# Primordial magnetic helicity evolution with a homogeneous magnetic field
from inflation

Axel Brandenburg  Nordita, KTH Royal Institute of Technology and Stockholm
University, Roslagstullsbacken 23, 10691 Stockholm, Sweden  Department of
Astronomy, AlbaNova University Center, Stockholm University, 10691 Stockholm,
Sweden  JILA and Laboratory for Atmospheric and Space Physics, University of
Colorado, Boulder, Colorado 80303, USA  McWilliams Center for Cosmology and
Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue,
Pittsburgh, Pennsylvania 15213, USA  Faculty of Natural Sciences and Medicine,
Ilia State University, 3-5 Cholokashvili Street, 0194 Tbilisi, Georgia  Ruth
Durrer  Département de Physique Théorique and Center for Astroparticle
Physics, Université de Geneve, Quai Ernest Ansermet 24, 1211 Genéve 4,
Switzerland  Yiwen Huang  McWilliams Center for Cosmology and Department of
Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh,
Pennsylvania 15213, USA  Department of Physics, University of California San
Diego, 9500 Gilman Drive 1110-115, La Jolla, California 92093  Tina
Kahniashvili  McWilliams Center for Cosmology and Department of Physics,
Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania
15213, USA  Faculty of Natural Sciences and Medicine, Ilia State University,
3-5 Cholokashvili Street, 0194 Tbilisi, Georgia  Abastumani Astrophysical
Observatory, 47 Kostava Street, 0179 Tbilisi, Georgia  Department of Physics,
Laurentian University, Ramsey Lake Road, Sudbury, Ontario P3E 2C,Canada  Sayan
Mandal 111Corresponding author; the authors are listed alphabetically.
<EMAIL_ADDRESS>McWilliams Center for Cosmology and Department of
Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh,
Pennsylvania 15213, USA  Faculty of Natural Sciences and Medicine, Ilia State
University, 3-5 Cholokashvili Street, 0194 Tbilisi, Georgia  Shinji Mukohyama
Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), The University of Tokyo Institutes for
Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

###### Abstract

Motivated by a scenario of magnetogenesis in which a homogeneous magnetic
field is generated during inflation, we study the magnetohydrodynamic
evolution of the primordial plasma motions for two kinds of initial conditions
– (i) a spatially homogeneous field with an unlimited correlation length, and
(ii) a zero flux scale-invariant statistically homogeneous magnetic field. In
both cases, we apply, for a short initial time interval, monochromatic forcing
at a certain wave number so that the correlation length is finite, but much
smaller than the typical length scale of turbulence. In particular, we
investigate the decay of nonhelical and helical hydromagnetic turbulence. We
show that, in the presence of a homogeneous magnetic field, the decay of
helical and nonhelical small-scale fields can occur rapidly. This is a special
property of a system with a perfectly homogeneous magnetic field, which is
sometimes considered as a local approximation to a slowly varying background
field. It can never change and acts as an imposed magnetic field. This is in a
sharp contrast to the case of a statistically homogeneous magnetic field,
where we recover familiar decay properties: a much slower decay of magnetic
energy and a faster growth of the correlation length, especially in the case
with magnetic helicity. The result suggests that a homogeneous magnetic field,
if generated during inflation, should persist under the influence of small-
scale fields and could be the origin of the large-scale magnetic field in the
Universe.

††preprint: NORDITA-2020-044, YITP-20-47, IPMU20-0042

##  I Introduction

One of several open problems in cosmology and astrophysics is the
understanding of the origin of large-scale magnetic fields in the Universe
Widrow:2002ud ; Durrer:2013pga . There are two widely considered approaches to
understand the origin of intercluster, large-scale correlated magnetic fields
– (i) an astrophysical scenario Kulsrud:2007an , where weak seed fields
generated by local sources are amplified and transferred to large scales by
various astrophysical processes, and (ii) a cosmological (or primordial)
scenario Kandus:2010nw , where a strong seed magnetic field generated in the
early universe evolves through magnetohydrodynamic (MHD) coupling with the
primordial plasma. Neronov and Vovk Neronov:1900zz  used the nonobservation of
GeV photons from TeV blazars to put a lower limit on the strength of magnetic
fields on extragalactic scales, obtaining a lower bound of $\sim
10^{-15}\,\mathrm{G}$ at $1\,\mathrm{Mpc}$. The limits were later revised to
$\sim 10^{-18}\,\mathrm{G}$ after considering that the observation period of
the sources was limited to only a few years Dermer:2010mm ; Taylor:2011bn .
The Fermi-LAT and the VERITAS collaborations have improved this limit again to
$\sim 10^{-15}\,\mathrm{G}$ at $1\,\mathrm{Mpc}$ Biteau:2018tmv ;
Archambault:2017hvo , based on ten years of observations of the TeV blazar
emission spectra. These observational limits favor the cosmological
(primordial) scenario of magnetogenesis Dolag:2010ni ; see Ref. Arlen:2012iy
for discussions on possible uncertainties in these lower limits based on
blazar spectra and Refs. Broderick:2018nqf ; AlvesBatista:2019ipr  on possible
impacts of plasma instabilities.

There are several scenarios for generating primordial magnetic fields in the
early universe; see Ref. Subramanian:2015lua  for a review. Here we consider
two main ideas. First, magnetic fields can be generated during inflation or
through processes related to it, like reheating or preheating. Reheating is
the epoch at the end of inflation when the energy in the hypothesized inflaton
field decays into the fields of the standard model and the temperature of the
universe rises sufficiently. The decay of the inflaton into bosons can be very
rapid owing to processes such as a parametric resonance or a tachyonic
instability. Such a rapid decay is called preheating. Primordial magnetic
fields can also be generated during cosmological phase transitions. The
evolution of these primordial fields in the expanding universe has been
studied by several authors by solving the MHD equations for the magnetic
field, the density, and the velocity of the plasma; see Ref.
Kahniashvili:2018mzl  for a brief review and references within. The fields
used are generally modeled either as homogeneous, or as statistically
homogeneous and isotropic random Gaussian stochastic fields. Statistical
homogeneity implies that the two-point correlation function of the magnetic
field is independent of the position in space. In this paper, we show that
these two approaches can result in very different dynamics of the induced
turbulent motions in the early universe. In particular for the former case,
small-scale helical and nonhelical fields decay in a way very different from
the case of statistically homogeneous fields, as discussed below.

While there were earlier ideas suggesting the presence of a homogeneous
magnetic field Zeldovich:1965 ; Doroshkevich:1965 ; Thorne:1967zz , those
papers study just cosmological consequences without specifying or discussing a
generation mechanism. While Ref. Harrison:1969  discusses the generation of a
homogeneous magnetic field, it assumes the existence of “protogalaxies” with
angular momentum in the radiation dominated epoch, which contradicts with the
current understanding of our Universe. Reference Bertolami:1998dn  also
discusses a generation mechanism but it assumes a tachyonic mass for a gauge
field, which means that the corresponding Higgs field is a ghost and thus the
model is unstable. While the model in Ref. Maeda:2012eg  is similar to gauge-
flation or chromo-natural inflation, what is called a magnetic field there is
simply due to the nonlinear part of the field strength and is thus actually an
electric field, if a part of the non-Abelian gauge field is projected onto an
Abelian gauge field via a spontaneous symmetry breaking (e.g., ${\rm
SU}(2)\times{\rm U}(1)\to{\rm U}(1)$) or if their ansatz of the gauge
potential is applied to an Abelian gauge field. As far as the authors know, no
stable generation mechanism of homogeneous magnetic fields has been proposed
until recently. (In open Universes, while one might hope to find such a
mechanism through supercurvature modes, but it is known that there is actually
no supercurvature mode for vector fields [14].) However, the lack of a
generation mechanism does not necessarily mean a lack of interest in
homogeneous magnetic fields. Indeed, as already mentioned above, the studies
of cosmological consequences of a homogeneous magnetic field date back to the
seminal works by Zel’dovich, Doroshkevich, and Thorne in 1960’=s.

Recently, one of the authors Mukohyama:2016npi  proposed a stable generation
mechanism of homogeneous magnetic field, based on a $U(1)$ gauge theory of
electromagnetism with a coupling to Horndeski type scalar-tensor gravity,
where gravity is described by the metric tensor field and an additional scalar
field222See, e.g., Fujii:2003pa  for more details about scalar-tensor
gravity.. During inflation, the model admits a stable (quasi) de Sitter
solution with a homogeneous magnetic field as an attractor of the system.
Therefore the model provides a classical generation mechanism of homogeneous
magnetic fields during inflation. After inflation and the stabilization of the
scalar field at the minimum of its potential, on the other hand, gravity is
effectively described by general relativity. Later in the present paper, we
show that the action after inflation is the Einstein-Maxwell action,
supplemented with a nonminimal coupling between curvature and
electromagnetism. We also show that, upon imposing observational constraints,
the nonminimal coupling can be ignored for the analysis of the magnetic field
evolution. This in particular means that the no-minimal coupling does not
introduce new instabilities in the homogeneous magnetic field background in
the late-time cosmology.

It was already known that in the additional presence of primordial small-scale
turbulence, the magnetic energy spectrum changes only very little at large
length scales Kahniashvili:2012vt . This led one of the authors
Mukohyama:2016npi  to expect that this also applies to the case of a
homogeneous magnetic field, but this remained to be verified by numerical MHD
simulations. It is therefore important to study the MHD evolution of
primordial plasma motions in the presence of these homogeneous magnetic
fields, which is what we focus on in this work.

We are particularly interested in the evolution of magnetic helicity, which is
known not to be conserved in a periodic domain in the presence of a
homogeneous magnetic field Ber97 . However, if we were to consider a perfectly
homogeneous magnetic field as a local approximation to a slowly varying
background magnetic field, magnetic helicity conservation would be restored.
To illuminate the remarkable properties of a perfectly homogeneous magnetic
field, we also discuss the alternative approach of working instead with a
statistically homogeneous magnetic field, which does not impose any
constraints on the magnetic helicity evolution. The presence of magnetic
helicity substantially changes the decay rate for MHD turbulence BM99 . In
this paper we compare the decay dynamics for homogeneous and statistically
homogeneous magnetic fields with a scale-invariant spectrum. In previous
works, we have studied only statistically homogeneous magnetic fields induced
by the turbulence dynamics Brandenburg:2016odr  but did not include the
turbulence in the presence of a homogeneous magnetic field.

This paper is arranged as follows. The model is described in Sec. II, where we
discuss the formalism for how a spatially homogeneous magnetic field is
realized during inflation, and after that until recombination. In Sec. III, we
describe in detail the setup of our simulations, discussing, in particular,
various initial conditions to examine peculiar features associated with the
use of an imposed magnetic field. We present numerical solutions in Sec. IV
and in Sec. V, we present our conclusions. Throughout this paper we work in
natural units where $\hbar=c=1$, and our metric signature is $(-,+,+,+)$. For
the electromagnetic quantities we use Lorentz-Heaviside units.

##  II Homogeneous magnetic fields

In this section we briefly describe a theoretical framework in which a
spatially homogeneous magnetic field background can be realized during and
after inflation in the early Universe. In the inflationary stage, the
background spacetime is not only homogeneous but also isotropic despite the
existence of the preferred spatial direction defined by the homogeneous
magnetic field. This is made possible by a nonlinear kinetic action for the
$U(1)$ gauge field nonminimally coupled to a scalar-tensor theory of gravity.
In the postinflationary stage, on the other hand, the scalar field is
stabilized around a minimum of a potential and thus the theory is reduced to
the Einstein-Maxwell theory supplemented with the Horndeski’s nonminimal
coupling. Therefore, after inflation the spacetime becomes anisotropic and the
homogeneous magnetic field adiabatically decays. If we are interested in the
postinflationary evolution of the $U(1)$ gauge field at subhorizon scales for
timescales sufficiently shorter than the cosmological time then the
gravitational effects of and on the gauge field can be neglected and the
system is described by the standard Maxwell theory expanded around the
homogeneous magnetic field background in Minkowski spacetime. As we shall see
in the next sections, the existence of the homogeneous magnetic field
significantly affects the evolution of the gauge field at subhorizon scales.

###  II.1 General action

We consider a metric $g_{\mu\nu}$, a $U(1)$ gauge field $A_{\mu}$ and a scalar
field $\phi$ in four dimensional spacetime described by the action

$I=\int d^{4}x\sqrt{-g}\left[L+L_{3}+L_{4}+L_{5}+L_{\rm H}\right]\,,$ (1)

where $L=L(\phi,X,W,Y,Z)$ is an arbitrary function of $\phi$,

$\displaystyle
X\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\,,\quad
W\equiv-\frac{1}{4}\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}\,,$ $\displaystyle
Y\equiv\mathcal{F}_{\mu\nu}\tilde{\mathcal{F}}^{\mu\nu}\,,\quad
Z\equiv\mathcal{F}^{\rho\mu}\mathcal{F}_{\rho}^{\
\nu}\partial_{\mu}\phi\partial_{\nu}\phi\,;$ (2)

$\mathcal{F}_{\mu\nu}$ and $\tilde{\mathcal{F}}^{\mu\nu}$ are defined by

$\displaystyle\mathcal{F}_{\mu\nu}\equiv
e^{\phi}F_{\mu\nu}\,,\quad\tilde{\mathcal{F}}^{\mu\nu}\equiv
e^{\phi}\tilde{F}^{\mu\nu}\,,$ $\displaystyle
F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,,\quad\tilde{F}^{\mu\nu}\equiv\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}\,,$
(3)

and $\epsilon^{0123}=-1/\sqrt{-g}$;

$\displaystyle L_{3}=$ $\displaystyle-G_{3}(\phi,X)\Box\phi\,,$ $\displaystyle
L_{4}=$ $\displaystyle
G_{4}(\phi,X)R+G_{4X}(\phi,X)\left[(\Box\phi)^{2}-(\nabla^{\mu}\nabla_{\nu}\phi)(\nabla^{\nu}\nabla_{\mu}\phi)\right]\,,$
$\displaystyle L_{5}=$ $\displaystyle
G_{5}(\phi,X)G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi-\frac{1}{6}G_{5X}(\phi,X)\left[(\Box\phi)^{3}\right.$
$\displaystyle-3(\Box\phi)(\nabla^{\mu}\nabla_{\nu}\phi)(\nabla^{\nu}\nabla_{\mu}\phi)$
$\displaystyle\left.+2(\nabla^{\mu}\nabla_{\nu}\phi)(\nabla^{\nu}\nabla_{\rho}\phi)(\nabla^{\rho}\nabla_{\mu}\phi)\right]$
(4)

are Horndeski scalar terms Horndeski:1974wa ; Deffayet:2011gz ; and

$L_{\rm
H}=\xi(\phi)\tilde{\mathcal{F}}^{\mu\nu}\tilde{\mathcal{F}}^{\rho\sigma}R_{\mu\nu\rho\sigma}$
(5)

is a simple modification of Horndeski’s nonminimal coupling of the $U(1)$
gauge field to the Riemann tensor $R^{\mu}_{\ \nu\rho\sigma}$ of the metric
$g_{\mu\nu}$ Horndeski:1976gi . Here, the scalar field $\phi$ and the gauge
field $A_{\mu}$ are normalized so that their mass dimensions are zero,
$G_{3,4,5}(\phi,X)$ are arbitrary functions of $\phi$ and $X$, the subscript
$X$ denotes derivative with respect to $X$, and $\xi(\phi)$ is an arbitrary
function of $\phi$. The action is invariant under the $U(1)$ gauge
transformation,

$A_{\mu}\to A_{\mu}+\partial_{\mu}\lambda\,,$ (6)

where $\lambda$ is an arbitrary function, and the equations of motion are
second-order differential equations. In principle it is possible to consider a
more general form of $L$ that depends on the second covariant derivatives of
$\phi$ and $A_{\mu}$ without introducing higher derivatives in the equations
of motion. For simplicity, however, we restrict our consideration to the above
form of $L$ that depends on only up to first derivatives of $\phi$ and
$A_{\mu}$. Also, the inclusion of the factor $e^{\phi}$ in the definitions of
$\mathcal{F}_{\mu\nu}$ and $\tilde{\mathcal{F}}^{\mu\nu}$ is redundant since
we allow for the explicit $\phi$-dependence of $L(\phi,X,W,Y,Z)$ and
$\xi(\phi)$. We nonetheless adopt the above definitions of
$\mathcal{F}_{\mu\nu}$ and $\tilde{\mathcal{F}}^{\mu\nu}$ including the factor
$e^{\phi}$ in order to make it easy to implement a scaling-type symmetry for
the description of the system during the inflationary stage [see Eqs. (7] and
(8) in the next subsection).

###  II.2 Stealth magnetic field during inflation

Following the discussion in Sec. V of Mukohyama:2016npi , we suppose that the
main source of curvature perturbations is not $\phi$ but something else. For
example, one can introduce another scalar field as an inflaton or a curvaton.
For simplicity we approximate the geometry during inflation by a de Sitter
spacetime. Then the effective cosmological constant induced by the field
responsible for curvature perturbations simply amounts to a constant shift of
$L(\phi,X,W,Y,Z)$.

In order to simplify the analysis and also to allow for an exact solution that
represents a de Sitter spacetime with a homogeneous magnetic field, we require
that the action is invariant under not only the $U(1)$ gauge transformation
(6) but also the following scaling-type global transformation for the range of
$\phi$ that is relevant for the inflationary epoch.

$\phi\to\phi+\phi_{0}\,,\quad A_{\mu}\to e^{-\phi_{0}}A_{\mu}\,,$ (7)

where $\phi_{0}$ is an arbitrary constant that is not too large to eject
$\phi$ from the inflationary range. Then for the range of $\phi$, the explicit
$\phi$-dependence of the functions $L(\phi,X,W,Y,Z)$, $G_{3,4,5}(\phi,X)$ and
$\xi(\phi)$ is forbidden so that

$\displaystyle L(\phi,X,W,Y,Z)$ $\displaystyle=$
$\displaystyle\bar{L}(X,W,Y,Z)\,,$ $\displaystyle G_{3,4,5}(\phi,X)$
$\displaystyle=$ $\displaystyle\bar{G}_{3,4,5}(X)\,,$ $\displaystyle\xi(\phi)$
$\displaystyle=$ $\displaystyle\bar{\xi}\,,$ (8)

where $\bar{L}(X,W,Y,Z)$ is an arbitrary function of ($X$, $W$, $Y$, $Z$),
$\bar{G}_{3,4,5}(X)$ are arbitrary functions of $X$ and $\bar{\xi}$ here is a
constant. We also impose the parity invariance so that the function
$\bar{L}(X,W,Y,Z)$ is even with respect to $Y$.

$\bar{L}(X,W,Y,Z)=\bar{L}(X,W,-Y,Z)\,.$ (9)

This is the system studied in Mukohyama:2016npi ; Mukohyama:2018obj .

For this system, we adopt the ansatz of the form

$\displaystyle g_{\mu\nu}dx^{\mu}dx^{\nu}=-N(t)^{2}dt^{2}$
$\displaystyle\qquad\qquad+a(t)^{2}\left[e^{4\sigma(t)}dx^{2}+e^{-2\sigma(t)}(dy^{2}+dz^{2})\right]\,,$
$\displaystyle\phi=\phi(t)\,,$ $\displaystyle A_{t}=0\,,\quad
A_{x}=\int^{t}\frac{N(t^{\prime})e^{4\sigma(t^{\prime})}}{a(t^{\prime})}E(t^{\prime})dt^{\prime}\,,$
$\displaystyle A_{y}=\frac{1}{2}Bz\,,\quad A_{z}=-\frac{1}{2}By\,,$ (10)

where $B$ is a constant. It was found in Mukohyama:2016npi  that the equations
of motion admit solutions of the form

$\displaystyle
H=\mathrm{const.}>0\,,\quad\Sigma=\mathrm{const.}\,,\quad\chi=\mathrm{const.}>0\,,$
$\displaystyle E=\mathrm{const.}\,,\quad B\neq 0\,,$ (11)

where

$H\equiv\frac{\dot{a}}{Na}\,,\quad\Sigma\equiv\frac{\dot{\sigma}}{N}\,,\quad\chi\equiv\frac{e^{\phi}e^{2\sigma}}{a^{2}}\,.$
(12)

By tuning one parameter in the action, the solution is reduced to a de Sitter
spacetime with magnetic field but without electric field Mukohyama:2016npi ,
i.e.

$\displaystyle
H=\mathrm{const.}>0\,,\quad\Sigma=0\,,\quad\chi=\mathrm{const.}>0\,,$
$\displaystyle E=0\,,\quad B\neq 0\,.$ (13)

The reason why fine-tuning of just one parameter leads to two equalities,
$\Sigma=0$ and $E=0$, is that we have imposed the discrete symmetry (9).
Reference Mukohyama:2016npi  also found the condition under which the de
Sitter solution with magnetic field but without electric field is an attractor
of the system within the ansatz (10). Reference Mukohyama:2018obj  then
analyzed general linear perturbations around the attractor solution and found
the condition under which the system of linear perturbations is free from
instabilities.

In the present paper we consider the stable attractor de Sitter solution with
magnetic field but without electric field as the origin of magnetic fields
that are observed in the late-time Universe. We denote the (approximately)
constant value of $H$ during inflation as $H_{\rm inf}$ 333In Refs.
Mukohyama:2016npi ; Mukohyama:2018obj  it was denoted as $H_{0}$..

###  II.3 Postinflationary system

Following again the discussion in Sec. V of Mukohyama:2016npi , we suppose
that the scaling-type global symmetry (7) is not respected for the range of
$\phi$ that is relevant for the postinflationary epoch so that the scalar
field $\phi$ is stabilized at a local minimum of a potential, which we denote
as $\phi_{f}$. The action of the system is still supposed to be of the general
form considered in Sec. II.1. Assuming that the mass of $\phi$ around the
local minimum of the potential is large enough, we integrate out $\phi$ by
setting $\phi=\phi_{f}$ (and thus $X=0$ and $\nabla_{\mu}\nabla_{\nu}\phi=0$)
in the general action. We then end up with the following action for the system
after inflation.

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[G_{4}(\phi_{f},0)R+L(\phi_{f},0,W_{f},Y_{f},0)\right.$
(14)
$\displaystyle\left.+\xi(\phi_{f})e^{2\phi_{f}}F_{\mu\nu}F_{\rho\sigma}R^{\mu\nu\rho\sigma}\right]\,,$

where

$W_{f}\equiv-\frac{1}{4}e^{2\phi_{f}}F_{\mu\nu}F^{\mu\nu}\,,\quad Y_{f}\equiv
e^{2\phi_{f}}F_{\mu\nu}\tilde{F}^{\mu\nu}\,.$ (15)

By Taylor expanding $L(0,W_{f},Y_{f},0)$ with respect to $W_{f}$ and $Y_{f}$
up to first order and using the discrete symmetry (9), we obtain the low-
energy effective action

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[\frac{M_{\rm
Pl}^{2}}{2}(R-2\Lambda)-\frac{1}{4}F^{\rm(post)}_{\mu\nu}F^{{\rm(post)}\mu\nu}\right.$
(16) $\displaystyle\left.+\frac{\lambda}{4M_{\rm
Pl}^{2}}\tilde{F}^{\rm(post)}_{\mu\nu}\tilde{F}^{\rm(post)}_{\rho\sigma}R^{\mu\nu\rho\sigma}\right]\,,$

where we have assumed that

$G_{4}(\phi_{f},0)>0\,,\quad L_{W}(\phi_{f},0,0,0,0)>0\,,$ (17)

and introduced

$M_{\rm
Pl}\equiv\sqrt{2G_{4}(\phi_{f},0)}\,,\quad\Lambda\equiv-\frac{L(\phi_{f},0,0,0,0)}{M_{\rm
Pl}^{2}}\,,$ (18) $\lambda\equiv\frac{4M_{\rm
Pl}^{2}\xi(\phi_{f})}{L_{W}(\phi_{f},0,0,0,0)}\,,\
F^{\rm(post)}_{\mu\nu}=e^{\phi_{f}}\sqrt{L_{W}(\phi_{f},0,0,0,0)}F_{\mu\nu}\,,$

and

$\tilde{F}^{\rm(post)}_{\mu\nu}\equiv\frac{1}{2}{\epsilon_{\mu\nu}}^{\rho\sigma}F^{\rm(post)}_{\rho\sigma}\,.$
(19)

Here, the subscript $W$ denotes partial derivative with respect to $W$. So
far, we have not yet fixed the overall normalization of $F_{\mu\nu}$ except
that the mass dimension of $A_{\mu}$ is zero. We now fix the normalization as

$e^{2\phi_{f}}L_{W}(\phi_{f},0,0,0,0)=M_{\rm Pl}^{2}\,,$ (20)

so that

$\lambda\equiv 4e^{2\phi_{f}}\xi(\phi_{f})\,,\quad
F^{\rm(post)}_{\mu\nu}=M_{\rm Pl}F_{\mu\nu}\,.$ (21)

The postinflationary system described by the action (16) is nothing but the
Einstein-Maxwell system supplemented with the Horndeski’s nonminimal coupling.

Hereafter, we omit the superscript “${\rm(post)}$” so that the action for the
postinflationary system is

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[\frac{M_{\rm
Pl}^{2}}{2}(R-2\Lambda)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right.$ (22)
$\displaystyle\left.+\frac{\lambda}{4M_{\rm
Pl}^{2}}\tilde{F}_{\mu\nu}\tilde{F}_{\rho\sigma}R^{\mu\nu\rho\sigma}\right]\,.$

###  II.4 Observational bounds on $\lambda$, $H_{\rm inf}$, and $\sigma$

In Ref. Mukohyama:2016npi , assuming that the stabilization of $\phi$ to the
constant value $\phi_{f}$ occurs immediately after inflation and that the
reheating process is instantaneous, the present amplitude of the large-scale
magnetic field was estimated as

$\mathcal{B}_{\rm today}\simeq e^{-\phi_{f}}|b|\times 10^{-6}\,\mathrm{G}\,,$
(23)

where $b\equiv B/H_{\rm inf}$. Also, Mukohyama:2018obj  found several examples
of parameters for which the system of linear perturbations is free from
instabilities. In those examples, both $b$ and $g_{h}$ are nonvanishing and of
order unity, where

$g_{h}\equiv\xi\frac{H_{\rm inf}^{2}}{M_{\rm Pl}^{2}}\,,$ (24)

and $\xi$ is the constant value of $\xi(\phi)$ for the range of $\phi$
relevant for the inflationary epoch as already stated around (8). Under the
assumption of immediate stabilization of $\phi$ after inflation, we have
$\xi(\phi_{f})=\xi$. Combining all these and the definition of $\lambda$ given
in (21), one obtains

$\lambda\simeq 4\times\left(\frac{\mathcal{B}_{\rm today}}{|b|\times
10^{-6}\,\mathrm{G}}\right)^{-2}\left(\frac{H_{\rm inf}}{M_{\rm
Pl}}\right)^{-2}g_{h}\,.$ (25)

The upper bound on the large scale magnetic field is roughly
$10^{-9}\,\mathrm{G}$ Shaw:2010ea  and the lower bound from the blazar
observations is roughly

$10^{-15}\,\mathrm{G}\lesssim\mathcal{B}_{\rm today}\lesssim
10^{-9}\,\mathrm{G}\,.$ (26)

On the other hand, constraints on $\lambda$ can be obtained by demanding that
the nonminimal coupling term is less important than the standard Maxwell term
Barrow:2012ay . Reference Allahyari:2020jkn  applied this idea to neutron
stars and found a conservative bound on $\lambda$ as

$|\lambda|\ll 10^{70}\,.$ (27)

Combining (27) with (25), one obtains a lower bound on the inflation scale,

$H_{\rm inf}\gg|b|\,|g_{h}|^{1/2}\,\left(\frac{\mathcal{B}_{\rm
today}}{10^{-9}\,\mathrm{G}}\right)^{-1}\times 10^{-15}\,\mathrm{GeV}\,.$ (28)

For the range (26) of $\mathcal{B}_{\rm today}$ and $\mathcal{O}(1)$ values of
$b$ and $g_{h}$, this is not a strong constraint. Under the assumption of
instantaneous reheating ($T_{\rm reh}\sim\sqrt{M_{\rm Pl}H_{\rm inf}}$), Eq.
(28) can be rewritten as a lower bound on the reheating temperature.

$T_{\rm reh}\gg|b|^{1/2}\,|g_{h}|^{1/4}\,\left(\frac{\mathcal{B}_{\rm
today}}{10^{-9}\,\mathrm{G}}\right)^{-1/2}\times 100\,\mathrm{GeV}\,.$ (29)

One can also obtain limits on the parameter $\sigma$ which characterizes the
degree of axisymmetry of the Bianchi-I spacetime from its contribution to the
quadrupole component $C_{2}$ of the power spectrum of temperature anisotropies
of the cosmic microwave background (CMB). This contribution can be written as
$C_{2}=16\pi(\sigma_{\rm dec}-\sigma_{0})^{2}/25$ (see Ref. Adamek:2011pr  for
an outline of the calculation), where $\sigma_{\rm dec}$ and $\sigma_{0}$ are
values of $\sigma$ at the decoupling and at the present, respectively. From
the observed CMB quadrupole of $C_{2}^{\rm
obs}=230\,\mu\mathrm{K}^{2}/T_{0}^{2}$, where $T_{0}$ is the CMB temperature
today. We can always normalize our coordinates such that
$\sigma_{0}=\sigma(t_{0})=0$ so that $C_{2}$ provides an upper bound on
$|\sigma_{\rm dec}|$,

$|\sigma_{\rm dec}|\lesssim 4\times 10^{-6}\,.$ (30)

###  II.5 Subhorizon description of postinflationary system

In general the effects of the nonminimal coupling can be ignored if

$\frac{(\mbox{curvature})}{M_{\rm Pl}^{2}}\ll\frac{1}{|\lambda|}\,.$ (31)

For the Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology, we have
$(\mbox{curvature})\sim T^{4}/M_{\rm Pl}^{2}$, where $T$ is the temperature of
the Universe, and thus the nonminimal coupling can be ignored if

$T\ll\left|\frac{\lambda}{10^{70}}\right|^{-1/4}\times 10\,\mathrm{GeV}\,.$
(32)

Therefore, imposing the conservative bound (27), we conclude that the
evolution of the FLRW background cosmology during and after nucleosynthesis
can be described by the standard Einstein-Maxwell theory without the
nonminimal coupling. For a local magnetic field with the amplitude
$\mathcal{B}_{\rm local}$, the induced curvature is of order
$(\mbox{curvature})\sim\mathcal{B}_{\rm local}^{2}/(8\pi M_{\rm Pl}^{2})$ and
thus the nonminimal coupling can be ignored if

$\mathcal{B}_{\rm local}\ll\left|\frac{\lambda}{10^{70}}\right|^{-1/2}\times
10^{21}\,\mathrm{G}\,.$ (33)

Assuming that the conservative bound (27) is satisfied, the right-hand side is
larger than $10^{21}\,\mathrm{G}$ and thus the maximum amplitude of the
magnetic field in the simulations studied in the next sections satisfies this
condition. Therefore, we can safely ignore the effects of the nonminimal
coupling and the theory is reduced to the standard Einstein-Maxwell theory
without the nonminimal coupling.

For the standard Einstein-Maxwell theory in a radiation dominated Universe
without the nonminimal coupling, the propagation speed of all physical degrees
of freedom is of order unity and the Jeans scale is of order the Hubble scale.
If we are interested in phenomena whose length and timescales are sufficiently
shorter than the Jeans scales and the cosmological scales then the evolution
of the system can be well described without taking into account the metric
perturbation and the background cosmological expansion. On these scales, the
system is well described by the standard Maxwell theory expanded around the
homogeneous magnetic field background in Minkowski spacetime.

##  III Magnetic field evolution

In the previous section, we have discussed how a spatially homogeneous
magnetic field can be realized during inflation, and more importantly, after
the end of inflation. We now turn our attention to the study of the MHD
evolution of such fields.

###  III.1 Basic equations

We study the time evolution in the presence of a homogeneous magnetic field
right after inflation. In particular, we study the evolution of an additional
field with some typical wave number $k_{*}$, which we induce by a random
forcing term present during a short initial time interval. In the radiation
dominated era, the primordial plasma is a relativistic, isothermal gas with
energy density $\rho$ and equation of state $w=1/3$. In Lorentz-Heaviside
units, the MHD equations for such a gas are Brandenburg:1996fc ;
Kahniashvili:2016bkp ; Brandenburg:2017neh

$\displaystyle\frac{\partial\ln\rho}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{4}{3}\left(\nabla\cdot\mathbf{u}+\mathbf{u}\cdot\nabla\ln\rho\right)$
(34)
$\displaystyle+\frac{1}{\rho}\left[\mathbf{u}\cdot(\mathbf{J}\times\mathbf{B})+\eta\mathbf{J}^{2}\right],$
$\displaystyle\frac{\partial\mathbf{u}}{\partial t}$ $\displaystyle=$
$\displaystyle-(\mathbf{u}\cdot\nabla)\mathbf{u}+\frac{\mathbf{u}}{3}\left(\nabla\cdot\mathbf{u}+\mathbf{u}\cdot\nabla\ln\rho\right)$
(35)
$\displaystyle-\frac{1}{4}\nabla\ln\rho+\frac{3}{4\rho}\mathbf{J}\times\mathbf{B}+\frac{2}{\rho}\nabla\cdot\left(\rho\nu\mbox{\boldmath${\sf
S}$}{}\right)$
$\displaystyle-\frac{\mathbf{u}}{\rho}\left[\mathbf{u}\cdot(\mathbf{J}\times\mathbf{B})+\eta\mathbf{J}^{2}\right]+\mbox{\boldmath${\cal
F}$}{}_{0},$ $\displaystyle\frac{\partial\mathbf{B}}{\partial t}$
$\displaystyle=$
$\displaystyle\nabla\times(\mathbf{u}\times\mathbf{B}-\eta\mathbf{J})+\mbox{\boldmath${\cal
E}$}{}_{0},$ (36)

where
$\mathsf{S}_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})-\frac{1}{3}\delta_{ij}\nabla\cdot\mathbf{u}$
are the components of the traceless rate-of-strain tensor, $\nu$ is the
kinematic viscosity, $\eta$ is the magnetic diffusivity,
$\mbox{\boldmath${\cal F}$}{}_{0}={\cal F}_{0}\bm{f}$ and
$\mbox{\boldmath${\cal E}$}{}_{0}={\cal E}_{0}\bm{f}$ are forcing terms, and

$\bm{f}(\bm{x},t)={\rm Re}\\{{\cal
N}\tilde{{\bm{f}}}(\bm{k},t)\exp[i\bm{k}\cdot\bm{x}+i\phi]\\},$ (37)

is a forcing function that consists of random, white-in-time, plane waves with
a certain average wave number $k_{*}$ Brandenburg:2000wm . Here, $\bm{x}$ is
the position vector and ${\cal N}=\sqrt{c_{\rm s}^{3}k_{*}}$ is a
normalization factor with $c_{\rm s}=\sqrt{w}=1/\sqrt{3}$ being the speed of
sound; see Ref. Brandenburg:2000wm  for details. At each time step, we select
randomly the phase $-\pi<\phi\leq\pi$, the direction of a unit vector
$\hat{\mbox{\boldmath$e$}}{}$, and the components of the wave vector $\bm{k}$
from many possible discrete wave vectors in a certain range around a given
value of $k_{*}$. The Fourier amplitudes are

$\tilde{{\bm{f}}}({\bm{k}})=\mbox{\boldmath${\sf
R}$}{}\cdot\tilde{{\bm{f}}}({\bm{k}})^{\rm(nohel)}\quad\mbox{with}\;\;{\sf
R}_{ij}=\frac{\delta_{ij}-i\sigma\epsilon_{ijk}\hat{k}}{\sqrt{1+\sigma^{2}}},\;$
(38)

where the parameter $\sigma$ characterizes the fractional helicity of
$\bm{f}$, and

$\tilde{{\bm{f}}}({\bm{k}})^{\rm(nohel)}=\left(\bm{k}\times\hat{\mbox{\boldmath$e$}}{}\right)/\sqrt{\bm{k}^{2}-(\bm{k}\cdot\hat{\mbox{\boldmath$e$}}{})^{2}}$
(39)

is a nonhelical forcing function. We use only those
$\hat{\mbox{\boldmath$e$}}{}$ that are not aligned with $\bm{k}$. Note that
$|\tilde{{\bm{f}}}|^{2}=1$. We consider both $\sigma=0$ and $\sigma=1$,
corresponding to the nonhelical and maximally helical cases. The forcing is
only enabled during the time interval $0\leq t\leq t_{*}$. In this sense, this
forcing procedure can be considered as part of the initial condition.

In this section and henceforth, we use $t$ to refer to conformal time, as
opposed to coordinate time in Sec. II. All other quantities are comoving
quantities, scaled by exploiting the conformal symmetry of Maxwell’s
equations; see Brandenburg:1996fc  for details. We solve Eqs. (34)–(36) using
the Pencil Code, a public MHD code444https://github.com/pencil-code, which is
well suited for studying and simulating turbulence. The simulations are
performed in a periodic domain of size $L$. Except for the homogeneous imposed
magnetic field at wave number $k=0$, the smallest nonvanishing wave number in
the domain is $k_{1}\equiv 2\pi/L$. Spatial derivatives are computed using
sixth order accurate finite differences and a third order accurate time
stepping scheme is used. The magnetic vector potential is advanced in time to
preserve solenoidality (the divergence-free condition) of the magnetic field.
We use a numerical resolution of $1152^{3}$ meshpoints for all simulations
presented in this paper.

###  III.2 Peculiarities connected with imposed fields

In a periodic domain, the case of an imposed magnetic field is in many ways
pathological, since it will always be present and can never decay. It can be
amplified linearly in time by a flow – even in two dimensions where no dynamo
effect is possible Mos70 . In addition, magnetic helicity associated with the
induced magnetic field based on the deviations of the magnetic field from the
imposed field is not conserved Ber97 . This is because it interacts with the
imposed field, which, owing to its constancy in space, cannot have magnetic
helicity. On the other hand, if we replace the imposed field by a large-scale
field with zero net flux, the magnetic helicity becomes well defined. The
total field can then decay to zero, and the magnetic helicity is now a
perfectly defined quantity that obeys the usual conservation law. We can
therefore ask how the presence of a large-scale magnetic field affects the
evolution of magnetic helicity of a field of much smaller length scale.

To better understand the aforementioned peculiarities, we note that in the
presence of an imposed magnetic field, a generalized quantity can be defined
that is still conserved MG82 , but that quantity is not gauge invariant and
hence not uniquely defined Brandenburg:2003pe . Let us discuss this here in
more detail. In the presence of an imposed field,
$\mathbf{B}_{0}=\mbox{const}$, one splits the magnetic field into a mean and a
fluctuating component, $\mathbf{B}=\mathbf{B}_{0}+\mathbf{b}$. The mean of
$\mathbf{b}$ is vanishing. Using $\mathbf{b}=\mathbf{\nabla}\times\mathbf{a}$,
the time derivative of the volume-averaged quantity
$\langle\mathbf{a}\cdot\mathbf{b}\rangle$, is found to have a term
$-2\alpha\mathbf{B}_{0}^{2}$, in addition to the Spitzer term
$-2\eta\langle\mathbf{j}\cdot\mathbf{b}\rangle$; see Appendix A for the
derivation. Here, $\alpha$ refers to the $\alpha$ effect and it models the
component of the electromotive force,
$\bm{\mathcal{E}}=\langle\mathbf{u}\times\mathbf{b}\rangle$, parallel to the
mean magnetic field. The $\alpha$ effect is responsible for the fact that the
mean magnetic helicity density ${\cal H}_{\rm
M}=\langle\mathbf{a}\cdot\mathbf{b}\rangle$ is no longer conserved Ber97 .

The presence of an imposed magnetic field was found to influence the sign of
the magnetic helicity and the inverse cascade Haugen:2004zh . For weak (or
zero) imposed fields, magnetic helicity and energy cascade strongly from the
forcing scale to large length scales, and the magnetic helicity has an
opposite sign to the kinetic helicity. For stronger fields, the inverse
cascade of magnetic helicity to larger scales is suppressed, and the sign of
the magnetic helicity flips over. The threshold strength of the imposed
magnetic field depends inversely on the square root of the magnetic Reynolds
number. This is understood to be a consequence of the $\alpha$ effect.

It was also found that, in the presence of an imposed magnetic field, the
induced magnetic field can undergo a certain enhancement around the forcing
wave number. Furthermore, small-scale dynamo action helps to lower the energy
density in the inertial region in $k$-space (Haugen:2004zh, ). In addition,
during the initial time interval $0\leq t\leq t_{*}$, we drive turbulence
either through the $\mbox{\boldmath${\cal F}$}{}_{0}$ or
$\mbox{\boldmath${\cal E}$}{}_{0}$ terms.

###  III.3 Initial conditions

We consider the following types of initial conditions. First, we consider a
homogeneous (imposed) magnetic field Haugen:2004zh , where the corresponding
correlation length is infinite. Second, we consider the case with no imposed
field, but with a zero flux initial scale-invariant magnetic field, so the
correlation length is finite, but much longer than the scale of turbulence.555
The finite value of the correlation length is determined by the cutoff scale
imposed to the scale-invariant spectrum at the low wave numbers region
Kahniashvili:2016bkp  We construct such a field in Fourier space as

$\tilde{B}_{i}({\bf k})=B_{\rm
ini}\left(\delta_{ij}-{\hat{k}}_{i}{\hat{k}}_{j}-i\sigma\epsilon_{ijl}{\hat{k}}_{l}\right)g_{j}({\bf
k})\,|{\bf k}|^{-3/2},$ (40)

where ${\bf g}({\bf k})$ is the Fourier transform of a Gaussian distributed
random vector field that is $\delta$-correlated in all three dimensions. The
degree of helicity is controlled by the parameter $\sigma$, which is $\pm 1$
for maximally helical fields with positive or negative helicity, and zero in
the nonhelical case. The magnetic field in real space is given by
$\mathbf{B}(\bm{x})=\int\tilde{\mathbf{B}}(\bm{k})e^{i\bm{k}\cdot\bm{x}}d^{3}k/(2\pi)^{3}$.
In all cases, we have initially $\rho=\mbox{const}$.

The forcing applied during $0\leq t\leq t_{*}$ consists of monochromatic
forcing (see, for example, Ref. Kahniashvili:2010gp ) at a wave number
$k=k_{\star}$. This forcing wave number corresponds to a fraction of the
Hubble scale $H_{\star}$ after inflation. One can think of this as the epoch
of reheating.

In either case, we consider the relativistic fluid to have an initial
turbulent velocity field $\mathbf{u}(\mathbf{x})$. Physically, turbulence can
be induced at reheating by energy injection from the inflaton into the
standard model particles and fields, or from bubble collisions during some
(yet unknown) phase transition – the spectral energy density $E_{\rm M}(k)$
has a $k^{4}$ subinertial range at large scales due to causality requirements
(see Refs. Durrer:2003ja ; Brandenburg:2018ptt ), while in the inertial range,
it tends to have a Kolmogorov spectrum proportional to $k^{-5/3}$.

Table 1:  Summary of the parameters of the simulations discussed in this
paper. Some characteristic parameters, including the final values of $p$ and
$q$ as defined below and the direction of evolution in the $pq$ diagram are
also indicated.

panel initial field $B_{0}$ $B_{\rm ini}$ $v_{\rm A}^{\max}$ ${\cal F}_{0}$
${\cal E}_{0}$ $\sigma$ $k_{*}$ $~{}\tau_{\rm A}$ $p$ $q$ evolution along (i)
homogeneous 0.1 0 0.08 0.02 0 0 60 0.19 10/7 2/7 $\quad\beta=4$ to
$(p,q)\to(10/7,\,2/7)$ (ii) homogeneous 0.1 0 0.08 0.02 0 1 60 0.19 10/7 2/7
$\quad\beta=4$ to $(p,q)\to(10/7,\,2/7)$ (a) homogeneous 0.03 0 0.46 0 0.0005
1 180 0.21 2 0 $\quad\beta=4$ to $(p,q)\to(2/3,\,2/3)$ (b) homogeneous 0.10 0
0.41 0 0.0005 1 180 0.06 2 0 $\quad p=2(1-q)$ to $(p,q)\to(2,0)$ (c)
homogeneous 0.16 0 0.31 0 0.0005 1 180 0.04 4 0 $\quad\beta=1$–$2$ to
$(p,q)\to(1,\,0.5)$ (d) homogeneous 0.20 0 0.25 0 0.0005 1 180 0.03 4 0
$\quad\beta=3$–$4$ to $(p,q)\to(0.1,\,0.8)$ (e) homogeneous 1.00 0 0.21 0
0.0005 1 180 0.006 4 0 $\quad\beta=3$–$4$ to $(p,q)\to(0,0)$ (A) $1/k$
spectrum 0 $10^{-3}$ 0.47 0 0.0005 1 180 6.4 0.6 0.6 $\quad\beta=0$ to
$(p,q)\to(0.6,\,0.6)$ (B) $1/k$ spectrum 0 $3\times 10^{-2}$ 0.37 0 0.0005 1
180 0.22 0.2 0.2 $\quad\beta=0$ to $(p,q)\to(0.2,\,0.2)$

We recall that, in the absence of a large-scale magnetic field, a small-scale
helical magnetic field undergoes inverse cascading such that the magnetic
energy at small wave numbers increases with time Christensson:2000sp ;
Banerjee:2004df . The characteristic length scale of the turbulence, $\xi_{\rm
M}$, increases with time like $t^{2/3}$, and the magnetic energy ${\cal
E}_{\rm M}$ decreases like $t^{-2/3}$, which is slower than in the nonhelical
case where ${\cal E}_{\rm M}\propto t^{-1}$ and $\xi_{\rm M}\propto t^{1/2}$.

One often considers the magnetic field evolution in a diagram of ${\cal
E}_{\rm M}$ versus $\xi_{\rm M}$. The nonobservation of GeV cascade photons
from the interaction of TeV photons from blazars with the extragalactic
background light, as mentioned above, has often been argued to imply the
presence of a lower limit on the product ${\cal E}_{\rm M}\xi_{\rm M}$ of
about $(10^{-15}\,{\rm G})^{2}\,{\rm Mpc}$. In a diagram of ${\cal E}_{\rm M}$
versus $\xi_{\rm M}$, the line corresponding to this lower limit has a slope
of $-1$, which is also the slope of the line representing the magnetic field
decay in the fully helical case, because ${\cal E}_{\rm M}\propto
t^{-2/3}\propto\xi_{\rm M}^{-1}$. For a nonhelical field, on the other hand,
we have ${\cal E}_{\rm M}\propto t^{-1}\propto\xi_{\rm M}^{-2}$. For this
reason, a nonhelical field will eventually drop below the line demarcating the
lower observational limit Brandenburg:2017neh . We now study how these decay
properties are affected by the presence of either an imposed or an initial
large-scale magnetic field.

Figure 1: The evolution of the magnetic (red) and kinetic (blue) energy
spectra for (i) nonhelical and (ii) helical turbulence. The thick lines are
the configurations at the latest times. Panels (i) and (ii) correspond to runs
(i) and (ii) in Table LABEL:Summary.

###  III.4 Parameters and analysis tools

By default, we measure lengths in units of $k_{1}^{-1}=L/2\pi$ and wave
numbers in units of $k_{1}$. Since $c=1$, time is measured in units of the
light travel time, $(ck_{1})^{-1}$, and viscosity or magnetic diffusivity are
measured in units of $c/k_{1}$. Furthermore, since $\rho=1$ initially, the
magnetic field is measured in units of $c/\sqrt{\rho}$. Our main control
parameters are $k_{*}$, the amplitudes of the imposed or initial fields,
$B_{0}$ and $B_{\rm ini}$, respectively, the amplitudes of the forcing
functions ${\cal E}_{0}$ and ${\cal F}_{0}$, and the values of $\nu$ and
$\eta$. For $k_{*}$, we consider the values 60 and 180, $B_{0}$ and $B_{\rm
ini}$ are varied between 0 and 1, while ${\cal E}_{0}$ and ${\cal F}_{0}$ are
varied 0 and 0.02, such that the energy density of the turbulence does not
exceed the radiation energy density by more than 10% after the duration of
turbulent driving, which we have chosen to be $t_{*}=5$ in the normalized
units defined below. In all cases, we use a resolution of $1152^{3}$
meshpoints and we found that $\nu=\eta=10^{-5}$ is sufficiently small to
dissipate the energy of the turbulence at the smallest length scale. A summary
of parameters of all runs is given in Table LABEL:Summary.

Figure 2: $pq$ diagrams for the cases of nonhelical (i) and helical (ii)
turbulence. The dots denote the instantaneous values of $(p,q)$ for the
magnetic (red) and kinetic (blue) fields. Bigger circles denote later times.
Panels (i) and (ii) correspond to runs (i) and (ii) in Table LABEL:Summary.
The solid lines correspond to the self-similarity line, $p=2(1-q)$, the dotted
lines denote $\beta=\mbox{const}$, and the dashed lines denote
$p=\mbox{const}\approx 0.58$, whose relevance is explained in the text.

It is sometimes convenient to express time in units of the Alfvén time,
$\tau_{\rm A}=(v_{\rm A}k_{*})^{-1}$, where $v_{\rm
A}^{2}=B_{0}^{2}/({\textstyle\frac{4}{3}}\rho)$ for cases with an imposed
magnetic field and $v_{\rm A}^{2}=B_{\rm
ini}^{2}/({\textstyle\frac{4}{3}}\rho)$ for cases with a zero flux large-scale
magnetic field. To specify the strength of the fluctuating magnetic field in
cases with $B_{0}\neq 0$, we also specify the quantity $v_{\rm
A,f}^{\max}=|\mathbf{B}-\mathbf{B}_{0}|/({\textstyle\frac{4}{3}}\rho)^{1/2}$.
The kinetic and magnetic energy densities are defined as ${\cal E}_{\rm
K}=\langle\rho\bm{u}^{2}\rangle/2$ and ${\cal E}_{\rm
M}=\langle\mathbf{B}^{2}\rangle/2$, respectively, and the kinetic and magnetic
energy spectra, $E_{\rm K}(k,t)$ and $E_{\rm M}(k,t)$, are normalized such
that

$\int E_{\rm K}(k,t)\,dk={\cal E}_{\rm K}\quad\mbox{and}\quad\int E_{\rm
M}(k,t)\,dk={\cal E}_{\rm M},$ (41)

respectively. We define the magnetic correlation length $\xi_{\rm M}$ as

$\xi_{\rm M}(t)=\left.\int k^{-1}E_{\rm M}(k,t)\,dk\right/\\!\\!\int E_{\rm
M}(k,t)\,dk.$ (42)

Finally, we define the instantaneous exponents describing the growth of
$\xi_{\rm M}(t)$ and the decay of ${\cal E}_{\rm M}(t)$ as

$q_{i}(t)=d\ln\xi_{i}/d\ln t,\quad p_{i}(t)={\color[rgb]{0,0.7,0.0}-}d\ln{\cal
E}_{i}/d\ln t.$ (43)

Those play important roles in describing the nature of the turbulence in
different cases Brandenburg:2016odr .

The various solutions are characterized by certain lines in the $pq$ diagram.
It was found that the point $(p,q)$ ultimately settles somewhere on what was
called the self-similarity line Brandenburg:2016odr , where

$p=2(1-q).$ (44)

Moreover, this evolution occurs along a line with

$\beta=p/q-1=\mbox{const},$ (45)

where the value of $\beta$ is determined by the nature of certain relevant
conservation laws. Eliminating $p$ from Eqs. (44) and (45), we find
$\beta=2/q-3$, where $q$ can be obtained from dimensional arguments in terms
of the dimensions of length $L$ and time $T$. We recall that $q$ characterizes
the scaling of the correlation length with time as $\xi_{\rm M}\sim t^{q}$.
Magnetic helicity has dimensions $L^{3}T^{-2}$, so $q=2/3$, and therefore
$\beta=0$. The mean squared vector potential, which is arguably relevant to
magnetically dominated turbulence Brandenburg:2014mwa , has dimensions
$L^{4}T^{-2}$, so $q=1/2$, and therefore $\beta=1$. The Saffman integral Saf67
has dimensions $L^{5}T^{-2}$, so $q=2/5$, and therefore $\beta=2$, while the
Loitsiansky integral BP56  has dimensions $L^{7}T^{-2}$, so $q=2/7$, and
therefore $\beta=4$.

Under certain conditions, the evolution may not be self-similar for extended
periods of time. In fact, for finite resolution and finite domain size, a
truly self-similar behavior is generally difficult to obtain. A prolonged
evolution along the line $p=\mbox{const}\approx 0.58$ was obtained
Brandenburg:2017rnt  when there is a complex interplay between kinetic and
current helicities. In the present work, we find examples of several of the
aforementioned relations.

##  IV Results

###  IV.1 Helical and nonhelical decay with imposed field

We consider decaying turbulence produced during a short initial period through
forcing at small scales with $k_{*}=60$ together with an imposed magnetic
field. We find that in the subinertial range, the magnetic energy spectrum
goes approximately as $k^{2}$, while the kinetic energy spectrum is shallower.
In Figs. 1(i) and 1(ii), we show the evolution of the magnetic and kinetic
energy spectra for the nonhelical and helical cases, respectively. We see that
the winding up of the initially uniform field by turbulence causes a Saffman
spectrum for the magnetic energy of the form $E_{\rm M}\sim k^{2}$, which is
shallower than the Batchelor $k^{4}$ spectrum. There is no inverse cascade in
the sense that, even at small $k$, the magnetic energy always decays. The
decay is faster at larger $k$, which causes $\xi_{\rm M}$ to increase, but
this is not due to the usual inverse cascade.

Figure 3:  Dependences of $\xi_{\rm M}(t)$, ${\cal E}_{\rm M}(t)$, ${\cal
E}_{\rm M}(t)/v_{\rm A}$, and ${\cal E}_{\rm M}(t)$ against $\xi_{\rm M}(t)$.
In the last panel, the slope ${\cal E}_{\rm M}\propto\xi_{\rm M}^{-2/3}$ is
shown for comparison. The time interval $0\leq t\leq t_{*}\equiv 5$ is marked
with thin lines, while later times are marked with thick lines. Blue (red)
lines denote cases with a perfectly homogeneous (statistically homogeneous)
magnetic field. Solid (dashed) lines correspond to cases with a weak (strong)
magnetic field; compare labels (a)–(d) with the corresponding runs in Table
LABEL:Summary. The filled symbols on each curve denote the five instances for
which the spectra below are shown.

To quantify the decay further, we now show in Figs. 2(i) and 2(ii) the
evolution of the instantaneous scaling exponents $p_{i}(t)$ versus $q_{i}(t)$
for $i={\rm M}$ and ${\rm K}$, where ${\mathcal{E}}_{i}$ is the energy density
and $\xi_{i}$ is the integral length scale for the magnetic and fluid fields.
We see that for both the helical and the nonhelical cases, the evolution of
the point $(p,q)$ tends to be close to the $\beta=4$ line, which implies the
conservation of the Loitsiansky integral BP56 . This evolution is similar to
that of nonhelical and nonmagnetic turbulence, which is quite surprising: in
the presence of a sufficiently strong constant magnetic field, magnetic
helicity seems to have no effect, and the decay is very different from that in
magnetically dominated turbulence, where $\beta=1$–$2$ has been found
Brandenburg:2014mwa ; Brandenburg:2016odr .

Figure 4:  Magnetic (red) and kinetic (blue) energy spectra for
$k_{*}/k_{1}=180$ with an imposed field, $B_{0}=0.03$, $0.1$, $0.16$, and
$0.2$ in panels (a)–(d), respectively. These panels correspond to runs (a)–(d)
in Table LABEL:Summary. Dotted, dashed, solid, dash-dotted, and dash-triple-
dotted lines indicate later times, denoted by filled symbols in Fig. 3. The
last time is also shown as a fat line.

Figure 5:  Similar to Fig. 4, but for an initial large-scale field, $B_{\rm
ini}=10^{-3}$ and $3\times 10^{-2}$ in panels (A) and (B), respectively.

Figure 6:  $pq$ diagrams for the magnetic field (red) and the velocity field
(blue) for $k_{*}/k_{1}=180$ with an imposed field, $B_{0}=0.03$, $0.1$,
$0.16$, and $0.2$ in panels (a)–(d), respectively. Again, these panels
correspond to runs (a)–(d) in Table LABEL:Summary. Later times are shown as
larger symbols. The arrows in each panel indicate the tentative direction of
evolution.

Figure 7:  Similar to Fig. 6, but for an initial large-scale field, $B_{\rm
ini}=10^{-3}$ and $3\times 10^{-2}$ in panels (A) and (B),

###  IV.2 Inverse cascade

We know that, in the absence of a large-scale magnetic field, a small-scale
helical magnetic field decays more slowly than a nonhelical one, and also its
correlation length increases faster than for a nonhelical field. It is
therefore of interest to study how the magnetic decay is affected by the
presence of this large-scale magnetic field. One may also ask whether some of
the magnetic energy of this large-scale field can be transferred to the
smaller scale field.

In all cases, we produce a small-scale helical magnetic field by driving the
system with a turbulent small-scale electromotive force for a short time
interval $0\leq t\leq t_{\ast}=5$. This driving is then turned off, leaving
the system to decay freely, except for the presence of the imposed magnetic
field. The runs are summarized in the lower part of Table LABEL:Summary.

The time evolution of $\xi_{\rm M}(t)$ and ${\cal E}_{\rm M}(t)$ is shown in
Figs. 3(a) and 3(c). In Fig. 3(b) we plot the evolution $\xi(t/\tau_{\rm A})$
versus normalized time Fig. 3(d). Our results allow us to show ${\cal E}_{\rm
M}(t)$ against $\xi_{\rm M}(t)$ in a parametric fashion; see Fig. 3. Note that
we have not included in ${\cal E}_{\rm M}(t)$ the additional presence of the
imposed magnetic field, i.e., the magnetic energy is defined solely based on
the magnetic field with nonvanishing wave numbers.

In Figs. 4 and 5, we present magnetic energy spectra for cases with an imposed
and an initial magnetic field, respectively. In both cases, we see inverse
cascading of the magnetic energy when the imposed or initial magnetic fields
are weak. However, when the field is increased, the inverse cascade eventually
stalls; see especially Fig. 4(c), where inverse cascading has stopped after
the peak of the spectrum traversed the $k$ axis by about a factor of 10.

Interestingly, in the presence of an initial (nonimposed) magnetic field, the
evolution of ${\cal E}_{\rm M}(t)$ versus $\xi_{\rm M}(t)$ follows the same
line in Fig. 3(d). This line corresponds to ${\cal E}_{\rm M}\propto\xi_{\rm
M}^{-1}$ and its height in that diagram characterizes the strength of magnetic
helicity Brandenburg:2017neh . Although the magnetic field was initially of
small scale only, at the end of the evolution, it has reached the scale of the
system. This is true for both weak and strong initial (nonhelical) magnetic
fields.

For an imposed magnetic field, on the other hand, the magnetic field is always
below the line ${\cal E}_{\rm M}\propto\xi_{\rm M}^{-1}$, which corresponds to
the evolution of a fully helical magnetic field. This is simply because
magnetic helicity is no longer conserved in that case; see the blue lines in
the last panel of Fig. 3. The marked segregation between the blue and red
lines in all panels of Fig. 3 is an impressive and very visual demonstration
of the tremendous difference between the cases of an initial and an imposed
magnetic field. Only when the imposed magnetic field is weak enough are the
two cases in mutual in agreement with each other; compare the red (A) and blue
(a) lines in all panels of Fig. 3.

Finally, we show in Figs. 6 and 7 the evolution of the instantaneous scaling
exponents $p(t)$ and $q(t)$ in a $pq$ diagram. We see that in the case with an
imposed magnetic field of moderate strength in panel (a) the point $(p,q)$
appears to evolve along the line $p=2(1-q)$ toward $(p,q)=(2,0)$. This is
indeed consistent with Fig. 3, where ${\cal E}_{\rm M}(t)$ is seen to decay
like $t^{-2}$ and $\xi_{\rm M}(t)$ is approximately flat. For a stronger
imposed field in panel (b), there seems to be an evolution along $\beta=3$–$4$
toward $(p,q)\to(0,0)$, but this is not consistent with Fig. 3, where ${\cal
E}_{\rm M}(t)$ is seen to decay like $t^{-4}$, while $\xi_{\rm M}(t)$ is still
approximately flat. Indeed, the points in Fig. 6(b) have a similar size,
suggesting that the evolution along the line $\beta=3$–$4$ is an intermediate
stage before later evolving toward $(p,q)\to(4,0)$, which is obviously outside
the plot range.

On the other hand, for an additional large-scale nonhelical magnetic field, in
addition to the small-scale helical one, the evolution of the point $(p,q)$
always occurs along the $\beta=0$ line. As time goes on, the point $(p,q)$
evolves further along the $\beta=0$ line toward the left to smaller values of
$p(t)$ and $q(t)$. For the weak large-scale magnetic field of panel (c), the
evolution stalls near the point $(p,q)=(0.6,\,0.6)$. Several intermediate
points cluster along the line $p=0.58$, which was identified in earlier work
Brandenburg:2017rnt , but this may be coincidental. Indeed, for the stronger
large-scale magnetic field of panel (d), the evolution continues toward the
point $(p,q)=(0.2,\,0.2)$. For sufficiently weak imposed magnetic fields, the
cases of imposed and initial magnetic fields again agree with each other;
compare Fig. 6(a) for run (a) with Fig. 7(A) for run (A).

These investigations have demonstrated the dramatic difference between imposed
and initial large-scale magnetic fields. When the imposed fields are weak, it
only affects the evolution of the small-scale helical magnetic field at later
times once its field strength approaches the value of the imposed field. In
the presence of a large-scale nonhelical magnetic field – here one with a
$k^{-1}$ spectrum – the inverse cascade is not suppressed. Both for weak and
strong magnetic fields, there is a spectral peak moving from large to small
wave numbers; see Fig. 5. Also the evolution in the $pq$ diagram is along the
$\beta=0$ line in both cases; see Fig. 7.

We emphasize again that the presence of an imposed field is pathological, if
the interest is to simulate an approximation to the case with a large-scale
magnetic field. We have demonstrated this here with an irregular large-scale
field with a $k^{-1}$ spectrum. Starting with an initially sinusoidal magnetic
field is probably similar in many ways, but this would introduce anisotropies,
which have not yet been studied in the context of decaying turbulence.

##  V Conclusions

We have discussed the viability of a homogeneous magnetic field after
inflation. Our work therefore extends the earlier work of one of the authors
Mukohyama:2018obj , which addressed only the stability of the magnetic field
in the inflationary stage. In this work, we have addressed the phenomenology
of the primordial plasma after inflation in the presence of a homogeneous
magnetic field. Our results apply to the early epochs of the Universe all the
way from the time when the inequality in Eq. (32) is satisfied until matter
radiation equality.

Our simulations have verified that, in the presence of an imposed magnetic
field, magnetic helicity is not conserved. Moreover, and this was not
previously known, our results demonstrate that the decay of magnetic energy in
the fluctuations is faster the stronger the imposed magnetic field. We have
also compared the magnetic field evolution with an alternative way of
simulating a cosmological large-scale magnetic field, namely to treat it as a
statistically homogeneous field with a scale-invariant spectrum. It is no
longer the stealth magnetic field considered in the scenario of
Mukohyama:2016npi , but one that could emerge at the end of inflation. Such a
magnetic field can be either helical Kahniashvili:2016bkp ;
Brandenburg:2018ptt  or nonhelical Kahniashvili:2012vt . In these cases,
magnetic helicity conservation is unaffected by the large-scale magnetic
field, and it decays just like without imposed magnetic field and thus much
more slowly than with a constant imposed magnetic field.

Conservation of magnetic helicity (and correspondingly its presence until
recombination) can have important observational consequences. In particular,
primordial magnetic helicity (as a manifestation of the possible violation of
parity in the early Universe) can leave traces in (i) the cosmic microwave
background fluctuations, resulting in nonzero temperature $B$-polarization,
and $E$\- and $B$-polarization cross correlations (see Kahniashvili:2014dfa ;
Ballardini:2014jta  and references therein), and (ii) the circular
polarization of gravitational waves generated in the early Universe through
helical hydrodynamical and MHD turbulence (see Pol:2019yex  and references
therein).

As for the backreaction of small-scale fields to the background magnetic field
at large scale, a priori there could be three possibilities: (i) small-scale
fields inverse cascade and deplete the background magnetic field at large
scale; (ii) small-scale fields inverse cascade and enhance the background
magnetic field at large scale; or (iii) small-scale fields do not affect the
background magnetic field at large scale. The result of the present paper
suggests that (iii) is the case. Therefore, a homogeneous magnetic field, if
generated during inflation, should persist [i.e., simply decay as $\propto
1/a^{2}$, as assumed in the derivation of (23)] under the influence of small-
scale fields and could be the origin of the large-scale magnetic field in the
Universe today. Depending on the strength of the background magnetic field,
however, the small-scale magnetic field can be significantly suppressed. The
low power at small scales (see the blue lines in Fig. 3) means that the
homogeneous background may dominate the magnetic fields in the Universe not
only at large scales but also at small scales. This implies that, depending on
its strength, the background magnetic field may be responsible not only for
the blazar observations, but also for the seeds of MHD processes at
astrophysical scales such as galactic dynamo.

###### Acknowledgements.

We thank Arthur Kosowsky and Alexander Tevzadze for useful discussions. T.K.
and S.M. (CMU) are especially grateful for the hospitality provided by the
University of Geneva, where initial discussions on this project were made.
Support through the NSF Astrophysics and Astronomy Grant Program (Grants No.
1615940 and No. 1615100), and the Shota Rustaveli NSF (Georgia) (Grants No.
FR/18-1462 and No. FR/19-8306) are gratefully acknowledged. The work of S.M.
(YITP) was supported by Japan Society for the Promotion of Science Grants-in-
Aid for Scientific Research No. 17H02890, No. 17H06359, and by World Premier
International Research Center Initiative, MEXT, Japan. R.D. is supported by
the Swiss National Science Foundation. 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.

##  Appendix A Magnetic helicity evolution in the presence of a homogeneous
magnetic field

The purpose of this appendix is to recall why magnetic helicity is not a
conserved quantity in the presence of a homogeneous or imposed magnetic field,
$\mathbf{B}_{0}$. We write $\mathbf{B}=\mathbf{B}_{0}+\mathbf{b}$, where
$\mathbf{b}=\bm{\nabla}\times\mathbf{a}$, with $\mathbf{a}$ being the vector
potential of $\mathbf{b}$. Equation (36) can then be written in terms of
$\mathbf{a}$ as

$\frac{\partial\mathbf{a}}{\partial
t}=\mathbf{u}\times\mathbf{B}_{0}+\mathbf{u}\times\mathbf{b}-\eta\mathbf{j}+\mbox{\boldmath${\cal
E}$}{}_{0}.$ (46)

We consider times $t>t_{\ast}$ when $\mbox{\boldmath${\cal E}$}{}_{0}=\bm{0}$.
Making use of the periodic boundary conditions for $\mathbf{a}$, the evolution
equation for $\langle\mathbf{a}\cdot\mathbf{b}\rangle$ is then

$\frac{d}{dt}\langle\mathbf{a}\cdot\mathbf{b}\rangle=2\langle(\mathbf{u}\times\mathbf{B}_{0})\cdot\mathbf{b}\rangle-2\eta\langle\mathbf{j}\cdot\mathbf{b}\rangle.$
(47)

Evidently, the first term on the right-hand side of Eq. (47) breaks magnetic
helicity conservation in the limit $\eta\to 0$, because

$\langle(\mathbf{u}\times\mathbf{B}_{0})\cdot\mathbf{b}\rangle=-\langle\mathbf{u}\times\mathbf{b}\rangle\cdot\mathbf{B}_{0}=-\alpha\mathbf{B}_{0}^{2}\neq
0$ (48)

for a helical magnetic field, where $\alpha\neq 0$. We recall an important
result from mean-field electrodynamics Mof78 ; KR80

$\langle\mathbf{u}\times\mathbf{b}\rangle_{i}=\alpha_{ij}\langle
B_{j}\rangle+\beta_{ijk}\partial\langle B_{j}\rangle/\partial x_{k}+\cdots,$
(49)

which reduces to
$\langle\mathbf{u}\times\mathbf{b}\rangle=\alpha\mathbf{B}_{0}$ for periodic
boundary conditions. Here, the ellipsis refers to higher order terms. The
nonconservation of magnetic helicity is not an artifact of having adopted
periodic boundary conditions, because they are just a tool for us to compute
averages over infinitely large length scales.

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