Noether Symmetry Approach in Eddington-inspired Born-Infeld gravity

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 underlying theory based on the use of Noether gauge symmetries (NGS). We study a Hessian matrix and quantify Euler-Lagrange equations of EiBI universe. We discuss the NGS approach for the Eddington-inspired Born-Infeld theory and show that there exists the de Sitter solution in this gravity model.


Introduction
Various cosmological observations make a strong evidence that the expansion of the universe is presently accelerating. These experimental results include Type Ia Supernovae [1,2], cosmic microwave background (CMB) radiation [3][4][5][6][7][8][9], large scale structure [10], baryon acoustic oscillations (BAO) [11] as well as weak lensing [12]. An expansion phase can be 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 to describe the late-time cosmic acceleration. The first one assumes the introduction of the so-called "dark energy (DE)" in the context of conventional general relativity. Another convincing approach is to engineer Einstein gravity on the large-scale methodology (see for reviews on not only dark energy problem but also modified gravity theories, e.g., [16][17][18][19][20]). However, the DE sector remains still unknown and possesses one of the unsolved problems in physics.
Therefore, it opens opportunities to search for modified theories of gravity to deal with such problems. By modifying the geometrical part of Einstein field equations or adding scalar field to the right-hand side of the Einstein field equations, both alternatives are able to explain effects of dark ingredients with acceptable assertions [21]. One of the simplest modifications to the standard general relativity is the f (R) theories of gravity in which the Lagrangian density f is an arbitrary function of the scalar curvature R [22,23]. Among numerous alternatives, these theories include higher order curvature invariants, see rigorous reviews on f (R) theories [24,25].
In cosmological framework, the Noether symmetry (NS) approach has revealed a useful tool not only to fix physically viable cosmological models with respect to the conserved quantities, but also to reduce dynamics and achieve exact solutions [26,27]. Moreover, the existence of Noether symetries plays crucial roles when studying quantum cosmology [28].

Eddington-inspired Born-Infeld Gravity
In 2009, Máximo Bañadoz [50] proposed a new form of the Born-Infeld action under Palatini formalism. This is the so-called Eddington-inspired-Born-Infeld (EiBI) gravity. This action is written as follows: where λ = 1 + κΛ is a dimensionless constant displaying the relation between the EiBI free parameter κ (with a dimension of M 2 P ) and the cosmological constant Λ (with a dimension of M −2 P ); whilst S m (g µν , Ψ) and S φ (g µν , φ) represent the matter field action and the scalar field action, respectively. Throughout this work, we set 8πG N c 4 = 1. Performing variation of Eq.(2.1) with respect to Γ λ µν 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 With the help of two ansatz forms of a spatially flat FLRW metric the EiBI action (2.2) can be expressed in terms of the cosmological variables as follows: where υ 0 is the spatial volume obtaining after a proper compactification for spatial flat section. l = +1 and l = −1 denote an ordinary scalar field and phantom scalar field, respectively. It is probably impossible to get rid of all second order derivative in Eq.(2.5) by performing an integration by parts. Therefore we neglect to write an explicit form of the point-like Lagrangian. In 2012, Delsate and Steindoff [49], however, wisely proposed the bi-metric form of the EiBI action under the metric formulation written as Notice that only g µν interacts with matter and scalar fields whereas q µν defines the background metric regarded as the fundamental reference frame of the universe [51]. Because the alternative action yields identical field equations as provided in the Bañadoz action, this indicates that two action forms are equivalent. Using a relation, S = ν 0 dtL [52] and the relation between two metric tensors, q µν = g µν + κR µν (Γ), the point-like Lagrangian can be extracted from equation(2.6) as follows Here the number of configuration space (or the minisuperspace) variables equal to five due to the appearance of variables {a(t), b(t), M (t), N (t), φ(t)} in Eq.(2.7). Apart from the kinetic part of the Lagrangian, we can set

Hessian matrix and Euler-Lagrange equations of EiBI universe
In the absent of {ȧ,Ṁ ,Ṅ } in the EiBI point-like Lagrangian, the EiBI Hessian matrix can be written as (3.1) Clearly, the determinant of the Hessian matrix of the EiBI point-like Lagrangian equals zero indicates again that Eq.(2.7) is a singular Lagrangian. Accordingly, variables {a, N, M } do not 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 can set 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-Lagrange equations, we can show that [54] d dt Here the configuration space variables are q i = {a, b, M, N, φ} and their time derivative on the tangent space areq i = dq i dt = {ȧ,ḃ,Ṁ ,Ṅ ,φ}. Because ∂ 2 L ∂q j ∂q i = 0 and ∂ 2 L ∂q j ∂q i for variables {a, M, N } in EiBI gravity, the Euler-Lagrange equations of these variables can be reduced to ∂L ∂a = ∂L ∂M = ∂L ∂N = 0 whereȧ,Ṁ ,Ṅ andä,M ,N can be set arbitrarily [54]. As expected,b andφ are determined by performing variation equation(2.7) with respect to the dynamical variables and their time derivative as expressed on the right-hand side of equation(3.2). As expected,b andφ are determined from taking variation Lagrangian with respect to the dynamical variables and their time derivative as shown on the righthand side of Eq.(3.3). We have to keep in mind that a crucial concept of a gauge theory is the general solution of the equations of motion which contains arbitrary functions of time and the canonical variables are not all independent but relate among each others via the constraint equations [54]. As the results of the vanishing of

4)
where λ a (t), λ M (t),and λ N (t) stand for Lagrange multipliers that are arbitrary functions of time. The total EiBI Hamiltonian (H EiBI ) can be used to evaluate an evolution invoking the Hamiltonian equations of motion as follows: For the EiBI Hamiltonian, it should be noted thatȧ = λ a ,Ṅ = λ N ,Ṁ = λ M are the primary constraints andṗ a =ṗ N =ṗ M = 0 are the secondary constraints that must be valid at all times [53] resulting in p a = p M = p N = 0. In order to obtain the dynamic solutions, we have to calculate the Euler-Lagrange equations for a(t), b(t), M (t), N (t), and φ(t) as shown below: where the conservation equation [52], dρ m /da = −3(ρ m + p m )/a, has been used to yield Eq.(3.7).

Noether Gauge Symmetries in EiBI Gravity
Noether vector (X N GS ) and the first prolongation vector field (X [1] NGS ) related to EiBI Lagrangian, as shown in Eq.(2.7), can be constructed as follows: 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 )} are possibly functioned by {t, q i }={t, a, b, N, M, φ}. Their time derivative can be defined aṡ 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 defined as It is worth noting that terms like, for example, ∂V (φ(t)) ∂t = ∂ȧ(t) 2 ∂t = 0, because there is not time variable(t) shown explicitly in L EiBI . The vector field X 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 the condition for Noether symmetry [48]. After using the Noether gauge symmetries condition with the EiBI point-like Lagrangian, this provides us with eighty terms for X [1] N GS L+LD t τ = D t B as shown below: If the Noether symmetry condition £ X NGS L EiBI = 0 is satisfied, then the function Σ 0 = α i ∂L ∂q i is a constant of motion [56]. This gives where two unknown functions β and ϕ will be studied in the next section. Up to this point, it is worth mentioning a dimension analysis of each variable, i.e.

Remarks on exact cosmological solutions
After a separation of monomials, we can quantify the system equations to yield In order to write τ (t) in a single form, we have to set c 6 = c 7 and 3c 3 − c 4 + 4c 2 = 5c 1 − c 5 .
There are three equations contributed of gauge function, where we use β t = ∂β ∂b db dt = c 1ḃ and ϕ t = ∂ϕ ∂φ dφ dt = c 2φ to get Eq.(5.18) and (5.19). We hence expect that the rest of the expression for boundary term, i.e. − 12c 1 λbḃ M , may relate with B t . From Eq.(5.4), Eq.(5.9) and Eq.(5.8), it is easy to see that This gives the following relation, 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) and Eq. (5.19), that is It is worth noting that c 1 is just an arbitrary constant and we can redefine it by replacing c 1 → 2c 1 . Interestingly, this is exactly matched with the constant of motions of EiBI gravity by this setting.
The relation between the the constant of motion and the boundary term has shown in [55]. Eq.(5.20) can be rewritten as where it is very naturally to set c 4 = c 5 and c 3 = − 2 3 c 2 . It is quite remarkable that most terms of B t become zero with the simple setting, i.e.
If we further set that for the power laws potential. Here we interest to examine further by setting V 0 e −φ ≃ V 0 (1− φ(t)) where φ(t) ≪ 1 and using the limit ranged of κΛ is 1.12 × 10 −4 κΛ 2.10 × 10 −3 , M = √ 1 + κΛ, N (t) = 1, a 2 b 2 = 1 1+κΛ as shown in [57]. This gives the de Sitter solution in EiBI gravity model in the eye of Noether gauge symmetry, Clearly from Eq.(5.30), if there is no contribution of matter fields, i.e. ρ m = 0, the scalar field φ(t) → 0, this allows H b =˙b b → const. This is the de Sitter phase of EiBI Universe.

Conclusion
We revisited a formal framework of the Eddington-inspired Born-Infeld (EiBI) theory of gravity and derived the point-like Lagrangian for underlying theory based on the use of Noether gauge symmetries (NGS). A Hessian matrix and quantify Euler-Lagrange equations of EiBI universe have been explicitly quantified. We also discussed the NGS approach for the Eddington-inspired Born-Infeld theory and comment on exact cosmological solutions. We end this work by providing some remarks. As expected, the NGS method can simplify the complication of constraint equations and also helps us to simplify further the gauge function equations with the linear forms of β(b), ϕ(φ), α(a), γ(N ), ξ(M ) and τ (t). By assuming 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 that there exists the de Sitter solution in this gravity model.