX G 5 ] 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,

6 6 Advances in High Energy Physics deceleration, jerk, and snap parameters are, respectively, defined as H= a a, q = a H a, j = a H 3 a, s= a H 4 a. (6) In order to have a more precise picture of these conditions (5), we use the relations of time derivatives of Hubble parameter in terms of cosmological quantities like deceleration, snap, and jerk parameters as H= H ( + q), H=H 4 (s j 5q 3). H=H 3 (j + 3q + ), (7) Moreover, we assume that the scalar field evolves as a power of scale factor, that is, (t) a(t) β [5, 5], which leads to G 4 [ρ m +3p m +K+3β 3 a 3β H 4 G3X 4β H a β G 3 3β 3 a 3β H 4 (β q)g 3X 6H 4 β a β G 4X +3H 6 β 4 a 4β G 4XX 8H 4 β 3 a 3β G X +6H β a β G 4 +6H βa β (β q)g 4 H 6 β 3 a 3β G 5X +H 8 β 5 a 5β G 5XX 3H 6 β 4 a 4β G 5X +6H 6 β 4 a 4β (β q)g 5X 6H 4 β a β ( + q) G 5 + H 4 β a β (β q)g 5 +6H 4 β 3 a 3β G 5 3H 8 β 5 a 5β (β q) G 5XX +H 6 β 3 a 3β ( + q) G 5X 9H 6 β 3 a 3β (β q)g 5X βha β, βa β ( H+βH )=βa β H (β q), +6H 4 β 3 a 3β (β q)g 4X X β H a β, X β H 3 a β (β q). (8) H 4 β 3 a 3β G 4X +6H β a β G 4 H 6 β 4 a 4β (β q)g 4XX + H 4 β a β Here β is a nonzero parameter (β =0yields constant scalar field, β > 0 yields expanding scalar field, and β < 0 corresponds to contracting scalar field). Clearly, remains as a positive quantity. Introducing these quantities in the energy conditions given by (5), it follows that G 4 [K X G 3, +6H G 4X 6H G 5 +G 4 ] β H a β +[3HG 3X 0HG 4X +3H 3 G 5X +HG 5 ] β 3 H 3 a 3β +(6H G 4XX 4H G 5X ) β 4 H 4 a 4β H βa β G 4 +H 8 β 5 a 5β G 5XX β 3 H 4 a 3β (β q)g 3X 4H 4 a β β (β q) G 4X +4H 4 a ββ ( + q) G 4X 4H 6 β 4 a 4β (β q)g 4XX +H a β (β q)βg 4 +β 3 H 4 a 3β (β q)g 4X +H 6 ( + q) β 3 a 3β G 5X 3H 6 β 3 a 3β (β q)g 5X H 8 β 5 a 5β (β q)g 5XX +H 6 β 4 a 4β (β q)g 5X +4H 4 β a β (β q)g 5 H 4 ( + q) β a β G 5 +(ρ m +p m ) 0, (+q)g 4X H 4 β a β (β q)g 4X 8H 4 β a β G 4X +β a β H K X ] 0, [H β a β K G X K+3H 4 β 3 a 3β G 3X H β a β G H 4 β a β G 4X +6H 6 β 4 a 4β G 4XX 6H 4 β 3 a 3β G X 6H βa β G 4 +5H 6 β 3 a 3β G 5X +H 8 β 5 a 5β G 5XX 9H 4 β a β G 5 3H 6 β 4 a 4β G 4X +ρ m ] 0, G 4 [ρ m p m +β H a β K X K+3H 4 β 3 a 3β G 3X +β H 3 a β (β q ) G 3X + 8β H 4 a β G 4X + H 4 β a β G 4XX H 4 β 3 a 3β G 4X 0H βa β G 4 +7H 6 β 3 a 3β G 5X +H 8 β 5 a 5 βg 5XX H 4 β a β G 5 4H 4 β a β G 5 +H 4 β a β ( + q) G 5 H 4 β 3 a 3β G 5 4H 4 β a β G X +4 H 4 β a β (β q )G 4X 4H 4 β a β ( + q) G 4X +4H 6 β 4 a 4β (β q)g 4XX H a β β(β q)g 4

7 Advances in High Energy Physics 7 H β a β G 4 H 4 β 3 a 3β (β q )G 4X H 6 β 3 a 3β (+q) G 5X +3H 6 β 3 a 3β (β q)g 5X +H 8 β 5 a 5β (β q)g 5XX H 6 β 4 a 4β (β q)g 5X ] 0. (9) 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 G 4 is a positive quantity. However, we have discussed earlier that G 4, being a gravitational constant, would be positive in allcases(ifitisnotso,thenweimposethisconditionand restrict the free parameters). Clearly, these conditions are only dependent on the Hubble, deceleration parameters and arbitrary functions, namely, K, G 3, G 4,andG 5.Oncethese 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 H 0 = 0.78, the scale factor a 0 =, 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, G 3,andG 5 zero with G 4 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 scalartensor theory based on tensor and scalar perturbations is available in literature [3]. By introducing perturbed metric, it has been shown that for the avoidance of ghost and gradient instabilities, the tensor perturbations suggest while scalar perturbations impose F S = d a dt ( a Θ G T ) F T >0, G S = Σ Θ G T +3G T >0, () where the quantities Σ and Θ are defined in [3]. 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.. k-essence Models in General Relativity. The k-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(, X) + (M pl /)R + L m ]d 4 x with the following choice of the functions: K=K(,X), G 4 = M pl, G 3 =G 5 =0, () where M pl is Planck mass. The k-essence models can be classified into three forms: (i) K(, X) = K (X) (Kinetic case), (ii) K(, X) = K (X)B(), (iii) K(, X) = K (X) + B(). For the choice of arbitrary functions given by (), the energy conditions (5)take the following forms: NEC : M pl WEC :ρ eff +p eff 0, M pl SEC :ρ eff +p eff 0, M pl [XK X +ρ m +p m ] 0, [XK X K+ρ m ] 0, [XK X +K+ρ m +3p m ] 0, DEC :ρ eff +p eff 0, ρ eff 0, M pl [XK X K+ρ m p m ] 0. (3) Here K(, X) is arbitrary. In order to see how the function K(, X) can be constrained by using the previous energy conditions, we choose aparticularmodel ofk-essence [55] as follows: F T =[G 4 X( G 5X +G 5 )] > 0, G T =[G 4 XG 4X X(H G 5X G 5 )] > 0, (0) K(,X)= [C C B ]+ (C +C )X + M4 0 (X ), (4)

8 8 Advances in High Energy Physics where C, C, B,andM 0 are arbitrary constants. In this case, WEC requires the following conditions: M pl M pl [ X(C +C )+XM 4 0 (X ) M4 0 (X ) (C C B )+ρ m ] 0, [X (C +C )+M 4 0 X (X ) +ρm +p m ] 0, (5) where M pl > 0. For the interpretation of the previous inequalities, we consider the power law ansatz for the scalar field a β ; β =0, which further yields X (β a β H /). Consequently, the WEC (5)turnsouttobe [ β a β H M4 0 (β a β H [ β a β H ( β a β H (C +C )+ β a β H M 4 a β H 0 (β ) ) (C C B )+ρ m ] 0, (C +C )+M 4 β a β H 0 )+ρ m +p m ] 0. (6) (7) It is difficult to find the admissible ranges of all constants C, C, B, M 0,andβ 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, ρ m > 0 and ρ m +p m > 0. Moreover, we take the present value of Hubble parameter and choose some particular values of the constants C and C to find the ranges of B, M 0,andβ, consistent with the WEC. It turns out from the graphs that we can take the parameter β as follows β >.4, 0 < β <.4 and β<0, while B and can be positive or negative. For the consistency of condition (7), we restrict the parameter M 0 as 0<M 0 <.Fromthecondition(6),itcanbeobservedthat energy conditions are satisfied only when we take β >.4, with arbitrary B and β <.4 with B>0only. Other choices of these parameters lead to violation of WEC. Figures (a) and (b) show that WEC is satisfied with these fixed input parameters by taking β >.4, 0<M 0 <,and0<b<. 4.. Brans-Dicke Theory. Brans-Dicke gravity with action g[(m pl Xω/) V() + (/)M pl R + L m ]d 4 x can be defined by the following choice of functions: K= M plxω V(), G 3 =G 5 =0, G 4 = M pl. (8) 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() instead of in G 4.Inthiscase,theenergy conditions take the form NEC : [ M pl M plω H M pl + M pl + ρm +p m ] 0, WEC :ρ eff +p eff 0, [ ωm pl M pl SEC :ρ eff +p eff 0, [ M pl M plω + V() V () 3H M pl + ρm ] 0, + 3H +3 M pl + ρm +3p m ] 0, DEC :ρ eff +p eff 0, ρ eff 0, [ V() M pl 5 H M pl M pl + ρm p m ] 0. M pl (9) Asuitablechoiceofthefieldpotentialhasalwaysbeena matter of debate in BD gravity. We use power laws for the scalar field and potential as = 0 a β and V = V 0 m, respectively, where m and β are nonzero parameters. Consequently, the WEC restricts the parameters as V 0 3H 0 β ω β H 0, β H 0 ω H 0 β+h 0 aβ β a β βh 0 ( + q 0) 0. (30) For accelerated expanding universe, the observed range of BD parameter is < ω < 3/ [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 (a) shows that the condition ρ eff 0 leads to β < 0,which further yields positive present value of the BD field potential V 0. Moreover, the second condition will be satisfied if we take β<0witharbitrary ω and β>0with ω>0only. Figure (b) indicates that ρ eff +p eff 0issatisfied for a particular choice of β<0and <ω<.5. Clearly, the energy condition ρ eff +p eff 0is violated for β>0with ω<0.itiseasy tocheckthatthischoiceofparameters(β=,ω=.8)

9 Advances in High Energy Physics 9 ρ eff + p eff β B 0. ρ eff β B (a) ρ eff +p eff 0 (b) ρ eff 0 Figure : Plot (a) shows ρ eff +p eff 0versus β and M 0. Plot (b) represents ρ eff 0versus parameters β and B with M 0 = 0..Inbothcases, we take the present values of cosmological parameters with C =, C =. ρ eff + p eff β ω.9 0 V β ω.9 (a) ρ eff +p eff 0 (b) ρ eff 0(V 0 0) Figure : Plots (a) and (b) show WEC versus parameters ω and β for H 0 = 0.78 and q 0 = also leads to the violation of condition imposed by scalar perturbations given by the expression 4.3. f(r) Gravity. The action for f(r) gravity described by S= g[(m pl /)f(r) + L m]d 4 x can be obtained from the action (6)for H +4 (H+ H + ) H + (H + ) >a, 4 ( Xω/ 3H H/ (H + ) )+3>0 (3) K= M pl (Rf R f), G 3 =G 5 =0, G 4 = M pl, = M plf,r. (3) Here f(r) is an arbitrary function of the Rici scalar. Using thesevaluesinenergyconditions(5), we obtain NEC :( R RH) f (R) + R f (R) +ρ m +p m 0, and consequently yields the ghost instabilities for the model. However, the conditions imposed by tensor perturbation are trivially satisfied. WEC :ρ eff +p eff 0, ρ m (f (R) Rf (R)) 3H Rf (R) 0,

10 0 Advances in High Energy Physics SEC :ρ eff +p eff 0, ρ eff 0 (f (R) Rf (R))+3( R+ RH) f (R) +3 R f (R) +ρ m +3p m 0, DEC :ρ eff +p eff 0, ρ eff 0 (5HR+ R) f (R) R f (R) (f(r) Rf (R))+ρ m p m 0, (33) ρ eff q α.8 where prime denotes the derivative with respect to R. Letus consider the example of logarithmic model in f(r) gravity defined by [57] Figure 3: Plot shows WEC versus free parameters q and α.herewe take q 0 = 0.64, H 0 = 0.78,andj 0 =.4. f (R) =R[log (αr)] q R; α<0, q>0. (34) 4.4. Kinetic Gravity Braiding Model. Kinetic gravity braiding is defined by the action [58] In this case, the WEC (ρ eff 0)leads to S= d 4 x g (K (, X) G 3 (, X) + M pl R+M pl 6H 0 ( q 0)[log ( 6αH 0 ( q 0))] q +6H 0 ( q 0) +6H 0 ( q 0) [(log ( 6αH 0 ( q 0))) q +q(log ( 6αH 0 ( q 0))) q ] 36H 4 0 (j 0 q 0 ) q [ 6H0 ( q 0) (log ( 6αH 0 ( q 0)) q ) + q(q ) 6H 0 ( q 0) (log ( 6αH 0 ( q 0))) q ] 0, (35) where we have used R= 6H ( q) and the present values of respective parameters. Notice that we have taken only the condition ρ eff 0as the other condition involves the present value of snap parameter s which is not correctly estimated in theliteratureyet.figure 3 showsthatthewecissatisfiedfor a suitable range of both the parameters q and α. [( ) ( μ ν ) ( μ ν )] + L m ), (36) where the functions K and G 3 are arbitrary (while other functionsaretakentobezeroinaction(6)). A particular choice of these functions proposed by Dvali and Turner [59] is given by K = X and G 3 = cx n,wherecand n are constants. For this choice of model, the respective energy conditions turn out to be NEC : M pl WEC :ρ eff +p eff 0, SEC :ρ eff +p eff 0, M pl [ X + 6ncHX n cnx n +ρ m +p m ] 0, M pl [ X+6ncH X n +ρ m ] 0, [ 4X+6ncHX n 6cnX n +ρ m +3p m ] 0, DEC :ρ eff +p eff 0, ρ eff 0, M pl [6ncHX n +cnx n +ρ m p m ] 0. (37)

11 Advances in High Energy Physics ρ eff β n 0. ρ eff + p eff β n (a) ρ eff 0 (b) ρ eff +p eff 0 Figure 4: Plots (a) and (b) show WEC, namely, ρ eff 0and ρ eff +p eff 0versus parameters n and β. By taking the power law evolution of the scalar field, WEC yields [ β a β H [ β a β H +6ncH( β a β H +6ncH( β a β H n ) n ) ] 0, n ) (38) cn( β a β H ] 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, wherewehavetaken0<n<and < β <.Inthiscase,scalarperturbationsleadtothe following constraints: H ( + q) cnhxn +cn X n X a +cnx n >(H cnx n ), ( X + cnhxn +6nc(n ) HX n 3H M (39) pl (HMpl cnxn ) ) M pl > 3. It is easy to check that the energy conditions are violated for 0.6 < n < and negative range of β (e.g., β = 0). 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 M pl > Covariant Galilean Model. In the absence of potential, the covariant Galilean model [59] is defined by the following choice of parameters in action (6): K= c X, G 3 = c 3 M 3 X, G 5 = 3c 5 M 9 X, G 4 = M pl c 4 M 6 X, (40) where c, c 3, c 4,andc 5 are dimensionless constants, while M is constant with dimensions of mass. Using these values in (5), it follows that NEC : ((M pl /) (c 4/M 6 )X ) [ Xc +6XH c 3 M 3 7H X c 4 M H 3 X c 5 M 9 X c 3 M 3 + 4HX X c 4 M 6 +6HX c 4 M 6 X (HH +3H ) c 5 4H X c 5 M 9 +ρm +p m ] 0, WEC :ρ eff +p eff 0, ((Mpl /) (c 4/M 6 )X ) [ Xc +6XH c 3 M 3 96H X c 4 M 6 +84H 3 X c 5 M 9 +ρm ] 0, M 9

12 Advances in High Energy Physics SEC :ρ eff +p eff 0, ((Mpl /) (c 4/M 6 )X ) [ 4Xc +6XH c 3 M 3 6X c 3 M 3 + 4XH c 4 M 6 48H X c 4 M 6 + H 3 X c 5 M 9 80H X c 5 M 9 7X HH c 5 M X XH c 4 M X H c 4 M 6 + 4HX X c 4 M 6 + 7H X c 4 M 6 +ρm +3p m ] 0, DEC : ρ eff +p eff 0, ρ eff 0, ((Mpl /) (c 4/M 6 )X ) [6XH c 3 M 3 +X c 3 M 3 7H X c 4 c 4 M 6 48XH + 08H 3 X c 5 M 6 4HX X c 4 M 6 6 HX c 4 M 6 M 6 + X (HH +H ) c 5 M 9 + 4H X c 5 M 9 +ρ m p m ] 0. (4) By taking the power law ansataz for scalar field and consequently, for kinetic term X, the WEC in terms of presentvaluesoftheinvolvedparameterrequirethefollowing inequalities: ( (c 4 β 4 H0 4/M6 )) [ c H 0 β + 3H4 0 c 3β 3 M 3 8H6 0 β4 c 4 M 6 + 5H7 0 βc 5 M 9 β 3 H 4 0 (β q 0) c 3 M 3 + 4H 6 0 β4 (β q 0 ) c 4 M 6 6H6 0 β4 (+q) c 4 M 6 3H 4 0 β4 (3H 4 0 β(β q 0) H 4 0 β(+q)) c 5 M 9 6H 8 0 β5 (β q 0 ) c 5 M 9 ] 0, ( (c 4 β 4 H 4 0 /M6 )) [ H 0 β c + 3c 3H 4 0 β3 M 3 4H6 0 β4 c 4 M 6 +ρ m 0 ] 0. (4) Clearly, these conditions are satisfied when both G 4 and terms inside the brackets are positive. Since G 4 >0requires β > (7.5M 6 /c 4 ) /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 c remains negative as shown in Figure 5. Herewehave taken 5 < c < or 50 < c < 0, and 5.5 < β < 8. In this case, tensor perturbations suggest the following conditions for the avoidance of ghost instabilities: F T =M c 4 pl X M 6 X c 5 M 9 >0, G T =M c 4 pl +6X M 6 X H c 5 M 9 >0. (43) Clearly, the energy conditions are violated for β > 8 with 5 < c <. 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. 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 extradegrees 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 haveexpressedthestrong,weak,null,anddominantenergy 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, G 3, G 4,andG 5 that correspond to different DE proposals. For the application of these energy conditions, we have taken different choices of the functions K, G 3,G 4,andG 5 and have deduced the energy conditions for k-essence model, BD gravity, f(r) theory, kinetic gravity braiding, and covariant

13 Advances in High Energy Physics 3 ρ eff + p eff β c ρ eff β c (a) ρ eff +p eff 0 (b) ρ eff 0 Figure 5: Plots (a) and (b) show WEC versus parameters c and β with c 3 =, c 4 =4, c 5 =3,andM=. 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 in k-essence model is satisfied only if the free parameters satisfy 0 < M 0 <, β >.4 with arbitrary B and β <.4 with B>0. (ii) In the literature [40 4], using the equivalence between BD and f(r) gravity (ω = 0), ithasbeen shown that V 0 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, V 0 > 0 whichisphysicallycorrect.however,the condition ρ eff +p eff 0issatisfiedonlywhenwetake β<0witharbitrary ω and β>0with 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) In f(r) gravity, many authors have used the energy conditions to find the constraints on the models like f(r) = αr n or R+αR n [40 4]. In the present case,wehavefoundtheconstraintsonthelogarithmic f(r) model. It is seen that the restrictions on free parameters α<0and q>0are consistent with WEC. (iv) In kinetic gravity braiding, the WEC is satisfied for the presented model only when β > 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 satisfy β > (7.5M 6 /c 4 ) /4 and c <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. [3]. 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. Appendix T () μν = K X μ ν + Kg μν G 3X ν μ (μ G 3 ν) + g μν λ G 3 λ + G 4XR μ ν + G 4XX [( ) ( α β ) ] μ ν +G 4X μ ν G 4X λ μ λ ν λ G 4X λ (μ ν) + λ G 4X λ μ ν g μν (G 4 XG 4 ) g μν [ G 4X α β α β +G 4XX α λ β λ α β + G 4X [( ) ( α β ) ]] [G 4X R λ(μ ν) λ (μ G 4X ν) ] +g μν [G 4X R αβ α β λ G 4X λ ] G 4X R μαβνβ α β +G 4 μ ν +G 4 μ ν G 4X λ λ (μ ν) +G 4XX α α μ β β ν G 5X R αβ α β (μ ν) +G 5X

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