New conservation laws and exact cosmological solutions in Brans-Dicke cosmology with an extra scalar field

The derivation of conservation laws and invariant functions is an essential procedure for the investigation of nonlinear dynamical systems. In this study we consider a two-field cosmological model with scalar fields defined in the Jordan frame. In particular we consider a Brans-Dicke scalar field theory and for the second scalar field we consider a quintessence scalar field minimally coupled to gravity. For this cosmological model we apply for the first time a new technique for the derivation of conservation laws without the application of variational symmetries. The results are applied for the derivation of new exact solutions. The stability properties of the scaling solutions are investigated and criteria for the nature of the second field according to the stability of these solutions are determined.


INTRODUCTION
The detailed analysis of recent cosmological observations indicates that the universe has been through two accelerating phases [1][2][3][4]. The current acceleration era is assumed to be driven by an unknown source known as dark energy, whose main characteristic is the negative pressure which provides an anti-gravity effect [5]. Furthermore, the early-universe acceleration era, known as inflation, is described by a scalar field, the inflaton, which is used to explain the homogeneity and isotropy of the present universe. In particular, the scalar field dominates the dynamics and explains the expansion era [6,7]. Nevertheless, the scalar field inflationary models are mainly defined on homogeneous spacetimes, or on background spaces with small inhomogeneities [8,9]. In [10] it was found that the presence of a positive cosmological constant in Bianchi cosmologies leads to expanding Bianchi spacetimes, evolving towards the de Sitter universe. That was the first result to support the cosmic "no-hair" conjecture [11,12]. This latter conjecture states that all expanding universes with a positive cosmological constant admit as asymptotic solution the de Sitter universe. The necessity of the de Sitter expansion is that it provides a rapid expansion for the size of the universe such that the latter effectively loses its memory on the initial conditions, which implies that the de Sitter expansion solves the "flatness", "horizon" and the monopole problem [13,14].
In the literature scalar fields have been introduced in the gravitational theory in various ways. The simplest scalar field model is the quintessence model, which consists of a scalar field minimally coupled to gravity [15,16]. Another family of scalar fields are those which belong to the scalar-tensor theory. In this theory the scalar field is non-minimally coupled to gravity which makes it essential for the physical state of the theory. Another important characteristic of the scalar-tensor theories is that they are consistent with Mach's principle.
According to the cosmological principle in large scale the universe is assumed to be homogeneous, isotropic and spatially flat. This implies that the background space is described by the Friedmann -Lemaître -Robertson -Walker (FLRW) spacetime. This spacetime is characterized by the scale factor which defines the radius of the three-dimensional (3d) Euclidean space. Since General Relativity is a second order theory the field equations involve second order derivatives of the scale factor. For simple cosmological fluids like the ideal gas or the cosmological constant, the field equations can be solved explicitly [26]. However, when additional degrees of freedom are introduced, like a scalar field, the field equations cannot be solved with the use of closed-form functions and techniques of analytic mechanics and one looks for First Integrals (FIs) which establish their (Liouville) integrability [27][28][29][30].
The standard method for the determination of FIs is Noether's theory [31]. However, there have appeared alternative geometric methods [32][33][34][35][36][37] which use the symmetries of the metric defined by the kinetic energy in order to determine the FIs of the dynamic equations. In the following we shall make use of one such approach in order to determine the FIs (conservation laws) of the field equations.
In the present study we consider a cosmological model in which the gravitational Action Integral is that of Brans-Dicke theory with an additional scalar field minimally coupled to gravity [38,39]. This two-scalar field model belongs to the family of multi-scalar field models which have been used as unified dark energy models [40][41][42] or as alternative models for the description of the acceleration phases of the universe [43][44][45][46]. Furthermore, multi-scalar field models can attribute the additional degrees of freedom provided by the alternative theories of gravity [47][48][49]. The structure of the paper is as follows.
In Section 2, we define the cosmological model and we present the gravitational field equations. In Section 3, we present some important results on the derivation of quadratic first integrals (QFIs) for a family of second order ordinary differential equations (ODEs) with linear damping and perform a classification according to the admitted conservation laws. The results are applied to the cosmological model we consider in Section 4 where we construct the conservation laws for the gravitational field equations. Due to the nonlinearity of the field equations it is not possible to write the general solution of the field equation in closed-form. However, we find some exact closed-form solutions with potential interest for the description of the cosmological history. The stability of these exact solutions is investigated in section 5. Finally, in Section 6 we summarize our results and we draw our conclusions.

COSMOLOGICAL MODEL
For the gravitational Action Integral we consider that of Brans-Dicke scalar field theory with an additional matter source leading to the expression [17,18] where φ (x κ ) denotes the Brans-Dicke scalar field and ω BD is the Brans-Dicke parameter.
The action S m is assumed to describe an ideal gas with constant equation of state parameter and the Lagrangian function L ψ (ψ, ψ ;µ ) corresponds to the second scalar field ψ (x κ ) which is assumed to be that of quintessence and minimally coupled to the Brans-Dicke scalar field.
With these assumptions the Action Integral (1) takes the following form The gravitational field equations follow from the variation of the Action Integral (2) with respect to the metric tensor. They are where G µν = R µν − 1 2 Rg µν is the Einstein tensor. The energy-momentum tensor T µν = ψ T µν + m T µν where m T µν corresponds to the ideal gas and ψ T µν provides the contribution of the field ψ x k in the field equations.
Concerning the equations of motion for the matter source and the two scalar fields, we find m T µν;σ g µσ = 0, while variation with respect to the fields φ (x κ ) and ψ (x κ ) provides the second order differential equations We assume the background space to be the Friedmann -Lemaître -Robertson -Walker (FLRW) spacetime with line element where a(t) is the scale factor of the universe and H (t) =˙a a is the Hubble function. We note that a dot indicates derivative with respect to the cosmic time t.
From the line element (6) follows that the Ricci scalar is R = 6 ä a + ȧ a 2 . Replacing in the gravitational field equations (3) we obtain where ρ m , p m are the mass density and the isotropic pressure of the ideal gas; and for the quintessence field For the equations of motion for the scalar fields we find andψ Finally, for the matter source the continuity equation m T µν;σ g µσ = 0 readṡ For an ideal gas the equation of state is p m = w m ρ m , where w m is an arbitrary constant.
Substituting in equation (12) we find the solution where ρ m0 is an arbitrary constant.
The system of the ODEs that should be solved consists of the differential equations (7), (10) and (11).

QUADRATIC FIRST INTEGRALS FOR A CLASS OF SECOND ORDER ODES WITH LINEAR DAMPING
Consider the second order ODEẍ where the constant n = −1. In the following we shall determine the relation between the functions ω(t), Φ(t) for which the ODE (14) admits a quadratic first integral (QFI). The case of linear first integrals (LFIs) is also included in our study.
(17) in [51]) and has been answered partially using different methods. In [50] the author used the Hamiltonian formalism where one looks for a canonical transformation to bring the Hamiltonian in a time-separable form. In [51] the author used a direct method for constructing FIs by multiplying the equation with an integrating factor. In [51] it is shown that both methods are equivalent and that the results of [51] generalize those of [50]. In the following we shall generalize the results of [51].
Equation (14) is equivalent (see e.g. [52]) to the equation where the functionω(τ ) and the new independent variable τ are defined as We assume that equation (15) admits the general quadratic first integral where the unknown coefficients K, K 1 , K 11 are arbitrary functions of τ, x. We impose the Replacing the second derivatives d 2 x dτ 2 , whenever they appear using equation (15) we find that the function K 11 = K 11 (τ ) and the following system of equations must be satisfied where b 1 (τ ), b 2 (τ ) are arbitrary functions.
We consider the solution of the latter system (19) - (21) for various values of the power n.
As will be shown for the values n = 0, 1, 2 there results a family of 'frequencies'ω(τ ) parameterized with functions, whereas for the values n = −1 results a family of 'frequencies' ω(τ ) parameterized with constants.

Case n = 2
For n = 2, we derive the functionω = K −5/2 11 and the QFI where c 4 , c 5 are arbitrary constants and the function K 11 (τ ) is given by Using the transformation (16) the above results become and ...
We note that for n = 2 equation (14), or to be more specific its equivalent (15), arises in the solution of Einstein field equations when the gravitational field is spherically symmetric and the matter source is a shear-free perfect fluid (see e.g. [55][56][57][58][59][60]).
The QFI (17) is and the functionω It has been checked that (36), (37) for n = 0, 1, 2 give results compatible with the ones we found for these values of n. Using the transformation (16) we deduce that the original system (14) is integrable iff the functions ω(t), Φ (t) are related as follows In this case the associated QFI (36) is These expressions generalize the ones given in [51]. Indeed if we introduce the notation

COSMOLOGICAL EXACT SOLUTIONS
We can use the above results as an alternative to the Euler-Duarte-Moreira method of integrability of the anharmonic oscillator [61] in order to find exact solutions in the modified Brans-Dicke (BD) theory.
We note that for b 1 = 0 we find the results of the subsection 4.4 below when n = 0.
In the special case with c 5 = 0, we find for equation (55) Therefore the Klein-Gordon equation (40) becomes The latter equation can be solved by quadratures. In particular admits the Lie symmetries By using the vector field Γ 1 we find the reduced equation 1 2 d dλ f 2 + 2λ d dλ f + 12f + λ 2 = 0 in which f (λ) = t 3ψ , λ = t 2 ψ. The latter equation is an Abel equation of second type.
Moreover if we assume that λ is a constant, λ = λ 0 then we find ψ = λ 0 t −2 where by replacing in equation (57) it follows λ 0 = 24. Therefore we end up with the solution ψ = 24 t 2 . Let us now find the complete solution for the gravitational field equations for this particular exact solution.
Replacing these results in the rest of the field equations for dust fluid source, that is, p m = 0 and ρ m = ρ 0 a −3 where ρ 0 is a constant, the evolution equation for the Brans-Dicke field becomesφ which admits the general solution where k 1 is an arbitrary constant. Finally by replacing in the constraint equation (7) follows We conclude that the gravitational field equations for this model with the use of the QFI for equation (40) admit the following exact solution with physical quantities For the solution (58) the transformation (41) gives Then the transformed field equations (42) and (44)  In this case the associated QFI (36) becomes and the function a(τ ) is Substituting the given functions ω(t), Φ(t) in the relation (38) we find equivalently that and the associated QFI (39) becomes We consider the following special cases for which equation (40)   For small values of |τ | (i.e. c 1 = c 3 = 0) the scale factor (62) is approximated as a (τ ) τ − n+3 12 , therefore it follows where B 0 = − c 2 (n−1) 4 n+3 3(n−1) and t 0 is an arbitrary constant.
For this asymptotic solution the equation of motion (40) for the second field ψ becomes For the latter equation the QFI (64) is This QFI for the scale factor (65) together with the results of the cases n = 0, 1, 2 produce new solutions ψ(t) which have not found before.
Furthermore, for the scale factor (65) the closed-form solution for the scalar field ψ (t) from (66) is derived whereas for the BD field φ (t) it follows that n = 3, φ(t) = φ 0 (t−t 0 ) 2 and ω BD = − 3 2 . However, this value for the BD parameter ω BD is not physically acceptable. Hence we do not have any close-form solution. In all discussion above we have considered ρ m = 0.
The closed-form solution found in this section is not the general solution of the field equations. That is easy to be seen since they have less free parameters from the degrees of freedom of the dynamical system. However, this form of solutions are of special interest in cosmological studies because they can describe various phases of the cosmological evolution, such as the early inflationary epoch.

STABILITY OF SCALING SOLUTIONS
According to the methods in [62][63][64] let be F (ψ,ψ, ψ) = 0 (79) a second order ODE in one dimension which admits a singular power law solution where ψ 0 is an arbitrary constant. To examine the stability of the solution ψ c , the logarithmic time T through t = e T is introduced, such that t → 0 as T → −∞ and t → +∞ as T → +∞.
We use ψ ≡ dψ dT in the following discussion. The following dimensionless function is introduced and the stability analysis in translated into the analysis of the stability of the equilibrium point u = 1 of a transformed dynamical system. To construct the aforementioned system the following relations are useful: In this section we use a similar procedure for analyzing stability of the scaling solutions obtained in section 4.4.
The origin O is a source, and the orbits diverge to infinity.
When n < 1 the power law solution P is a sink, whereas in the other cases is a saddle given |n| > 1.

CONCLUSIONS
In this work we considered a cosmological model consisted by a Brans-Dicke field and a minimally coupled quintessence field in a spatially flat FLRW background space. For this cosmological model the gravitational field equations consist a Hamiltonian system of six degrees of freedom. The dynamical variables correspond to the scale factor and to the two scalar fields.
In order to study the integrability of the field equations we have applied a direct method which determines the FIs of a dynamical system without the use of Noether's theorem.
In this approach one assumes a generic form for the FIs, say I, and applies directly the condition dI/dt = 0 using the dynamical equations. These considerations resulted in a system of partial differential equations involving the unknown coefficients defining I and the dynamical quantities which characterize the dynamical system. The resulting system of equations is solved in terms of the symmetries and the Killing tensors of the kinetic metric and its solution provides the considered FIs.
For a power law scalar field potential function of the quintessence field we found conservation laws quadratic in the first order derivatives. Using the conservation laws we were able to find exact solutions for the field equations. In particular we found scaling solutions for the scale factor which describe ideal gas solutions. The stability properties of these solutions was investigated. We were able to recover previous published results in the literature and also to find new QFIs.
Using methods in [62][63][64] we have studied second order ODE in one dimension which admits a singular power law solution ψ c (t) = ψ 0 t β where ψ 0 is an arbitrary constant. To examine the stability of the solution ψ c , the logarithmic time T through t = e T was introduced, such that t → 0 as T → −∞ and t → +∞ as T → +∞. According to our analysis, the scaling solution (73) is transformed to the equilibrium point P := (x, y) = (1, 0), which is a sink for −1 < n < 1 or a saddle for n < −1, or n > 1. The dynamical system also admits the trivial solution O : (x, y) = (0, 0) as an equilibrium point and in case that n is odd, the symmetrical point P given byP := (x, y) = (−1, 0) is also an equilibrium point.
The origin is unstable for n > 1. If n is odd number, the pointP exists and it is a saddle.
Until now, the majority of this kind of studies, for the investigation of conservation laws, have been done mainly with the application of variational symmetries. Our approach is more general and does not required the existence of a point-like Lagrangian, that is, of a minisuperspace description. Therefore, this generic approach can be applied in other gravitational models without minisuperspace such are the Class B Bianchi spacetimes.