Is interacting vacuum viable?

We study the asymptotic dynamics of dark energy as a mixture of pressureless matter and an interacting vacuum component. We find that the only dynamics compatible with current observational data favors an asymptotically vanishing matter-vacuum energy interaction in a model where dark energy is simulated by a generalized Chaplygin gas cosmology


I. INTRODUCTION
In recent years most of the cosmological studies have been focused on variations of General Relativity and modifications of the Standard Cosmological model [1]. This is done in order to provide a more reliable framework to explain the present physical evidence of the universe. It is observed today that only 5% of the matter content is of baryonic form and additional evidence coming from the high redshift surveys of type I supernovae [2,3] indicate that we currently live in a universe that undergoes an accelerated expansion.
In the present literature many cosmological models involve the presence of an exotic type of matter component that lies beyond the framework of standard cosmology, cf. [4][5][6][7], in an attempt to explain the present acceleration of the observed universe. Additionally, the case of coupling and energy transfer between dark energy and dark matter leads to research efforts that try to alleviate the so-called coincidence problem [8,9]. One unified dark energy model that has attracted the interest for research is the Generalized Chaplygin Gas model (GCG in short). This model has a dual character since at early times it satisfies the properties of a matter-dominated universe whereas at late times it approaches the limiting behavior of dark-energy dominated universe [10].
In previous works [11][12][13], we have studied the asymptotic dynamics near finite-time singularities of flat and curved universes filled with two interacting fluids using an interaction term that was first introduced by Barrow and Clifton, cf. [14,15]. In the present work, we consider the case of energy exchange between dark matter (as pressureless dust) and dark energy (as vacuum) with the local energy transfer being associated with the energy density of the vacuum (that is ρ V ) so that Q µ = −∇ µ ρ V [16].
In the limit of zero energy exchange (Q µ = 0), or equivalently if the vacuum energy is covariantly conserved (∇ µ ρ V = 0), then the vacuum energy must be homogeneous in spacetime and equal to a cosmological * email: georgia.Kittou@aum.edu.kw constant [17]. Under these conditions, we address the question of the viability and stability of the interacting vacuum model on approach to the finite-time singularity by studying the asymptotic properties of solutions of the scale factor, the total energy density and total pressure of the universe.
The cosmological model is expected to be stable, and therefore acceptable, if asymptotically it reproduces the dominant features of dark matter and dark energy at both early and late times respectively. We show that asymptotically at early times the energy exchange is vanishing and the energy density of the vacuum is approximately zero, in contrast to what occurs in the standard cosmological model. Hence, at early times and in the absence interaction, our model is indistinguishable from the CDM model [18].
The asymptotic analysis of the solutions is carried out using the method of asymptotic splittings, cf. [19,20]. The analysis provides a complete description of all possible dominant features that the solution possesses as it is driven to a blow-up.
The plan of this paper is as follows. In the next section, we write down all possible asymptotic decompositions of the basic differential equations of our problem describing the GCG model. Sections A, B and C present a detailed study of the various asymptotic solutions. In the last section we discuss our results and point out some interesting open problems in this field.

II. DECOMPOSED DARK ENERGY MODELS
We study the case of the generalized Chaplygin gas model in flat FRW universe as a mean to explain the accelerated expansion of the universe [21]. In the GCG approach the exotic cosmological fluid is defined by the barotropic equation of state where A is a positive constant and 0 < α ≤ 1. This leads to a cosmological solution for the density where a is the scale factor of the universe and B is a positive integration of constant for a well defined ρ cgc at all times. From Eq. (2), one can conclude that at early times the asymptotic solution for the energy density reproduces the ΛCDM model as described by in the limit of vanishing constant α → 0 [22,23]. At late times the solution (2) implies that the fluid behaves as a cosmological constant This interpolation of the model between two different fluids at different stages of the evolution of the universe suggests that the GCG model can be interpreted as a mixture of two cosmological fluids with mutual interaction. Now, any unphysical oscillations or exponential blowup in the matter spectrum produced by such a unified model [22] can be avoided, if one excludes coupling with phantom fields [18]. Therefore, the unique coupling between dark matter (pressureless dust) and dark energy (cosmological constant) makes the GCG model a wellbehaved model both at early times (approach the successful CDM model) and at late times (approach de Sitter Universe).
The Einstein equations for a flat Friedman universe filled with pressureless dust (ρ m ) and vacuum (ρ v ), scale factor a(t) and Hubble expansion rate H =ȧ/a reduce to the Friedman equation If we assume a fluid interaction of the form [16] and given Eq. (2) then the energy density for dark matter is given by the form [18] the energy density for the vacuum satisfies the form The continuity equations for matter and vacuum are given byρ respectively. Equations (5),(9) and (10) describe a 3dimensional system with unknowns (a, ρ m , ρ V ) satisfying the constraint given by Eq. (5). After some manipulations it is proved that the above set of equations leads to the following master differential equation It will be very useful for our calculations to rewrite the master equation (11) in a suitable dynamical system form. In this respect, we rename H = x and find the 2-dimensional systeṁ Equivalently, we have the vector field The vector field can split [19] in three different ways namely In the following sections we apply the method of asymptotic splittings, analytically expounded in [19,20], to describe the asymptotic properties of the solutions of the dynamical system (12) in the vicinity of its finite-time singularities.
A. Approaching the CDM model In this section, we give necessary conditions in terms of the parameter α for the existence of generalized Fuchsian series type solutions [20] towards the finite-time singularity of the first decomposition To do so, we look for possible dominant balances by substituting the forms x(t) = θt p , y(t) = ξt q in the dominant part of the decomposition described by the system belowẋ = ẏ y = −3(α + 1)xy.
We assume here that Ξ = (θ, ξ) ∈ C and p = (p, q) ∈ Q. This leads to the unique balance for 0 < α ≤ 1. The subdominant part of the splitting (14) satisfies and is asymptotically subdominant [11] in the sense that only if α → 0. We therefore conclude that in the neighbourhood of the finite-time singularity the asymptotic solution is meaningful only in the limit of vanishing α, that is in the absence of interaction between the two fluids.
Next we calculate the Kovalevskaya matrix given by, where Df is the Jacobian matrix of the decomposition. For this case the Kovalevskaya matrix reads As discussed in [19] the number of non-negative Kexponents equals the number of arbitrary constants expected to appear in the series solution, while the −1 exponent corresponds to the arbitrary constant relevant to the position of the singularity (for notational convenience taken to be t = 0). Therefore, if the balance is to correspond to a general solution, two arbitrary constants are expected to appear in the series expansion (since the original system (12) is two dimensional). Here we find with a corresponding eigenvector v T 2 = (1, −1).
Hence, it is expected that the balance B I will correspond to a general solution. Substituting the series expansions in the system (12) we arrive after manipulations at the following asymptotic solution around the singularity The y-expansion is derived from the above by differentiation. As a final test for the validity of this solution, a compatibility condition has to be satisfied for every positive K-exponent [11]. For the positive eigenvalue 2 and an associated vector v T 2 = (1, −1) it reads and this is indeed true based on previous recursive calculations. It follows from Eq. (26) that all solutions are dominated by the x = H ∼ 2 3 t −1 solution which in terms of the scale factor reads a(t) ∼ t 2/3 as t → 0, α → 0.
A comment about the asymptotic results of our solution is in order. After applying some manipulations on Eqs. (5-9) and Eq. (10) one can derive the following equations for the evolution of energy densities Having in mind the dominant form of the solution for the scale factor given by Eq. (28), it follows that in the vicinity of the finite-time singularity and in the absence of interaction the energy densities for both fluids satisfy the forms The result above can be also derived from Eqs. (7)- (8) for the same limits as above. Now, the geometric character of the singularity is completely described in terms of the asymptotic behavior of the total energy density and pressure of the model, the asymptotic behavior of the scale factor and the Hubble parameter [12]. For the solutions above it occurs that as t → 0 and α → 0. The asymptotic conditions above describe the case of a Big Bang type of singularity. Consequently, the singularity is necessarily placed at early times. It is discussed in cf. [18] that the contribution from the cosmological constant is negligibly small at early times, hence we conclude that our decomposition describes a model that is indistinguishable from a CDM dominated universe in the past.

B. Quasi de Sitter Universe
Let us now move on to the asymptotic analysis of the decomposition with dominant part given by the vector field Now by substituting in the asymptotic system (ẋ,ẏ) = [y, −2α(y 2 /x)] the forms x(t) = θt p and y(t) = ξt q we find the following dominant balance where θ is an arbitrary constant. The candidate subdominant part of the vector field, namely f (sub) II (x, y) = [0, −3(α + 1)xy] is vanishing asymptotically without any restrictions on the values of the parameter α nor the constant θ. Hence the decomposition is acceptable. To continue with, the Kovalevskaya matrix is given by with corresponding eigenvalues and an eigenvector v T 2 = 1, We note here that the second K-exponent is zero. Hence the arbitrary constants at the j = 0 level of expansion (cf. [11][12][13]20] for this terminology) are the coefficients given by the dominant balance (34), that is (c 01 , c 02 ) = θ, θ 2α+1 . Therefore, the asymptotic solution is general since two arbitrary constants appear in the asymptotic solution as described below for 0 < α ≤ 1. Since we are interested in expanding universes (H > 0), it follows that the arbitrary constant θ attains only positive values. By integrating the solution above one obtains asymptotically the general solution for the scale factor described by the expression where C = (2α + 1)/(2α + 2). A comment about the asymptotic behavior of the scale factor is in order. The specific form of Eq. (39) describes an exponential evolution of the universe, with slower rate of expansion than the de Sitter universe, valid for a time interval. Clearly, as interaction takes place the transfer of energy from dark matter to dark energy (described by Eqs. (9)-(10)) results in an important growth of the energy density of the vacuum. However the presence of dark matter decelerates the rate of expansion.
As shown in the asymptotic solution (39) the α parameter determines the asymptotic states of the universe. For 0 < α ≤ 1 the universe enters (for a time interval) a quasi de Sitter space where the total energy density, total pressure, scale factor and Hubble parameter are asymptotically equal to respectively as t → 0, while higher derivatives of H diverge. This is a new type of singularity, a combination of Type IV [24] and Type II (sudden) singularity placed at late times. For α → 0 (limit of no interaction) it is expected that the universe asymptotically (as t → 0) will approach the CDM model. This is indeed true since for α → 0 the asymptotic analysis is identical to the one performed for the first decomposition in section II A. Consequently, the particular decomposition successfully reproduces the CDM model (as t → 0 and α → 0).
In addition, it is also expected at late times that the dominance of dark energy will drive the evolution of the universe towards de Sitter space. This is feasible in two ways; in the limit as α → ∞ and at a specific time t f of the cosmic evolution. For the first case we deduce from Eq. (39) the following asymptotic forms for the total energy density, total pressure, scale factor and the Hubble parameter as t → 0, α → ∞. The forms above describe a dark energy dominated universe with a sudden type singularity placed at late times. This particular value of the parameter α describes the case where energy is transferred from dark matter to dark energy without bound. However, this rapid rate of energy exchange is unphysical and questions the stability of the asymptotic solutions (38) as well as the asymptotic system (33).
For the second case, an interesting result arises when we compare the asymptotic solution given by Eq. (39) with the known solution of the scale factor describing a de-Sitter Universe, that is We assume here for purposes of notation that the constant θ is positive and plays the role of the cosmological constant. In particular, it can be proved that our solution (39) describes a de-Sitter Universe at a finite time t f = 0 (which depends on the parameter α), before approaching the singularity (40) at late times. Moreover, if we assume a cut-off in the energy exchange at t f then GCG model becomes indistinguishable from de-Sitter Universe at late times.

C. Interacting Vacuum
We now focus on the asymptotic analysis of all-termsdominant case, that is the decomposition (10), or equivalently described by the asymptotic systeṁ The subdominant vector field is the zero field in this case and there is one distinct balance given by The Kovalevskaya matrix is given by with corresponding eigenvalues Even though the parameter α is present in the second K-exponent, the form of the dominant the balance (44) indicates that on approach to the finite-time singularity the dominant part of asymptotic solution x ∼ (2/3)t −1 is independent from the choice of the parameter α.
For the whole series expansion though, the choice of the parameter α will determine the level of expansion at which the second arbitrary constant is expected to appear. For purposes of illustration we choose α = 1/2 so that spec(K III ) = (−1, 3). Then the associated eigenvector reads v T 2 = (1, −1).
The candidate asymptotic solution is expected to be general if two arbitrary constants (the position of the singularity and one constant at the j = 3 level of expansion) appear in the series expansion. After substituting the forms (25) into the asymptotic system (43), and for α = 1/2, we find the following asymptotic solution The compatibility condition at the j = 3 level reads and it is indeed satisfied after recursive calculations. Hence the asymptotic solution found above is general. In particular, the dominant behavior of the solution (48) on approach to the finite-time singularity is identical to the one of the decomposition f I (x, y). We conclude here that at early times, even when interaction takes place (α = 0), the model has the same asymptotic features as in the case where the interaction is switched off asymptotically and the universe is matter dominated. Hence, at early times the contribution of dark energy is negligible.

III. DISCUSSION
In this paper we analysed the stability of the singular flat space solutions that arise in the content of a unified dark energy model (GCG model) on approach to the finite-time singularity. We have shown that spacetime evolves from a phase that is initially dominated by dark matter to a phase that is asymptotically de-Sitter under some restrictions. The transition period in our model, between dark matter and dark energy domination corresponds to a quasi-inflationary regime that posses a new type of singularity asymptotically.
We conclude that the current observational data are supportive towards an asymptotically vanishing interaction in a model where dark energy is simulated by a generalized Chaplygin gas cosmology. In particular, it is shown that for such unified model the interaction is asymptotically vanishing at early times and the contribution of dark energy (as cosmological constant) is negligible. Hence, the model is indistinguishable from CDM universe. Such a model attains a pole-like [13] type of singularity and it is proved in previous works [11,12] that such a dominant behavior is an attractor of all possible asymptotic solutions on approach to the finite-time singularity.
An interesting era of expansion arises at late times when the vector field decomposition admits a quasi de-Sitter solution on approach to the finite-time singularity for 0 < α ≤ 1. In particular, the decomposition reproduces the successful CDM model at early times (in the limit as α → 0) and approaches de-Sitter Universe at some finite time away from the finite-time singularity. This intermediate phase of evolution is in alignment with the predictions of the GCG model for both early and late times. The validity of the solution though is questioned and is a part of future work [25].
To conclude with, it would be interesting to apply the central projection technique of Poincarè to the dominant part of each of the asymptotic solutions on approach to the finite-time singularity to discuss the asymptotic stability of the model at infinity. This is examined in [25].

A. Acknowledgments
We thank Prof. David Wands and Prof. Elias C. Vagenas for discussions and useful comments.