Electromagnetic, Gravitational, and Plasma-Related Perturbations of Locally Rotationally Symmetric Class II Spacetimes ^†

Abstract

We investigate electromagnetic, gravitational, and plasma-related perturbations to the first order on homogeneous and hypersurface orthogonal locally rotationally symmetric (LRS) class II spacetimes. Due to the anisotropic nature of the studied backgrounds, we are able to include a non-zero magnetic field to the zeroth order. As a result of this inclusion, we find interesting interactions between the electromagnetic and gravitational variables already of the first order in the perturbations. The equations governing these perturbations are found by using the Ricci identities, the Bianchi identities, Einstein’s field equations, Maxwell’s equations, particle conservation, and a form of energy-momentum conservation for the plasma components. Using a $1+1+2$ covariant split of spacetime, the studied quantities and equations are decomposed with respect to the preferred directions on the background spacetimes. After linearizing the decomposed equations around an LRS background, performing a harmonic decomposition, and imposing the cold magnetohydrodynamic (MHD) limit with a finite electrical resistivity, the system is then reduced to a set of ordinary differential equations in time and some constraints. On solving for some of the harmonic coefficients in terms of the others, the system is found to decouple into two closed and independent subsectors. Through numerical calculations, we then observe some mechanisms for generating magnetic field perturbations, showing some traits similar to previous works using Friedmann–Lemaître–Robertson–Walker (FLRW) backgrounds. Furthermore, beat-like patterns are observed in the short wave length limit due to interference between gravitational waves and plasmonic modes.

1. Introduction

Using the standard model of cosmology, based on the homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes, most astronomical observations on a cosmological scale can be explained. However, some unsolved mysteries remain. One of these mysteries pertains to the origin of the large-scale magnetic fields that can be observed on the level of galaxies and clusters of galaxies. Although several mechanisms have been proposed for generating and amplifying these magnetic fields, the topic has seemingly been rather controversial [1]. Amidst the controversy, a common explanation appears to be that the magnetic fields were originally created very early in the history of the universe. It is then suggested that these primordial seed fields have been amplified later on through galactic dynamo mechanisms, creating the fields we observe today [1]. Although the galactic dynamo could greatly amplify the seeds, the proposed mechanisms that generate the seeds may be too weak, or, in some cases, lie uncomfortably close to the minimum strength requirement. In addition to the strength requirement, there are also stringent requirements on the coherence length of the seeds, further limiting the possible candidates1.

To increase the number of possible candidates, one way around the stringent requirements would be if some mechanism existed for amplifying the seeds after they have been created, but acting before the galactic dynamo starts. Some possible mechanisms of this kind have already been proposed. Examples include mechanisms based on gravitational wave interactions [2,3,4] and velocity perturbations in the cosmological plasma [4]. Although some of the mechanisms for amplifying the seed fields have been suggested to be too weak [5,6,7], the prospect of a mechanism based on the dynamics of a self-gravitating plasma and classical general relativity is still enticing, since it would not explicitly depend on any additional exotic physics. Therefore, it is still of great interest to study the interactions between plasma and general relativistic effects.

The studies of self-gravitating plasmas in a cosmological scenario seem to have mainly used a perturbative approach with the FLRW spacetimes as zeroth order backgrounds [2,3,4,5,6,7,8]. However, due to the isotropy of the FLRW spacetimes, it is essentially impossible to have a non-zero magnetic field 3-vector in the background, since that field would otherwise define a preferred spatial direction. To circumvent this problem, several different approaches have been used. One approach, which is usually referred to as the weak-field approximation, is to assume that the background magnetic field is sufficiently small so that it does not contribute to the energy-momentum tensor, which involves squares of the magnetic field [2]. With this assumption, the magnetic field should not affect the underlying geometry, and hence, in a sense, not violate the isotropy of the background spacetime. However, this approach runs into some formal problems, since the generated magnetic field perturbations that are explicitly studied are not gauge invariant in the strict mathematical sense of the Stewart–Walker lemma [3,9].

To better avoid interference from gauge related degrees of freedom, second-order schemes have also been used. The magnetic field is then introduced as a first-order perturbation rather than in the background [3,4,7]. The interaction between the gravitational effects and the electromagnetic fields then becomes a second-order quantity, since this involves products of the electromagnetic fields and other first-order variables. However, some care is still needed, as it is not strictly correct to integrate the second-order results for the gauge-invariant variables to obtain explicit expressions for the generated magnetic field, since this quantity is not gauge invariant [7]. Hence, one is restricted to interpreting gauge-invariant quantities to draw any conclusion about the generated magnetic fields [7].

Finally, a more drastic approach, which is the one that will be used here, is to use an anisotropic background model instead of the FLRW spacetimes. Then there is no longer any isotropy problem, and a non-zero magnetic field of the zeroth order is allowed without causing severe problems with the background geometry. In this paper, the aim is therefore to investigate if it is possible to get interesting gauge-invariant interactions already of the first order in the perturbations if we use a certain class of anisotropic backgrounds instead of the FLRW spacetimes. More specifically, we will consider backgrounds that are homogeneous, hypersurface orthogonal, and belong to the locally rotationally symmetric (LRS) class II of spacetimes, which allow for a preferred spatial direction.

First-order perturbations on LRS class II spacetimes have been studied previously, both in the perfect fluid case [10,11,12,13] and when including dissipative effects [14]. However, electromagnetic perturbations on general LRS class II spacetimes have, to our knowledge, mainly been considered as test fields, without any electromagnetic fields on the background [15,16]. Hence, it is still interesting to investigate electromagnetic perturbations on LRS class II spacetimes when including more interactions with gravitational and matter-related perturbations.

Our approach to studying these interactions will follow the same steps as in [14], but instead of specializing the equations to describe one-component dissipative fluids, we will consider self-gravitating plasmas. Thus, we will make use of the results for general energy-momentum tensors presented in [14], but we will then close the general system by imposing Maxwell’s equations, particle conservation, and equations describing energy-momentum conservation for the plasma components. In doing so, this article reproduces the methods and equations from [17] and an upcoming contribution to the proceedings from the sixteenth Marcel Grossmann meeting on general relativity [18], but extends these accounts with additional numerical results and discussions.

As the backgrounds are here assumed to be homogeneous, so will the zeroth order magnetic field be by construction. The existence of magnetic fields of this kind, defining a preferred spatial direction, in the early universe is constrained by observations of the CMB, as their inclusion would lead to modifications of the widely accepted inflationary predictions. The upper limits for the magnetic field strength derived from the CMB observations depend on the nature of free streaming particles, but is generally stated to be of the order of ${10}^{-9}$ Gauss when redshifted to today, corresponding roughly to ${10}^{-7}$ of the CMB energy density (see [19,20,21] and references therein). However, even if the zeroth order magnetic field is constrained to be rather small, this should not cause any fundamental consistency problems with the formalism used here, provided that the magnitude of the perturbations is kept sufficiently small.

More severely, with the current construction, the background magnetic field cannot readily be used to represent stochastic seed fields with non-trivial power spectra, which are often found from primordial magnetogenesis scenarios (see [19,20] and references therein). Hence, the formalism used here may not be the one that is best suited for studying every aspect of the magnetogenesis problem, but could still give some insights regarding the interactions between electromagnetic, gravitational, and plasma-related perturbations. The approximation of homogenous magnetic fields have in fact already been used in the context of amplifying primordial seeds, although at first-order in a second-order scheme [3]. More recently, a stable mechanism for generating homogeneous magnetic fields in the early universe has also been proposed and investigated [22,23,24]. Thus, it is still relevant to consider homogeneous magnetic fields in a cosmological context. In addition to the cosmological scenario, the formalism developed here could also possibly be used on slightly smaller, but still very large, scales to study magnetized plasmas, where large background magnetic fields could be more plausible.

Ultimately, the aim and main contribution of this article is not to provide precise numerical estimates, but rather to highlight how the present formalism can be used to study interactions between electromagnetic, gravitational, and plasma related perturbations. What is of great interest is then how the results from this formalism differ qualitatively from previous works in this context, such as [2,3,4,7].

2. Spacetime Dynamics

To determine the behavior of both the zeroth-order background and the first-order perturbations, we will follow [25] and make use of the Ricci identities, the Bianchi identities, and Einstein’s field equations. In doing so, we will assume that the studied spacetimes have two preferred directions as specified by the vector fields $u{}^a$ and $n{}^a$. The field $u{}^a$ is assumed to be timelike and normalized as $u{}^{a}u{}_a=-1$, whilst $n{}^a$ is spacelike with $n{}^{a}n{}_a=1$2. As for the physical interpretation of these directions, $u{}^a$ denotes the 4-velocity of some fundamental observer, whilst $n{}^a$ defines the direction with respect to which the backgrounds are assumed to be locally rotationally symmetric. It is then natural to make use of the Ricci identities for these preferred vector fields,

$$\begin{array}{cc}\hfill 2\nabla {}_{[a}\nabla {}_{b]}u{}_c-R{}_{abc}du{}^d& =0,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill 2\nabla {}_{[a}\nabla {}_{b]}n{}_c-R{}_{abc}dn{}^d& =0.\hfill \end{array}$$

(2)

Together with the contracted Bianchi identities

$$\begin{array}{cc}\hfill 2\nabla {}_{[e}R{}_{d]}{}_b+\nabla {}_a{R}^a{}_{bde}& =0,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \nabla {}^a{G}_{ab}& =0,\hfill \end{array}$$

(4)

and Einstein’s field equations

$$G_{ab}=T_{ab}-\Lambda g_{ab},$$

(5)

these relations provide most of the necessary equations. Here $G_{ab}\equiv R_{ab}-g_{ab}R/2$ is the usual Einstein tensor, whilst $T_{ab}$ is the energy-momentum tensor, which could be general at this point. However, with a general energy-momentum tensor, these equations will not be enough to get a closed system [14]. Therefore, to arrive at a closed system, we will here specialize the energy-momentum tensor to describe a plasma together with some electromagnetic fields. Hence we assume that

$$T^{ab}={T_{\left(F\right)}}^{ab}+{T_{\left(EM\right)}}^{ab},$$

(6)

where ${T_{\left(F\right)}}^{ab}$ is due to the plasma, whilst ${T_{\left(EM\right)}}^{ab}$ comes from the macroscopic electromagnetic fields. The electromagnetic part can, in turn, be written as

$${T_{\left(EM\right)}}^{ab}={F^a}_c{F}^{bc}-\frac{1}{4}{g}^{ab}{F}^{cd}{F}_{cd},$$

(7)

in terms of the Faraday tensor $F_{ab}$, which satisfies the usual Maxwell equations

$$\begin{array}{cc}\hfill \nabla {}_{[a}{F}_{bc]}& =0,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \nabla {}_b{F}^{ab}& =j{}^a,\hfill \end{array}$$

(9)

where $j{}^a$ is the 4-current density.

As for the plasma contribution, we will follow the formalism in [26] and assume that this contribution consists of separate contributions from each plasma component, so that

$${T_{\left(F\right)}}^{ab}=\sum _i{T_{\left(i\right)}}^{ab},$$

(10)

where the ith component satisfies the following energy-momentum conservation equation

$$\begin{array}{cc}\hfill \nabla {}_b{T_{\left(i\right)}}^{ab}& =F^{ab}{j}_{\left(i\right)}{}_b+I_{\left(i\right)}{}^a,\hfill \end{array}$$

(11)

with

$$\sum _i{I}_{\left(i\right)}{}^a=0.$$

(12)

Here, $I_{\left(i\right)}{}^a$ includes interactions such as collisions between the different fluid components, whilst $j_{\left(i\right)}{}^a$ is the 4-current density of the ith component. The current density satisfies the relations

$$\begin{array}{cc}\hfill j_{\left(i\right)}{}^a& ={\rho }_{\left(i\right)}{u}_{\left(i\right)}{}^a,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill j{}^a& =\sum _i{j}_{\left(i\right)}{}^a,\hfill \end{array}$$

(14)

where ${\rho }_{\left(i\right)}=q_{c\left(i\right)}{\mathcal{N}}_{\left(i\right)}$ is the charge density, $q_{c\left(i\right)}$ is the charge, ${\mathcal{N}}_{\left(i\right)}$ is the number density, and $u_{\left(i\right)}{}^a$ is the the 4-velocity of the ith component. In the following, we will also write the interaction terms and the total current density as

$$\begin{array}{cc}\hfill I_{\left(i\right)}{}^a& ={\epsilon }_{\left(i\right)}u{}^a+f_{\left(i\right)}{}^a,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill j{}^a& =\rho u{}^a+J{}^a,\hfill \end{array}$$

(16)

where $J{}^a$ and $f_{\left(i\right)}{}^a$ are orthogonal to $u{}^a$. Furthermore, we will assume that each fluid component can be described as a perfect fluid, so that

$$T_{\left(i\right)}{}^{\mathit{ab}}=\left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right)u_{\left(i\right)}{}^a{u}_{\left(i\right)}{}^b+p_{\left(i\right)}{g}^{ab},$$

(17)

where ${\mu }_{\left(i\right)}$ and $p_{\left(i\right)}$ are the energy density and pressure relative to the component rest frame, as defined by the 4-velocity $u_{\left(i\right)}{}^a$. This 4-velocity can in turn be written as

$$u_{\left(i\right)}{}^a={\gamma }_{\left(i\right)}\left(u{}^a+v_{\left(i\right)}{}^a\right),$$

(18)

where

$$u{}^a{v}_{\left(i\right)}{}_a=0,{\gamma }_{\left(i\right)}=\frac{1}{\sqrt{1-{v_{\left(i\right)}}^2}},{v_{\left(i\right)}}^2\equiv v_{\left(i\right)}{}_a{v}_{\left(i\right)}{}^a.$$

(19)

We then also impose particle conservation for each fluid component by requiring that

$$\begin{array}{cc}\hfill \nabla {}_a\left({\mathcal{N}}_{\left(i\right)}{u}_{\left(i\right)}{}^a\right)& =0.\hfill \end{array}$$

(20)

Finally, after specifying a microscopic description of $I_{\left(i\right)}{}^a$, equations of state $p_{\left(i\right)}=p_{\left(i\right)}({\mu }_{\left(i\right)},{\mathcal{N}}_{\left(i\right)})$, and choosing a specific frame, we now have enough equations to obtain a closed system. To obtain a simple, yet still informative, system, we will later on assume that the plasma is sufficiently cold so that $p_{\left(i\right)}=0$. Then we will also assume that the interactions between the fluid components can be described with a single scalar electrical resistivity, $\eta$, in the magnetohydrodynamic (MHD) approximation. In this approximation, the plasma is described in terms of some collective variables instead of the individual component variables. Therefore, after linearizing and harmonically decomposing the equations, we will write them in terms of the harmonic coefficients of the following total fluid energy density and average 3-velocity

$$\begin{array}{cc}\hfill {\mu }_{\left(F\right)}& \equiv \sum _i{\mu }_{\left(i\right)},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\mu }_{\left(F\right)}{v}_{\left(F\right)}{}^a& \equiv \sum _i{\mu }_{\left(i\right)}{v}_{\left(i\right)}{}^a.\hfill \end{array}$$

(22)

Using Ohm’s law, the total 3-current density can then be written as

$$J{}^a=\frac{1}{\eta }\left(E{}^a+ϵ{}^{\mathit{abc}}{v}_{\left(F\right)}{}_{b}B{}_c\right),$$

(23)

where $E{}^a$ and $B{}^a$ are the electric and magnetic fields defined through

$$F{}_{a}b=ϵ{}_{\mathit{abc}}B{}^c+2u{}_{[a}E{}_{b]},$$

(24)

with $ϵ{}_{\mathit{abc}}\equiv \eta {}_{\mathit{dabc}}u{}^d\equiv 4!\sqrt{-g}\delta {}_{[d}^0\delta {}_a^1\delta {}_b^2\delta {}_{c]}^{3}u{}^d$. However, the cold MHD assumption will only be imposed after linearizing and harmonically decomposing the equations. Until then, we will continue using the more general multifluid description with non-zero $p_{\left(i\right)}$ and with interactions and current densities determined by Equations (12)–(16).

3. Covariant Splits of Spacetime

As the studied spacetimes are assumed to have two preferred directions, it is natural to decompose the studied quantities with respect to those directions. Since the vector fields defining these directions are physical objects, the decompositions respect the tensorial properties of the decomposed variables. Hence they are usually referred to as covariant splits of spacetime.

3.1. 1 + 3

Starting with a $1+3$ split with respect to the timelike vector field $u{}^a$, we write the metric as

$$g{}_{\mathit{ab}}=U{}_{\mathit{ab}}+h{}_{\mathit{ab}},$$

(25)

where $U{}_{\mathit{ab}}=-u{}_{a}u{}_b$ is a projection tensor along $u{}^a$ whilst $h{}_{\mathit{ab}}$ projects onto the 3-dimensional hypersurfaces orthogonal to $u{}^a$ [27]. Using Equation (25), all relevant objects can then be decomposed relative to $u{}_a$ and the 3-dimensional hypersurfaces. Some of the quantities from Section 2 have already been decomposed in this manner, such as the component 4-velocities, the total current density, and the Faraday tensor. However, it remains to fully decompose the total energy momentum tensor and the Riemann tensor. Performing this decomposition, we encounter the following projected variables

$$\begin{array}{cc}\hfill \mu & \equiv T{}^{\mathit{cd}}u{}_{c}u{}_d,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill q{}_a& \equiv -T{}^{\mathit{cd}}u{}_{c}h{}_{\mathit{da}},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill p& \equiv \frac{1}{3}T{}^{\mathit{cd}}h{}_{\mathit{cd}},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \pi {}_{\mathit{ab}}& \equiv \left(h{{}_{(a}}^{c}h{{}_{b)}}^d-\frac{1}{3}h{}^{\mathit{cd}}h{}_{\mathit{ab}}\right)T{}_{\mathit{cd}}\equiv T{}_{\langle ab\rangle },\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill E{}_{\mathit{ab}}& \equiv C{}_{\mathit{abcd}}u{}^{c}u{}^d,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill H{}_{\mathit{ab}}& \equiv \frac{1}{2}{{ϵ}_a}^{\mathit{ef}}C{}_{\mathit{efbd}}u{}^d,\hfill \end{array}$$

(31)

where $\mu$ is the energy density, $q{}^a$ is the energy flux orthogonal to $u{}^a$, p is the isotropic pressure, and $\pi {}_{\mathit{ab}}$ is the trace-free anisotropic pressure, whilst $E{}_{\mathit{ab}}$ and $H{}_{\mathit{ab}}$ are the electric and magnetic parts of the Weyl tensor $C{}_{\mathit{abcd}}$ [27]. When defining the anisotropic pressure $\pi {}_{\mathit{ab}}$, we have introduced a notation with angular brackets to denote the projected, symmetric, and trace-free (PSTF) part of a tensor relative to the projection tensor $h{}_{\mathit{ab}}$.

We also obtain some new projected variables when considering the covariant derivative of the preferred vector field. Defining the time derivative

$$\dot{T}{}^{\mathit{ab}\dots }{}_{cd\dots }\equiv u{}^e\nabla {}_{e}T{}^{ab\dots }{}_{cd\dots },$$

(32)

and the spatial derivative fully projected onto the 3-dimensional hypersurfaces

$$D{}_{e}T{}^{ab\dots }{}_{cd\dots }\equiv {h^a}_f{h^b}_g\cdots {h_c}^h{h_d}^i\cdots {h_e}^j\nabla {}_{j}T{}^{fg\dots }{}_{hi\dots },$$

(33)

we can write the covariant derivative of $u{}^a$ as

$$\nabla {}_{a}u{}_b=-u{}_a\dot{u}{}_b+\frac{1}{3}\Theta h{}_{\mathit{ab}}+\omega {}_{\mathit{ab}}+\sigma {}_{\mathit{ab}}.$$

(34)

Here, $\Theta \equiv D{}^{a}u{}_a$ is the expansion rate, $\omega {}_{\mathit{ab}}\equiv D{}_{[a}u{}_{b]}$ is the vorticity, and $\sigma {}_{\mathit{ab}}\equiv D{}_{\langle a}u{}_{b\rangle }$ is the rate of shear [27]. Instead of working directly with the vorticity tensor, we will write it in terms of the vorticity vector

$$\omega {}^a\equiv \frac{1}{2}ϵ{}^{\mathit{abc}}\omega {}_{\mathit{bc}}.$$

(35)

3.2. 1 + 1 + 2

After having decomposed all relevant variables with respect to $u{}^a$, we can go one step further and also decompose the 3-dimensional hypersurfaces with respect to $n{}^a$, leading to a so-called $1+1+2$ covariant split [25]. For this purpose, we write the projection tensor $h{}_{\mathit{ab}}$ as

$$h{}_{\mathit{ab}}=N{}_{\mathit{ab}}+n{}_{a}n{}_b.$$

(36)

where $N{}_{\mathit{ab}}$ projects onto the 2-dimensional sheets orthogonal to both $u{}^a$ and $n{}^a$. Using Equation (36), the relevant 3-vectors can then be decomposed as

$$\begin{array}{cccc}\hfill q{}^a& =Qn{}^a+Q{}^a,\hfill & \hfill B{}^a& =\mathfrak{B}n{}^a+\mathfrak{B}{}^a,\hfill \end{array}$$

(37)

$$\begin{array}{cccc}\hfill \omega {}^a& =\Omega n{}^a+\Omega {}^a,\hfill & \hfill E{}^a& =\mathfrak{E}n{}^a+\mathfrak{E}{}^a,\hfill \end{array}$$

(38)

$$\begin{array}{cccc}\hfill \dot{u}{}^a& =\mathcal{A}n{}^a+\mathcal{A}{}^a,\hfill & \hfill J{}^a& =\mathcal{J}n{}^a+\mathcal{J}{}^a,\hfill \end{array}$$

(39)

$$\begin{array}{cccc}\hfill \dot{n}{}^a& =\mathcal{A}u{}^a+\alpha {}^a,\hfill & \hfill f_{\left(i\right)}{}^a& ={\mathcal{F}}_{\left(i\right)}n{}^a+{\mathcal{F}}_{\left(i\right)}{}^a,\hfill \end{array}$$

(40)

$$\begin{array}{cccc}\hfill v_{\left(i\right)}{}^a& ={\mathcal{V}}_{\left(i\right)}n{}^a+{\mathcal{V}}_{\left(i\right)}{}^a,\hfill & \hfill v_{\left(F\right)}{}^a& ={\mathcal{V}}_{\left(F\right)}n{}^a+{\mathcal{V}}_{\left(F\right)}{}^a,\hfill \end{array}$$

(41)

Occasionally, we will use a notation with a bar over an index to denote a projection with $N{}_{\mathit{ab}}$, so that $\psi {}_{\overline{a}}={N_a}^b{\psi }_b$.

As for PSTF 3-tensors $\psi {}_{\mathit{ab}}$, these are decomposed as

$$\psi {}_{\mathit{ab}}=\Psi \left(n{}_{a}n{}_b-\frac{1}{2}N{}_{\mathit{ab}}\right)+2n{}_{(a}\Psi {}_{b)}+\Psi {}_{\mathit{ab}},$$

(42)

where $n{}^a\Psi {}_a=0$ and

$$\Psi {}_{\mathit{ab}}=\left(N{{}_{(a}}^{c}N{{}_{b)}}^d-\frac{1}{2}N{}_{\mathit{ab}}N{}^{\mathit{cd}}\right)\psi {}_{\mathit{cd}}\equiv \psi {}_{\left\{ab\right\}}.$$

(43)

Here we have used a notation with curly brackets to denote the PSTF-part of a tensor with respect to the projection tensor $N{}_{\mathit{ab}}$. For the PSTF 3-tensors relevant here, we therefore write

$$\begin{array}{cc}\hfill \pi {}_{\mathit{ab}}=& \Pi \left(n{}_{a}n{}_b-\frac{1}{2}N{}_{\mathit{ab}}\right)+2n{}_{(a}\Pi {}_{b)}+\Pi {}_{\mathit{ab}},\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill E{}_{\mathit{ab}}=& \mathcal{E}\left(n{}_{a}n{}_b-\frac{1}{2}N{}_{\mathit{ab}}\right)+2n{}_{(a}\mathcal{E}{}_{b)}+\mathcal{E}{}_{\mathit{ab}},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill H{}_{\mathit{ab}}=& \mathcal{H}\left(n{}_{a}n{}_b-\frac{1}{2}N{}_{\mathit{ab}}\right)+2n{}_{(a}\mathcal{H}{}_{b)}+\mathcal{H}{}_{\mathit{ab}},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \sigma {}_{\mathit{ab}}=& \Sigma \left(n{}_{a}n{}_b-\frac{1}{2}N{}_{\mathit{ab}}\right)+2n{}_{(a}\Sigma {}_{b)}+\Sigma {}_{\mathit{ab}}.\hfill \end{array}$$

(47)

In a similar manner as before, some new projected variables are also obtained when considering the spatial derivative of $n{}^a$. Defining the spatial gradient along $n{}^a$,

$${{\widehat{T}}^{ab\dots }}_{cd\dots }\equiv n{}^{e}D{}_e{T^{ab\dots }}_{cd\dots },$$

(48)

and the spatial derivative fully projected onto the 2 sheets,

$$\delta {}_e{T^{ab\dots }}_{cd\dots }\equiv {N^a}_f{N^b}_g\cdots {N_c}^h{N_d}^i\cdots {N_e}^{j}D{}_j{T^{fg\dots }}_{hi\dots },$$

(49)

the spatial derivative of $n{}^a$ can be written as

$$D{}_{a}n{}_b=n{}_{a}a{}_b+\frac{1}{2}\varphi N{}_{\mathit{ab}}+\xi ϵ{}_{\mathit{ab}}+\zeta {}_{\mathit{ab}},$$

(50)

where $ϵ{}_{\mathit{ab}}\equiv ϵ{}_{\mathit{abc}}n{}^c$. Here $\varphi \equiv \delta {}^{c}n{}_c$ describes an expansion of the 2 sheets, $\xi ϵ{}_{\mathit{ab}}\equiv \delta {}_{[a}n{}_{b]}$ describes a twist of these sheets, $\zeta {}_{\mathit{ab}}\equiv \delta {}_{\{a}n{}_{b\}}$ is the so-called distortion, and $a{}_a\equiv \widehat{n}{}_a$ [25].

After having decomposed the relevant variables, we also need to decompose the equations from Section 2. This involves decomposing the covariant derivatives of scalars, 2-vectors, PSTF 2-tensors, $ϵ{}_{\mathit{ab}}$ and $N{}_{\mathit{ab}}$. After inserting all of the decompositions into the equations, they are then contracted with various combinations of $u{}^a$,$n{}^a$, $ϵ{}_{\mathit{ab}}$ and $N{}_{\mathit{ab}}$ to extract the individual equations for the projections. However, since this is a rather long process, we will omit the details here and continue with the perturbative approach.

4. Background Spacetimes

We begin by considering the equations to the zeroth order. As such, we need to specify the properties of the background spacetimes to be studied. These are chosen to be homogeneous and hypersurface orthogonal members of LRS class II, which is characterized by that $\xi$, $\omega {}_{\mathit{ab}}$, and $H{}_{\mathit{ab}}$ are all identically zero [15,28]. To be able to use the same type of harmonic decomposition for all of the studied spacetimes, we will also impose the requirement that $\varphi =0$ [12]. Additionally, it is also assumed that the fluid velocities ${\mathcal{V}}_{\left(i\right)}$, the interaction terms ${\mathcal{F}}_{\left(i\right)}$ and ${\epsilon }_{\left(i\right)}$ all vanish identically to the zeroth order. Due to all of these assumptions, the only relevant non-zero dynamical variables to the zeroth order are

$$S^{\left(0\right)}=\left\{\Theta ,\Sigma ,{\mu }_{\left(i\right)},{\mathcal{N}}_{\left(i\right)},\mathfrak{B},\mathcal{E},\mu ,p,\Pi ,p_{\left(i\right)}\right\}.$$

(51)

Setting all other variables to zero in the equations from the previous section and using the spatial homogeneity of the background, the quantities in $S^{\left(0\right)}$ are found to satisfy the relations

$$\begin{array}{cc}\hfill \dot{\Theta }& =-\frac{{\Theta }^2}{3}-\frac{3{\Sigma }^2}{2}-\frac{1}{2}\left(\mu +3p\right)+\Lambda ,\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill \dot{\Sigma }& =\frac{2}{3}\left(\mu +\Lambda \right)+\frac{{\Sigma }^2}{2}-\Sigma \Theta -\frac{2{\Theta }^2}{9}+\Pi ,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {\dot{\mu }}_{\left(i\right)}& =-\left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right)\Theta ,\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill {\dot{\mathcal{N}}}_{\left(i\right)}& =-{\mathcal{N}}_{\left(i\right)}\Theta ,\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill \dot{\mathfrak{B}}& =\left(\Sigma -\frac{2\Theta }{3}\right)\mathfrak{B},\hfill \end{array}$$

(56)

where

$$\begin{array}{cc}\hfill 3\mathcal{E}& =-2\left(\mu +\Lambda \right)-3{\Sigma }^2+\frac{2{\Theta }^2}{3}+\Sigma \Theta -\frac{3\Pi }{2},\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill \mu & =\sum _i{\mu }_{\left(i\right)}+\frac{{\mathfrak{B}}^2}{2},p=\sum _i{p}_{\left(i\right)}+\frac{{\mathfrak{B}}^2}{6},\Pi =-\frac{2{\mathfrak{B}}^2}{3},\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill 0& =\sum _i{q}_{c\left(i\right)}{\mathcal{N}}_{\left(i\right)},\hfill \end{array}$$

(59)

It should be noted here that we need an additional equation for the pressures $p_{\left(i\right)}$ to obtain a closed system. Hence, on specifying equations of state $p_{\left(i\right)}=p_{\left(i\right)}({\mu }_{\left(i\right)},{\mathcal{N}}_{\left(i\right)})$ and a set of initial conditions, the behavior of the background spacetime is fully determined by the system above.

Furthermore, on introducing coordinates, the line element for this class of spacetimes can be written as [12,28]

$$d{s}^2=-d{t}^2+a_1^2\left(t\right)d{z}^2+a_2^2\left(t\right)\left(d{\vartheta }^2+f_{\kappa }\left(\vartheta \right)d{\phi }^2\right),$$

(60)

where the scale factors $a_1$ and $a_2$ are related to the expansion rate and the shear through the relations

$$\Theta =\frac{{\dot{a}}_1}{a_1}+2\frac{{\dot{a}}_2}{a_2},\Sigma =\frac{2}{3}\left(\frac{{\dot{a}}_1}{a_1}-\frac{{\dot{a}}_2}{a_2}\right),$$

(61)

or

$$\frac{{\dot{a}}_1}{a_1}=\Sigma +\frac{\Theta }{3},\frac{{\dot{a}}_2}{a_2}=-\frac{1}{2}\left(\Sigma -\frac{2\Theta }{3}\right).$$

(62)

As for the precise nature of the function $f_{\kappa }\left(\vartheta \right)$, this will depend on the 2D curvature [11]

$$\mathcal{R}=\frac{2\kappa }{a_2^2}=2\left(\mu +\Lambda \right)+\frac{3{\Sigma }^2}{2}-\frac{2{\Theta }^2}{3}.$$

(63)

For $\kappa =0$, the 2-sheets are flat with either $f_0\left(\vartheta \right)=1$ or $f_0\left(\vartheta \right)={\vartheta }^2$. If instead $\kappa =1$, the sheets are closed with $f_1\left(\vartheta \right)={sin}^2\vartheta$. Lastly, if $\kappa =-1$, the sheets are open with $f_{-1}\left(\vartheta \right)={sinh}^2\vartheta$.

5. Perturbations

With the zeroth-order spacetime specified, we now consider first-order perturbations on this background. To avoid unphysical modes due to the implicit mapping between the background spacetime and the perturbed manifold, we will describe the perturbations in terms of gauge invariant variables [27]. Making use of the Stewart–Walker lemma, the gauge-invariant variables of the first order are chosen to be quantities that vanish in the background [9]. Hence, quantities such as $\mathcal{H}{}_{\mathit{ab}}$ and $\mathfrak{B}{}_a$, which are naturally zero in the background, are gauge invariant to the first order. However, not all quantities are zero in the background. Thus, to represent the perturbations of the quantities in $S^{\left(0\right)}$, we will follow [11,12,13,14] and make use of the spatial gradients of these quantities, as these gradients vanish on the background due to the assumed homogeneity. We therefore introduce the following gauge-invariant variables

$$\begin{array}{cccccccc}\hfill X{}_a& \equiv \delta {}_a\mathcal{E},\hfill & \hfill V{}_a& \equiv \delta {}_a\Sigma ,\hfill & \hfill W{}_a& \equiv \delta {}_a\Theta ,\hfill & \hfill Y{}_a& \equiv \delta {}_a\Pi ,\hfill \\ \hfill \mu {}_a& \equiv \delta {}_a\mu ,\hfill & \hfill p{}_a& \equiv \delta {}_{a}p,\hfill & \hfill \mathcal{B}{}_a& \equiv \delta {}_a\mathfrak{B},\hfill & \hfill {\mathcal{Z}}_{\left(i\right)}{}_a& \equiv \delta {}_a{\mathcal{N}}_{\left(i\right)},\hfill \\ & \hfill & \hfill {\mu }_{\left(i\right)}{}_a& \equiv \delta {}_a{\mu }_{\left(i\right)},\hfill & \hfill p_{\left(i\right)}{}_a& \equiv \delta {}_a{p}_{\left(i\right)}.\hfill & & \hfill \end{array}$$

(64)

Note that we do not introduce any new gauge-invariant variables using the “$\widehat{}$” gradients here, as these can be given in terms of the “$\delta$” gradients and the vorticity after performing the intended harmonic decompositions later on [13,14]. Our complete set of gauge-invariant first-order variables therefore becomes

$$\begin{array}{cc}\hfill S^{\left(1\right)}\equiv \{& X{}_a,V{}_a,W{}_a,\mu {}_a,p{}_a,Y{}_a,\mathcal{A},\mathcal{A}{}_a,\Sigma {}_a,\Sigma {}_{\mathit{ab}},\mathcal{E}{}_a,\mathcal{E}{}_{\mathit{ab}},\hfill \\ \hfill & \mathcal{H},\mathcal{H}{}_a,\mathcal{H}{}_{\mathit{ab}},\alpha {}_a,a{}_a,\xi ,\varphi ,\zeta {}_{\mathit{ab}},\Omega ,\Omega {}_a,Q,Q{}_a,\Pi {}_a,\hfill \\ \hfill & {\mu }_{\left(i\right)}{}_a,p_{\left(i\right)}{}_a,{\mathcal{Z}}_{\left(i\right)}{}_a,{\mathcal{V}}_{\left(i\right)},{\mathcal{V}}_{\left(i\right)}{}_a,{\epsilon }_{\left(i\right)},{\mathcal{F}}_{\left(i\right)},{\mathcal{F}}_{\left(i\right)}{}^a,\hfill \\ \hfill & \mathfrak{E},\mathfrak{E}{}_a,\mathfrak{B}{}_a,\mathcal{B}{}_a,\rho ,\mathcal{J},\mathcal{J}{}_a\}.\hfill \end{array}$$

(65)

To obtain linearized equations governing the variables in $S^{\left(1\right)}$, we write the decomposed equations from Section 2 in terms of these quantities by using commutation relations for the projected derivatives, which can be found in Appendix A. In doing so, we also omit every term that is of second order or higher. This yields a large set of equations involving derivatives of the projected variables with respect to both time and space. Since the equations originating from the Ricci and Bianchi identities can be found in [14], we will here only state the equations related to the electromagnetic fields, the individual plasma components, and the decomposition of the total energy momentum quantities in terms of their constituents.

The linearized equations for the variables related to the total energy-momentum tensor are

$$\begin{array}{cc}\hfill \mu {}^a& =\sum _i{{\mu }_{\left(i\right)}}^a+\mathfrak{B}\mathcal{B}{}^a,\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill p{}^a& =\sum _i{p_{\left(i\right)}}^a+\frac{\mathfrak{B}}{3}\mathcal{B}{}^a,\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill Q& =\sum _i\left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right){\mathcal{V}}_{\left(i\right)},\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill Q{}^a& =\sum _i\left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right){\mathcal{V}}_{\left(i\right)}{}^a+ϵ{}^{\mathit{ab}}\mathfrak{B}\mathfrak{E}{}_b,\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill Y{}^a& =-\frac{4\mathfrak{B}}{3}\mathcal{B}{}^a,\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill \Pi {}^a& =-\mathfrak{B}\mathfrak{B}{}^a.\hfill \end{array}$$

(71)

The first order equations for the fluid components are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {{\dot{\mu }}_{\left(i\right)}}^{\overline{a}}=& \mathcal{A}{}^a{\dot{\mu }}_{\left(i\right)}+\frac{1}{2}\left(\Sigma -\frac{2\Theta }{3}\right){{\mu }_{\left(i\right)}}^a-\left({{\mu }_{\left(i\right)}}^a+{p_{\left(i\right)}}^a\right)\Theta \hfill \\ \hfill & -\left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right)\left(\delta {}^a{\widehat{\mathcal{V}}}_{\left(i\right)}+\delta {}^a\delta {}^b\mathcal{V}{}_{\left(i\right)}b+W{}^a\right)+\delta {}^a{\epsilon }_{\left(i\right)},\hfill \end{array}\end{array}$$

(72)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right)\delta {}^a{\dot{\mathcal{V}}}_{\left(i\right)}=& -{{\widehat{p}}_{\left(i\right)}}^{\overline{a}}+\left(2ϵ{}^{\mathit{ab}}\Omega {}_b-\delta {}^a{\mathcal{V}}_{\left(i\right)}\right){\dot{p}}_{\left(i\right)}+{\rho }_{\left(i\right)}\delta {}^a\mathfrak{E}\hfill \\ \hfill & -\left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right)\left(\delta {}^a\mathcal{A}+\left(\Sigma +\frac{\Theta }{3}\right)\delta {}^a{\mathcal{V}}_{\left(i\right)}\right)+\delta {}^a{\mathcal{F}}_{\left(i\right)},\hfill \end{array}\end{array}$$

(73)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right){{\dot{\mathcal{V}}}_{\left(i\right)}}^{\overline{a}}=& -{\dot{p}}_{\left(i\right)}{\mathcal{V}}_{\left(i\right)}{}^a-p_{\left(i\right)}{}^a+{\rho }_{\left(i\right)}\left(\mathfrak{E}{}^a+\mathfrak{B}ϵ{}^{a}b\mathcal{V}{}_{\left(i\right)}b\right)\hfill \\ \hfill & -\left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right)\left(\mathcal{A}{}^a-\frac{1}{2}\left(\Sigma -\frac{2\Theta }{3}\right){\mathcal{V}}_{\left(i\right)}{}^a\right)+{{\mathcal{F}}_{\left(i\right)}}^a,\hfill \end{array}\end{array}$$

(74)

$$\begin{array}{cc}\hfill {{\dot{\mathcal{Z}}}_{\left(i\right)}}^{\overline{a}}=& \mathcal{A}{}^a{\dot{\mathcal{N}}}_{\left(i\right)}+\left(\frac{\Sigma }{2}-\frac{4\Theta }{3}\right){{\mathcal{Z}}_{\left(i\right)}}^a\hfill \\ \hfill & -{\mathcal{N}}_{\left(i\right)}\left(W{}^a+\delta {}^a{\widehat{\mathcal{V}}}_{\left(i\right)}+\delta {}^a\delta {}^b\mathcal{V}{}_{\left(i\right)}b\right).\hfill \end{array}$$

(75)

Finally, the linearized versions of Maxwell’s equations are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{\mathfrak{E}}-ϵ{}_{\mathit{ab}}\delta {}^a\mathfrak{B}{}^b=& 2\xi \mathfrak{B}+\left(\Sigma -\frac{2\Theta }{3}\right)\mathfrak{E}-\mathcal{J},\hfill \end{array}\end{array}$$

(76)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \widehat{\mathfrak{E}}+\delta {}_a\mathfrak{E}{}^a=& 2\mathfrak{B}\Omega +\rho ,\hfill \end{array}\end{array}$$

(77)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{\mathcal{B}}{}^{\overline{a}}+ϵ{}_{\mathit{bc}}\delta {}^a\delta {}^b\mathfrak{E}{}^c=& \mathcal{A}{}^a\dot{\mathfrak{B}}+\frac{3}{2}\left(\Sigma -\frac{2\Theta }{3}\right)\mathcal{B}{}^a+\left(V{}^a-\frac{2}{3}W{}^a\right)\mathfrak{B},\hfill \end{array}\end{array}$$

(78)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \widehat{\mathcal{B}}{}^{\overline{a}}+\delta {}^a\delta {}^b\mathfrak{B}{}_b=& 2ϵ{}^{\mathit{ab}}\Omega {}_b\dot{\mathfrak{B}}-\mathfrak{B}\delta {}^a\varphi ,\hfill \end{array}\end{array}$$

(79)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{\mathfrak{E}}{}_{\overline{a}}+ϵ{}_{\mathit{ab}}\left(\widehat{\mathfrak{B}}{}^b-\mathcal{B}{}^b\right)=& -\frac{1}{2}\left(\Sigma +\frac{4\Theta }{3}\right)\mathfrak{E}{}_a+\mathfrak{B}ϵ{}_{\mathit{ab}}\left(\mathcal{A}{}^b-a{}^b\right)-\mathcal{J}{}_a,\hfill \end{array}\end{array}$$

(80)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{\mathfrak{B}}{}_{\overline{a}}-ϵ{}_{\mathit{ab}}\left(\widehat{\mathfrak{E}}{}^b-\delta {}^b\mathfrak{E}\right)=& -\frac{1}{2}\left(\Sigma +\frac{4\Theta }{3}\right)\mathfrak{B}{}_a+\mathfrak{B}\left(-\alpha {}_a+\Sigma {}_a+ϵ{}_{\mathit{ab}}\Omega {}^b\right),\hfill \end{array}\end{array}$$

(81)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \delta {}^a\rho =& \sum _i{q}_{c\left(i\right)}{\mathcal{Z}}_{\left(i\right)}{}^a,\hfill \end{array}\end{array}$$

(82)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{J}=& \sum _i{q}_{c\left(i\right)}{\mathcal{N}}_{\left(i\right)}{\mathcal{V}}_{\left(i\right)},\hfill \end{array}\end{array}$$

(83)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{J}{}_a=& \sum _i{q}_{c\left(i\right)}{\mathcal{N}}_{\left(i\right)}{\mathcal{V}}_{\left(i\right)}{}_a.\hfill \end{array}\end{array}$$

(84)

6. Harmonic Decomposition

To simplify the linearized equations, they are then harmonically decomposed using the harmonics $P^{k_{\Vert }}\left(z\right)$ and $Q^{k_{\perp }}(\vartheta ,\phi )$, which are suited for homogeneous and hypersurface orthogonal LRS class II backgrounds with $\varphi =0$ [12]. These harmonics are defined on the background as functions satisfying the equations

$$\begin{array}{cccccc}\hfill \widehat{\Delta }{P}^{k_{\Vert }}& =-\frac{k_{\Vert }^2}{a_1^2}{P}^{k_{\Vert }},\hfill & \hfill \delta {}_a{P}^{k_{\Vert }}& =0,\hfill & \hfill {\dot{P}}^{k_{\Vert }}& =0,\hfill \end{array}$$

(85)

$$\begin{array}{cccccc}\hfill {\delta }^2{Q}^{k_{\perp }}& =-\frac{k_{\perp }^2}{a_2^2}{Q}^{k_{\perp }},\hfill & \hfill {\widehat{Q}}^{k_{\perp }}& =0,\hfill & \hfill {\dot{Q}}^{k_{\perp }}& =0,\hfill \end{array}$$

(86)

where the differential operators are defined as $\widehat{\Delta }\equiv n{}^a\nabla {}_{a}n{}^b\nabla {}_b$, and ${\delta }^2=\delta {}_a\delta {}^a$, whilst $k_{\Vert }$ and $k_{\perp }$ are comoving dimensionless wave numbers. Using these harmonics, scalars can be decomposed as

$$\Psi =\sum _{k_{\Vert },k_{\perp }}{\Psi }_{k_{\Vert }{k}_{\perp }}^S{P}^{k_{\Vert }}{Q}^{k_{\perp }},$$

(87)

where the coefficients ${\Psi }^S$ now only depend on time. To decompose 2-vectors $\Psi {}_a$ and PSTF 2-tensors $\Psi {}_{\mathit{ab}}$, we also introduce vector and tensor harmonics following

$$\begin{array}{cccc}\hfill Q^{k_{\perp }}{}_a& =a_2\delta {}_a{Q}^{k_{\perp }},\hfill & \hfill Q^{k_{\perp }}{}_{\mathit{ab}}& =a_2^2\delta {}_{\{a}\delta {}_{b\}}{Q}^{k_{\perp }},\hfill \end{array}$$

(88)

$$\begin{array}{cccc}\hfill {\overline{Q}}^{k_{\perp }}{}_a& =a_2ϵ{}_{\mathit{ab}}\delta {}^b{Q}^{k_{\perp }},\hfill & \hfill {\overline{Q}}^{k_{\perp }}{}_{\mathit{ab}}& =a_2^2ϵ{}_c\{a\delta {}^c\delta {}_{b\}}{Q}^{k_{\perp }}.\hfill \end{array}$$

(89)

Using these harmonics, we may write

$$\begin{array}{cc}\hfill \Psi {}_a& =\sum _{k_{\Vert },k_{\perp }}{P}^{k_{\Vert }}\left({\Psi }_{k_{\Vert }{k}_{\perp }}^V{Q}^{k_{\perp }}{}_a+{\overline{\Psi }}_{k_{\Vert }{k}_{\perp }}^V{\overline{Q}}^{k_{\perp }}{}_a\right),\hfill \end{array}$$

(90)

$$\begin{array}{cc}\hfill \Psi {}_{\mathit{ab}}& =\sum _{k_{\Vert },k_{\perp }}{P}^{k_{\Vert }}\left({\Psi }_{k_{\Vert }{k}_{\perp }}^T{Q}^{k_{\perp }}{}_{\mathit{ab}}+{\overline{\Psi }}_{k_{\Vert }{k}_{\perp }}^T{\overline{Q}}^{k_{\perp }}{}_{\mathit{ab}}\right).\hfill \end{array}$$

(91)

Here, we have used bars to denote the so-called odd parts of the decompositions, whilst the even parts are written without bars. In the following, when considering the relations for the individual harmonic coefficients, we will omit their subscripts, as this should not lead to any ambiguities.

Inserting the harmonic decompositions into the linearized equations and using the properties of the harmonics, described in Appendix B, the system can be reduced significantly by solving for some of the harmonic coefficients in terms of the others. The harmonic decompositions of the linearized equations, which were presented in Section 5, can be found in Appendix C. However, here we encounter a difficulty which was not present in previous papers. At this point in previous works, the system effectively divides into two separate parts, one containing the odd coefficients and another containing the even parts [12,13,14]. One exception to the division into even and odd equations, which is also present in previous works, are quantities defined using a Levi–Civita psuedo-tensor, such as the vorticity and the magnetic part of the Weyl tensor. For these quantities, it is generally the even coefficients that are present in the odd system and vice versa, but the systems still decouple from each other. However, when including a non-zero magnetic field of the zeroth order, the division into even and odd subsystems does not seem possible in general, as the pseudoscalar $\mathfrak{B}$ serves as a coupling that connects the systems together. This can be seen when considering the harmonically decomposed momentum equations for the fluid components

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right){{\dot{\mathcal{V}}}_{\left(i\right)}}^V=& -{\dot{p}}_{\left(i\right)}{{\mathcal{V}}_{\left(i\right)}}^V-{p_{\left(i\right)}}^V+{\rho }_{\left(i\right)}\left({\mathfrak{E}}^V-\mathfrak{B}{{\overline{\mathcal{V}}}_{\left(i\right)}}^V\right)\hfill \\ \hfill & -\left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right)\left({\mathcal{A}}^V-\frac{1}{2}\left(\Sigma -\frac{2\Theta }{3}\right){{\mathcal{V}}_{\left(i\right)}}^V\right)+{{\mathcal{F}}_{\left(i\right)}}^V,\hfill \end{array}\end{array}$$

(92)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right){{\dot{\overline{\mathcal{V}}}}_{\left(i\right)}}^V=& -{\dot{p}}_{\left(i\right)}{{\overline{\mathcal{V}}}_{\left(i\right)}}^V-{{\overline{p}}_{\left(i\right)}}^V+{\rho }_{\left(i\right)}\left({\overline{\mathfrak{E}}}^V+\mathfrak{B}{{\mathcal{V}}_{\left(i\right)}}^V\right)\hfill \\ \hfill & -\left({\mu }_{\left(i\right)}+p_{\left(i\right)}\right)\left({\overline{\mathcal{A}}}^V-\frac{1}{2}\left(\Sigma -\frac{2\Theta }{3}\right){{\overline{\mathcal{V}}}_{\left(i\right)}}^V\right)+{{\overline{\mathcal{F}}}_{\left(i\right)}}^V.\hfill \end{array}\end{array}$$

(93)

Here, it can be explicitly observed that ${{\mathcal{V}}_{\left(i\right)}}^V$ is coupled to ${{\overline{\mathcal{V}}}_{\left(i\right)}}^V$ through the terms proportional to $\mathfrak{B}$. To avoid this problem and simplify matters even further, we will now apply the cold MHD approximation.

7. Cold MHD Approximation

In the cold MHD approximation, we assume that ${\mu }_{\left(i\right)}\approx m_{\left(i\right)}{\mathcal{N}}_{\left(i\right)}$ and $p_{\left(i\right)}\approx 0$ are of both zeroth and first order, and write the harmonic equations in terms of the collective variables

$$\begin{array}{cccc}\hfill {\mu }_{\left(F\right)}& \equiv \sum _i{\mu }_{\left(i\right)},\hfill & \hfill {\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S& \equiv \sum _i{\mu }_{\left(i\right)}{{\mathcal{V}}_{\left(i\right)}}^S,\hfill \\ \hfill {{\mu }_{\left(F\right)}}^V& \equiv \sum _i{{\mu }_{\left(i\right)}}^V,\hfill & \hfill {\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V& \equiv \sum _i{\mu }_{\left(i\right)}{{\mathcal{V}}_{\left(i\right)}}^V,\hfill \\ \hfill {{\overline{\mu }}_{\left(F\right)}}^V& \equiv \sum _i{{\overline{\mu }}_{\left(i\right)}}^V,\hfill & \hfill {\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V& \equiv \sum _i{\mu }_{\left(i\right)}{{\overline{\mathcal{V}}}_{\left(i\right)}}^V.\hfill \end{array}$$

(94)

The equations governing these variables are found by adding the equations for the individual fluid components. Using charge neutrality to the zeroth order and assuming that the plasma consists of electrons and a much more massive ion counterpart, it can then be shown that the collective velocities satisfy the equations

$$\begin{array}{cc}\hfill {\mu }_{\left(F\right)}{{\dot{\mathcal{V}}}_{\left(F\right)}}^V=& -\frac{\mathfrak{B}}{\eta }\left({\overline{\mathfrak{E}}}^V+\mathfrak{B}{{\mathcal{V}}_{\left(F\right)}}^V\right)-{\mu }_{\left(F\right)}\left({\mathcal{A}}^V-\frac{1}{2}\left(\Sigma -\frac{2\Theta }{3}\right){{\mathcal{V}}_{\left(F\right)}}^V\right),\hfill \end{array}$$

(95)

$$\begin{array}{cc}\hfill {\mu }_{\left(F\right)}{{\dot{\overline{\mathcal{V}}}}_{\left(F\right)}}^V=& \frac{\mathfrak{B}}{\eta }\left({\mathfrak{E}}^V-\mathfrak{B}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V\right)-{\mu }_{\left(F\right)}\left({\overline{\mathcal{A}}}^V-\frac{1}{2}\left(\Sigma -\frac{2\Theta }{3}\right){{\overline{\mathcal{V}}}_{\left(F\right)}}^V\right),\hfill \end{array}$$

(96)

where we have written the currents as

$$\begin{array}{cc}\hfill {\mathcal{J}}^V& =\frac{1}{\eta }\left({\mathfrak{E}}^V-\mathfrak{B}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V\right),\hfill \end{array}$$

(97)

$$\begin{array}{cc}\hfill {\overline{\mathcal{J}}}^V& =\frac{1}{\eta }\left({\overline{\mathfrak{E}}}^V+\mathfrak{B}{{\mathcal{V}}_{\left(F\right)}}^V\right),\hfill \end{array}$$

(98)

using the linearised and harmonically decomposed version of Equation (23). The other equations obtained on rewriting the multifluid system in this manner can be found in Appendix D. The momentum equations, Equations (95) and (96), should be compared to Equations (92) and (93). At the cost of introducing an electrical resistivity $\eta$, we see that we no longer have any coupling between the even and odd velocities in Equations (95) and (96) in contrast to Equations (92) and (93). This simplifies matters, and the system is ultimately seen to decouple into two subsectors, as shown in Section 8. However, we will need some model for the electrical resistivity if we want to completely close the system. Therefore, when performing numerical calculations in Section 9, we will make the simplified assumption that $\eta$ follows a Spitzer-like expression so that

$$\eta \propto {T_{\left(e\right)}}^{-3/2},$$

(99)

where $T_{\left(e\right)}$ is the electron temperature [29]. Assuming that the electron pressure, although neglected in the main equations, follows a polytropic equation of state and an ideal gas law,

$$p_{\left(e\right)}\propto {{\mathcal{N}}_{\left(e\right)}}^{\gamma }\propto {\mathcal{N}}_{\left(e\right)}{T}_{\left(e\right)},$$

(100)

it follows from the electron number conservation and Equation (99) that

$$\dot{\eta }=\frac{3\left(\gamma -1\right)}{2}\Theta \eta .$$

(101)

8. Final System

The harmonically decomposed system of equations in the cold MHD approximation can be significantly reduced by solving for some of the coefficients in terms of the others. However, before arriving at the final system, we should note that there is still some freedom in specifying the dyad $\{u^a,n^a\}$ to the first order. At the zeroth order, this dyad is fixed. The vector field $u^a$ is then assumed to be orthogonal to the hypersurfaces of homogeneity, and coincides with the plasma velocities. As for the field $n^a$, this is fixed since it defines the direction of the local rotational symmetry. However, when going to the first order, the homogeneity and rotational symmetry are broken, and the vector field $u^a$ does not need to coincide with the plasma velocities. As a result, there is some freedom in defining $u^a$ and $n^a$ to the first order. We will use this freedom to set $a^V$, ${\overline{a}}^V$, ${\mathcal{A}}^S$, ${\mathcal{A}}^V$, and ${\overline{\mathcal{A}}}^V$ to zero. For details on the effect and use of infinitesimal transformations of the dyad $\{u^a,n^a\}$, the reader is referred to [11].

Finally, after specifying the dyad to the first order, we arrive at two subsystems for the first order quantities. These systems are seen to close independently of each other and consist of the odd sector $\{{\Omega }^S,{\overline{\mathcal{E}}}^T,{\mathcal{H}}^T,{{\overline{\mathcal{V}}}_{\left(F\right)}}^V,{\mathfrak{E}}^S,{\mathfrak{E}}^V,{\overline{\mathfrak{B}}}^V\}$ and the even sector $\{{\overline{\Omega }}^V,{\overline{\mathcal{H}}}^T,{\mathcal{E}}^T,{\Sigma }^T,{{\mu }_{\left(F\right)}}^V$, ${{\mathcal{V}}_{\left(F\right)}}^S,{{\mathcal{V}}_{\left(F\right)}}^V,{\mathcal{B}}^V,{\overline{\mathfrak{E}}}^V\}$. All other coefficients can be given algebraically in terms of these sets.

8.1. Odd Sector

The coefficients in the odd sector satisfy the following evolution equations

$$\begin{array}{cc}\hfill {\dot{\Omega }}^S=& \left(\Sigma -\frac{2\Theta }{3}\right){\Omega }^S,\hfill \end{array}$$

(102)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\overline{\mathcal{E}}}}^T=& -\frac{3}{2}\left(F+\Sigma D\right){\overline{\mathcal{E}}}^T+\frac{i{k}_{\Vert }}{a_1}\left(1-D\right){\mathcal{H}}^T\hfill \\ \hfill & +P{\Omega }^S+\frac{1}{a_2}\left(1+D\right)\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right),\hfill \end{array}\end{array}$$

(103)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\mathcal{H}}}^T=& \left(\frac{3i{a}_1\Pi }{k_{\Vert }{k}_{\perp }^2}\left(\Sigma +\frac{\Theta }{3}\right)+S\right){\Omega }^S+\frac{1}{a_2}\mathfrak{B}{\overline{\mathfrak{B}}}^V\hfill \\ \hfill & +\frac{i{a}_1}{a_2{k}_{\Vert }B}[\left(\Sigma +\frac{\Theta }{3}\right)\left(3\Pi +\frac{k_{\perp }^2}{a_2^2}\right)\hfill \\ \hfill & -\frac{k_{\Vert }^2}{a_1^2}\left(\Sigma -\frac{2\Theta }{3}\right)]\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right)\hfill \\ \hfill & +\frac{i{a}_1}{2k_{\Vert }}\left(\frac{2k_{\Vert }^2}{a_1^2}-CB+9\Sigma E+3\Pi \right){\overline{\mathcal{E}}}^T-\frac{3}{2}\left(2E+F\right){\mathcal{H}}^T,\hfill \end{array}\end{array}$$

(104)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\overline{\mathcal{V}}}}_{\left(F\right)}^V=& \frac{\mathfrak{B}}{\eta {\mu }_{\left(F\right)}}\left({\mathfrak{E}}^V-\mathfrak{B}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V\right)+\frac{1}{2}\left(\Sigma -\frac{2\Theta }{3}\right){{\overline{\mathcal{V}}}_{\left(F\right)}}^V,\hfill \end{array}\end{array}$$

(105)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\mathfrak{E}}}^S=& \frac{k_{\perp }^2}{a_2}{\overline{\mathfrak{B}}}^V-\frac{2i{a}_1}{k_{\Vert }}\mathfrak{B}\left(\Sigma +\frac{\Theta }{3}\right){\Omega }^S+\left(\Sigma -\frac{2\Theta }{3}-\frac{1}{\eta }\right){\mathfrak{E}}^S,\hfill \end{array}\end{array}$$

(106)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\mathfrak{E}}}^V=& \frac{i{k}_{\Vert }}{a_1}{\overline{\mathfrak{B}}}^V-\frac{1}{2}\left(\Sigma +\frac{4\Theta }{3}\right){\mathfrak{E}}^V-\frac{1}{\eta }\left({\mathfrak{E}}^V-\mathfrak{B}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V\right)-\frac{2a_2\dot{\mathfrak{B}}}{k_{\perp }^2}{\Omega }^S,\hfill \end{array}\end{array}$$

(107)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\overline{\mathfrak{B}}}}^V=& \frac{i{k}_{\Vert }}{a_1}{\mathfrak{E}}^V-\frac{1}{a_2}{\mathfrak{E}}^S-\frac{1}{2}\left(\Sigma +\frac{4\Theta }{3}\right){\overline{\mathfrak{B}}}^V\hfill \\ \hfill & -\frac{i{a}_1}{k_{\Vert }}\mathfrak{B}[\frac{2\left(\mathcal{R}{a}_2^2-k_{\perp }^2\right)}{a_{2}B}\left(\frac{3\Sigma }{2}{\overline{\mathcal{E}}}^T+\frac{i{k}_{\Vert }}{a_1}{\mathcal{H}}^T\right)\hfill \\ \hfill & -\frac{2}{B}\left(3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)+\frac{2k_{\Vert }^2}{a_1^2}\right)\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right)\hfill \\ \hfill & -\frac{a_2}{k_{\perp }^2}\left(\mu +p-\frac{\Pi }{2}-3\mathcal{E}+2N+3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)+\frac{2k_{\Vert }^2}{a_1^2}\right){\Omega }^S],\hfill \end{array}\end{array}$$

(108)

where the auxiliary variables are defined as

$$\begin{array}{cc}\hfill {\tilde{k}}^2=& \frac{k_{\perp }^2}{a_2^2}+\frac{2k_{\Vert }^2}{a_1^2},\hfill \end{array}$$

(109)

$$\begin{array}{cc}\hfill B=& {\tilde{k}}^2+\frac{9}{2}{\Sigma }^2+3\left(\mathcal{E}+\frac{1}{2}\Pi \right),\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill CB=& \Sigma \left(\Theta -\frac{3\Sigma }{2}\right)-\frac{k_{\perp }^2}{a_2^2},\hfill \end{array}$$

(111)

$$\begin{array}{cc}\hfill F=& \Sigma +\frac{2\Theta }{3},\hfill \end{array}$$

(112)

$$\begin{array}{cc}\hfill D=& C+\frac{\mu +p-2\Pi }{B},\hfill \end{array}$$

(113)

$$\begin{array}{cc}\hfill P=& \frac{2}{k_{\perp }^{2}B}\left(\mu +p-\frac{1}{2}\Pi \right)\left(B+3\mathcal{E}+\mu +p-\frac{1}{2}\Pi +\mathcal{R}-\frac{k_{\perp }^2}{a_2^2}\right),\hfill \end{array}$$

(114)

$$\begin{array}{cc}\hfill S=& \frac{2i{a}_1}{3k_{\Vert }{k}_{\perp }^{2}B}\left(\mu +p-\frac{1}{2}\Pi \right)\left(3\Sigma \left(\frac{k_{\perp }^2}{a_2^2}-\frac{k_{\Vert }^2}{a_1^2}+3\Pi \right)+\Theta \left({\tilde{k}}^2+3\Pi \right)\right),\hfill \end{array}$$

(115)

$$\begin{array}{cc}\hfill EB=& \frac{\Sigma }{2}\left(CB-\mathcal{E}-\frac{5}{2}\Pi \right)+\frac{\Theta }{3}\left(\mathcal{E}-\frac{1}{2}\Pi \right),\hfill \end{array}$$

(116)

$$\begin{array}{cc}\hfill N=& \left(\mu +p-\frac{1}{2}\Pi \right)\left(1+\frac{2}{a_2^{2}B}\left(\mathcal{R}{a}_2^2-k_{\perp }^2\right)\right).\hfill \end{array}$$

(117)

We obtain the remaining relevant harmonic coefficients as algebraic expressions in terms of ${\Omega }^S,{\overline{\mathcal{E}}}^T,{\mathcal{H}}^T,{{\overline{\mathcal{V}}}_{\left(F\right)}}^V,{\mathfrak{E}}^S,{\mathfrak{E}}^V$, and ${\overline{\mathfrak{B}}}^V$. The remaining scalar coefficients are

$$\begin{array}{cc}\hfill {\rho }^S=& \frac{i{k}_{\Vert }}{a_1}{\mathfrak{E}}^S-\frac{k_{\perp }^2}{a_2}{\mathfrak{E}}^V-2\mathfrak{B}{\Omega }^S,\frac{i{k}_{\Vert }}{a_1}{\xi }^S=\left(\Sigma +\frac{\Theta }{3}\right){\Omega }^S,\hfill \end{array}$$

(118)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{i{a}_2{k}_{\Vert }}{a_1{k}_{\perp }^2}{\mathcal{H}}^S=& \frac{\left(\mathcal{R}{a}_2^2-k_{\perp }^2\right)}{a_{2}B}\left(\frac{3\Sigma }{2}{\overline{\mathcal{E}}}^T+\frac{i{k}_{\Vert }}{a_1}{\mathcal{H}}^T\right)-\frac{a_2}{k_{\perp }^2}\left(\mu +p-\frac{\Pi }{2}+N\right){\Omega }^S\hfill \\ \hfill & -\frac{1}{B}\left(3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)+\frac{2k_{\Vert }^2}{a_1^2}\right)\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right).\hfill \end{array}\end{array}$$

(119)

The remaining vector coefficients are in turn given by

$$\begin{array}{cc}\hfill {\Omega }^V=& \frac{i{a}_2{k}_{\Vert }}{a_1{k}_{\perp }^2}{\Omega }^S,\hfill \end{array}$$

(120)

$$\begin{array}{cc}\hfill {\overline{Q}}^V=& {\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V,\hfill \end{array}$$

(121)

$$\begin{array}{cc}\hfill {{\overline{\mu }}_{\left(F\right)}}^V=& \frac{2a_2{\dot{\mu }}_{\left(F\right)}}{k_{\perp }^2}{\Omega }^S,{\overline{\mathcal{B}}}^V=\frac{2a_2\dot{\mathfrak{B}}}{k_{\perp }^2}{\Omega }^S,{\overline{\mu }}^V={{\overline{\mu }}_{\left(F\right)}}^V+\mathfrak{B}{\overline{\mathcal{B}}}^V,\hfill \end{array}$$

(122)

$$\begin{array}{cc}\hfill {\overline{W}}^V=& \frac{2a_2\dot{\Theta }}{k_{\perp }^2}{\Omega }^S,{\overline{X}}^V=\frac{2a_2\dot{\mathcal{E}}}{k_{\perp }^2}{\Omega }^S,{\overline{V}}^V=\frac{2a_2\dot{\Sigma }}{k_{\perp }^2}{\Omega }^S,\hfill \end{array}$$

(123)

$$\begin{array}{cc}\hfill {\overline{p}}^V=& \frac{\mathfrak{B}}{3}{\overline{\mathcal{B}}}^V,{\overline{Y}}^V=-\frac{4\mathfrak{B}}{3}{\overline{\mathcal{B}}}^V,{\overline{\Pi }}^V=-\mathfrak{B}{\overline{\mathfrak{B}}}^V,\hfill \end{array}$$

(124)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{i{k}_{\Vert }}{a_1}{\overline{\Sigma }}^V=& -\frac{a_2}{k_{\perp }^2}\left(\mu +p-\frac{\Pi }{2}+N+3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)+\frac{k_{\Vert }^2}{a_1^2}\right){\Omega }^S\hfill \\ \hfill & +\frac{(\mathcal{R}{a}_2^2-k_{\perp }^2)}{a_{2}B}\left(\frac{3\Sigma }{2}{\overline{\mathcal{E}}}^T+\frac{i{k}_{\Vert }}{a_1}{\mathcal{H}}^T\right)\hfill \\ \hfill & -\frac{1}{B}\left(3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)+\frac{2k_{\Vert }^2}{a_1^2}\right)\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right),\hfill \end{array}\end{array}$$

(125)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\overline{\mathcal{E}}}^V=& \frac{i{a}_1{a}_2}{k_{\Vert }{k}_{\perp }^2}(\left(3\mathcal{E}+\frac{\Pi }{2}+2\mu +2p\right)\left(\Sigma +\frac{\Theta }{3}\right)\hfill \\ \hfill & -\frac{3\Sigma }{B}\left(\mu +p-\frac{\Pi }{2}\right)\left(\frac{2k_{\Vert }^2}{a_1^2}+3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)\right)){\Omega }^S\hfill \\ \hfill & -\frac{\mathcal{R}{a}_2^2-k_{\perp }^2}{2a_2}\left(\frac{i{a}_1}{k_{\Vert }}\left(1-\frac{9{\Sigma }^2}{2B}\right){\overline{\mathcal{E}}}^T+\frac{3\Sigma }{B}{\mathcal{H}}^T\right)+\frac{\mathfrak{B}}{2}{\overline{\mathfrak{B}}}^V\hfill \\ \hfill & +\frac{i{a}_1}{k_{\Vert }B}[\left(\Sigma +\frac{\Theta }{3}\right)\left({\tilde{k}}^2-\mathcal{R}-\frac{3\Sigma }{2}\left(\Sigma -\frac{2\Theta }{3}\right)\right)\hfill \\ \hfill & -\frac{3\Sigma k_{\Vert }^2}{a_1^2}]\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right),\hfill \end{array}\end{array}$$

(126)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{H}}^V=& \frac{a_2}{k_{\perp }^2}\left(3\mathcal{E}-N\right){\Omega }^S+\frac{\mathcal{R}{a}_2^2-k_{\perp }^2}{a_{2}B}\left(\frac{3\Sigma }{2}{\overline{\mathcal{E}}}^T+\frac{i{k}_{\Vert }}{a_1}{\mathcal{H}}^T\right)\hfill \\ \hfill & -\frac{1}{B}\left(3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)+\frac{2k_{\Vert }^2}{a_1^2}-\frac{B}{2}\right)\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right),\hfill \end{array}\end{array}$$

(127)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{i{k}_{\Vert }}{a_1}{\overline{\alpha }}^V=& -\frac{a_2}{k_{\perp }^2}\left(3\mathcal{E}-N\right){\Omega }^S-\frac{\mathcal{R}{a}_2^2-k_{\perp }^2}{a_{2}B}\left(\frac{3\Sigma }{2}{\overline{\mathcal{E}}}^T+\frac{i{k}_{\Vert }}{a_1}{\mathcal{H}}^T\right)\hfill \\ \hfill & +\frac{1}{B}\left(3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)+\frac{2k_{\Vert }^2}{a_1^2}\right)\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right).\hfill \end{array}\end{array}$$

(128)

Finally, the algebraic tensor coefficients are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{B}{2}{\overline{\Sigma }}^T=& -\frac{2}{k_{\perp }^2}\left(\mu +p+\frac{B-\Pi }{2}\right){\Omega }^S+\frac{3\Sigma }{2}{\overline{\mathcal{E}}}^T+\frac{i{k}_{\Vert }}{a_1}{\mathcal{H}}^T\hfill \\ \hfill & -\frac{1}{a_2}\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right),\hfill \end{array}\end{array}$$

(129)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{B}{2}{\overline{\zeta }}^T=& \frac{2i{a}_1}{k_{\Vert }{k}_{\perp }^2}\left(\Sigma +\frac{\Theta }{3}\right)\left(\mu +p+\frac{B-\Pi }{2}\right){\Omega }^S-\frac{i{a}_1}{2k_{\Vert }}\left(\mathcal{R}-{\tilde{k}}^2\right){\overline{\mathcal{E}}}^T\hfill \\ \hfill & +\left(\Sigma +\frac{\Theta }{3}\right){\mathcal{H}}^T+\frac{i{a}_1}{a_2{k}_{\Vert }}\left(\Sigma +\frac{\Theta }{3}\right)\left({\mu }_{\left(F\right)}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V+\mathfrak{B}{\mathfrak{E}}^V\right).\hfill \end{array}\end{array}$$

(130)

8.2. Even Sector

The evolution equations for the coefficients in the even sector are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {{\dot{\mu }}_{\left(F\right)}}^V=& \left(\frac{\Sigma }{2}-\frac{4\Theta }{3}-\frac{3\Sigma {\mu }_{\left(F\right)}}{2B}\right){{\mu }_{\left(F\right)}}^V-\frac{3\Sigma {\mu }_{\left(F\right)}\mathfrak{B}}{2B}{\mathcal{B}}^V\hfill \\ \hfill & +\frac{i{a}_1{\mu }_{\left(F\right)}}{2a_2{k}_{\Vert }}\left(\left(1-\frac{3\Sigma }{B}\left(\Sigma +\frac{\Theta }{3}\right)\right){\mu }_{\left(F\right)}-\frac{2k_{\Vert }^2}{a_1^2}\right){{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & -{\mu }_{\left(F\right)}\left(\left(1+\frac{3\Sigma }{2B}\left(\Sigma -\frac{2\Theta }{3}\right)\right){\mu }_{\left(F\right)}-\frac{k_{\perp }^2}{a_2^2}\right){{\mathcal{V}}_{\left(F\right)}}^V\hfill \\ \hfill & +\frac{3a_2\Sigma {\mu }_{\left(F\right)}}{2}\left(C-1\right){\mathcal{E}}^T-\frac{a_{2}B{\mu }_{\left(F\right)}}{2}\left(C-1\right){\Sigma }^T\hfill \\ \hfill & -\frac{i{a}_2{k}_{\Vert }{\mu }_{\left(F\right)}}{2a_1}\left(J-2\right){\overline{\mathcal{H}}}^T-\frac{i{a}_1{\mu }_{\left(F\right)}}{k_{\Vert }}\left(\frac{G}{B}+\frac{2k_{\Vert }^2}{a_1^2}\right){\overline{\Omega }}^V\hfill \\ \hfill & +\frac{\mathfrak{B}{\mu }_{\left(F\right)}}{2B}\left(2B+3\Sigma \left(\Sigma -\frac{2\Theta }{3}\right)\right){\overline{\mathfrak{E}}}^V,\hfill \end{array}\end{array}$$

(131)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\mathcal{B}}}^V=& \frac{i{a}_1\mathfrak{B}{\mu }_{\left(F\right)}}{a_2{k}_{\Vert }B}\left(B-3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)\right){{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & -\frac{3\mathfrak{B}\Sigma }{B}{{\mu }_{\left(F\right)}}^V+3a_2\mathfrak{B}\Sigma C{\mathcal{E}}^T-a_2\mathfrak{B}BC{\Sigma }^T-\frac{i{a}_2{k}_{\Vert }\mathfrak{B}J}{a_1}{\overline{\mathcal{H}}}^T\hfill \\ \hfill & -\frac{2i{a}_1\mathfrak{B}G}{k_{\Vert }B}{\overline{\Omega }}^V+\left(\frac{3\Sigma {\mathfrak{B}}^2}{B}\left(\Sigma -\frac{2\Theta }{3}\right)-\frac{k_{\perp }^2}{a_2^2}\right){\overline{\mathfrak{E}}}^V\hfill \\ \hfill & +\frac{3}{2}\left(\Sigma -\frac{2\Theta }{3}-\frac{2\Sigma {\mathfrak{B}}^2}{B}\right){\mathcal{B}}^V-\frac{3\mathfrak{B}\Sigma {\mu }_{\left(F\right)}}{B}\left(\Sigma -\frac{2\Theta }{3}\right){{\mathcal{V}}_{\left(F\right)}}^V,\hfill \end{array}\end{array}$$

(132)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\overline{\mathcal{H}}}}^T=& -\frac{i{k}_{\Vert }}{a_1{a}_{2}B}\left(1-\frac{2a_2^2{\mathfrak{B}}^2}{k_{\perp }^2}\right)\left({{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V\right)+\frac{i{a}_2{k}_{\Vert }}{a_1{k}_{\perp }^2}\mathfrak{B}{\mathcal{B}}^V\hfill \\ \hfill & +\frac{2}{a_{2}B}\left(\mu +p+\Pi \right)\left(\Sigma +\frac{\Theta }{3}\right)\left(1-\frac{2a_2^2{\mathfrak{B}}^2}{k_{\perp }^2}\right){\overline{\Omega }}^V\hfill \\ \hfill & +\frac{1}{a_2^{2}B}\left(1-\frac{2a_2^2{\mathfrak{B}}^2}{k_{\perp }^2}\right)\left(\Sigma +\frac{\Theta }{3}\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & -\frac{i{k}_{\Vert }}{a_1{a}_{2}B}\left(1-\frac{2a_2^2{\mathfrak{B}}^2}{k_{\perp }^2}\right)\left(\Sigma -\frac{2\Theta }{3}\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right)\hfill \\ \hfill & +\frac{i{a}_1}{k_{\Vert }}\left(\frac{a_2^2{\mathfrak{B}}^{2}BL}{3k_{\perp }^2}+\frac{3\Pi }{2}\left(\Sigma +\frac{\Theta }{3}\right)\right){\Sigma }^T\hfill \\ \hfill & -\frac{i{a}_1}{k_{\Vert }}\left(\frac{3\Pi }{2}+\left(1-C\right)\frac{k_{\Vert }^2}{a_1^2}-\frac{{\mathfrak{B}}^2}{a_2}\left(\frac{\mathcal{R}{a}_2^2-k_{\perp }^2}{a_{2}B}-\frac{a_2^3\Sigma L}{k_{\perp }^2}\right)\right){\mathcal{E}}^T\hfill \\ \hfill & -\left(\frac{2a_2^2{\mathfrak{B}}^{2}L}{3k_{\perp }^2}+\frac{3}{2}\left(F+\frac{M}{B}\right)\right){\overline{\mathcal{H}}}^T,\hfill \end{array}\end{array}$$

(133)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {{\dot{\mathcal{V}}}_{\left(F\right)}}^S=& -\left(\Sigma +\frac{\Theta }{3}\right){{\mathcal{V}}_{\left(F\right)}}^S,\hfill \end{array}\end{array}$$

(134)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {{\dot{\mathcal{V}}}_{\left(F\right)}}^V=& \frac{1}{2}\left(\Sigma -\frac{2\Theta }{3}-\frac{2{\mathfrak{B}}^2}{\eta {\mu }_{\left(F\right)}}\right){{\mathcal{V}}_{\left(F\right)}}^V-\frac{\mathfrak{B}}{\eta {\mu }_{\left(F\right)}}{\overline{\mathfrak{E}}}^V,\hfill \end{array}\end{array}$$

(135)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\mathcal{E}}}^T=& -\frac{3}{2}\left(F+\Sigma C\right){\mathcal{E}}^T+\frac{i{k}_{\Vert }}{2a_1}\left(J-2\right){\overline{\mathcal{H}}}^T\hfill \\ \hfill & -\frac{1}{2}\left(\mu +p-2\Pi \right){\Sigma }^T+\frac{3\Sigma }{2a_{2}B}\left({{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V\right)+\frac{i{a}_{1}G}{a_2{k}_{\Vert }B}{\overline{\Omega }}^V\hfill \\ \hfill & -\frac{i{a}_1}{2a_2^2{k}_{\Vert }B}\left(B-3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & -\frac{1}{a_{2}B}\left(B-\frac{3\Sigma }{2}\left(\Sigma -\frac{2\Theta }{3}\right)\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right),\hfill \end{array}\end{array}$$

(136)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\overline{\mathfrak{E}}}}^V=& \frac{2a_2^2{k}_{\Vert }^2\mathfrak{B}}{a_1^2{k}_{\perp }^{2}B}{{\mu }_{\left(F\right)}}^V+\frac{2i{a}_2{k}_{\Vert }\mathfrak{B}{\mu }_{\left(F\right)}}{a_1{k}_{\perp }^{2}B}\left(\Sigma +\frac{\Theta }{3}\right){{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & +\left(\frac{2a_2^2{k}_{\Vert }^2{\mu }_{\left(F\right)}}{a_1^2{k}_{\perp }^{2}B}\left(\Sigma -\frac{2\Theta }{3}\right)-\frac{1}{\eta }\right)\mathfrak{B}{{\mathcal{V}}_{\left(F\right)}}^V\hfill \\ \hfill & +\left(\frac{\mathcal{R}{a}_2^2-k_{\perp }^2}{a_{2}B}-\frac{a_2^3\Sigma L}{k_{\perp }^2}\right)\mathfrak{B}{\mathcal{E}}^T+\frac{a_2^3\mathfrak{B}BL}{3k_{\perp }^2}{\Sigma }^T+\frac{2i{a}_2^3{k}_{\Vert }\mathfrak{B}L}{3a_1{k}_{\perp }^2}{\overline{\mathcal{H}}}^T\hfill \\ \hfill & +\frac{4i{a}_2^2{k}_{\Vert }\mathfrak{B}}{a_1{k}_{\perp }^{2}B}\left(\mu +p+\Pi \right)\left(\Sigma +\frac{\Theta }{3}\right){\overline{\Omega }}^V\hfill \\ \hfill & -\left(\frac{\Sigma }{2}+\frac{2\Theta }{3}+\frac{1}{\eta }+\frac{2a_2^2{k}_{\Vert }^2{\mathfrak{B}}^2}{a_1^2{k}_{\perp }^{2}B}\left(\Sigma -\frac{2\Theta }{3}\right)\right){\overline{\mathfrak{E}}}^V\hfill \\ \hfill & +\left(1+\frac{a_2^2{k}_{\Vert }^2}{a_1^2{k}_{\perp }^2}\left(1+\frac{2{\mathfrak{B}}^2}{B}\right)\right){\mathcal{B}}^V,\hfill \end{array}\end{array}$$

(137)

$$\begin{array}{cc}\hfill {\dot{\Sigma }}^T=& \left(\Sigma -\frac{2\Theta }{3}\right){\Sigma }^T-{\mathcal{E}}^T,\hfill \end{array}$$

(138)

$$\begin{array}{cc}\hfill {\dot{\overline{\Omega }}}^V=& -\left(\frac{2\Theta }{3}+\frac{\Sigma }{2}\right){\overline{\Omega }}^V,\hfill \end{array}$$

(139)

where the additional auxiliary variables are defined as

$$\begin{array}{cc}\hfill M=& 2\left(\mathcal{E}+\frac{\Pi }{2}\right)\left(\Sigma +\frac{\Theta }{3}\right)+\frac{\Sigma }{a_2^2}\left(\mathcal{R}{a}_2^2-k_{\perp }^2\right),\hfill \end{array}$$

(140)

$$\begin{array}{cc}\hfill JB=& \frac{a_1^2{k}_{\perp }^2}{a_2^2{k}_{\Vert }^2}\left(\mathcal{R}-{\tilde{k}}^2\right)-3\Sigma \left(\Sigma -\frac{2\Theta }{3}\right),\hfill \end{array}$$

(141)

$$\begin{array}{cc}\hfill G=& \left(\mu +p+\Pi \right)\left(\mathcal{R}-{\tilde{k}}^2\right),\hfill \end{array}$$

(142)

$$\begin{array}{cc}\hfill LB=& 3\Sigma \left(\frac{k_{\perp }^2}{a_2^2}-\frac{k_{\Vert }^2}{a_1^2}\right)+\Theta {\tilde{k}}^2.\hfill \end{array}$$

(143)

The remaining relavant coefficients can be written in the following way in terms of ${\overline{\Omega }}^V$, ${\overline{\mathcal{H}}}^T$, ${\mathcal{E}}^T$, ${\Sigma }^T$, ${{\mu }_{\left(F\right)}}^V$, ${{\mathcal{V}}_{\left(F\right)}}^S$, ${{\mathcal{V}}_{\left(F\right)}}^V$, ${\mathcal{B}}^V$, and ${\overline{\mathfrak{E}}}^V$. The scalar coefficients are

$$\begin{array}{cc}\hfill Q^S=& {\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S,\hfill \end{array}$$

(144)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{i{k}_{\Vert }}{a_1{a}_2^2}{\varphi }^S=& -\frac{BL}{3}{\Sigma }^T-\frac{2k_{\Vert }^2}{a_1^2{a}_{2}B}\left({{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V\right)-\frac{2i{k}_{\Vert }L}{3a_1}{\overline{\mathcal{H}}}^T\hfill \\ \hfill & -\frac{2i{k}_{\Vert }}{a_1{a}_2}\left(\Sigma -\frac{2\Theta }{3}+\frac{2\left(\mu +p+\Pi \right)}{B}\left(\Sigma +\frac{\Theta }{3}\right)\right){\overline{\Omega }}^V\hfill \\ \hfill & +\left(\Sigma L-\frac{k_{\perp }^2}{a_2^{4}B}\left(\mathcal{R}{a}_2^2-k_{\perp }^2\right)\right){\mathcal{E}}^T-\frac{2i{k}_{\Vert }}{a_1{a}_2^{2}B}\left(\Sigma +\frac{\Theta }{3}\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & -\frac{2k_{\Vert }^2}{a_1^2{a}_{2}B}\left(\Sigma -\frac{2\Theta }{3}\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right).\hfill \end{array}\end{array}$$

(145)

The vector coefficients are in turn given by

$$\begin{array}{cc}\hfill {\mu }^V=& {{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V,\hfill \end{array}$$

(146)

$$\begin{array}{cc}\hfill p^V=& \frac{\mathfrak{B}}{3}{\mathcal{B}}^V,Y^V=-\frac{4\mathfrak{B}}{3}{\mathcal{B}}^V,\hfill \end{array}$$

(147)

$$\begin{array}{cc}\hfill Q^V=& {\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V,\hfill \end{array}$$

(148)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathfrak{B}}^V=& \frac{2i{k}_{\Vert }{a}_2^2\mathfrak{B}}{a_1{k}_{\perp }^{2}B}\left({{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V\right)+\frac{i{k}_{\Vert }{a}_2^2}{a_1{k}_{\perp }^2}{\mathcal{B}}^V\hfill \\ \hfill & -\frac{4a_2^2\mathfrak{B}}{k_{\perp }^{2}B}\left(\mu +p+\Pi \right)\left(\Sigma +\frac{\Theta }{3}\right){\overline{\Omega }}^V\hfill \\ \hfill & -\frac{2a_2^3\mathfrak{B}L}{3k_{\perp }^2}{\overline{\mathcal{H}}}^T+\frac{i{a}_1{a}_2^3\mathfrak{B}BL}{3k_{\Vert }{k}_{\perp }^2}{\Sigma }^T\hfill \\ \hfill & +\frac{i{a}_1\mathfrak{B}}{k_{\Vert }}\left(\frac{\mathcal{R}{a}_2^2-k_{\perp }^2}{a_{2}B}-\frac{a_2^3\Sigma L}{k_{\perp }^2}\right){\mathcal{E}}^T-\frac{2a_2\mathfrak{B}}{k_{\perp }^{2}B}\left(\Sigma +\frac{\Theta }{3}\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & +\frac{2i{k}_{\Vert }{a}_2^2\mathfrak{B}}{k_{\perp }^2{a}_{1}B}\left(\Sigma -\frac{2\Theta }{3}\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right),\hfill \end{array}\end{array}$$

(149)

$$\begin{array}{cc}\hfill {\Pi }^V=& -\mathfrak{B}{\mathfrak{B}}^V,\hfill \end{array}$$

(150)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{V^V}{a_2}=& -\frac{B}{3}\left(2C+1\right){\Sigma }^T-\frac{2i{k}_{\Vert }}{3a_1}\left(1+J\right){\overline{\mathcal{H}}}^T+\Sigma \left(2C+1\right){\mathcal{E}}^T\hfill \\ \hfill & -\frac{2\Sigma }{a_{2}B}\left({{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V\right)+\frac{4i{a}_1}{3a_2{k}_{\Vert }}\left(\frac{k_{\Vert }^2}{a_1^2}-\frac{G}{B}\right){\overline{\Omega }}^V\hfill \\ \hfill & +\frac{2i{a}_1}{3a_2^2{k}_{\Vert }B}\left(B-3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & +\frac{2}{3a_{2}B}\left(B-3\Sigma \left(\Sigma -\frac{2\Theta }{3}\right)\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right),\hfill \end{array}\end{array}$$

(151)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{W^V}{a_2}=& \frac{B}{2}\left(C-1\right){\Sigma }^T+\frac{3\Sigma }{2a_{2}B}\left({{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V\right)+\frac{i{k}_{\Vert }}{2a_1}\left(J-2\right){\overline{\mathcal{H}}}^T\hfill \\ \hfill & +\frac{i{a}_1}{a_2{k}_{\Vert }}\left(\frac{2k_{\Vert }^2}{a_1^2}+\frac{G}{B}\right){\overline{\Omega }}^V+\frac{3\Sigma }{2}\left(1-C\right){\mathcal{E}}^T\hfill \\ \hfill & -\frac{i{a}_1}{2a_2^2{k}_{\Vert }B}\left(B-3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & +\frac{1}{a_{2}B}\left(B+\frac{3\Sigma }{2}\left(\Sigma -\frac{2\Theta }{3}\right)\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right),\hfill \end{array}\end{array}$$

(152)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{i{k}_{\Vert }}{a_1{a}_2}{\mathcal{E}}^V=& \frac{k_{\Vert }^2}{a_1^2{a}_{2}B}\left(1-\frac{a_2^2{\mathfrak{B}}^2}{k_{\perp }^2}\right)\left({{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V\right)-\frac{a_2{k}_{\Vert }^2}{2a_1^2{k}_{\perp }^2}\mathfrak{B}{\mathcal{B}}^V\hfill \\ \hfill & -\left(\frac{a_2^2{\mathfrak{B}}^{2}BL}{6k_{\perp }^2}+\frac{3}{2}\left(\mathcal{E}+\frac{\Pi }{2}\right)\left(\Sigma +\frac{\Theta }{3}\right)\right){\Sigma }^T\hfill \\ \hfill & +\frac{2i{k}_{\Vert }}{a_1{a}_{2}B}\left(\mu +p+\Pi \right)\left(\Sigma +\frac{\Theta }{3}\right)\left(1-\frac{a_2^2{\mathfrak{B}}^2}{k_{\perp }^2}\right){\overline{\Omega }}^V\hfill \\ \hfill & -\frac{i{k}_{\Vert }}{a_1}\left(\frac{3}{2B}\left(\frac{\Sigma }{a_2^2}\left(\mathcal{R}{a}_2^2-k_{\perp }^2\right)+2\left(\mathcal{E}+\frac{\Pi }{2}\right)\left(\Sigma +\frac{\Theta }{3}\right)\right)+\frac{a_2^2{\mathfrak{B}}^{2}L}{3k_{\perp }^2}\right){\overline{\mathcal{H}}}^T\hfill \\ \hfill & +\left(\frac{3}{2}\left(\mathcal{E}+\frac{\Pi }{2}\right)-\frac{k_{\Vert }^2}{a_1^2}C-\frac{{\mathfrak{B}}^2}{2}\left(\frac{\mathcal{R}{a}_2^2-k_{\perp }^2}{a_2^{2}B}-\frac{a_2^2\Sigma L}{k_{\perp }^2}\right)\right){\mathcal{E}}^T\hfill \\ \hfill & +\frac{i{k}_{\Vert }}{a_1{a}_2^{2}B}\left(1-\frac{a_2^2{\mathfrak{B}}^2}{k_{\perp }^2}\right)\left(\Sigma +\frac{\Theta }{3}\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & +\frac{k_{\Vert }^2}{a_1^2{a}_{2}B}\left(1-\frac{a_2^2{\mathfrak{B}}^2}{k_{\perp }^2}\right)\left(\Sigma -\frac{2\Theta }{3}\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right),\hfill \end{array}\end{array}$$

(153)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{X^V}{a_2}=& \frac{3}{2}\left(\Sigma -\frac{2\Theta }{3}\right)\left(\mathcal{E}+\frac{\Pi }{2}\right){\Sigma }^T+\frac{1}{a_2}\left(1-\frac{1}{B}\left(\frac{k_{\perp }^2}{a_2^2}+3\left(\mathcal{E}+\frac{\Pi }{2}\right)\right)\right)\mathfrak{B}{\mathcal{B}}^V\hfill \\ \hfill & +\frac{1}{3a_2}\left(1-\frac{3}{B}\left(\frac{k_{\perp }^2}{a_2^2}+3\left(\mathcal{E}+\frac{\Pi }{2}\right)\right)\right){{\mu }_{\left(F\right)}}^V\hfill \\ \hfill & +\frac{3i{a}_1}{k_{\Vert }B}\left(\frac{k_{\Vert }^2}{a_1^2}\left(\mathcal{E}+\frac{\Pi }{2}\right)\left(\Sigma -\frac{2\Theta }{3}\right)+\frac{k_{\perp }^2\Sigma }{2a_2^4}\left(\mathcal{R}{a}_2^2-k_{\perp }^2\right)\right){\overline{\mathcal{H}}}^T\hfill \\ \hfill & +\frac{4i{k}_{\Vert }}{a_1{a}_{2}B}\left(\left(\mu +p+\Pi \right)\left(\Sigma +\frac{\Theta }{3}\right)+\frac{3a_1^2\Sigma G}{4k_{\Vert }^2}\right){\overline{\Omega }}^V\hfill \\ \hfill & +C\left(\frac{k_{\perp }^2}{a_2^2}+3\left(\mathcal{E}+\frac{\Pi }{2}\right)\right){\mathcal{E}}^T\hfill \\ \hfill & +\frac{i{a}_1}{a_2^2{k}_{\Vert }B}\left(\frac{3\Sigma }{2a_2^2}\left(\mathcal{R}{a}_2^2-k_{\perp }^2\right)-\frac{k_{\Vert }^2}{a_1^2}\left(\Sigma -\frac{2\Theta }{3}\right)\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & +\frac{1}{a_{2}B}\left(\Sigma -\frac{2\Theta }{3}\right)\left(\mathcal{R}-\frac{k_{\perp }^2}{a_2^2}+\frac{3\Sigma }{2}\left(\Sigma -\frac{2\Theta }{3}\right)\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right),\hfill \end{array}\end{array}$$

(154)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{i{k}_{\Vert }}{a_1{a}_2}{\Sigma }^V=& \frac{i{k}_{\Vert }}{a_1{a}_2}{\overline{\Omega }}^V-\frac{1}{2}\left(\mathcal{R}-\frac{k_{\perp }^2}{a_2^2}+B\right){\Sigma }^T\hfill \\ \hfill & +\frac{3\Sigma }{2}{\mathcal{E}}^T-\frac{i{k}_{\Vert }}{a_1}{\overline{\mathcal{H}}}^T,\hfill \end{array}\end{array}$$

(155)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{2}{3a_2}{\overline{\mathcal{H}}}^V=& -\left(\mathcal{E}+\frac{\Pi }{2}\right){\Sigma }^T-\frac{\Sigma }{a_{2}B}\left({{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V\right)-\frac{i{k}_{\Vert }J}{3a_1}{\overline{\mathcal{H}}}^T-\frac{2i{a}_{1}G}{3a_2{k}_{\Vert }B}{\overline{\Omega }}^V\hfill \\ \hfill & +\Sigma C{\mathcal{E}}^T-\frac{i{a}_1}{3a_2^2{k}_{\Vert }B}\left(3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)-B\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & -\frac{1}{3a_{2}B}\left(3\Sigma \left(\Sigma -\frac{2\Theta }{3}\right)-B\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right),\hfill \end{array}\end{array}$$

(156)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{i{k}_{\Vert }}{a_1{a}_2}{\alpha }^V=& -\frac{3}{2}\left(\mathcal{E}+\frac{\Pi }{2}\right){\Sigma }^T-\frac{3\Sigma }{2a_{2}B}\left({{\mu }_{\left(F\right)}}^V+\mathfrak{B}{\mathcal{B}}^V\right)-\frac{i{k}_{\Vert }}{2a_1}J{\overline{\mathcal{H}}}^T-\frac{i{a}_{1}G}{a_2{k}_{\Vert }B}{\overline{\Omega }}^V\hfill \\ \hfill & +\frac{3\Sigma C}{2}{\mathcal{E}}^T+\frac{i{a}_1}{2a_2^2{k}_{\Vert }B}\left(B-3\Sigma \left(\Sigma +\frac{\Theta }{3}\right)\right){\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^S\hfill \\ \hfill & +\frac{1}{a_{2}B}\left(B-\frac{3\Sigma }{2}\left(\Sigma -\frac{2\Theta }{3}\right)\right)\left({\mu }_{\left(F\right)}{{\mathcal{V}}_{\left(F\right)}}^V-\mathfrak{B}{\overline{\mathfrak{E}}}^V\right).\hfill \end{array}\end{array}$$

(157)

Finally, the only remaining tensorial coefficient is

$$\frac{i{k}_{\Vert }}{a_1}{\zeta }^T=\left(\Sigma +\frac{\Theta }{3}\right){\Sigma }^T-{\mathcal{E}}^T.$$

(158)

9. Analyzing the Final System

Since the final equations from the previous section are still rather difficult to deal with analytically, we will mainly consider numerical examples. In doing so, we will focus on analyzing how the perturbations depend on the chosen length scales. However, before moving on to the numerical calculations, we should make some notes about the inherent wave length dependencies introduced when defining the gauge-invariant variables and the harmonics.

9.1. Rescaling the Harmonics

When investigating the dependence on length scale, it is important to note that, due to our definition of the vector and tensor harmonics as gradients of the scalar harmonics, these will naturally contain factors of $k_{\perp }$. Hence, the vector and tensor coefficients will contain inherent factors of $k_{\perp }$ relative to the scalar coefficients. Defining a new set of harmonics, where the inherent factors of $k_{\perp }$ have been removed,

$$\begin{array}{cccc}\hfill {Q_{*}}^{k\perp }{}_a& =\frac{a_2}{k_{\perp }}\delta {}_a{Q}^{k_{\perp }},\hfill & \hfill {Q_{*}}^{k\perp }{}_{\mathit{ab}}& =\frac{a_2^2}{k_{\perp }^2}\delta {}_{\{a}\delta {}_{b\}}{Q}^{k_{\perp }},\hfill \end{array}$$

(159)

$$\begin{array}{cccc}\hfill {{\overline{Q}}_{*}}^{k\perp }{}_a& =\frac{a_2}{k_{\perp }}ϵ{}_{\mathit{ab}}\delta {}^b{Q}^{k_{\perp }},\hfill & \hfill {{\overline{Q}}_{*}}^{k\perp }{}_{\mathit{ab}}& =\frac{a_2^2}{k_{\perp }^2}ϵ{}_c\{a\delta {}^c\delta {}_{b\}}{Q}^{k_{\perp }}.\hfill \end{array}$$

(160)

we note that

$$\begin{array}{cccc}\hfill \Psi {}^V& =k_{\perp }^{-1}{{\Psi }_{*}}^V,\hfill & \hfill \overline{\Psi }{}^V& =k_{\perp }^{-1}{{\overline{\Psi }}_{*}}^V,\hfill \end{array}$$

(161)

$$\begin{array}{cccc}\hfill \Psi {}^T& =k_{\perp }^{-2}{{\Psi }_{*}}^T,\hfill & \hfill \overline{\Psi }{}^T& =k_{\perp }^{-2}{{\overline{\Psi }}_{*}}^T,\hfill \end{array}$$

(162)

where the ${\Psi }_{*}$ are the coefficients relative to the new harmonics. These harmonics have the benefit that the scalar, vector and tensor coefficients do not differ with factors of $k_{\perp }$ simply due to the definition of the harmonics.

However, there are still some inherent factors of $k_{\perp }$ that need to be addressed. These factors occur for the gauge-invariant variables that are defined as gradients of quantities that are non-zero on the background spacetime. If we let ${\Phi }_a={\delta }_a\Phi$, then the $\delta$ gradient implies that the coefficients $\{{{\Phi }_{*}}^V,{{\overline{\Phi }}_{*}}^V\}$ will naturally be one order higher in the factor $k_{\perp }/a_2$ than the coefficients for the corresponding scalar perturbation of $\Phi$. Hence, when investigating the length scale dependencies, we will focus on coefficients of the form $\{{\Psi }^S,{{\Psi }_{*}}^{V,T},{{\overline{\Psi }}_{*}}^{V,T},a_2{k}_{\perp }^{-1}{{\Phi }_{*}}^V,a_2{k}_{\perp }^{-1}{{\overline{\Phi }}_{*}}^V\}$ to avoid the aforementioned inherent factors of $k_{\perp }$ due to our definitions.

9.2. Dependence on Perturbation Length Scale

Equipped with the full system of ordinary differential equations and a rescaled set of harmonic coefficients, we now consider specific numerical examples. In these calculations, we will follow a similar convention as in [10] and set $\Lambda =1$, which will fixate the remaining length dimension. As such, we will, in the following, treat all quantities as being dimensionless. In our calculations, we will use the background spacetime specified by the following initial conditions and parameter values:

$$\begin{array}{cccccc}\hfill \Sigma \left(t_0\right)& =0,\hfill & \hfill {\mu }_{\left(F\right)}\left(t_0\right)& =(1-r_{\mathfrak{B}})\mu \left(t_0\right),\hfill & \hfill a_1\left(t_0\right)& =1,\hfill \\ \hfill \Theta \left(t_0\right)& =0.6,\hfill & \hfill \mathfrak{B}\left(t_0\right)& =\sqrt{2r_{\mathfrak{B}}\mu \left(t_0\right)},\hfill & \hfill a_2\left(t_0\right)& =1,\hfill \\ \hfill \mu \left(t_0\right)& =0.36,\hfill & \hfill \gamma & =\frac{5}{3},\hfill & & \hfill \end{array}$$

(163)

where $r_{\mathfrak{B}}$ is the fraction between the initial magnetic energy density and the total energy density. The initial conditions for $\mu$, $\Sigma$, and $\Theta$ are identical to those for a dust background presented in [10], which ends at an expanding de Sitter solution. When not stated otherwise, we will use $r_{\mathfrak{B}}=0.1$ and the initial resistivity $\eta \left(t_0\right)={10}^{-3}$. We will also choose the k-space direction $k_{\Vert }=2k_{\perp }=2k\left(t_0\right)/\sqrt{5}$, where

$$k^2=\frac{k_{\Vert }^2}{a_1^2}+\frac{k_{\perp }^2}{a_2^2},$$

(164)

is the square of the norm of the physical wave vector. For all numerical calculations, we solve Equations (52), (53), (56), (102)–(108), (131)–(139) and (A62) together with Equations (62) and (101). When not stated otherwise, the equations are solved using ode45 in MATLAB with a relative tolerance of ${10}^{-9}$, an absolute tolerance of ${10}^{-34}$, and an initial time $t_0=0$.

9.2.1. Tensor Perturbations

As a first example, we consider the generated magnetic fields due to perturbations in some tensorial coefficients. Following a similar line as in [4], we will let the initial tensorial shear perturbation represent some initial gravitational wave content. The aim is then to investigate how the generated magnetic fields depend on the chosen length scale. Defining the initial Hubble length $L_H=3/\Theta \left(t_0\right)$ and a characteristic length scale for the perturbations $L=2\pi /k\left(t_0\right)$, we solve the final system of equations for values of $L/L_H$ in the range $\left[{10}^{-3},{10}^3\right]$. As for the initial perturbations, we choose

$$\begin{array}{c}\hfill {{\Sigma }_{*}}^T\left(t_0\right)={{\overline{\Sigma }}_{*}}^T\left(t_0\right)={10}^{-19}{k}_m^2\sim {10}^{-13},\end{array}$$

(165)

or equivalently

$$\begin{array}{c}\hfill \Sigma {}^T\left(t_0\right)=\overline{\Sigma }{}^T\left(t_0\right)={10}^{-19}\frac{k_m^2}{k_{\perp }^2},\end{array}$$

(166)

where $k_m$ is the maximum value of $k\left(t_0\right)$ in the studied range. Requiring ${\mathfrak{B}}^V\left(t_0\right)=0$ and using Equations (129) and (146), we also let

$${\mathcal{H}}^T\left(t_0\right)=-\frac{i{a}_{1}B}{2k_{\Vert }}{\overline{\Sigma }}^T{|}_{t_0},{\overline{\mathcal{H}}}^T\left(t_0\right)=\frac{i{a}_{1}B}{2k_{\Vert }}{\Sigma }^T{|}_{t_0},$$

(167)

whilst $\{{\Omega }^S$, ${\overline{\mathcal{E}}}^T$, ${{\overline{\mathcal{V}}}_{\left(F\right)}}^V$, ${\mathfrak{E}}^S$, ${\mathfrak{E}}^V$, ${\overline{\mathfrak{B}}}^V$, ${\overline{\Omega }}^V$, ${\mathcal{E}}^T$, ${{\mu }_{\left(F\right)}}^V$, ${{\mathcal{V}}_{\left(F\right)}}^S$, ${{\mathcal{V}}_{\left(F\right)}}^V$, ${\mathcal{B}}^V$, ${\overline{\mathfrak{E}}}^V\}$ are chosen to vanish initially. Hence, in particular, the perturbations of the electromagnetic fields and the plasma velocities are all zero at $t_0$. Solving the system, we then get the results shown in Figure 1a–c. For super-horizon scales, $L\gg L_H$, the curves in the logarithmic plot are linear with a slope close to two. Decreasing L below $L_H$, the values extracted at the fixed time $t=8$ begin to oscillate. In the region around $L/L_H={10}^{-1}$, the curves displaying the maximum values are almost linear, with a slope of approximately one. Eventually when decreasing L even further, it can be seen that the curves approach a constant value.

Finally, it should be remarked that the noise-like spikes at values of $L/L_H$ slightly larger than unity, as can be seen in Figure 1c, are most likely due to numerical problems encountered when the auxiliary variable B, defined in Equation (110), passes through zero for $t>t_0$. As the final system of ordinary differential equations has been derived under the assumption that B is non-zero, a zero-crossing for B clearly poses a problem, as the derivatives involving factors of $B^{-1}$ are not well defined when $B=0$. In addition to the derivatives not being well defined, we also encounter problems with the coefficients that are given as algebraic expressions in terms of the coefficients in the final system, as their corresponding expressions may involve factors of $B^{-1}$. Therefore, we also have to be careful when specifying initial conditions for the perturbations when $B\left(t_0\right)\sim 0$, since the coefficients with algebraic expressions, which should remain small relative to the background, may become too large even if the initial conditions for the variables in the ODE system appear small.

9.2.2. Velocity Perturbations with Vanishing Initial Ohmic Current

As a second example, we now consider how the generated magnetic field is affected by perturbations in the plasma velocity. To avoid aforementioned problems with the initial conditions when $B\left(t_0\right)\sim 0$, we write the initial velocity perturbation as3

$$\mathcal{V}{}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)=\overline{\mathcal{V}}{}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)={10}^{-19}{k}_{m}B\left(t_0\right).$$

(168)

Since the electrical resistivity is rather small, we also introduce the following initial perturbations in the electric field

$${\mathfrak{E}}^V\left(t_0\right)=\mathfrak{B}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V{|}_{t_0},{\overline{\mathfrak{E}}}^V\left(t_0\right)=-\mathfrak{B}{{\mathcal{V}}_{\left(F\right)}}^V{|}_{t_0}$$

(169)

as to avoid large initial ohmic currents. Finally, requiring that ${\mathfrak{B}}^V\left(t_0\right)=0$, we use Equation (149) and choose

$${{\mathcal{V}}_{\left(F\right)}}^S\left(t_0\right)=\frac{i{k}_{\Vert }{a}_2}{a_1}\left(\Sigma -\frac{2\Theta }{3}\right){\left(\Sigma +\frac{\Theta }{3}\right)}^{-1}\left({{\mathcal{V}}_{\left(F\right)}}^V-\frac{\mathfrak{B}}{{\mu }_{\left(F\right)}}{\overline{\mathfrak{E}}}^V\right){|}_{t_0}.$$

(170)

The other perturbations, $\{{\Omega }^S$, ${\overline{\mathcal{E}}}^T$, ${\mathcal{H}}^T$, ${\mathfrak{E}}^S$, ${\overline{\mathfrak{B}}}^V$, ${\overline{\Omega }}^V$, ${\overline{\mathcal{H}}}^T$, ${\mathcal{E}}^T$, ${\Sigma }^T$,${{\mu }_{\left(F\right)}}^V$, ${\mathcal{B}}^V\}$, that are explicitly present in the ODE system are set to zero initially. Performing the calculations over the same range of length scales as in the previous example, the results in Figure 1d–f are obtained. There, it can be seen that for superhorizon scales, the maximum and end values at $t=8$ follow each other closely with a slope of approximately unity in the logarithmic scale. However, for subhorizon scales, the curves seemingly diverge. The maximum curve continues with a slope of roughly $-1.7$, whilst the curve representing the values at $t=8$ follows a slope closer to $-1$. Finally, it should be noted that the values at $t=8$ begin oscillating when decreasing the length scale below the Hubble scale. However, in contrast to the behavior in the odd sector shown in Figure 1f, the oscillations of the quantities in the even sector, displayed in Figure 1d,e, have become unnoticeable for length scales below ${10}^{-2}{L}_H$.

Finally, as we are considering initial velocity perturbations, it is also interesting to investigate the generation of perturbations in the tensorial coefficients. In Figure 2a,b we display the generated perturbations in the coefficients ${{\Sigma }_{*}}^T$ and ${{\overline{\mathcal{H}}}_{*}}^T$. There, it can be seen that the generated fields increase with decreasing length scale. For subhorizon scales, both ${{\Sigma }_{*}}^T$ and ${{\overline{\mathcal{H}}}_{*}}^T$ display straight lines in the logarithmic scale with a slope of approximately $-3$ and $-2$, respectively. On superhorizon scales, the curves for ${{\Sigma }_{*}}^T$ have a slope of $-1$, while the curves for ${{\overline{\mathcal{H}}}_{*}}^T$ have a slope of about $-2$.

9.2.3. Velocity Perturbations with Vanishing Initial Energy Flux

In the previous example, both velocity and electric field perturbations were introduced to ensure that the ohmic currents and magnetic fields vanished initially. However, in doing so, we introduce an initial energy flux that will affect many gravitationally related variables. For instance, this flux appears in both the differential equations and algebraic relations pertaining to the electric and magnetic parts of the Weyl tensor. Hence, the previously chosen initial values may implicitly contain gravitational effects. Therefore, to better isolate the plasma related effects, we also investigate the case where the initial conditions are chosen so that the energy flux vanishes initially. For this purpose, we introduce

$$\begin{array}{cc}\hfill \mathcal{V}{}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)& =\overline{\mathcal{V}}{}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)={10}^{-19}{k}_m\hfill \end{array}$$

(171)

$$\begin{array}{cc}\hfill {\mathfrak{E}}^V\left(t_0\right)& =-\frac{{\mu }_{\left(F\right)}}{\mathfrak{B}}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V{|}_{t_0},\hfill \end{array}$$

(172)

$$\begin{array}{cc}\hfill {\overline{\mathfrak{E}}}^V\left(t_0\right)& =\frac{{\mu }_{\left(F\right)}}{\mathfrak{B}}{{\mathcal{V}}_{\left(F\right)}}^V{|}_{t_0}.\hfill \end{array}$$

(173)

where we seemingly no longer need the factor $B\left(t_0\right)$ in the velocity perturbations. It should be noted that these initial conditions for the electric field coefficients are incompatible with the ideal MHD limit, as they would in general imply an infinite initial ohmic current for $\eta \to 0$. To avoid similar large initial ohmic currents in our calculations, we increase the resistivity to $\eta \left(t_0\right)=1$. Finally, we set $\{{\Omega }^S$, ${\overline{\mathcal{E}}}^T$, ${\mathcal{H}}^T$, ${\mathfrak{E}}^S$, ${\overline{\mathfrak{B}}}^V$, ${\overline{\Omega }}^V$, ${\overline{\mathcal{H}}}^T$, ${\mathcal{E}}^T$, ${\Sigma }^T$, ${{\mu }_{\left(F\right)}}^V$, ${{\mathcal{V}}_{\left(F\right)}}^S$, ${\mathcal{B}}^V\}$ to zero at $t_0$. Then, on performing the calculations in the same range of length scales as in previous examples, we obtain the results shown in Figure 1g–i. There, it can be seen that, on superhorizon scales, the curves for the maximum value and the value at $t=8$ decay with a slope close to $-1$. For subhorizon scales, the maximum curve and the general trend of the oscillating $t=8$ curve both appear rather independent of the length scale. In Figure 1h,i, the values at $t=8$ are notably smaller than the maximum values, which is also true for subhorizon scales in Figure 1g. For superhorizon scales, the curves in Figure 1g seemingly coincide.

With these initial conditions, it is also interesting to again examine the effect of the velocity perturbations on the tensorial quantities. In Figure 2c,d, we display the generated perturbations in the coefficients ${{\Sigma }_{*}}^T$ and ${{\overline{\mathcal{H}}}_{*}}^T$. There, it can be seen that the generated tensorial perturbations increase with decreasing $L/L_H$, with different slopes on super and subhorizon scales. On superhorizon scales, the curves for ${{\Sigma }_{*}}^T$ have a slope of $-3$, while ${{\overline{\mathcal{H}}}_{*}}^T$ have a slope of about $-2$. On subhorizon scales, the curves for ${{\Sigma }_{*}}^T$ and ${{\overline{\mathcal{H}}}_{*}}^T$ both have a slope of $-1$. Hence, on attributing generated fields in Figure 1g–i and Figure 2 to plasma-related effects, it can be seen that these effects become more important on subhorizon scales.

9.3. Beat Waves

To further investigate the plasma-related effects on subhorizon scales, we perform similar calculations as in Section 9.2.2, but while varying the magnitude of the background magnetic field rather than the length scale of the perturbations. Hence, we introduce the initial conditions

$$\begin{array}{cc}\hfill \mathcal{V}{}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)& =\overline{\mathcal{V}}{}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)={10}^{-10},\hfill \end{array}$$

(174)

$$\begin{array}{cc}\hfill {\mathfrak{E}}^V\left(t_0\right)& =\mathfrak{B}{{\overline{\mathcal{V}}}_{\left(F\right)}}^V{|}_{t_0},{\overline{\mathfrak{E}}}^V\left(t_0\right)=-\mathfrak{B}{{\mathcal{V}}_{\left(F\right)}}^V{|}_{t_0},\hfill \end{array}$$

(175)

but set ${{\mathcal{V}}_{\left(F\right)}}^S$ to zero. To highlight the sought effects, we decrease the initial resistivity drastically to $\eta \left(t_0\right)={10}^{-12}$ and choose the length scale to be $L={10}^{-5}{L}_H$. Since the smallness of the chosen electrical resistivity appears to make the system quite stiff, we use ode15s instead of ode45 in this section, although without specifying an analytical expression for the Jacobian matrix. As for the initial magnitude of the background magnetic field, this is, for reasons made more clear in the next section, specified through the Alfvén velocity $v_A$

$${v_A}^2\equiv \frac{\frac{{\mathfrak{B}}^2}{{\mu }_{\left(F\right)}}}{1+\frac{{\mathfrak{B}}^2}{{\mu }_{\left(F\right)}}}.$$

(176)

On using that

$$\begin{array}{c}\hfill r_{\mathfrak{B}}=\frac{v_A^2}{2-v_A^2}{|}_{t_0},\end{array}$$

(177)

in Equation (163) and calculating for $v_A\left(t_0\right)=0.95$, the results in Figure 3 are obtained for the tensorial coefficients of the magnetic part of the Weyl tensor. There, it can be seen that a clear beat pattern emerges.

To get an analytical explanation of this beat pattern, we now seek wave equations for the plasma velocities and the magnetic part of the Weyl tensor in the limit of large wave numbers. Looking at Equation (134), we see that ${{\mathcal{V}}_{\left(F\right)}}^S$ will not give rise to any wave equation. Using Equations (62) and (134), we find that the evolution of ${{\mathcal{V}}_{\left(F\right)}}^S$ is simply determined by the scale factors on the background

$$\begin{array}{c}\hfill \frac{{{\dot{\mathcal{V}}}_{\left(F\right)}}^S}{{{\mathcal{V}}_{\left(F\right)}}^S}=-\left(\Sigma +\frac{\Theta }{3}\right)=-\frac{{\dot{a}}_1}{a_1}\Rightarrow {{\mathcal{V}}_{\left(F\right)}}^S=\frac{{{\mathcal{V}}_{\left(F\right)}}^S\left(t_0\right)}{a_1},if{a}_1\left(t_0\right)=1.\end{array}$$

(178)

Thus, to obtain any interesting wave equations, we instead look at $\mathcal{V}{}_{{\left(F\right)}_{*}}{}^V$ and $\overline{\mathcal{V}}{}_{{\left(F\right)}_{*}}{}^V$. In doing so, we will assume that the plasma is ideal ($\eta \to 0$), and that the wave lengths considered are much smaller than the scales on the background, so that

$$k^2,\frac{k_{\Vert }^2}{a_1^2},\frac{k_{\perp }^2}{a_2^2}\gg {\Sigma }^2,{\Theta }^2,{\mathfrak{B}}^2,{\mu }_{\left(F\right)},\mathcal{E}.$$

(179)

We will then assume that the frequencies for the variables $\mathcal{V}{}_{{\left(F\right)}_{*}}{}^V$, $\overline{\mathcal{V}}{}_{{\left(F\right)}_{*}}{}^V$, ${{\mathfrak{E}}_{*}}^V$, ${{\overline{\mathfrak{E}}}_{*}}^V$, ${{\mathcal{H}}_{*}}^T$ and ${{\overline{\mathcal{H}}}_{*}}^T$ are of order k in this limit, so that, when applying a time derivative on one of these coefficients, the result is one order higher in k. The idea is then to truncate the equations, keeping only the terms with the highest orders in k. However, to be able to compare the orders of various terms in the equations, we first have to consider the inherent difference in order between the harmonic coefficients.

With the chosen initial conditions in this section, ${\Omega }^S,{{\overline{\Omega }}_{*}}^V$ and ${{\mathcal{V}}_{\left(F\right)}}^S$ are identically zero, and will hence be omitted in the following. Motivated by numerical results using the aforementioned initial conditions, we will treat the magnitudes of ${{\overline{\mathcal{H}}}_{*}}^T,{{\mathcal{E}}_{*}}^T,{{\overline{\mathfrak{E}}}_{*}}^V$ and $\mathcal{V}{}_{{\left(F\right)}_{*}}{}^V$ as being of the same order in k, up to factors of order 10. The coefficients $\mu {}_{{\left(F\right)}_{*}}{}^V$ and ${{\mathcal{B}}_{*}}^V$ are in turn treated as being one order higher in k, whilst ${{\Sigma }_{*}}^T$ is seen to be one order lower in k than $\mathcal{V}{}_{{\left(F\right)}_{*}}{}^V$. With a similar numerical motivation, we assume that the magnitudes of ${{\overline{\mathcal{E}}}_{*}}^T,{{\mathcal{H}}_{*}}^T,{{\mathfrak{E}}_{*}}^V,{{\overline{\mathfrak{B}}}_{*}}^V$ and $\overline{\mathcal{V}}{}_{{\left(F\right)}_{*}}{}^V$ are all of the same order in k, whilst the magnitude of ${\mathfrak{E}}^S$ is significantly smaller. The smallness of ${\mathfrak{E}}^S$ is consistent with the ideal limit $\eta \to 0$, as ${\mathfrak{E}}^S$ should then tend to zero.

On using these relative orders in k and keeping only the terms of highest order in k, the following wave equations can be derived

$$\begin{array}{cc}\hfill \ddot{\mathcal{V}}{}_{{\left(F\right)}_{*}}{}^V+{v_A}^2\left(\frac{k_{\Vert }^2}{a_1^2}+\frac{k_{\perp }^2}{a_2^2}\right)\mathcal{V}{}_{{\left(F\right)}_{*}}{}^V& =0,\hfill \end{array}$$

(180)

$$\begin{array}{cc}\hfill \ddot{\overline{\mathcal{V}}}{}_{{\left(F\right)}_{*}}{}^V+{v_A}^2\frac{k_{\Vert }^2}{a_1^2}\overline{\mathcal{V}}{}_{{\left(F\right)}_{*}}{}^V& =0,\hfill \end{array}$$

(181)

where the previously defined Alfvén velocity $v_A$ appears. When deriving Equations (180) and (181), we used Equations (107) and (137) to write the currents in Equations (105) and (135) in terms of derivatives of the electric field. The electric field coefficients were then eliminated by noting that, in the ideal limit $\eta \to 0$, we must have

$$\begin{array}{cc}\hfill {\mathfrak{E}}^S& \to 0,\hfill \end{array}$$

(182)

$$\begin{array}{cc}\hfill {{\mathfrak{E}}_{*}}^V& \to \mathfrak{B}\overline{\mathcal{V}}{}_{{\left(F\right)}_{*}}{}^V,\hfill \end{array}$$

(183)

$$\begin{array}{cc}\hfill {{\overline{\mathfrak{E}}}_{*}}^V& \to -\mathfrak{B}\mathcal{V}{}_{{\left(F\right)}_{*}}{}^V,\hfill \end{array}$$

(184)

for the currents in Equations (97), (98) and (A69) to remain finite.

The wave equations for the plasma velocities can be compared with non-relativistic results for ideal MHD waves in magnetized plasmas [29]. It can then be seen that Equation (180) is consistent with a fast magnetosonic mode in the limit of the vanishing speed of sound. The fact that ${{\mathcal{V}}_{\left(F\right)}}^S$ shows no wave-like behavior is also consistent with a slow magnetosonic mode in this limit. This is reasonable since we have neglected the fluid pressures and, hence, also the acoustic modes. As for ${{\overline{\mathcal{V}}}_{\left(F\right)}}^V$, Equation (181) is seen to be consistent with a shear Alfvén mode.

Continuing with the tensor coefficients of the magnetic part of the Weyl tensor, these can, to the highest order in k, be found to satisfy the equations

$$\begin{array}{cc}\hfill {{\ddot{\overline{\mathcal{H}}}}_{*}}^T+k^2{{\overline{\mathcal{H}}}_{*}}^T& =\frac{i{k}_{\Vert }{k}_{\perp }}{a_1{a}_2}\left({\mu }_{\left(F\right)}+2{\mathfrak{B}}^2\right)\mathcal{V}{}_{{\left(F\right)}_{*}}{}^V,\hfill \end{array}$$

(185)

$$\begin{array}{cc}\hfill {{\ddot{\mathcal{H}}}_{*}}^T+k^2{{\mathcal{H}}_{*}}^T& =\frac{i{k}_{\Vert }{k}_{\perp }}{a_1{a}_2}\left({\mu }_{\left(F\right)}+2{\mathfrak{B}}^2\right){\overline{\mathcal{V}}}_{{\left(F\right)}_{*}}{}^V.\hfill \end{array}$$

(186)

Assuming that ${\overline{\mathcal{V}}}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)$ and ${\mathcal{V}}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)$ are real constants, ${{\mathcal{H}}_{*}}^T\left(t_0\right)={{\overline{\mathcal{H}}}_{*}}^T\left(t_0\right)=0$, and that the scale factors and the background variables are approximately constant on the time scales that we are considering, Equations (180)–(186) have the solutions

$$\begin{array}{cc}\hfill {\mathcal{V}}_{{\left(F\right)}_{*}}{}^V& ={\mathcal{V}}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)cos\left(v_{A}kt\right),{\overline{\mathcal{V}}}_{{\left(F\right)}_{*}}{}^V={\overline{\mathcal{V}}}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)cos\left(v_A{k}_{p}t\right),\hfill \end{array}$$

(187)

$$\begin{array}{cc}\hfill {{\overline{\mathcal{H}}}_{*}}^T& =\frac{2i{k}_{\Vert }{k}_{\perp }\left({\mu }_{\left(F\right)}+2{\mathfrak{B}}^2\right){\mathcal{V}}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)}{k^2{a}_1{a}_2\left(1-v_A^2\right)}sin\left(\frac{1-v_A}{2}kt\right)sin\left(\frac{1+v_A}{2}kt\right),\hfill \end{array}$$

(188)

$$\begin{array}{cc}\hfill {{\mathcal{H}}_{*}}^T& =\frac{2i{k}_{\Vert }{k}_{\perp }\left({\mu }_{\left(F\right)}+2{\mathfrak{B}}^2\right){\overline{\mathcal{V}}}_{{\left(F\right)}_{*}}{}^V\left(t_0\right)}{k^2{a}_1{a}_2\left(1-{\left(v_A{k}_p/k\right)}^2\right)}sin\left(\frac{1-\left(v_A{k}_p/k\right)}{2}kt\right)sin\left(\frac{1+\left(v_A{k}_p/k\right)}{2}kt\right).\hfill \end{array}$$

(189)

Here $k_p=|k_{\Vert }/a_1|$, and all scale factors and background quantities should be evaluated at $t_0=0$. It should here be noted that we have set the first-order derivatives to zero at $t_0$. This is consistent with the first-order differential equations, as they imply that the derivatives of these harmonic coefficients are of order $k^0$ at $t_0$. As the amplitudes of the derivatives should be of order k, their initial values of order $k^0$ are neglected to the highest order.

Studying the solutions for ${{\mathcal{H}}_{*}}^T$ and ${{\overline{\mathcal{H}}}_{*}}^T$, we clearly see the cause of the beat pattern in Figure 3, which emerges due to the interaction between gravitational wave modes and the magnetized MHD waves. It should also be noted that a resonance can occur when the Alfvén velocity approaches unity. For ${{\overline{\mathcal{H}}}_{*}}^T$, the resonance is independent of the precise relation between $k_{\Vert }$ and $k_{\perp }$, whereas the factor $k_p/k$ limits the resonance for ${{\mathcal{H}}_{*}}^T$.

Comparing the analytical solution for ${{\overline{\mathcal{H}}}_{*}}^T$ with the numerical data, we see a good agreement initially in Figure 4a. However, in Figure 4b, it can be seen that, as time increases, the numerical results become shifted relative to the analytical expression, indicating a change of frequency for the numerical results. The amplitude of the numerical results also seems to decrease slightly in comparison to the analytical expression, but this effect is less noticeable than the frequency shift for the chosen parameters. These differences between the numerical and analytical results are expected, as we, when deriving the analytical expression, have neglected frequency redshifts and damping effects due to the evolution of the background spacetime.

10. Discussion

Some of the numerical results presented here agree rather well with analytical expressions presented in [4], which were found using second-order gauge-invariant perturbation theory on spatially flat FLRW backgrounds in the ideal and cold MHD limit. In accordance with Equation (50) in [4], it was found that, in the superhorizon limit, the dominant late-time contribution to the generated magnetic fields due to gravitational waves depended quadratically on the ratio between the length scale of the magnetized region to first order and the Hubble scale. This contribution was also found to be proportional to the initial shear perturbation [4]. Similar quadratic dependencies on the characteristic length scale of the magnetic field have also been reported in [2,3]. Replacing the length scale of the magnetized region in [4] with the perturbation length scale L used here, we find, despite the differences in approach and background spacetimes, a similar quadratic dependence in Figure 1a–c, where we have assumed that the initial tensorial shear perturbations are independent of the length scale. It should also be noted that the final result in [4] for the generated magnetic field, which was obtained by integrating the equations, has been argued to not be gauge invariant [7]. Since our generated gauge-invariant magnetic field variables ${{\overline{\mathfrak{B}}}_{*}}^V,{{\mathfrak{B}}_{*}}^V$, and $a_2{k}_{\perp }^{-1}{{\mathcal{B}}_{*}}^V$ show similar quadratic dependencies on the characteristic length scale, despite the differences, this result seems rather robust and independent of the precise details.

However, focusing on the subhorizon scale, we note some differences between our results and Equation (50) in [4]. Setting the initial velocity perturbation to zero in Equation (50) from [4], it predicts that the dominant late time contribution in the subhorizon limit should depend linearly on the characteristic length scale of the magnetic field. When including velocity perturbations, the dominant contribution is instead predicted to be proportional to the initial velocity perturbation and inversely proportional to the length scale of the magnetic field [4]. However, with the initial conditions, integration times, and wave lengths considered here, the generated magnetic fields in Figure 1a–c,g–i are found to be rather independent of the length scale of the perturbations in the subhorizon limit. Although the values at $t=8$ oscillate in this limit, the general trend of these values and the calculated maximum amplitudes seem to be independent of $L/L_H$. When considering Figure 1d–f, it must be noted that $B\left(t_0\right)$ introduced in the initial conditions for this case contributes with a factor $L^{-2}$ through the initial velocity perturbation in the subhorizon limit. Therefore, if we assume that the generated field is proportional to the initial velocity perturbation and remove the additional factors of L due to $B\left(t_0\right)$ by multiplying with $L^2$, we would expect the maximum values to be almost independent of the length scale in this limit whilst the values at $t=8$ show a slope of unity. These subhorizon results are in contrast to the predictions from [4], from which we would expect Figure 1a–c to show a slope of unity and Figure 1g–i a slope of negative unity. However, looking at superhorizon scales in Figure 1g–i, we see the inverse dependence on length scale that [4] predicts for subhorizon scales. This may be due to the fact that we, in some sense, have removed some gravitational effects from the initial conditions for Figure 1g–i, so that the contributions from these effects no longer overshadow the velocity contribution in the superhorizon limit.

The exact explanation behind the difference between our subhorizon results and those in [4] is, however, hard to pinpoint without detailed analytical solutions. Possible, but speculative, explanations could be differences in the studied backgrounds, the initial conditions for the perturbations, the chosen model for the electrical resistivity, or that the studied wave lengths were simply too large. It may also be due to some decaying non-dominant terms, similar to the implicit terms in [4], that have not yet completely vanished at the integration times that our results were extracted. Due to all the possible differences, it is not surprising that our results differ from those in [4]. It is, however, interesting that they qualitatively still agree rather well in the superhorizon limit.

Due to the similarities in the superhorzion limit, the results obtained here in this limit are expected to yield estimates for magnetic seed field amplification similar to those in [2,3], in the sense of the magnitude of the generated magnetic perturbations. The homogeneous background magnetic field is then to be interpreted as the seed field in our case. As highlighted in [2,3], the superhorizon limit for the shear driven amplification, shown in Figure 1a–c, looks especially promising, as the amplification can seemingly become quite large for large length scales due to the quadratic behavior. Although promising, similar amplifications have been found to be crucially dependent on the estimate of the shear anisotropy [3]. Assuming that the amplifications in Figure 1a–c are proportional to the shear anisotropy, which is suggested by further numerical calculations, we can get a rough estimate of the amplification by multiplying the shear anisotropy with the square of the perturbation length scale. Extrapolating our results to significantly larger scales than in Figure 1a–c, and using $L/L_H={10}^{20}$ at the end of inflation as in [3] for a field spanning a comoving scale of 10 kpc today, a shear anisotropy ${\Sigma }^T/\Theta \gtrsim {10}^{-40}$ would be needed at the end of inflation to get any significant amplification, as was concluded in [3]. In [2], a shear anisotropy of the order of ${10}^{-26}$ was used, leading to an amplification by as much as 14 orders of magnitude, putting the studied seed fields within the range of galactic dynamo requirements. However, in [3], the estimate of the shear anisotropy was revised to ${10}^{-45}$, making the amplification negligibly small.

As for the amplifications from velocity perturbations in the subhorizon limit, these are also expected to be rather small. In [4], velocity perturbations interacting with the preexisting magnetic field, starting somewhat after matter-radiation equality, were found to, at best, lead to a doubling of the initial magnetic field strength. In that case, the amplification was increased due to an inverse dependence on the subhorizon length scale. However, in our case, we do not see any clear length scale dependence in the subhorizon limit in Figure 1g–i. To get a rough estimate of the amplification here, we can use the same parameters as in [4], extrapolating our results to length scales of the order of $L/L_H={10}^{-4}$. Assuming that the amplification is linear in the velocity perturbation, which further numerical calculations suggest, the amplification is governed mainly by the magnitude of the velocity perturbation, which was estimated to be about ${10}^{-4}$ in [4].

Thus, from very crude estimates, we do not expect any significant amplification, but this depends on factors such as the length scales considered and the magnitude of the initial shear anisotropy. To determine the precise size and relevance of the amplifications encountered in this work, in the sense of the magnitude of the generated magnetic perturbations, a more detailed investigation regarding initial conditions and spectra would be needed. However, due to the assumption of a cold plasma with vanishing pressure, the present formalism is not ideal for considering the evolution of magnetic seeds in the early universe. In a cosmological context, the formalism could have relevance at the end of inflation, where the equation of state effectively could be that of dust [30], or at the early stages of the matter dominated epoch, but not when the equation of state is that of radiation. Hence, we refrain from making any detailed estimates at this point, noting that the aim of this work is mainly to highlight the perturbative formalism.

Finally, it should be noted that the interactions we observe are crucially dependent on the background magnetic field. As an example, it can be seen in Equation (188) that the magnitude of the generated perturbations in the magnetic part of the Weyl tensor increases with increasing $\mathfrak{B}\left(t_0\right)$, especially as $v_A\to 1$. Furthermore, looking at the equations in Section 8, we note that if we were to set $\mathfrak{B}=0$, this would imply that the equations for the electromagnetic fields decouple from the others, at least to the first order. Thus, the interactions we see here of the first order are fundamentally dependent on the anisotropic nature of the background spacetime, as this is what allows us to have a non-zero background magnetic field.

11. Conclusions

In this paper, we have studied linear electromagnetic, gravitational, and plasma-related perturbations on homogeneous and orthogonal LRS class II spacetimes, where we have allowed for a non-zero magnetic field on the background. Using a $1+1+2$ covariant formalism and ultimately applying the cold MHD limit, we have derived a closed set of equations governing the harmonic coefficients of the perturbations. On analyzing the system, we have observed the generation of magnetic field perturbations due to initial perturbations in tensorial quantities related to gravitational waves. The generated fields were then found to depend quadratically on the characteristic length scale in the superhorizon limit, in agreement with previous works using FLRW backgrounds. However, in the subhorizon limit, the similarities were not as apparent.

In addition to the generation of magnetic field perturbations, we also observed beat waves arising due to interference between gravitational waves and the magnetized MHD modes. Thus, replacing the background FLRW spacetime with an anisotropic LRS class II spacetime and allowing for a zeroth-order magnetic field, we were indeed able to see interesting interactions already of the first order in the perturbations.