Cosmological Solutions in Multiscalar Field Theory

We consider a cosmological model with two scalar fields minimally coupled to gravity which have a mixed kinetic term. Hence, Chiral cosmology is included in our analysis. The coupling function and the potential function, which depend on one of the fields, characterize the model we study. We prove the existence of exact solutions that are of special interest for the cosmological evolution. Furthermore, we derive the necessary conditions that connect the two free functions of the theory so that these solutions can be admitted. We find that unified dark matter and dark energy solutions are allowed by the theory in various scenarios involving the aforementioned functions.


INTRODUCTION
From the detailed analysis of the recent cosmological data [1][2][3] it has been made clear the necessity for the introduction of an exotic fluid term with negative pressure in Einstein's General Relativity. The role of this exotic matter source is to explain the acceleration phases of the universe. So far its nature is unknown and it has only been observed indirectly; it is thus referred in the literature as dark energy. The contribution of the dark energy in the observable universe is approximately ∼ 70% while the rest ∼ 30% corresponds mostly to dark matter (with a smaller contribution by baryonic matter and radiation). Dark matter is a pressureless fluid source (dust) that also has not been observed directly and which was introduced in order to explain the observed rotation curves of the galaxies.
Although the cosmological constant is the most simple dark energy candidate, Λ-cosmology suffers by two major drawbacks: a) fine tuning and b) the coincidence problem [4,5]. A quite wide-known alternative approach on the cosmological constant is scalar field cosmology [6]. Scalar fields can be introduced either minimally or nonminimally coupled to gravity. In what regards the minimally coupled case, the general solution for an arbitrary potential function has been obtained for all types of Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes: open, closed and spatially flat [7]. Scalar fields can also be used to incorporate -in an equivalent picture -the new degrees of freedom provided by the higher-order derivatives in extended theories of gravity [8][9][10][11][12][13]. Apart from the application of scalar fields in order to account for the late-time acceleration of the universe, they have been additionally used as unified dark energy models [14][15][16][17][18][19][20][21][22][23][24].
With the term scalar field cosmology usually someone refers to quintessence theories. However, there are many alternatives like, phantom fields, Hordenski gravity [10], or even multi-scalar field models. The latter have a broad range of applications as they can provide alternative models of inflation, namely: hybrid inflation, α−attractors, double inflation and many others [25][26][27][28][29][30][31][32][33]. Quintom scalar field models consist of two scalar fields, a quintessence field and a phantom field, for a review we refer the reader in [38]. The main characteristic of the quintom models is that in general the two scalar fields are minimally coupled and only a few models have been proposed where an interaction term between the two fields is introduced in the potential [38]. While a quintom model with a mixed-kinetic term has been introduced before [39], due to the fact that the geometry of the space where the kinetic terms of the scalar field form is flat there can always be defined new scalar fields so as to neglect the mixed kinetic term.
Multi-scalar field models where interaction exists in the kinetic term of the Lagrangian is where this work focuses. A specific family of these models, where the two scalar fields defined into a two-dimensional space of constant nonvanishing curvature, are the Chiral cosmological models [33,40,41] which are directly related with the non-linear sigma cosmological model [42][43][44]. In this work we consider two-scalar field cosmological model where the one field is quintessence and the second field interact with the quintessence only in the kinetic term. The model consists of two unknown functions, the potential which drives the dynamics of the quintessence and the interaction between the fields. For this cosmological model we prove the existence of particular solutions of great interest and determine the conditions that they impose in the functions characterizing the theory. Among the solutions we study are: scaling solutions, de Sitter solutions and that of asymptotic de Sitter spacetimes with or without cosmological singularity. The last cosmological scenario is very interesting since ΛCDM cosmology is recovered. What is more, the determination of exact or analytical solutions is important because the main properties of a given model can be determined without suffering from the problem of selecting the initial conditions. Moreover, when we use numerical methods to solve a given dynamical system, we know that the numerical simulations describe real trajectories of the problem. Some results with exact and analytic solutions in multi scalar field cosmology are given in [40,41,[45][46][47][48][49][50][51][52][53][54][55][56][57].
The outline of the paper is as follows: In Section 2 we briefly present the cosmological model that we consider and derive the gravitational field equations in the case of a spatially flat FLRW spacetime. In Section 3 we prove the existence, under conditions, of four important families of scale factor solution of cosmological interest. More specifically, we determine solutions for scaling solutions, de Sitter solution, asymptotically de Sitter spacetimes and exponentials of arbitrary powers of time. There exists a case where the scalar field potential and the interaction function are exponential with different exponents, where the scalar factor is given by a power-law function in which the acceleration of the universe can be described, including the inflationary era. For that specific cosmological model in Section 4 we perform a detailed analysis on the dynamics of field equations by studying the existence and the stability of critical points. Finally, in Section 5 we discuss our results and draw our conclusions.

Let us consider the action integral
characterized by the Lagrangian density where we have adopted the geometric units 8πG = c = 1. As usual, g is used to symbolize the determinant of the space-time metric, while R is reserved for the scalar curvature. From (2.2) it is obvious that we consider two scalar fields φ (x µ ) and ψ (x µ ) as the matter content of the theory. At the same time we assume an interacting function F φ x k and a potential V (φ (x µ )) that depend solely on the scalar field φ (x µ ).
The field equations produced in this setting, by variation with respect to the metric g µν (x µ ) and the scalars φ (x µ ), ψ (x µ ) are respectively: where R µν is the Ricci tensor and the prime denotes a derivative with respect to φ. The energy momentum tensor that appears in (2.3a) is given by Due to the fact that we choose the functions F (φ (x µ )), V (φ (x µ )) to depend only on φ (x µ ), the action (2.1) is bound to remain form invariant under the global transformation ψ (x µ ) → ψ (x µ ) + ε, in which ε is a constant. The corresponding conservation law is already obvious from the equation of motion (2.3) which implies where the term inside the bracket is the Noether current corresponding to the aforementioned symmetry transforma-tion. We consider a spatially flat FLRW spacetime with line element where a (t) is the scale factor of the universe and N (t) is the lapse function. For the scalar fields we assume that they inherit the symmetries of the spacetime which means φ = φ(t) and ψ = ψ(t). Under this assumption the gravitational plus matter field equations reduce to the following set of ordinary differential equations where the dot signifies derivative with respect to the parameter t. From the last equation the corresponding conserved quantity of the reduced system becomes apparent since it implies with c 1 being a constant of integration. It is interesting to note that the same situation can be reproduced in the context of a minisuperspace approximation through the Lagrangian function whose Euler-Lagrange equations are equivalent to (2.7) and of course its symmetries lead to the same conserved charge Q = ∂L ∂ψ . We now proceed with our analysis by investigating the conditions under which certain interesting gravitational solutions may emerge. In what follows we adopt the gauge choice N = 1 so that the results are expressed directly in a cosmological time.

EXACT SOLUTIONS
The gravitational field equations form a constraint system of nonlinear second-order differential equations for the three dependent variables {a (t) , φ (t) , ψ (t)}. There are many alternative ways to study the dynamics and the evolution of the field equations. In this work we are interested in the existence of exact solutions. More specifically, we assume that the physical space is specified by particular functional forms for the scalar factor a (t) which describe various eras in the evolution of the universe. Such an analysis is important because it provides information for the existence of various cosmological eras for the specific cosmological model. In addition to this, the cosmological validity of each model can be studied directly from such an analysis.
From the conserved quantity (2.8) we easily deduce thaṫ .
By adding (2.7a) and (2.7b) the relation is obtained (recall that we have set N = 1). At this point we have acquired the potential V (φ) as a pure function of t. We continue our analysis by considering several interesting situations regarding the scale factor.

Power law solutions
In this case we are interesting in studying the conditions that a solution of the form can set over the theory. The power-law solution (3.3) is important, because it can describe the matter dominated era solution σ = 2 3 , the radiation era solution σ = 1 2 , or the early accelerated phase of the universe when σ > 1. For this choice of scale factor the space-time exhibits a curvature singularity at t = 0 for any value of σ. With the help of (3.3) relation (3.2) results in This leads us to consider two distinct situations depending on the value of σ.
We start by considering that the potential V (φ) cannot be zero, which according to (3.4) corresponds to σ = 1 3 . By inverting equation (3.4) we obtain t as a function of φ, i.e.
If we additionally take the derivative of (3.4) with respect to time, the relation emerges. Direct substitution of (3.1), (3.5) and (3.6) into the constraint equation (2.7a) leads to the expression which involves only the two functions of φ. This is how F (φ) and V (φ) have to be related for a power law solution of the form (3.3) with σ = 1 3 to be admitted by the system. Any choice for the potential V (φ) leads to the corresponding expression of the coupling F (φ) needed for such a solution. Let us proceed with the study of two particular examples.
a. Example 1: Power law potential. Let us set V (φ) = V 0 φ κ , with V 0 and κ constants. Then, (3.7) implies where with F 0 we re-parametrized the constant of integration c 1 corresponding to the conserved charge Q. The two constants are related through the expression By setting (3.1), (3.3), (3.8) and V (φ) = V 0 φ κ into the constraint equation (2.7a) we can solve the latter with respect toφ. In particular, we obtain: (3.10) Substitution of theφ we derived and of all the rest of the previous mentioned expressions into equation (2.7b) leads to an algebraic solution for the scalar field φ: Consequently, the scalar field ψ can be easily obtained now from (3.1) with a simple integration. The solution reads: where ψ 0 is a constant of integration. One can easily verify that the φ(t) and ψ(t) as given by (3.11) and (3.12), together with the scale factor (3.3) satisfy the field equations (2.7) for the power law potential under consideration and the coupling function (3.8). We have thus obtained the solution for the theory, which admits a power law solution for the scale factor and for a power law potential V with respect to φ.
b. Example 2: Exponential potential. We repeat the same procedure only this time we use a potential of the form V (φ) = V 0 e λφ with λ a constant. The compatible (for a power law a(t)) coupling function F (φ) now becomes where F 0 is connected to the c 1 that we saw previously through (3.14) We notice that unlike the power law potential case, here, an exponential potential needs also an exponential coupling function to produce a scale factor of the form (3.3). As previously done, we substitute (3.1), (3.3), (3.13) and V (φ) = V 0 e λφ into the constraint equation (2.7a). From the latter we obtain the expression forφ. Subsequently, we substitute the latter into equation (2.7b) and we get an algebraic solution for φ(t) which this time reads The direct integration of (3.1) leads to Again, it is easy to check that the above expressions satisfy the field equations for the choice of potential and the corresponding coupling function (3.13). After these two examples we can proceed with the case that we excluded from the power law solution of the scale factor.
This means that a(t) = t 1 3 , which corresponds to a potential V (φ) = 0. The function F (φ) remains arbitrary with the solution being described by the equationsψ The latter set implies that the following relation exists between the two scalar fields with ψ 0 being a constant. The function F (φ) can be chosen arbitrary and the solution can be obtained through a simple integration since (3.18) yields (3.20)

De Sitter universe
In this case we introduce a scale factor of the form a(t) = e λt , where λ is a constant. Later in our analysis we shall treat the more general case a(t) = e λt µ , but here we want to distinguish the particular solution corresponding to µ = 1 due to its importance in cosmology. The choice a(t) = e λt for the scale factor leads (3.2) to become Hence, the potential plays the role of a pure cosmological constant.
The constraint equation with the use of (3.1), which now readṡ where F (φ) can be any arbitrary function of φ. As in the special case of the previous section, we can deduce a relation between ψ and φ sinceψ We notice that the previous expressions forψ andφ from (3.23) and (3.22) are such so that the sum of the kinetic terms corresponding to the matter content of the Lagrangian (2.9) is zero, i.e.
Thus, it is necessary for one of the kinetic terms to change its sign. This was expected from the moment that we noticed that the choice a(t) = e λt led to a potential that is a constant. We already know that such a space-time is a pure cosmological constant solution, hence any other contributions should in principle cancel each other in the Lagrangian.

Asymptotically de Sitter space-time
In this section we examine a couple of cases where the space-time can be considered as being asymptotically de Sitter in the sense that, locally, the metric becomes that of a positive cosmological constant solution in the limit t → ±∞.

Space-time without singularity
We adopt for the scale factor the choice a(t) = (cosh(t)) α . The resulting spacetime has its curvature scalars regular both at t → 0 and t → ±∞. We can distinguish two cases depending on whether the constant α takes the value 1 3 or not. We start from the generic case, so we assume a = 1 3 . Under this assumption (3.2) becomes which can be inverted to For large values of t, the potential (3.27) becomes V (φ) ≃ 3α 2 from where the de Sitter solution a (t) ≃ e αt is recovered.
If we take the total derivative with respect to time, equation (3.27) leads tȯ Substitution of (3.28), (3.29) and (3.1) into the constraint equation (2.7a) provides us with a relation between the two functions of φ For completeness, we need to also consider the case α = 1 3 . When this value for the power of cosh(t) is adopted, relation (3.2) results in a constant potential V (φ) = 1 3 . The constraint equation (2.7a) and the first integral Q yield: From the above we extract easily the relatioṅ What is more, equation (3.31) is separable and we can easily derive t in terms of φ for any (admissible) function F (φ), If we consider a scale factor of the form a(t) = (sinh(t)) α we obtain a space-time that possesses a curvature singularity at t → 0, as is evident by the Ricci scalar that reads Once more we have to distinguish cases depending on the parameter α. We start by considering α = 1 3 . The same analysis that we followed previously results in the potential (3.36) which in its turn leads to With the help of the above the constraint equation (2.7a) gives rise to which for integer values of α can be either equal to (3.30) or opposite of it. In particular, if α is an odd number (3.39) is the opposite of (3.30). On the other hand, if it is even, then the two expressions yield the same result. An interesting value for the parameter α is that of α = 2 3 , where the model mimics the Λ-cosmology. Finally, we conclude this section by considering the α = 1 3 case. Once more the potential assumes the constant value V (φ) = 1 3 , while for the two scalar fields we havė By combining the last two relations we geṫ In this section we examine the effect of a scale factor of the form a(t) = e λt µ with µ = 1 (remember that the µ = 1 case was studied separately in section 3.2). For λ < 0 the scale factor can be said that follows a generalized normal distribution (the particular constant factor needed to multiply the exponent can be provided by scaling appropriately the spatial distance r in the metric). In what regards the singularities of the space-time, it can be observed from the scalar curvature R = 6λµt µ−2 (µ − 1 + 2λµt µ ) , (3.44) that the latter diverges at t → ∞ when µ > 1 and at t = 0 when µ ∈ (−∞, 2) − {1}. With this choice of a scale factor equation (3.2) is written as We need to distinguish two cases depending on whether µ = 2 or not, something that is going to become obvious in the subsequent analysis. In this setting it is interesting to observe that the above equation can be solved algebraically with respect to t even for an arbitrary power µ. The solution can be given in terms of nested radicals in a manner similar that the solution of the more general equation of the form [58] αt 2µ + βt µ + γ = t ν .
In our case we have ν = 2 and γ = 0. This value for γ leads us to a small modification of the process of deriving the solution in comparison to what is presented in [58]. We first notice that (3.45) can be written as We raise this last relation to the power 2 − µ (we remind that we assume µ = 2) and we obtain This means that the t in (3.46) may now be given in terms of an infinite number of radicals through successive substitutions of (3.47). In particular we get: By obtaining the expression forφ from the time derivative of (3.45) and by using equation (3.1) into the constraint (2.7a) we are led to where t is understood to be given in terms of V (φ) from (3.48). So F (φ) and V (φ) need to also be related with an expression involving nested radicals in order to acquire a solution of the form a(t) = e λt µ with µ = 2. We now proceed to complete this section with the study of the system when µ = 2.
This situation corresponds to a normal distribution scale factor whenever λ < 0. However, in what follows we make no particular assumption over the sign of λ. By setting µ = 2 in (3.45) we immediately see that it results in At the same time from its time derivative we getφ By following the exact same procedure as in the previous cases, the above relations help us arrive at .
We study a quick example that demonstrates the application of relation (3.52).
a. Example: Power Law potential. We choose to set V (φ) = V 0 φ κ + Λ, where Λ is a constant. According to (3.52) the compatible F (φ) function, for the scale factor we want, is .
The above result leads (3.1) to yield Thus, we derived the solution that allows for a(t) = e λt 2 when a power law potential is present. We summarise all the results obtained in this section, involving the effect that the various scale factors have in the relation between F (φ) and V (φ), in table (I).
We introduce the new dimensionless variables [59] x where H =ȧ a . The field equations are written as the following system in which the new independent variable τ is defined as τ = ln (a (t)). From (4.2) it is clear that the variables {x, y, z} evolve on a three-dimensional sphere with y ≥ 0. Moreover, every critical point of the latter system corresponds to a phase where the scale factor a (t) is given by the expressions a (t) = t 2 3(1+w tot ) when w tot = −1 or a (t) = e H0t when w tot = −1 in which w tot is the equation of state parameter for the effective fluid defined as (4.6) The critical points of the system (4.2)-(4.5) are determined to be Points P (±) A and P B correspond to the scaling solution where the field ψ does not contribute to the evolution of the universe. Points P (±) A and P B are defined in the surface z = 0 and are given in [59]. Points P (±) C are the new points that emerge from our analysis. We continue with the discussion of the physical quantities of the critical points as also with their stability. In order to study the latter, we reduce the dynamical system into a two-dimensional system given by the equations (4.3) and (4.4), where from (4.2) it follows z 2 = 1 − x 2 − y 2 .
As far as the points P (±) A are concerned, the universe is dominated by the kinetic term of the scalar field φ, i.e. w tot = 1. The eigenvalues of the linearized system are determined to be from where we infer that the points are always unstable because e 1 P (±) A is always positive.
Point P B exists only when λ 2 < 6 and describes a scaling solution where w tot = −1 + λ 2 3 . The eigenvalue of the linearized system are determined to be from where we conclude that the solution at point P B is stable when λ 2 < 3. The universe is accelerated when λ 2 < 2.   Fig. 1 See Fig. 2 In what regards the new points P C , attained from our analysis, they describe a scaling solution with w tot = −1 + 2λ λ+µ which corresponds to an accelerated universe when µ < 0, µ 2 < λ < |µ| or µ > 0, − µ < λ < µ 2 . This is the solution we previously derived. It is important to mention that the points exist when λ = −µ and µ < 0, λ ≤ − √ 6 and − √ 6 < λ < 0, µ < 6−λ 2 λ from where we infer that w tot < − 1 3 when λ ≤ − √ 3, µ < λ As far as the stability is concerned, because of the nonlinearity dependence of the eigenvalues e 1 (P C ) , e 2 (P C ) on the free parameters µ, and λ, we solve numerically the conditions where e 1 (P C ) < 0, e 2 (P C ) < 0, and derive when the points are attractors. The region in the space {λ, µ} where the points are attractors are given in Fig. 1.
In Fig. 2 we present the phase space diagram in the variables {x, y, z} for the dynamical system of our consideration for two set of the values λ and µ. More specifically, the plots are for the set (λ, µ) = (−1, 2) where the only stable point is P B and for (λ, µ) = (−3, −4) where points P (±) C are the two attractors of the dynamical system. We summarize the results of this section in in Table II.

CONCLUSION
We considered a two-scalar field cosmological model with a mixed kinetic term. More specifically, we assume that the matter source of the Einstein's General Relativity is described by two scalar field which they have an interacting term in the kinetic part of the scalar field Lagrangian. For this model, we investigate the existence of some exact solutions for the scale factor which describe different phases in the evolution of the universe.
In particular, we proved the existence of power-law solutions, de Sitter universe, asymptotic de Sitter universe with or without initial singularity and for a scale factor that depends on the exponential function of arbitrary powers of the time variable. For each specific solution we found the constraining condition between the two unknown functions of the theory so that it would be admissible. These unknown functions are the scalar field potential V (φ) and the one that provides the interaction in kinetic term F (φ). A special case of the model which we considered is the Chiral cosmology, which can be seen as a unified dark energy and dark matter model. We also demonstrated the validity of our result by providing several examples in which we choose the potential function V (φ) and derive the compatible F (φ) for which the space-time of our choice is admissible.
For the Chiral cosmological model and for the exponential potential we apply the critical point analysis in order to determine the stability of the scaling solutions. We found that the dynamical system admits five critical points where the three of them correspond to the quintessence model of General Relativity. The two new critical points describe scaling solutions and by our analysis we were able to study the stability of the exact solutions.
In a future work we plan to investigate the case where the two scalar fields are interacting also in the potential term, as also when they are nonminimally coupled with the gravity [36].