Energy Conditions in f(G) Modified Gravity with Non-minimal Coupling to Matter
aa r X i v : . [ phy s i c s . g e n - ph ] N ov Energy Conditions in f ( G ) Modified Gravity withNon-minimal Coupling to Matter
A. Banijamali a , B. Fazlpour b and M. R. Setare c a Department of Basic Sciences, Babol University of Technology, Babol, Iran b Babol Branch, Islamic Azad University, Babol, Iran c Department of Science, Campus of Bijar, University of KurdistanBijar, IRAN.
Abstract
In this paper we study a model of modified gravity with non-minimal coupling be-tween a general function of the Gauss-Bonnet invariant, f ( G ), and matter Lagrangianfrom the point of view of the energy conditions. Such model has been introducedin Ref. [21] for description of early inflation and late-time cosmic acceleration. Wepresent the suitable energy conditions for the above mentioned model and then, weuse the estimated values of the Hubble, deceleration and jerk parameters to apply theobtained energy conditions to the specific class of modified Gauss-Bonnet models. PACS numbers:
Keywords:
Modified Gauss-Bonnet gravity, Non-minimal coupling, Energy conditions [email protected] [email protected] [email protected] Introduction
Nowadays it is strongly believed that the universe is experiencing an accelerated expansion,and this is supported by many cosmological observations [1-4]. This accelerated expansioncan be explained in terms of the so called dark energy ( for reviews see [5]) in the frameworkof general relativity or by modification of general relativity.The simplest type of modified gravity models is well known as f ( R ) gravity where the Ricciscalar in the Einstein-Hilbert action is replaced by a general function of the scalar curvature(see [6, 7] for reviews).There are also other modified gravity models which are the generalization of f ( R ) gravityand among them, the modified Gauss-Bonnet (GB) gravity i.e. f ( G ) gravity, is more inter-esting [8, 9]. The GB combination, G , is a topological invariant in four dimensions, so inorder to play some roles in the field equations, one needs either couple GB term to a scalarfield like f ( φ ) G , or choose f ( G ) gravity where f is an arbitrary function of G .If we compare f ( G ) gravity with other theories of modified gravity we find some advan-tages in the Gauss-Bonnet gravity. For example in the context of f ( G ) gravity there existsa de-Sitter point that can be used for cosmic acceleration [8, 9, 21]. Note that in f ( G )gravity, there are no problems [8, 9] with the Newton law, instabilities and the anti-gravityregime. In comparing with the simple f ( R ) modified gravity we should mentioned that itis not generally easy to construct viable f ( R ) models that are consistent with cosmologicaland local gravity constraints. The main reason for this is that f ( R ) gravity gives rise to astrong coupling between DE and a non-relativistic matter in the Einstein frame [37]. How-ever there is no conformal transformation separating G from scalar field, unlike the f ( R )theory to obtain and Einstein frame action with a canonical scalar field coupled to matter.Furthermore the f ( G ) models might be less constrained by local gravity constraints relativeto the f ( R ) models. The main reason is that even in the vacuum spherically symmetricbackground the Gauss-Bonnet scalar takes a non-vanishing value. This property is differentfrom f ( R ) gravity in which the Ricci scalar R vanishes in the vacuum spherically symmetricbackground [38]. Moreover, In considering alternative higher-order gravity theories, one isliable to be motivated in pursuing models consistent and inspired by several candidates ofa fundamental theory of quantum gravity. Indeed, motivations from string/M- theory pre-dict that scalar field couplings with the Gauss- Bonnet invariant, G , are important in theappearance of non-singular early time cosmologies [33]. In summary, modified f ( G ) gravityrepresents a quite interesting gravitational alternative to dark energy with more freedom ifcompare with f ( R ) gravity (for recent review see [14]).The f ( G ) gravity is an enrichment theory of modified gravity. A number of its abilities areas follows: it has the possibility to describe the inflationary era, a transition from a deceler-ation phase to an acceleration phase, crossing the phantom divide line and passing the solarsystem tests for a reasonable defined function f [10-13].Moreover, a gravitational source of the inflation and dark energy may be the non-minimalcoupling of some geometrical invariants function with matter Lagrangian. Such non-minimalmodified gravity has been introduced in Refs. [15, 16] for the study of gravity assisted darkenergy occurrence. It may be also applied for realization of dynamical cancellation of cos-2ological constant [17]. The viability criteria for such theory was recently discussed in Refs.[18-20]. Non- minimal coupling of f ( G ) modified gravity with matter Lagrangian has beeninvestigated in [21]. It is shown [21], that such a model can easily unify the early inflationwith late-time cosmic acceleration for the special choise of gravitational functions. Clearly,as any model in f ( G ) theory, there are particular conditions which have to be satisfied inorder to ensure that the model is viable and physically meaningful [22-24]. In the presentwork we study the above mentioned model of f ( G ) gravity with non-minimal coupling tomatter from the viewpoint of the energy conditions.Furthermore, the energy conditions are essential for studying the singularity theorems aswell as the theorems of black hole thermodynamics. For example the Hawking-Penrose sin-gularity theorems invoke the weak and strong energy conditions, whereas the proof of thesecond law of black hole thermodynamics needs the null energy condition [25]. Various formsof energy conditions namely, strong, weak, dominant and null energy conditions are obtainedusing Raychaudhuri equation along with attractiveness property of gravity [25, 26]. Energyconditions have been widely studied in the literature. Phantom field potential [27], expan-sion history of the universe [28] and evolution of the deceleration parameter [29] have beeninvestigated using classical energy conditions of general relativity. Energy conditions in thecontext of f ( R ) gravity have been derived in [30] and following the formalism developed in[30], several authors have studied some cosmological issues in modified f ( R ) gravity fromthis point of view (see for example [31, 32]). Also, appropriate energy conditions for the f ( R ) gravity non-minimally coupled with matter Lagrangian has been investigated in [22].In a recent paper [33], the general energy conditions for modified Gauss-Bonnet gravity or f ( G ) gravity have been presented and viability of specific realistic forms of f ( G ) proposedin [21] have been analyzed by imposing the weak energy condition.In this paper we generalize procedure developed in [33] for f ( G ) theories to the gravity modelwith non-minimal coupling of f ( G ) to the matter. Then, we use the estimated values of theHubble, deceleration and jerk parameters to apply the obtained energy conditions to thespecific class of models.An outline of this paper is as follows: in the next section we review the field equations ofmodified f ( G ) theory non-minimally coupled with matter. In section 3 we present the suit-able energy conditions for such a model. In section 4 we examine the weak energy conditionfor two specific class of f ( G ) models. Section 5 is devoted to our conclusions. Our starting action for modified Gauss-Bonnet gravity non-minimally coupled with matteris as follows [21]: S = Z d x √− g (cid:2) κ R + f ( G ) L m (cid:3) , (1)where g is the determinant of the metric tensor g µν , R is the Ricci scalar and f ( G ) isa general function of Gauss-Bonnet invariant G . One notes that there is a non-minimal3oupling between f ( G ) gravity and the matter Lagrangian L m in (1).The theory of kind (1) has been previously investigated in order to unify inflation and late-time cosmic acceleration [21].The Gauss-Bonnet term is given by, G = R − R µν R µν + R µνρσ R µνρσ . (2)Varying the action (1) with respect to g µν leads to:0 = 12 κ (cid:0) R µν − g µν R (cid:1) + 2 f ′ ( G ) L m R R µν − f ′ ( G ) L m R µ ρ R νρ + 2 f ′ ( G ) L m R µρσλ R ν ρσλ + 4 f ′ ( G ) L m R µρσν R ρσ + 2 (cid:16)(cid:0) g µν ⊔⊓ − ∇ µ ∇ ν (cid:1) f ′ ( G ) L m (cid:17) R + 4 (cid:0) ∇ ρ ∇ µ f ′ ( G ) L m (cid:1) R νρ + 4 (cid:0) ∇ ρ ∇ ν f ′ ( G ) L m (cid:1) R µρ + 4 (cid:0) ⊔⊓ f ′ ( G ) L m (cid:1) R µν − (cid:0) g µν ∇ λ ∇ ρ f ′ ( G ) L m (cid:1) R λρ + 4 (cid:0) ∇ ρ ∇ λ f ′ ( G ) L m (cid:1) R µρνλ + f ( G ) T µν , (3)where, T µν = 2 √− g δδg µν (cid:16) Z d x √− g L m (cid:17) , (4)is the energy-momentum tensor of the ordinary matter and prime stands for derivative withrespect to G . For a flat Friedman-Robertson-Walker (FRW) metric with scale factor a ( t ), ds = − dt + a ( t )( dr + r d Ω ) , (5)and assuming matter as a perfect fluid, the field equation (3) has the following simple forms,3 κ H + 24 H ddt (cid:0) f ′ ( G ) L m (cid:1) − Gf ′ ( G ) L m = f ( G ) ρ, (6)1 κ (cid:0) H + 3 H (cid:1) + 8 H d dt (cid:0) f ′ ( G ) L m (cid:1) + 16 H ddt (cid:0) f ′ ( G ) L m (cid:1) ( ˙ H + H ) − Gf ′ ( G ) L m = − f ( G ) p, (7)where ρ and p are the energy density and pressure, respectively. The overdot denotes a timederivative.In order to write the field equations as those in general relativity, one can define the effectiveenergy density and pressure, ρ eff = 3 κ H , (8)and p eff = − κ (cid:0) H + 3 H (cid:1) , (9)where ρ eff and p eff are the effective energy density and pressure, respectively and given by, ρ eff = f ( G ) h G f ′ ( G ) f ( G ) L m − H ˙ G f ′′ ( G ) f ( G ) L m + ρ i , (10)4nd p eff = f ( G ) " − G f ′ ( G ) f ( G ) L m +8 H (cid:16) ¨ G f ′′ ( G ) f ( G ) + ˙ G f ′′′ ( G ) f ( G ) (cid:17) L m +16 H ˙ G f ′′ ( G ) f ( G ) (cid:0) ˙ H + H (cid:1) L m + p , (11)where, we have used the equations (6) and (7). The energy conditions originate from the Raychaudhury equation together with the attrac-tiveness of the gravity [26]. The Raychaudhury equation in the case of a congruence of nullgeodesics defined by the vector field k µ is given by, dθdτ = − θ − σ µν σ µν + ω µν ω µν − R µν k µ k ν , (12)where R µν is the Ricci tensor, θ is the expansion parameter, σ µν and ω µν are the shear andthe rotation associate to the congruence respectively.For any hypersurface of orthogonal congruences ( ω µν = 0), the condition of attractivenessof gravity ( dθdτ <
0) yields to R µν k µ k ν ≥ σ µν σ µν ≥ T µν k µ k ν ≥ R µν k µ k ν is straight-forward. So, using the effective gravitation field equations, the energy conditions in the caseof our model are given by, ρ eff + p eff ≥ f or null energy condition ( N EC ) , (13) ρ eff ≥ and ρ eff + p eff ≥ f or weak energy condition ( W EC ) , (14) ρ eff + 3 p eff ≥ and ρ eff + p eff ≥ f or strong energy condition ( SEC ) , (15)and ρ eff ≥ and ρ eff ± p eff ≥ f or dominant energy condition ( DEC ) . (16)Inserting equations (10) and (11) into equations (13)-(16) lead to the following respectiveforms, N EC : ρ + p − H ˙ G f ′′ ( G ) f ( G ) L m + 8 H (cid:16) ¨ G f ′′ ( G ) f ( G ) + ˙ G f ′′′ ( G ) f ( G ) (cid:17) L m + 16 H ˙ H ˙ G f ′′ ( G ) f ( G ) L m ≥ , (17)5 EC : ρ + G f ′ ( G ) f ( G ) L m − H ˙ G f ′′ ( G ) f ( G ) L m ≥ , ρ eff + p eff ≥ . (18) SEC : ρ + 3 p − G f ′ ( G ) f ( G ) L m + 24 H ˙ G f ′′ ( G ) f ( G ) L m + 24 H (cid:16) ¨ G f ′′ ( G ) f ( G ) + ˙ G f ′′′ ( G ) f ( G ) (cid:17) L m + 48 H ˙ H ˙ G f ′′ ( G ) f ( G ) L m ≥ , ρ eff + p eff ≥ . (19) DEC : ρ − p + 2 G f ′ ( G ) f ( G ) L m − H ˙ G f ′′ ( G ) f ( G ) L m − H (cid:16) ¨ G f ′′ ( G ) f ( G ) + ˙ G f ′′′ ( G ) f ( G ) (cid:17) L m − H ˙ H ˙ G f ′′ ( G ) f ( G ) L m ≥ , ρ eff + p eff ≥ , ρ eff ≥ . (20)In order to analyze the model of type (1) with non-minimal coupling between f ( G ) gravityand matter from the point of view of energy conditions, we use the standard terminologyin studying energy conditions for modified gravity theories. To this end, according to theHubble parameter H = ˙ aa we define the deceleration ( q ), jerk ( j ), and snap ( s ) parametersas, q = − H ¨ aa , j = 1 H ... aa , and s = 1 H .... aa . (21)In terms of above parameters, time derivatives of the Hubble parameter can be expressedas, ˙ H = − H (1 + q ) , (22)¨ H = H ( j + 3 q + 2) , (23)... H = H ( s − j − q − . (24)The Gauss-Bonnet invariant in FRW background is as G = 24 H (cid:0) H + ˙ H (cid:1) . So, G and itstime derivatives can be written as, G = − H q , (25)˙ G = 24 H (cid:0) q + 3 q + j (cid:1) , (26)¨ G = − H (cid:0) q + 22 q + 6 qj + 15 q + 4 j − s − (cid:1) . (27)By inserting relations (22)-(27) into equations (17)-(20) one can obtain the following formsof energy conditions ρ + p − H (cid:0) q + 34 q + 8 q j + 24 q + 7 j − s − (cid:1) f ′′ ( G ) f ( G ) L m + 4608 H (cid:0) q + 3 q + j (cid:1) f ′′′ ( G ) f ( G ) L m ≥ , (28)6or null, ρ − H q f ′ ( G ) f ( G ) L m − H (cid:0) q + 3 q + j (cid:1) f ′′ ( G ) f ( G ) L m ≥ , (29)for weak, ρ + 3 p + 48 H q f ′ ( G ) f ( G ) L m − H (cid:0) q + 30 q + 8 q j + 18 q + 5 j − s − (cid:1) f ′′ ( G ) f ( G ) L m + 13824 H (cid:0) q + 3 q + j (cid:1) f ′′′ ( G ) f ( G ) L m ≥ , (30)for strong and ρ − p − H q f ′ ( G ) f ( G ) L m + 192 H (cid:0) q + 22 q + 8 q j + 6 q + j − s − (cid:1) f ′′ ( G ) f ( G ) L m − H (cid:0) q + 3 q + j (cid:1) f ′′′ ( G ) f ( G ) L m ≥ , (31)for dominant energy condition respectively. The subscript 0 stands for the present value ofquantities. These forms of energy conditions are suitable to impose bounds on a given f ( G )model by using the estimate values of the q , j and s . Now, let us see how the above energy conditions work for specific type of f ( G ) models. Inwhat follows we focus just on WEC inequality (29). Our reason apart from the simplicityis that all other energy conditions depend on the present value of snap parameter s and asmentioned in [30] until now no reliable measurement of this parameter has been reported.Also, for simplicity we only examine the case in which p = ρ = 0, although this is not aphysically interesting case, but this is easily corrected, since one can always add a positiveenergy density or pressure from matter and/or radiation to any model satisfying the WEC, and it will still satisfy the WEC [33]. Case 1:
Our first example is coming from Ref. [9], f ( G ) = α G n . (32)It has been shown in [9] that this type of f ( G ) theory can produce quintessence, phantomor cosmological constant cosmology and has the possibility of realizing transition from thedeceleration to the acceleration era. It was shown in [39] that the model of type (32) with n <
0, is not cosmologically viable because of separatrices between radiation and dark energydominations. In [38] it was found that the model (32) with n >
0, can be consistent with7olar system tests for n ≤ .
074 if the Gauss-Bonnet term is responsible for dark energy.Inserting f ( G ) from (32) and their derivatives into equation (29) for WEC yields to( − n α [ An − ( A + 1) n ] L m ≤ , (33)where A = q +3 q + jq .The following estimated values of deceleration and jerk parameters [34, 35]; q = − . ± . j = 2 . +0 . − . allow us to further analyze equation (33). Note that the roots of[ An − ( A + 1) n ] = 0 are (0 , A ), so it has positive values for 0 > n > .
63 and negativevalues for 0 ≤ n ≤ .
63. In addition, the condition required for attractiveness of gravity i.e. f ( G ) > α ( − n (cid:0) . H (cid:1) n >
0. Now, using the above explanations, one can studythe inequality (33) as follows: (i) for α >
0, one concludes from the attractivness property of gravity that n should be evenand then the equation (33) tell us that 0 ≤ n ≤ .
63, and therefore there is no allowed valuefor n . (ii) for α < n should be odd and equation (33) leads to 0 > n > .
63 so the allowed setsof n s are n = { , , , ... } and n = {− , − , − , ... } . Case 2:
In the second example we consider a class of models as follows: f ( G ) = αG n (cid:0) βG m (cid:1) . (34)The models of kind (34) have been proposed in Ref. [21] in order to address the late-timecosmic acceleration. It has been shown in [40] that the model (34) is consistent with localtests and cosmological bounds. The typical property of such theory is the presence of theeffective cosmological constant epochs in such a way that early-time inflation and late-timecosmic acceleration are naturally unified within single model. It is shown that classicalinstability does not appear here and Newton law is respected. Some discussion of possibleanti-gravity regime appearance and related modification of the theory is also done in [40].Furthermore, it is shown in [36] that four types of future singularities can be cured in sucha models if one considers n > m < f ( G ) and their derivatives into equation (29), WEC reads h α ( − n a n + αβ ( − n + m ( a n + a m + 2 Amn ) (cid:0) H q (cid:1) m i L m ≤ , (35)where a n = An − ( A + 1) n and a n + a m + 2 Amn = A ( n + m ) − ( A + 1)( n + m ). For thenext purposes, we mention the roots of a n = 0 and a n + a m + 2 Amn = 0 are n = (0 , A )and n + m = (0 , A ) respectively. So, they have positive values for 0 > n > .
63 and0 > n + m > .
63 and negative values for 0 ≤ n ≤ .
63 and 0 ≤ n + m ≤ .
63 respectively.In addition, the condition f ( G ) > α ( − n (cid:0) . H (cid:1) n (cid:16) β ( − m (cid:0) . H (cid:1) m (cid:17) > i) In the first situation we assume that a n > a n + a m + 2 Amn > a n < a n + a m + 2 Amn < (i1) for α ( − n >
0, if | a n a n + a m +2 Amn | (cid:0) . H (cid:1) − m ≤ β < (cid:0) . H (cid:1) − m then m should beodd and if − (cid:0) . H (cid:1) − m < β < −| a n a n + a m +2 Amn | (cid:0) . H (cid:1) − m then m should be even. (i2) for α ( − n <
0, it is impossible to realize the conditions (35) and f ( G ) > (ii) Here, in the second situation we present the conditions required for WEC fulfilmentfor a n < a n + a m + 2 Amn > a n > a n + a m + 2 Amn < (ii1) for α ( − n >
0, if β ≥ | a n a n + a m +2 Amn | (cid:0) . H (cid:1) − m then m should be even and if β ≤ −| a n a n + a m +2 Amn | (cid:0) . H (cid:1) − m then m should be odd. (ii2) for α ( − n <
0, if β > (cid:0) . H (cid:1) − m then m should be odd and if β < − (cid:0) . H (cid:1) − m then m should be even. In this work we investigated a model of modified gravity with non-minimal coupling betweenmodified Gauss-Bonnet gravity, f ( G ), and matter Lagrangian described by action (1) fromthe viewpoint of the energy conditions. We derived the suitable energy conditions inequali-ties for such a model as equations (17)- (20).We examined the WEC inequality equation (29) for two class of viable models of f ( G ) grav-ity presented in equations (32) and (34). We concluded that the model f ( G ) = α G n obeythe WEC only for α <
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