# Metamaterial Model of Tachyonic Dark Energy

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

_{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}= E

_{z}obeying the following wave equation [9,10]:

_{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}plays the role of effective energy, while vector (k

_{x},k

_{y}) plays the role of momentum. The effective mass squared m*

^{2}appears to be positive. Note that components of metamaterial dielectric tensor define the effective metric g

_{ik}of this spacetime: g

_{00}= ε

_{1}and g

_{11}= g

_{22}= −ε

_{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 [16], while “gravitational bending” of the effective spacetime may lead to an experimental model of the big bang [10]. In the weak gravitational field limit the effective Einstein equation:

_{00}is identified with –ε

_{1}, Equation (8) must be translated as:

^{(2)}is the 2D Laplacian operating in the xy plane, γ* is the effective “gravitational constant”, and σ

_{zz}is the zz component of the Maxwell stress tensor of the electromagnetic field in the medium [18]:

**Figure 1.**Typical geometries of hyperbolic metamaterials: (

**a**) metal wire array structure, and (

**b**) multilayer metal-dielectric structure. The role of time in the effective 3D Minkowski spacetime is played by z coordinate aligned with the optical axis of the metamaterial, while k

_{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 (

**c**).

_{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 [8].

_{1}< 0 while ε

_{2}> 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 g

_{ik}of this spacetime change to g

_{00}= ε

_{1}and g

_{11}= g

_{22}= ε

_{2}, and the dispersion law of extraordinary photons changes to:

^{eff}

_{ik}which coincides with the Maxwell stress tensor:

_{ik}which are made by a single extraordinary plane wave propagating inside the hyperbolic metamaterial may be calculated similar to [8]. 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:

_{zz}and σ

_{xx}from a single plane wave are:

_{yy}= 0, leading to tachyonic negative TrT

^{eff}=−B

^{2}/4π.

_{00}= ε

_{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 [8], second order nonlinear susceptibilities χ

^{(2)}

_{ijl}of the metamaterial are assumed to be zero). These nonlinear corrections look like the Kerr effect assuming that the extraordinary photon wave vector components are large compared to ω/c:

^{(3)}of the hyperbolic metamaterial. Similar to the “bradyonic case” considered in [8], the sign of χ

^{(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 [20], 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.

_{1}= ε

_{xy}= nε

_{m}+ (1 − n)ε

_{d}

_{m}and ε

_{d}are the dielectric permittivities of the metal and dielectric phase, respectively. A suitable choice of n is known to lead to hyperbolic behavior in such metamaterials [21]. However, ordinary metals cannot be used in a hyperbolic metamaterial design having closely spaced tachyonic and bradyonic frequency bands due to their broadband metallic (ε

_{m}< 0) behavior. This difficulty may be overcome by using such materials as SiC, which have narrow Restsrahlen metallic bands in the long wavelength infrared (LWIR) range. Silicon carbide may be used in combination with such material as Si, which exhibits broadband dielectric behavior in the LWIR having ε

_{d}~ 10.9. Indeed, our calculations presented in Figure 2 indicate that around n ~ 0.3 a multilayer SiC-Si metamaterial does have pronounced closely spaced tachyonic and bradyonic frequency bands, while having relatively low losses. The calculated values of ε

_{1}and ε

_{2}are based on the measured optical properties of SiC reported in [22]. A somewhat similar result has been reported in [21] for a SiC-Si wire array structure, but unlike our multilayer design, fabrication of SiC-Si wire arrays may prove extremely difficult. We should also point out that such materials as SiO

_{2}, Al

_{2}O

_{3}, etc. also have pronounced Restsrahlen bands in the LWIR range, so that they can be used in the layered design with equal success. If native undoped Si is used in the metamaterial design, its nonlinearity will be dominated by negative thermo-optic coefficient due to thermal expansion. Thus, the desired negative χ

^{(3)}metamaterial will be obtained resulting in positive γ*.

^{(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 [23]. 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 [3] will become amenable to direct experimental testing.

**Figure 2.**(

**a**) Calculated diagonal components of the dielectric permittivity tensor of a multilayer SiC-Si metamaterial in the long wavelength infrared (LWIR) frequency range. The volume fraction of SiC equals n = 0.3. Two closely spaced tachyonic and bradyonic hyperbolic bands arise near λ = 11 μm; (

**b**) Calculated Im(ε)/|ε| for both directions indicate low loss hyperbolic behavior inside the bands.

## 3. Conclusions

## Conflicts of Interest

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Smolyaninov, I.I.
Metamaterial Model of Tachyonic Dark Energy. *Galaxies* **2014**, *2*, 72-80.
https://doi.org/10.3390/galaxies2010072

**AMA Style**

Smolyaninov II.
Metamaterial Model of Tachyonic Dark Energy. *Galaxies*. 2014; 2(1):72-80.
https://doi.org/10.3390/galaxies2010072

**Chicago/Turabian Style**

Smolyaninov, Igor I.
2014. "Metamaterial Model of Tachyonic Dark Energy" *Galaxies* 2, no. 1: 72-80.
https://doi.org/10.3390/galaxies2010072