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Galaxies 2014, 2(1), 72-80; doi:10.3390/galaxies2010072
Abstract: Dark energy with negative pressure and positive energy density is believed to be responsible for the accelerated expansion of the universe. Quite a few theoretical models of dark energy are based on tachyonic fields interacting with itself and normal (bradyonic) matter. Here, we propose an experimental model of tachyonic dark energy based on hyperbolic metamaterials. Wave equation describing propagation of extraordinary light inside hyperbolic metamaterials exhibits 2 + 1 dimensional Lorentz symmetry. The role of time in the corresponding effective 3D Minkowski spacetime is played by the spatial coordinate aligned with the optical axis of the metamaterial. Nonlinear optical Kerr effect bends this spacetime resulting in effective gravitational force between extraordinary photons. We demonstrate that this model has a self-interacting tachyonic sector having negative effective pressure and positive effective energy density. Moreover, a composite multilayer SiC-Si hyperbolic metamaterial exhibits closely separated tachyonic and bradyonic sectors in the long wavelength infrared range. This system may be used as a laboratory model of inflation and late time acceleration of the universe.
Recent observational data have revealed accelerated expansion of the universe which cannot be explained by gravitational dynamics of ordinary matter. One of the possible explanations of these observations involves existence of a dark energy with negative pressure and positive energy density. It is assumed that gravitational repulsion due to dark energy accelerates present day expansion of the universe. Among various theoretical models of dark energy proposed so far, such as cosmological constant , quintessence , etc., tachyon-based models play a very prominent role [3,4,5,6,7]. A typically used relativistic tachyonic Lagrangian:
Our proposal is built on the recently developed hyperbolic metamaterial-based model of 2 + 1 dimensional gravity . It appears that wave equation describing propagation of extraordinary light inside hyperbolic metamaterials exhibits 2 + 1 dimensional Lorentz symmetry . The role of time in the corresponding effective 3D Minkowski spacetime is played by the spatial coordinate aligned with the optical axis of the metamaterial . Nonlinear optical Kerr effect “bends” this spacetime resulting in effective gravitational interaction between extraordinary photons. In order for the effective gravitational constant to be positive, negative self-defocusing Kerr medium must be used . These results are quite interesting given the fact that physical vacuum itself may behave as a hyperbolic metamaterial when subjected to very strong magnetic field [11,12]. Moreover, negative self-defocusing Kerr effect typically arises due to thermal expansion of the medium, which makes the effective 2 + 1 dimensional gravity a thermal effect obeying basic laws of thermodynamics. This feature of our model appears to be very attractive since recent theoretical developments strongly indicate thermodynamic origins of gravitational interaction [13,14].
2. Results and Discussion
Before we proceed to a model of interacting tachyonic fields, let us recall basic properties of hyperbolic metamaterials and their description using effective 2 + 1 dimensional Minkowski spacetime. Recent advances in electromagnetic metamaterials enable design of novel physical systems which can be described by effective space-times having very unusual metric and topological properties . In particular, hyperbolic metamaterials (see Figure 1) offer an interesting experimental window into physics of Minkowski spacetimes, since propagation of extraordinary light inside a hyperbolic metamaterial is described by wave equation exhibiting 2 + 1 dimensional Lorentz symmetry. A detailed derivation of this result can be found in [9,10]. Assuming that the metamaterial in question is uniaxial and non-magnetic, electromagnetic field inside the metamaterial may be separated into ordinary and extraordinary waves: vector of the extraordinary light wave is parallel to the plane defined by the k–vector of the wave and the optical axis of the metamaterial. In the frequency domain (in some frequency band around ω = ω0) the metamaterial may be described by anisotropic dielectric tensor having opposite signs of the diagonal components εxx = εyy = ε1 and εzz = ε2, while all the non-diagonal components are assumed to be zero in the linear optics limit. Propagation of extraordinary light in such a metamaterial may be described by a coordinate-dependent wave function φω = Ez obeying the following wave equation [9,10]:
Detailed analysis performed in  indicates that nonlinear corrections to ε1 due to Kerr effect lead to effective gravitational interaction between the extraordinary photons, and the sign of the third order nonlinear susceptibility χ(3) of the hyperbolic metamaterial must be negative for the effective gravity to be attractive. It is also interesting to note that in the strong gravitational field limit this model contains 2 + 1 dimensional black hole analogs in the form of subwalength solitons .
Let us analyze how the basic framework outlined above can be extended to the tachyonic case. Very recently it has been noted  that depending on the frequency range and materials used, extraordinary photons in both hyperbolic metamaterial configurations shown in Figure 1a,b may exhibit a tachyonic dispersion relation. Let us analyze solutions of Equation (4) in the case where ε1 < 0 while ε2 > 0. It is clear that extraordinary photon propagation through such a metamaterial may still be described using an effective 2 + 1 dimensional Minkowski spacetime. However, the effective metric coefficients gik of this spacetime change to g00 = ε1 and g11 = g22 = ε2, and the dispersion law of extraordinary photons changes to:
The contributions to σik which are made by a single extraordinary plane wave propagating inside the hyperbolic metamaterial may be calculated similar to . Assuming without a loss of generality that the B field of the wave is oriented along y direction, the other field components may be found from Maxwell equations as:
Taking into account the dispersion law Equation (11) of the extraordinary wave, the contributions to σzz and σxx from a single plane wave are:
Similar to , nonlinear optical Kerr effect leads to gravity-like self-interaction of the tachyonic field. Taking into account that g00 = ε1, the Einstein Equation (8) translates into:
This assumption has to be the case if extraordinary photons may be considered as classic “particles”. Equation (17) establishes connection between the effective gravitational constant γ* and the third order nonlinear susceptibility χ(3) of the hyperbolic metamaterial. Similar to the “bradyonic case” considered in , the sign of χ(3) must be negative for the effective gravity to be attractive (since ε2 > 0). Since most liquids exhibit large and negative thermo-optic coefficient resulting in large and negative χ(3), and there exist readily available ferrofluid-based hyperbolic metamaterials , laboratory experiments with gravitationally self-interacting tachyonic fields appear to be realistic in the near future. Moreover, as we will demonstrate below, even more curious case of coexisting mutually-interacting tachyonic and bradyonic fields seems to be no more difficult to realize. Since gravitational dynamics of mutually interacting tachyonic and bradyonic fields may have contributed to inflation and late time acceleration of our universe [3,5], such experiments would be very interesting.
We should also point out that ε and χ(3) tensors of the metamaterial do not need to stay coordinate independent. Spatial behavior of the dielectric permittivity tensor components may be engineered so that the background metric may closely emulate metric of the universe during inflation . On the other hand, engineered higher order nonlinear susceptibility terms χ(n) may be used to emulate the desired functional form of the tachyonic potential V(ϕ) (see Equations (1–3)). As a result, various scenarios of tachyonic inflation  will become amenable to direct experimental testing.
In conclusion, we have demonstrated that extraordinary photons in a composite multilayer SiC-Si hyperbolic metamaterial exhibit closely separated tachyonic and bradyonic frequency bands around λ = 11 μm. Nonlinear optical Kerr effect leads to effective gravitational interaction of photons in these bands. This interaction may be used to study gravitational dynamics of tachyonic and bradyonic fields, which is responsible for inflation and late time acceleration of the universe in the tachyonic models of dark energy. While metamaterial losses constitute an important performance-limiting issue for this model, loss compensation using gain media  is known to be able to overcome this problem. In our particular case loss compensation is simplified by the fact that in our metamaterial design the tachyonic and the bradyonic bands are located very close to each other.
Conflicts of Interest
The author declares no conflict of interest.
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