Processing math: 100%

f(ˉR,L(X))-gravity in the context of dark energy with power law expansion and energy conditions

Figures(4) / Tables(1)

Get Citation
Goutam Manna, Arijit Panda, Aninda Karmakar, Saibal Ray and Md. Rabiul Islam. f(ˉR,L(X))-gravity in the context of dark energy with power law expansion and energy conditions[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac9fbe
Goutam Manna, Arijit Panda, Aninda Karmakar, Saibal Ray and Md. Rabiul Islam. f(ˉR,L(X))-gravity in the context of dark energy with power law expansion and energy conditions[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac9fbe shu
Milestone
Received: 2022-08-20
Article Metric

Article Views(1105)
PDF Downloads(31)
Cited by(0)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

f(ˉR,L(X))-gravity in the context of dark energy with power law expansion and energy conditions

    Corresponding author: Saibal Ray, saibal.ray@gla.ac.in
  • 1. Department of Physics, Prabhat Kumar College, Contai, Purba Medinipur 721404, India
  • 2. Department of Physics, Prabhat Kumar College, Contai, Purba Medinipur 721404, India & Department of Physics, Raiganj University, Raiganj, Uttar Dinajpur 733134, West Bengal, India
  • 3. Department of Physics, Prabhat Kumar College, Contai, Purba Medinipur 721404, India
  • 4. Centre for Cosmology, Astrophysics and Space Science (CCASS), GLA University, Mathura 281406, Uttar Pradesh, India
  • 5. Department of Physics, Raiganj University, Raiganj, Uttar Dinajpur 733 134, West Bengal, India

Abstract: The objective of this work is to generate a general formalism of f(ˉR,L(X))gravity in the context of dark energy under the framework of K-essence emergent geometry with the Dirac-Born-Infeld (DBI) variety of action, where ˉR is the familiar Ricci scalar, L(X) is the DBI type non-canonical Lagrangian with X=12gμνμϕνϕ, and ϕ is the K-essence scalar field. The emergent gravity metric (ˉGμν) and the well known gravitational metric (gμν) are not conformally equivalent. We have constructed a modified field equation using the metric formalism in f(ˉR,L(X))-gravity incorporating the corresponding Friedmann equations into the framework of the background gravitational metric, which is of Friedmann-Lemaître-Robertson-Walker (FLRW) type. The solution of the modified Friedmann equations have been deduced for the specific choice of f(ˉR,L(X)), which is of Starobinsky-type, using the power law expansion method. The consistency of the model with the accelerating phase of the universe has been shown when we restrict ourselves to consider the value of the dark energy density as ˙ϕ2=8/9=0.888<1, which indicates that the present universe is dark-energy dominated. Graphical plots for the energy density (ρ), pressure (p), and equation of state parameter (ω) with respect to (w.r.t.) time (t) based on parametric values are interestingly consistent with the dark energy domination theory, and hence the accelerating features. We also highlight the corresponding energy conditions and constraints of the f(ˉR,L(X)) theory with a basic example.

    HTML

    I.   INTRODUCTION
    • Weyl [1] in 1919 and Eddington [2] in 1923included higher order constant terms concerned with the curvature in the action of General Relativity (GR) first introduced by Albert Einstein. We can quantize GR by the non-renormalizability of the theory in a conventional way. Utiyama and De Witt [3] in 1962 showed that we can renormalize GR in one loop if the Einstein-Hilbert (EH) action is constructed with higher-order curvature terms. In addition, when quantum corrections, or string theory, entered into the scenario, the action of effective gravitational field with low energy required higher-order curvature terms [4]. Inspired by this argument, scientists have tried to modify the EH action in applied higher order theories of gravity.

      Therefore, higher-orders of Ricci scalar should be incorporated into the action. We should keep in mind that the modified theories of gravity are important at scales only in the range of the Planck scale, i.e., in the newborn universe or at black hole (BH) singularities, e.g., inflation due to curvature case [5]. Sotiriou and Felice [6, 7] reviewed the well-known "f(R) theory of gravity." This f(R) gravity was reconstructed by Dunsby et al. [8] considering the background as Friedman-Lemaitre-Robertson-Walker (FLRW) type. Particularly, the authors predicted the only Lagrangian with a real value for which f(R) can generate a true Lambda cold dark matter (ΛCDM) expansion for a dust-like matter-filled universe and is equivalent to the EH Lagrangian with a cosmological constant, which is positive. A general formalism was proposed by Mukherjee and Banerjee [9], using which we can observe the later time dynamics of the universe for a given analytic model of f(R) gravity considering cold dark matter. The outcomes of energy conditions of the f(R) gravity for various cosmological cases have also been investigated in Ref. [1014]. The energy conditions using the Raychaudhuri equations in the expanding universe were also studied in Refs. [1517].

      Harko and Lobo [18] considered the action as an arbitrary function of not only the Ricci scalar (R) but also the matter Lagrangian (Lm). With the help of the metric formalism, the authors achieved the gravitational field equations of f(R,Lm) gravity, along with the equations of motion (EOM) of test particles. Subsequently, Wang et al. [19] achieved the general energy conditions of the f(R,Lm) gravity. In Refs. [2022], the authors described the power-law cosmic expansion in higher derivative gravity. Harko et al. [23] included the trace of the Stress-energy tensor in the action along with the Ricci scalar and developed a new theory named `f(R,T) theory'. This type of modified gravity has been extensively used recently to study various cosmological phenomena [2427].

      The development of the K-essence model was carried out in Refs. [2836]. On the basis of the Dirac-Born-Infeld (DBI) model [3739], Manna et al. [4042] derived a K-essence emergent gravity metric (ˉGμν), which has a different significance than the usual gravitational metric, gμν. The K-essence model [2836] is essentially a scalar field theory where the kinetic energy of the K-essence field dominates the potential energy of the field. The Lagrangian corresponding to the K-essence field is of non-canonical type. The difference between K-essence theory, which incorporates non-canonical kinetic parts, and canonical relativistic field theories lies in the non-trivial dynamical solutions of the K-essence EOM. In addition to the spontaneous breaking of Lorentz invariance, it also changes the perturbed metric near the solutions. Hence, these perturbations move along the formal emergent or analogue curved space-time with the perturbed metric. The Lagrangian for the K-essence model can be written as,L=V(ϕ)F(X), where ϕ is the K-essence scalar field and X=12gμνμϕνϕ [43].

      A modified version of f(R) gravity was also studied by Nojiri et al. [44]. They considered the higher order kinetic terms of a scalar field, which was incorporated in vacuum f(R)gravity's action part. The authors included a general class of the K-essence Lagrangian, G(X) with the action of vacuum f(R) gravity. In the background of slow-roll approximation, the authors investigated the inflationary sides of their theory. Odintsov et al. [45] analyzed the consequences of K-essence geometry in the f(R) gravity when cold dark matter and radiation are present. Perfect fluids with a similar model were considered in Ref. [44]. Their findings included several cosmological quantities, such as the dark energy equation of state parameter (ω), the dark energy density parameter (Ωd), and some state finder quantities. Recently, Oikonomou et al. [46] discussed the phase space of a simple K-essence f(R) gravity theory.

      Next, let us discuss the importance of the K-essence theory with specific choice of the DBI type Lagrangian. It is now unavoidable to admit the acceleration of the universe after analyzing the observations of Large-Scale Structure, type Ia Supernovae's observations, and measurements of anisotropy of Cosmic Microwave Background [47]. It has also been acknowledged that our universe is now dominated by a component named dark energy, which imposes negative pressure. Scientists proposed cosmological constant or vacuum density to be the candidates for such an exotic component of the universe. However, the barrier of the cosmological coincidence problem created hurdles for us and raised the question, "why does the strange dark energy component possess a tiny energy density (O(meV4)) compared with the simple expectation based on quantum field theory?" Additionally, "the occurrence of the acceleration at such a late stage of evolution" continuously poked cosmologists for better theories. The problem with most of the dark energy models (e.g., cosmological constant) is that they require extraordinary fine tuning of the initial energy density, which is on the order of 100 or more smaller than the initial matter-energy density.

      A new model, known as K-essence theory, with a scalar field and excellent dynamic properties that can avoid the long awaited fine-tuning problem has been developed [28]. The most promising feature of this model is that it brings the negative pressure from its nonlinear kinetic energy of the scalar field. It has already been investigated for a wide class of theories. There exist attractor solutions [29, 48] in which the scalar field propagates with different evolution speeds to achieve the required equation of state of the K-essence theory in different epochs with changing background equation of state. While our universe was going through the epoch of radiation domination, K-essence field was dominant, and by imitating the equation of state (EOS) of the radiation, the ratio of K-essence field and radiation density is held constant. At the time of dust domination, the K-essence theory failed to mimic the EOS of dust-like phase for its dynamical characteristics. It also significantly decreased its energy value and settled upon a constant value. Subsequently, the field outgrew the matter density and took the universe into cosmic acceleration at a time roughly corresponding to the current age of the universe. Finally, the EOS of K-essence theory gradually settled to a value between 0 and 1.

      The well-known Quintessence Trackers models [4955] almost give the same result as that of the K-essence theory, but with one problem. Although it can mimic the state equation of matter and radiation for background EOS, it requires an adjustable parameter, which needs to be fine tuned to get the preferable energy density that can produce the negative pressure at present age.

      K-essence theory is different in the sense that it traces the background energy density when the universe was in radiation epoch only. The sharp transition of positive pressure to negative pressure at the matter-radiation equality occurs automatically because of its dynamics. K-essence theory was unable to dominate before matter-radiation equality as it was busy tracking the radiation background. Since the energy density inevitably drops to a small value at the transition to dust domination, it is impossible to dominate immediately just after the dust domination. On the other hand, as the matter density drops more rapidly than the energy density with expanding universe, the K-essence field came into control at an age roughly around the current epoch. Thus, the whole Cosmic Coincidence Problem (i.e., why we live in the era of dark matter and dark energy density's equality) vanishes owing to the fact that we came to observe the universe at the right time after matter-radiation equality.

      K-essence models also ensures the production of the dark energy component where the sound speed (cs) does not exceed light speed. There exists a difference between these models and scalar field quintessence models from the observational background with a canonical kinetic term (for which cs=1), and this may be one of the ways to reduce cosmic microwave background (CMB) fluctuations measured on large angular scales [5658]. Though, there may be some stages where the fluctuations of the field can propagate superluminally (cs>1) [32, 33, 59].

      Some cosmological behaviors and the stability of the K-essence model in FLRW spacetime has been studied by Yang et al. [60]. Some opposite results have been obtained for small sound speeds of scalar perturbations, which implies clustering of dark energy and an increase in cosmological perturbations [61, 62].

      Historically, Born and Infeld [37] introduced a non-canonical kinetic theory to overcome the infinite self-energy of the electron. Some more non-canonical theories were also studied in literature, like [38, 39]. The studies [6371], focusing on topics such as string theory, brane cosmology, and D-branes have also used the DBI type non-canonical Lagrangian.

      Motivated by this importance of the K-essence theory, which prescribes a way to investigate the effects of the presence of the dark energy component in the cosmological framework, in this paper, we study f(R)gravity in the context of K-essence emergent gravity, i.e., dark energy in a general manner. We have made the generalization of the f(ˉR,L(X)) theory with the help of the metric formalism, where ˉR is the Ricci scalar of the K-essence geometry and L(X) is the DBI type non-canonical Lagrangian. Panda et al. [72] modified the f(R,T) theory in the context of dark energy using the K-essence model. The process of studying these two papers is fundamentally different, i.e., the consideration of actions differs.

      Further, we have calculated the energy conditions and modified Friedman equations in f(ˉR,L(X)) gravity and considered the flat FLRW-type metric as the background gravitational metric. The modified field equation, Friedmann equations, and energy conditions for the new f(ˉR,L(X)) gravity theory are different from the usual f(R,Lm) [18] and f(R) [6] gravity theories. In addition, we have solved the modified Friedmann equations using the power law cosmic expansion method.

      The remainder of this paper is organized as follows. In Sec. II, we briefly discuss the K-essence emergent geometry based on the following works [3034, 4042]. In Sec. III, we formulate f(ˉR,L(X)) gravity in the context of the K-essence emergent geometry. We also derive the modified field equations and the requirement condition of the energy-momentum tensor conservation in f(ˉR,L(X)) gravity. The modified Friedmann equations are introduced in Sec. IV, considering the background gravitational metric as flat FLRW and the K-essence scalar field as a function of time only. The solution of the Friedmann equations is solved for specific choice of f(ˉR,L(X)) using the power law method in Sec. V, whereas in Sec. VI, we develop the energy conditions and constraints of f(ˉR,L(X)) gravity with an example. The last section, Sec. VII, contains some general discussion and key conclusions of our work. Additionally, we briefly discuss the f(R)-gravity and f(R,Lm)gravity and corresponding energy conditions [1014, 19, 73] in the Appendix.

    II.   K-ESSENCE THEORY: A GLIMPSE OF BACKGROUND AND DEVELOPMENT
    • In this section, we will discuss the development of the modified metric corresponding to the emergent spacetime, which is related with a general background geometry and a very general K-essence scalar field. The K-essence scalar field, ϕ, has action [3034]

      Sk[ϕ,gμν]=d4xgL(X,ϕ),

      (1)

      which has a minimal coupling with the background space-time metric, gμν, and X=12gμνμϕνϕ represents the canonical kinetic term. The energy-momentum tensor is

      Tμν2gδSkδgμν=2Lgμν+gμνL=LXμϕνϕ+gμνL,

      (2)

      with LX=dLdX,LXX=d2LdX2,Lϕ=dLdϕ, and the symbol μ standing for the covariant derivative with respect to the gravitational metric, gμν.

      The EOM of a scalar field is

      1gδSkδϕ=Gμνμνϕ+2XLXϕLϕ=0,

      (3)

      where

      GμνcsL2X[LXgμν+LXXμϕνϕ],

      (4)

      with 1+2XLXXLX>0 and c2s(X,ϕ)(1+2XLXXLX)1.

      The inverse metric, Gμν, can be written in the following form:

      Gμν=LXcs[gμνc2sLXXLXμϕνϕ].

      (5)

      Applying a conformal transformation further [40, 41], ˉGμνcsLXGμν gives

      ˉGμν=gμνLXXLX+2XLXXμϕνϕ.

      (6)

      Using Eq. (2), the effective emergent metrics (6) can be written as [32, 33]

      ˉGμν=[1LLXXLX(LX+2XLXX)]gμν+LXXLX(LX+2XLXX)Tμν.

      (7)

      We should always keep it in mind that, LX0 when c2s is positive, and only then, Eqs. (1) – (4) will yield meaningful physics.

      Evidently, if ϕ has a non-trivial space-time configuration, then usually the emergent metric, ˉGμν, is not conformally equivalent to gμν. So ϕ has dissimilar characteristics as ompared with canonical scalar fields with the locally defined causal structure. Further, if there is no explicit dependency of L on ϕ, the reformed EOM (3) becomes

      1gδSkδϕ=ˉGμνμνϕ=0.

      (8)

      The authors [33, 34, 3742, 74] take the Dirac-Born-Infeld (DBI) type non-canonical Lagrangian as L(X,ϕ)=V(ϕ)[112X], where V(ϕ) is a constant potential and kinetic energy of the K-essence scalar field and is much greater than the potential part of the Lagrangian. In this article, we choose the DBI type non-canonical Lagrangian to be an explicit function of X only as L(X,ϕ)L(X) [43], without any loss of generality and dimensionality since in the K-essence theory, the kinetic energy dominates over the potential energy of the system. Therefore, we can write the Lagrangian as

      L(X,ϕ)L(X)=V[112X],

      (9)

      where V is a constant potential term.

      Then c2s(X,ϕ)=12X, and hence, the effective emergent metric (6) turns out to be

      ˉGμν=gμνμϕνϕgμνμϕνϕ,

      (10)

      since ϕ is a scalar.

      Eqs. (2) and (10) can be rewritten in terms of Tμν and μϕ as

      ˉGμνL=LXμϕνϕ+TμνLμϕνϕ.

      (11)

      Following [40, 75], the relation between the new Christoffel symbols and the old ones is

      ˉΓαμν=Γαμν+(12X)1/2ˉGαγ×[ˉGμγν(12X)1/2+ˉGνγμ(12X)1/2ˉGμνγ(12X)1/2]=Γαμν12(12X)[δαμνX+δαμνX].

      (12)

      Therefore, using the new Christoffel connections, ˉΓ, the geodesic equation for the K-essence becomes

      d2xαdλ2+ˉΓαμνdxμdλdxνdλ=0,

      (13)

      where λ is an affine parameter.

      Now introducing the covariant derivative, Dμ [32, 33], corresponding to the emergent metric ˉGμν (DαˉGαβ=0) as

      DμAν=μAνˉΓλμνAλ,

      (14)

      and the inverse of the emergent metric is ˉGμν such that ˉGμλˉGλν=δνμ.

      Ultimately, the "emergent" Einstein's equation becomes

      ˉEμν=ˉRμν12ˉGμνˉR=κTμν,

      (15)

      where κ=8πG is a constant, ˉRμν is emergent gravity's Ricci tensor, and ˉR(=ˉRμνˉGμν)is the Ricci scalar of the emergent space-time.

    III.   f(ˉR,L(X))GRAVITY IN THE CONTEXT OF K-ESSENCE EMERGENT SPACE-TIME
    • We now consider the action of modified gravity in the context of K-essence emergent space-time, which takes the following form (κ=1)

      S=d4xˉGf(ˉR,L(X)),

      (16)

      where f(ˉR,L(X)) is an arbitrary function of the Ricci scalar ˉR, the non-canonical Lagrangian density, L(X), corresponding to the K-essence theory, and

      ˉG=det(ˉGμν).

      Based on the following work [18], varying the action, S, with respect to the K-essence emergent gravity metric, ˉGμν, we obtain

      δS=[fˉR(ˉR,L)δˉR+fL(ˉR,L)δLδˉGμνδˉGμν12ˉGμνf(ˉR,L)δˉGμν]×ˉGd4x,

      (17)

      where we have denoted fˉR(ˉR,L)=f(ˉR,L)ˉR and fL(ˉR,L)=f(ˉR,L)L.

      Now, we obtain the variation of the Ricci scalar for the K-essence emergent gravity metric

      δˉR=δ(ˉRμνˉGμν)=δˉRμνˉGμν+ˉRμνδˉGμν=ˉRμνδˉGμν+ˉGμν(DλδˉΓλμνDνδˉΓλμλ),

      (18)

      where

      ˉRμν=μˉΓααναˉΓαμν+ˉΓαβμˉΓβανˉΓααβˉΓβμν,

      (19)

      ˉΓαμν=12ˉGαβ[μˉGβν+νˉGμββˉGμν],

      (20)

      and the variation of δˉΓλμν is

      δˉΓλμν=12ˉGλα[DμδˉGνα+DνδˉGμαDαδˉGμν].

      (21)

      Thus, the expression for the variation of the Ricci scalar, δˉR, is

      δˉR=ˉRμνδˉGμν+ˉGμνDαDαδˉGμνDμDνδˉGμν.

      (22)

      Therefore, variation of Eq. (17) is

      δS=[fˉR(ˉR,L)ˉRμνδˉGμν+fˉR(ˉR,L)ˉGμνDαDαδˉGμνfˉR(ˉR,L)DμDνδˉGμν+fL(ˉR,L)δLδˉGμνδˉGμν12ˉGμνf(ˉR,L)δˉGμν]ˉGd4x.

      (23)

      After partially integrating second and third terms of the above Eq. (23), we get

      δS=[fˉR(ˉR,L)ˉRμν+ˉGμνDμDμfˉR(ˉR,L)DμDνfˉR(ˉR,L)+fL(ˉR,L)δLδˉGμν12ˉGμνf(ˉR,L)]δˉGμνˉGd4x.

      (24)

      Therefore, using the principle of least action, i.e. δS=0, we have the modified field equation for f(ˉR,L(X)) theory

      fˉR(ˉR,L)ˉRμν+ˉGμνDαDαfˉR(ˉR,L)DμDνfˉR(ˉR,L)12ˉGμνf(ˉR,L)+fL(ˉR,L)δLδˉGμν=0.

      (25)

      Now, we evaluate the term δLδˉGμν as

      δLδˉGμν=δLδXδXδgμνδgμνδˉGμν=12LXDμϕDνϕ(1+DαϕDαϕ),

      (26)

      since for the scalar field μϕμϕDμϕ.

      Using Eqs. (11), (25), and (26), we obtain the expression for the modified field equation for the f(ˉR,L(X)) theory in terms of Tμν as

      fˉR(ˉR,L)ˉRμν+(ˉGμνˉDμDν)fˉR(ˉR,L)12[f(ˉR,L)LfL(ˉR,L)(1+DαϕDαϕ)]ˉGμν+12LfL(ˉR,L)DμϕDνϕ[1+DαϕDαϕ]

      =12fL(ˉR,L)Tμν[1+DαϕDαϕ]=12fL(ˉR,L)ˉTμν,

      (27)

      where ˉ=DμDμ and ˉTμν=Tμν[1+DαϕDαϕ].

      The above Eq. (27) is different from the usual Eq. (104) of f(R,Lm) theory in the presence of the K-essence scalar field (vide the Appendix). If we consider the emergent gravity metric,ˉGμν, is conformally equivalent to the gravitational metric, gμν, and L can be matter Lagrangian, then we get back to the usual f(R,Lm) theory in the absence of the K-essence scalar field. Further, if we consider f(ˉR,L(X))f(R,Lm)12R+Lm, i.e., the Hilbert-Einstein Lagrangian form, then from Eq. (27), we lead to the standard Einstein field equation Rμν12gμνR=Tμν.

      Contracting the above field equation, Eq. (27), with ˉGμν, we have the modified trace equation for the f(ˉR,L(X)) theory as

      fˉR(ˉR,L)ˉR+3ˉfˉR(ˉR,L)2[f(ˉR,L)fL(ˉR,L)L(1+DαϕDαϕ)]+12LfL(ˉR,L)DμϕDμϕ[1+DαϕDαϕ]=12fL(ˉR,L)ˉT,

      (28)

      where ˉT=ˉTμμ is trace of the energy-momentum tensor.

      Subtracting Eq. (28)×ˉGμν from Eq. (27)×3, we get

      fˉR(ˉR,L)(ˉRμν13ˉGμνˉR)+16ˉGμν[f(ˉR,L)LfL(ˉR,L)(1+DαϕDαϕ)]+13LfL(ˉR,L)DμϕDνϕ[1+DαϕDαϕ]=12fL(ˉR,L)(ˉTμν13ˉGμνˉT)+DμDνfˉR(ˉR,L),

      (29)

      which is an another form of the modified field equation in the presence of the K-essence scalar field, ϕ.

      By taking covariant divergence with respect to Dμ of Eq. (27), we have

      Dμ[fˉR(ˉR,L)]ˉRμν(ˉDνDνˉ)fˉR(ˉR,L)+fˉR(ˉR,L)Dμ(ˉRμν12ˉGμνˉR)+12ˉGμνfˉR(ˉR,L)Dμ(ˉR)12Dμ[f(ˉR,L)]ˉGμν+12Dμ[LfL(ˉR,L)(1+DαϕDαϕ)]ˉGμν+12Dμ[LfL(ˉR,L)DμϕDνϕ(1+DαϕDαϕ)]=12Dμ[fL(ˉR,L)ˉTμν].

      (30)

      Using identities on purely geometrical grounds [7678] , Dμ(ˉRμν12ˉGμνˉR)=DμˉEμν=0, Dμ[fˉR(ˉR,L)]ˉRμν=(ˉDνDνˉ)fˉR(ˉR,L), and also Eqs. (11) and (26), the above Eq. (30) becomes

      Dμ[fL(ˉR,L)ˉTμν]=fL(ˉR,L)Dμ[ˉGμνL]+Dμ[LfL(ˉR,L)(1+DαϕDαϕ)(ˉGμν+DμϕDνϕ)]DμˉTμν=Dμln[fL(ˉR,L)]×[LXDμϕDνϕ(1+DαϕDαϕ)]+Dμ[LDμϕDνϕ(1+DαϕDαϕ)+LˉGμνDαϕDαϕ]DμˉTμν=2Dμln[fL(ˉR,L)]δLδˉGμν+Dμ[LDμϕDνϕ(1+DαϕDαϕ)+LˉGμνDαϕDαϕ].

      (31)

      Thus, the requirement of the conservation of the energy-momentum tensor (DμˉTμν=0) for the K-essence Lagrangian, gives an effective functional relation as

      2Dμln[fL(ˉR,L)]δLδˉGμν+Dμ[LDμϕDνϕ(1+DαϕDαϕ)+LˉGμνDαϕDαϕ]=0.

      (32)

    IV.   MODIFICATION OF THE FRIEDMANN EQUATIONS
    • We consider the gravitational metric, gμν, to be a flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric and the line element for this is

      ds2=dt2a2(t)3i=1(dxi)2,

      (33)

      with a(t) being the scale factor, as usual.

      From Eq. (10), we have the components of the emergent gravity metric as

      ˉG00=(1˙ϕ2);ˉGii=[a2(t)+(ϕ)2];ˉG0i=˙ϕϕ=ˉGi0,

      (34)

      where we consider ϕϕ(t,xi), ˙ϕ=ϕt, and ϕ=ϕxi.

      So, the line element of the FLRW emergent gravity metric is

      dS2=(1˙ϕ2)dt2[a2(t)+(ϕ)2]3i=1(dxi)22˙ϕϕdtdxi.

      (35)

      Now, from the emergent gravity equation of motion, Eq. (8), we have

      ˉG00(00ϕΓ0000ϕΓi00iϕ)+ˉGii(iiϕΓ0ii0ϕΓiiiiϕ)+ˉG0i(0iϕΓ00i0ϕΓioiiϕ)+ˉGi0(i0ϕΓ0i00ϕΓii0iϕ)=0.

      (36)

      For simplification, we consider the homogeneous K-essence scalar field, ϕ, i.e., ϕ(t,xi)ϕ(t) then ˉG00=(1˙ϕ2), ˉG0i=ˉGi0=0=iϕ, ˉGii=a2(t), and X=12gμνμϕνϕ=12˙ϕ2. This consideration is possible in this case, since the dynamical solutions of the K-essence scalar fields spontaneously break Lorentz symmetry. Therefore, the flat FLRW emergent gravity line element (35) and the equation of motion (36) become

      dS2=(1˙ϕ2)dt2a2(t)3i=1(dxi)2,

      (37)

      and

      ˙aa=H(t)=¨ϕ˙ϕ(1˙ϕ2),

      (38)

      where H(t)=˙aa is the usual Hubble parameter (always ˙a0). Eq. (38) gives the relation between the Hubble parameter and the time derivatives of the K-essence scalar field. Note that in the above spacetime, Eq. (37) always is ˙ϕ2<1. If ˙ϕ2>1, the signature of this spacetime will be ill-defined. Moreover, the ˙ϕ20 condition holds, instead of ˙ϕ2=0, which leads to non-applicability of the K-essence theory. Additionally, ˙ϕ21 because Ωmatter+Ωradiation+Ωdarkenergy=1 and we can measure ˙ϕ2 as dark energy density in units of the critical density, i.e., it is nothing but Ωdarkenergy [4042, 74]. Therefore ˙ϕ2 takes a value between 0 and 1.

      The Ricci tensors and Ricci scalar of the emergent gravity space-time are

      ˉRii=a21˙ϕ2[¨aa+2(˙aa)2+˙aa˙ϕ¨ϕ1˙ϕ2]=a21˙ϕ2[¨aa+(˙aa)2(2˙ϕ2)]=a21˙ϕ2[˙H+H2(3˙ϕ2)],

      (39)

      ˉR00=3¨aa+3˙aa˙ϕ¨ϕ1˙ϕ2=3¨aa+3(˙aa)2˙ϕ2=3[˙H+H2(1˙ϕ2)],

      (40)

      and

      ˉR=61˙ϕ2[¨aa+(˙aa)2+˙aa˙ϕ¨ϕ1˙ϕ2]=61˙ϕ2[¨aa+(˙aa)2(1˙ϕ2)]=61˙ϕ2[˙H+H2(2˙ϕ2)],

      (41)

      where we have used the relation Eq. (38) and ˙HHt=a¨a˙a2a2.

      Combining Eqs. (39) and (40) with (41), we get

      ˉR00=12(1˙ϕ2)ˉR3H2,

      (42)

      ˉRii=a2(1˙ϕ2)[16ˉR(1˙ϕ2)+H2].

      (43)

      We assume that the energy-momentum tensor is an ideal fluid type, which is

      Tνμ=diag(ρ,p,p,p)=(ρ+p)uμuνδνμpTμν=ˉGμαTαν,

      (44)

      where p is the pressure and ρ is the matter density of the cosmic fluid. In the comoving frame, we have u0=1 and uα=0; α=1,2,3 in the K-essence emergent gravity spacetime.

      Now, the question is whether this type of energy-momentum tensor is valid or not in the case of a perfect fluid model when the kinetic energy (˙ϕ2) of the K-essence scalar field is present. The answer is "yes" since our Lagrangian is L(X)=112X. This class of models is equivalent to perfect fluid models with zero vorticity, and the pressure (Lagrangian) can be expressed through the energy density only [33, 74].

      Now, we evaluate the ii and 00components of the modified field equation (Eq. 27) using Eq. (44) and considering ϕϕ(t) only:

      FˉRii+(ˉGiiˉDiDi)F12[fLfL(1+˙ϕ2)]ˉGii=12fLa2(t)ˉp

      (45)

      and

      FˉR00+(ˉG00ˉD0D0)F12[fLfL(1+˙ϕ2)]ˉG00+12LfL˙ϕ2(1+˙ϕ2)=12fL(1˙ϕ2)ˉρ,

      (46)

      with F=fˉR(ˉR,L)f(ˉR,L)ˉR, ˉp=p(1+˙ϕ2), and ˉρ=ρ(1+˙ϕ2).

      Now, we calculate the terms ˉG00ˉF and ˉGiiˉF using the determinant of the flat FLRW emergent gravity metric, ˉG=a31˙ϕ2, and Eq. (38):

      ˉG00ˉF=¨F+3˙aa˙F+˙F˙ϕ¨ϕ(1˙ϕ2)=¨F+H˙F(3˙ϕ2),

      (47)

      and

      ˉGiiˉF=DiDiFa2(1˙ϕ2)[¨F+2H˙F(1˙ϕ2)],

      (48)

      where we have used, (it)2=a21˙ϕ2 for the flat FLRW emergent gravity metric.

      Now, we substitute Eqs. (42) and (47) into Eq. (46) to obtain the first modified Friedmann equation as

      3H2=1F[12ˉρfL(1˙ϕ2)+3H˙F+(1˙ϕ2)12(FˉRf)+12LfL(1+˙ϕ2)]=1F[12ˉρfL(1˙ϕ2)+3H˙ˉRFˉR+3HFLLX˙X+(1˙ϕ2)12(FˉRf)+12LfL(1+˙ϕ2)].

      (49)

      We also substitute Eqs. (39), (41), and (48) into the ii-components of Eq. (45), and after rearranging, we get the second modified Friedmann equation for the flat FLRW K-essence emergent gravity spacetime under f(ˉR,L(X)) theory. Hence,

      2˙H+H2(32˙ϕ2)=1F[12ˉpfL(1˙ϕ2)+¨F+2H˙F(1˙ϕ2)12(1˙ϕ2)(fˉRF)+12LfL(1˙ϕ2)(1+˙ϕ2)]=1F[12ˉpfL(1˙ϕ2)+¨ˉRFˉR+(˙ˉR)2FˉRˉR+2H˙ˉRFˉR(1˙ϕ2)12(1˙ϕ2)(fˉRF)+12LfL(1˙ϕ2)(1+˙ϕ2)]+1F[2H(1˙ϕ2)FLLX˙X+FLL(LX˙X)2+FLLXX(˙X)2+FLLX¨X].

      (50)

      Since the Lagrangian (L) of the K-essence theory is a function of X(=12gμνμϕνϕ), we can write

      ˙F=FˉR˙ˉR+FLLX˙Xand¨F=FˉR¨ˉR+(˙ˉR)2FˉRˉR+FLL(LX˙X)2+FLLXX(˙X)2+FLLX¨X.

      (51)

      The above Friedmann equations in the presence of the kinetic energy of the K-essence scalar field are different from the usual f(R) gravity model. Notably, if we consider f(ˉR,L(X))f(R) and ˉGμνgμν, then, the above modified Friedmann equations (49) and Eq. (50) reduces to the usual Friedmann equations of f(R) gravity, with κ=1 and Tμν replaced by 12Tμν [6, 7] as

      3H2=1F[ρ2+RFf23H˙RFR],

      (52)

      and

      2˙H+3H2=1F[p2+(˙R)2FRR+2H˙RFR+¨RFRfRF2].

      (53)
    V.   SOLUTION TO THE FRIEDMANN EQUATION USING POWER LAW
    • We choose a Starobinsky type model [5, 6] to investigate our theory and evaluate some cosmological values of the universe. This model has achieved popularity as the inflationary predictions produced by this theory seem very much consistent with the observational data. The coefficient of the R2 curvature term single-handedly obtains the slow-roll inflation with tremendous success, without the introduction of outside inflation field by hand. Other reasons for the Starobinksy model to be treated as an important model have been discussed in [79]. Now, we write f(ˉR,L) as

      f(ˉR,L)=ˉR+αˉR2+L,

      (54)

      and L is the DBI type Lagrangian mentioned in Eq. (9).

      Therefore, we get

      fL=fL=1,F=fˉR=1+2αˉR,FˉR=2α,FL=0.

      (55)

      Using these values and after some algebraic calculations we can write Friedmann equation (49) as

      (1+2αˉR)3H2=12ˉρ(1˙ϕ2)+12(1˙ϕ2)αˉR2+6αH˙ˉR+˙ϕ2(11˙ϕ2).

      (56)

      Analogous to [2022], let us now assume there exists an exact power–law solution to the field equations, i.e., the scale factor behaves as

      a(t)=a0tm,

      (57)

      where m(>0) is a fixed real number.

      The definition of H gives us

      H=˙aa=mt,˙H=mt2,¨H=2mt3,˙¨H=6mt4.

      (58)

      Now, taking Eq. (41) into consideration, we can evaluate the value of Ricci scalar as

      ˉR=6(1˙ϕ2)t2[m+m2(2˙ϕ2)],

      (59)

      and then using Eq. (38) we get

      ˙ˉR=61˙ϕ2[¨H+4H˙H(1˙ϕ2)2H3˙ϕ2]=12t3(1˙ϕ2)[mm3˙ϕ22m2(1˙ϕ2)].

      (60)

      Now, putting the values of Eqs. (57)–(60) into (56), we simply get

      12ˉρ(1˙ϕ2)=90αm2t4(1˙ϕ2)180αm3t4(1˙ϕ2)+180αm3˙ϕ2t4(1˙ϕ2)108αm4˙ϕ2t4(1˙ϕ2)+18αm4˙ϕ4t4(1˙ϕ2)3m2t2+˙ϕ2(11˙ϕ2).

      (61)

      Now, from the second Friedmann equation (50), we have

      (1+2αˉR)[2˙H+H2(32˙ϕ2)]=12ˉp(1˙ϕ2)+2α¨ˉR+4αH˙ˉR(1˙ϕ2)+12αˉR2(1˙ϕ2)+˙ϕ2(11˙ϕ2)

      (62)

      or

      12ˉp(1˙ϕ2)=2mt2+3m2t2+2m˙ϕ2t25m2˙ϕ2t2186αm2t4(1˙ϕ2)+240αm2˙ϕ2t4(1˙ϕ2)+84αm3t4(1˙ϕ2)252αm3˙ϕ2t4(1˙ϕ2)+12αm4˙ϕ2t4(1˙ϕ2)+72αmt4(1˙ϕ2)42αm4˙ϕ4t4(1˙ϕ4)+96αm3˙ϕ4t4(1˙ϕ2)12˙ϕ2(11˙ϕ2)+12˙ϕ4(11˙ϕ2).

      (63)

      For our case, the energy-momentum conservation relation is

      DμˉTμν=0,

      (64)

      with ˉTμν=Tμν[1+DαϕDαϕ].

      Now, using Eqs. (44) and (64), we have the conserving equation as

      ˙ˉρ=3˙aa(ˉρ+ˉp),

      (65)

      where ˉρ and ˉp already have been defined. It is essential to mention here that ˉρ and ˉp are not the same as the normal ρ and p.

      Now, considering the power law, we get the following from Eq. (65):

      ˉρ=ρ(1+˙ϕ2)=ρ0t3m(1+ω),

      (66)

      where ω=ˉpˉρ=pρ.

      Now, putting the value of ˉρ into the Friedmann equation (61), we have

      12ρ0t3m(1+ω)=90αm2t4(1˙ϕ2)2180αm3t4(1˙ϕ2)2+180αm3˙ϕ2t4(1˙ϕ2)2108αm4˙ϕ2t4(1˙ϕ2)2+18αm4˙ϕ4t4(1˙ϕ2)23m2t2(1˙ϕ2)+˙ϕ2(1˙ϕ2)(11˙ϕ2).

      (67)

      On the other hand, to maintain the energy-momentum conservation, the relation (32) must be satisfied. So, the effective functional relation (32) for homogeneous K-essence scalar field reduces to

      3˙ϕ22=21˙ϕ2,

      (68)

      where we have used Eqs. (9) and (55).

      Solving Eq. (68), we have either ˙ϕ2=0, which is not acceptable for our case, or

      ˙ϕ2=89=0.888=constant.

      (69)

      It should be noted that the exact solution of field equations (Eq. (2) in Ref. [22]) are already obtained by the assumption of the power law form of the scale factor using the Starobinsky Model in [22]. The results of that case are

      ρϕ=3n2t2ρ0t3n(1+ω)+54αn2(2n1)t4,

      (70)

      and

      pϕ=n(23n)t2+18αn(2n1)(43n)t4ωρm0t3n(1+ω),

      (71)

      where ρϕ and pϕ is the energy density and pressure of the scalar field, and n is synonymous to m for our case.

      Rearranging Eq. (70), we get

      ρ0t3n(1+ω)=3n2t2+108αn3t454αn2t412˙ϕ2V(ϕ),

      (72)

      where they have defined, ρϕ=12˙ϕ2+V(ϕ), pϕ=12˙ϕ2V(ϕ), and V(ϕ) is the scalar potential.

      Singh et al. [22] used a canonical Lagrangian and the usual field equations of f(R)-gravity, but, in our case, we have used a non-canonical Lagrangian and the corresponding field equations (27). This is the basic difference between these two studies. Notably, the scalar field of each is not identical with the K-essence scalar field.

      Now, let us concentrate upon the deceleration parameter using the expression

      q=1H2¨aa=1m1,

      (73)

      where we have used Eq. (57).

      From the above expression, it is clear that for our present epoch, the deceleration parameter should have a negative value to support the acceleration of the universe. Therefore, we can conclude from Eq. (73) that the m takes a value greater than1. A negative value of m cannot be considered since observations show the universe is expanding.

      As we know, the value of ˙ϕ2 is less than 1, so neglecting the higher order terms O(˙ϕ4) in Eq. (61) and (63), and using Eq. (69), we get the equation of state (EOS) parameter of this scenario as:

      ω=(2+13m)t2+α(648+246m1260m296m3)3mt2+αm(810180m864m2).

      (74)

      The variation of ρ and p (using Eqs. (61) and (63) and omitting O(˙ϕ4) terms) with time (t) has been plotted in Fig. 1 for different choices of the positive power law parameter (m=1.5,2) and the positive coefficient of ˉR2 in the Starobinsky model (α=1,3,5,7,9). Figure 2 shows the variation of the EOS parameter, ω, with t for the aforementioned values of m and α. As we know, the values of ρ and p should differ in signature for a dark-energy dominated era and simultaneously the value of the EOS parameter (ω) should approach a value close to1.Therefore, the above two figures conclude that the choice of positive m and positive α is ruled out for our model to produce dark energy conditions. The time, t, here is the cosmological time, i.e., the time corresponding to the FLRW metric.

      Figure 1.  (color online) Variation of ρ and p with t for different values of m (=1.5,2) and α (=1,3,5,7,9).

      Figure 2.  (color online) Variation of ω with t for different values of m (=1.5,2) and α (=1,3,5,7,9).

      The negative values of α hav already been considered in [80] for f(R,T) gravity. So, let's check the results of our model for a positive value of m(=2,3) and negative values of α(=0.9,0.7,0.5,0.3,0.1). Figure 3 depicts the variation of ρ and p with time (t) for the above parameter values. Figure 4 shows the variation of the EOS parameter (ω) with time (t). From Fig. 3, it is evident that the value of ρ and p have the expected nature at a particular region of time. Simultaneously, Fig. 4 produces the anticipated value of ω, which is 1 for the dark energy epoch. We discuss the results more elaborately obtained in Figs. 3 and 4 in the following subsection.

      Figure 3.  (color online) Variation of ρ and p with t for different values of m (=3,2) and α(=0.9,0.7,0.5,0.3,0.1).

      Figure 4.  (color online) Variation of ω with t for different values of m (=2,3) and α(=0.1,0.3,0.5,0.7,0.9).

    • A.   Observational verification of the model

    • Before entering into this section, we would like to discuss two significant works, one was done by Tripathi et al. [81] and the other one was done by Moraes et al. [80]. In [81], they constrained the dark energy models for low redshifts and compared the data with the observations of Supernova Type Ia data, Baryon Acoustic Oscillation data, and Hubble parameter measurements. On the other hand, in [80], the authors studied various cosmological aspects with the help of the Starobinsky model in the framework off(R,T) gravity. They found the nature of material content of the universe, i.e. ρ and p in both decelerated and accelerated regimes of the universe.

      The variation of ρ and p with time obtained in Fig. 3 is quite similar with the variation obtained in [80], though our models differ from each other. Figure 3 shows that at early time (t0), the pressure was positive. But, after a certain value of time, it takes negative value, which may be correlated with the effect of the negative pressure fluid responsible for the accelerating universe. We have shown a table which depicts that for different choices of the positive m and the negative α, and we get a range of t (from Fig. 4) where the value of ω agrees with the observational data of Supernova Type Ia data, Baryon Acoustic Oscillation data, and Hubble parameter measurements (Observational data are taken from [81]). Furthermore, if we concentrate on the range of t that has been shown in Table 1 and match those values with Fig. 3, then can be observed that at those particular time regions, the value of p takes the negative sign, whereas the ρ is positive.

      mαtω (3σ confidence)Observation
      20.914.9615.50.95ω1.13SNIa+ BAO+ H(z)
      322.823.87
      20.713.213.67
      320.121.05
      20.511.1611.55
      317.0617.8
      20.38.648.95
      313.2213.78
      20.14.985.16
      37.637.97

      Table 1.  Table for Observational Verification of the Model.

    VI.   ENERGY CONDITIONS IN f(ˉR,L(X))-GRAVITY
    • With the help of the modified field equation (27) for f(ˉR,L(X)) theory, the emergent Einstein's equation (15) can be written as (κ=1)

      ˉRμν12ˉGμνˉR=Teffμν,

      (75)

      where

      Teffμν=1F[12fLˉTμν12ˉRFˉGμν(ˉGμνˉDμDν)F+12ˉGμν(fLfL(1+DαϕDαϕ))12LfLDμϕDνϕ(1+DαϕDαϕ)]=1F[12fLˉTμν+12ˉGμν(fFˉR)(ˉGμνˉDμDν)F12LfLˉGμν(1+DμϕDμϕ)2],

      (76)

      with F=fˉR=f(ˉR,L(X))ˉR.

      The trace of the effective energy momentum tensor (76) is

      Teff=1F[12fLˉT+2(fFˉR)3ˉF2LfL(1+DμϕDμϕ)2].

      (77)

      Now, from Eq. (75), we have the emergent Ricci tensor in terms of the effective energy momentum tensor as

      ˉRμν=Teffμν12ˉGμνTeff.

      (78)

      Let ˉuμ be the tangent vector field to a congruence of time-like geodesics in the K-essence emergent space-time manifold endowed with the metric ˉGμν (ˉGμνˉuμˉuν=1), then the strong energy condition (SEC) (107) in f(ˉR,L(X)) modified gravity can be expressed as

      ˉRμνˉuμˉuν=(Teffμνˉuμˉuν12Teff)0.

      (79)

      On the other hand, if we consider ˉkμ be the tangent vector along the null geodesic congruence (ˉGμνˉkμˉkν=0), then the null energy condition (NEC) (109) in f(ˉR,L(X)) gravity is

      ˉRμνˉkμˉkν=Teffμνˉkμˉkν0.

      (80)

      So, considering an additional condition [19] fL(ˉR,L)fˉR(ˉR,L)>0, and the K-essence scalar field to be homogeneous, i.e., ϕ(xi,t)ϕ(t), and using the perfect fluid energy momentum tensor (44), we have the SEC and NEC in the f(ˉR,L(X)) gravity are

      SEC:ˉρ+3ˉp2fL(fFˉR)+2L(1+˙ϕ2)2+6fL(1˙ϕ2)[¨F(123˙ϕ2)+H˙F(1˙ϕ2+23˙ϕ4)]0,

      (81)

      NEC:ˉρ+ˉp+2fL(¨FH˙F˙ϕ2)0,

      (82)

      where ˉρ=ρ(1+˙ϕ2) and ˉp=p(1+˙ϕ2).

      To evaluate the effective density, ˉρeff, and effective pressure, ˉpeff, in the K-essence emergent f(ˉR,L(X)) gravity, we consider the two following equations

      Teffμνˉuμˉuν12ˉGμνTeffˉuμˉuν=ˉρeff+3ˉpeff,

      (83)

      and

      Teffμνˉkμˉkν=ˉρeff+ˉpeff.

      (84)

      Solving these Eqs. (49) and (50), we get

      ˉρeff=ˉρ+1fL(fFˉR)6fL(1˙ϕ2)[13¨F˙ϕ2+H˙F(113˙ϕ4)]L(1+˙ϕ2)2,

      (85)

      ˉpeff=ˉp1fL(fFˉR)+3fL(1˙ϕ2)[13¨F(2˙ϕ2)+H˙F(123˙ϕ2+13˙ϕ4)]+L(1+˙ϕ2)2.

      (86)

      From the above Eqs. (85) and (86), we have WEC and DEC, respectively, for the K-essence emergent f(ˉR,L(X)) gravity as

      WEC:ˉρ+1fL(fFˉR)6fL(1˙ϕ2)[13¨F˙ϕ2+H˙F(113˙ϕ4)]L(1+˙ϕ2)20,

      (87)

      DEC:ˉρˉp+2fL(fFˉR)2fL(1˙ϕ2)[¨F(1+12˙ϕ2)+3H˙F(3213˙ϕ216˙ϕ4)]2L(1+˙ϕ2)20.

      (88)

      These energy conditions (81), (82), (87), and (88) of the K-essence emergent f(ˉR,L(X)) gravity are different from the usual f(R,Lm)-gravity (111) and f(R)-gravity (110) in the presence of the K-essence scalar field, ϕ. Also, note that if we consider f(ˉR,L(X))R and ˉGμνgμν, then we can get back to the usual energy conditions of GR, i.e., SEC:ρ+3p0; NEC:ρ+p0; and WEC:ρ0 and DEC:ρ|p|.

      One may notice that we briefly have discussed the energy conditions of f(R) and f(R,Lm) gravity in the Appendix.

    • A.   Constraints on K-essence emergent f(ˉR,L(X))-gravity

    • The inequalities of the energy conditions (81), (82), (87), and (88) can also be expressed in terms of the deceleration (q), jerk (j), and snap (s) parameters such that the Ricci scalar and its derivatives for a spatially flat K-essence emergent FLRW geometry (37) are

      ˉR=61˙ϕ2[˙H+H2(2˙ϕ2)]=6H21˙ϕ2[1q˙ϕ2],

      (89)

      ˙ˉR=61˙ϕ2[¨H+4H˙H(1˙ϕ2)2H3˙ϕ2]=6H31˙ϕ2[(jq2)+2˙ϕ2(12q)],

      (90)

      ¨ˉR=61˙ϕ2[(˙¨H+4˙H2+4H¨H)2˙ϕ2(2˙H2+3H¨H2H4+3H2˙H]=6H41˙ϕ2[(s+q2+8q+6)2˙ϕ2(3+3j+10q+2q2)],

      (91)

      where [11, 19, 8285]

      q=1H2¨aa;j=1H3˙¨aa;s=1H4¨¨aa.

      (92)

      Now, from Eq. (51), we evaluate the values of ˙F and ¨F (using Eq. (38)) in terms of q,j,ands as

      ˙F=6H31˙ϕ2FˉR[(jq2)+2˙ϕ2(12q)]HFLLXX˙ϕ2(1˙ϕ2),

      (93)

      ¨F=6H41˙ϕ2FˉR[(s+q2+8q+6)2˙ϕ2(3+3j+10q+2q2)]+36H6(1˙ϕ2)2FˉRˉR[(jq2)+2˙ϕ2(12q)]2+H˙ϕ4(1˙ϕ2)2(FLLL2X+FLLXX)FLLX˙ϕ2(1˙ϕ2)[˙H2H2(12˙ϕ2)].

      (94)

      Therefore, putting these values of ˙F and ¨F into the energy conditions (81), (82), (87), and (88), we have the energy conditions in terms of q, j, and s. We can easily check that these energy conditions in terms of q, j, and s are also different from the f(R,Lm)-gravity [19] in the presence of the K-essence scalar field, ϕ.

    • B.   An example of the energy conditions

    • Considering the Starobinsky Model, i.e., Eq. (54), we obtain the following results as fL=1, F=1+2αˉR, ˙F=2α˙ˉR, and ¨F=2α¨ˉR.

      Using these results we get the energy conditions from (81), (82), (87), and (88) as follows:

      SEC:ˉρ+3ˉp+2[αˉR2+L˙ϕ2(2+˙ϕ2)]+12α¨ˉR1˙ϕ2(123˙ϕ2)+12αH˙ˉR1˙ϕ2(1˙ϕ2+23˙ϕ4)0

      (95)

      NEC:ˉρ+ˉp+4α(¨ˉRH˙ˉR˙ϕ2)0

      (96)

      WEC:ˉραˉR24α1˙ϕ2[¨ˉR˙ϕ23˙ˉR(113˙ϕ4)]L˙ϕ2(2˙ϕ2)0

      (97)

      DEC:ˉρˉp2αˉR24α1˙ϕ2[¨ˉR(112˙ϕ2)3H˙ˉR(3213˙ϕ216˙ϕ4)]2L˙ϕ2(2+˙ϕ2)0

      (98)

      Again, if we put the values of ˉR, ˙ˉR, and ¨ˉR from (89), (90), and (91) into the above equations (95), (96), (97), and (98), we easily reconstruct the energy conditions in terms of the deceleration (q), jerk (j), and snap (s) parameters.

    VII.   DISCUSSION AND CONCLUSION
    • In this work, we present a new type of modified theory, viz. f(ˉR,L(X))-gravity, with a general formalism in the context of dark energy (using K-essence emergent geometry) where ˉR is the Ricci scalar of this geometry, L(X) is the DBI type non-canonical Lagrangian with X=12gμνμϕνϕ, and ϕ is the K-essence scalar field. The K-essence emergent metric, ˉGμν, is not conformally equivalent to the gravitational metric, gμν. This new type of modified theory is a general mixing between f(R) gravity and K-essence emergent gravity based on the DBI model.

      Let us discuss some salient features of the present study which are as follows:

      (1) It is to be noted that the modified field equation (27) is different from the usual f(R) and f(R,Lm) gravities. If we consider f(ˉR,L(X))R and ˉGμνgμν, then we can easily get back to the standard Einstein field equation. The effective functional relation for the requirement of the conservation of the energy-momentum tensor is also different from f(R,Lm)-gravity. We derive the modified Friedmann equations for the f(ˉR,L(X))-gravity considering the background gravitational metric as flat FLRW and the K-essence scalar field, ϕ, being simply a function of time only, which are quite different from the Friedmann equations of the standard f(R) gravity.

      (2) For the particular choice (viz. Starobinksy-type), Eq. (54) of f(ˉR,L) and from the requirement of the energy-momentum conservation (32), the kinetic energy of the K-essence scalar field is a constant. This value of ˙ϕ2(=0.888) is less than unity, which is comparable with the range of ˙ϕ2. It is also noted that the K-essence theory can be used to investigate the effects of the presence of dark energy on cosmological scenarios. In this context, if we consider ˙ϕ2 to be dark energy density in unit of critical density as [4042], then the value of dark energy density, i.e., ˙ϕ2=89=0.888, indicates that the present universe is dark-energy dominated.

      It is well known that the present observational value [86, 87] of dark energy density is approximately 0.75. Therefore, we note that in the context of the dark energy regime, our result is in good agreement with observational data. Nowadays, people believe that dark energy is one of the reasons for the accelerating universe. So, our value of dark energy density may indicate that the universe is more accelerating. From Figs. 3 and 4, we also observe that our model is observationally verified for certain values of parameters. According to Fig. 3, the negativity of pressure can be achieved after a certain value of time, which may be responsible for the accelerating universe. The variation of EOS (ω) with time (t) in Fig. 4 shows that ω approaches negative values, which corresponds to observational results [81] of the present universe.

      Also, we would like to put here the following two special aspects which emerge from the present investigation: (i) This model can open up an alternative window to explore the current cosmic acceleration without a stringent condition of invoking an exotic component as the dark energy. In other words, this theory seems interesting from a purely gravitational theory standpoint, rather than the cosmological context of dark energy whose very existence is still an issue of doubt [88] according to the latest analysis of data from the Planck consortium [86, 87]. However, the arbitrariness in the choice of different functional forms of f(ˉR,L(X)) based on a DBI Lagrangian gives rise to the problem of how to constrain the many possible f(ˉR,L(X)) gravity theories on physical grounds. In this context, we have shed some light on this issue by discussing some constraints on general f(ˉR,L(X)) gravity from the so-called energy conditions. (ii) Also, we have derived the null, strong, weak, and dominant energy conditions in the framework of f(ˉR,L(X)) gravity from the Raychaudhuri equations. These energy conditions are different from the usual f(R,Lm) and f(R) theories. With the help of the specific form of f(ˉR,L(X)), we also have derived these energy conditions in f(ˉR,L(X))-gravity.

      However, we intend to report on this interesting theory in the near future encompassing the multifarious aspects of cosmology.

    ACKNOWLEDGEMENTS
    • All of the authors would like to thank the referee for illuminating suggestions to improve the manuscript.

    APPENDIX

      A.   The modified field equation in f(R,Lm)-Gravity

    • Let us consider here the well known f(R) and f(R,Lm) gravity, where R is the Ricci scalar with respect to the gravitational metric, gμν, and Lm is the matter Lagrangian. The total action for the f(R) gravity is [6, 7]

      S=12κd4xgf(R)+SM(gμν,ψ),

      where SM is the matter term, ψ denotes the matter fields, κ=8πG, G is the gravitational constant, g is the determinant of the gravitational metric, and R(=gμνRμν) is the Ricci scalar.

      Varying with respect to the gravitational metric, we achieve the modified field equation as

      f(R)Rμν12f(R)gμν[μνgμν]f(R)=κTμν,

      with

      Tμν=2gδSMδgμν,

      where f(R)=f(R)R, μ is covariant derivative with respect to the gravitational metric, and μμ.

      On the other hand, the action for the f(R,Lm) gravity is [18, 19]

      S=12κd4xgf(R,Lm),

      where f(R,Lm) is an arbitrary function of the Ricci scalar R, and the Lagrangian density corresponding to matter, Lm. The energy-momentum tensor is

      Tμν=2gδ(gLm)δgμν=2Lmgμν+gμνLm,

      where the Lagrangian density, Lm, is only matter dependent on the metric tensor components, gμν.

      The modified field equations of the f(R,Lm)-gravity model is

      fR(R,Lm)Rμν+(gμνμν)fR(R,Lm)12[f(R,Lm)LmfLm(R,Lm)]gμν=12fLm(R,Lm)Tμν,

      where fR(R,Lm)=f(R,Lm)/R and fLm(R,Lm)=f(R,Lm)/Lm. However, if f(R,Lm)=R/2+Lm, then the above Eq. (A6) reduces to the usual field equation Rμν(1/2)gμνR=κTμν.

    • B.   Brief review of energy conditions in General Relativity

    • Following most of the techniques of [1014, 19, 73], we will derive the energy conditions for modified (f(R), f(R,Lm), etc.) gravities. From these theories we can approach the Null Energy Condition (NEC) and Strong Energy Condition (SEC) in the context of GR. The origin of these energy conditions comes from the Raychaudhuri equations. Let uμ be the tangent vector field to a congruence of time-like geodesics in a space-time manifold endowed with a metric, gμν. Therefore, the Raychaudhuri equation [8994] is

      dθdτ=13θ2σμνσμν+ωμνωμνRμνuμuν,

      where Rμν is the Ricci tensor corresponding to the metric gμν, and θ, σμν, and ωμν are the expansion, shear, and rotation associated with the congruence, respectively. While in the case of a congruence of null geodesics defined by the vector field, kμ, the Raychaudhuri equation [92] is given by

      dθdτ=12θ2σμνσμν+ωμνωμνRμνkμkν.

      These equations are purely based on geometric statements, and as such it makes no reference to any gravitational field equations. In other words, the Raychaudhuri equation can be thought of as geometrical identities, which do not depend on any gravitational theory. These equations provide the evolution of the expansion of a geodesic congruence. However, since the GR field equations relate Rμν to the energy-momentum tensor, Tμν, the combination of Einstein and Raychaudhuri equations can be used to restrict energy-momentum tensors on physical grounds. Indeed, the shear is a "spatial" tensor given by σ2σμνσμν0.

      Thus, it is clear from the Raychaudhuri equation that for any hypersurface orthogonal congruences (ωμν0), the condition for attractive gravity (convergence of timelike geodesics or geodesic focusing) reduces to (Rμνuμuν0), which by virtue of Einstein’s equation implies

      Rμνuμuν=(TμνT2gμν)uμuν0,

      where T is the trace of the energy momentum tensor, Tμν (κ=1). Here, Eq. (B3) is nothing but the SEC stated in a coordinate-invariant way in terms of Tμν and vector fields of fixed (time-like) character. Thus, in the context of GR, the SEC ensures the fact that the gravity is attractive. In particular, for a perfect fluid of density, ρ, and pressure, p,

      Tμν=(ρ+p)uμuνpgμν

      and the restriction given by Eq. (B3) takes the familiar form for the SEC, i.e., ρ+3p0.

      On the other hand, the condition for the convergence (geodesic focusing) of hypersurface orthogonal (ωμν0) congruences of null geodesics along with Einstein’s equation implies

      Rμνkμkν=Tμνkμkν0

      which is the condition for NEC written in a coordinate-invariant way.

      Thus, in GR the NEC ultimately encodes the null geodesic focusing due to the gravitational attraction. For the energy-momentum tensor of a perfect fluid (B4), the above condition (B5) reduces to the well-known form of the NEC, i.e., ρ+p0.

      The Weak Energy Condition (WEC) states that Tμνuμuν0 for all time-like vectors, uμ, or equivalently for perfect fluid it is ρ>0 and ρ+p>0. The Dominant Energy Condition (DEC) includes the WEC, as well as the additional requirement that Tμνuμ is a non space-like vector, i.e., TμνTνλuμuλ0. For a perfect fluid, these conditions, together, are equivalent to the simple requirement that ρ|p|, the energy density must be non-negative, and greater than or equal to the magnitude of the pressure.

      In f(R)-gravity [10, 11], the energy conditions for perfect fluid are given by

      SEC:ρ+3pf+Rf+3(¨R+˙RH)f"+3˙R2f"0,NEC:ρ+p+(¨R˙RH)f"+˙R2f"0,WEC:ρ+12(fRf)3˙RHf"≥0,

      DEC:ρp+fRf(¨R+5˙RH)f"˙R2f"0,

      where f=f(R)R.

      In f(R,Lm)-gravity [19], the energy conditions are

      SEC:ρ+3p2fLm[fRf]+6fLm[˙R2f"+¨Rf"+H˙Rf"]2Lm0.NEC:ρ+p+2fLm[˙R2f"+¨Rf"]0,WEC:ρ+1fLm[fRf]6fLmH˙Rf"+Lm0,DEC:ρp+2fLm[fRf]2fLm[˙R2f"+¨Rf"+6H˙Rf"]+2Lm0,

      where f=f(R,Lm)R.

Reference (94)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return