# Correspondence between dark energy quantum cosmology and Maxwell equations

## Abstract

A Friedmann–Robertson–Walker cosmology with dark energy can be modelled using a quintessence field. That system is equivalent to a relativistic particle moving on a two-dimensional conformal spacetime. When the quintessence behaves as a free massless scalar field in a Universe with cosmological constant, the quantized version of that theory can lead to a supersymmetric Majorana quantum cosmology. The purpose of this work is to show that such quantum cosmological model corresponds to the Maxwell equations for electromagnetic waves propagating in a medium with specific values for its relative permittivity and relative permeability. The form of those media parameters are calculated, implying that a Majorana quantum cosmology can be studied in an analogue electromagnetic system.

## 1 Introduction

The aim of this work is to show that there exists a correspondence between the seemingly different physical models of a cosmological model using dark energy and Maxwell equations. The link between these two formalisms appears when one considers the quantum version of the cosmological model [1] using the Breit prescription [2] for spin particles.

The representation of dark energy using a quintessence field with a potential, allows us to describe the cosmological dynamics in fashion which is analogous to the description of the dynamics of a relativistic particle. In the case of quintessence described by a free massless scalar field in a Universe with cosmological constant, the resultant theory may be quantized by using a Klein–Gordon scheme, giving rise to the Wheeler–DeWitt equation [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. However, using the Breit prescription [2], the same model can be quantized as a spinorial theory. This procedure yields a Majorana version for the quantum cosmology which happens to be supersymmetric [1].

In addition of the cosmological implications of such theory, the aim of this article is to show its direct correspondence with the description of propagating electromagnetic fields in a medium using Maxwell equations. We can identify the relative permittivity and permeability of the medium with parameters of the quantum cosmological model. As we show, this implies that the supersymmetric Majorana quantum cosmology can be studied in an analogue electromagnetic system using either normal materials or negative-index metamaterials (NIMs) [14, 15].

## 2 Quantum cosmology with dark energy

*a*(

*t*) is the scale factor, and the curvature constant \(k=\pm 1,0\). The evolution of a FRW cosmology with cosmological constant \(\Lambda \), interacting with a quintessence (massless scalar) field \(\phi (x^\beta )\) characterized by a potential \({{\mathcal {V}}}(\phi )\), can be found by using Einstein equations [1] (\(8\pi G/c^4=1\), where

*G*is the gravitational constant and

*c*is the speed of light)

## 3 Correspondence to Maxwell equations

*B*and

*D*are much larger than the variation range of \(\lambda \), i.e., \(\lambda (\partial \Phi /\partial z)\ll (d\lambda /dz)\Phi \). Using these approximations, Maxwell equation (19) can be written in an approximated form as

There are materials with relative permittivity and relative permeability that can have the form (21). Composite ferrites [31, 32, 33, 34] can, under appropriated conditions, achieve almost matching permeability and permittivity values by shining radiation of different frequencies on the material. This implies that the above results can be tested in an analogue fashion using those materials. For example, for a spatially flat cosmology, the Maxwell equations and the Supersymmetric Majorana quantum cosmology coincide for \(\lambda (\xi )=\sqrt{-3 {V}/32} \exp (2\xi )\), with constant \(V<0\), and refraction index \(n(\xi )\approx \pm [1+3 V\exp (4\xi )/64]\). For this case, Maxwell equations can only describe a quantum cosmology for \(\xi \ll \ln (11/|V|)\). Furthermore, the boundary conditions for \(\Phi \) given in Eq. (18) (which depend on \(\Omega \)) should be suitable for simulating a quantum cosmology. For \(k=0\), the boundary conditions for \(\xi \rightarrow \infty \) (\(a\rightarrow \infty \)) can be established to obtain a vanishing magnetic field at infinity in one spatial dimension (see Ref. [1]).

Our proposal is in the same spirit than similar ones for analogue optical systems for quantum cosmologies [35, 36], and for gravity in general (see for example Refs. [37, 38, 39, 40]). However, our result establishes an “optical” analogue for a new kind of spinor quantized cosmological model. The proposed analogue electromagnetic media that correspond to the quantum cosmology is time-independent but space-dependent, which is an approach opposite to previous attempts [35]. Relations (21) are satisfied by certain tunable metamaterials [41, 42] and composite ferrites [31, 32, 33, 34] used to operate at a wider range of frequencies. All of the above makes of this Majorana Supersymetric quantum cosmological model a system worth to be studied by studying wave propagation in Maxwell equations in the appropriated media.

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