The Brans-Dicke field in Non-metricity gravity: Cosmological solutions and Conformal transformations

We consider the Brans-Dicke theory in non-metricity gravity, which belongs to the family of symmetric teleparallel scalar-tensor theories. Our focus lies in exploring the implications of the conformal transformation, as we derive the conformal equivalent theory in the Einstein frame, distinct from the minimally coupled scalar field theory. The fundamental principle of the conformal transformation suggests the mathematical equivalence of the related theories. However, to thoroughly analyze the impact on physical variables, we investigate the spatially flat Friedmann--Lema\^{\i}tre--Robertson--Walker geometry, defining the connection in the non-coincidence gauge. We construct exact solutions for the cosmological model in one frame and compare the physical properties in the conformal related frame. Surprisingly, we find that the general physical properties of the exact solutions remain invariant under the conformal transformation. Finally, we construct, for the first time, an analytic solution for the symmetric teleparallel scalar-tensor cosmology.


INTRODUCTION
Symmetric Teleparallel General Relativity (STGR) [1] represents an alternative gravitational theory, considered equivalent to General Relativity (GR).In STGR, the fundamental geometric elements consist of the metric tensor g µν and the symmetric, flat connection Γ λ µν with the covariant derivative ∇ λ , leading to ∇ λ g µν ̸ = 0.While GR defines autoparallels using the Levi-Civita connection for the metric tensor g µν , STGR emphasizes the non-metricity component, crucial for the theory's description.The equivalence between these two gravitational theories becomes evident upon a study of the gravitational Lagrangians [2].In GR, the Lagrangian function involves the Ricci scalar constructed by the Levi-Civita connection R, whereas in STGR, the corresponding Lagrangian is defined by the non-metricity scalar Q.The Ricci scalar and the non-metricity scalar differ by a boundary term B = R−Q [1][2][3].
Consequently, the variation of the two distinct Lagrangians yields the same physical theory.
However, this equivalence breaks down when introducing matter non-minimally coupled to gravity [4][5][6], or nonlinear terms of the gravitational scalars in the Action Integral [7,8].
In f (Q)-gravity [7,8], a straightforward extension of the STGR theory, the gravitational Lagrangian takes the form of a nonlinear function f of the non-metricity scalar Q.These nonlinear terms introduce additional degrees of freedom, leading to modifications in the gravitational field equations that give rise to new phenomena [9].In the context of cosmology, f (Q) has been proposed as a solution to the dark energy problem [10][11][12][13][14][15] and has been utilized to explain cosmic acceleration [16][17][18][19].
In the symmetric teleparallel theory of gravity, the presence of a flat geometry defined by the connection Γ λ µν allows for the existence of a coordinate system known as the coincidence gauge, where the covariant derivative can be represented as a partial derivative.This implies that in the symmetric teleparallel theory of gravity, the inertial effects can be distinguished from gravity.Consequently, the choice of the connection as the starting point in the symmetric teleparallel theory leads to the formulation of distinct gravitational theories [3].As a result, self-accelerating solutions can naturally emerge both in the early and late universe [20].The impact of different connections on the existence of cosmological solutions has been extensively explored in [20], while the scenario of static spherically symmetric spacetimes has been considered in [21,22].The reconstruction of the cosmological history was derived in [23][24][25].Specifically, the phase-space analysis was studied, for the field equations for the four different connections which describe the Friedmann-Lemaître-Robertson-Walker (FLRW) geometry [3].For similar studies see also [26,27].Quantum cosmology in f (Q)gravity investigated in [28], while in [29] a minisuperspace description is presented from where it follows that the f (Q)-theory can be described by two scalar fields.The first scalar field corresponds to the degrees of freedom associated with the higher-order derivatives of the theory, whereas the second scalar field is linked to the connection defined in the noncoincidence gauge.For further investigations into f (Q)-gravity and its generalizations, we recommend referring to the works cited in [30][31][32][33][34][35][36][37][38][39][40] and the references provided therein.Scalar fields non-minimally coupled to gravity have found extensive application in gravitational physics within the framework of General Relativity, such as in scalar-curvature theories [41,42], or in the context of teleparallelism, specifically scalar-tensor theories [43,44].The Brans-Dicke theory [45] represents one of the earliest scalar-curvature theories, formulated with the intention of establishing a gravitational theory that adheres to Mach's principle.This model is defined in the Jordan frame [46], where the presence of a matter source is essential for the existence of physical space.In contrast, General Relativity is defined in the Einstein frame, enabling the existence of physical space even in the absence of a matter term.The Brans-Dicke parameter is a characterized constant of the theory which indicates the coupling between the scalar and the gravitation Lagrangian [47].When the Bans-Dicke parameter vanishes, the theory is equivalent to the f (R)-gravity, where the non-minimally coupled scalar field attributes the higher-order degrees of freedom [48].
Although the scalar-curvature theory is initially defined in the Jordan frame, a geometrical mapping exists that enables the transformation of the theory into the Einstein frame.
Consequently, the scalar-curvature theory can be interpreted in the form equivalent to General Relativity, involving a minimally coupled scalar field.This geometric mapping is a conformal transformation, establishing a connection between the solution trajectories of the two frames [50].However, the physical properties of the solution trajectories are not invariant under the application of the conformal transformation.For example singular solutions does not remain singulars after the application of the conformal transformation, for more details see the discussion [51][52][53] and references therein.More recently, the Hamiltonian inequivalence between the Jordan and Einstein frames has been explored in [54][55][56].
In this study we are interested to study the effects of the conformal transformation on the physical properties of cosmological solutions on the Brans-Dicke analogue in symmetric teleparallel scalar-tensor theory [4].It is known that f (Q)-gravity is equivalent to a specific family of symmetric teleparallel scalar-tensor models, and we use the analogy of the Brans-Dicke model with the f (R)-gravity in order to introduce the non-metricity Brans-Dicke theory.We focus in the cosmological scenario of a spatially flat FLRW geometry.Moreover, we consider the case in which the connection is defined in the non-coincidence gauge and the gravitational theory is equivalent to a multiscalar field model.While the mathematical application of the conformal transformation in non-metricity theory has been previously explored in [5], no concrete conclusions were drawn regarding the physical properties of the solutions under the conformal transformation.More recently, in [57], several exact cosmological solutions were identified in the non-metricity scalar-tensor theory for the noncoincidence gauge.Within this work, we aim to determine exact and analytic solutions for the non-metricity Brans-Dicke cosmological theory, subsequently comparing the physical properties of the solutions between the Jordan and the Einstein frames.The structure of the paper is outlined as follows.
In Section 2 we discuss the fundamental properties and definitions of symmetric teleparallel gravity.Additionally, we explore f (Q)-theory and the symmetric teleparallel scalartensor theory of gravity.We demonstrate that f (Q)-theory can be reformulated as a nonmetricity scalar-tensor theory.Furthermore, we present the utilization of conformal transformations and the derivation of the conformal equivalent theory in Section 3. In Section 4, we introduce the extension of the Brans-Dicke field in non-metricity gravity.Here, we introduce a novel parameter ω, akin to the Brans-Dicke parameter of scalar-curvature theory.As ω → 0, the gravitational Action characterizes the f (Q)-theory, similarly to how the Brans-Dicke field characterizes f (R)-gravity in the same limit.Within this gravitational model, we consider a spatially flat FLRW background geometry, and for the connection defined in the non-coincidence gauge, we present the field equations in both the Jordan frame and the Einstein frame.
To explore the effects of the conformal transformation on the physical properties of solution trajectories within the conformal equivalent theories, Section 5 is dedicated to deriving precise solutions for the field equations.We conduct a comparative analysis of the physical properties between the two frames.It is observed that singular scaling solutions in one frame correspond to singular scaling solutions in the other frame, displaying identical asymptotic behaviour.Additionally, for the non-singular de Sitter solution, it is established that the asymptotic behaviour of physical properties remains unchanged under the application of the conformal transformation.Moreover, in Section 6, we introduce an analytical solution for the scalar-tensor theory in non-metricity gravity for the first time.The analysis reveals that this universe originates from a Big Rip singularity, transitions into an era characterized by an ideal gas, and ultimately converges towards a de Sitter universe as a future attractor.Notably, the observed behaviour of the physical parameters remains consistent regardless of the frame in which the theory is defined.Finally, our findings are summarized in Section 7.

SYMMETRIC TELEPARALLEL GRAVITY
Let M n be a manifold defined by the metric tensor, g µν , and the derivative ∇ λ , defined by the generic connection Γ λ µν with conditions, the Γ λ µν to inherit the symmetries of the metric tensor g µν ; that is, if X is a Killing vector of g µν , i.e.L X g µν , then L X Γ λ µν = 0, in which L X is the Lie derivative with respect the vector field X.Furthermore, for the connection Γ λ µν it holds that the Riemann tensor R κ λµν and torsion tensor T λ µν are always zero; that is, In symmetric teleparallel theory of gravity only the non-metricity tensor survives, defined as [1] that is, We define the disformation tensor and the non-metricity conjugate tensor [3] where now the non-metricity vectors Q λ and Q ′λ are defined as and The non-metricity scalar is defined as and the gravitational Action Integral in STGR is given by the following expression [1] The non-metricity scalar, Q, and the Ricciscalar R for the Levi-Civita connection Γλ µν of the metric tensor g µν differ by a boundary term B, that is, [4] where and ∇λ denotes covariant derivative with respect to the Levi-Civita connection, Γλ µν .

f (Q)-theory
An extension of STGR which has drawn the attention recently is the f (Q)-gravity.In this theory, the gravitational Lagrangian is a nonlinear function The resulting gravitational field equations are where G µν is the Einstein-tensor.
Moreover, connection Γ λ µν satisfies the equation of motion When equation ( 12) is satisfied for a given connection, we designate the connection as defined in the coincidence gauge.Conversely, if the equation is not satisfied, the connection is said to be defined in the non-coincidence gauge.Furthermore, in the limit at which f (Q) becomes linear, the field equations are reduced to those of symmetric teleparallel gravity (STGR).
Last but not least, in the presence of a matter source minimally coupled to gravity, the field equations ( 11) are modified as follows with the energy-momentum tensor T µν to give the degrees of freedom for the matter source.

Symmetric teleparallel scalar-tensor theory
The symmetric teleparallel scalar-tensor theory is a Machian gravity, that is, it satisfies Mach's principle, for which a scalar field non-minimally coupled to gravity exists.
The gravitational Action Integral is [4] where V (ϕ) is the scalar field potential, which drives the dynamics and F (ϕ) is the coupling function between the scalar field and the gravitational scalar Q.The function, ω (ϕ), can be eliminated with the introduction of the new scalar field dΦ = ω (φ)dφ.Hence, the Action Integral (14) becomes The field equations which follow from the gravitational Action (14) are It is important to observe that for ω (φ) = 0, F (φ) = φ, the latter field equations take the functional form of f (Q)-theory [4], where now

CONFORMAL TRANSFORMATION
The symmetric teleparallel scalar-tensor theory satisfies Mach's principle, that is, the gravitational theory is defined in the Jordan frame.A similar result holds for the f (Q)theory.The Jordan frame is related to the Einstein frame through a conformal transformation.This transformation relates theories which are conformal equivalent.This equivalence it has to do with the trajection solutions for the field equations, but it is not a physical equivalence; since the physical properties of the theories do not remain invariant under a conformal transformation.Conformal transformations for the four-dimensional manifold were investigated in [5].Below we consider a n-dimensional space.
Therefore, for the nonmetricity tensor we find and Therefore, for the non-metricity scalar we find Consider now the Action Integral ( 14) for the n-dimensional conformally related metric ḡµν , that is, With respect to the metric g µν and the conformal factor Ω, the latter Action Integral is We select F (φ) e (n−2)Ω = 1, that is Ω = 1 2−n ln F (φ). Therefore, the latter Action reads The second terms become We end with the gravitational Lagrangian

BRANS-DICKE COSMOLOGY IN SYMMETRIC TELEPARALLEL THEORY
Similarly to the consideration of the Brans-Dicke field in the scalar-curvature theory, we take into account the following Action Integral within a four-dimensional manifold in the context of symmetric teleparallel theory.Indeed, in Action ( 14) we assume F (φ) = φ and ω (φ) = ω φ , ω = const.. Thus, we arrive at the Lagrangian Parameter ω play a similar role as that of the Brans-Dicke parameter.
We define the new field φ = e ϕ , in order to write the latter Action in the form of the Dilaton field On the other hand, in the Einstein frame, the equivalent Action integral is The solution trajectories of the field equations for the two gravitational theories described by the Action integrals ( 28), ( 29) are linked by the conformal transformation.However, no definitive conclusion can be drawn concerning the relationship of the physical properties of the solutions under the application of the conformal transformation.
The objective of this study is to examine how the conformal transformation impacts the physical properties of the trajectory solutions in symmetric teleparallel theory.To conduct such an analysis, we consider the background geometry which describes an isotropic and homogeneous spatially flat FLRW universe, with the line element in which a (t) is the scale factor and N (t) is the lapse function.We derive the field equations for the two conformally related models, namely S D and SD .
We obtain exact and analytic solutions for one of the models and thoroughly examine the physical properties of these solutions.Subsequently, we apply the conformal transformation to ascertain the corresponding exact and analytic solutions for the second model, delineating the specific physical properties of these solutions.Finally, we conduct a comparative analysis of the physical properties between the solutions of the two conformally related theories.
For the spatially flat FLRW geometry described by the line element (30) there are three families of symmetric connections which describe a flat geometry and inherit the symmetries of the background space [20].One family is defined in the coincidence gauge, for this family the non-metricity scalar Q has the same factional form with the torsion scalar of teleparallelism.Thus, for the connection in the coincidence gauge the symmetric teleparallel scalar-tensor theory is equivalent to the scalar-torsion theory and f (Q)-theory is equivalent to f (T )-theory.The remaining two families of connections are defined in the non-coincidence gauge where, as it was found in [29], a scalar field is introduced into the gravitational theory which describes the connection.
In this piece of study we select to work in the framework of the connection with nonzero components in which ψ = dψ dt , and without loss of generality we have assumed that N (t) = 1.Thus, the non-metricity scalar is calculated We substitute into (28) and subsequently derive the cosmological field equations in the Jordan frame, yielding: φ + φ2 + 3H φ = 0, (38) where H = ȧ a is the Hubble function.Equations ( 35)-( 38) constitute a Hamiltonian dynamical system described by the pointlike Lagrangian, in which equation ( 35) is the constraint equation describing the conservation law of "energy" for the classical Hamiltonian system.Recall that for ω = 0, the latter Lagrangian reduces to that of f (Q)-gravity for the same connection.
The field equations for the conformal equivalent theory (29) are where we have assumed N (t) = 1, and Last but not least, the point-like Lagrangian for the field equations is At this juncture, it is crucial to highlight that, for ω = 0, the latter gravitational theory is equivalent to the non-metricity theory with boundary term, specifically with the f (Q, B) = Q + f (B) theory of gravity [39].

EXACT SOLUTIONS
In this Section we determine the existence of exact solutions for the field equations that hold special significance.Additionally, we investigate the physical properties of the solutions within the conformal equivalent theory.Our focus is on determining the prerequisites for the existence of singular solutions, corresponding to universes dominated by an ideal gas, as well as identifying the conditions for a de Sitter solution.Subsequently, we utilize the conformal transformation to deduce the exact solution in the second frame, subsequently studying the physical properties and conducting a comparative analysis of the solutions between the two frames.

Singular solution in the Jordan frame
We assume the scaling solution, a (t) = a 0 t p , H (t) = p t , for the cosmological model defined in the Jordan frame.This scale factor describes a universe dominated by an ideal gas with equation of state parameter w ef f = 2−3p 3p .Thus, for p = 2 3 , the solution describes a universe dominated by a pressureless fluid, i.e. dust.For p = 1 2 it describes a universe dominated by radiation.Moreover, for p > 1 or p < 0, the exact solution describes acceleration.
For the power-law singular solution a (t) = a 0 t p and from the equation of motion (38) it follows that and from the remaining of the field equations we derive In the special limit for which ϕ 1 = 0 and V (ϕ) = 0, the latter expression becomes weff = 1, and easily we can write the scale factor in terms of the new parameter τ as α (τ ) ≃ τ 1 3 .
Therefore for ϕ 1 = 0, any scaling solution in the Jordan frame corresponds to the scaling solution which describes a stiff fluid in the Einstein frame.
On the other hand, for ϕ 1 ̸ = 0 and for large values of t, it follows that weff (t) ≃ −1+ 2 3p for p > 1  3 .This means that, in the asymptotic limit, the solution in the Einstein and in the Jordan frames has the same physical properties.We recall that τ (t → ∞) → ∞ for ϕ 1 > 0 and p > 1  3 .Hence, as far as we move from the singularity the two frames describe the same physical universe.In the contrary, near to the singularity, that is t → 0, weff (t) ≃ 1.
For p = 1 3 , the solution at the Einstein frame is Thus, for large values of time, the asymptotic solution resembles that of a stiff fluid, similar to the scenario in the Jordan frame.

de Sitter universe in the Jordan frame
Consider now the de Sitter universe with a (t) = a 0 e H 0 t , H (t) = H 0 .Then from the field equations in the Jordan frame we derive and This means that the de Sitter solution exists for constant potential function V (ϕ).

Einstein frame
Now we transform the solutions in the Einstein frame.Indeed, the scale factor and the Hubble function becomes while the effective equation of state parameter reads Hence for large values of t → ∞, it follows that weff (t) ≃ −1, while for small values of where for e 3H 0 ϕ 1 → 0, the limit weff (t) ≃ −1 follows, and for e 3H 0 ϕ 1 → ∞ we derive weff (t) ≃ 1.

Singular solution in the Einstein frame
Consider now the scaling solution α (τ ) = α 0 τ q , then from equations ( 41)- (44) we derive and or equivalently In the case of q = 1 3 , the exact solution follows, that is,

.1. Jordan frame
For q ̸ = 1 3 , the solution at the Jordan frame is We remark that w ef f (τ → 0) ≃ − 2 3 and w ef f (τ → ∞) ≃ −1 + 2 3q .Hence, far from the singularity, the physical properties of the solution remain unchanged under the influence of the conformal transformation.
The case q = 1 3 was studied before.Thus we omit it.

de Sitter universe in the Einstein frame
For the exponential scale factor α (τ ) = α 0 e H0 τ , from the field equations ( 41)- (44) in the Einstein frame we determine the exact solution Therefore, the scalar field potential is (78) From these expressions we have the limits and We conclude that the de Sitter universe is the asymptotic solution in the two frames.
The above discussion highlights that the solutions exhibit identical physical properties in both the Jordan and Einstein frames at the asymptotic limits.This observation is significant and sets it apart from the scalar-curvature or scalar-torsion theories of gravity, where such equivalence does not hold true.

ANALYTIC SOLUTION
In the preceding Section, we explored the existence of exact solutions for the field equations.The derived solutions exhibit fewer degrees of freedom compared to the original dynamical system, rendering them special or asymptotic solutions.Subsequently, we proceed to establish the analytic solution for the field equations.Specifically, for the Brans-Dicke field with the potential function V (ϕ) = V 0 exp ((λ − 1) ϕ), we derive the analytic solution for the field equations ( 35)- (38).The field equations form a three-dimensional Hamiltonian system with six degrees of freedom, enabling the application of the Hamilton-Jacobi method to simplify the field equations and to construct the analytic solution.
We consider the point transformation in which the Lagrangian function of the field equations is We have considered the lapse function N (t) to be a non-constant function, we see below that this necessary in order to write the closed-form solution of the field equations.
From Lagrangian function (80) we can define the momentum that is, Therefore, the Hamiltonian function H = p q ∂L ∂ q − L can be written where H = 0, follows from the constraint equation (35).
Consequently, Hamilton's equations are ṗu = 0 , ṗψ = 0 (86) and from which we infer that p u and p ψ are constants, that is p u = p 0 u , and p ψ = p 0 ψ .Let S = S (u, Φ, ψ) be the Action, then from (85) we can write the Hamilton-Jacobi The Hubble function and the equation of state parameter w ef f are expressed as It is observed that the universe initiates from a big rip singularity, subsequently progresses towards a saddle point characterized by an ideal gas, representing the matter-dominated era and finally transitions to the de Sitter point.This behaviour is consistent across solution trajectories in both frames, mirroring the findings for the asymptotic solutions in the preceding section.While previously, the resemblance in the evolution of physical parameters was noted at the asymptotic limits, Figure 1 demonstrates that this similarity persists throughout the global evolution of the cosmological solution.

CONCLUSIONS
We performed an extensive analysis on the influence of the conformal transformation on the physical properties of cosmological solution trajectories within symmetric teleparallel gravity's conformal equivalent theories.To undertake this analysis, we introduced the Brans-Dicke model in the context of non-metricity gravity, alongside an analogue of the Brans-Dicke parameter.Notably, when this parameter approaches zero, the non-metricity scalar-tensor theory is reduced to the f (Q)-theory.
Regarding the background geometry, we focused on the isotropic and homogeneous spatially flat FLRW metric.Concerning the theory's connection, we specifically examined a connection defined within the non-coincidence gauge.It is worth recalling that in the coincidence gauge, the cosmological field equations simplify to those of scalar-torsion theory, limiting the new information that could be deduced from this study.For this particular cosmological model, we derived the field equations in both the Jordan and the Einstein For all the plots we consider the initial conditions p 0 u , p 0 ψ , u 0 = (1, 0.8, −10), and V 0 = 1.We observe that the behaviour for the equation of state parameter is similar in the two frames and the de Sitter solution is a common future solution.

frames.
We derived exact solutions of particular significance in one frame, illustrating both singular and non-singular solutions.Subsequently, we utilized the conformal transformation to reconstruct the exact solutions for the conformal equivalent theory.Our analysis involved a thorough comparison of the physical properties for the two theories, each defined within different frames.Notably, we discovered that the physical properties remained invariant under the influence of the conformal transformation.
Consequently, singular solutions in one frame corresponded to singular solutions in the other frame, displaying similar properties in the asymptotic limit.Furthermore, we observed that the non-singular de Sitter solution remained a de Sitter solution in the alternate frame as well.Furthermore, we constructed for the fist time an analytic solution for the cosmological field equations in non-metricity scalar-tensor theory.This solution describes an cosmological model with Big Rip singularity, which involves to a matter dominated solution and the final state of the universe is that of the de Sitter universe.Surprisingly this specific cosmological history describes the conformal equivalent theory.Hence, the physical equivalence of the physical solutions between the two frames extends the asymptotic limits of the solutions.
In a future study we plan to investigate further such analysis by investigate the case of compact objects.

)Figure 1
Figure 1 illustrates the qualitative evolution of the equation of state parameter, w ef f (Φ (a)), for the above mentioned analytical solution, considering various values of the free parameters.Additionally, we calculate and display the evolution of the equation of state parameter weff (Φ (α)) for the conformal equivalent theory as defined in the Einstein frame.The plots in both frames utilize identical values for the free parameters, reflecting corresponding initial conditions.

FIG. 1 :
FIG. 1: Qualitative evolution of the effective equation of state parameter in the Jordan frame w ef f (a) and in the Einstein frame weff (α) for different values of the free parameters.For all the