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Dynamics of Tachyonic Dark Matter

Usually considered highly speculative, tachyons can be treated via straightforward Einsteinian dynamics. Kinetic theory and thermodynamics for a gas of ``dark'' tachyons are readily constructed. Such a gas exhibits density and pressure which, for the dominant constituent of a suitable Friedmann-Robertson-Walker spacetime, can drive cosmic evolution with features both similar to and distinct from those of a standard dark-energy/dark-matter model. Hence, tachyons might bear further consideration as a cosmic dark-matter candidate.


I. INTRODUCTION
Tachyons-faster-than-light particles [1], with spacelike four-momenta-are usually considered either unphysical or rather fanciful speculation. (But see, e.g., Clay and Crouch [2]. The extant literature on tachyons is extensive and diverse. A more thorough review than can be accommodated here is in preparation. [3]) It is possible, however, to construct straightforward, consistent Einsteinian dynamics for such particles. It is further possible to treat kinetic theory and thermodynamics for a gas of"dark" (noninteracting) tachyons. The density and pressure-i.e., the equation of state-for such a gas have interesting consequences if the gas is taken to be the dominant mass/energy constitutent of a Friedmann-Robertson-Walker cosmological model. For suitable choices of parameters, such a model exhibits rapid expansion from an initial singularity, decelerating to a minimum expansion rate, followed by accelerating expansion. This is like the behavior of the dark-energy/dark-matter model which has become the standard model of cosmology, although the late-time behavior of the two models is different enough that they might be distinguished by observation or even experiment. While challenges remain, it may be that tachyons deserve further investigation as a potential dark-matter candidate.
The mechanics of a single tachyon interacting with simple electric and magnetic fieldsthe simplest interaction consistent with Einsteinian special relativity-is described in Sec. II, following. As such dynamics undergirds all that follows, these elementary results are presented in some detail. The kinetic theory and thermodynamics of a gas of "dark" or noninteracting tachyons is worked out in Sec. III.The consequences of the resulting properties for a tachyon-dominated homogeneous and isotropic cosmological spacetime are shown in Sec. IV. Conclusions and further questions are discussed in Sec. V. For conciseness, units with speed of light c = 1 are used throughout.

II. SINGLE-TACHYON DYNAMICS
The simplest equation of motion, consistent with Einsteinian dynamics, for a single tachyon is the Lorentz force law [4] with Faraday field tensor Here the tachyon has invariant mass[10] m = iµ-imaginary, because the tachyon has spacelike four-momentum-real electric charge q, and "proper time" dτ = i dλ, with λ a real affine parameter on the spacelike world line of the tachyon. With four-momentum p α = iµ u α , this equation can be written dp with E = p 0 the energy and p x = p 1 the momentum of the tachyon. The solution to these, choosing λ = 0 at E = 0 and p x = µ, is For α > 0, this is the upper branch of an hyperbola, describing a tachyon coming in from

B. Tachyon in electric field-two dimensions
With two electric-field components and two momentum components, the equation of motion takes the form These imply These have solution This result describes three distinct types of motion.

Simplest two-dimensional case:
so the tachyon asymptotically approaches the speed of light from above. This corresponds to trajectory show that a tachyon pushed in the direction it is moving slows down, approaching the speed of light from above with increasing energy. But solutions (2.12a)-(2.12c) show that a tachyon pushed perpendicular to its direction of motion does not "back up"; it accelerates in the direction it is pushed.

C. Tachyon in magnetic field
For a tachyon in a uniform, purely magnetic field, say in the z direction, equation of motion (2.1a) reduces to For suitable choice of the zero of λ, this has general solution The tachyon executes uniform helical (or circular, for p (0) z = 0) motion at constant speed about the magnetic field lines. Again the tachyon accelerates in the direction it is pushed, perpendicular to its trajectory, not the opposite direction.
The existence of electrically charged tachyons raises several conundra: Such a tachyon should emit Cerenkov radiation in vacuum. This should be readily detectable [1], and the associated loss of energy should cause the tachyon to accelerate toward infinite speed and zero energy. Furthermore, it would be energetically possible for a free photon to decay into a tachyon/antitachyon pair, if the pair coupled to the electromagnetic field. Unless some other principle suppresses this process, the propagation of photons across cosmic distances should severely constrain the existence of charged tachyons. It is possible, however, to sidestep such questions by considering electrically neutral or "dark" tachyons.

III. KINETIC THEORY AND THERMODYNAMICS OF TACHYON GAS
It is possible to determine thermodynamic properties and an equation of state for a tachyon gas, i.e.,for an ensemble of (neutral, dark) tachyons of mass m = iµ and nonnegative energies E with probability distribution where N is a normalization factor to be determined and β = 1/(k B T ) is the inverse temperature of the distribution. This can be done by evaluating the energy-momentum tensor with n the number density of the particles in the rest frame of the ensemble. The tensor takes the matrix form angled brackets denoting ensemble averages.

A. Kinetic theory
Thermodynamic averages for this tachyon gas can be expressed as a related sequence of integrals. The normalization constant N for the energy distribution is determined by the where the energy/momentum relation E 2 − p 2 = −µ 2 implies p dp = E dE. The average energy is given by The mean squared energy is the mean squared momentum is found from p 2 = E 2 + µ 2 . These can be expressed in

5c)
and These results can be compared with those for a gas of ordinary-slower-than-light or bradyonic-particles with real invariant mass m b . In this case the normalization constant is given by the average energy by the mean squared energy by and the mean squared momentum by here with subscript b distinguishing these bradyonic quantities, and featuring modified Bessel Results (3.5a)-(3.5d) and (3.6a)-(3.6d) are general, applicable at any temperature T .
At low temperatures-k B T ≪ µ and k B T ≪ m b -averages (3.5b)-(3.5d) take the limiting for cold tachyons. Averages (3.6b)-(3.6d) take the familiar forms for cold bradyons. At high temperatures-k B T ≫ µ and k B T ≫ m b -the averages approach The behaviors of both hot tachyons and hot bradyons approach that of photons.

Energy-momentum tensor (3.3) implies tachyon-gas equation of state
with ρ the energy density and n the particle-number density of the gas. The corresponding bradyon-gas equation of state is differing only in the sign of the mass term. The low-temperature limit of Eq. (3.10) is (3.12) the pressure approaches finite limit 1 3 µn, while the energy density approaches zero. In contrast, the low-temperature limit of Eq. (3.11) is This can actually be interpreted in exactly the same way as the energy equation used to determine the speed of a car on a roller coaster, neglecting friction and drag: The terms on the left correspond to kinetic and potential energy, respectively; that on the right is the conserved total energy. The behavior of the "potential energy" term is illustrated in Fig. 1.
It can be parametrized as All of these behaviors are exhibited in exact solutions of the Friedmann Equation (4.4).
In terms of conformal time η defined via dη = dt/a(t), closed models with k = +1 satisfy for 0 ≤ η ≤ 2 tan −1 (A/B). Spatially flat k = 0 models satisfy for 0 ≤ η ≤ 2A/B. Open k = −1 models with B > A are described either by for η ∈ (−∞, +∞). For B = A the five possibilites are: for η ∈ [0, +∞), or its time reversal for η ∈ (−∞, +∞), or its time reversal for η ∈ (−∞, +∞). For B < A the solution again takes the form with Hubble parameter H ≡ (1/a)(da/dt), density ratios Ω V ≡ ρ V /ρ tot , etc., and subscript 0 again denoting present-time values. This can be integrated: to display the behavior a(t) of the model. In Fig. 2 an open tachyon-dominated model given by Eqs. to accelerated expansion, the so-called "cosmic jerk." At late times the ΛCDM shows the exponential expansion associated with vacuum-energy domination; this model approaches de Sitter space asymptotically. In constrast, the tachyon-dominated model shows the linear behavior a(t) ∼ t at late times; the model approaches the Milne [5] version of flat spacetime.

V. CONCLUSIONS
It is possible to construct straightforward Einsteinian dynamics and thermodynamics for tachyons. Tachyons subject to electromagnetic forces behave exactly as expected, although the existence of charged tachyons may raise several conundra. Kinetic theory for a gas of "dark" tachyons implies an equation of state similar in form to that for a gas of bradyons, save that the invariant-mass density term appears with positive rather than negative sign, corresponding to much greater pressure in a tachyon gas than in a bradyon gas of the same energy density and temperature.
For a constituent of a Friedmann-Roberston-Walker spacetime, the tachyonic equation of state implies a density which varies with scale factor in a manner similar to that of a bradyon gas, but again with invariant-mass density contributions of the opposite sign. As a dominant source in such a spacetime, this density can give rise to cosmic evolution similar-but not identical-to that in models containing ordinary bradyonic matter and radiation. An open (hyperbolic) tachyon-dominated model can even evolve with decelerating, then accelerating expansion, similar-but not( identical-to that of the "standard model" containing bradyonic matter, radiation, and "dark energy." Whether a tachyon-dominated cosmological model is competitive with the standard model depends on whether detailed features of the tachyonic model can be matched to the extensive array of astronomical observations on cosmological scales, including supernova distances, the cosmic microwave background, and large-scale structure. Such a project is the subject of several ongoing works [6], [7], [8].
As noted, the evolution of a tachyon-dominated model can be similar but not identical to that of the standard model: The latter continues to accelerate exponentially in time, while the former asymptotically approaches expansion at a constant rate. This difference might even affect the dynamics of elementary-particle fields [9], making it possible to test the two models via experiments at the scale of such particles.
Of course, the actual existence of tachyons would raise many other questions. A proper, quantum-field-theoretic description of tachyons would involve a number of subtleties, such as tachyon "spin," statistics, and interactions with other quantum fields. But such a possibility might not be quite as far fetched as it is usually regarded.