###### Abstract

Conversion of electromagnetic energy into magnetohydrodynamic energy occurs
when the electric conductivity changes from negligible to finite values. This
process is relevant during the epoch of reheating in the early Universe at the
end of inflation and before the emergence of the radiation-dominated era. We
find that the conversion into kinetic and thermal energies is primarily the
result of electric energy dissipation, while magnetic energy only plays a
secondary role in this process. This means that, since electric energy
dominates over magnetic energy during inflation and reheating, significant
amounts of electric energy can be converted into magnetohydrodynamic energy
when conductivity emerges early enough, before the relevant length scales
become stable.

###### keywords:

Electric energy; cosmological inflation; emergence of conductivity

25 9 1270 Academic Editor: Johan Anderson, 1 August 2023 24 August 2023 29
August 2023 https://doi.org/10.3390/e25091270 Electromagnetic Conversion into
Kinetic and Thermal Energies Electromagnetic Conversion into Kinetic and
Thermal Energies Axel Brandenburg 1,2,3,4,†,‡, and Nousaba Nasrin Protiti1,2
Axel Brandenburg and Nousaba Nasrin Protiti Brandenburg, A.; Protiti, N.N.
Correspondence<EMAIL_ADDRESS>
##  1 Introduction

In hydrodynamic turbulence, dissipation of energy is in principle
straightforward: it must be equal to the energy input accomplished through
forcing; see Figure 1. But when magnetic fields are involved, energy can be
transferred from kinetic energy to magnetic by performing work against the
Lorentz force, $W_{\rm L}$. In that case, the situation is more complicated,
because there are now two exit channels, and it is a priori not clear, which
of the two takes the lion’s share in specific situations; see Figure 2. A
related problem may also occur when the electric energy reservoir is involved,
and especially when this energy reservoir is later absent due to high
conductivity. Before getting to that, let us first recall the different
situations in hydrodynamic and hydromagnetic turbulence.

Figure 1:  Kinetic energy dissipation, $\epsilon_{\rm K}$, of forced
turbulence with kinetic energy density $\langle\rho\bm{u}^{2}\rangle/2$, where
$\rho$ is the density and $\bm{u}$ is the velocity: in the steady state,
everything that gets in does get out.

Figure 2 presumes that kinetic energy can be tapped by dynamo action and
converted into magnetic energy (Moffatt, 1978). This is a generic process that
we now know works in virtually all types of turbulent systems provided the
electric conductivity is large enough (Tobias, 2021). And here comes already
the first problem. Large conductivity means small magnetic diffusivity and
therefore also less dissipation (Brandenburg, 2014). Looking at Figure 2
however, this seems puzzling: In the steady state, the dynamo term $W_{\rm L}$
must be just as large as the resistive term, $\epsilon_{\rm M}$. Thus, if the
dynamo is efficient, also the dissipation must be large, which is not expected
(and also not true).

Figure 2:  Dissipation in dynamos: there are now two exit channels,
$\epsilon_{\rm K}$ and $\epsilon_{\rm M}$, and it is not clear who takes the
lion’s share. Dynamo action corresponds to $W_{\rm L}<0$ (work done against
the Lorentz force, arrow to the right), although energy can also go the other
way around when an initial magnetic field decays.

The puzzle of efficient dynamo action, but inefficient dissipation was solved
by realizing that at large conductivity (and especially large magnetic Prandtl
number, which is the ratio $\mbox{\rm Pr}_{\rm M}\equiv\nu/\eta$ of kinematic
viscosity $\nu$ to magnetic diffusivity $\eta$, is much larger than unity), a
second conversion occurs at smaller length scales where magnetic energy can be
converted back into kinetic energy. This process was termed a reversed dynamo
(Brandenburg and Rempel, 2019), and it happens at small scales when $\mbox{\rm
Pr}_{\rm M}\gg 1$. The concept of a reversed dynamo was already introduced
previously in the context of large-scale flows in two-fluid systems driven by
microscopic fields and flows Mahajan et al. (2005). However, in the context of
Ref. (Brandenburg and Rempel, 2019), the focus was on small-scale dynamos that
drive small-scale flows by the Lorentz force when $\mbox{\rm Pr}_{\rm M}\gg
1$.

While the conversion between magnetic and kinetic energies is reasonably well
understood, not much is known about the conversion from electromagnetic
energy, i.e., the sum of electric and magnetic energies, into magnetic and
hydrodynamic energies when the electric conductivity gradually increases. Such
a process is important at the end of cosmological inflation Field and Carroll
(2023). A stochastic electromagnetic field may have been produced during
inflation and reheating (Sharma et al., 2017). At the end of reheating, the
electric conductivity of the Universe increased. As discussed in Ref.
Brandenburg and Sharma (2021), significant magnetic field losses can occur if
the increase in conductivity is slow, and especially when the magnetic
diffusivity is at an intermediate level for a long time. In the two extreme
cases of very large diffusivity (corresponding to a vacuum with undamped
electromagnetic waves), and very small diffusivity (corresponding to nearly
perfect conductivity), no significant losses are expected. It is only during
the period when the magnetic diffusivity is at an intermediate level that
significant resistive losses can occur.

Once the conductivity has reached large values, i.e., when the magnetic
diffusivity is small, strong turbulent flows will be driven. In that regime,
the Faraday displacement current can be neglected and the equations reduce to
those of magnetohydrodynamics Alfvén (1942). The resulting turbulent flows
cause the magnetic field to undergo turbulent decay with inverse cascading, as
has been studied intensively since the mid 1990s (Brandenburg et al., 1996;
Christensson et al., 2001; Banerjee and Jedamzik, 2004; Kahniashvili et al.,
2013; Brandenburg et al., 2015; Brandenburg and Kahniashvili, 2017; Hosking
and Schekochihin, 2022). At some point around the time of recombination, the
photon mean free path becomes very large, and a process called Silk damping
becomes important (Silk, 1968). It results from the interactions between
photons and the gas and damps out all inhomogeneities in the photon–baryon
plasma (Kolb and Turner, 1990). In Ref. Brandenburg et al. (1997), this was
modeled as a strongly increased viscosity, thereby making the magnetic Prandtl
number even larger. However, a more physical approach is to add a friction
term of the form $-\bm{u}/\tau$ on the right hand side of the momentum
equation (Banerjee and Jedamzik, 2004). It is generally taken for granted that
magnetic fields just survive Silk damping without much additional loss, and
that they are simply frozen into the plasma. However, the details of this
process have not yet been modeled. It is clear, however, that the assumption
of a well-conducting Universe is an excellent one, even after the epoch of
recombination some 380,000 years after the Big Bang, when there were very few
charged particles. As we emphasize below, the electric conductivity was then
still large enough that the electric field was negligible—even in the voids
between galaxy clusters. In cosmology, it is only near the end of inflation
that the electric field can play a significant role. The electric energy
density was then comparable to or in excess of the magnetic energy density.

The goal here is to understand more quantitatively how much magnetic energy
survives during the conversions from electromagnetic fields to
magnetohydrodynamic fields as the conductivity increases. We also consider, in
more detail, the conversion from magnetic fields to electric fields at the end
of the cosmological reheating phase, which is when both fields are still
growing and not yet equal to each other—unlike the situation when
electromagnetic waves are already established and there is no longer any
growth.

##  2 Energetics during the Emergence of Conductivity

The evolution of the electric and magnetic fields, $\bm{E}$ and $\bm{B}$,
respectively, is given by the Maxwell equations, written here in SI units:

$\frac{1}{c^{2}}\frac{\partial\bm{E}}{\partial
t}={\bm{\nabla}}\times\bm{B}-\mu_{0}\bm{J},\qquad{\bm{\nabla}}\cdot\bm{E}=\rho_{\rm
e}/\epsilon_{0},$ (1) $\frac{\partial\bm{B}}{\partial
t}=-{\bm{\nabla}}\times\bm{E},\qquad{\bm{\nabla}}\cdot\bm{B}=0,$ (2)

where $c$ is the speed of light, $\mu_{0}$ is the vacuum permeability,
$\epsilon_{0}\equiv 1/(\mu_{0}c^{2})$ is the vacuum permittivity, and
$\rho_{\rm e}$ is the charge density. To close the equations, we use Ohm’s
law,

$\bm{J}=\sigma\,(\bm{E}+\bm{u}\times\bm{B}),$ (3)

where $\sigma$ is the electric conductivity and $\bm{u}$ is the velocity.

In the very early Universe, inflation dilutes the plasma to the extent that
there are virtually no particles, and hence the electric conductivity
vanishes. Eventually, a phase of reheating must have occurred. One possibility
is that the stretching associated with the cosmological expansion leads to
electromagnetic field amplification until the electric field begins to exceed
the critical field strength for the Schwinger effect Kobayashi and Afshordi
(2014) to lead to the production of charged particles, and thereby to the
emergence of electric conductivity. This change in $\sigma$ implies the
existence of a phase when $\sigma$ has an intermediate value for a certain
duration. This leads to a certain electromagnetic energy loss given by
$\bm{J}\cdot\bm{E}$. This is a well-known result in magnetohydrodynamics,
where the displacement current is ignored, so we have
${\bm{\nabla}}\times\bm{B}=\mu_{0}\bm{J}$. This is then used when deriving the
magnetic energy equation by taking the dot product of Equation (2) with
$\bm{B}$, so we have

$\frac{\partial}{\partial
t}\left(\bm{B}^{2}/2\mu_{0}\right)=-\bm{B}\cdot{\bm{\nabla}}\times\bm{E}/\mu_{0}=\bm{J}\cdot\bm{E}-{\bm{\nabla}}\cdot(\bm{E}\times\bm{B}/\mu_{0}),$
(4)

where we have introduced the Poynting vector $\bm{E}\times\bm{B}/\mu_{0}$. In
the following, we often adopt volume averaging, which we denote by angle
brackets. They depend just on time $t$, but not on position $\bm{x}$. We also
adopt periodic boundary conditions in all three directions, so we call the
domain triply periodic. Since a divergence under triply-periodic volume
averaging vanishes, we just have

$\frac{{\rm d}{}}{{\rm
d}{}t}\left\langle\bm{B}^{2}/2\mu_{0}\right\rangle=-\left\langle\bm{J}\cdot\bm{E}\right\rangle\quad\mbox{(ignoring
the displacement current)}.$ (5)

The $\langle\bm{J}\cdot\bm{E}\rangle$ term, in turn, has two contributions.
Using Ohm’s law in the form

$\bm{E}=\bm{J}/\sigma-\bm{u}\times\bm{B},$ (6)

we find
$\langle\bm{J}\cdot\bm{E}\rangle=\langle\bm{J}^{2}\rangle/\sigma-\langle\bm{J}\cdot(\bm{u}\times\bm{B})\rangle$,
or, using $-\bm{J}\cdot(\bm{u}\times\bm{B})=\bm{u}\cdot(\bm{J}\times\bm{B})$,
we have

$\langle\bm{J}\cdot\bm{E}\rangle=\langle\bm{J}^{2}/\sigma\rangle+\langle\bm{u}\cdot(\bm{J}\times\bm{B})\rangle,$
(7)

so part of the electromagnetic energy turns into Joule (or magnetic) heating,
$\epsilon_{\rm M}\equiv\langle\bm{J}^{2}/\sigma\rangle$, and another part is
converted into kinetic energy through work done by the Lorentz force, $W_{\rm
L}\equiv\langle\bm{u}\cdot(\bm{J}\times\bm{B})\rangle$, which eventually also
becomes converted into heat through viscous (kinetic) heating, $\epsilon_{\rm
K}$. In the case of dynamo action discussed in the introduction, of course,
$W_{\rm L}$ is negative, so work is done against the Lorentz force. This is
why the direction of the arrow in Figure 2 is reversed. Force-free magnetic
fields have $W_{\rm L}=0$ and can therefore not be sustained against
dissipation, but they can be long lived if the current density is small
enough; see Ref. Henriksen (2019) for examples.

In the scenario where reheating is caused by the feedback from the Schwinger
effect, there would be thermal energy supply both from $\epsilon_{\rm K}$ and
$\epsilon_{\rm M}$, leading therefore to a direct coupling between the
resulting heating and the emergence of $\sigma$. The flows of energy between
magnetic, electric, and kinetic energy reservoirs is illustrated in Figure 3.
We denote those by

${\cal E}_{\rm M}\equiv\langle\bm{B}^{2}/2\mu_{0}\rangle,\quad{\cal E}_{\rm
E}\equiv\langle\epsilon_{0}\bm{E}^{2}/2\rangle,\quad\mbox{and}\quad{\cal
E}_{\rm K}\equiv\langle\rho\bm{u}^{2}/2\rangle,$ (8)

respectively. Their evolution equations can be obtained from Equations (1) and
(2), along with the momentum and continuity equations,

$\rho\frac{{\rm D}{}\bm{u}}{{\rm
D}{}t}=-{\bm{\nabla}}p+\bm{J}\times\bm{B}+{\bm{\nabla}}\cdot(2\rho\nu\mbox{\boldmath${\sf
S}$}{}),$ (9) $\frac{{\rm D}{}\ln\rho}{{\rm D}{}t}=-{\bm{\nabla}}\cdot\bm{u},$
(10)

where ${\rm D}{}/{\rm D}{}t\equiv\partial/\partial t+\bm{u}\cdot{\bm{\nabla}}$
is the advective derivative, $p=\rho c_{\rm s}^{2}$ is the pressure for an
isothermal equation of state with sound speed $c_{\rm s}$, which is constant,
$\nu$ is the viscosity, and ${\sf
S}_{ij}=(\partial_{i}u_{j}+\partial_{j}u_{i})/2-\delta_{ij}{\bm{\nabla}}\cdot\bm{u}/3$
are the components of the rate-of-strain tensor ${\sf S}$.

Figure 3:  Energy conversion from magnetic to kinetic energies via the
electric energy reservoir.

Note that, unlike the cases depicted in Figures 1 and 2, there is no energy
input in the system shown in Figure 3. This would change if we were to add
forcing in the momentum equation in Equation (9). We allude to this
interesting possibility at the end of the conclusions in Section 5. Another
possibility that we do discuss in some detail is energy input during the
reheating phase at the end of inflation. We turn to this aspect in Section
4.5.

Taking the dot product of Equation (9) with $\bm{u}$, using Equation (10),
integration by part, and the facts that $\partial_{i}u_{j}$ can be written as
the sum of a symmetric and an antisymmetric tensor, but that the
multiplication with ${\sf S}$ (a symmetric and trace-free tensor) gives no
contribution when $\delta_{ij}{\bm{\nabla}}\cdot\bm{u}/3$ is added, we find
that ${\sf S}_{ij}\partial_{i}u_{j}=\mbox{\boldmath${\sf S}$}{}^{2}$, and thus
obtain the evolution equation for the kinetic energy in the form

$\frac{{\rm d}{}}{{\rm
d}{}t}\left\langle\rho\bm{u}^{2}/2\right\rangle=-\left\langle\bm{u}\cdot{\bm{\nabla}}p\right\rangle+\left\langle\bm{u}\cdot(\bm{J}\times\bm{B})\right\rangle-\left\langle
2\rho\nu\mbox{\boldmath${\sf S}$}{}^{2}\right\rangle,$ (11)

which we can also write more compactly as $\dot{\cal E}_{\rm K}=W_{\rm
P}+W_{\rm L}-\epsilon_{\rm K}$, where $W_{\rm
P}=-\langle\bm{u}\cdot{\bm{\nabla}}p\rangle$ has been defined as the work done
by the pressure force, $\epsilon_{\rm K}=\langle 2\rho\nu\mbox{\boldmath${\sf
S}$}{}^{2}\rangle$ is the viscous heating, and the dot on the kinetic energy
${\cal E}_{\rm K}$ (not to be confused with $\epsilon_{\rm K}$) denotes a time
derivative. Here we have made use of the fact that the divergence
${\bm{\nabla}}\cdot(p\bm{u})=\bm{u}\cdot{\bm{\nabla}}p+p{\bm{\nabla}}\cdot\bm{u}$
has a vanishing volume average for a triply-periodic domain, and therefore
$-\langle\bm{u}\cdot{\bm{\nabla}}p\rangle=\langle
p{\bm{\nabla}}\cdot\bm{u}\rangle$, making it clear that this term leads to
compressional heating and was found to be important in gravitational collapse
simulations Brandenburg and Ntormousi (2022). We will see later that, when
energy is supplied through $W_{\rm L}$, the energy is used to let the kinetic
energy grow ($\dot{\cal E}_{\rm K}>0$) and to drive viscous heating, i.e., we
have

$W_{\rm L}=\dot{\cal E}_{\rm K}+\epsilon_{\rm K}-W_{\rm P}.$ (12)

We recall that the dot on ${\cal E}_{\rm K}$ denotes a time derivative. The
term $W_{\rm P}$ is usually small and negative and thus also contributes (but
only little) to increasing thermal energy. In the present simulations, we used
an isothermal equation of state and thus ignored the evolution of thermal
energy, ${\cal E}_{\rm T}=\langle\rho e\rangle$, where $e=c_{\rm v}T$ is the
internal energy, $c_{\rm v}$ is the specific heat at constant volume, and $T$
is the temperature. If we had included it, we would have had

$\dot{\cal E}_{\rm T}=\epsilon_{\rm M}+\epsilon_{\rm K}-W_{\rm P}.$ (13)

This thermal evolution is important in simulations of thermal magneto-
convection (Brandenburg et al., 1996), where it facilitates buoyancy
variations, or in simulations of the magneto-rotational instability, where
potential energy is converted into kinetic and magnetic energies that then
dissipate as heat and radiation Brandenburg et al. (1995). For our purposes,
however, it suffices to integrate instead the kinetic and magnetic
contributions in time, i.e., to compute $\int\epsilon_{\rm K}\,{\rm d}{}t$ and
$\int\epsilon_{\rm M}\,{\rm d}{}t$, respectively.

Let us now discuss the interplay between electric and magnetic energies. This
interplay is usually ignored in magnetohydrodynamics, where the evolution of
the electric field, i.e., the Faraday displacement current, is ignored Alfvén
(1942). Taking the dot product of Equation (1) with $\bm{E}/\mu_{0}$ and using
$1/(\mu_{0}c^{2})=\epsilon_{0}$, we obtain

$\frac{\partial}{\partial
t}\left(\epsilon_{0}\bm{E}^{2}/2\right)=\frac{\bm{E}}{\mu_{0}c^{2}}\cdot\frac{\partial\bm{E}}{\partial
t}=\bm{E}\cdot{\bm{\nabla}}\times\bm{B}/\mu_{0}-\bm{J}\cdot\bm{E},$ (14)

so, after averaging, we have

$\frac{{\rm d}{}}{{\rm
d}{}t}\left\langle\epsilon_{0}\bm{E}^{2}/2\right\rangle=\left\langle\bm{E}\cdot{\bm{\nabla}}\times\bm{B}/\mu_{0}\right\rangle-\left\langle\bm{J}\cdot\bm{E}\right\rangle.$
(15)

Next, taking the dot product of Equation (2) with $\bm{B}/\mu_{0}$, we obtain

$\frac{\partial}{\partial
t}\left(\bm{B}^{2}/2\mu_{0}\right)=-\bm{B}\cdot{\bm{\nabla}}\times\bm{E}/\mu_{0}.$
(16)

In view of the $\langle\bm{E}\cdot{\bm{\nabla}}\times\bm{B}/\mu_{0}\rangle$
term in Equation (15), it is convenient to rewrite Equation (16) in the form

$\frac{\partial}{\partial
t}\left(\bm{B}^{2}/2\mu_{0}\right)=-\bm{E}\cdot{\bm{\nabla}}\times\bm{B}/\mu_{0}-{\bm{\nabla}}\cdot(\bm{E}\times\bm{B}/\mu_{0}).$
(17)

Again, given that the Poynting flux divergence vanishes under a triply-
periodic volume averaging, we have

$\frac{{\rm d}{}}{{\rm
d}{}t}\left\langle\bm{B}^{2}/2\mu_{0}\right\rangle=-\left\langle\bm{E}\cdot{\bm{\nabla}}\times\bm{B}\right\rangle/\mu_{0}.$
(18)

Note here the difference to Equation (5), which ignores the displacement
current. An equation similar to Equation (5) can only be recovered for the sum
of electric and magnetic energies, which yields

$\frac{{\rm d}{}}{{\rm
d}{}t}\left(\left\langle\bm{B}^{2}/2\mu_{0}\right\rangle+\left\langle\epsilon_{0}\bm{E}^{2}/2\right\rangle\right)=-\left\langle\bm{J}\cdot\bm{E}\right\rangle.$
(19)

An important property of well-conducting media that are considered in
magnetohydrodynamics is that the electric energy is negligible compared to the
magnetic energy. In that limit, Equations (5) and (19) do indeed become
equivalent.

More compactly, we can then write Equation (18) in the form $\dot{\cal E}_{\rm
M}=-Q_{\rm E}$, where $Q_{\rm
E}=\langle\bm{E}\cdot{\bm{\nabla}}\times\bm{B}\rangle$ acts as a source in
$\dot{\cal E}_{\rm E}=Q_{\rm E}-\epsilon_{\rm M}-W_{\rm L}$. Thus, we clearly
see that the electric energy reservoir is not a secondary one whose energy
content is small because of inefficient coupling, but it is an unavoidable
intermediate one through which magnetic energy is channeled efficiently
further to kinetic and thermal energies. This raises the question how safe is
the neglect of the displacement current when prior to the emergence of
conductivity the electric energy dominates over magnetic. This is a typical
situation in inflationary magnetohydrodynamics scenarios that we consider
later in this paper. Before that, we first discuss the nonconducting case
where electric and magnetic energy densities are equally large.

##  3 Numerical Experiments with Different Temporal Conductivity Variations

To illustrate the conversion from electromagnetic energy to
magnetohydrodynamic and thermal energies during the emergence of electric
conductivity, let us consider here a simple one-dimensional experiment.

###  3.1 Electromagnetic Waves and Their Suppression by Conductivity

In one dimension with $\partial/\partial x\neq 0$ and $\sigma=0$, we can have
electromagnetic waves, for example $B_{y}^{\pm}(x,t)=B_{0}\sin k(c\mp ct)$ and
$E_{z}^{\pm}(x,t)=\mp kB_{0}\sin k(c\mp ct)$, traveling in the positive
(negative) $x$ direction. Note that the electric and magnetic energies are
here equal to each other. However, when $\sigma$ becomes large, $|\bm{E}|$
becomes suppressed. To understand this suppression, let us look at Equation
(1). When $\sigma$ becomes large, the ${\bm{\nabla}}\times\bm{B}$ term no
longer needs to be balanced by the displacement current, but by the actual
current. Inserting $\bm{J}=\sigma\bm{E}$ (for the comoving current density),
we find $\bm{E}=\eta{\bm{\nabla}}\times\bm{B}$, so we expect
$|\bm{E}|/|c\bm{B}|=O(\eta k/c)$. Thus, once $\sigma$ becomes large,
$|\bm{E}|/c$ becomes suppressed relative to $|\bm{B}|$ by a factor, $\eta
k/c=k/(c\mu_{0}\sigma)$. Additionally, as mentioned above, both $|\bm{E}|$ and
$|\bm{B}|$ become suppressed due to the intermediate phase when $\sigma$ is
neither small nor large yet. This was discussed in the appendix of Ref.
Brandenburg and Sharma (2021), who found that for a linearly increasing
conductivity profile $\sigma(t)=\sigma_{\max}\,t/t_{\rm trans}$ during a
certain time interval $t_{0}\leq t\leq t_{0}+t_{\rm trans}$ of duration
$t_{\rm trans}$ and starting at $t=t_{0}$, there was an amplitude drop, whose
value increases approximately inversely proportional to
$\eta_{\min}k^{2}t_{\rm trans}$, where $\eta_{\min}=1/\mu_{0}\sigma_{\max}$.

To specify the temporal variation of the conductivity profile, we define a
piecewise linear function that goes from 0 (for $t\leq t_{0}$) to 1 (for
$t\geq t_{0}+\tau_{0}$) through

$\Theta=\max\left[\min\left(\frac{t-t_{0}}{\tau_{0}},\,1\right),\,0\right].$
(20)

The linear $\sigma$ profile used in Ref. Brandenburg and Sharma (2021) is
given by

$\sigma(t)=\sigma_{\min}+(\sigma_{0}-\sigma_{\min})\,\Theta(t),$ (21)

where $\sigma_{0}=1/\mu_{0}\eta_{0}$ and $\sigma_{\min}=1/\mu_{0}\eta_{\max}$.
Here, we also study a profile whose logarithm is linearly varying. We
therefore refer to it as a logarithmic profile, which is of the form

$\sigma(t)=\sigma_{0}\exp\left\\{\ln(\sigma_{\max}/\sigma_{0})\,\left[1-\Theta(t)\right]\right\\},$
(22)

which allows us to specify the duration, during which $\sigma$ transits by an
order of magnitude for any value of $\sigma$. For the linear $\sigma$ profile,
by contrast, the duration would be different for different $\sigma$ ranges,
and it would be very short for large values of $\sigma$.

For the simulations, we use the Pencil Code Pencil Code Collaboration et al.
(2021), which employs the magnetic vector potential $\bm{A}$, so that
$\bm{B}={\bm{\nabla}}\times\bm{A}$ is always divergence-free. The evolution
equation for $\bm{A}$ can then be written as

$\frac{1}{c^{2}}\frac{\partial^{2}\bm{A}}{\partial
t^{2}}-\nabla^{2}\bm{A}+{\bm{\nabla}}{\bm{\nabla}}\cdot\bm{A}+\frac{1}{\eta(t)}\left(\frac{\partial\bm{A}}{\partial
t}+\bm{u}\times\bm{B}\right)=0,$ (23)

where ${\bm{\nabla}}\cdot\bm{A}=0$ if the Coulomb gauge is used. In the Pencil
Code, different gauges are possible, but the Weyl gauge with
$\partial\bm{A}/\partial t=-\bm{E}$ is the one used in that code. In that
case, the ${\bm{\nabla}}{\bm{\nabla}}\cdot\bm{A}$ term must be retained.
Equation (23) shows that in a vacuum, where $1/\eta\to 0$, one recovers a
standard wave equation for waves with propagation speed $c$. In the opposite
limit, where $\eta\to 0$, one can neglect the
$(\eta/c^{2})\,\partial^{2}\bm{A}/\partial t^{2}$ term and one recovers the
usual induction equation, where $\eta\nabla^{2}\bm{A}$ acts as a diffusion
term.

###  3.2 Transition to the High-Conductivity Regime for Different Parameters

The transition to the high-conductivity regime involves the conversion of
electromagnetic waves to magnetohydrodynamic waves Alfvén (1942). One can
imagine that this process is more efficient when the frequencies of both waves
are equal. In the high conductivity regime, the frequency of
magnetohydrodynamic waves depends on the strength of the imposed magnetic
field, $B_{0}$, which determines the nominal Alfvén speed, $v_{\rm
A0}=B_{0}/\sqrt{\rho\mu_{0}}$. Since Alfvén waves propagate along the magnetic
field, and since $\partial/\partial x\neq 0$, we impose the magnetic field
also in the $x$ direction, i.e., we write
$\bm{B}=\hat{\mbox{\boldmath$x$}}{}B_{0}+{\bm{\nabla}}\times\bm{A}$, where
${\bm{\nabla}}\times\bm{A}$ is the departure from the imposed magnetic field.
In the following numerical experiments, we choose $t_{0}=0$.

Figure 4:  Evolution of $B_{y}(x,t)$ for the logarithmic $\sigma$ profile with
(a) $v_{\rm A0}=1$, (b) $v_{\rm A0}=0.3$, and (c) $v_{\rm A0}=0.1$, and
$t_{\rm trans}=10$ in all cases.

The result is shown in Figure 4, where we compare $B_{y}(x,t)$ as a colored
contour plot in the $xt$ plane. We compute the solution in a domain of size
$L$, so the lowest wave number is $k_{1}=2\pi/L$. The density is initially
uniform and equal to $\rho_{0}$. In the following, we use units where
$c=k_{1}=\rho_{0}=1$. We see that for $v_{\rm A0}=1$, the wave propagates
almost unaffectedly by the switch to high conductivity. Here, the frequency of
the electromagnetic wave is $ck_{1}=1$ and the nominal frequency of the Alfvén
wave, $v_{\rm A0}k$, is also unity but the actual frequency is slightly less
than that. This is because of special relativity effects forcing the wave
speed to be always less than $c$. In fact, the actual wave speed is $v_{\rm
A}=v_{\rm A0}/(1+v_{\rm A0}^{2}/c^{2})^{1/2}$ Gedalin (1993).

Figure 5:  Evolution of $B_{y}(x,t)$ for the logarithmic $\sigma$ profile with
$v_{\rm A}=0.3$ and (a) the logarithmic $\sigma$ profile with $t_{\rm
trans}=10$, (b) the linear $\sigma$ profile with $t_{\rm trans}=10$, and (c)
the linear $\sigma$ profile with $t_{\rm trans}=500$.

For smaller values of $v_{\rm A0}$, we see that not only the wave speed is
less, as seen from the shallower inclination of the pattern in Figures 4(b)
and (c), but there is also a certain drop of the wave amplitude, and there is
also an additional modulation resulting from an effective initial condition at
$t=0$ that does not match the eigenfunction for an Alfvén wave.

In Figure 5, we compare the logarithmic $\sigma$ profile with the linear one
using $v_{\rm A0}=0.3$. For the logarithmic profile, the drop in amplitude is
clearly larger than that for the linear $\sigma$ profile. To obtain a similar
drop with the linear $\sigma$ profile, one would need to increase $t_{\rm
trans}$ to about 500; see Figure 5(c).

To be more quantitative, we compare in Figure 6(a) the evolution of $B_{y}$ at
one specific point $x=x_{*}$ for the three runs of Figures 5(a) and (c). Note
that the drop of the wave amplitude after $t=0$ is similar for runs a and c,
but much less for run b.

In Figure 6(b), we also show how $\sigma$ varies. We do this by plotting the
nondimensional resistivity

$R(t)\equiv\eta(t)k/c,$ (24)

which decreases from $10^{4}$ to $5\times 10^{-4}$. We recall that it is also
this ratio that we identified in the beginning of this section as the one that
characterizes the value of $|E|/|cB|$. We see that most of the decay happens
when it transits through unity. Owing to the logarithmic nature of the
profile, the quantity $R(t)$ spends a time interval of about $ckt_{\rm
trans}=5$ while $R(t)$ changes from 10 to 0.1. By contrast, for the linear
profile, the time interval is virtually non-existing. For $t_{\rm trans}=500$,
on the other hand, $ckt_{\rm trans}$ is similar to what led to the to a
similar decay for the logarithmic profile. This is also confirmed by the inset
of panel b, which shows that $R(t)$ traverses unity by a margin of one order
of magnitude for a and c, but not for b. The results discussed above confirm
that the relevant time interval is indeed that where $R(t)$ is within an order
of magnitude around unity.

Figure 6:  (a) Evolution of $B_{y}$ at one specific point $x=x_{*}$ in the
three runs of Figures 5(a) and (c). Note that the drop of the wave amplitude
after $t=0$, and specifically at $t=50$, is similar for runs a and c, but much
less for b. (b) Dependence of the nondimensional resistivity $R(t)=\eta(t)k/c$
for the logarithmic profile with $t_{\rm trans}=10$ in run a (black), and the
linear profile with $t_{\rm trans}=10$ in run b (red) and $t_{\rm trans}=500$
in run c (blue). The inset shows a blow-up of a narrow strip around $R=1$
using a logarithmic time axis. We see from the inset that the time spent in
$R(t)$ traversing unity by a margin of one order of magnitude (marked by the
thick part of the black line) is similar for a and c, but virtually non-
existing for b.

Looking at Figure 7, we see that the electric energy was initially equal to
the magnetic one, but as the conductivity increases, there is a rapid decline
of electric energy ($\dot{{\cal E}_{\rm E}}<0$), and most of it dissipates
thermally, while only a small fraction ($<10\%$ for $t_{\rm trans}=10$) is
transferred to kinetic energy. It turns out that the mean magnetic and
electric energy densities decay like $\exp(-\nu k^{2}t)$, i.e., without a
factor of $2$ in the exponent, reflecting therefore not a change in the
kinetic energy, but rather in the velocity, which enters through the work term
$W_{\rm L}$.). Furthermore, the ratio is here ${\cal E}_{\rm E}/{\cal E}_{\rm
M}\approx 10$ for $\mbox{\rm Pr}_{\rm M}=20$. Note that the oscillations in
${\cal E}_{\rm M}+{\cal E}_{\rm K}$ (orange lines) are compensated mostly
entirely by those in ${\cal E}_{\rm E}$ (blue lines).

Figure 7:  Initially, all the energy is in electromagnetic energy, ${\cal
E}_{\rm E}+{\cal E}_{\rm M}$ for $\nu=0.01$ and $\eta_{\rm fin}=5\times
10^{-4}$. In the end, all the energy is converted into heat. The red lines
give the integrated Ohmic and viscous energy gains, $\int\epsilon_{\rm
M}\,{\rm d}{}t$ and $\int\epsilon_{\rm K}\,{\rm d}{}t$, respectively. At
intermediate times, this energy is distributed to equal amounts among kinetic
energy ${\cal E}_{\rm K}$ (green lines) and magnetic energy ${\cal E}_{\rm M}$
(gray lines). The orange lines shows their sum, ${\cal E}_{\rm K}+{\cal
E}_{\rm M}$. The blue lines represent ${\cal E}_{\rm E}$. The inset shows a
blow-up of the same graph around the origin. We see that ${\cal E}_{\rm E}$
varies in phase with ${\cal E}_{\rm K}$, but an anti-phase both with ${\cal
E}_{\rm M}$ and the residual ${\cal E}_{\rm M}+{\cal E}_{\rm K}$.

In Figure 8, we show the evolution of various energy fluxes. We see that the
magnetic energy decays and gives off energy to the electric energy reservoir
through the term $Q_{\rm
E}=\langle\bm{E}\cdot{\bm{\nabla}}\times\bm{B}\rangle>0$. The magnetic heating
is thus composed of the following terms:

$\epsilon_{\rm M}=-\dot{\cal E}_{\rm E}+Q_{\rm E}-W_{\rm L}.$ (25)

For rapid transits, $t_{\rm
trans}\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr<\cr\sim\cr}}}}5$,
$Q_{\rm E}$ is small compared with $-\dot{\cal E}_{\rm E}$, so $\epsilon_{\rm
M}$ is mostly entirely the result of exhausting electric energy, i.e.,
$\dot{\cal E}_{\rm E}<0$. For longer transits, $t_{\rm trans}>10$, $Q_{\rm
E}\approx-\dot{\cal E}_{\rm E}$, so $\epsilon_{\rm M}$ is supplied to about
50% through $Q_{\rm E}$ and to another 50% through $-\dot{\cal E}_{\rm E}$.
These differences are summarized in Table 3.2.

Figure 8:  Evolution of energy fluxes for the model with the logarithmic
conductivity profile with $\eta=5\times 10^{-4}=\nu$ at late times. In all
cases, the initial diffusivity is $\eta_{\rm ini}=10^{4}$. The main difference
to the run with a larger viscosity is that $\epsilon_{\rm K}$ is larger.

[t!] The two regimes of energy transfer for short and long transits.  rapid
transits long transits criterion $t_{\rm trans}<10$ $t_{\rm trans}>10$ Lorentz
work $W_{\rm L}/\epsilon_{\rm M}\approx 0.1$ $W_{\rm L}/\epsilon_{\rm
M}\approx 5\,t_{\rm trans}^{-1.6}$ heating $\epsilon_{\rm M}\approx-\dot{\cal
E}_{\rm E}$ and $Q_{\rm E}\ll\epsilon_{\rm M}$ $\epsilon_{\rm
M}\approx-0.5\,\dot{\cal E}_{\rm E}\approx 0.5\,Q_{\rm E}$

In connection with Figure 4, we noted that there is a certain drop of the wave
amplitude after the transit to large conductivity. This drop was larger for a
larger ratio of the electromagnetic to Alfvén wave speeds. When the nominal
Alfvén speed was equal to the speed of light, the drop was small. In Figure 9
we quantify this by plotting ${\cal E}_{\rm M}$ at $t=100$, i.e., after the
conductivity has increased to a large value, vs $t_{\rm trans}$ for different
values of $v_{\rm A0}/c$. We confirm the results of Ref. Brandenburg and
Sharma (2021) where the logarithmic drop was found to depend linearly on the
value of $t_{\rm trans}$. However, we now also see that the slope of this
curve decreases with increasing Alfvén wave speed.

Figure 9:  ${\cal E}_{\rm M}$ at $t=100$, i.e., after the conductivity has
increased to large value, vs $t_{\rm trans}$ for $v_{\rm A0}/c=1$ (orange),
$v_{\rm A0}/c=0.3$ (red), and $v_{\rm A0}/c=0.1$ (blue).

##  4 Cosmological Application Prior to Radiation Domination

As alluded to above, the end of inflation might provide an opportunity to
illustrate electromagnetic energy conversion, because in that case, the
electric energy can greatly exceed the magnetic one.

###  4.1 Magnetic Fields in Cosmology

In the present Universe, magnetic fields are constantly being regenerated by
dynamo action on all scales up to those of galaxy clusters. The energy source
is here gravitational, which is released through accretion or direct collapse.
Magnetic fields may also be present on even larger scales. However, in the
locations between galaxy clusters, i.e., in what is often referred to as
voids, it is generally thought impossible to produce magnetic fields through
contemporary dynamo action; see Refs. Durrer and Neronov (2013); Subramanian
(2016) for reviews on the subject. Nevertheless, indirect evidence for the
existence of magnetic fields in voids, and more specifically for lower limits
of the magnetic field strength, comes from the non-observation of secondary
photons in the halos to blazars, which are active galactic nuclei producing
TeV photons. These photons interact with those of the extragalactic background
light through inverse Compton scattering to produce GeV photons. Those
secondary GeV photons are not observed. Their non-observation could be
explained by an intervening magnetic field of about $10^{-16}\,{\rm G}$ on a
megaparsec scale Neronov and Vovk (2010); Taylor et al. (2011). This field
would deflect electrons and positrons in opposite directions, preventing them
from recombining and thereby disrupting the energy cascade toward the lower
GeV photons.

The non-observation of GeV photons might have other reasons, for example
plasma instabilities that disrupt the electron–positron beam Broderick et al.
(2012, 2018). Nevertheless, even then, a certain fraction of the plasma beam
disruption might still be caused by magnetic fields Sironi and Giannios
(2014), which could explain the GeV halos of at least some blazars Alves
Batista et al. (2019). If magnetic fields really do exist on very large
cosmological scales, they may be primordial in origin. This may mean that they
have been created during or before the radiation-dominated era of the
Universe, for example during one of the cosmological phase transitions or
during inflation. Inflation was a phase where the conversion from
electromagnetic fields to magnetohydrodynamic fields played an important role,
which is what we are interested in here.

###  4.2 Use of Comoving Variables and Conformal Time

The Universe is expanding with time, as described by the scale factor $a(t)$.
The equations of magnetohydrodynamics therefore contain additional terms with
factors of $a(t)$ and its time derivatives. However, by using scaled
variables, $\tilde{\bm{A}}=a\bm{A}$, $\tilde{\bm{B}}=a^{2}\bm{B}$,
$\tilde{\bm{E}}=a^{2}\bm{E}$, $\tilde{\bm{J}}=a^{3}\bm{J}$,
$\tilde{\bm{x}}=\bm{x}/a$, along with conformal time, $\tilde{t}=\int{\rm
d}{}t/a(t)$, all $a(t)$ factors and other terms involving $a(t)$ disappear
from the magnetohydrodynamic equations Brandenburg et al. (1996). The velocity
is the same in both frames, i.e., $\tilde{\bm{u}}=\bm{u}$.

Given that the equations with tilded variables are equal to the ordinary ones
in a non-expanded Universe, it is convenient to skip all tildes from now on.
However, when discussing the evolution of the scale factor, for example, we
again need physical time, which will then be denoted by $t_{\rm phys}$, while
$t$ then still denotes conformal time. Here is where we have a notational
dilemma, because in cosmology, derivatives with respect to physical (or
cosmic) time are often denoted by dots, while those with respect to conformal
time are denoted by primes. We therefore decided here to follow the same
convention, so $a^{\prime}={\rm d}{}a/{\rm d}{}t$ and $a^{\prime\prime}={\rm
d}{}^{2}a/{\rm d}{}t^{2}$ denote derivatives with respect to conformal time.

###  4.3 Inflationary Magnetogenesis

Inflationary magnetogenesis models assume the breaking of conformal invariance
through a coupling to a scalar field such as the inflaton. Another possible
coupling is through an axion field, which would result in helical
magnetogenesis, but this will not be considered here. The dynamics of the
scalar field is interesting in its own right; see Refs. Adshead et al. (2015,
2016, 2020) for numerical investigations. To simplify the model, one commonly
replaces this coupling by a prefactor $f^{2}$, where $f$ depends on the scale
factor of the Universe. This factor $f^{2}$ enters in the electromagnetic
energy contribution to the Lagrangian density $f^{2}F^{\mu\nu}F_{\mu\nu}$,
where $F_{\mu\nu}$ is the Faraday tensor Ratra (1992). Early approaches to
inflationary magnetogenesis exposed specific problems: the strong coupling and
the backreaction problems Demozzi et al. (2009), as well as the Schwinger
effect constraint, which can lead to a premature increase in the electric
conductivity. This shorts the electric field and prevents further magnetic
field growth Kobayashi and Afshordi (2014). This is particularly important for
models that solve the backreaction problem by choosing a low energy scale
inflation (Ferreira et al., 2013), but could be avoided if charged particles
attain sufficiently large masses by some mechanism in the early Universe
Kobayashi and Sloth (2019). The three problems are avoided by requiring the
function $f$ to obey certain constraints Sharma et al. (2017, 2018).

Successful models of inflationary magnetogenesis are thus possible, but this
does not mean that the underlying cosmological models are also physically
preferred options. Nevertheless, for the purpose of discussing the
electromagnetic energy conversion, which is the goal of this paper, those
models are a useful choice.

Three-dimensional simulations of inflationary magnetogenesis have been
performed by assuming an abrupt switch from electromagnetism without currents
and magnetohydrodynamics where the displacement current is already neglected
Brandenburg and Sharma (2021); Brandenburg et al. (2021). They solved the
evolution equations for the scaled magnetic vector potential,
$\mbox{\boldmath${\cal A}$}{}\equiv f\bm{A}$, in the Coulomb gauge:

$\left[\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}-\nabla^{2}-k_{*}^{2}(t)\right]\mbox{\boldmath${\cal A}$}{}=0.$ (26)

where $k_{*}^{2}(t)=f^{\prime\prime}/f$ is a generation term, because it
destabilizes the field at large length scales for wavenumbers $k<k_{*}(t)$.
Analogous to the primes on $a(t)$, primes on $f(t)$ also denote conformal time
derivatives. Toward the end of the reheating phase, where $f\to 1$, we expect
$k_{*}(t)\to 0$.

Our aim here is to present calculations where the transit from vacuum to high
conductivity is continuous. In particular, to calculate the generation term
$k_{*}^{2}(t)$, one commonly uses a power law representation in terms of
$a(t)$ of the form $f\propto a^{\alpha}$ with $\alpha>0$ during inflation and
$f\propto a^{-\beta}$ with $\beta>0$ during reheating Subramanian (2010). We
are here only interested in the reheating phase where $a(t)\propto t^{2}$
Sharma et al. (2017, 2018) such that it is unity when the radiation-dominated
era begins, and therefore $f=1$ and $k_{*}^{2}(t)=0$ for $a>1$. For $a<1$, by
contrast, we have

$k_{*}^{2}(t)=\beta\left[(\beta+1)(a^{\prime}/a)^{2}-a^{\prime\prime}/a\right].$
(27)

Note that for $a=t^{2}$, we have $a^{\prime}=2t$ and $a^{\prime\prime}=2$, so
$(a^{\prime}/a)^{2}=4/t^{2}$ and $a^{\prime\prime}/a=2/t^{2}$, and therefore
$f^{\prime\prime}/f=2\beta(2\beta+1)/t^{2}$.

Contrary to the earlier numerical work Brandenburg and Sharma (2021);
Brandenburg et al. (2021), the displacement current is now included at all
times. However, there is still a problem in that $k_{*}^{2}(t)$ has a
discontinuity from $k_{*}^{2}(1)=\beta(\beta+1)\neq 0$ to zero at the moment
when the conductivity is turned on. In the simulations, this did not seem to
have any serious effect on the results, because the magnetic field at the end
of the electromagnetic phase only acted as an initial condition for the
magnetohydrodynamic calculation after the switch. In a continuous calculation
without switch, however, this problem must be avoided. This will be addressed
next.

###  4.4 Continuous Version of the Generation Term

An instructive way of obtaining a smooth transition from a quadratic to a
linear growth profile of $a(t)$ is obtained by solving the Friedmann equations
for a piecewise constant equation of state, $w(a)$, which relates the pressure
with the density through $p=w\rho$. Under the assumption of zero curvature,
i.e., the Universe is conformally flat, but expanding, the Friedmann equations
can be written as a single equation which, in physical time, takes the form

$a^{-1}\,{\rm d}{}^{2}a/{\rm d}{}t_{\rm
phys}^{2}=-\textstyle{\frac{1}{2}}H^{2}\left[1+3w(a)\right],$ (28)

where $H=a^{-1}{\rm d}{}a/{\rm d}{}t_{\rm phys}$ is the standard Hubble
parameter. Here, $w(a)=1/3$ during the radiation-dominated era and $w(a)=0$
during reheating when there were no photons, which is therefore equivalent to
the matter-dominated era that also occurs later after recombination and before
the Universe began to accelerate again. The accelerated exponential expansion
of the Universe during inflation, and also the late acceleration of the
present Universe, correspond to $w=-1$, but this will not be considered in the
present paper.

It is convenient to solve the Friedmann equation with zero curvature in
conformal time. It then takes the form
$a^{\prime\prime}/a=\textstyle{\frac{1}{2}}{\cal H}^{2}\,(1-3w)$, where ${\cal
H}=a^{\prime}/a$ is the conformal Hubble parameter. It is related to the usual
one, $H$, through ${\cal H}={\rm d}{}a/\partial t_{\rm phys}=aH$. Note the
opposite sign of the terms on the right-hand side and the opposite sign in
front of $3w(a)$ compared to the formulation in terms of physical time. The
equation for $a^{\prime\prime}$ is easily solved by splitting it into two
first-order equations and introducing a new variable $b(t)$ and solving for

$a^{\prime}=b,\quad b^{\prime}=(b^{2}/2a)(1-3w);$ (29)

see also Ref. He et al. (2023) for similar work in another context.

Figure 10 shows the solution for $a(t)$ and the ratios $a^{\prime}/a$ and
$a^{\prime\prime}/a$ compensated by $t$ and $t^{2}$, respectively, which
allows us to see more clearly how $a^{\prime}/a$ changes from $2/t$ to $1/t$
and $a^{\prime\prime}/a$ changes from $2/t^{2}$ to zero as we go from the
reheating era to the radiation-dominated Universe after reheating.

Figure 10:  $t$ dependence of (a) the scale factor $a(t)$, (b) the compensated
Hubble coefficient ${\cal H}t=t\,a^{\prime}/a$, and (c) the compensated left-
hand side of the Friedmann equation, $t^{2}\,a^{\prime\prime}/a$. In (a), the
asymptotic dependences $a=t^{2}$ and $2t$ for $t\ll 1$ and $\gg 1$ are
overplotted as dashed-dotted orange lines. In (c), the function $3w(a(t))$ is
overplotted as a dotted red line.  Figure 11:  $t$ dependence of ${\cal
E}_{\rm M}$ (red), ${\cal E}_{\rm E}$ (blue), and ${\cal E}_{\rm K}$ (green)
for runs with $\beta=1$ (dotted lines), $2$ (dashed lines), and $4$ (solid
lines) for Set (i) with $k=10$, $t_{0}=1$, and $t_{\rm trans}=10$. The initial
amplitudes have been arranged such that $B_{\rm rms}=0.01$ at $t=1$. From the
double-logarithmic representation, we see that the growth of ${\cal E}_{\rm
M}$ and ${\cal E}_{\rm E}$ is algebraic, and much faster for the models with a
larger value of $\beta$. Before $t=1$, ${\cal E}_{\rm E}$ dominates over
${\cal E}_{\rm M}$, but drops immediately after $t=1$, when resistivity
emerges and kinetic energy is being generated. Both ${\cal E}_{\rm M}$ and
${\cal E}_{\rm K}$ are larger for larger values of $\beta$.

The generation term $k_{*}^{2}(t)\equiv f^{\prime\prime}/f$ determines the
wavenumber below which the solution is still unstable. However, since
$k_{*}^{2}(t)=2\beta(2\beta+1)/t^{2}$, we have $ctk={\rm const}{}\approx
2\beta+1/2$; see Table 4.4. In Figure 11, we plot the evolution of ${\cal
E}_{\rm E}$, ${\cal E}_{\rm M}$, and ${\cal E}_{\rm K}$ for $k=10$ for all
three values of $\beta$: 1, 2, and 4. Here and below, the initial amplitudes
have been arranged such that ${\cal E}_{\rm M}=10^{-4}$ at $t=t_{0}$. In all
cases, the solution has become stable by the time $t=t_{0}=1$, and we see
electromagnetic oscillations toward the end of the reheating phase before
conductivity turns on at $t_{0}=1$. This is here referred to as Set (i).

[H] Parameters relevant for the models with different values of $\beta$.
$\beta$ $2\beta+1$ $2\beta+1/2$ $k_{*}(1)$ 1 3 2.5 2.45 2 5 4.5 4.47 4 9 8.5
8.49

It is easy to see that on large length scales, when the $\nabla^{2}$ operator
in Equation (26) is negligible compared with $k_{*}^{2}(t)$, we have

$\displaystyle{\cal A}_{z}(x,t)=A_{0}t^{2\beta+1}k^{-1}\cos kx,\quad
A_{z}(x,t)={\cal A}_{z}/f=A_{0}t^{\beta+1}k^{-1}\cos kx,$ (30) $\displaystyle
B_{y}(x,t)=A_{0}t^{\beta+1}\sin kx,\quad E_{z}(x,t)=-\partial A_{z}/\partial
t=-(\beta+1)A_{0}t^{\beta}k^{-1}\cos kx.$ (31)

Thus, for $ckt\ll 1$, corresponding to super-horizon scales, where and when
the modes are still unstable, we have $tE_{\rm rms}/B_{\rm
rms}\approx\beta+1$. On smaller length scales, i.e., for larger $k$ values,
the modes become stable and we have the usual electromagnetic waves.

Figure 12:  Similarly to Figure 11, but for Set (ii) with $k=10$, $t_{0}=0.1$,
and $t_{\rm trans}=1$.  Figure 13:  Similarly to Figure 11, but for Set (iii)
with $k=1$, $t_{0}=1$, and $t_{\rm trans}=10$.

When modeling the transition from a vacuum to that of high conductivity and
the corresponding Joule heating, we still need to make a choice as to when
$\sigma$ would begin to increase, i.e., we need to choose values of $t_{0}$
and $t_{\rm trans}$. If we choose the value of $t_{0}$ to be too large, we
obtain solutions where electromagnetic waves have already been established;
see Figure 11. The smallest wavenumber in our one-dimensional domain is
$k=10$, so by the time $t=1$, even the largest modes in the domain are stable.
We also see that at early times, ${\cal E}_{\rm E}$ and ${\cal E}_{\rm M}$
grow in an algebraic fashion and then become oscillatory when $k_{*}(t)$ has
dropped below $k$. At $t=t_{0}=1$, when conductivity turns on, the electric
energy decreases rapidly, while the magnetic energy diminishes only very
slowly. The generated hydrodynamic energy is however small. This is similar to
what we studied in Section 3.2.

Our objective here is to study cases that are different from what was studied
in Section 3.2. Therefore we now choose Set (ii) with $t_{0}=0.1$ and $t_{\rm
trans}=1$ (Figure 12) and another Set (iii) with $k_{1}=1$ and $t_{0}=1$
(Figure 13). Again, the electric energy drops significantly when conductivity
turns on, but now there is a much larger spread in the resulting maximum
magnetic energies for the three cases with $\beta=1$, 2, and 4. For $\beta=4$,
${\cal E}_{\rm K}$ reaches about one percent of ${\cal E}_{\rm M}$ at $t=0.2$,
for example.

When increasing the wavenumber to $k=1$, the largest modes are still unstable
for the three cases with $\beta=1$, 2, and 4; see Figure 13. Here, $k=1$ and
$t_{0}=1$ and $t_{\rm trans}=10$. The spread in the magnetic energy is
similar, but the maximum kinetic energy is now much larger; see Table 4.4.

[b!] Summary of various extrema for each of the three sets of models and
values of $\beta$.  Set $k$ $t_{0}$ variable $\beta=1$ $\beta=2$ $\beta=4$ (i)
10 1 $\max{\cal E}_{\rm E}$ $1.3\times 10^{-4}$ $4.1\times 10^{-5}$ $8.3\times
10^{-5}$ (ii) 10 0.1 $1.8\times 10^{-4}$ $6.4\times 10^{-4}$ $5.8\times
10^{-3}$ (iii) 1 1 $2.7\times 10^{-3}$ $1.4\times 10^{-2}$ $2.1\times 10^{-1}$
(i) 10 1 $\max{\cal E}_{\rm M}$ $1.6\times 10^{-4}$ $1.1\times 10^{-4}$
$1.8\times 10^{-4}$ (ii) 10 0.1 $1.5\times 10^{-4}$ $4.3\times 10^{-4}$
$4.2\times 10^{-3}$ (iii) 1 1 $3.2\times 10^{-4}$ $9.6\times 10^{-4}$
$1.2\times 10^{-2}$ (i) 10 1 $\max{\cal E}_{\rm K}$ $2.5\times 10^{-9}$
$9.0\times 10^{-10}$ $3.9\times 10^{-9}$ (ii) 10 0.1 $6.9\times 10^{-8}$
$4.2\times 10^{-7}$ $6.6\times 10^{-5}$ (iii) 1 1 $9.8\times 10^{-7}$
$2.3\times 10^{-5}$ $7.7\times 10^{-3}$ (i) 10 1 $\max Q_{\rm G}$ $1.4\times
10^{-4}$ $2.0\times 10^{-4}$ $1.4\times 10^{-3}$ (ii) 10 0.1 $4.4\times
10^{-3}$ $3.8\times 10^{-2}$ $8.6\times 10^{-1}$ (iii) 1 1 $8.4\times 10^{-3}$
$7.8\times 10^{-2}$ $2.7\times 10^{0}$ (i) 10 1 $\max(-Q_{\rm E})$ $2.0\times
10^{-3}$ $9.3\times 10^{-4}$ $2.0\times 10^{-3}$ (ii) 10 0.1 $5.3\times
10^{-3}$ $2.2\times 10^{-2}$ $2.7\times 10^{-1}$ (iii) 1 1 $1.3\times 10^{-3}$
$4.7\times 10^{-3}$ $5.9\times 10^{-2}$ (i) 10 1 $\max\epsilon_{\rm M}$
$2.0\times 10^{-3}$ $4.1\times 10^{-4}$ $2.5\times 10^{-3}$ (ii) 10 0.1
$2.1\times 10^{-2}$ $1.1\times 10^{-1}$ $1.5\times 10^{0}$ (iii) 1 1
$3.5\times 10^{-2}$ $2.5\times 10^{-1}$ $5.8\times 10^{0}$ (i) 10 1
$\max(-\dot{\cal E}_{\rm E})$ $3.3\times 10^{-3}$ $6.3\times 10^{-4}$
$2.7\times 10^{-3}$ (ii) 10 0.1 $1.6\times 10^{-2}$ $5.9\times 10^{-2}$
$5.6\times 10^{-1}$ (iii) 1 1 $2.4\times 10^{-2}$ $1.3\times 10^{-1}$
$1.9\times 10^{0}$ (i) 10 1 $\max\dot{\cal E}_{\rm E}$ $1.2\times 10^{-3}$
$3.4\times 10^{-4}$ $8.0\times 10^{-4}$ (ii) 10 0.1 $8.6\times 10^{-3}$
$5.4\times 10^{-2}$ $5.4\times 10^{-1}$ (iii) 1 1 $1.8\times 10^{-2}$
$1.3\times 10^{-1}$ $2.0\times 10^{0}$

###  4.5 Energy Conversions during Reheating

During reheating, there is an additional source of energy resulting from the
generation term $k_{*}^{2}(t)$. The term $k_{*}^{2}(t)$ appeared in Equation
(26) for $\mbox{\boldmath${\cal A}$}{}=f\bm{A}$. However, to write down the
relevant equation for $\bm{E}=-\partial\bm{A}/\partial t$, we have to revert
to the original equation for $\bm{A}$, which reads Subramanian (2010)

$\left[\frac{1}{c^{2}}\left(\frac{\partial^{2}}{\partial
t^{2}}+2\frac{f^{\prime}}{f}\frac{\partial}{\partial
t}\right)-\nabla^{2}\right]\bm{A}=0.$ (32)

Thus, Equation (1) with the current density term restored, now becomes

$\frac{1}{c^{2}}\left(\frac{\partial\bm{E}}{\partial
t}+2\frac{f^{\prime}}{f}\bm{E}\right)={\bm{\nabla}}\times\bm{B}-\mu_{0}\bm{J},$
(33)

and therefore, Equation (15) for the electric energy now has an extra term and
reads

$\frac{{\rm d}{}}{{\rm
d}{}t}\left\langle\epsilon_{0}\bm{E}^{2}/2\right\rangle=-2(f^{\prime}/f)\left\langle\epsilon_{0}\bm{E}^{2}\right\rangle+\left\langle\bm{E}\cdot{\bm{\nabla}}\times\bm{B}/\mu_{0}\right\rangle-\left\langle\bm{J}\cdot\bm{E}\right\rangle.$
(34)

During reheating with $f\propto a^{-\beta}\propto t^{-2\beta}$, we have
$f^{\prime}/f=-2\beta/t$, so the first term on the right-hand side of Equation
(34) is positive for $\beta>0$, so there is growth of the electric energy.
Similarly to what was performed in Section 3.2, we can write the electric
energy equation more compactly as $\dot{\cal E}_{\rm E}=Q_{\rm G}+Q_{\rm
E}-\epsilon_{\rm M}-W_{\rm L}$, where $Q_{\rm G}=-4(f^{\prime}/f){\cal E}_{\rm
E}$ is now the dominant source, but here $Q_{\rm E}$ plays the role of a sink
during the first part of the evolution. This equation generalizes Equation
(25) to the case with electromagnetic field generation during reheating; see
also Figure 14. Unlike the earlier case of Figure 3, where there was no energy
input, we here have a system that it driven by energy input through the
$Q_{\rm G}$ term.

Figure 14:  Similar to Figure 3, but now with inflationary magnetogenesis
energy generation and energy exchange between electric and magnetic energies
in both directions.  Figure 15:  $t$ dependence of $Q_{\rm G}$ (red), $Q_{\rm
E}$ (green), $\epsilon_{\rm M}$ (black), $\dot{\cal E}_{\rm E}$ (blue), and
$W_{\rm L}$ (orange) for the runs of Set (ii) in Figure 12 with (a) $\beta=1$,
(b) $\beta=2$, and (c) $\beta=4$ for $t_{0}=0.1$ and $t_{\rm trans}=1$.
Figure 16:  Similarly to Figure 15, but for the runs of Set (i) in Figure 11
with $t_{0}=1$ and $t_{\rm trans}=10$. Note that $Q_{\rm G}$ and $Q_{\rm E}$
vary in anti-phase.

The evolution of $Q_{\rm G}$, $Q_{\rm E}$, $\dot{\cal E}_{\rm E}$, and
$\epsilon_{\rm M}$ is shown in Figure 15 during magnetic field generation in
case (ii) for all three values of $\beta$. It is instructive to write the
electric energy equation as

$Q_{\rm G}=\epsilon_{\rm M}+\dot{\cal E}_{\rm E}+W_{\rm L}-Q_{\rm E}.$ (35)

Comparing the three panels of Figure 15, we thus see that for $\beta=1$, there
is a slow generation phase starting much before $t_{0}$. It should be noted,
however, that the ranges on the vertical axes are different for the different
panels.

At $t=t_{0}=0.1$, there is a rise of conductivity, and therefore a sharp rise
in Ohmic heating, $\epsilon_{\rm M}$. This is also the time when $\dot{\cal
E}_{\rm E}$ reaches a maximum and becomes negative shortly thereafter. For
large values of $\beta$, this moment happens a bit later, at $t=0.11$ compared
to $t=0.10$ for $\beta=1$. Note that, while for $\beta=4$ the maxima of
$Q_{\rm G}$ and $\epsilon_{\rm M}$ are similar, for smaller values of $\beta$,
the maxima of $\epsilon_{\rm M}$ are much larger than those of $Q_{\rm G}$.
Instead, for $\beta=1$, for example, we have $\epsilon_{\rm
M}\approx-\dot{\cal E}_{\rm E}$, i.e., almost the entire heating is here
caused by dissipation of electric energy.

In Figure 16, we show a plot similar to Figure 15, but for the case (i), where
all modes were already oscillatory at $t=t_{0}=1$, when conductivity turned
on. The Ohmic heating now plays a minor role in the sense that its maximum
value is much less than the extrema of $Q_{\rm G}$, $Q_{\rm E}$, and
$\dot{\cal E}_{\rm E}$. For $\beta=1$ and 2, we see that $\dot{\cal E}_{\rm
E}$ and $Q_{\rm E}$ are nearly in phase shortly before conductivity turns on.
This means that the electric and magnetic energies are strongly coupled and a
flow of energy from magnetic to electric energy ($Q_{\rm E}>0$) leads to an
increase in electric energy ($\dot{\cal E}_{\rm E}>0$). This is expected,
because there is only an oscillatory exchange between electric and magnetic
energies. For $\beta=4$, on the other hand, the oscillatory phase just started
to develop shortly before $t=1$, but the curves are similar to those for
$\beta=1$ and 2, although shifted toward earlier times. The time of the first
maximum of $Q_{\rm G}$ is at $t=0.9$ for $\beta=4$, while for $\beta=2$, it is
at $t=0.46$ and for $\beta=1$ it is at $t=0.25$, and we see that the profiles
of all curves are indeed very similar around those times.

##  5 Conclusions

In this paper, we have studied the conversion of electromagnetic energy into
kinetic and thermal energies as the electric conductivity transits from zero
(vacuum) to large values. This problem has relevance to the reheating phase at
the end of cosmological inflation and before the emergence of a relatively
long radiation-dominated era before the time of recombination, which is much
later (on a logarithmic time scale). While not much is known about the
physical processes leading to reheating and the emergence of conductivity, a
lot can now be said about the general process of such an energy conversion.

Already in the absence of cosmological expansion, we have seen that the
transition to conductivity involves an oscillatory exchange between electric
and magnetic energies. It is mainly the electric energy reservoir that
delivers energy to the kinetic and thermal energy reservoirs, and not the
magnetic energy directly, as in magnetohydrodynamics. We knew already from
earlier work that the duration of the transit plays a significant role in
causing a drop in magnetic energy. We now also see that this drop depends on
the magnetic field strength and thus the typical Alfvén speed. The drop can
become small if the Alfvén speed becomes comparable to the speed of light.
Furthermore, for short transits, we have seen that energy transfer between
electric and magnetic energies is small and that the initial electric energy
goes directly into thermal energy. For longer transits, however, the mutual
exchange with magnetic energy becomes approximately equal to the thermal
energy loss, so thermalization now also involves the magnetic energy
reservoir.

When applying electromagnetic energy conservation to the problem of reheating,
we have a new quality in the model in that there is now also energy transfer
through conformal invariance breaking, which may occur during inflation and
reheating. This is obviously speculative Ratra (1992); Kronberg (1994); Widrow
(2002); Giovannini (2004); Kandus et al. (2011); Durrer and Neronov (2013);
Subramanian (2016); Vachaspati (2021), but a very promising scenario for the
generation of large-scale magnetic fields in the early Universe and for
explaining the observed lower limits of the intergalactic magnetic field on
megaparsec length scales Neronov and Vovk (2010); Taylor et al. (2011).

The present study has shown that significant work can be done by the Lorentz
force when the electromagnetic energy conversion happens early and on scales
large enough so that the modes are still growing in time. This is because
there is then significant excess of electric energy over magnetic. This is an
effect that was ignored in previous simulations of inflationary magnetogenesis
and, in particular, in studies of the additional contributions to the
resulting relic gravitational wave production Brandenburg and Sharma (2021);
Brandenburg et al. (2021).

It will be useful to extend our studies to turbulent flows and magnetic
fields. This requires that one solves for the evolution of $\rho_{\rm e}$ and
that ${\bm{\nabla}}\cdot\bm{E}=\rho_{\rm e}/\epsilon_{0}$ is obeyed at all
times. This constraint was automatically obeyed in our one-dimensional models.
In future, it would also be interesting to study dynamo action in situations
of moderate magnetic conductivity where coupling with the electric energy
reservoir could reveal new aspects.

Conceptualization, A.B. and N.N.P.; methodology, A.B.; software, A.B. and
N.N.P. All authors have read and agreed to the published version of the
manuscript.

This research was funded by Vetenskapsrådet grant number 2019-04234 and NASA
ATP award number 80NSSC22K0825.

The source code used for the simulations of this study, the Pencil Code
(Pencil Code Collaboration et al., 2021), is freely available on
https://github.com/pencil-code/ (accessed on 1 August 2023). The DOI of the
code is https://doi.org/10.5281/zenodo.2315093 (accessed on 1 August 2023).
The simulation setups and the corresponding secondary data are available on
http://norlx65.nordita.org/~brandenb/projects/EMconversion (accessed on 1
August 2023) and on https://doi.org/10.5281/zenodo.8203242 (accessed on 1
August 2023).

###### Acknowledgements.

The authors thank Johan Anderson for suggesting this contribution to the
special issue on Energy Transfer and Dissipation in Plasma Turbulence. We are
also indebted to the comments and suggestions of the two referees, which have
helped to improve the presentation. We acknowledge the allocation of computing
resources provided by the Swedish National Allocations Committee at the Center
for Parallel Computers at the Royal Institute of Technology in Stockholm and
Linköping. The authors declare no conflicts of interest. References yes

## References

  * Moffatt (1978) Moffatt, H.K.  Magnetic Field Generation in Electrically Conducting Fluids; Cambridge University Press: Cambridge, 1978.
  * Tobias (2021) Tobias, S.M.  The turbulent dynamo.  J. Fluid Mech. 2021, 912, P1, [arXiv:physics.flu-dyn/1907.03685].  doi:black10.1017/jfm.2020.1055.
  * Brandenburg (2014) Brandenburg, A.  Magnetic Prandtl Number Dependence of the Kinetic-to-magnetic Dissipation Ratio.  Astrophys. J. 2014, 791, 12, [arXiv:astro-ph.SR/1404.6964].  doi:black10.1088/0004-637X/791/1/12.
  * Brandenburg and Rempel (2019) Brandenburg, A.; Rempel, M.  Reversed Dynamo at Small Scales and Large Magnetic Prandtl Number.  Astrophys. J. 2019, 879, 57, [arXiv:astro-ph.SR/1903.11869].  doi:black10.3847/1538-4357/ab24bd.
  * Mahajan et al. (2005) Mahajan, S.M.; Shatashvili, N.L.; Mikeladze, S.V.; Sigua, K.I.  Acceleration of Plasma Flows Due to Reverse Dynamo Mechanism.  Astrophys. J. 2005, 634, 419–425, [arXiv:astro-ph/astro-ph/0503431].  doi:black10.1086/432867.
  * Field and Carroll (2023) Field, G.B.; Carroll, S.M.  Axions and Cosmic Magnetic Fields.  arXiv e-prints 2023, p. arXiv:2307.05425, [arXiv:hep-ph/2307.05425].  doi:black10.48550/arXiv.2307.05425.
  * Sharma et al. (2017) Sharma, R.; Jagannathan, S.; Seshadri, T.R.; Subramanian, K.  Challenges in inflationary magnetogenesis: Constraints from strong coupling, backreaction, and the Schwinger effect.  Phys. Rev. D 2017, 96, 083511, [arXiv:astro-ph.CO/1708.08119].  doi:black10.1103/PhysRevD.96.083511.
  * Brandenburg and Sharma (2021) Brandenburg, A.; Sharma, R.  Simulating Relic Gravitational Waves from Inflationary Magnetogenesis.  Astrophys. J. 2021, 920, 26, [arXiv:astro-ph.CO/2106.03857].  doi:black10.3847/1538-4357/ac1599.
  * Alfvén (1942) Alfvén, H.  Existence of Electromagnetic-Hydrodynamic Waves.  Nature 1942, 150, 405–406.  doi:black10.1038/150405d0.
  * Brandenburg et al. (1996) Brandenburg, A.; Enqvist, K.; Olesen, P.  Large-scale magnetic fields from hydromagnetic turbulence in the very early universe.  Phys. Rev. D 1996, 54, 1291–1300, [arXiv:astro-ph/astro-ph/9602031].  doi:black10.1103/PhysRevD.54.1291.
  * Christensson et al. (2001) Christensson, M.; Hindmarsh, M.; Brandenburg, A.  Inverse cascade in decaying three-dimensional magnetohydrodynamic turbulence.  Phys. Rev. E 2001, 64, 056405.  doi:black10.1103/PhysRevE.64.056405.
  * Banerjee and Jedamzik (2004) Banerjee, R.; Jedamzik, K.  Evolution of cosmic magnetic fields: From the very early Universe, to recombination, to the present.  Phys. Rev. D 2004, 70, 123003, [arXiv:astro-ph/astro-ph/0410032].  doi:black10.1103/PhysRevD.70.123003.
  * Kahniashvili et al. (2013) Kahniashvili, T.; Tevzadze, A.G.; Brandenburg, A.; Neronov, A.  Evolution of primordial magnetic fields from phase transitions.  Phys. Rev. D 2013, 87, 083007, [arXiv:astro-ph.CO/1212.0596].  doi:black10.1103/PhysRevD.87.083007.
  * Brandenburg et al. (2015) Brandenburg, A.; Kahniashvili, T.; Tevzadze, A.G.  Nonhelical Inverse Transfer of a Decaying Turbulent Magnetic Field.  Phys. Rev. L 2015, 114, 075001, [arXiv:astro-ph.CO/1404.2238].  doi:black10.1103/PhysRevLett.114.075001.
  * Brandenburg and Kahniashvili (2017) Brandenburg, A.; Kahniashvili, T.  Classes of Hydrodynamic and Magnetohydrodynamic Turbulent Decay.  Phys. Rev. Lett. 2017, 118, 055102, [arXiv:physics.flu-dyn/1607.01360].  doi:black10.1103/PhysRevLett.118.055102.
  * Hosking and Schekochihin (2022) Hosking, D.N.; Schekochihin, A.A.  Cosmic-void observations reconciled with primordial magnetogenesis.  arXiv e-prints 2022, p. arXiv:2203.03573, [arXiv:astro-ph.CO/2203.03573].
  * Silk (1968) Silk, J.  Cosmic Black-Body Radiation and Galaxy Formation.  Astrophys. J. 1968, 151, 459.  doi:black10.1086/149449.
  * Kolb and Turner (1990) Kolb, E.W.; Turner, M.S.  The early universe; Vol. 69, Avalon Publishing: New York, NY, USA, 1990.
  * Brandenburg et al. (1997) Brandenburg, A.; Enqvist, K.; Olesen, P.  The effect of Silk damping on primordial magnetic fields.  Phys. Lett. B 1997, 392, 395–402, [arXiv:hep-ph/hep-ph/9608422].  doi:black10.1016/S0370-2693(96)01566-3.
  * Kobayashi and Afshordi (2014) Kobayashi, T.; Afshordi, N.  Schwinger effect in 4D de Sitter space and constraints on magnetogenesis in the early universe.  J. High Energy Phys. 2014, 2014, 166, [arXiv:hep-th/1408.4141].  doi:black10.1007/JHEP10(2014)166.
  * Henriksen (2019) Henriksen, R.  Solar and Galactic Magnetic Halo Structure: Force-Free Dynamos?  Galaxies 2019, 7, 53.  doi:black10.3390/galaxies7020053.
  * Brandenburg and Ntormousi (2022) Brandenburg, A.; Ntormousi, E.  Dynamo effect in unstirred self-gravitating turbulence.  MNRAS 2022, 513, 2136–2151, [arXiv:astro-ph.GA/2112.03838].  doi:black10.1093/mnras/stac982.
  * Brandenburg et al. (1996) Brandenburg, A.; Jennings, R.L.; Nordlund, Å.; Rieutord, M.; Stein, R.F.; Tuominen, I.  Magnetic structures in a dynamo simulation.  J. Fluid Mech. 1996, 306, 325–352.  doi:black10.1017/S0022112096001322.
  * Brandenburg et al. (1995) Brandenburg, A.; Nordlund, A.; Stein, R.F.; Torkelsson, U.  Dynamo-generated Turbulence and Large-Scale Magnetic Fields in a Keplerian Shear Flow.  Astrophys. J. 1995, 446, 741.  doi:black10.1086/175831.
  * Pencil Code Collaboration et al. (2021) Pencil Code Collaboration.; Brandenburg, A.; Johansen, A.; Bourdin, P.; Dobler, W.; Lyra, W.; Rheinhardt, M.; Bingert, S.; Haugen, N.; Mee, A.; Gent, F.; Babkovskaia, N.; Yang, C.C.; Heinemann, T.; Dintrans, B.; Mitra, D.; Candelaresi, S.; Warnecke, J.; Käpylä, P.; Schreiber, A.; Chatterjee, P.; Käpylä, M.; Li, X.Y.; Krüger, J.; Aarnes, J.; Sarson, G.; Oishi, J.; Schober, J.; Plasson, R.; Sandin, C.; Karchniwy, E.; Rodrigues, L.; Hubbard, A.; Guerrero, G.; Snodin, A.; Losada, I.; Pekkilä, J.; Qian, C.  The Pencil Code, a modular MPI code for partial differential equations and particles: multipurpose and multiuser-maintained.  J. Open Source Softw. 2021, 6, 2807.  doi:black10.21105/joss.02807.
  * Gedalin (1993) Gedalin, M.  Linear waves in relativistic anisotropic magnetohydrodynamics.  Phys. Rev. E 1993, 47, 4354–4357.  doi:black10.1103/PhysRevE.47.4354.
  * Durrer and Neronov (2013) Durrer, R.; Neronov, A.  Cosmological magnetic fields: their generation, evolution and observation.  Astron. Astrophys. Rev. 2013, 21, 62, [arXiv:astro-ph.CO/1303.7121].  doi:black10.1007/s00159-013-0062-7.
  * Subramanian (2016) Subramanian, K.  The origin, evolution and signatures of primordial magnetic fields.  Rep. Prog. Phys. 2016, 79, 076901, [arXiv:astro-ph.CO/1504.02311].  doi:black10.1088/0034-4885/79/7/076901.
  * Neronov and Vovk (2010) Neronov, A.; Vovk, I.  Evidence for Strong Extragalactic Magnetic Fields from Fermi Observations of TeV Blazars.  Science 2010, 328, 73, [arXiv:astro-ph.HE/1006.3504].  doi:black10.1126/science.1184192.
  * Taylor et al. (2011) Taylor, A.M.; Vovk, I.; Neronov, A.  Extragalactic magnetic fields constraints from simultaneous GeV-TeV observations of blazars.  Astron. Astrophys. 2011, 529, A144, [arXiv:astro-ph.HE/1101.0932].  doi:black10.1051/0004-6361/201116441.
  * Broderick et al. (2012) Broderick, A.E.; Chang, P.; Pfrommer, C.  The Cosmological Impact of Luminous TeV Blazars. I. Implications of Plasma Instabilities for the Intergalactic Magnetic Field and Extragalactic Gamma-Ray Background.  Astrophys. J. 2012, 752, 22, [arXiv:astro-ph.CO/1106.5494].  doi:black10.1088/0004-637X/752/1/22.
  * Broderick et al. (2018) Broderick, A.E.; Tiede, P.; Chang, P.; Lamberts, A.; Pfrommer, C.; Puchwein, E.; Shalaby, M.; Werhahn, M.  Missing Gamma-ray Halos and the Need for New Physics in the Gamma-ray Sky.  Astrophys. J. 2018, 868, 87, [arXiv:astro-ph.HE/1808.02959].  doi:black10.3847/1538-4357/aae5f2.
  * Sironi and Giannios (2014) Sironi, L.; Giannios, D.  Relativistic Pair Beams from TeV Blazars: A Source of Reprocessed GeV Emission rather than Intergalactic Heating.  Astrophys. J. 2014, 787, 49, [arXiv:astro-ph.HE/1312.4538].  doi:black10.1088/0004-637X/787/1/49.
  * Alves Batista et al. (2019) Alves Batista, R.; Saveliev, A.; de Gouveia Dal Pino, E.M.  The impact of plasma instabilities on the spectra of TeV blazars.  MNRAS 2019, 489, 3836–3849, [arXiv:astro-ph.HE/1904.13345].  doi:black10.1093/mnras/stz2389.
  * Adshead et al. (2015) Adshead, P.; , John T. Giblin, J.; Scully, T.R.; Sfakianakis, E.I.  Gauge-preheating and the end of axion inflation.  J. Cosmol. Astropart. Phys. 2015, 2015, 034–034, [arXiv:astro-ph.CO/1502.06506].  doi:black10.1088/1475-7516/2015/12/034.
  * Adshead et al. (2016) Adshead, P.; Giblin, John T., J.; Scully, T.R.; Sfakianakis, E.I.  Magnetogenesis from axion inflation.  J. Cosmol. Astropart. Phys. 2016, 2016, 039, [arXiv:astro-ph.CO/1606.08474].  doi:black10.1088/1475-7516/2016/10/039.
  * Adshead et al. (2020) Adshead, P.; Giblin, J.T.; Pieroni, M.; Weiner, Z.J.  Constraining axion inflation with gravitational waves from preheating.  Phys. Rev. D 2020, 101, 083534, [arXiv:astro-ph.CO/1909.12842].  doi:black10.1103/PhysRevD.101.083534.
  * Ratra (1992) Ratra, B.  Cosmological “Seed” Magnetic Field from Inflation.  Astrophys. J. Lett. 1992, 391, L1.  doi:black10.1086/186384.
  * Demozzi et al. (2009) Demozzi, V.; Mukhanov, V.; Rubinstein, H.  Magnetic fields from inflation?  J. Cosmol. Astropart. Phys. 2009, 8, 025, [arXiv:astro-ph.CO/0907.1030].  doi:black10.1088/1475-7516/2009/08/025.
  * Ferreira et al. (2013) Ferreira, R.J.Z.; Jain, R.K.; Sloth, M.S.  Inflationary magnetogenesis without the strong coupling problem.  J. Cosmol. Astropart. Phys. 2013, 2013, 004, [arXiv:astro-ph.CO/1305.7151].  doi:black10.1088/1475-7516/2013/10/004.
  * Kobayashi and Sloth (2019) Kobayashi, T.; Sloth, M.S.  Early cosmological evolution of primordial electromagnetic fields.  Phys. Rev. D 2019, 100, 023524, [arXiv:astro-ph.CO/1903.02561].  doi:black10.1103/PhysRevD.100.023524.
  * Sharma et al. (2018) Sharma, R.; Subramanian, K.; Seshadri, T.R.  Generation of helical magnetic field in a viable scenario of inflationary magnetogenesis.  Phys. Rev. D 2018, 97, 083503 (SSS18), [arXiv:astro-ph.CO/1802.04847].  doi:black10.1103/PhysRevD.97.083503.
  * Brandenburg et al. (2021) Brandenburg, A.; He, Y.; Sharma, R.  Simulations of Helical Inflationary Magnetogenesis and Gravitational Waves.  Astrophys. J. 2021, 922, 192, [arXiv:astro-ph.CO/2107.12333].  doi:black10.3847/1538-4357/ac20d9.
  * Subramanian (2010) Subramanian, K.  Magnetic fields in the early Universe.  Astron. Nachr. 2010, 331, 110, [0911.4771].  doi:black10.1002/asna.200911312.
  * He et al. (2023) He, Y.; Roper Pol, A.; Brandenburg, A.  Modified propagation of gravitational waves from the early radiation era.  J. Cosmol. Astropart. Phys. 2023, 2023, 025, [arXiv:gr-qc/2212.06082].  doi:black10.1088/1475-7516/2023/06/025.
  * Kronberg (1994) Kronberg, P.P.  Extragalactic magnetic fields.  Rep. Prog. Phys. 1994, 57, 325–382.  doi:black10.1088/0034-4885/57/4/001.
  * Widrow (2002) Widrow, L.M.  Origin of galactic and extragalactic magnetic fields.  Rev. Mod. Phys. 2002, 74, 775–823, [arXiv:astro-ph/astro-ph/0207240].  doi:black10.1103/RevModPhys.74.775.
  * Giovannini (2004) Giovannini, M.  The Magnetized Universe.  Int. J. Mod. Phys. D 2004, 13, 391–502, [arXiv:astro-ph/astro-ph/0312614].  doi:black10.1142/S0218271804004530.
  * Kandus et al. (2011) Kandus, A.; Kunze, K.E.; Tsagas, C.G.  Primordial magnetogenesis.  Phys. Rep. 2011, 505, 1–58, [arXiv:astro-ph.CO/1007.3891].  doi:black10.1016/j.physrep.2011.03.001.
  * Vachaspati (2021) Vachaspati, T.  Progress on cosmological magnetic fields.  Rep. Prog. Phys. 2021, 84, 074901, [arXiv:astro-ph.CO/2010.10525].  doi:black10.1088/1361-6633/ac03a9.