Dynamical analysis and cosmological evolution in Weyl integrable gravity

We investigate the cosmological evolution for the physical parameters in Weyl integrable gravity in a Friedmann--Lema\^{\i}tre--Robertson--Walker universe with zero spatially curvature. For the matter component, we assume that it is an ideal gas, and of the Chaplygin gas. From the Weyl integrable gravity a scalar field is introduced by a geometric approach which provides an interaction with the matter component. We calculate the stationary points for the field equations and we study their stability properties. Furthermore, we solve the inverse problem for the case of an ideal gas and prove that the gravitational field equations can follow from the variation of a Lagrangian function. Finally, variational symmetries are applied for the construction of analytic and exact solutions.


INTRODUCTION
The cosmological constant component in the Einstein-Hilbert Action Integral, is the simplest dark energy candidate to describe of the recent acceleration phase of the universe, as it is provided by the cosmological observations [1]. In the so-called ΛCDM cosmology the universe is considered to be homogeneous and isotropic, described by the Friedmann-Lemaître-Robertson-Walker (FLRW) geometry with spatially flat term, where the matter * Electronic address: anpaliat@phys.uoa.gr component consists of the cosmological constant and a pressureless fluid source which attributes the dark matter component of the universe. The gravitational field equations are of second-order and can be integrated explicitly. Indeed, the field equations can be reduced to that of the one-dimensional "hyperbolic oscillator". However, as the cosmological observations are improved, Λ-cosmology loses the important position in the "armoury" of cosmologists. For an interesting discussion on the subject we refer the reader to the recent review [2]. Furthermore, because of the simplicity of the field equations in Λ-cosmology, the cosmological constant term cannot provide a solution for the description of the complete cosmological evolution and history.
In order to solve these problems, cosmologists have introduced various solutions in the literature by introducing new degrees of freedom in the field equations. Time-varying Λ term, scalar fields and fluids with time-varying equation of state parameters, like the Chaplygin gases have been proposed to modify the energy-momentum tensor of the field equations [3][4][5][6][7][8]. On the other hand, a different approach is inspired by the modification of the Einstein-Hilbert Action integral, and leads to the family of theories known as alternative/modified theories of gravity [9][10][11]. Another interesting consideration is the interaction between the various components consisted of the energy momentum tensor [12]. Interaction in the dark components of the cosmological model, that is, between, the dark energy and the dark matter terms is supported by cosmological observations [13][14][15][16].
For a given proposed dark energy mode model, there are systematic methods for the investigation of the physical properties of the model. The derivation of exact and analytic solutions is an essential approach because analytic techniques can be used for the investigation of the cosmological viability of the model [17][18][19][20]. Furthermore, from the analysis of the asymptotic dynamics, that is, of the determination of the stationary points, the complete cosmological history can be constructed [21][22][23]. Indeed, constraints for the free parameters of a given model can be constructed through the analysis of the stationary points and the specific requirements for the stability of the stationary points [24][25][26][27][28].
In this piece of work, we study the evolution of the cosmological dynamics for the theory known as Weyl integrable gravity (WIG) [29][30][31][32][33][34][35]. In WIG a scalar field is introduced into the Einstein-Hilbert Action Integral by a geometric construction approach. Indeed, in Riemannian geometry the basic geometric object is the covariant derivative ∇ µ and the metric tensor g µν , such that it has no metricity component,i.e. ∇ κ g µν = 0 [36]. In Weyl geometry the fundamental geometric objects are the gauge vector field ω µ and the the metric tensor g µν , such that∇ κ g µν = ω κ g µν , where now∇ µ notes the covariant derivative with respecto the affine connectionΓ κ µν which is defined asΓ κ µν = Γ κ µν − ω (µ δ κ ν) + 1 2 ω κ g µν . When ω µ is defined by a scalar field φ,Γ κ µν describes the affine connection for the conformal metric g µν = φg µν . The field equations of the WIG in the vacuum are equivalent to that of General Relativity with a massless scalar field, with positive or negative energy density. However, when a matter source is introduced, interaction terms appear as a natural consequence of the geometry of the theory [36]. In geometric terms of interaction context , we investigate the dynamics of the cosmological field equations so that we construct the cosmological history and investigate the viability of the theory. Furthermore, the integrability property for the field equations is investigated by using the method of variational symmetries for the determination of conservation laws.
In Section 2 we present the basic elements for the WIG theory. Furthermore, we write the field equations for our cosmological model in a spatially flat FLRW background space.
In Section 3 we present the main results of our analysis in which we discuss the asymptotic dynamics for the field equations in the cases for which the matter source is an ideal gas, or a Chaplygin gas. Moreover, we investigate the dynamics in the presence of the cosmological constant term. In Section 4 we show that the field equations have a minisuperspace description when the matter source is an ideal gas. Specifically, we solve the inverse problem and we construct a point-like Lagrangian which describes the cosmological field equations. With the use of the variational symmetries we determine a conservation law and we present the analytic solution for the field equations by using the Hamilton-Jacobi approach. Our results are summarized in Section 5.

WEYL INTEGRABLE GRAVITY
Consider the two conformal related metric tensors g µν ,g µν such thatg µν = φg µν . The Christoffel symbols of the two conformal related metrics are related as In Weyl geometry the fundamental objects are the metric tensor g µν and the covariant derivative∇ µ defined by the Christoffel symbolsΓ κ µν . Hence, the curvature tensor is defined Consequently, the Ricci tensors of the two conformal metrics are related as follow thus the Ricci scalar In WIG the the foundamental Action Integral is defined by using the Weyl Ricci scalar R and the scalar field φ by the expression where ξ is a coupling constant. From (5) we observe that φ is a massless scalar field.
However, in a more general consideration a potential function may be considered.
From the Action Integral (5) the Weyl-Einstein equations are as [36] whereG µν is the Weyl Einstein tensor. By using the Riemannian Einstein tensor G µν , the Weyl-Einstein field equations (6) become [36] where λ is defined as 2λ ≡ 4ξ − 3. Equations (7) are nothing else than the field equations of Einstein's General Relativity with a massless scalar field. When λ > 0, the scalar field φ is a quintessence while, when λ < 0, φ is a phantom field [36] Moreover, for the equation of motion of the scalar field φ, the Klein-Gordon equation is or by using the Riemannian covariant derivative ∇ µ , expression (8) is written in the usual form g µν ∇ ν ∇ µ φ = 0.
As it was found in [36], the introduction of a perfect fluid in the gravitational model leads to the following set of gravitational field equations [36] that is, where T (m) Equation (11) follows from the identity G µν ;ν = 0, which provides the conserve of the effective energy-momentum tensor.

FLRW spacetime
Following the cosmological principle, in very large scales the universe is considered to be isotropic and homogeneous. Hence, the physical space is described by the FLRW spacetime, where the three-dimensional surface is a maximally symmetric space and admits six isometries. However, from cosmological observations the spatial curvature is very small, which means that we can consider as background space the spatially flat FLRW metric Moreover, we assume the comoving observer u µ = δ t µ , with expansion rate θ = 3˙a a , for the line element (12) and for a scalar field φ = φ (t), the gravitational field equations are From the modified Friedmann equations we observe the existence of a nonzero interacting term for scalar field φ and the matter component ρ m . When λ > 0, energy decays from scalar field to the ρ m , while for λ < 0 energy decays from ρ m to the field φ. Furthermore, the effective equation of state parameter for the effective cosmological matter is defined as Finally, for the nature of the matter source ρ m in the following we consider that ρ m is an ideal gas, or a Chaplygin gas.

COSMOLOGICAL DYNAMICS
We continue our analysis with the investigation of the stationary points for the cosmological field equations. In order to proceed with the study we define the new dimensionless variables in the context of θ-normalization where for the equation of state parameter for the matter source we consider (i) ideal gas Moreover, Ω m is bounded as 0 ≤ Ω m ≤ 1, such that the solution is physically acceptable, that is, from (18) it follows that there are physical stationary points only when λ > 0.
The stationary points of equation (19) are Points x ± 1 describe asymptotic solutions where only the scalar field contributes to the cosmological fluid. The effective equation of state parameter is derived to be w ef f x ± 1 = 1, from which we infer that the solution is that of a stiff fluid. On the other hand, the point x 2 is physically acceptable when λ ≥ 1 6(γ−2) 2 , and the point describes a scaling solution with which means that the asymptotic solution describes an accelerated universe, where in the limit λ = 1 8(1−2γ)+6γ 2 , the asymptotic solution is that of the de Sitter universe. We proceed with the investigation of the stability properties for the stationary points.
We linearize equation (19) and we find the eigenvalues e 1 x ± . Thus, point x − 1 is always a source, x + 1 is an attractor when λ < 1 6(γ−2) 2 , while x 2 is the unique attractor when it exists.

Ideal gas with Λ = 0
In the presence of the cosmological constant, that is, Λ = 0, and when the matter term is that of the ideal gas, the field equations are written as follows and Furthermore, we assume that |Ω Λ | ≤ 1, from which we infer that x is also bounded, and we do not have to study the dynamical system for the existence of stationary points at infinity.
The stationary points of the dynamics system (22), (23) are defined in the plane {x, Ω Λ }, , Ω Λ (B)). The points are Points B ± 1 , B 2 are actually the stationary points A ± 1 and A 2 respectively for which the cosmological constant component is zero. The physical properties are the same as before.
x ′ = 1 12 and where the new variable Y is defined as Y = Ae − 1 2 (1+α)φ θ −(2+α) . The stationary points C = (x (C) , Y (C) )of the dynamical system (27), (28), with Ω m > 0 are and For point C 1 we derive Ω (C 1 ) = 1 − 6 λ , w ef f (C 1 ) = 1 6λ . The point is physically acceptable when λ ≥ 1 6 while it always describes a universe without acceleration. For λ = 1 6 , the asymptotic solution is that of dust, while for λ = 1 2 the asymptotic solution is that of radiation. The eigenvalues of the linearized system around the stationary point are calculated , e 2 (C 1 ) = 1−6λ 12λ , from which we can easily conclude that the stationary point is always a saddle point.
Point C 2 describes a universe for which Ω m (C 2 ) = 1 2 −3λ+ 3λ (1 + 3λ) and w ef f ( . The point is well defined when λ > 0, while for large values of λ it follows that w ef f (C 2 ; λ >> 1) ≃ −1, which means that point C 2 can describe a solution near to the de Sitter point. On the other hand, point C 3 is physical acceptable for 0 < λ ≤ 1 24 , while we derive Ω m = −3λ + 3λ (1 + 3λ) and The eigenvalues of the linearized system near to the stationary points C 2 and C 3 are determined. Numerically we find that e 1 (C 2 ) , e 2 (C 2 ) have always negative real parts for λ > 0 and α ≥ 1; on the other hand Re (e 1 (C 3 )) > 0, Re (e 2 (C 3 )) > 0 for α ≥ 1 , 0 < λ ≤ 1 24 . Hence, point C 2 is always an attractor while point C 3 is always a source.

Chaplygin gas with Λ = 0
In the presence of a nonzero cosmological constant term, the field equations are reduced to the following dynamical system x ′ = 1 12 The physically acceptable stationary points D = (x (D) , Y (D) , Ω Λ (D)) are where D 1 , D 2 and D 3 have the same physical properties as points C 1 , C 2 and C 3 respectively.
For the point D 4 we find Ω m (D 4 ) = −12λ and w ef f (D 4 ) = −1, which means that the asymptotic solution is physically acceptable when − 1 12 ≤ λ < 0, while the asymptotic solution is that of the de Sitter universe.
The eigenvalues of the linearized system near D 1 are e 1 (D 1 ) = (1+α)(1+3λ) 3λ , e 2 (D 1 ) = 1−6λ 12λ and e 3 (D 1 ) = 1+6λ 6λ , which means that point D 1 is always a saddle point. For the points D 2 and D 3 we find numerically that D 2 is always an attractor while D 3 is always a source.

MINISUPERSPACE DESCRIPTION AND CONSERVATION LAWS
For an ideal gas p m = (γ − 1) ρ m , from equation (16) it follows ρ m (t) = ρ m0 a −3γ e φ in which ρ m0 is a constant of integration.
We substitute this into the rest of the field equations and we end with the following dynamical system For the second-order differential equations (39), (40) in the space of variables {a, φ}, the inverse problem for the determination of a Lagrangian function, provides that the function is an autonomous Lagrangian function for the field equations, while equation (38) is conservation law of "energy", i.e. the Hamiltonian H, constraint H = 0.
In general, the field equations for the cosmological model in WIG theory with an ideal gas, for the metric follow from the singular point-like Lagrangian

Integrability property and analytic solution
Because the field equations admit a point-like Lagrangian various techniques inspired by analytic mechanics be applied for the study of the dynamical system. Indeed, variational symmetries and conservation laws can be determined by using Noether's theorems [37]. That approach has been widely used in various gravitational systems. New integrable cosmological models as also new analytic and exact solutions were found through the use of variational symmetries, see for instance [38].
We investigate for variational symmetries which have point transformations as generators and provide conservation laws linear in the velocities. Hence, for the Lagrangian function (41) and for ρ m0 = 0, we find that the variational symmetry X = 2 3 a∂ a + 4 (γ − 2) ∂ φ exists for Λ = 0, and the corresponding conservation law is Function F a,ȧ, φ,φ , dF dt = 0, is the second-conservation law for the dynamical system, which means that the field equations form an integrable dynamical system.
In order to reduce the field equations and determine exact solutions, we apply the Hamilton-Jacobi approach. We define the momentum p a = −6aȧ, and p φ = λa 3φ ., thus the Hamiltonian function H (a, φ, p a , p φ ) = 0, reads while the Hamilton-Jacobi equation is written in the following form where now p a = ∂S ∂a and p φ = ∂S ∂φ . Moreover, the conservation law (44) provides the constraint equation for the Action We define the new variable φ = 6 (γ − 2) ln a + Φ, such that the constraint equation This new set of variables {a, Φ} are the normal coordinates for the dynamical system.
Consequently, in the normal variables the analytic expression for the Action as provided by the Hamilton-Jacobi equation is However, in the new coordinates the momentum are defined as which give the following expressions for the scale factor and the scalar field Hence, by using the Action (49) and expressions (53), (54), the cosmological field equations can be written into an equivalent system. We summarize the results in the following proposition. dt and λ = 1 6 . Thus, the field equations areȧ with exact solution For this exact solution the background space is The later solution describes a universe dominated by a perfect fluid source with constant equation of state parameter. This specific solution is descibed by the stationary points A 2 , thus. the results are in agreement with the asymptotic analysis for the dynamics.

CONCLUSIONS
In this work we considered WIG to describe the cosmological evolution for the physical parameters in FLRW spacetime with zero spatially curvutare. The gravitational field equations in WIG are of second-order and Einstein's theory, with the presence of the of a scalar field, is recovered. Scalar field plays the role for conformal factor which relates the connection of Weyl theory with the Levi-Civita connection of Riemannian geometry. However, the field equations differ when matter is introduced in the gravitational model. Indeed, in WIG the matter source interacts with the scalar field. The interaction term is introduced naturally from the geometric character of the theory.
In our study we considered the matter source to be described by that of an ideal gas, that is p m = (γ − 1) ρ m , or by the Chaplygin gas p m = − A 0 ρ α m . We defined new dimensionless variables based on the Hubble-normalization in order to write the field equations as a system of first-order algebraic differential system. In each model, we determined the stationary points for the latter system and we determined their dynamical properties as also the physical properties of the asymptotic solutions. In our analysis we considered also a nonzero cosmological constant.
For the ideal gas, we found that there exists an attractor with an asymptotic solution that of an ideal gas, but with different parameter for the equation of state. For instance, we can consider the matter source to be that of radiation while the attractor to describe an accelerated universe. In the presence of the cosmological constant, we find two asymptotic solutions which can describe the past acceleration phase of the universe known as inflation, as also the late time acceleration. The future attractor describes the de Sitter universe.
When the matter component is that of a Chaplygin gas the stationary points as also the cosmological evolution are similar with the previous case.
Moreover, for the ideal gas case, we solved the inverse problem and determined a Lagrangian function, and a minisuperspace description, which generates the cosmological equations under a variation. We applied Noether's theorems for point transformations in order to construct a nontrivial conservation law when the cosmological constant term is zero. Hence, the cosmological field equations form a Liouville integrable dynamical system. The closedform expression for the Hamilton-Jacobi equation derived. Finally, for a specific values for the free parameters we were able to construct an exact solution which is in agreement with the asymptotic analysis.
In a subsequent analysis we plan to investigate further the field equations as a Hamilton system and understand how a nonzero cosmological constant affects the integrability