Stability of accelerating cosmology in two scalar-tensor theory: Little Rip versus de Sitter

We develop the general reconstruction scheme in two scalar model. The quintom-like theory which may describe (different) non-singular Little Rip or de Sitter cosmology is reconstructed. (In)stability of such dark energy cosmologies as well as the flow to fixed points is studied. The stability of Little Rip universe which leads to dissolution of bound objects sometime in future indicates that no classical transition to de Sitter space occurs.


I. INTRODUCTION
The observational data clearly indicate that current universe experiences the cosmic acceleration (dark energy epoch). We are still far away from complete understanding of dark energy which is often associated with some fluid with equation of state parameter w being close to −1, or with modified gravity. The observational data favor the ΛCDM model whose equation of state parameter w is equal to −1 exactly. Nevertheless, it is quite possible that the universe evolution is governed by phantom (w < −1) or quintessence (−1/3 > w > −1) dark energy. If so then the future universe can evolve to finite-time singularity like Big Rip [1] or quintessence-related soft singularity of one of three types according to classification [2]. The natural prescription to cure finite-time future singularity may be found in frames of number of viable models of modified gravity [3]. However, it is still very interesting to understand if the solution of singularity problem may be found within fluid dark energy.
Recently, a new scenario to avoid future singularity has been proposed in [4] (for further development see Refs. [5][6][7][8][9]). According to this scenario the universe equation of state parameter w is less than −1, so that dark energy density increases with time but w sufficiently rapidly approaches −1 asymptotically. In this way, the finite-time singularity is avoided. However, such proposed non-singular cosmology leads to dissolution of bound objects some when in future, similarly to Big Rip singularity. That is why the scenario was called Little Rip cosmology. The scalar models to describe Little Rip were introduced in Ref. [5], they turn out to be phantom-like scalars. It is known that one scalar models are stable in phantom phase and instable in non-phantom phase. Moreover, the large instability occurs when crossing the phantom-divide (or cosmological constant border). In order to understand Little Rip cosmology better, as well as its relation with asymptotically de Sitter universe and possible transitions between these two spaces, the more realistic description of Little Rip may be necessary.
In the present paper we develop such description in terms of two-scalar tensor theory which represents kind of quintom model [10,11] (for review, see [12]). Indeed, one of two scalars is taken to be phantom.
In Section II, we consider a general formulation of reconstruction in two scalar model and investigate the stability of the solution. In this formulation, we construct a model which has a stable cosmological solution describing the phantom-divide crossing. In Section III, we reconstruct a model which describes the cosmological solutions with and without Little Rip and investigate the (in)stability of the solutions. The existence of the solution describing de Sitter space-time and the stability of the de Sitter solution when it exists as well as possible transition of Little Rip cosmology to de Sitter one are investigated. In Section IV, we also consider the reconstruction of the two scalar model in terms of the e-foldings N and investigate the flow of the solution in terms of dimensionless variables, which give the fixed points for some solutions. Some summary and outlook are given in Discussion section.

II. RECONSTRUCTION OF TWO SCALAR MODEL AND (IN)STABILITY
For the model with one scalar, the solution is stable in the phantom phase but unstable in the non-phantom phase. The instability becomes very large when crossing the cosmological constant line w = −1. In order to avoid this problem, one may consider two scalar model [13,14].
We now consider the following two scalar model Here ω(φ) and η(χ) are functions of the scalar field φ and χ, respectively. The FRW equations give Then if the explicit solution follows One may choose that ω should be always positive and η be always negative, for example Here α is an arbitrary real function. We now define a new functionf (φ, χ) bỹ which givesf If V (φ, χ) is given by usingf (φ, χ) as the FRW and the following scalar field equations are also satisfied: In case of the one scalar model, the instability becomes infinite at the crossing w = −1 point. One may expect that such a divergence of the instability does not occur for model with two scalars.
By introducing the new quantities, X φ , X χ , andỸ as the FRW equations and the scalar field equations (9) are rewritten as Here d/dN ≡ H −1 d/dt. In the solution (4), X φ = X χ =Ỹ = 1. The following perturbation may be considered Hence d dN The eigenvalues of the matrix M are given by The eigenvalues (15) for the two scalar model are clearly finite. Hence, the instability, if any, could be finite and by choosing α in (5) properly, the instability can be removed, in general. In fact, right on the transition point wherė H = f ′ (t) = 0 and therefore f ′ (φ) = f ′ (χ) = 0, for the choice in (5) with constant α, α(φ) = α(χ) = α > 0, we find Then the eigenvalues (14) reduce to Then as long asḦ αH < 3, all the eigenvalues are negative and therefore the solution (4) is stable. Hence, we gave general formulation of reconstruction in two scalar model and investigated the stability of the solution. By using this formulation, we can construct a model which has a stable cosmological solution corresponding to the phantom-divide crossing.

III. RECONSTRUCTION OF LITTLE RIP COSMOLOGY
In this section, by using the formulation of the previous section, we construct a model which may generate a Little Rip cosmology [4].
As an example we consider the following Hubble rate which corresponds to the Little Rip. Here H 0 and λ are positive constants. Eq. (17) shows that there is no curvature singularity for finite t.
When we ignore the contribution from matter, the equation of state (EoS) parameter w of the dark energy can be expressed in terms of the Hubble rate H as Then ifḢ > 0, w < −1. By using Eq. (18), one finds and therefore w < −1 and w → −1 when t → +∞, and w is always less than −1 whenḢ is positive. The parameter A in [4] corresponds to 2λ/ √ 3 in (17) and is bounded as 2.74 × 10 −3 Gyr −1 ≤ A ≤ 9.67 × 10 −3 Gyr −1 by the results of the Supernova Cosmology Project [16].
In the model (17), H is always finite but increases exponentially, what generates the strong inertial force. The inertial force becomes larger and larger and any bound object is ripped. This phenomenon is called a "Little Rip" [4].
For the model (17) with (20), the eigenvalues (14) are given by corresponding expressions when λt ≫ 1, which are negative. Therefore the solution is stable. Let us consider the possibility that the universe could evolve to the de Sitter space-time. In order for the solution corresponding to the de Sitter space-time to exist, there should be an extremum in the potential and the potential should be positive there. If the extremum is local minimum with respect to φ, which is the canonical scalar and local maximum with respect to χ, which is non-canonical or phantom scalar, the solution is stable. For the potential (23), there is an extremum when where the value of the potential V (φ, χ) is given by which is negative and therefore there does not exist the solution corresponding to the de Sitter space-time. Hence, the universe does not evolve into the de Sitter space-time.
We now show that the (asymptotically) Little Rip solution (17) is always asymptotically stable. For large t, one assumes the solution behaves as (17). In three eigenvalues (14), for the asymptotically Little Rip solution which is in the phantom phaseḢ > 0, the eigenvalue MỸ is negative. If we write α(χ) in (5) as the eigenvalue M χ (14) can be expressed as a function of t = χ as follows, If M χ could be positive, q(t)q ′ (t) must be negative. Since q(t) 2 is positive, q(t) 2 goes to a constant q(t) 2 → Q 0 ≥ 0. Then due to the factor e λt in the denominator of the first term in (28), the first term goes to small value for large t and we find M χ → −3 < 0. Therefore M χ is asymptotically negative. For the eigenvalue M φ , one gets Then again in order that M φ could be positive, q(t)q ′ (t) must be negative and therefore q(t) 2 goes to a constant q(t) 2 → Q 0 ≥ 0. Due to the factor e λt in the denominator of the first term in (29), the first term goes to small value for large t and M φ → −3 < 0. Therefore, all the eigenvalues are negative and the Little Rip solution is asymptotically stable.
As another example, one can consider the following the model: Here H 0 , H 1 , and λ are positive constants and we assume H 0 > H 1 and t > 0. Since the second term decreases when t increases, the universe goes to asymptotically de Sitter space-time. Then from Eq. (18), we find As in the previous example (17), w < −1 and w → −1 when t → +∞. In this model, there does not occur the Little Rip. The inertial force generating the Little Rip is given by Here we consider two points separated by a distance l and assume there is a particle with mass m at each of the points, Since the magnitudes of H andḢ are bounded in the model (30), the Little Rip does not occur although the magnitudes of H andḢ become larger and larger in the model (17). For t → ∞, Eq. (31) gives the asymptotic behavior of w to be which is identical with (19) if we replace λH 1 /H 0 with λ. By choosing α in (5) as, with a constant α 0 , we find ω(φ) and η(χ) in (5) as follows Using (6), one getsf and the potential (8) is given by For the model (30) with (34), the eigenvalues (14) are given by Therefore as long as 3 > λ H0 , the solution (30) is stable. Since the solution can be unstable if 3 < λ H0 , we again consider the possibility that the universe could evolve to the de Sitter space-time. For the potential (37), there is an extremum when where the value of the potential V (φ, χ) is given by If 12 > λ H0 , which is consistent with the condition 3 < λ H0 that the solution (30) is stable, V (φ, χ) is positive and there is a solution corresponding to the de Sitter space-time. Therefore there is a possibility that the universe could evolve into the de Sitter space-time. Note, however, the solution (39) corresponds to the minimum with respect to both of φ and χ and therefore the de Sitter solution is not stable.
We may also consider the following the model: Here H 0 , H 1 , and λ are positive constants and we assume H 0 > H 1 and t > 0. Since the second term decreases when t increases, the universe goes to asymptotically de Sitter space-time. Then from Eq. (18), it follows Hence, the EoS parameter is always larger than −1 and w → −1 when t → +∞. Therefore the universe is in non-phantom phase. By choosing α in (5) as in (34), we find ω(φ) and η(χ) (5) as follows Using (6), one getsf and the potential (8) is given by For the model (41) with (34), the eigenvalues (14) are given by , there is a possibility that the universe could evolve to the de Sitter space-time as in the model (37). For the potential (45), there is an extremum when where the value of the potential V (φ, χ) is identical with that in (40) and positive if 12H 0 > λ. The solution (47) is not stable, again.
As one more example, we consider the realistic model which contains the inflation at t → −∞, phantom crossing at t = 0, and the Little Rip when t → ∞: Here H 0 and λ are positive constants. SinceḢ we findḢ < 0 when t < 0, that is, the universe is in non-phantom phase andḢ > 0 when t > 0, that is, the universe is in phantom phase. There occurs the phantom crossing at t = 0. Therefore the present universe corresponds to t ∼ 0. When λt ≫ 1, we find that the Hubble rate H behaves as and therefore there occurs the Little Rip. The EoS parameter w is now given by Hence, w < −1 when t > 0 and w > −1 when t < 0. In the limit t → ±∞, w → −1. Thus when t → −∞, there occurs the accelerating expansion, which may correspond to the inflation in the early universe. When w = − 1 3 , that is, there occurs the transition between non-accelerating expansion and accelerating expansion. There are two negative solutions in (52) in general. Let us denote the solution as t i and t l and assume t i < t l < 0. Then t = t i corresponds to the end of inflation and t = t l to the transition from the non-accelerating expansion to the late accelerating expansion in the present universe. More explicitly Let the present universe corresponds to t = t present . Since t i − t present = 137 × 10 8 ∼ 1 Hpresent = 138 × 10 8 years (H present ∼ 70km/s Mpc), if we assume t present = 0, however, Eq. (52) does not have a solution for t i . Then we may assume t present > 0, that is, the present universe is after the phantom crossing. In this case in principle, one can solve (52) with respect to λ. Then we can obtain the value of t l from the second equation in (53). Roughly one can expect the magnitude of the value could be t l ∼ 50 × 10 8 years and therefore the realistic cosmology follows.
The acceleration by the gravitational force between the sun and the earth is given by Here l is the distance between the sun and the earth and ω A is the angular speed If the acceleration a e of the inertial force by the expansion (32) exceeds a g , there occurs the rip between the earth and the sun, that is which tells e 2λt = 8.35 × 10 19 or λt = 22.9 .
If λ = O 10 −10 years −1 , t ∼ 10 11 years. For the model (48), by choosing α in (5) to be a constant α = H 0 λ, one finds Using (6), it followsf (φ, χ) = H 0 e −λφ + sinh λχ , For the model (50) with α = H 0 λ, the eigenvalues (14) are given by Therefore if 3 > λ H0 , all the eigenvalues are negative and therefore the solution is stable. For the potential (60), there is no extremum and therefore there does not exist the solution corresponding to the de Sitter space-time.
Thus, we constructed scalar models which describe the cosmological solutions with and without Little Rip and investigated the (in)stability of the solutions. We also investigated the existence of the solution describing de Sitter space-time and the stability of the de Sitter solution when it exists as well as possible transition of Little Rip cosmology to de Sitter one. The results are summarized in Table I.

IV. RECONSTRUCTION IN TERMS OF E-FOLDINGS AND SOLUTION FLOW
Here we consider the reconstruction of the two scalar model in terms of the e-foldings N . (For such a formulation in modified gravity, see [15].) We also investigate the flow of the solution by defining dimensionless variables, which give the fixed points for some solutions.
Let us consider two scalar model again. By using the e-foldings N the FRW equations (2) and the scalar field equations (9) are rewritten as where ′ denotes the derivative with respect to the e-foldings N ≡ ln a. New functionf (φ, χ) is defined bỹ Here f 0 is a dimensionless constant. We assume the potential V (φ, χ) is given by Then if the functions ω(φ) and η(χ) satisfy the following relations a solution of φ, χ and H is given by Then one can obtain a model which reproduces arbitrary expansion history of the universe given by H = f (N )/κ, by choosing ω(φ), η(χ) and V (φ, χ) by (67)  Let us introduce the dimensionless variables as follows: Eqs. (64) and (65) are rewritten as Now the Hubble rate H is given by In order for the Hubble rate to be real, the values of X, Y , Z and W are restricted to a region When ω(φ) and η(χ) satisfy (69), this system has two fixed points as follows: Here the solution is given by (70).
Point B (X, Y, Z, W ) = (β 1 , β 1 , 1, 0) We now define β(N ) by Here β 0 and β 1 are dimensionless constants which satisfy the identities This point exists only if there exist β 0 and β 1 which satisfy (79). In this point, the solution is given by .
Especially when β 1 = 0, this point describes de Sitter space-time.
We now choose ω(φ) and η(χ) as where α(N ) is an arbitrary function. If we choose α(N ) = α 0 f ′ (N ) and f (N ) being a monotonically increasing or decreasing function, ω(φ) and η(χ) are respectively given by Here α 0 is a dimensionless constant and ǫ ≡ f ′ (N )/|f ′ (N )|. Then V (φ, χ) has the following form: As an example, we consider where λ is a dimensionless constant. Eq. (84) has a solution H(N ) = f 0 e λN /κ. Then ω(φ) and η(χ) are given by and V (φ, χ) is: The EoS parameter of the point A is which is independent of N . If λ > 0, the point B exists and is located in (3/λ, 3/λ, 1, 0). Then the solution is given by Here N 0 is an arbitrary constant. The EoS parameter of this point is −3. In this model, the dynamics of X and Y are independent of Z and W . Therefore, we consider a small fluctuation from each fixed point by  Here X 0 and Y 0 are the values of X and Y in each fixed point. Then (72) and (73) have the following form: The eigenvalues of this matrix (91) are given by These indicate that the point A is stable if −3 < λ < 3 and unstable if λ < −3 or λ > 3. Similarly, the point B is stable if λ > 3 and unstable if 0 < λ < 3. The dynamics of X and Y are shown in FIG. 1 and FIG. 2. We now consider another example where γ is a dimensionless constant. We should note that the case γ = 1 corresponds to the Little Rip model (17).
Then ω(φ) and η(χ) are given by and V (φ, χ) has the following form: The EoS parameter of the point A is which becomes −1 when N → ∞. If γ < 1/2, the point B exists and is located in (0, 0, 1, 0). Then the solution is This point corresponds to the de Sitter space-time. We consider a small fluctuation from the point B by Then (72) and (73) have the following form: The solution of this equation (99) is given by Here C 1± and C 2± are arbitrary constants and σ 1± and σ 2± are given by These indicate that the point B is always unstable. On the other hand, when φ, χ → ∞, the solution of X and Y are given by Here X 1 and Y 1 are arbitrary constants. This indicates that the values of X and Y approach to (1, 1) if φ, χ → ∞ when N → ∞. Then the Hubble rate approaches to f 0 N γ /κ, which corresponds to the Little Rip universe if γ = 1.
The dynamics of X and Y are shown in FIG. 3 and FIG. 4. Then we have completed the formulation of the reconstruction in terms of the e-foldings. The e-foldings description is directly related with redshift and therefore, with the cosmological observations. We also investigated the flow for the solution, which shows the (in)stability of the reconstructed solution obtained in a large range. Even if the solution is stable, when the stable region is small, the evolution of the universe depends strongly on the initial conditions. If the The point A is located in (1, 1). The point B is located in (0, 0), which corresponds to the de Sitter universe. initial condition is out of the range, the universe does not always evolve to the solution obtained by the reconstruction.
On the other hand, if the stable region is large enough, even if the universe started from an initial condition in a rather large region, the universe evolves to the solution obtained by the reconstruction.

V. DISCUSSION
In summary, we gave a general formulation of reconstruction in two scalar model and investigated the stability of the solution. This formulation helps us to construct a model which has a stable cosmological solution describing the phantom-divide crossing. By using the formulation, we constructed a model which describe the cosmological solutions with and without Little Rip and investigated the (in)stability of the solutions. The existence of the solution describing de Sitter space-time was also investigated and furthermore the stability of the de Sitter solution when it exists as well as possible transition of Little Rip cosmology to de Sitter one was investigated. We also considered the reconstruction of the two scalar model in terms of the e-foldings N and investigated the flow of the solution by defining dimensionless variables, which give the fixed points for some solutions.
Finally, let us make several remarks about the relation of the qualitative behavior of the Universe evolution and the shape of the scalar potential.
In case of the usual canonical scalar field as φ in (1), when the field climbs up the potential, the kinetic energy decreases until the kinetic energy vanishes. Even in case of the phantom field as χ in (1) with non-canonical kinetic term, the kinetic energy decreases when the field climbs up the potential. In case of the phantom field, the kinetic energy is unbounded below and therefore the absolute value of the kinetic energy increases when the field climbs up the potential. The big rip or Little Rip occurs when the potential goes to infinity. If the potential tends to infinity in the finite future, the evolution corresponds to the big rip but if the potential goes to infinity in the infinite future, the evolution corresponds to the Little Rip. Then the necessary condition that the big or Little Rip could occur is 1. The potential does not have maximum and it goes to infinity.
2. There is a path in the potential that the potential becomes infinite but the kinetic energy of the canonical scalar field is vanishing.
Since we identify the scalar field φ and χ with the cosmological time, the second condition means ω(φ) in (1) goes to zero when φ goes to infinity and therefore the phantom field χ dominates. Conversely, if there is a maximum in the potential or there is no path in the potential that the potential becomes infinite but the kinetic energy of the canonical scalar field goes to zero, there does not occur big rip nor Little Rip. Let us suppose the case that there is a maximum in the potential. If the fields stay near the potential maximum, the universe becomes asymptotically de Sitter space-time. If the fields go through the maximum and the potential decreases, the kinetic energy of the canonical scalar field increases but the absolute value of the kinetic energy of the phantom field decreases. If the kinetic energy of the phantom field goes to zero, the canonical field becomes dominant and the Universe could enter the non-phantom (quintessence) phase and there might occur the deceleration phase in future.