Supernovae observations indicate that acceleration of the Universe started only recently around redshift 0.5, and that before this period the Universe was in deceleration [24]. Dark energy models have then to reproduce correctly this transition from a decelerating to an accelerating Universe. Most dark energy candidates require extraordinary fine-tuning of some of their parameters or of their initial conditions, to lead to a phase of acceleration precisely today. This problematic is called the coincidence problem. For example, the value of the cosmological constant, the simplest model of dark energy, has to be accurately tuned in order to become dominant exactly today. Equivalently, in quintessence dark energy [25]-[28], some parameter of the field potential has to be chosen very carefully to get acceleration at the required time.
K-essence provides an elegant solution to the coincidence problem. In this model pro-posed by Mukhanov, Steinhard and Picon [29, 30], the Universe contains in addition to usual matter and radiation, a scalar field with non-canonical kinetic terms. Such terms, which are motivated by string and supergravity theories, result in very interesting equations of motion. Indeed, under some conditions, field equations possess fixed points, i.e. solutions which remain constant as the Universe evolves. Some of these fixed points are attractors, and therefore every solution that enters their basin of attraction quickly approaches them and get trapped. They can therefore be used to construct a dynamical explanation to the coincidence problem. Each fixed point is characterized by two quantities, which are fixed by the background content of the Universe. One is the equation of state of k-essence wk= Pρk
k, and the other its energy density ratio Ωk= ρρk
tot. HerePk denotes the pressure of k-essence, ρk its energy density andρtot the total energy density in the Universe. The values of wk and Ωk differ when the Universe is dominated by radiation, matter or k-essence itself. In the early Universe, when radiation is dominant, the fixed point is characterized by wk= 13. Hence, k-essence mimics radiation. This point is therefore called the radiation fixed point.
The value of Ωk depends on the particular form of the non-canonical kinetic terms. When the Universe is dominated by matter, we find two other fixed points. The first one, called de Sitter fixed point, is only an approximative solution of the equation of motion. It is given by wk≃ −1 and Ωk≃0. The second one, called the matter fixed point has wk= 0, meaning that k-essence mimics matter. The value of Ωkat the matter fixed point is again dependant on the particular form of the non-canonical kinetic terms. Finally, when the Universe is dominated by k-essence, there exists another fixed point called the k-attractor characterized by wk <0 and Ωk ≃1. At the k-attractor, k-essence dominates completely the Universe.
Using these fixed points, it is then possible to construct an evolution of the k-essence field compatible with observations. In the early Universe, initial conditions are chosen in the basin of attraction of the radiation fixed point. Hence k-essence quickly reaches this attractor and remains there as long as the Universe is dominated by radiation. It is possible to choose kinetic terms such that at the radiation fixed point Ωk . 0.1, in order not to violate nucleosynthesis bounds. Then the transition from radiation to matter domination forces the field to leave the radiation fixed point, which is no more a solution of the equations of motion, and to evolve to the de Sitter fixed point. After some time, since the de Sitter fixed point is not an exact solution, the field naturally evolves to another stage. This can be either the matter fixed point or the k-attractor, depending on the particular form of the non-canonical kinetic terms. Those can also be adjusted such that today the field is on its way from the de Sitter fixed point to the matter or k-attractor, and has wk<−13 and Ωk≃0.7, in agreement with the observed values. Hence, in this scenario k-essence drives the acceleration of the Universe today: it plays the role of dark energy.
Moreover, the fact that acceleration takes place precisely today is not a coincidence. It is closely related to the transition from radiation to matter domination, which forces the field to leave the radiation fixed point, and consequently to reach acceleration only sometimes after the transition, i.e. today. Furthermore, the evolution of the field is insensitive to the particular choice of initial conditions, as long as they lay in the basin of attraction of the radiation fixed point. K-essence seems therefore a good candidate to explain the current acceleration of the Universe without fine-tuning.
However, in our paper [3] which is reproduced in Chapter 4, we demonstrate that perturbations of the k-essence field propagate superluminally during some stage of the Universe evolution, leading to causality problems. We determine the linearized evolution equations of k-essence perturbations which (in the limit of high wave-number) propagate with the k-essence sound speed. We then prove that this sound speed becomes larger than the speed of light, for all kind of models described above. We establish furthermore that superluminal propagation is not directly due the presence of non-canonical kinetic terms in the action. It follows rather from the transition from a decelerating to an accelerating Universe. Indeed, to evolve from the radiation fixed point, with wk = 13, to a period of acceleration with wk < −13, the field has to pass by the matter fixed point wk = 0.
However, if k-essence falls on the matter fixed point directly after leaving the radiation fixed point, it will stay there for ever and mimic matter. Consequently it can not drive the acceleration of the Universe. Hence, the action for k-essence has to be chosen such that the field avoids the matter attractor after leaving the radiation fixed point. We have demonstrated that this choice results in an increase of the sound speed above the value of
the speed of light shortly after the radiation-matter transition. Hence, the very property which makes k-essence valuable, i.e. its ability to evolve naturally from a decelerating to an accelerating Universe, is responsible for superluminal motion of the field perturbations.
Special Relativity states that nothing can propagate faster than the speed of light [31].
It is therefore important to investigate the consequences of superluminal propagation in order to decide if k-essence is ruled out or not as a good dark energy model. In our preprint paper [4] which is reproduced in Chapter 5, we study in details this problematic. We show that two different positions can be adopted regarding superluminal propagation. The first approach, which respects the principle of Einstein’s Relativity, leads to the existence of closed curves along which a signal can propagate, i.e. closed signal curves. This entails a clear violation of causality, therefore, superluminal motion rules out the theory. The other proposal enables one to save a theory with superluminal propagation by preventing the existence of closed signal curves. However, the consequence of this approach is violation of Lorentz invariance.
The origin of these two approaches resides in the ambiguity to define a time direction of propagation for superluminal signals. The equations of motion define the sound speed of perturbationscs, but not the time direction of propagation. The sound speed is indeed the same for a signal propagating into the future (dt >0) and to the right (dx >0) or into the past (dt <0) and to the left (dx <0). If the sound speed is smaller than the speed of light, we can simply impose that all signals have to propagate into the future. This choice can be made for all observers at once, independently of their motion. Past and future are indeed well defined notions and therefore, if dt >0 in one reference frame, after a Lorentz transformation dt′ > 0. On the contrary, if the sound speed is larger than the speed of light, it is no more possible to define past and future without ambiguity for all observers.
One can for example choose dt > 0 for an observer, and after a Lorentz transformation obtain dt′ < 0. Moving observers therefore disagree on the notion of past and future for a signal propagating at a speed cs > c. Hence, we need to define a new rule in order to describe the propagation of superluminal signals. The existence or not of closed signal curves depends then on the rule chosen.
One of the basic principle of Einstein’s relativity states that every reference frame is suitable to describe physics. All observers are indeed on equal footing and therefore they follow the same laws of physics, independently of their motion. The only way to respect this principle is then to construct a rule describing superluminal motion which is the same for all observers. Each observer can define his proper time, as the time given by a clock at rest with respect to himself. This notion of proper time allows him to determine his proper past and his proper future. We define then propagation of superluminal signals such that an observer always sends signal into his proper future. This rule removes the ambiguity about time direction of propagation while respecting Einstein’s principle: all observers behave in the same way. Of course an observer moving with respect to the sender can see the signal propagating into his past. But this does not contradict the rule, since only the observer who sends the signal has to see it propagating into his future. However, the direct consequence of this rule is the existence of closed signal curves. In figure 5.1 (see chapter 5), the construction of such a curve is presented. It involves two observers, one moving with
respect to the other, sending signals with two different superluminal velocities. Hence we conclude that, in the context of Einstein’s relativity, the existence of superluminal motion leads to violation of causality and therefore rules out the theory.
However, if we are ready to give up Einstein’s Relativity principle, it is possible to con-struct a rule governing superluminal propagation, which excludes the existence of closed signal curves. We can indeed choose a preferred reference frame (for example the cosmo-logical frame in which the k-essence background is homogeneous and isotropic) and define past and future with respect to this frame. All observers have then to send signals into the future as defined by this frame. Consequently no closed signal curve can be constructed, since no signal can propagate into the past in this preferred frame. This rule is in contra-diction with Einstein’s Relativity principle since it does not regard observers as equivalent.
Some moving observers can for example not use their proper time as a suitable time coor-dinate to describe propagation of signals. According to their proper time they indeed have to send superluminal signals into their proper past. They have then to use the proper time of the preferred observer to describe physics. It is therefore not possible to have a theory with superluminal motion which satisfies both causality and Lorentz invariance. Even if the Lagrangian is invariant under Lorentz transformation, the choice of a preferred frame breaks the symmetry. Hence it is no longer equivalent to solve the equations of motion in the preferred frame or in a moving frame. In the preferred frame, we can solve the equa-tions of motion and choose the retarded Green function as the physical solution, whereas in some moving frames this procedure does not work any more. The rule states indeed that the retarded Green function is not always the one which describes properly signal propagation. In some moving frame, we have to take a mixture of retarded and advanced Green function [32]. Hence there is only a class of reference frames in which the Green function is determined by initial conditions in the past.
The other strange consequence of this approach is the important role played by the cosmological (preferred) frame on small scales. Up to now we never observed any effect of this frame on our scale. It is in principle meaningful only on cosmological scales, where it is related to the homogeneous and isotropic background. In the context of superluminal propagation, the cosmological frame becomes however also crucial on small scales, since it defines in which time direction signals have to propagate. Hence, even on our scale Lorentz symmetry is broken. However boost symmetry is tested with very high accuracy in particles colliders, which implies that interactions of k-essence with Standard Model particles must be extremely small in order not to generate an observable breaking of this symmetry at the currently available energies.
Hence even if the second approach does not contain any logical contradiction and is con-sequently appealing to save a theory with superluminal propagation, it leads to non trivial consequences on the fundamental Relativity principle of Einstein and therefore seems to us very unphysical. In the end we will have to rely on observations to know if superlumi-nal motion exists or not. If Einstein’s principles are correct, we know that we will never observe superluminal motion, since it would mean that causality is violated. On the other hand, if we do observe superluminal motion, then we will know that different observers are not equivalent and that Lorentz invariance has to be given up.