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Influence of backreaction of electric fields and Schwinger effect

on inflationary magnetogenesis

O. O. Sobol,

1,2

E. V. Gorbar,

2,3

M. Kamarpour,

and S. I. Vilchinskii

2,4

Institute of Physics, Laboratory for Particle Physics and Cosmology,

École Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland

Physics Faculty, Taras Shevchenko National University of Kyiv, 64/13,

Volodymyrska Street, 01601 Kyiv, Ukraine

Bogolyubov Institute for Theoretical Physics, 14-b, Metrologichna Street, 03680 Kyiv, Ukraine

D´epartement de Physique Th´eorique, Center for Astroparticle Physics,

Universit´e de Gen`eve, 1211 Gen`eve 4, Switzerland

(Received 2 August 2018; published 26 September 2018)

We study the generation of electromagnetic fields during inflation when the conformal invariance

of Maxwell’s action is broken by the kinetic coupling f

ðϕÞF

μν

of the electromagnetic field to the

inflaton field ϕ. We consider the case where the coupling function fðϕÞ decreases in time during inflation

and, as a result, the electric component of the energy density dominates over the magnetic one. The system

of equations which governs the joint evolution of the scale factor, inflaton field, and electric energy density

is derived. The backreaction occurs when the electric energy density becomes as large as the product

of the slow-roll parameter ϵ and inflaton energy density, ρ

∼ ϵρ

inf

. It affects the inflaton field evolution and

leads to the scale-invariant electric power spectrum and the magnetic one which is blue with the spectral

index n

¼ 2 for any decreasing coupling function. This gives an upper limit on the present-day value of

observed magnetic fields below 10

−22

G. It is worth emphasizing that since the effective electric charge

of particles e

eff

¼ e=f is suppressed by the coupling function, the Schwinger effect becomes important

only at the late stages of inflation when the inflaton field is close to the minimum of its potential.

The Schwinger effect abruptly decreases the value of the electric field, helping to finish the inflation

stage and enter the stage of preheating. It effectively produces the charged particles, implementing the

Schwinger reheating scenario even before the fast oscillations of the inflaton. The numerical analysis is

carried out in the Starobinsky model of inflation for the powerlike f ∝ a

and Ratra-type f ¼ expðβϕ=M

coupling functions.

DOI: 10.1103/PhysRevD.98.063534

I. INTRODUCTION

One of the important problems of modern cosmology is

the origin of the magnetic fields which are present at all

scales in the Universe [1–8], especially of the magnetic

fields detected in the cosmic voids through the gamma-ray

observations of distant blazars [9–13] with very large

coherence scale λ

≳ 1 Mpc. Together with the observations

of the cosmic microwave background (CMB) [8,14–17]

and ultrahigh-energy cosmic rays [18] this implies the

following constraints on the strength of these fields:

−18

≲B

≲10

−9

G. If the correlation length of the magnetic

field is λ

≲ 1 Mpc, the minimal needed magnetic field

strength is larger by the factor ðλ

=1 MpcÞ

−1=2

[9,11–13].

Observed intergalactic magnetic fields can have either

astrophysical or primordial origin, and both magnetogen-

esis scenarios are currently under discussion. Although

astrophysical mechanisms based on a Biermann battery

[19–21] have been proposed to generate the “seed”

magnetic fields and different types of dynamo can enhance

them [22–25], it is problematic to embed the magnetic

fields with a large correlation length into the cosmic voids.

Therefore, the primordial origin of the large-scale magnetic

fields seems to be more realistic. In particular, the phase

transitions in the early Universe may lead to the magnetic

fields of the necessary strength [26–31]. However, their

coherence length is determined by the horizon size at the

moment of phase transition and is much less than Mpc

today. Then the most natural is the inflationary magneto-

genesis, proposed in Refs. [32,33], which can easily attain

very large coherence length.

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP

PHYSICAL REVIEW D 98, 063534 (2018)

2470-0010=2018=98(6)=063534(19) 063534-1 Published by the American Physical Society

Since Maxwell’s action is conformally invariant, the

fluctuations of the electromagnetic field do not undergo

enhancement in the conformally flat inflationary back-

ground [34]. In order to generate electromagnetic fields we

need to break the conformal invariance. This can be done

by introducing the interaction with scalar or pseudoscalar

inflaton fields or curvature scalar (see, e.g., the pioneer

works [32,33,35,36]). In our study we consider the kinetic

coupling of the electromagnetic field to the scalar inflaton

via the term f

ðϕÞF

μν

, which was first introduced by

Ratra [33] and has been revisited many times for different

types of coupling functions [37–45].

This model modifies the standard electromagnetic

Lagrangian, multiplying it by the function of the inflaton

field. As it was mentioned in [40], if one rescales the

electromagnetic potential in order to absorb f

, electric

charges of particles effectively will depend on f

−1

Therefore, to avoid the strong coupling problem during

inflation one needs to require f ≥ 1. Since the inflaton field

and the scale factor change monotonously during inflation

it is natural to assume that the coupling function is a

decreasing function during inflation which attains large

values in the beginning.

For decreasing coupling functions, it is well known that

the electric energy density dominates the magnetic one

[39,40,44] and if we try to generate the magnetic field

strong enough to be in accord with the observations, the

electric field appears to be even stronger and its energy

density exceeds that of the inflaton field. This is known as

the backreaction problem. In previous studies, the authors

tried to avoid this problem, because it does not allow us to

solve the background equations and Maxwell equation

separately [39,40,42,43]. However, it is interesting what

happens when the backreaction becomes important and

whether the amplification really stops in this regime. These

are open questions in the literature.

Since strong electric fields could be generated during

inflation, the pair creation in a strong electric field, which is

known as the Schwinger effect [46], can become important

and affect the magnetogenesis. The Schwinger effect in the

constant and homogeneous background electric field in de

Sitter space-time was investigated by many authors [47–60].

The cases of (1 þ 1)-dimensional [47,50,51],(2 þ 1)-

dimensional [49],and(3 þ 1)-dimensional [48,52,53,57,

58] de Sitter space-time with scalar [47–49,53,57–59] and

spinor charged fields [50–52,57,58], including also an

external magnetic field [56], were investigated. It is impor-

tant to note that constantelectric energy density is considered

rather than the case of a constant comoving electric field.

According to Ref. [61], maintaining this regime would

require the existence of ad hoc currents that could violate

the second law of thermodynamics.

The cosmological Schwinger effect relates the particle

production by an electric field and the exponential expan-

sion of the Universe and contains interesting features which

are absent in its flat-space counterpart, namely (i) the

infrared hyperconductivity in the bosonic case when the

conductivity becomes very large in the limit of small mass

of charged particles and (ii) the negative conductivity in the

weak-field regime eE ≪ H

which can, in principle, lead to

the enhancement of the electric field.

The induced current of created particles obtained by

direct averaging of the corresponding current operator

contains ultraviolet divergences, which can be regularized

using adiabatic subtraction [48,52] or the point-splitting

method [53]. Although these techniques remove the diver-

gent parts, the finite part is not uniquely defined and

depends on the applied subtraction scheme. Adiabatic

subtraction to the second order in the adiabaticity parameter

and point-splitting regularization define the minimal sub-

traction scheme which is commonly used. However, in the

massive limit, when the particle’s mass is much greater than

the Hubble parameter, the current and conductivity contain

exponentially unsuppressed terms, which is not consistent

with the standard Bogolyubov calculations. In order to

eliminate this discrepancy the authors of Ref. [57] proposed

a new maximal subtraction scheme, which normalizes the

behavior of the current in the massive case. However, the

main features in the small mass regime remain the same

and the infrared hyperconductivity in the scalar case is

observed [48,53].

In addition, the charged particles show negative con-

ductivity in the weak-field regime, which can lead to an

instability and an avalanchelike enhancement of electric

field up to a certain critical value [52,58]. In the very recent

article [59] Stahl considers a possibility to enhance the

quantum fluctuations of the electromagnetic field even

without interaction with an inflaton. However, the negative

conductivity is rather speculative [57,58] and may be an

artifact of used subtraction schemes.

An attempt to combine the generation of the electro-

magnetic field due to kinetic coupling with the inflaton, the

Schwinger effect and the backreaction into a consistent

picture was made in Ref. [62]. It was shown that the

expressions for the Schwinger current in the time-dependent

electric background in the strong-field regime have the

same functional dependence as in the case of a constant

electric field. However, the backreaction on the background

evolution was calculated perturbatively only in the first

order, which is valid only at the early stages of inflation.

The impact on the generated magnetic field was not

discussed as well.

This paper is organized as follows. We derive a self-

consistent system of equations which describes the joint

evolution of the scale factor, inflaton field, and electric field

energy density in Sec. II, where we take into account the

backreaction of generated fields on the background evo-

lution and the Schwinger effect which is important at the

late stages of inflation. In Sec. III, we study the regime

when the backreaction becomes important and analyze the

SOBOL, GORBAR, KAMARPOUR, and VILCHINSKII PHYS. REV. D 98, 063534 (2018)

063534-2

main features of the electromagnetic field power spectra

generated in this regime. The results of numerical calcu-

lations of the power spectra of generated fields and the

present value of the magnetic field in the Starobinsky

inflation model for two types of coupling functions are

represented in Sec. IV. The summary of the obtained results

is given in Sec. V.

II. SELF-CONSISTENT SYSTEM OF EQUATIONS

We consider a spatially flat Friedmann-Lamaître-

Robertson-Walker Universe with metric tensor

μν

¼ diagð1; −a

; −a

Þ;

ﬃﬃﬃﬃﬃﬃ

−g

¼ a

; ð1Þ

and use the natural system of units where ℏ ¼ c ¼ 1, M

ð8πGÞ

−1=2

¼ 2.4 × 10

GeV is a reduced Planck mass,

and e ¼

ﬃﬃﬃﬃﬃﬃﬃﬃ

4πα

≈ 0.3 is the absolute value of the electron’s

charge.

The action which describes inflaton field ϕ, electromag-

netic field A

, and charged field χ (either bosonic or

fermionic) reads

S ¼

ﬃﬃﬃﬃﬃﬃ

−g



μν

∂

ϕ∂

ϕ − VðϕÞ

−

ðϕÞg

μα

νβ

μν

αβ

þ L

charged

ðA; χÞ



; ð2Þ

where VðϕÞ is the inflaton effective potential; fðϕÞ is the

kinetic coupling function; and L

charged

ðA; χÞ is a gauge-

invariant Lagrangian of the charged field χ interacting with

the electromagnetic field A

The corresponding Euler-Lagrange equations have the

form

ﬃﬃﬃﬃﬃﬃ

−g

∂

ﬃﬃﬃﬃﬃﬃ

−g

μν

∂

ϕþ

dϕ

¼ −

μν

; ð3Þ

ﬃﬃﬃﬃﬃﬃ

−g

∂

ﬃﬃﬃﬃﬃﬃ

−g

μα

νβ

ðϕÞF

αβ

¼−j

; ð4Þ

where the 4-current is defined as usual

∂L

charged

ðA; χÞ

∂A

: ð5Þ

The right-hand side of Eq. (3) describes the backreaction of

created electric fields on the evolution of the inflaton field.

Evolution of the Universe is driven by the total energy

density of all fields. It can be calculated as the 00-component

of the stress-energy tensor. The latter is defined as usual as

μν

ﬃﬃﬃﬃﬃﬃ

−g

δS

δg

μν

¼ ∂

ϕ∂

ϕ − f

ðϕÞg

αβ

μα

νβ

− g

μν

þ T

ðchargedÞ

μν

; ð6Þ

where L

is the Lagrangian density of the inflaton and

electromagnetic fields which is represented by the first three

terms in the square brackets in Eq. (2).

In the simplest case the inflaton field is spatially

homogeneous. If the coupling function fðϕÞ always

decreases in time, then it is well known [39,40] that the

electric component of the created electromagnetic field

dominates the magnetic one and leads to the backreaction

problem. Therefore, we take into account the presence of

electric field F

¼ aðtÞE

ðtÞ and neglect the magnetic

component F

¼ a

ijk

≈ 0. Then the energy density

reads

ρ ¼



þ VðϕÞ



þ ρ

¼ ρ

inf

þ ρ

; ð7Þ

where ρ

is the energy density of the charged particles

produced by the Schwinger effect.

The Friedmann, Klein-Gordon, and Maxwell equations

take the following form:





ðρ

inf

þ ρ

Þ; ð8Þ



ϕ þ 3H

ϕ þ

dϕ

¼ fðϕÞf

ðϕÞE

ðtÞ; ð9Þ

∂

ða

Þ¼−aj

; ð10Þ

where overdots denote the derivatives with respect to

cosmic time and the prime denotes the derivative with

respect to the inflaton field.

It has to be mentioned that we consider the classical

electric field, which is generated from quantum fluctuations

due to interaction with the inflaton. It is useful to analyze

the mode composition of this field. When the modes are

inside the horizon they oscillate in time and have to be

treated as quantum fluctuations. However, when they cross

the horizon the mode functions start to behave monoton-

ically and can be chosen real. According to Ref. [63], this

corresponds to the quantum-to-classical transition and

these modes can be treated classically. Therefore, the

contribution to the electric field is made only by the

modes, which are outside the horizon. Since their wave-

length is larger than the horizon, i.e., the largest observable

scale, the corresponding electric field can be considered

homogeneous.

INFLUENCE OF BACKREACTION OF ELECTRIC FIELDS … PHYS. REV. D 98, 063534 (2018)

063534-3

A. Equation for electric field energy density

It is convenient to rewrite the Maxwell equation (10) in

terms of electric field energy density ρ

¼ f

=2:

þ 4Hρ

þ 2

¼ −

ðE · jÞ: ð11Þ

The “classical” part of the electric energy density at a

certain moment of time is determined by the modes, which

crossed the horizon from the beginning of inflation till the

moment under consideration:

ðtÞ¼

ðtÞ

dρ

ðtÞ;k

ðtÞ¼aðtÞHðtÞ; ð12Þ

where k

is the momentum of the mode which crosses the

horizon at the beginning of inflation.

However, Eq. (11) does not take into account the fact that

the number of relevant modes with wavelength larger than

the horizon changes in time. In order to deal with this, we

should introduce an additional term which describes the

modes crossing the horizon at a given time t and starting to

contribute to the total energy density of the electric field:

dρ



k¼k

: ð13Þ

The power spectra of electric and magnetic fields are

expressed through the mode function of the electromag-

netic field Aðk; tÞ in the standard way [39]:

dρ

2π



∂

∂t

Aðk; tÞ

fðtÞ



; ð14Þ

dρ

2π

jAðk; tÞj

: ð15Þ

This mode function satisfies the following equation (in

the conformal time η):

∂

∂η



−

∂

∂η



¼ 0; ð16Þ

and the initial conditions for the modes inside the horizon

have the form of the Bunch-Davies vacuum,

Aðk; tÞ¼

ﬃﬃﬃﬃﬃ

−ikηðtÞ

;kηðtÞ → −∞: ð17Þ

When a certain mode crosses the horizon it changes its

dependence on time from oscillatory to monotonous. We

can assume that at the moment of horizon crossing its

behavior is still approximately described by Eq. (17). Then



∂

∂t

Aðk; tÞ

fðtÞ



≈

2kf









ð18Þ

and the “boundary” term (13) takes the form

2π

2kf











k¼k

4π









: ð19Þ

Finally, the equation which governs the behavior of

electric field energy density is given by

þ 4Hρ

þ 2

¼ −

ðE · jÞþ

4π









ð20Þ

It is possible to derive Eq. (20) in an alternative way by

inspecting the time dependence of the electric power

spectrum. In the deeply subhorizon regime k ≫ aH the

mode function takes the value of the Bunch-Davies initial

conditions (17). Far outside the horizon k ≪ aH, the

solution of Eq. (16) is

¼ C

þ C

aðt

Þf

ðt

;

∂

∂t

: ð21Þ

Matching solutions (17) and (21) at the moment of

horizon crossing t

when k ¼ aH, we find

−ikη

ﬃﬃﬃﬃﬃ

¼ −

−ikη

ﬃﬃﬃﬃﬃ





; ð22Þ

where all the quantities with subscript k must be taken at

the moment of horizon crossing t

. Then, using Eqs. (14)

and (15), we can write the power spectra of generated fields

as follows:

dρ

d ln k

2π

ðkÞj

4π



1 þ







;

ð23Þ

dρ

d ln k

2π



ðkÞþC

ðkÞ

aðt

Þf

ðt



4π



1 −



i þ



aðt

Þf

ðt



ð24Þ

Integrating Eq. (23) over modes outside the horizon at a

given moment of time, we obtain the energy density of the

electric field:

SOBOL, GORBAR, KAMARPOUR, and VILCHINSKII PHYS. REV. D 98, 063534 (2018)

063534-4

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