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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 [

Our proposal is built on the recently developed hyperbolic metamaterial-based model of 2 + 1 dimensional gravity [

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 [_{0}) the metamaterial may be described by anisotropic dielectric tensor having opposite signs of the diagonal components ε_{xx} = ε_{yy} = ε_{1} and ε_{zz} = ε_{2}, 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 φ_{ω} = _{z} obeying the following wave equation [_{1} > 0 while ε_{2} < 0, this wave equation coincides with the Klein-Gordon equation for a massive scalar field φ_{ω} in 3D Minkowski spacetime:
_{z}_{x}_{y})^{2}_{ik}_{00}_{1} and _{11}_{22}_{2}. This spacetime may be made “causal” by breaking the mirror and temporal symmetries of the metamaterial, which results in one-way light propagation along the timelike spatial coordinate [_{00}_{1}^{(2)} is the 2D Laplacian operating in the _{zz}

Typical geometries of hyperbolic metamaterials: (_{z} behaves as an effective “energy”; depending on frequency range and materials used, both configurations may exhibit either bradyonic or tachyonic dispersion relations shown in (

Detailed analysis performed in [_{1} 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 [_{1}_{2}_{ik}_{00} =_{1} and _{11} = g_{22} = _{2}, and the dispersion law of extraordinary photons changes to:
^{eff}_{ik}

The contributions to σ_{ik}

Taking into account the dispersion law Equation (11) of the extraordinary wave, the contributions to σ_{zz}_{xx}_{yy}^{eff}=^{2}/4

Similar to [_{00} =_{1}, the Einstein Equation (8) translates into:
_{1} are assumed to be small, so that we can separate ε_{1} into the constant background value ε_{1}^{(0)} and weak nonlinear corrections (note that similar to [^{(2)}_{ijl}

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 [^{(3)} must be negative for the effective gravity to be attractive (since ε_{2} > 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 [

Let us consider a multilayer metal-dielectric metamaterial shown in _{1} = _{xy}_{m}_{d}_{m}_{d}_{m} <_{d}_{1}_{2}_{2}, Al_{2}O_{3}, ^{(3)} metamaterial will be obtained resulting in positive γ

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 [^{(n)} may be used to emulate the desired functional form of the tachyonic potential

(

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 [

The author declares no conflict of interest.