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Abstract
The particle horizon represents the boundary between observable and unobservable regions of the universe, which changes as the universe expands. Based on transformation optics, hyperbolic electromagnetic metamaterials can be utilized to simulate metrics with different signs due to their unique anisotropic properties. In this paper, we use hyperbolic metamaterials to visually depict the variation of the particle horizon under three models of an expanding universe (open, flat, and closed) by substituting one-dimensional time with one-dimensional space. The good agreement between theory and simulation confirms that hyperbolic metamaterials are excellent for simulating space-times, suggesting their potential as a new platform for cosmological analogies.
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1. Introduction
Einstein’s general relativity [1] established the equivalence between the energy tensor and the curvature of space-time, revealing the geometry characteristic of space-times. The simulation of propagation of light in curved space is thus meaningful and has always been a research hot spot. In optics, the emergence of transformation optics (TO) [2–4], which constructs an equivalent connection between the metric tensor and the electromagnetic material parameters, such as the permittivity and permeability tensors, has facilitated the production of numerous optical metamaterials to mimic astronomical phenomena. For example, the optical black hole was proposed [5] and the first optical black hole in microwave frequencies was experimentally demonstrated [6]. The equivalent permittivity tensor of Schwarzschild black hole was theoretically derived and simulated [7]. The cosmological redshift can also be obtained by using the equivalent time-dependent material characteristics [8]. The gravitational lensing effect was experimentally visualized [9]. A generalized analytical formalism for developing analogs of spherically symmetric static black holes was discussed [10]. Lately, a new way was proposed [11] by utilizing the TO theory to directly obtain the equivalent isotropic refractive index profiles of a class of static curved spaces, and has further been used to study the de Sitter space with a generalized Poincaré lens [12]. The above simulations and experiments have visualized large-scale astronomical phenomena in front of us and provided a new perspective for our understanding of fundamental physics.
Recently, hyperbolic metamaterials have been extensively investigated due to their unique property that the components of the permittivity tensor along different principal axes have different signs [13,14]. This material property leads to hyperbolic dispersion, which in principle can support infinitely large propagating momentum, giving rise to a great many applications including negative refraction, enhanced spontaneous emission, and hyperlens [15–17]. Moreover, they were also found useful in modeling cosmic effects [18–21]. The pioneering work focuses on metric signature transition [18], which proposes that hyperbolic metamaterials can be utilized to mimic two-time metrics and reveals the possibility of a spatial coordinate becoming time-like.
A further application was the analogy of expanding universe. The universe began with the big-bang and then experienced rapid expansion, and many researchers were dedicated to simulate this process. The plasmonic hyperbolic metamaterials were experimentally constructed to emulates a big-bang-like event [19], and then this model was extended to mimic the cosmological inflation where the spatial dispersion results in scale-dependent structure replicating hypothesized fractal structure of the real observable universe [21]. However, the variation of particle horizon in different space-times, especially in the expanding universes remains elusive and is less performed in hyperbolic metamaterials. In this paper, we directly replace the time aixs with one of the spatial axes and simulate the variation of particle horizon in the expanding universe with hyperbolic metamaterials within three different cosmological models.
2. Theory
The theory of transformation optics [4] demonstrates the relationship between space-time metric and the material parameters of the transformed medium:
where $\varepsilon ^{ij}$ and $\mu ^{ij}$ are the permittivity and permeability of the transformed medium, respectively. $g_{ij}$ represents the Robertson-Walker metric, and $g=det(g_{ij})$ is the determinant of $g_{ij}$. Next, we introduce the expression of the Robertson-Walker metric and derive the material parameters corresponding to this metric.2.1 Scale factor
The homogeneous and isotropic universe is described by the Robertson-Walker metric (RW), which written in spherical coordinates ($r, \theta, \varphi$) is
The scale factor $a(t)$ is a function of $t$ and always greater than zero ($a(t) > 0$) in an expanding universe, which is the main influence factor and can be obtained from the Friedamnn equations (obtained from Einstein’s equations):
The extreme case of a radiation-dominated universe, where energy and momentum are solely contributed by radiation, is considered to solve $a(t)$. In the radiation-dominated universe, the pressure $p$ and density $\rho$ obey the relationship $p=\rho /3$, substituting which into Eq. (4) and integrating over time we can derive
where $c_1$ is a constant. By substituting Eq. (5) in Eq. (3), we can finally derive the specific expression of the scale factor in the three cases of spatial curvature. where $B=8\pi c_1 /3$ is also a constant and here we set it as 1 without loss of generality. The variation of $a(t)$ in three different cases is shown in Fig. 1(a), where the temporal axis is replaced by $y$ axis.
Fig. 1. (a) The variation of the scale factor with $y$. The green, red, and blue lines correspond to $\kappa$ equals -1, 0, and 1, respectively. (b-d) The variation of particle horizon in three cosmology models ($\kappa =0,-1$ and $1$). In each plot, dashed, dotted, and dot-dashed lines indicate that the point source is placed at (0, 0.02), (0, 0.5), and (0, 1), respectively.
2.2 Equivalent hyperbolic material parameters
We now project the spatial portion of the metric onto the equatorial plane, which means $\theta =\pi /2$, thus the Robertson-Walker metric is simplified into
where $r^2=x^2+y^2$. The evolution of the universe is the same along the $x$-axis and $y$-axis because the universe is spatially homogeneous and isotropic, we only consider the effects contain one spatial dimension and one temporal dimension and replace t-axis with $y$-axis, thus the metric can be simply expressed in cartesian coordinates as where we use natural unit and set $c = 1$. In order to obtain the effective material tensor of three order, another spatial dimension is also required. Here, we add a z-axis as the additional spatial coordinate, and rewrite the line element of the RW metric as where the metric $g_{ij}$, the inverse metric $g^{ij}$ and the determinant $g=det(g_{ij})$ are obtained asAccording to Eq. (1), the permittivity and permeability tensors can be calculated as
Equation (13) shows that all tensor components are gradient with respect to spatial coordinates, which are parameters of a highly complicated metamaterial. Since we consider the 2D TM polarization in our study, where the magnetic field is polarized along z-axis ($H_z$), only the parameters $\varepsilon ^{xx}, \varepsilon ^{yy}$ and $\mu ^{zz}$ are dominant. Besides, the non-magnetic condition ($\mu ^{zz}=1$) should be required if the actually practical implementation is taken into account [3], thus the reduced material parameters can be obtained as
2.3 Geodesic equation
The ray trajectories in curved spacetimes can be determined by solving geodesic equations, which generally take the following form
3. Simulations and results
Now we demonstrate the change of particle horizon with time in the process of the expanding universe by using hyperbolic metamaterials. The commercial software Comsol Multiphysics is adopted for the simulations. In this paper, the point sources of TM wave are located at (0, 0.02), (0, 0.5), (0, 1), where the coordinates are in units of meters for instance. The wave optical results are shown in the upper panel of Fig. 2-4. Because perfectly matched layer (PML) in Comsol Multiphysics is not valid for simulations of hyperbolic materials, where massive reflection and scatterings will be produced [22], we add additional losses to the equivalent hyperbolic materials ($\varepsilon ^{xx}=Re(\varepsilon ^{xx}+0.03i),~\varepsilon ^{yy}=Re(\varepsilon ^{yy}+0.03i),~\mu ^{zz}=1$), which is similar to the practical metamaterials and natural van der Waals (vdW) materials [13,14].
Fig. 2. The field patterns (a-c) and the ray trajectories (d-f) in the hyperbolic metamaterials of open universe ($\kappa =-1$). The point sources are located at (0, 0.02), (0, 0.5) and (0, 1) in (a,d), (b,e), and (c,f), respectively.
The ray trajectories are determined by solving by geodesic equations. For example, in flat spacetime, the metric components are $g_{xx}=-2y,~g_{yy}=1,~g^{xx}=-1/2y,~g^{yy}=1$. According to Eq. (15), the Christoffel symbol can be obtained as: $\Gamma ^x_{xy}=\Gamma ^x_{yx}=1/2y,~\Gamma ^y_{xx}=1$, thus the geodesic equations are expressed as $\ddot {x}+\dot {x}\dot {y}/y=0,~\ddot {y}+\dot {x}\dot {y}=0$. Adding the light cone equation: $ds^2=0~(i.e.\ddot {y}-2y\ddot {x}=0)$, the world lines of the particles can be calculated as shown in the lower panel of Fig. 3. The other cases are calculated in the same way.
Fig. 3. The field patterns (a-c) and the ray trajectories (d-f) in hyperbolic mematerials for flat universe ($\kappa =0$). The point sources are respectively located on (0, 0.02), (0, 0.5) and (0, 1) in (a,d), (b,e) and (c,f), respectively.
In Fig. 1(a), we show the variation of the scale factor along $y$-axis in three cases of the universe models. The green, red, and blue lines correspond to $\kappa = -1,$ $0$, and $1$, respectively. The scale factor is monotonic increasing in $\kappa = -1$ and $0$, and increases first and then decreases at $\kappa =1$. The particle horizon varies accordingly under different universe models, as shown in Figs. 1(b-d). We theoretically calculate and display the variation of particle horizon with different $\kappa$, where the green, red, and blue lines are related to $\kappa$, where the green, red, and blue lines indicate $\kappa = -1$, $0$ and $1$, and the dashed, dotted, and dot-dashed lines denote that point source place at (0, 0.02), (0, 0.5), and (0, 1), respectively. Since the particle horizon is singular at ($x$, 0), we choose to start from (0, 0.02). It is observed that in cases where $\kappa = -1$ and $\kappa =0$, the particle horizon diminishes as the $y$ coordinate of the point source position increases, while the particle horizon is limited less than $x=1$ in case of $\kappa$ = 1.
Next, we visualize the universe models with three kinds of hyperbolic metamaterials in Eq. (13) and Eq. (14) from both geometrical and wave optical perspectives. In Fig. 2, we plot the $z$ component of the magnetic field patterns (Fig. 2(a-c)) and ray trajectories (Fig. 2(d-f)) in the open universe ($\kappa =-1$), where the point source is located at (0, 0.02), (0, 0.5), and (0, 1), respectively in Fig. 2(a, d), Fig. 2(b, e), and Fig. 2(c, f). We can observe from Fig. 2(a-c) that the magnetic fields are effectively confined, the edge of the fields shrinks with the increase of the position along $y$-axis and agrees well with the particle horizons in Fig. 2(d-f). The green dashed, dotted, and dot-dashed lines respectively denote the emitting rays in light cone with varying degrees of bending from different point sources. The singularity ($y=0$) is noteworthy both theoretically and in simulations as the beginning of the universe, where light will be emitted or halted. This property can be demonstrated intuitively from the equivalent material parameters in Eq (13) ($\varepsilon ^{xx}=\infty,~\varepsilon ^{yy}=0,~\mu ^{zz}=0$) obtained directly by TO, which are also singular at $y=0$, and can be verified by calculating the light rays from the geodesic equations, where the rays will infinitely approach the line $y=0$ but never get there. Although the changes of singularity in the reduced materials ($\varepsilon ^{xx}=1,~\varepsilon ^{yy}=0,~\mu ^{zz}=1$) allow field propagate through, there is a sharp boundary before and after the singularity in the field patterns, where obvious distinction of the waveforms can be seen. In addition, only the space-time after the singularity has physical significance, therefore the propagation properties of the waves and rays above $y=0$ are mainly concerned.
In Fig. 3 and Fig. 4, we show the fields simulation and light rays in the hyperbolic metamaterials of flat universe ($\kappa =0$) and closed universe ($\kappa =1$). We can observe that the field constraint effects and the trends of the world lines (red dashed, dotted, and dot-dashed lines) from different positions of point sources in flat universe (Fig. 3) resemble those in open universe (Fig. 2), with slight variations in bending degrees. What’s particular distinct is that the particle horizons are fully confined to a range less than $x=\pm 1$ in the closed universe, as illustrated by the field patterns in Fig. 4(a-c) and the ray trajectories in Fig. 4(d-f). Besides, it should be noted that the fields and rays are also restricted along $y$-axis at $y=0$, and $y=2$, where the former represents the discussed space-time singularity, while the latter serves as a boundary of the closed universe, indicating that extra boundaries occur and enclose the universe in a limited space-time. In fact, this phenomenon can be effectively presented from the expressions of the corresponding hyperbolic metamaterials ($\varepsilon ^{xx}=1, \varepsilon ^{yy}=-(2y-y^2)/(1-x^2 ), \mu ^{zz}=1$), where the value of $\varepsilon ^{yy}$ approaches zero when $y= 0$ and $2$. The blue dashed, dotted, and dot-dashed lines vividly depict the variation of the world lines and particle horizons.
Fig. 4. The field patterns (a-c) and the ray trajectories (d-f) in hyperbolic mematerials for close universe ($\kappa =1$). The point sources are respectively located on (0, 0.02), (0, 0.5) and (0, 1) in (a,d), (b,e) and (c,f), respectively.
4. Conclusion
In summary, based on the application of transformation optics in the RW metric, we use hyperbolic metamaterials to simulate the variation of particle horizons under three models of the expanding universes that dominated by radiation through replacing the time-axis with one space-axis, and compared the wave optical results with ray trajectories calculated by geodesic equations. The particle horizon shrinks as the point source position increases along $y$-axis in the case of open universe ($\kappa =-1$) and flat universe ($\kappa =0$). In the closed universe ($\kappa = 1$), it is limited to $x=\pm 1$ and $y=0$, $y=2$. These findings are effectively demonstrated using equivalent hyperbolic metamaterials.
The excellent properties of hyperbolic metamaterials in simulating space-time are revealed by the good agreement between geometrical and wave optical results. Our study provides a possibility to mimic the astronomical phenomenon related to horizon such as Bekenstein-Hawking radiation [23] and cosmological horizons radiation [24]. The equivalent hyperbolic metamaterials make it possible to practically implement such a large-scale astronomical phenomenon in such a small-scale device. Most recently, the hyperbolic responses have also been found in the 2D natural van de Waals materials, such as $\alpha$-phase molybdenum trioxide ($\alpha$–MoO$_3$) [25–27]. They support the propagation of natural hyperbolic phonon polaritons in mid-infrared band, and is immune from Ohmic loss as a result of the absence of electron–electron scattering. Thus, these materials possess a strong density of states and long lifetimes and offer tremendous opportunities for manipulation of light, such as visualizing the cosmological phenomena in nanoscale [27]. In the future, we hope to experimentally achieve the particle horizons of the expanding universes in these platforms.
Funding
National Natural Science Foundation of China (12173031, 12221003, 92050102); Jiangxi Provincial Natural Science Foundation (20224ACB201005); Fundamental Research Funds for the Central Universities (20720230102).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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