Noether Symmetry Approach in Eddington-inspired Born-Infeld gravity
Thanyagamon Kanesom, Phongpichit Channuie, Narakorn Kaewkhao
aa r X i v : . [ g r- q c ] F e b Noether Symmetry Approach in Eddington-inspiredBorn-Infeld gravity
Thanyagamon Kanesom, a Phongpichit Channuie, b,c,d
Narakorn Kaewkhao a a Department of Physics, Faculty of Science, Prince of Songkla University, Hatyai 90112, Thailand b School of Science, Walailak University, Nakhon Si Thammarat, 80160, Thailand c College of Graduate Studies, Walailak University, Nakhon Si Thammarat, 80160, Thailand d Research Group in Applied, Computational and Theoretical Science (ACTS),Walailak University, Nakhon Si Thammarat, 80160, Thailand
E-mail: [email protected] , [email protected] , [email protected] Abstract:
In this work, we take a short recap of a formal framework of the Eddington-inspired Born-Infeld (EiBI) theory of gravity and derive the point-like Lagrangian for un-derlying theory based on the use of Noether gauge symmetries (NGS). We study a Hessianmatrix and quantify Euler-Lagrange equations of EiBI universe. We discuss the NGS ap-proach for the Eddington-inspired Born-Infeld theory and show that there exists the deSitter solution in this gravity model.
Keywords:
Noether Symmetry Approach, Exact Solutions, Eddington-inspired Born-Infeld gravity ontents
Various cosmological observations make a strong evidence that the expansion of the universeis presently accelerating. These experimental results include Type Ia Supernovae [1, 2],cosmic microwave background (CMB) radiation [3–9], large scale structure [10], baryonacoustic oscillations (BAO) [11] as well as weak lensing [12]. An expansion phase canbe basically explained by the simplest model: the so-called Lambda cold dark matter(ΛCDM) [13]. However, the ΛCDM model is plagued by the cosmological problem [14]and the coincident problem [15]. There are at least two promising explanations to date todescribe the late-time cosmic acceleration. The first one assumes the introduction of theso-called “dark energy (DE)” in the context of conventional general relativity. Anotherconvincing approach is to engineer Einstein gravity on the large-scale methodology (see forreviews on not only dark energy problem but also modified gravity theories, e.g., [16–20]).However, the DE sector remains still unknown and possesses one of the unsolved problemsin physics.Therefore, it opens opportunities to search for modified theories of gravity to deal withsuch problems. By modifying the geometrical part of Einstein field equations or addingscalar field to the right-hand side of the Einstein field equations, both alternatives are ableto explain effects of dark ingredients with acceptable assertions [21]. One of the simplestmodifications to the standard general relativity is the f ( R ) theories of gravity in which theLagrangian density f is an arbitrary function of the scalar curvature R [22, 23]. Amongnumerous alternatives, these theories include higher order curvature invariants, see rigorousreviews on f ( R ) theories [24, 25].In cosmological framework, the Noether symmetry (NS) approach has revealed a usefultool not only to fix physically viable cosmological models with respect to the conservedquantities, but also to reduce dynamics and achieve exact solutions [26, 27]. Moreover, theexistence of Noether symetries plays crucial roles when studying quantum cosmology [28].– 1 –he Noether symmetry approach has been employed to various cosmological scenarios sofar including the f ( T ) gravity [29], the f ( R ) gravity [30], the alpha-attractors [31], sphericaland cylindrical solutions in f ( T ) gravity [32], f ( G ) gravity [33], non-local curvature andGauss–Bonnet cosmologies [34], and others cosmological scenarios, e.g. [35–41]. The studyof Palatini f ( R ) cosmology using the NS approach for the matter-dominated universe wascarried out in Ref.[42, 43]. Moreover, the exact solutions for potential functions, scalarfield and the scale factors in the Bianchi models have been investigated in [44, 45].Apart from the NS approach [26, 27], the Noether Gauge Symmetry (NGS) [46–48]is more generalized. In this work, we examine a formal framework of Eddington-inspiredBorn-Infeld (EiBI) gravity through the NGS approach and present a detailed calculationof the point-like Lagrangian. Notice that the point-like Lagrangian derived from the alter-native form of the EiBI action was proposed by Delsate and Steindoff [49] instead of usingthe original form of the EiBI action suggested by M´aximo Ba˜nadoz [50].This paper is organized as follows: We will start by making a short recap of a formalframework of the Eddington-inspired Born-Infeld theory of gravity in Sec.2. Here we derivethe point-like Lagrangian for underlying theory. In Sec.3, we study a Hessian matrix andquantify Euler-Lagrange equations of EiBI universe. In Sec.4, the NGS approach for theEddington-inspired Born-Infeld theory is discussed. We comment on exact cosmological so-lutions of the EiBI theory based on the use of Noether symmetries of point-like Lagrangiansin Sec.5. Finally, we conclude our findings in the last section. In 2009, M´aximo Ba˜nadoz [50] proposed a new form of the Born-Infeld action under Palatiniformalism. This is the so-called Eddington-inspired-Born-Infeld (EiBI) gravity. This actionis written as follows: S EiBI ( g, Γ) = 2 κ Z d x hq | g µν + κR µν (Γ) | − λ q | g µν | i + S m ( g µν , Ψ) + S φ ( g µν , φ ) , (2.1)where λ = 1 + κ Λ is a dimensionless constant displaying the relation between the EiBIfree parameter κ (with a dimension of M P ) and the cosmological constant Λ (with adimension of M − P ); whilst S m ( g µν , Ψ) and S φ ( g µν , φ ) represent the matter field action andthe scalar field action, respectively. Throughout this work, we set πG N c = 1 . Performingvariation of Eq.(2.1) with respect to Γ λµν gives the relation between two metric tensors, i.e. q µν = g µν + κR µν (Γ) . Hence equation (2.1) can be written in the bi-metric form as S EiBI ( g, Γ) = 2 κ Z d x "q | q µν | − λ q | g µν | + S m ( g µν , Ψ) + S φ ( g µν , φ ) . (2.2)With the help of two ansatz forms of a spatially flat FLRW metric ds g = g µν dx µ dx ν = − N ( t ) dt + a ( t ) d~x , (2.3) ds q = q µν dx µ dx ν = − M ( t ) dt + b ( t ) d~x , (2.4)– 2 –he EiBI action (2.2) can be expressed in terms of the cosmological variables as follows: S EiBI = 2 κ υ Z dt L EiBI = 2 υ κ Z dt "s a h N − κ ( ˙ M ˙ bM b − ¨ bb ) ih κa M ( − ¨ bb + 2˙ b − b ˙ b ˙ MM ) i − N a (1 + κ Λ) − N a h − ρ m ( a ) + l ˙ φ N − V ( φ ) i , (2.5)where υ is the spatial volume obtaining after a proper compactification for spatial flatsection. l = +1 and l = − S EiBI ( g, q ) = λ Z d x √− q " R ( q ) − λκ + 1 κ q αβ g αβ − r gq ! + S φ ( g, φ ) + S M ( g, Φ) . (2.6)Notice that only g µν interacts with matter and scalar fields whereas q µν defines the back-ground metric regarded as the fundamental reference frame of the universe [51]. Becausethe alternative action yields identical field equations as provided in the Ba˜nadoz action,this indicates that two action forms are equivalent. Using a relation, S = ν R dt L [52] andthe relation between two metric tensors, q µν = g µν + κR µν (Γ), the point-like Lagrangiancan be extracted from equation(2.6) as follows L EiBI = λM b " − b M b − λκ + 1 κ N M + 3 a b ! + N a l ˙ φ N − V ( φ ) − ρ m ( a ) ! . (2.7)Here the number of configuration space (or the minisuperspace) variables equal to five dueto the appearance of variables { a ( t ) , b ( t ) , M ( t ) , N ( t ) , φ ( t ) } in Eq.(2.7). Apart from thekinetic part of the Lagrangian, we can set V eff = λMb κ h λ − (cid:16) N M + 3 a b (cid:17)i + 2 N a (cid:16) V ( φ ) + ρ m ( a ) (cid:17) as an effective potential in the gravity model. It also notes that Eq.(2.7) is asingular Lagrangian due to the existence of two Lapse functions, N ( t ) and M ( t ) as shownin the denominators of Eq.(2.7). – 3 – Hessian matrix and Euler-Lagrange equations of EiBI universe
In the absent of { ˙ a, ˙ M , ˙ N } in the EiBI point-like Lagrangian, the EiBI Hessian matrix canbe written as[ W ij ] EiBI = ∂ L∂ ˙ a ∂ L∂ ˙ a∂ ˙ b ∂ L∂ ˙ a∂ ˙ N ∂ L∂ ˙ a∂ ˙ M ∂ L∂ ˙ a∂ ˙ φ∂ L∂ ˙ b∂ ˙ a ∂ L∂ ˙ b ∂ L∂ ˙ b∂ ˙ N ∂ L∂ ˙ b∂ ˙ M ∂ L∂ ˙ b∂ ˙ φ∂ L∂ ˙ N∂ ˙ a ∂ L∂ ˙ N∂ ˙ b ∂ L∂ ˙ N ∂ L∂ ˙ N∂ ˙ M ∂ L∂ ˙ N∂ ˙ φ∂ L∂ ˙ M∂ ˙ a ∂ L∂ ˙ M∂ ˙ b ∂ L∂ ˙ M∂ ˙ N ∂ L∂ ˙ M ∂ L∂ ˙ M∂ ˙ φ∂ L∂ ˙ φ∂ ˙ a ∂ L∂ ˙ φ∂ ˙ b ∂ L∂ ˙ φ∂ ˙ N ∂ L∂ ˙ φ∂ ˙ M ∂ L∂ ˙ φ = − λbM a lN . (3.1)Clearly, the determinant of the Hessian matrix of the EiBI point-like Lagrangian equals zeroindicates again that Eq.(2.7) is a singular Lagrangian. Accordingly, variables { a, N, M } donot contribute to dynamics and have to be considered as a further constraint equations.This tells us that a ( t ) , M ( t ) and N ( t ) are not independent variables anymore then we canset them to an arbitrary functions of time[53], i.e. F ( t ) = F ( a, M, N ). Variables b ( t ) and φ ( t ) however remain considered independently. With the definition of the Euler-Lagrangeequations, we can show that [54] ddt ∂ L ∂ ˙ q i − ∂ L ∂q i = ∂∂q j (cid:16) ∂ L ∂ ˙ q i (cid:17) dq j dt + ∂∂ ˙ q j (cid:16) ∂ L ∂ ˙ q i (cid:17) d ˙ q j dt − ∂ L ∂q i , = ¨ q j ∂ L ∂ ˙ q j ∂ ˙ q i + ˙ q j ∂ L ∂q j ∂ ˙ q i − ∂ L ∂q i = 0 , (3.2)¨ q j ∂ L ∂ ˙ q j ∂ ˙ q i = − ˙ q j ∂ L ∂q j ∂ ˙ q i + ∂ L ∂q i . (3.3)Here the configuration space variables are q i = { a, b, M, N, φ } and their time derivativeon the tangent space are ˙ q i = dq i dt = { ˙ a, ˙ b, ˙ M , ˙ N , ˙ φ } . Because ∂ L ∂ ˙ q j ∂ ˙ q i = 0 and ∂ L ∂q j ∂ ˙ q i forvariables { a, M, N } in EiBI gravity, the Euler-Lagrange equations of these variables canbe reduced to ∂ L ∂a = ∂ L ∂M = ∂ L ∂N = 0 where ˙ a, ˙ M , ˙ N and ¨ a, ¨ M , ¨ N can be set arbitrarily [54].As expected, ¨ b and ¨ φ are determined by performing variation equation(2.7) with respectto the dynamical variables and their time derivative as expressed on the right-hand sideof equation(3.2). As expected, ¨ b and ¨ φ are determined from taking variation Lagrangianwith respect to the dynamical variables and their time derivative as shown on the right-hand side of Eq.(3.3). We have to keep in mind that a crucial concept of a gauge theoryis the general solution of the equations of motion which contains arbitrary functions oftime and the canonical variables are not all independent but relate among each othersvia the constraint equations [54]. As the results of the vanishing of ∂ L ∂ ˙ a∂ ˙ a ≡ ∂ L ∂ ˙ a , ∂ L ∂ ˙ M and ∂ L ∂ ˙ N , the canonical momenta associated to a, N, and M yield p a = ∂ L /∂ ˙ a = 0 , p N = ∂ L /∂ ˙ N = 0 , p M = ∂ L /∂ ˙ M = 0, respectively. The Hamiltonian constraint equation canbe straightforwardly derived from the canonical momenta via the Lagrangian and theirsLagrange multipliers { λ i = λ a ( t ) , λ M ( t ) , λ N ( t ) } as follows: H EiBI = ∂ L EiBI ∂ ˙ q i ˙ q i − L EiBI = ∂ L EiBI ∂ ˙ b ˙ b + ∂ L EiBI ∂ ˙ φ ˙ φ − L EiBI , (3.4)– 4 – EiBI , tot = H EiBI + Σ λ i p i , = (cid:16) − λb ˙ bM (cid:17) ˙ b + (cid:16) a l ˙ φN (cid:17) ˙ φ − L EiBI + λ a p a + λ N p N + λ M p M = 0 , = h − λb ˙ b M + a l ˙ φ N i + 1 κ h λ M b − λb N M − λa M b +2 κN a V ( φ ) + 2 κρ ( a ) N a i + λ a p a + λ M p M + λ N p N = 0 , (3.5)where λ a ( t ) , λ M ( t ),and λ N ( t ) stand for Lagrange multipliers that are arbitrary functionsof time. The total EiBI Hamiltonian ( H EiBI ) can be used to evaluate an evolution invokingthe Hamiltonian equations of motion as follows:˙ a = ∂ H EiBI , tot ∂p a = λ a ( t ) , ˙ M = ∂ H EiBI , tot ∂p M = λ M ( t ) , ˙ N = ∂ H EiBI , tot ∂p N = λ N ( t ) , ˙ b = ∂ H EiBI , tot ∂p b = − p b M λb , ˙ φ = ∂ H EiBI , tot ∂p φ = p φ N a l , ˙ p b = − ∂ H EiBI , tot ∂b = 6 λ ˙ b M − κ h λ M b − λb N M − λa M i = 0 , ˙ p φ = − ∂ H EiBI , tot ∂φ = − κ h κN a V ′ ( φ ) i = − N a V ′ ( φ ) = 0 , ˙ p a = − ∂ H EiBI , tot ∂a = 3 a l ˙ φ N + 1 κ h λabM − κa N V ( φ ) + 6 κa p m ( a ) N i = 0 , ˙ p N = − ∂ H EiBI , tot ∂N = − a l ˙ φ N + 1 κ h λb NM − κa V ( φ ) − κρ m ( a ) a i = 0 , ˙ p M = − ∂ H EiBI , tot ∂M = − λb ˙ b M + 1 κ h − λ b − λb N M + 3 λa b i = 0 . (3.6)For the EiBI Hamiltonian, it should be noted that ˙ a = λ a , ˙ N = λ N , ˙ M = λ M are theprimary constraints and ˙ p a = ˙ p N = ˙ p M = 0 are the secondary constraints that must bevalid at all times [53] resulting in p a = p M = p N = 0. In order to obtain the dynamicsolutions, we have to calculate the Euler-Lagrange equations for a ( t ) , b ( t ) , M ( t ) , N ( t ), and φ ( t ) as shown below:3 a l ˙ φ N + 6 λabMκ − N a V ( φ ) + 6 N a p m ( a ) = 0 , (3.7)¨ b − ˙ b ˙ MM + ˙ b b − λM b κ + bN κ + a M κb = 0 , (3.8)6 λb ˙ b M − λ b κ + 3 λa bκ − λb κ N M = 0 , (3.9) − a l ˙ φ N + 2 λb NκM − a V ( φ ) − a ρ m ( a ) = 0 , (3.10)¨ φ + (cid:16) aa + ˙ NN (cid:17) ˙ φ + V ′ ( φ ) N l = 0 , (3.11)where the conservation equation [52], dρ m /da = − ρ m + p m ) /a, has been used to yieldEq.(3.7). – 5 – Noether Gauge Symmetries in EiBI Gravity
Noether vector (X
NGS ) and the first prolongation vector field (X [1]NGS ) related to EiBILagrangian, as shown in Eq.(2.7), can be constructed as follows:X
NGS = τ ∂∂t + α ∂∂a + β ∂∂b + γ ∂∂N + ξ ∂∂M + ϕ ∂∂φ , (4.1)X [1]NGS = X NGS + ˙ α ∂∂ ˙ a + ˙ β ∂∂ ˙ b + ˙ γ ∂∂ ˙ N + ˙ ξ ∂∂ ˙ M + ˙ ϕ ∂∂ ˙ φ , (4.2)where the undetermined parameters { τ ( t, q i ) , α ( t, q i ) , β ( t, q i ) , γ ( t, q i ) , ξ ( t, q i ) , ϕ ( t, q i ) } arepossibly functioned by { t, q i } = { t, a, b, N, M, φ } . Their time derivative can be defined as˙ α ( t, a, b, N, M, φ ) = D t α − ˙ a D t τ, (4.3)for variable α ( t ). This can be applied in the same way for other undetermined variables.The operator of a total differentiation (D t ) with respect to t in EiBI gravity can be definedas D t = ∂∂t + ˙ a ∂∂a + ˙ b ∂∂b + ˙ M ∂∂M + ˙
N ∂∂N + ˙ φ ∂∂φ . (4.4)It is worth noting that terms like, for example, ∂V ( φ ( t )) ∂t = ∂ ˙ a ( t ) ∂t = 0 , because there is nottime variable(t) shown explicitly in L EiBI . The vector field X [1]
NGS is a NGS of a Lagrangian L ( t, a, b, φ, M, N, ˙ b, ˙ φ ), if there exists a gauge function B( t, a, b, φ, M, N ) which obeys thefollowing condition (see Ref.[55] for explicit derivation):X [1] NGS L + L D t τ = D t B . (4.5)For NSG without gauge term and the prolongation part of the vector field, i.e. B( t, q i ) = 0,it requires that τ ( t, q i ) = 0 . Accordingly, Eq.(4.5) can be reduced to £ X NGS L = 0 that is thecondition for Noether symmetry [48]. After using the Noether gauge symmetries conditionwith the EiBI point-like Lagrangian, this provides us with eighty terms for X [1] NGS L + L D t τ =– 6 – t B as shown below:6 λabM ακ − λa N ακ − a N ρ ( a ) α − a N V ( φ ) α + 3 λa M βκ − λb M βκ + 3 λb N βκM − λa γκ + 2 λb N γκM − a ρ ( a ) γ − a V ( φ ) γ + 3 λa bξκ − λ b ξκ − λb N ξκM − λβ ˙ b M + 6 λbξ ˙ b M − a N αρ ′ ( a ) − a N ϕV ′ ( φ ) + 3 la α ˙ φ N − la γ ˙ φ N − λb ˙ bβ t M + 3 λa bM τ t κ − λ b M τ t κ − λa N τ t κ + λb N τ t κM − a N ρ ( a ) τ t − a N V ( φ ) τ t + 6 λb ˙ b τ t M − la ˙ φ τ t N + 2 la ˙ φϕ t N − λb ˙ b ˙ φβ φ M + 6 λa bM ˙ φτ φ κ − λ b M ˙ φτ φ κ − λa N ˙ φτ φ κ + 2 λb N ˙ φτ φ κM − a N ρ ( a ) ˙ φτ φ − a N V ( φ ) ˙ φτ φ + 12 λb ˙ b ˙ φτ φ M − la ˙ φ τ φ N + 4 la ˙ φ ϕ φ N − λb ˙ b ˙ M β M M + 6 λa bM ˙ M τ M κ − λ b M ˙ M τ M κ − λa N ˙ M τ M κ + 2 λb N ˙ M τ M κM − a N ρ ( a ) ˙ M τ M − a N V ( φ ) ˙ M τ M + 12 λb ˙ b ˙ M τ M M − la ˙ M ˙ φ τ M N + 4 la ˙ M ˙ φϕ M N − λb ˙ b ˙ N β N M + 6 λa bM ˙ N τ N κ − λ b M ˙ N τ N κ − λa N ˙ N τ N κ + 2 λb N ˙ N τ N κM − a N ρ ( a ) ˙ N τ N − a N V ( φ ) ˙ N τ N + 12 λb ˙ b ˙ N τ N M − la ˙ N ˙ φ τ N N + 4 la ˙ N ˙ φϕ N N − λb ˙ b β b M + 6 λa bM ˙ bτ b κ − λ b M ˙ bτ b κ − λa N ˙ bτ b κ + 2 λb N ˙ bτ b κM − a N ρ ( a )˙ bτ b − a N V ( φ )˙ bτ b + 12 λb ˙ b τ b M − la ˙ b ˙ φ τ b N + 4 la ˙ b ˙ φϕ b N − λb ˙ a ˙ bβ a M + 6 λa bM ˙ aτ a κ − λ b M ˙ aτ a κ − λa N ˙ aτ a κ + 2 λb N ˙ aτ a κM − a N ρ ( a ) ˙ aτ a − a N V ( φ ) ˙ aτ a + 12 λb ˙ a ˙ b τ a M − la ˙ a ˙ φ τ a N + 4 la ˙ a ˙ φϕ a N = B t + ˙ aB a + ˙ bB b + ˙ M B M + ˙ N B N + ˙ φB φ . If the Noether symmetry condition £ X NGS L EiBI = 0 is satisfied, then the function Σ = α i ∂ L ∂ ˙ q i is a constant of motion [56]. This givesΣ , EiBI = α i ∂ L ∂ ˙ q i = β ∂ L ∂ ˙ b + ϕ ∂ L ∂ ˙ φ = β h − λb ˙ bM i + ϕ h a l ˙ φN i , (4.6)where two unknown functions β and ϕ will be studied in the next section. Up to thispoint, it is worth mentioning a dimension analysis of each variable, i.e. [dimensionless] =[ λ ] = [l] = [a] = [b] = [ φ ] = [ N ] = [ M ]; [ α ] = [ β ] = [ γ ] = [ ξ ] = [ ϕ ] = [ τ ] = [ M − P ]; [ κ ] =[ M − P ]; [ ˙ a ] = [˙ b ] = [ ˙ φ ] = [ M p ]; [ ϕ t ] = [ β t ] = [ τ t ] = [ M − P ].– 7 – Remarks on exact cosmological solutions
After a separation of monomials, we can quantify the system equations to yield τ a = τ b = τ M = τ N = τ φ = 0 , (5.1) β a = β M = β N = 0 , (5.2) ϕ a = ϕ M = ϕ N = 0 , (5.3)0 = − λbβ φ M + la ϕ b N , (5.4)0 = 3 α − aγN − aτ t + 4 aϕ φ , (5.5)0 = − β + bξM + bτ t − bβ b , (5.6) B a = B M = B N = 0 (5.7) B b = − λbβ t M , (5.8) B φ = 2 la ϕ t N , (5.9) B t = α (cid:20) λabMκ − λa Nκ − a N ρ ( a ) − a N V ( φ ) − a N ρ ′ ( a ) (cid:21) + β (cid:20) λa Mκ − λ b Mκ + 3 λb N κM (cid:21) + γ (cid:20) − λa κ + 2 λb NκM − a ρ ( a ) − a V ( φ ) (cid:21) + ξ (cid:20) λa bκ − λ b κ − λb N κM (cid:21) + ϕ h − a N V ′ ( φ ) i + τ t (cid:20) λa bMκ − λ b Mκ − λa Nκ + λb N κM − a N ρ ( a ) − a N V ( φ ) (cid:21) . (5.10)From Eq.(5.2),Eq.(5.3 ) and Eq.(5.4), one found that β = β ( b ) and ϕ ( φ ) . If we choose β ( b ) = c b, (5.11) ϕ ( φ ) = c φ. (5.12)From Eq.(5.1), there is only one possibility left with τ t = 0 . Therefore the polynomialof α, β, ϕ, γ , and ξ no longer have hold in τ ( t ) . Proposing the linearity of the relationsexpressed in Eq.(5.5) and Eq.(5.6), we have to set α ( a ) = c a, (5.13) γ ( N ) = c N, (5.14) ξ ( M ) = c M. (5.15)With this setting, we can solve Eq.(5.5) and Eq.(5.6) to get τ ( t ) = (3 c − c + 4 c ) t + c , (5.16) τ ( t ) = (5 c − c ) t + c . (5.17)– 8 –n order to write τ ( t ) in a single form, we have to set c = c and 3 c − c + 4 c = 5 c − c . There are three equations contributed of gauge function, B b = − λbβ t M = − c λb ˙ bM , (5.18) B φ = 2 la ϕ t N = 2 c la ˙ φN , (5.19) B t = c (cid:20) λa bMκ − λa Nκ − a N ρ ( a ) − a N V ( φ ) − a N ρ ′ ( a ) (cid:21) + c (cid:20) λa bMκ − λ b Mκ + 3 λb N κM (cid:21) + c (cid:20) − λa Nκ + 2 λb N κM − a N ρ ( a ) − a N V ( φ ) (cid:21) + c (cid:20) λa bMκ − λ b Mκ − λb N κM (cid:21) +(3 c − c + 4 c ) (cid:20) λa bMκ − λ b Mκ − λa Nκ + λb N κM − a N ρ ( a ) − a N V ( φ ) (cid:21) − c a N φV ′ ( φ ) , (5.20)where we use β t = ∂β∂b dbdt = c ˙ b and ϕ t = ∂ϕ∂φ dφdt = c ˙ φ to get Eq.(5.18) and (5.19). We henceexpect that the rest of the expression for boundary term, i.e. − c λb ˙ bM , may relate with B t .From Eq.(5.4), Eq.(5.9) and Eq.(5.8), it is easy to see that6 λbM = − B b β t , la N = B φ ϕ t . (5.21)This gives the following relation, 6 λbβ φ M = la ϕ b N , (5.22) − B b β t β φ = B φ ϕ t ϕ b , (5.23)This confirms again that β φ = ϕ b = 0, but keeps β t = 0 and ϕ t = 0. That also means that B b = 0 and B φ = 0 . The boundary term can be also partly derived from Eq.(5.18) andEq.(5.19), that is B ( b,φ ) = − c λb ˙ bM + 2 c a lφ ˙ φN . (5.24)It is worth noting that c is just an arbitrary constant and we can redefine it by replacing c → c . Interestingly, this is exactly matched with the constant of motions of EiBIgravity by this setting. Σ , EiBI = − c λb ˙ bM + 2 c a lφ ˙ φN . (5.25)– 9 –he relation between the the constant of motion and the boundary term has shown in [55].Eq.(5.20) can be rewritten as B t = λa Nκ (cid:20) − c − c (cid:21) + a N ρ ( a ) (cid:20) − c − c (cid:21) + a N V ( φ ) (cid:20) − c − c (cid:21) + λa bMκ (cid:20) c + 12 c + 15 c − c + 3 c (cid:21) + λ b Mκ (cid:20) − c − c − c + 2 c − c (cid:21) + λb N κM (cid:20) c + 4 c + 3 c + c − c (cid:21) − a N (cid:20) c ρ ′ ( a ) + c V ′ ( φ ) (cid:21) . (5.26)where it is very naturally to set c = c and c = − c . It is quite remarkable that mostterms of B t become zero with the simple setting, i.e. B t = λa bMκ (cid:20) c + 2 c (cid:21) + λ b Mκ (cid:20) c + 2 c (cid:21) + λb N κM (cid:20) c + 2 c (cid:21) − c a N (cid:20) − ρ ′ ( a ) + V ′ ( φ ) (cid:21) . (5.27)If we further set that c = − c = − ( − ) c = c , it is worth seeing that B t = c a N (cid:20) − ρ ′ ( a ) + 3 V ′ ( φ ) (cid:21) , = 6 c a N (cid:20) ρ m ( a ) + P m ( a ) a + V ′ ( φ ) (cid:21) , (5.28)where the continuity equation, ρ ′ ( a ) = − ( ρ m + P m ) a ) , has been used to get Eq.(5.28).Due to the appearance of c on the right-hand side of Eq.(5.28), it is reasonable to set B t ≡ − c λb ˙ bM . This gives the relation between two scale factors, b ( t ) and a ( t ), as shownbelow b ˙ b = − a N ( t ) M ( t )1 + κ Λ (cid:20) ρ m ( a ) + P m ( a ) a + V ′ ( φ ) (cid:21) . (5.29)To explain the expansion universe at late time, the exponential potential, i.e. V ( φ ) = V e − φ is suitable for the model of gravity than the power law potential, V ( φ ) = V φ . For theexponential potential, this gives b ˙ b = − a N ( t ) M ( t )1 + κ Λ (cid:20) ρ m ( a ) + P m ( a ) a − V e − φ (cid:21) > . (5.30)whereas b ˙ b = − a N ( t ) M ( t )1 + κ Λ (cid:20) ρ m ( a ) + P m ( a ) a + 2 V φ ( t ) (cid:21) < . (5.31)– 10 –or the power laws potential. Here we interest to examine further by setting V e − φ ≃ V (1 − φ ( t )) where φ ( t ) ≪ κ Λ is 1 . × − . κ Λ . . × − , M = √ κ Λ , N ( t ) = 1, a b = κ Λ as shown in [57]. This gives the de Sitter solution inEiBI gravity model in the eye of Noether gauge symmetry,˙ bb ≃ V √ κ Λ , b ( t ) = e V t √ κ Λ . (5.32)Clearly from Eq.(5.30), if there is no contribution of matter fields, i.e. ρ m = 0 , the scalarfield φ ( t ) →
0, this allows H b = ˙ bb → const. This is the de Sitter phase of EiBI Universe. We revisited a formal framework of the Eddington-inspired Born-Infeld (EiBI) theory ofgravity and derived the point-like Lagrangian for underlying theory based on the use ofNoether gauge symmetries (NGS). A Hessian matrix and quantify Euler-Lagrange equa-tions of EiBI universe have been explicitly quantified. We also discussed the NGS approachfor the Eddington-inspired Born-Infeld theory and comment on exact cosmological solu-tions.We end this work by providing some remarks. As expected, the NGS method cansimplify the complication of constraint equations and also helps us to simplify further thegauge function equations with the linear forms of β ( b ) , ϕ ( φ ) , α ( a ) , γ ( N ) , ξ ( M ) and τ ( t ) . Byassuming the equality of B t and the constant of motion, the two scale factors a ( t ) and b ( t )are correlated through the matter fields and the scalar field. Interestingly, we show thatthere exists the de Sitter solution in this gravity model. Acknowledgments
T. Kanesom is financially supported by the Science Achievement Scholarship of Thailand(SAST) and Graduate School and the Department of Physics, Prince of Songkla University.P. Channuie acknowledged the Mid-Cereer Research Grant 2020 from National ResearchCouncil of Thailand under a contract No. NFS6400117.
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