Could dark energy be vector-like?

In this paper I explore whether a vector field can be the origin of the present stage of cosmic acceleration. In order to avoid violations of isotropy, the vector has be part of a ‘cosmic triad’, that is, a set of three identical vectors pointing in mutually orthogonal spatial directions. A triad is indeed able to drive a stage of late accelerated expansion in the universe, and there exist tracking attractors that render cosmic evolution insensitive to initial conditions. However, as in most other models, the onset of cosmic acceleration is determined by a parameter that has to be tuned to reproduce current observations. The triad equation of state can be sufficiently close to minus one today, and for tachyonic models it might be even less than that. I briefly analyse linear cosmological perturbation theory in the presence of a triad. It turns out that the existence of non-vanishing spatial vectors invalidates the decomposition theorem, i.e. scalar, vector and tensor perturbations do not decouple from each other. In a simplified case it is possible to analytically study the stability of the triad along the different cosmological attractors. The triad is classically stable during inflation, radiation and matter domination, but it is unstable during (late time) cosmic acceleration. I argue that this instability is not likely to have a significant impact at present.


Introduction
A combination of different cosmic probes that primarily involves supernova data [1,2] suggests that the universe is currently undergoing a stage of accelerated expansion. Little is known about the origin of this stage of cosmic acceleration [3]. It might be related to a breakdown of general relativity on large scales [4,5], or it could be the effect of 'dark energy' [6], a negative pressure component that causes the universe expansion to accelerate. The simplest possibility is that dark energy is merely a (tiny) cosmological constant. If dark energy is dynamical, it is mostly assumed to be a scalar field, quintessence [7].
On purely phenomenological grounds, within the context of general relativity, dark energy might be characterized by its equation of state w DE , its speed of sound c 2 s and its anisotropic stresses [8]. Conventional quintessence models [7] have an equation of state −1 ≤ w DE , a speed of sound c 2 s = 1 and no anisotropic stresses, whereas in kessence models [9,10] the speed of sound is in general different from one. The range of possible phenomenological properties of dark energy is not exhausted by the former models. Phantom dark energy [11] has an equation of state w DE < −1, and thus violates the dominant energy condition [12]. The origin of such an equation of state is the 'wrong' sign kinetic term of the phantom scalar field. Because of that, phantom dark energy is quantum-mechanically unstable upon decay of the vacuum into (positive

JCAP07(2004)007
Could dark energy be vector-like? energy) gravitons and (negative energy) phantom particles [12,13]. Hence, if future more precise observations confirm the current trend and favour a dark energy equation of state smaller than minus one [1,14], alternative (viable) models are needed to account for such a value [1,16].
In this paper I explore whether dark energy can be vector-like. Vector-like dark energy turns out to display a series of properties that make it particularly interesting phenomenologically. On one hand, it can also violate the dominant energy condition, w DE < −1, while possessing a conventional kinetic term. On the other hand, it has non-anisotropic stress perturbations and it leads to violations of the decomposition theorem [17], i.e. the decoupling of scalar, vector and tensor cosmological perturbations. As we shall see, other interesting features include the existence of tracking attractors, that render the evolution of a vector field in an expanding universe rather insensitive to initial conditions. In spite of this attractive feature, the onset of cosmic acceleration is determined by a parameter in the Lagrangian that has to be properly adjusted, as in most other models. In that respect, these forms of vector dark energy are similar to quintessence trackers [18,10].
Non-gravitational interactions are known to be mediated by vector fields. In addition, from a four-dimensional point of view, certain components of higher dimensional metrics behave like vectors [19]. It is hence natural to study the evolution of vector fields in a cosmological setting. However, the existence of a spatially non-vanishing vector breaks the isotropy of a Friedmann-Robertson-Walker (FRW) universe. From the point of view of gravity, such a breaking manifests itself in anisotropic stresses caused by the vector. If dark energy is such a vector, as long as it remains subdominant, this violation is likely to be observationally irrelevant [20]. Once dark energy comes to dominate though, one would expect an anisotropic expansion of the universe, in conflict with the significant isotropy of the cosmic microwave background (CMB) [21]. Because of that, in this paper I consider a 'cosmic triad', i.e. a set of three equal length vectors that point in three mutually orthogonal spatial directions. Remarkably, the existence of a triad turns out to be compatible with spatial isotropy, at least from the point of view of gravity. While the triad guarantees the isotropy of the background, it does not automatically imply the isotropy of its perturbations. Eventually, it might even be necessary to introduce fields that explicitly violate rotational symmetry, as there appear to be hints of (statistical) anisotropy in the CMB fluctuations [22]. Along these lines, I speculate below that a triad could provide a link between cosmic acceleration and some of the anomalies observed in the CMB radiation [22,23].
Mainly because they single out spatial directions, vector fields have received comparatively little attention in cosmology. Ford has proposed an inflationary model where a vector is responsible for a stage of inflation [24]. Our treatment here is to some extent similar to his proposal. Jacobson and Mattingly have studied the dynamics of a vector with of fixed length, with the specific purpose of studying violations of Lorentz invariance [25]. A vector-like form of quintessence has been also considered by Kiselev [26], though his vectors significantly differ from ours. Zimdahl et al have suggested that a (timelike) vector force could be responsible for the present acceleration of the universe [27]. Also, it has been noted that the addition of higher order powers of the Maxwell field strength to the Lagrangian of an electromagnetic field might cause the universe to accelerate [28]. The literature on magnetic fields in the early universe is more extensive; see [29] and references therein.

Vector dark energy
Consider a set of three self-interacting vector fields A a µ . Strictly speaking, this is really a set of three one-forms, but I shall call them vectors. Latin indices label the different fields (a, b, . . . = 1 · · · 3) and Greek indices their different spacetime components (µ, ν, . . . = 0 · · · 3). As we shall see below, this number of vector fields is dictated by the number of spatial dimensions and the requirement of isotropy. We would like to study the dynamics of such a 'cosmic triad' in the presence of gravity. Consider hence those vectors minimally coupled to general relativity, The action (1) thus contains three identical copies of the Lagrangian of a single vector field. The term V (A a2 ) is a self-interaction that explicitly violates gauge invariance. For completeness, in the appendix I show how a triad could naturally appear from a gauge theory with a single SU(2) gauge group. In the following, I use the Einstein summation convention throughout, where a sum is implied only over indices in opposite positions. The indices that label the different vectors are raised and lowered with the flat 'metric' δ ab .
The kinetic term F 2 in the action (1) is not unique in the following sense. Up to boundary terms, ∇ ν A µ ∇ ν A µ is the only additional diffeomorphism invariant quadratic term that contains two derivatives of the vector A µ [30,25]. Because the dynamics of vectors known to occur in nature are described by a Maxwell term, I consider an F 2 term only. Additional couplings of the vector are constrained by tests of gravity [30] and limits on possible violations of Lorentz symmetry 1 [33]. To avoid such violations, I assume that the matter Lagrangian L m only depends on the metric g µν and on the remaining matter fields ψ, but not on the cosmic triad A a . In that respect, the triad is analogous to conventional quintessence [34].
Varying the action (1) with respect to the metric one obtains the Einstein equations G µν = 8πGT µν , where the energy-momentum tensor of the triad is given by This energy-momentum tensor is the sum of the three different energy-momentum tensors of the decoupled vectors, (A) T µν = a (a) T µν , neither of which is of perfect-fluid form. By varying the action with respect to the vectors A a µ , one obtains their equations of motion, The four-divergence of the last equation yields a constraint equation for the vector. As a consequence, each vector has three dynamical degrees of freedom, as it should.
We shall study the dynamics of these vectors in a flat, homogeneous and isotropic FRW universe with metric An ansatz for the vectors that turns out to be compatible with the symmetries of this metric (homogeneity and isotropy) is Hence the three vectors point in three mutually orthogonal spatial directions and they share the same time-dependent length, A a2 ≡ A a µ A aµ = A 2 (t). Substituting the ansatz (5) and the metric (4) into the vector equations of motion (3) I find where a dot means a derivative with respect to cosmic time t. Note that the 0-component of equation (3) forces A a 0 to vanish, as in the ansatz (5). Substituting the metric (4) into the Einstein equations one obtains where H ≡ȧ/a is the Hubble 'constant'. The energy density of the universe is ρ ≡ −T 0 0 and its pressure is defined by p · δ i j ≡ T i j , where i and j run over the spatial spacetime components. Note that the energy-momentum tensor has to be compatible with the symmetries of the metric. For the FRW metric (4), G 0 i = 0, so T 0 i should also vanish. It can be easily verified that this is indeed the case for the ansatz (5). The requirement of isotropy is non-trivial for a single vector, since its energy-momentum tensor is Although this energy-momentum tensor is diagonal, its value along the j = a direction is different from the one along the directions perpendicular to it. Nevertheless, the total energy-momentum tensor (A) T µν ≡ a (a) T µν has isotropic stresses, and the corresponding energy density ρ A and (isotropic) pressure p A are given by Note that the equation of motion (6) can also be derived from the conditioṅ To conclude this section let me point out a remarkable property of the cosmic triad: its equation of state w A ≡ p A /ρ A can become less than −1 (with a positive energy density) if dV /dA 2 is negative [35]. Because a mass term for a vector A µ has the form V (A 2 ) = (m 2 /2)A µ A µ , I call such vectors 'tachyonic'. Tachyons (particles of negative squared mass) are usually associated with instabilities. In many cases, an instability merely signals the tendency of the system to evolve. In cosmology, those
instabilities are not particularly terrible. In fact, a universe in stable equilibrium would be pretty unexciting, as it would not even expand. In the absence of gravitational instability, structures would not form, and without the (effective) tachyonic mass of a scalar, it would be quite difficult to seed a scale invariant spectrum of perturbations during inflation [36,37]. Scalar tachyons 2 have indeed been widely considered in the literature. Other forms of instability are more worrisome, such as the quantum-mechanical instability of the vacuum in the presence of a phantom [12,13]. By simple analogy, any form of instability associated with a tachyonic vector is not expected to be of this second kind, as the vectors have a conventional kinetic term. This question is not only of academic interest, since analyses of observational data (marginally) favour a dark energy equation of state w DE < −1 [1,14].
I shall not deal here with the quantum mechanics of tachyonic particles, which even for scalars is not free of problems. Nevertheless, I also want to present some arguments that suggest that a tachyonic vector might be similar to a phantom field [38]. One of the arguments goes back to the Stueckelberg theory of massive vectors [39,40]. Consider for simplicity a vector field in Minkowski space, The field is massive for κ = 1 and tachyonic for κ = −1. Upon the substitution Note that (13) contains an additional scalar, the Stueckelberg field S. For a massive vector (κ = 1) S has a conventional kinetic term, but for a tachyonic vector (κ = −1), its kinetic term has the 'wrong' sign, like that of a phantom [11]. However, the additional field S turns out to be constrained in the quantum theory. Even though it describes a massive vector, the Lagrangian (13) has a gauge symmetry, where λ is a scalar function. Upon quantization, this gauge freedom is fixed by imposing the Stueckelberg subsidiary condition ∂ µ A µ − mS = 0, which relates S and the divergence of the vector [40]. Thus, strictly speaking, the field S is not simply a phantom scalar in the conventional sense. On the other hand, there are other properties that suggest phantomlike behaviour of a tachyonic vector, such as the opposite sign of the propagator in the high momentum limit, or the opposite sign in front of the squared longitudinal momentum in the Hamiltonian [41].
In the time-dependent situation that we are dealing with, where the triad vectors have a non-vanishing expectation value, the issue is slightly more complicated. Consider quantum fluctuations δA µ around one of the triad vectors in our classical solutions, A a µ → A a µ + δA µ . Expanding the vector Lagrangian in (1) to second order in δA µ and neglecting fluctuations in the gravitational field I get, for one of the triad vectors, Terms linear in the perturbations vanish if A aµ satisfies the classical equations of motion.
Note that in addition to the mass term proportional to dV /dA 2 , there is an additional
contribution proportional to d 2 V /d 2 A 2 that breaks Lorentz invariance (because of the coupling of δA µ to the classical, non-vanishing vector A µ ). Therefore, the quantization of δA µ is expected to be significantly different from that of the tachyonic vector in the Lagrangian (12).

Cosmic evolution
Our next task consists in studying the evolution of the cosmic triad in a universe that contains additional forms of 'matter', like an inflaton 3 (p inf ≈ −ρ inf ), radiation (p r = ρ r /3) or dust (p d = 0). Ideally, we would like the cosmic triad to remain subdominant during most of cosmic history, and just around red-shift z ≈ 1 come to dominate the energy density of the universe and trigger a stage of accelerated expansion. The vector equation of motion (6) is formally the same as the one for a self-interacting conformally coupled scalar field. Indeed, the term in parenthesis is proportional to the Ricci scalar R, which vanishes during radiation domination. Of course, the similarity between the equations of motion of a vector and a conformally coupled scalar arises from the conformal invariance of the Maxwell Lagrangian. In some instances it is going to be more convenient to deal with a set of two first-order differential equations, rather that with a single second-order one. Introducing the number of e-folds dN ≡ d log a as a time variable, and defining the vector equation of motion (6) can be recast as where the Hubble constant is given by equation (7). In the following I study the vector equations of motion in two limits: the limit where matter dominates (early stages of cosmic evolution) and the limit of triad domination (late stages). The evolution of the triad depends on the self-interaction V . In this paper I focus on a class of runaway power-law potentials where M is a positive parameter with dimensions of mass. This class turns out to be sufficiently general for our purposes. In five spacetime dimensions, a self-interaction of this form (with n = 1/2) leads to inflation along 'our' three spatial dimensions, while keeping the size of the remaining fifth dimension essentially constant [35]. These runaway potentials are also reminiscent of a class of 'tracker' quintessence [18,10]. Note that for n > 0, the case we are interested in, all these models are tachyonic. Non-tachyonic interactions do not appear to be particularly interesting.

Matter domination
Suppose that the energy density of the universe is dominated by the inflaton, radiation or dust, i.e. the contribution of the triad to the total energy density of the universe is negligible. The scale factor then grows like and w m is the matter equation of state. Thus, β 1 during nearly de Sitter inflation, β = 1/2 during radiation domination and β = 2/3 during dust domination. We shall keep β as a free parameter and consider solutions of the equation of motion where A also grows as a power in cosmic time. The reader can verify that, indeed, is a solution of the equation of motion (6) for an interaction given by equation (18). Actually, this solution is also an attractor. Perturbing A → A + δA and linearizing equation (6) for the given unperturbed background (19) I find The term in parenthesis is positive during dust domination, and it vanishes during radiation domination. At the same time, the derivative d 2 V /dA 2 is always positive for the potentials (18). Therefore, small deviations from the solution (20) oscillate and decay away. Note that along the attractor, the equation of state of the triad is constant, Hence, these solutions are analogous to the quintessence and trackers discussed in [18] and [10]. The triad equation of state is always smaller than that of the dominant component, and the closer the latter is to −1, the less their difference is. The triad equation of state as a function of n is plotted for various values of β in figure 1. Because in the present epoch the universe is not dominated by dust any more, the value of the triad equation of state today lies between the ones along the matter attractor and the de Sitter attractor that I will discuss next. Hence, experimental constraints [1,2] on the value of w DE restrict the possible values of n. In particular, in order to obtain w A 0.75 today, models with n 1/2 have to be considered. This restriction applies provided that initial conditions are chosen in the basin of attraction of the tracking solution.
For later purposes, let me also discuss an approximate solution of the system (16), (17). Suppose that B HA and dV /dA HB. Loosely speaking, these inequalities are attained in the limits of large A or large H. Then, equations (16), (17) have the approximate solution Along this solution the kinetic energy of the field B 2 ∝ e −4N decreases, whereas the potential energy V ∝ e 2nN increases. Hence, soon the potential energy dominates the kinetic one, B 2 V , so along this approximate solution the equation of state is
which is less than −1. However, this approximate solution only holds temporarily, as the assumptions that we have made finally become violated. Later on I will discuss an example where initial conditions place the triad on the approximate solution (23) for a significant period of time. Note that in the opposite limit, the limit where the kinetic energy dominates the potential one, B 2 V , the triad equation of state is radiation-like, w A ≈ 1/3. In both limits, the triad is quite different from a (canonical) scalar field (with positive potential), for which −1 ≤ w ≤ 1.

Triad domination
Along the attractor (20), the energy density of the triad decays more slowly than that of matter. Therefore, sooner or later the triad will come to dominate the universe. Consider thus the equation of motion (6) when the triad dominates the content of the universe. Using equations (7), (8) and (10), (11) the vector equation of motion (6) takes the form which is just like the equation of motion of a minimally coupled scalar with appropriate potential (we will not need the explicit form of W (A)). Solutions of equation (25) with constant A = A * can be easily identified by requiring dW/dA to have a zero at A * . This leads to the condition Those constant A solutions are expected to be stable (attractors) if d 2 W/dA 2 > 0, which implies dV dA 2 + 1 4πG Along the attractor the energy density is, from equations (7) and (10), i.e. a constant. Hence, these constant A solutions are de Sitter attractors with w A = −1; they are natural candidates for accommodating the present accelerated expansion of the universe. Figure 2 shows a phase diagram of the system (25) when the triad is the dominant component of the universe. Note that along the de Sitter attractor, the kinetic energy of the triad does not vanish. Inserting the potentials (18) into equation (26) and verifying condition (27) I find that there is a single (stable) de Sitter attractor at Substituting this value of A * into (28) and requiring it to be of the order of the present energy density, one can then estimate what is the required value of M to fit the present stage of accelerated expansion, Therefore, for n ≈ 0 the required value leads to the 'infamous' scale M ∼ 10 −3 eV, whereas larger values of n result in more reasonable energies. Note that we are fitting, rather that explaining, the time at which cosmic acceleration begins. As mentioned above, the exponent n determines the value of the equation of state today. The closer n is to 0, the closer w A is to −1 today.
where the prior on the density parameter of dust Ω M = 0.27 ± 0.04 has been assumed [1]. Because the uncertainties are correlated, the reader is advised to look at the constraints on the w DE −dw DE /dz plane in figure 10 of [1]. Note that the limits (31) are significantly weaker than the ones derived assuming that w DE is constant.
The first example has a potential and I am assuming that the Hubble constant today is H 0 = 70 km s −1 Mpc −1 . Note that in this model V is not an analytic function of A 2 . In particular, A 2 is constrained to be spacelike. Figure 3 shows the evolution of the energy density for different sets of initial conditions, and figure 4 shows the corresponding behaviour of the triad equation of state. As clearly seen in the figures, the late time evolution of the cosmic triad is quite insensitive to initial conditions. For this particular model, w A ≈ −0.87, dw A /dz ≈ 0. of dark energy (31); see also figure 10 in [1]. Larger values of n lead to violations of the limit on w DE today, whereas smaller values yield w A s closer to −1.
In the second example the interaction term is given by This corresponds to n = 1/2 in equation (18). In this case the triad evolution can be consistent with current observations if the initial value of A is fine-tuned. The initial value of B = (Ȧ + HA) can be quite arbitrary though. Cosmic evolution for such a model and tuned initial conditions is shown in figures 5 and 6. The different lines correspond to different initial values of B. Note that all of them yield the same final result, and in particular all of them join the approximate solution (23). As seen in figure 6 the equation of state along that solution remains constant at w A = −4/3 (equation (24)) all the way to today. By construction, in this particular example w A ≈ −1.33, dw A /dz ≈ 0 and Ω A = 0.72 today. For integer values of n and fine-tuned initial conditions it is also possible to obtain histories where the equation of state w A remains constant at a value significantly smaller than −1 for a long period of time, and just recently approaches −1.
For these models dw A /dz tends to violate the limit in (31). Certain analyses of supernova data [14] suggest that the equation of state has evolved from w DE ≈ 0 at z ≈ 1 to w DE < −1 today [14]. Within the class of models (18) I have not been able to obtain such a behaviour while keeping at the same time an early epoch of radiation domination.

JCAP07(2004)007
Could dark energy be vector-like?  (24), and remains at that value to the present epoch.

Perturbations
The properties of any dark energy candidate not only comprise its equation of state, but also the way the perturbations (if any) behave [8]. These perturbations are coupled through Einstein's equations to metric perturbations, which in turn affect observables such as CMB temperature anisotropies. The evolution of triad perturbations can also help to determine whether models with w A < −1 suffer from serious instabilities and are hence unviable. Unfortunately, cosmological perturbation theory with a triad turns out to be rather involved, as scalar, vector and tensor modes couple to each other. Therefore, in this section I only scratch the surface and mainly present qualitative features of the equations.
In scalar longitudinal gauge and vector gauge, the most general linearly perturbed spatially flat FRW metric is [42,17,43,8] where B i is a transverse vector, ∂ i B i = 0, and h ij is a transverse and traceless tensor, In a similar way, we can decompose the perturbations of the vector fields in the triad δA a µ into scalar and vector components, Here δA a 0 and χ a are scalars and δA a i is a transverse vector, ∂ i δA ai = 0. Note that for convenience we are using conformal time. In the following, spatial indices are raised and lowered with the metric δ ij .

Linearized triad equations
Because the vector fields in the triad do not couple to each other, we can study each of them at a time. Let me denote by A a i the spatial components of one of the background triad vectors and let me drop in the following the index a. The 0-component of the linearized equation of motion (3) in the spacetime (34) is where a prime denotes a derivative with respect to conformal time. Remarkably, even though this is a scalar equation, it contains the vector perturbation B i . In that respect, non-vanishing vector fields lead to violations of the decomposition theorem [17]. In the absence of a background vector quantity, the only way to obtain a scalar linear in a vector B i is to compute its divergence ∂ i B i , which vanishes for a transverse vector. However, if the spacetime contains a non-vanishing background vector A i , it is possible to construct a scalar linear in a vector perturbation through the combination A i B i . The reason that the decomposition theorem is expected to hold in FRW universes is that in general they do not have any spatial direction singled out.
The i-components of the linearized equations have the form of the gradient of an equation that involves scalars plus an equation that involves transverse vector quantities. The scalar equation is where δ is implicitly defined by the decomposition into a scalar δ and a transverse vector δ i . The perturbation of A 2 is given by
and, in particular, does not contain δA 0 . The remaining vector equation takes the form Therefore, again, scalar, vector and tensor perturbations are not decoupled, since there exists a non-vanishing vector in the background. Note that (36) is not a dynamical equation for δA 0 , but a constraint. It seems that for tachyonic models, dV /dA 2 < 0, one cannot solve for δA 0 , since in that case the operator ∂ i ∂ i − 2a 2 dV /dA 2 is not invertible. This would point to a potential inconsistency of tachyonic models. However, as we shall see next, this difficulty is only apparent. Taking the Laplacian of equation (37) and using the time derivative of the constraint (36) to substitute for the value of ∂ i ∂ i δA 0 , one obtains a first-order differential equation in time for δA 0 that does not contain its spatial derivatives, Here, f is a function of the specified variables and we shall not be concerned with its derivatives. What is important is that if a set of initial conditions satisfying the constraint (36) is specified, then the solution of equations (41) and (37) is guaranteed to satisfy equation (36) at all times. Hence, it is in fact possible to solve for δA 0 .
To conclude this part let us count how many 'degrees of freedom' (per vector) the triad perturbations contain. We have seen that δA 0 is constrained, so it is not dynamical. Equation (37) contains the second time derivative of χ (1 dof) and equation (40) the second time derivative of the transverse vector δ A (2 dof). Thus, there are three degrees in freedom in total, as pertains to a massive vector.

Perturbed energy-momentum tensor
The perturbations in the triad induce perturbations in its energy-momentum tensor, which in turn are responsible for sourcing metric perturbations. In this subsection I shall deal with vector and tensor perturbations, which cannot be sourced in conventional cosmological models.
The perturbations in the energy-momentum tensor of the triad vectors A a µ can be decomposed into an isotropic pressure δp A and a traceless anisotropic stress Π i The anisotropic stress itself can be decomposed into scalar, vector and tensor components, , and (t) Π i j is transverse and traceless.
The equations of motion for vector (metric) perturbations are [8,17] where (v) Π i j is the vector part of the triad anisotropic stress tensor. In the absence of anisotropic stress sources, equation (42) implies that vector perturbations decay away, B i ∝ 1/a 2 . So even if they are generated in the early stages of the universe, they are not expected to be significant today. The transverse and traceless anisotropic stress sources tensor perturbations, i.e. gravitational waves, As opposed to the vector perturbations case, in the absence of sources the amplitude of long wavelength gravitational waves remains constant. Hence, if they are primordially produced, say, during an inflationary stage, they could still have a sizable amplitude today. I shall not write down all the terms that the perturbed spatial components of the energy-momentum tensor of a single triad vector contain. They straightforwardly (but tediously) follow from the insertion of the perturbations (34) and (35) into equation (2). Instead, for the sake of illustration I shall consider only where the dots denote the multiple terms that I am not explicitly writing down. In order to study the evolution of vector and tensor perturbations we have to compute the vector and tensor components of the previous expression. In Fourier space, these are given by [17] p which as required are transverse and traceless. Note that we sum over the three triad vectors to obtain the total energy-momentum tensor perturbation. Because the triad perturbations δA ai are a priori totally independent from each other, the anisotropic stress (t) Π i j does not vanish in general. Thus, vector fields are expected to source not only vector perturbations, but also tensor perturbations. Again, the reason is that the decomposition theorem is violated. With the aid of the background vectors it is possible to construct traceless and transverse quantities linear in the perturbations.
If the perturbations have certain symmetries, (t) Π i j does indeed vanish (for the particular term in the energy-momentum tensor that we are considering). Because A ai and k i are the only vectors in the problem, the triad perturbation δA ai is expected to be a function of A ai and k i . Since δA a is transverse, it has to be of the form where α and β are two scalar functions and ε ijk is totally antisymmetric. Assume that α and β do not depend on the index a (that is, they do not depend on k i A ai ). Then, substituting equation (47) in (46) and using (5), one finds not only that (t) Π i j = 0, but also that (v) Π i = 0.

Stability
The main worry that one faces when dealing with tachyonic fields is as regards their stability. In order to work out to what extent the solutions that we have found in section 3 are stable, we should solve the system of cosmological perturbation equations just presented. Obviously, the coupling between scalar, vector and tensor perturbations makes this task quite formidable. In this section we dramatically simplify the equations by neglecting metric perturbations and concentrating on a particular set of vector modes. The hope is that this drastic simplification captures the qualitative features of the equations.
So let us set Φ = Ψ = B i = h ij = 0 and consider the triad perturbation equations in Fourier space. From equation (38), for modes for which k · A = 0, it follows that δ = 0. Then, χ = δA 0 = 0 is a solution of equations (36) and (37). Because δ i ∝ (A j δA j )A i , if in addition we consider perturbations such that δ A · A = 0, the remaining vector equation (40) reads where for convenience I have gone back to cosmic time. Note how 2dV /dA 2 plays the role of a mass term in the last equation. This is why I call interactions with dV /dA 2 < 0 'tachyonic'. Generically, if dV /dA 2 is negative, one expects growing modes, that is, instabilities.
In order to check the stability of the triad, it suffices to consider long wavelength modes, k = 0, in equation (48). For sufficiently high k, the gradient dominates the interaction and solutions are stable. Hence, any form of instability is an 'infrared' effect, rather than an ultraviolet one. For a given expansion, equation (19), along the attractor (20), the long wavelength solution of equation (48) is Here, C + and C − are two integration constants and for convenience I have divided by the scale factor to obtain the length of the perturbation. We should compare these solutions with the behaviour of the background, equation (20). Recall than A is the common length of the background triad vectors. The C + mode in equation (49) grows as fast as A and the C − mode grows less rapidly than A if n(1 − 3β) < 1 + 3β. Therefore, the C − mode decays relative to A during inflation, radiation and dust domination. Hence, within the scope of our analysis, the system is not unstable throughout the period where the triad is subdominant 4 . Note however that the triad would be unstable for certain values of n if there had been a period of cosmic history during which β < 1/3. When the triad becomes dominant, the vector evolves towards the de Sitter attractor (29). The solution of equation (48) along the de Sitter attractor is where H * is the value of the Hubble constant along the de Sitter solution. Note that along the latter A itself is constant. Thus, the de Sitter attractor is unstable, in the sense that δA i /a (the length of the vector perturbation) grows relative to A (the length of
the background vector). Because during the previous stages of cosmic history (when the triad was subdominant) the relative amplitude of the perturbations δA/A has remained unaltered, the time perturbations becoming relevant will depend on early universe initial conditions. If the primordial vector amplitude agrees for instance with the (scalar) amplitude of density fluctuations, δA/A ≈ 10 −5 , δA/A becomes of order one about 12 e-folds after the onset of triad domination. By then the universe has presumably grown anisotropic, because there is no reason to expect perturbations in the three different triad vectors to be correlated. We will have to wait several billion years for that to happen. At present the effect is still small, since the relative amplitude has increased by at most a factor of order one. In fact, it is tempting to speculate on whether some of the anomalies observed in the CMB radiation [23], especially in the quadrupole and octopole (see however [44]) and the related hints of statistical anisotropy [22], might be due to such an instability, which sets in once the universe starts to accelerate and mostly affects large scales.

Summary and conclusions
In this paper I have considered whether a vector field could be responsible for the current stage of cosmic accelerated expansion. The existence of a vector with non-vanishing spatial components turns out to be compatible with the isotropy of a Friedmann-Robertson-Walker universe provided that the vector is part of a 'cosmic triad', i.e. a set of three identical vector fields pointing in mutually orthogonal directions. A set of three identical self-interacting vectors naturally arises for instance in a gauge theory with SU(2) or SO(3) gauge group. A distinctive property of a cosmic triad is that its equation of state of can become less than −1, even though its kinetic terms have the conventional form. The necessary condition is that the self-interaction is 'tachyonic', i.e. it naively gives rise to a negative squared vector mass. Although the simple analogy with scalar tachyons suggests that the study of tachyonic vectors is justified, there are also arguments that connect tachyonic vectors to phantom particles [38]. In analogy to tracking quintessence models [18], in this paper I have explored a particular class of tachyonic models, where the interaction is an inverse power law. For appropriate initial conditions, there exist seemingly viable cosmologies where dark energy has an equation of state w < −1 throughout cosmic history, including today. On top of that, these models have attractors that render cosmic evolution quite insensitive to initial conditions and yield equations of state sufficiently close to −1 today. The experimental constraints on the value of the dark energy equation of state restrict the power of the vector self-interaction, while the time at which cosmic acceleration sets in determines its energy scale. Within the class of models that I have considered, the current limits on the equation of state of dark energy require the vector self-interaction to be a non-analytic function of its squared length.
Finally, I have scratched the surface of cosmological perturbation theory in the presence of a cosmic triad. The most remarkable feature is the violation of the decomposition theorem. In the presence of a vector which has non-vanishing spatial components, scalar, vector and tensor perturbations do not decouple from each other. In particular, metric vector perturbations show up in scalar equations, and in the absence of particular symmetries in the perturbations vectors are able to source tensors. The (scalar)

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Could dark energy be vector-like? time components of the triad vectors are not dynamical, i.e. they are constrained. Despite an apparent difficulty, it is possible to solve the constraint also for tachyonic models. I have also considered solutions of the perturbation equations for a particular set of modes, under the assumption that metric perturbations are negligible. During inflation, radiation and dust domination, the relative perturbations in the triad remain constant. However, there is a long wavelength instability during the late time stage of de Sitter acceleration, where triad perturbations grow relative to the background. Hence, the time at which the universe becomes anisotropic depends on early universe initial conditions. For reasonable primordial perturbation amplitudes, the universe is expected to become anisotropic a long time after the onset of cosmic acceleration. The instability of the triad during this epoch also suggests a possible relation between the large angle anomalies in the CMB sky and the onset of cosmic acceleration, but further work is needed to test this idea. A more careful investigation is also needed to establish whether tachyonic vectors are fully stable, and how the inclusion of metric perturbations affects the behaviour of the triad perturbations (and vice versa).
To conclude, at the level of the present analysis it seems that vector fields could indeed be responsible for the present stage of late time cosmic acceleration, although it is as yet unclear how quantum mechanics constrains the tachyonic models that I have studied here.

Acknowledgments
This paper is dedicated to Isabel. It is a pleasure to thank Sean Carroll, Sergei Dubovsky, Vikram Duvvuri, Wayne Hu, Lam Hui, Eugene Lim, Geraldine Servant and Paul Steinhardt for useful discussions, and the KICP for software assistance. This work was supported by the US DOE grant DE-FG02-90ER40560.

Appendix: A non-Abelian triad realization
In this appendix, I show that a cosmic triad can arise from a non-Abelian SU(2) gauge theory. For that purpose, consider the gauge invariant Yang-Mills Lagrangian where F a µν is the non-Abelian field strength The totally antisymmetric tensor ε a bc encodes the structure constants of the SU(2) Lie algebra. The equations of motion of the fields in an arbitrary spacetime are 3) The ansatz (5) satisfies these equations of motion in the FRW universe (4)  It can be verified that T 0 i = 0, whereas − T 0 0 = 3 2 (Ȧ + HA) 2 + 3V, The self-interaction V is again given by equation (A.5). Therefore, the energy density and pressure of the non-Abelian gauge fields agree with those of the triad, equations (10), (11). This equivalence between the triad and the non-Abelian gauge fields in the symmetric case that we are considering is confirmed by substituting the ansatz (5) into the actions (1) and (A.1). Note that for the triad vectors, as opposed to scalar fields or perfect fluids, the Lagrangian density is not the pressure or the energy density. The existence and some properties of non-trivial solutions of general relativity coupled to an SU(2) gauge field in a FRW universe have been considered in [45].