Imperfect Dark Energy from Kinetic Gravity Braiding

# Imperfect Dark Energy from Kinetic Gravity Braiding

Cédric Deffayet
AstroParticule & Cosmologie, UMR7164-CNRS, Université Denis Diderot-Paris 7, CEA, Observatoire de Paris, 10 rue Alice Domon et Léonie Duquet,
F-75205 Paris Cedex 13, France
Oriol Pujolàs
CERN, Theory Division, CH-1211 Geneva 23, Switzerland
Ignacy Sawicki and Alexander Vikman
Center for Cosmology and Particle Physics, New York University,
New York, NY 10003, USA

###### Abstract:

We introduce a large class of scalar-tensor models with interactions containing the second derivatives of the scalar field but not leading to additional degrees of freedom. These models exhibit peculiar features, such as an essential mixing of scalar and tensor kinetic terms, which we have named kinetic braiding. This braiding causes the scalar stress tensor to deviate from the perfect-fluid form. Cosmology in these models possesses a rich phenomenology, even in the limit where the scalar is an exact Goldstone boson. Generically, there are attractor solutions where the scalar monitors the behaviour of external matter. Because of the kinetic braiding, the position of the attractor depends both on the form of the Lagrangian and on the external energy density. The late-time asymptotic of these cosmologies is a de Sitter state. The scalar can exhibit phantom behaviour and is able to cross the phantom divide with neither ghosts nor gradient instabilities. These features provide a new class of models for Dark Energy. As an example, we study in detail a simple one-parameter model. The possible observational signatures of this model include a sizeable Early Dark Energy and a specific equation of state evolving into the final de-Sitter state from a healthy phantom regime.

dark energy theory; modified gravity; scalar-tensor models; galileon
preprint: CERN-PH-TH/2010-166

## 1 Introduction, Summary and Future Directions

Our purpose in this article is to abstract these developments in modified gravity and to consider the addition of second-derivative couplings as a generalisation of scalar-tensor theories. Specifically, we explore a rather large family of theories where the Lagrangian for the scalar field takes the form

 L=K(ϕ,X)+G(ϕ,X) □ϕ , (1)

with and some generic functions, with proposed already as k-essence [24, 25, 26]. We will show that for any form of the functions the equations of motion are of second order.222Generic choices of give equations of motion with explicit dependence on the first derivative . Hence, the Lagrangians (1) do not have the Galilean symmetry which characterizes the interactions of [7] and [8].

Hence, this Lagrangian can encode perfectly valid interactions for a single scalar degree of freedom. As we will see, a theory like (1) has rather unfamiliar properties which we will elucidate. We will concentrate on applying such constructions to model Dark Energy.

The Lagrangian (1) can be viewed as reminiscent of a partial resummation of a gradient expansion commonly used in fluid dynamics. As is well known, the stress tensor for k-essence (the case) takes the perfect-fluid form. As we will elaborate on in a future work [27], the interactions give rise to deviations of a certain type from the perfect-fluid picture. Hence, we will refer to the scalar field described by (1) as an imperfect scalar.

Let us also remark on another interesting property of the models of type (1). Once gravity is introduced, the interaction term contains a scalar-tensor kinetic coupling, schematically , due to the Christoffel symbols present in the covariant D’Alembertian operator. A consequence of these couplings is a novel kind of mixing which we have named kinetic braiding. It implies that the scalar equation of motion contains the second derivative of the metric—and vice-versa—in an essential way: there exists no Einstein frame where the kinetic terms are diagonalised333An ordinary scalar with nonminimal coupling to curvature, e.g. , also leads to kinetic mixing. However, in this case the mixing is non-essential in the sense that it can be removed by a field redefinition (in the form of a conformal transformation). Note that kinetic braiding is present in the higher-order terms of the covariant Galileon [9, 20].. This kinetic braiding manifests itself either at nonlinear level or on non-trivial backgrounds. In this sense, this leads to a novel modification of gravity that perhaps has not been fully appreciated. Another consequence is the fact that this operator may be non-zero and actually important on backgrounds where the second derivatives vanish, e.g. the sound speed can be non-vanishing in an exact de-Sitter spacetime. We will give some examples of these in the text.

An obvious domain in which models of type (1) can have interesting applications is cosmology. As a consequence of kinetic braiding, the Friedmann equation has remarkable properties, such as the presence of a term linear in , similarly to the DGP model. Moreover, even in the absence of any direct coupling between the and external matter in the action, the evolution of the scalar generically monitors any form of external energy density present. In the shift-symmetric case, where is a Goldstone boson, there exist attractor solutions on which the shift charge exactly vanishes such that the energy density stored in the scalar is a certain model-dependent function of the external energy density, : . Furthermore, for appropriately chosen (whenever ), the attractor tends to de Sitter space in the asymptotic future. This is similar to ghost condensation [3], the difference being that on the attractor the ghost condensate behaves like a cosmological constant. On the other hand, the imperfect scalar has a nontrivial equation of state dependent on the external matter content.

The existence of this monitoring behaviour opens up a number of applications, the most obvious of which is that can play the role of dark energy. Under very general circumstances, the on its attractor behaves as a phantom, . We will show that when evolution is started away from the attractor, the imperfect scalar initially redshifts in the usual manner. Once the attractor is approached, crosses the phantom divide and ends up dominating and accelerating the universe at late times. All of this can be accomplished without generating ghosts or gradient instabilities.

The existence of a physically plausible theory that can cross the phantom divide, i.e. violate the Null Energy Condition, opens a Pandora’s box full of exotic phenomena such as wormholes, bouncing universes, etc. However, currently, is not observationally excluded, see e.g. [28].

We analyse in detail a simple one-parameter example that exhibits the features above, where the imperfect scalar on the attractor evolves from a subdominant phantom (with ) during matter domination to a dominating late-time de Sitter phase. In addition, the energy density of this fluid while off-attractor, has a local maximum at matter-radiation equality, allowing us to propose this model as a simple example of a form of early dark energy.

This article is organized as follows: We introduce the class of models (1) in section §2, discussing their equations of motion and coupling to gravity along with the effective metric for the propagation of perturbations. We specialise our results to the cosmological background in section §3, and discuss the stability conditions for background solutions: positivity of the sound speed and of the kinetic term of perturbations. In section §4, we specialise to the cosmology of the shift-symmetric case. We discuss the dynamical attractors in the presence of matter and in the late universe—once matter has diluted away. In section §5, we discuss a simple one-parameter model to demonstrate explicitly the sort of phenomenology one can obtain within the new framework proposed. We invite those readers more interested in dark-energy phenomenology to start there. Finally, in Appedix A, we present a derivation of the quadratic action for cosmological perturbations for theories described by (1).

This work has a certain overlap with Refs [29, 30, 31, 32, 33] where the cosmology of the decoupling limit of DGP and some of its simple modifications are considered. These references concentrate on models with and non-minimally coupled to the Ricci scalar, à la Brans-Dicke gravity. In this work, we extend the above class of theories to a general function while restricting to a minimal coupling to gravity. As such we make it clear that the new interesting phenomenology is a result of kinetic braiding. In [34, 35] the authors take the effective field theory approach to discuss the ability of generalisation of the decoupling limit of DGP to stably violate the Null Energy Condition and propose a new mechanism to replace inflation as the initial stage of the universe. In addition, in Refs [36, 37], the cosmology of higher-order terms of the general covariant galileon is analysed.

We believe that the models described by Eq. (1) provide many avenues for exploration. In particular, the reinterpretation of these models as imperfect fluids yields many interesting insights, as we will show in our upcoming work, Ref. [27]. It is well known that the classical results of single-field inflation can be obtained entirely in terms of perfect-fluid variables [38]. The implementation of inflation using imperfect scalars provides another rich arena for model-building, with potentially non-trivial effects on cosmological perturbations. We will return to a thorough investigation of cosmological perturbations in the future.

## 2 General Properties of Kinetic Gravity Braiding Theories

In this paper we consider a class of scalar field theories minimally coupled to gravity, which is described by the action 444We use the metric signature convention .

 S=SEH+Sϕ=∫d4x√−g[−R2+K+G×B], (2)

where we denote

 X≡12gμν∇μϕ∇νϕ, and B≡gμν∇μ∇νϕ≡□ϕ,

and and are arbitrary functions of the scalar field and its standard kinetic term . Here and in most of the paper, we use reduced Planck units where . In further discussion we will also use the corresponding Lagrangian for the scalar

 L(ϕ,X,B)=K(ϕ,X)+G(ϕ,X)×B,

which we will consider as a function of three variables: , and ; whenever we differentiate with respect to or , we are keeping constant. Our class of models extends those studied up to now by generalising the interaction term of DGP decoupling limit and the galileon models to a function , significantly enriching the phenomenology.

One can naively motivate our interest in non-canonical field theories containing the new term by a similar argument to that for k-essence. In k-essence, we treat the canonical kinetic term as a first term in an effective-field-theory expansion, since no symmetries exist that would prevent additional contributions, i.e.

 K(ϕ,X)∼X(1+c1(ϕ)X+c2(ϕ)X2+...).

Then after resummation of these corrections one obtains a non-canonical Lagrangian with a derivatively self-coupled scalar field, e.g. as in the case of the Dirac-Born-Infeld theories. However, one could equally well start from the canonical kinetic term and integrate it by parts with no effect on the observed dynamics. In that case, it would not be unreasonable to expect there to exist higher-order corrections similar to those discussed above,

 G(ϕ,X)□ϕ∼−ϕ□ϕ(1+~c1(ϕ)X+~c2(ϕ)X2+...).

Note that both these expansions are in . Below we will show that it is not possible to recast such corrections to as a modification of the function and surface terms in the action, and therefore these theories are outside of the k-essence scheme. We stress that, in the Einstein-Hilbert action, the second derivative enters the Lagrangian only linearly and, despite the presence of these higher derivatives in the Lagrangian, the equations of motion remain of the second order. Therefore, similarly to the Einstein’s general relativity, there are no new (ghosty) degrees of freedom which would appear as the result of the Ostrogradsky theorem555There is another important similarity with General Relativity: a rigorous formulation of the variational principle requires a boundary term similar to the Gibbons-Hawking-York term in General Relativity. In this work we do not consider this boundary term assuming that it exists. The term appropriate for the decoupling limit of DGP was found in [39].. Thus, at least for some functions and and some general backgrounds, there are no ghosts and pathologies associated with them—the theories containing are on the same footing as k-essence or DBI theories.

The form of the action in Eq.(2) can be further generalised by introducing a non-minimal coupling of the scalar field to gravity, or a modification of the type. An appropriate change of variables will transform the action to the Einstein frame, where the usual Einstein-Hilbert term is restored. In the case of a theory containing only gravity and the scalar, this is equivalent to choosing different functions and . However, if external matter is present, this transformation introduces an additional coupling between the matter and the scalar field666Following the same logic one could also consider additional derivative couplings of the scalar field to the Einstein tensor. In that case, equations of motion would still be of the second order, see e.g. recent works [40, 41, 42, 43, 44], contrary to [45, 46].. Such theories can actually evade Solar-System tests of general relativity because of the presence of the Vainshtein mechanism [14, 47]. However, in this work we will concentrate on the minimally coupled models. This minimal coupling to gravity is the crucial difference of our work from the recent studies of the Galileon scalar-tensor models [29, 30, 31, 32, 33, 48]. In particular, we have to stress that the most unexpected and interesting features of these theories are a result not only of the possible non-minimal coupling of the scalar to gravity but of the presence of in the action.

In our discussion we will frequently use a Lagrangian equivalent to : integrating by parts the scalar-field contribution to the action Eq. (2), we obtain

 P=K−[(∇λϕ)∇λ]G=K−2XGϕ−GX∇λϕ∇λX, (3)

where the subscripts and denote partial differentiation with respect to these independent variables. It is clear that it is the dependence of on the field’s gradient, , that prevents this theory from being recast as a k-essence model, whereas the derivative of with respect to can be recast as an additional k-essence term. We will therefore only consider such non-trivial models where . The above form of the Lagrangian also implies that no quantities will ever be dependent on the function itself, but only on its derivatives.

### 2.1 Equation of Motion for the Scalar Field

The equation of motion for the scalar field can be obtained in the usual way by varying the action

 1√−gδSϕδϕ=Pϕ−∇μ((LX−Gϕ)∇μϕ−∇μG),

where . In particular, this presentation of the result makes explicit the existence of a Noether current for Lagrangians invariant under constant shifts of the field, ,

 Jμ=(LX−2Gϕ)∇μϕ−GX∇μX. (4)

In terms of this current, the equation of motion takes the elegant form

 ∇μJμ=Pϕ. (5)

The fully expanded equation of motion is somewhat unwieldy, but its structure is important for future discussion:

 (6)

where is the Ricci tensor and , and are constructed out of the metric , field and its first derivatives only. The term which is zero order in the second derivatives is the field derivative of

 E=2X(KX−Gϕ)−K, (7)

the terms linear in the second derivatives are contracted with

 Lμν=(KX−2Gϕ+2XGXϕ)gμν+(KXX−2GϕX)∇μϕ∇νϕ,

and the terms quadratic in the second derivatives are contracted with

 Qαβμν=12(gαβHμν+Hαβgμν)−14(gμβHνα+gνβHμα+gμαHνβ+gναHμβ), (8)

where

 Hμν=GXgμν+GXX∇μϕ∇νϕ. (9)

Most importantly, the equation of motion contains at most second-order derivatives, and therefore the theories with kinetic braiding do not contain any hidden degrees of freedom. Moreover, the structure of the tensor implies that appears only linearly in the equation of motion and therefore the equation of motion is solvable with respect to , for any time coordinate777Lorentz symmetry implies that this statement is true for the second derivatives with respect to any coordinate . This is sufficient to ensure that this equation is of the normal type required by the Cauchy-Kowalewski theorem and that the Cauchy problem has a unique local solution, at least for analytic functions.

The presence of a third-order derivative might have been expected owing to the d’Alamber-tian term in the Lagrangian; however, these terms arise in the combination and can be commuted away, leaving behind a term coupling the scalar to the Ricci tensor, . This is the key feature of theories with kinetic braiding. Namely a seemingly minimally coupled theory has an equation of motion depending explicitly on the curvature so that the scalar equation of motion includes the second derivatives of both the scalar field and the metric.

It is interesting to note that Eq. (6) can be considered as a generalisation of the Ampère-Monge equation for four-dimensional manifolds with Lorentzian signature.

### 2.2 Energy-Momentum Tensor

The energy-momentum tensor (“EMT”) for the scalar field is most easily derived in the standard way using the Lagrangian presented in Eq.(3)

 Tμν≡2√−gδSϕδgμν=LX∇μϕ∇νϕ−gμνP−∇μG∇νϕ−∇νG∇μϕ. (10)

It is key to observe that the EMT also contains second derivatives of appearing in , and . Therefore both the Einstein’s equations and the equation of motion for the scalar field, Eq. (6), contain second derivatives of both the metric and the scalar field , so that the system is not diagonal in second derivatives. This kinetic mixing is essential and cannot be undone through a conformal transformation. We shall refer to such a mixing as kinetic braiding and show that it leads to a number of surprising properties for this system.

In particular, for general configurations with timelike field derivatives the structure of the EMT is not that of a perfect fluid, contrary to minimally coupled k-essence theories, including canonical scalar fields. Moreover, for some of such configurations, the EMT does not even have a timelike eigenvector. The imperfect nature of the fluid is directly related to kinetic braiding. In the paper [27], we discuss the corresponding imperfect-fluid picture in detail. For cosmological solutions, which are exactly homogeneous and isotropic, the EMT is forced to be of perfect-fluid type. However, cosmological solutions including small perturbations possess an EMT with small terms deviating from the perfect fluid. This changes the standard picture of cosmological perturbations where the intuition was gained from the perfect fluid case.

Using the Einstein equations including the EMT contribution of fields different from ,

 Rμν−12gμνR=Tμν+Textμν, (11)

we can eliminate the second derivatives of the metric in the scalar field equation of motion Eq.(6). In this way we obtain

 ~Lμν∇μ∇νϕ+(∇α∇βϕ)Qαβμν(∇μ∇νϕ)+Z− (12) −GX(Tμνext−12gμνT%ext)∇μϕ∇νϕ=0,

where the new term which does not contain second derivatives is

 Z=Eϕ−2XGX(X(KX−4Gϕ)+K),

while the terms linear in the second derivatives are now contracted with

 ~Lμν=Lμν−2XG2X(Xgμν−2∇μϕ∇νϕ). (13)

The tensor remains the same. Note that the new contributions modifying and are suppressed by . In Eq. (12) we have obtained a form of equations of motion which contains the second derivatives only of the scalar field but not of the metric888Here we assume that, as usual, other matter fields do not contain second derivatives in their EMT.. This is important for the investigation of causality and stability in this class of models, see section §2.3.

Let us finally emphasise the fact that the scalar inevitably becomes coupled to external matter in the non-linear fashion resulting from the last term of Eq. (12). This occurs in a quite peculiar manner: unless an additional explicit direct scalar/matter interaction is introduced, this coupling is one way and external matter does not become coupled to the scalar, only feeling it through its gravitational effects. This coupling will play a central role in the following sections, is a consequence of the kinetic braiding. A proper analysis of the constraints on the model arising as a result of this coupling will be deferred to future investigation, but for some headway in this direction see Ref. [33]. Let us only remark here that the self-interactions of the scalar can give rise to a Vainshtein mechanism that helps to evade the observational constraints.

### 2.3 Effective Metric

Let us perturb the system comprising the scalar equation of motion and Einstein equations around a given solution and . Without loss of generality, we will neglect the perturbations in the external fields, so that . Now let us find the characteristic surfaces (wave-fronts) for the linearized system. This is crucial for the investigation of causality and stability with respect to the high-frequency perturbations, see e.g. [49].

The linearized equation of motion for the scalar field Eq. (12) does not contain second derivatives of the metric. Therefore the matrix of the principal part of the differential operator for this system is lower triangular in coordinates. This implies that the characteristic equation for the system (see e.g. [49, p. 580]) becomes just a product of a purely gravitational and a purely scalar part and the second derivatives of the scalar field still present in the linearised Einstein equations do not play any role for characteristics. The causal structure for the propagation of the scalar perturbations can therefore be directly read from just equation (12). Note that the same procedure for the original equation of motion (6) would give an incorrect result.

From the standard eikonal ansatz with a slowly varying amplitude , taking the formal limit , we obtain

 Gμν∂μS∂νS=0, (14)

where the effective contravariant metric for propagation of perturbations is

 Gμν≡~Lμν+2Qαβμν∇α∇βϕ. (15)

Using formulae (8) and (13), we can expand this expression as

 Gμν=Ωgμν+Θ∇μϕ∇νϕ−∇μ(GX∇νϕ)−∇ν(GX∇μϕ), (16)

where we have introduced the notation

 Θ≡LXX+4XG2X, (17)

and

 Ω≡LX−2Gϕ+∇λ(GX∇λϕ)−2X2G2X. (18)

The velocity of the wavefront and the corresponding causal structure of the acoustic spacetime should be inferred from the inverse or covariant metric for which , for details see e.g. Appendix A of Ref. [50]. The effective metric for the decoupling limit of the DGP model (where ) was first obtained in Ref. [51], see also e.g. [34]. The result (16) is more general, because it is derived for a generic function .

The main obstacle for the analysis of this metric for general backgrounds is that contains not only second derivatives in and but also an additional tensorial structure which is a Lie derivative of the metric with respect to . This structure prevents from being an eigenvector of the metric for a general background as was the case for k-essence.

In this paper, we will restrict our analysis to the cosmological case. In particular, in section §3.2, we calculate the sound speed and present the condition for the absence of ghosts using the metric (16). The same results are confirmed by a direct calculation of the action for cosmological perturbations, presented in Appendix  A. The advantage of this effective-metric formalism is that it allows for the study of causality and stability for high-frequency perturbations for any given background without calculating the action for perturbations, which may be a very complicated task for general, less symmetric backgrounds.

## 3 Cosmology

### 3.1 Background Evolution

In this section, we will discuss cosmology in braided models (1). We will first consider the homogeneous and isotropic background solutions for a generic model and, in Section 4, we will specialize to the shift-symmetric case, where the integration of the equations of motion is almost straightforward.

For simplicity, we will restrict our discussion to the spatially flat Friedman-Robertson-Walker (“FRW”) metric

 ds2=dt2−a2(t)dx2. (19)

On this background, we have , where the dot represents the time derivative. In the current Eq. (4) only the “charge density”

 J≡J0=(KX−2Gϕ+3H˙ϕGX)˙ϕ, (20)

survives, so that the equation of motion (5) takes the form

 ˙J+3HJ=Pϕ, (21)

where, as usual, denotes the Hubble parameter. Note that this equation of motion not only contains but also . As a result of the symmetry of the cosmological setup, the energy-momentum tensor takes the perfect-fluid form and the only non-vanishing stress-tensor components, Eqs (10), reduce to the energy density

 ϵ≡T0i0=˙ϕJ+2XGϕ−K, (22)

and the pressure

 P≡−13Tiii=K−2XGϕ−2XGX¨ϕ, (23)

which turns out to be given by the Lagrangian (3). The two unusual properties arising as a result of kinetic braiding are that the pressure contains while the energy density depends on, , the Hubble parameter999Note that this situation is different from the so-called Inhomogeneous Equation of State where pressure is postulated to be a function of both the energy density and [52]..

It is convenient to introduce the variable “measuring” the strength of the kinetic braiding

 κ=XGX; (24)

its physical meaning will be further elucidated in the paper [27].

The part of energy density (22) which is not braided and which does not depend on the Hubble parameter looks similar to k-essence:

 E=2X(KX−Gϕ)−K, (25)

and its field derivative appears already in the equation of motion (6), so that

 ϵ(ϕ,˙ϕ,H)=E+6κ˙ϕH. (26)

The Friedmann equation becomes not the usual complete square but a general quadratic expression for ,

 H2=2κ˙ϕH+13(E+ρext), (27)

where we have introduced the energy density from any additional sector external to . This could be dust, radiation, a bare cosmological constant term or any other standard cosmological fluid.

It is useful to compare this equation with the Friedmann equation arising in the DGP scenario [53, 54]. In particular, we see that plays the role of the crossover scale beyond which gravity starts probing the extra dimension in DGP, while corresponds to the energy density localized on the brane.

We can solve for in Eq. (27) to obtain

 H=κ˙ϕ+σ√(κ˙ϕ)2+13(E+ρext), (28)

with denoting a choice of branch. We will see in section §4.3 that only one choice of (which depends on the forms of , ) is compatible with stable fluctuations around the solution.

The presence of the term linear in in (27) is responsible for a mismatch between the two branches of solutions (labelled by ) and expansion or contraction of the cosmologies. Only a simultaneous transformation gives the time-reserved solution.

For future reference, it is useful to also write down the acceleration equation,

 ˙H=−12(ϵ+ρext+P+pext)=κ¨ϕ−12J˙ϕ−12(ρext+pext), (29)

where is the pressure of the external matter; we will also sometimes use its equation-of-state parameter, .

Finally we present the expanded form of the equation of motion (12) in a cosmological context

 D¨ϕ+3J(H−κ˙ϕ)+ϵϕ=3κ(ρext+pext), (30)

where

 D=EX+6˙ϕHκX+6κ2. (31)

In the absence of kinetic braiding, , this reduces to the k-essence equation of motion, c.f. Ref. [55, Eq. (II.13)]. One notices that the external matter content explicitly filtered down into Eq. (30) despite the absence of any direct coupling between and matter in the Lagrangian. This is traced back to the explicit dependence of on and is a consequence of the braiding between the scalar and gravity in these models. The normalization factor in Eq. (31) will appear repeatedly below and it plays a prominent role since it determines whether the fluctuations of around the background are ghosty () or not (), see section §3.2.

Notice the presence of a term quadratic in . Once the factors are restored, we see that this term is suppressed by one more power of , reflecting that it is a type of backreaction effect implied by the scalar-gravity braiding. This term arises from the dependence of the equation of motion for , Eq. (6), on and the simultaneous dependence of on as implied by the Friedmann equation.

In the absence of external matter, close to the point where , will be small provided is a slowly-varying function of , through Eq. (30). Then Eq. (29) implies that we are dealing with a generalised slow-roll, quasi-de-Sitter state.

### 3.2 Stability

Now let us apply the machinery of the effective metric to cosmological solutions. Plugging in the cosmological background into the general expression for the metric (16) we obtain for the contravariant metric

 Gμν=δμ0δν0D−δμiδνia−2(Ω−4κ˙ϕH). (32)

From this expression, it follows that ghosts are absent if . Indeed, it is which is contracting the derivatives in the kinetic term for perturbations101010In the high wave-number limit, the gauge-invariant canonical variable is proportional to the gauge invariant . Also note that the full calculation of the quadratic action gives the expression for the normalisation of the cosmological perturbations, Eq. (97). This is modified compared to the standard case and therefore is likely impact predictions of inflation., see (96). Here it is important that the sign in front of be chosen correctly—so that this condition continuously transforms to that of k-essence.

This contravariant metric is diagonal. Therefore it is easy to find its inverse (or covariant) metric along with the corresponding acoustic line element

 dS2=G−1μνdxμdxν=(Ω−4κ˙ϕH)−1(c2sdt2−a2(t)dx2), (33)

where the sound speed is given by

 c2s=Ω˙ϕ−4κH˙ϕD. (34)

Thus for the stability with respect to high-frequency perturbations we should require that111111Note that the tensor modes are not affected in our model, as can be seen in the full calculation of the quadratic action Eq. (96), and therefore do not provide any new conditions for stability.

 D>0 and Ω−4κ˙ϕH>0. (35)

Only those models for which during the whole evolution history these two conditions are satisfied can be considered as plausible from a physical point of view. Further we calculate for the cosmological background and obtain

 Ω−4κ˙ϕH=J+2(˙κ+Hκ−κ2˙ϕ)˙ϕ, (36)

while the definition (31) can be rewritten as where the partial derivative of the energy density (which is considered as a function of three independent variables, see Eq. (26)) is taken keeping , as if the Hubble parameter were externally fixed.

Using these expressions, the formula for the sound speed (34) can be written as

 (37)

This expression depends on which can be eliminated using the equation of motion (30). It is important to note that

 c2s=P˙ϕ+4˙κ+2κ(4H−κ˙ϕ)ϵ˙ϕ−6κ(H−κ˙ϕ)≠˙P˙ϵ, (38)

even in the shift-symmetric case, contrary to k-essence [38]. Here the derivative of the total pressure is taken keeping . The sound speed in these models is arbitrary and can even be larger than the speed of light. In particular, this is exactly the case in the model discussed in the section §5, see e.g. Fig. 4. In this respect the situation is not very different from k-essence121212However, contrary to the case of k-essence and the conclusions of Ref. [56], in the model presented in section §5 there is a period during which the null-energy condition is violated on an isotropic background while the speed of sound is sublumninal, see Figs 4 or 5.. From the expression for the acoustic interval Eq. (33), it follows that there are no causal pathologies (Closed Causal Curves) even in the presence of superluminal propagation. We follow the discussion of this issue presented in Ref. [50] (see also the most recent paper [57] and older works [58, 59, 60, 61]) and claim that this superluminality does not imply any inconsistencies. A different issue is a possible UV completion of the theories with superluminal propagation. Currently, there are good arguments [62] that this UV completion cannot be realized within a Lorentz-invariant and renormalizable local QFT or perturbative string theory framework. However, on the level of an effective-field-theory description there are no inconsistencies. This is also accepted by some of the authors of [62], see [34, 56]. This possibility to realise superluminal propagation of perturbations on nontrivial backgrounds can lead to interesting cosmological scenarios in the context of inflation [63] and some exotic alternatives, e.g. [34, 64, 65, 66]. Moreover, this superluminality raises important questions regarding black hole physics [67, 68, 69, 70].

## 4 Cosmology of Shift-Symmetric Gravity-Braided Models

In this section, we will restrict our attention to models that realise an exact shift symmetry

 ϕ→ϕ+const,

in terms of an appropriate choice of the scalar-field variable . In practice, this implies that must be independent, that is the scalar Lagrangian takes the form131313Shift-invariance allows for somewhat more freedom in choosing Lagrangians. Obviously, a term in linear in is compatible with this symmetry. However, it can be absorbed in the form of by integration by parts. Similarly, a change of variable will trivially introduce -dependence but will still realize the same symmetry, even though not as a shift in .

 L=K(X)+G(X) □ϕ. (39)

Imposing such a symmetry, i.e. requiring that the field only be derivatively coupled, means that can be interpreted as a Goldstone boson of some broken symmetry. This naturally prevents such a field from acquiring a mass. As we will see, the dynamics of such kinetically braided Goldstone bosons make them a very compelling model for dark-energy dynamics. In addition, the scalar would have to be derivatively coupled to matter, if at all, which would make evading fifth-force constraints relatively easy141414Generic derivative couplings to matter can still give rise to sizeable effects, but typically the most problematic couplings can be suppressed by additional symmetries [3]. In fact, in this sense the situation in the present model quite similar to ghost condensation [3] and Hořava gravity [71, 72, 73].

In the shift-symmetric case, the shift-current (4) is conserved, and the scalar equation of motion is precisely this statement, . In a homogeneous and isotropic background, this reduces to simply

 ˙J+3HJ=0, (40)

that is, the shift charge in a comoving volume is constant. Furthermore, the equation of motion Eq. (40) can be trivially integrated,

 J=˙ϕ(KX+3˙ϕHGX)=consta3(t). (41)

Hence, the expansion of the universe drives to zero, and the locus of in configuration space represents a future attractor for expanding FRW space-times151515The appearance of de-Sitter attractors in shift-symmetric theories with higher derivatives was also used in the so-called the B-Inflation [74]. This behaviour was first noticed by Alexei Anisimov, one of the authors of the B-Inflation paper, who regrettably passed away after this preprint was submitted to the arXiv, RIP.. As we will see, even on the attractor one can have rather interesting behaviour, basically because the value of on the attractor depends on the amount of external matter density at every moment of time (because of the appearance of in ).

It is convenient and illuminating to split the energy density of the scalar into the contribution from the attractor and that of the departure from it,

 ϵ=ϵ∗+ϵJ , (42) withϵ∗≡ϵ|J=0 (43) andϵJ≡ϵ−ϵ∗ . (44)

Both ‘components’ behave like fluids with a particular equation of state on the background. By definition, the (or ‘off-attractor’) component dilutes away with expansion, so it behaves like a rather standard fluid. As we will see, the attractor component can be quite exotic, as it can easily exhibit phantom behaviour without generating instabilities. In fact, we will find that requiring that the energy density in the attractor be positive and that the field fluctuation be healthy (not a ghost) implies that generically the attractor displays phantom behaviour. Hence, the composition of the - and attractor components (that is, the situation with a generic initial condition) is such that the energy density in first dilutes away and then grows. That is, we will find that can cross the phantom divide without leading to instabilities.

At this point, we note that since the action contains second derivatives, therefore possibility of smoothly crossing of the barrier does not contradict the statement proved in [55] and rederived in different ways later in [75, 76, 77, 78, 79, 80, 50]. Therefore kinetic braiding provides a working example of the so-called Quintom scenario of Ref. [81], see also reviews [82, 83, 84]. Thus in this respect, kinetic braiding exhibits similar phenomenology to models with explicit nonminimal coupling to gravity, which also allow one to have a classically stable crossing of the phantom divide in scalar-tensor theories [85, 86, 87, 88]. Another theory comprising a single degree of freedom which is able to penetrate the phantom divide without classical instabilities is the so-called -fluid (“Angel Dust”) [89]. However, this is a very non-standard theory where issues related to the strong-coupling scale and quantisation remain to be addressed. In the case of kinetic braiding, the quantisation of perturbations is standard and the crossing occurs in a regime free of negative energy perturbations and which is completely under control. Models which exhibit healthy phantom behaviour within the Galileon framework were realised in Refs [34, 35]. Moreover, in Ref. [90] a healthy violation of the Null Energy Condition was achieved by including on the level of perturbations within the effective-field-theory framework.

A central issue in the significance of the split (42) is if and when the off-attractor component dilutes away sufficiently fast so that the attractor regime is actually reached. We discuss this among other points in §4.2. But first, in the next subsection, we shall analyse the generic properties of attractor solutions.

### 4.1 The Phantom Attractor

First of all, since the Friedmann equation fixes as a function of and the external matter energy density , one can view as a function of the two variables and ,

 J=J(X,ρext) ,

whose form depends only on the choice of . The attractor solutions are the roots of

 J(X∗,ρext)=0 (45)

We will furnish quantities evaluated on the attractor with an asterisk (‘*’) sub-/superscript161616In principle, one should introduce a label to distinguish among the several possible attractors each of which has its own non-overlapping basin of attraction. In addition, the form of also depends on the branch for the Friedman equation (), so the attractors in general also depend on . For the sake of clarity, we will spare the reader these labels with no risk of introducing ambiguities.. In particular, for any form of external matter, the attractor behaviour of the scalar is such that becomes a certain local function of ,

 X∗=X∗(ρext) , (46)

the form of which is solely determined by the dynamics, i.e. by the form of . Notice that this holds irrespective of the time dependence of (i.e. its equation of state), so implicitly Eq. (46) also represents the time evolution of on the attractor, .

From (22) it follows that on the attractor the energy density stored in the scalar, , is also only a function of ,

 ϵ∗=ϵ∗(ρext)≡−K(X∗(ρext)). (47)

Hence, the attractor has the remarkable property of responding to the external energy density in a way determined by the form of and .

We should emphasize that since is a function of only, this effectively allows one to modify the Friedman equation (viewed as the relation between and ) in almost any way in these models. and can be picked in such a way so as to replicate any evolution history, once the shift current dilutes and the attractor is reached. The attractor present in our kinetically braided model can therefore be considered as a concrete realisation of the phenomenological “Cardassian” scenario proposed in Ref. [91].

Of course, one must make sure that the perturbations are healthy, which will not be the case for a generic choice. It should be noted that these modifications are the result of the fact that scalar-gravity braiding modifies the Hamiltonian constraint in gravity.

The behaviour of the attractor can be easily understood in terms of the function . For example, if is a decreasing function (such as the one in the model of section §5), then as matter dilutes the attractor energy density grows, implying a phantom equation of state. In the presence of usual external matter (), will eventually dominate the expansion of the universe and, as long as , the final state will asymptote to de Sitter. We will see shortly that it is precisely this kind of setup, with phantom behaviour for the attractor, which is free of instabilities.

At this stage, the form of the function is inevitably implicit—below we will elaborate on how to extract it in general once and are given. However, for most purposes its explicit form is not really necessary.

For instance, the effective equation of state of the attractor can be written as

 1+w∗X=(1+wext)dlnϵ∗dlnρext=(1+wext)ρext6κ2∗K∗D∗ (48)

The last equality follows from the chain rule and reading off from the scalar equation of motion on the attractor,

 D∗¨ϕ∗=−κ∗˙ρextH. (49)

Since the normalisation factor , so that the fluctuation not be a ghost we find that (assuming ):

• the attractor is phantom () whenever its energy density is positive, and vice-versa;

• as the external energy density dilutes away, the attractor approaches de Sitter with from below—the phantom region, provided ;

• the equation of state of the attractor contribution is tied to the equation of state of the external matter and therefore will change, for example, at matter-radiation equality.

It is quite simple to imagine that it is the behaviour of the attractor that is relevant for the observations of dark energy. Therefore, provided that the off-attractor contribution has diluted away early enough during the history of the universe, it is a generic prediction of the models of Eq. (39) that dark energy will be seen to violate the NEC and approach de Sitter from below—the phantom region.

Equations (47) and (48) encode the most important feature of the behaviour of the attractor, namely that responds to the presence of any other form of matter, and that a stable quasi-de Sitter stage is reached only if the attractor is phantom. So far, we only used the positivity of the energy density () and . An additional constraint required for stability is discussed in §4.3.

Let us close this subsection with some gymnastics that can ease the task of finding the attractor, in particular the function , for an arbitrary and . The technical difficulty is that the explicit expression of found from Eqs (20) and (28) is rather involved and may not allow to solve easily for its roots. However, on the attractor, the equations simplify considerably—for example, the Friedman equation becomes

 3H2∗=−K∗+ρext. (50)

From the equation itself we see that we can also write . Thus, it is easy to see that solving is equivalent to solving171717Eq. (51) needs to be supplemented with (52) because the l.h.s. of Eq (51) contains less information than . Specifically, Eq (51) does not involve and hence is independent of the choice of branch of the Friedman equation, . Eq. (52) provides that information.

 K2X∗6X∗G2X∗+K∗=ρextand (51) sgn(K∗−X∗K∗XK∗X)=σsgn˙ϕ∗. (52)

Clearly, this form of the equations for the attractor is considerably simpler. In particular, Eq. (51) directly gives as a function of , which is just the inverse of the function in which we are interested, . Also, it is straightforward to obtain the relation between the attractor energy density and in parametric form,

 (ϵ∗(X),ρext(X))=(−K(X),KX(X)26XGX(X)2+K(X)),

in terms of the ‘parameter’ . This curve in the plane encodes how the attractor responds to the external matter for given (the branches corresponding to consecutive segments of the curve).

Finally, let us mention that the roots of

 K2X∗6X∗G2X∗+K∗=0 (53)

represent the pure-de-Sitter attractors (when ), corresponding to the asymptotic state of expanding solutions, once all matter and shift-current density, , dilute away. These are kinetic condensates similar to ghost condensation [3] in that the non-trivial condensate mimics a cosmological constant. However, let us emphasize that the present situation is qualitatively different from ghost condensation for two reasons: First, in ghost condensation (the case) the attractor can only behave like a pure cosmological constant. Instead, with it has a nontrivial (phantom) equation of state in the presence of external matter. Second, the fluctuations around the attractor behave very differently in ghost condensation and the model with kinetic braiding (39). Even on the pure de Sitter condensate the sound speed squared of the scalar fluctuation can be positive, . This is a consequence of the presence of the of the term and the resulting imperfection of the effective fluid. As we will see in §4.2, around the de Sitter condensate, the equation of state corresponding to a homogeneous perturbation (the off-attractor component due to ) vanishes. Hence, the off-attractor component behaves like a pressureless fluid. For a perfect fluid this would imply that , as indeed happens in ghost condensation (which is why one needs to resort to higher space-derivative terms in Ref. [3]). In our case, despite having for the off-attractor component, one finds (as long as ). This can also be seen from the EFT point of view by including operators like in the Lagrangian for perturbations [90].

### 4.2 Approach to the Attractor

The background dynamics away from the attractor are given by Eq. (41). The quantities that evolve with time in a simple way are and . In principle, then, to explicitly work out the time evolution of the system we need to express everything in terms of and . Unfortunately, this may not be possible even in concrete examples. Still, one can extract all the information from the functional form of . For simplicity let us concentrate on the -dependence and let us consider a generic form of such as depicted in Fig. 1.181818The inclusion of nontrivial matter in the discussion is straightforward, it only adds a direction to the plot of Fig. 1. The attractors so far discussed correspond to the cuts of with the axis.

It is easy to check that the normalization factor for the perturbations, , introduced in Eq. (31) in the shift symmetric case is nothing but

 D=(1−˙ϕκH)∂J(˙ϕ,ρext)∂˙ϕ∣∣ρext=const. (54)

Therefore, we can readily identify non-ghosty regions in the plot of as those where

 sgn(D)=σsgn(H)sgn(J˙ϕ)=+1, (55)

Therefore, given a fixed choice of branch, , the perturbations will alternate between ghosty and healthy whenever or the slope change sign. The roots of separate the basins of attraction of the different attractors, representing edges between them where . This implies the presence of a pressure singularity which the dynamics cannot penetrate and that the sound speed diverges ( for .

In the following, we shall concentrate on the evolution close enough to a attractor. Our approximation requires that two conditions be met: firstly, , where we are using the splitting of the fluid introduced in Eq. (42); secondly, we must be close enough to the attractor such that we are away from any extrema in , where the backwards-in-time evolution would meet a pressure singularity.

We will therefore study the equation of state of the fluid representing the excess over the attractor, with energy density . Assuming that one can invert to find , we can simply expand the total energy density for small to find, at lowest order in ,

 ϵJ≃Ξ˙ϕ∗J,whereΞ≡1−K∗XD∗(H−˙ϕ∗κ∗H).

Notice that the factor in front of is a function of , implying that does not dilute like dust as could have been naively expected and as is seen in the case of ghost condensates. Rather, the equation of state of the off-attractor component is found to be

 wJ=(1+wext)KX2D∗Ωext(1+2∂lnΞ∂lnX∣∣∣∗), (56)

where we have defined the contribution to total energy density of the external matter as . Hence, we see that the off-attractor contribution is also sensitive to the external matter, though differently from the attractor. In particular, we see that behaves like dust only when external matter is either absent or when it is just a cosmological constant.

It is interesting to note that, assuming the -derivative in the parentheses of Eq. (56) is not negative enough to change the sign, the energy density stored in the current will redshift away more slowly than the external matter. This is irrelevant for the purpose of the discussion of the cosmological dynamics, since our approximation assumed that the energy density stored in the imperfect scalar will be dominated by its attractor part. However, it is true that the minimum of the ratio will be reached around the time of the transition from the -dominated to attractor-dominated behaviour for the scalar fluid.

Finally, this discussion has only covered the case when is small (as compared to either or ). The opposite regime, when is dominated by the shift current will have significantly different behaviour. The nonlinearity of the equations make a general analysis somewhat difficult, so we defer an explicit analysis of this regime to the example of §5.

### 4.3 Stability on Attractor

Specialising to the shift-symmetric case allows us to concentrate on the attractor, where significant simplification occurs since . In particular, using Eq. (49), the speed of sound can be expressed as

 c2s∗=2κ∗(H−˙ϕ∗κ∗)˙ϕ∗D∗+6κ∗κX∗(ρext+pext)D2∗ (57)

The second term results from the coupling of the scalar to the other matter content and will dilute away as the scalar becomes the dominant source of energy density. When the scalar on the attractor dominates the energy density, we can express the sound speed positivity condition as

 c2s∗=KX∗(K∗−X∗KX∗)9H2D∗>0 (58)

which combined with Eq. (52) implies that only one of the branch of the Friedman equation, that for which , is stable. This equation also displays another difference between this theory and ghost condensation: even on the pure-de Sitter condensates the sound speed in the braided model is nonzero, and can be adjusted to be positive. The observation that an operator like can lead to in an expanding universe was also seen in [90] in the effective-field-theory language. It should be noted that the linear model presented in section §5 does have a vanishing sound speed on the de-Sitter attractor once all the external matter dilutes away. In general, this will not be the case, however.

## 5 Example: Imperfect Dark Energy

In this section, we will describe the properties of the cosmological solutions of the simplest model with kinetic braiding which exhibits the interesting behaviour described up to this point. We will choose functions and to be shift symmetric and linear in and choose the background Minkowski space () to be ghosty in order to make the de-Sitter attractor the final and stable point of the evolution. The simplest such choice is

 K =−X, (59) G =μX,

where is a coefficient with mass dimension .

Beneath we will in turn discuss the phase space of this theory and the regions of interest, the behaviour of the attractor solution, where we will find that while the sound speed squared remains positive. We will then detail how this attractor is approached and the behaviour of the energy density of the scalar during this time. Finally, we will discuss the regions of parameter space for this model which are not excluded by observations. We invite the reader to familiarise themselves with Fig. 2 and its caption: it provides an overview of the dynamics of the system which are described in detail in the rest of this section.

We have named this model Imperfect Dark Energy. The background EMT for cosmological solution has, of course, a perfect-fluid form. However, the EMT for the solution including small perturbations can no longer be so described; the fluid is imperfect, as we have shown in section §2.2. It is exactly this modification of the fluid’s properties which allows this model to circumvent certain no-go theorems and stably violate the Null-Energy Condition.

Other authors have also considered phenomenological models of dark energy which include deviations of the energy-momentum tensor away from the perfect-fluid form such as anisotropic stress [92, 93, 94].

### 5.1 Phase Space

The phase-space plot for for an FRW cosmology, extended to show and is presented in figure 3. We will argue in this section that only the region of to the right of the fixed point , where is positive, is healthy and relevant for the discussion of DE.

Figure 3 was obtained assuming the branch choice of

 H =μ˙ϕX+√2μ2X3+13(ρ% ext−X) (60) J =˙ϕ(3μH˙ϕ−1). (61)

Choosing the other branch is equivalent to mapping , and . Therefore, for every expanding cosmology on this phase plot, there exists an equivalent contracting one with the opposite velocity of .

The evolution of the background will proceed toward one of the two attractors which can be located by solving Eq. (61) for . We obtain for their positions

 3˙ϕ∗Hμ=1X∗ =(18μ2H2)−1, (62) and˙ϕ =0. (63)

On this attractor the Friedmann equation can be written in terms of the external energy density only

 H2=16(ρext+√ρ2ext+23μ−2). (64)

An observer who is not aware of the existence of and the attractor would be highly confused by this modification of the cosmological dynamics.

Since must approach zero in an expanding universe, the basins of attraction will be delimited by the extrema of . The system cannot penetrate these separatrix lines in the phase space since pressure is singular there: as already discussed in section §4.2, the normalisation of the kinetic term for perturbations, , is proportional to and vanishes on these extrema, causing both and the sound speed to diverge. Also, this implies that the basin of attraction around , since has a negative gradient there, has ghosty perturbations.

For all the values of (to the left of the blue line in Fig. 3), the energy density of the imperfect scalar is negative and we will not consider these phase-space regions further, since this cannot give the desired dark-energy phenomenology. The remainder of the negative- region does have positive energy density but the energy that can be stored in the scalar in the far past can only be very small and to all intents and purposes the dynamics would have been indistinguishable from the attractor for all of the observable history of the universe.

Hence, our discussion is going to concentrate on the region where both and the scalar energy density, , are positive, the perturbations are not ghostly and the background evolution in an expanding universe occurs toward from above. We will show that in this region there are no gradient instabilities and the scalar field’s evolution provides a viable and interesting model for dark energy.

### 5.2 Attractor Behaviour

In this section, we will assume that the initial value of was small enough such that the behaviour of the scalar at times relevant for our observations of the cosmology is effectively that of the attractor. As discussed in section §5.1, we will focus solely on the healthy attractor at .

On the attractor, since , the Friedmann equation simplifies. From Eq. (50),

 3H2=X∗+ρext, (65)

allowing us to define the cosmological quantities

 ΩX≡X∗3H2=(54μ2H4)−1Ωext≡ρext3H2, (66)

representing the contributions to the total energy density of the imperfect scalar and external matter, respectively.

The Friedmann equation Eq. (60) implies that when the Universe empties and , the requirement to keep the content of the square root positive bounds from below,

 X2∗>16μ2. (67)

this puts a lower limit on , i.e

 H2>13√6μ. (68)

The somewhat counterintuitive result is that a small parameter , i.e. if the operator is suppressed by a large mass scale, results in a universe with a large effective cosmological constant and therefore is excluded by observation. The mass scale suppressing the new term must in fact be small191919It is of course quite possible to have a very large mass scale suppressing provided that the final vacuum is the trival Minkowski one and dark energy is a result of a cosmological constant. The only way of evading this constraint is by introducing an external cosmological constant which will relax the bound, also from below. If there is no external cosmological constant, and the current acceleration of the universe is being driven by an attractor such as the one discussed then the energy scale suppressing the term is

 μ−1/3∼(H20MPl)1/3∼10−13eV. (69)

Despite being small this mass scale has the benefit of being technically natural: the radiative corrections to this term should be small since it is an irrelevant operator, as opposed to quintessence where the scalar mass suffers from quadratic divergences.

We can explicitly confirm that the perturbations on the attractor have a positive kinetic term. Evaluating Eq. (31) for the linear model at we obtain

 D∗=1+154μ2H4=1+ΩX. (70)

Now we turn to the study of the equation of state of the imperfect scalar. The assumption that moves on the attractor implies that

 ˙X∗=((6μ2H2)−1˙)=−2X∗˙HH. (71)

The presence of the coupling to the Ricci tensor in the scalar equation of motion Eq. (6) resulting from gravity braiding implies that the scalar is sensitive to the presence of external matter. This is manifested by the dependence of the position of the attractor on the evolution of the Hubble parameter. Differentiating Eq. (65) with respect to time leads to

 ˙HH2=−Ωext2(1+wext)−ΩX˙HH2. (72)

Solving for , we can write down the equation-of-state parameter for the scalar:

 1+w∗X=−(1+wext)1−ΩX1+ΩX (73)

which matches the result we could have obtained from Eq. (48). In particular, we find the following limits

 1+w∗X ≈−(1+wext) ΩX≪1 (74) 1+w∗X ≈−(1+wext)Ωext2≈0 Ωext≪1 (75)

As we discussed in section §4.1, while the imperfect scalar is on its attractor, it behaves as a phantom and its energy density grows with time. In particular, while subdominant, it anti-tracks the equation of state of the dominant energy source, by which we mean that its equation of state is related to that of the external matter by Eq. (74). As the dark energy begins to dominate the energy budget, it approaches the de-Sitter equation of state from below, where it remains as a pure de-Sitter condensate.

It is important to stress that while this behaviour is phantom, the perturbations to the fluid with kinetic braiding are not ghosty, by virtue of Eq. (70). In addition, the fluid is free of gradient instabilities: the speed of sound squared is positive. Using the results of section §4.3, we obtain for the sound speed on the attractor

 c2s∗ =1−ΩX3(1+ΩX)+(1+wext)1−ΩX(1+ΩX)2, (76)

with the following limits in the early and late universe

 c2s∗ =43+wext ΩX≪1, (77) c2s∗ =(53+wext)Ωext4 Ωext≪1. (78)

The speed of sound at the attractor is always positive and is determined by the external energy density.

The history of evolution of the dark energy is presented in figure 4 while the hydrodynamical properties of the attractor of the fluid of the imperfect scalar are summarised in table 1.