# Baryon isocurvature constraints on the primordial hypermagnetic fields

Kohei Kamada  Fumio Uchida  and Jun’ichi Yokoyama

###### Abstract

It has been pointed out that hypermagnetic helicity decay at the electroweak
symmetry breaking may have produced the observed baryon asymmetry of the
Universe through the chiral anomaly in the standard model of particle physics.
Although fully helical magnetic field that can adequately produce the observed
baryon asymmetry is not strong enough to explain the origin of the
intergalactic magnetic field inferred by the Fermi satellite, the mixture of
helical and nonhelical primordial magnetic fields may explain both
baryogenesis and the intergalactic magnetic fields simultaneously. We first
show that such a scenario is ruled out by the constraint on the amplitude of
baryon isocurvature perturbations produced by the primordial magnetic fields
to avoid overproduction of deuterium at the big bang nucleosynthesis. Then we
show that any attempt to explain the origin of intergalactic magnetic field by
primordial magnetogenesis before the electroweak symmetry breaking does not
work due to the above constraint irrespective of the helicity and baryogenesis
mechanism.

##  1 Introduction

The excess of baryons over anti-baryons in the present universe, or the baryon
asymmetry of the universe (BAU), is a key ingredient in modern cosmology and
essential for our very existence. It is quantified by the baryon-to-entropy
ratio, precisely determined by the observation of the cosmic microwave
background (CMB) [1] as $\eta_{B}\equiv n_{B}/s=(8.718\pm 0.054)\times
10^{-11}$, which is in good agreement with the one required for the success of
the Big Bang Nucleosynthesis (BBN) [2]. The Standard Model of particle physics
(SM) is hard to explain this tiny but non-zero asymmetry, especially in the
context of inflationary cosmology, and its origin has been regarded as a
fundamental problem in particle physics and cosmology. A dynamical mechanism,
baryogenesis, thus should have taken place in the early Universe after the end
of inflation before the BBN, since inflation dilutes away the preexisting
asymmetry.

To generate non-vanishing baryon asymmetry we must satisfy the Sakharov
criteria [3]: baryon-number violation, C and CP violation, and departure from
thermal equilibrium. Since they are hardly satisfied in the realms of the SM,
new physical degrees of freedom beyond the SM are often introduced, e.g., new
fields to make the electroweak symmetry breaking (EWSB) strongly first order
and to give the source of CP violation for the electroweak baryogenesis [4],
or the right-handed neutrinos for leptogenesis [5]. In this regard, a
baryogenesis scenario from hypermagnetic helicity decay [6, 7, 8, 9, 10, 11]
is remarkable since it does not require any new degrees of freedom beyond the
SM in the mechanism itself. It utilizes the chiral anomaly in the SM [12, 13,
14], which relates the changes in the global baryon (B) and lepton (L) number
($Q_{B}$ and $Q_{L}$) to the changes in the SU(2)L Chern-Simons number
($N_{\mathrm{CS}}^{L}$) and the U(1)Y hypermagnetic helicity
($\mathcal{H}_{Y}$),

$\begin{split}\Delta Q_{B}=\Delta Q_{L}=N_{\rm g}\left(\Delta
N_{\mathrm{CS}}^{L}-\frac{g^{\prime
2}}{16\pi^{2}}\Delta\mathcal{H}_{Y}\right),\end{split}$ (1.1)

where $N_{\mathrm{g}}=3$ is the number of fermion generations and $g^{\prime}$
is the gauge coupling for the U(1)Y hypergauge interaction. Thus baryon
asymmetry can be generated if the hypermagnetic helicity decay occurs. This
idea can be realized in realistic cosmological scenarios such as the axion
inflation [15, 16, 17], the SU(5) Grand Unified Theory baryogenesis [18, 19,
20] through the chiral plasma instability [21, 22, 23], or the Affleck-Dine
mechanism [24, 25], in which the long-range U(1)Y hypermagnetic fields are
generated before the EWSB with helicity (See, e.g., Refs. [26, 27, 28, 29, 30,
31, 32, 33]). The hypermagnetic helicity decay occurs mainly at the EWSB when
the hypermagnetic fields are converted into the U(1)EM magnetic fields. The
Sakharov criteria are satisfied by the SM chiral anomaly (B-violation) and the
existence of the long-range helical hypermagnetic fields (C and CP-violation
as well as the deviation from thermal equilibrium). It has been shown that in
the electroweak crossover suggested by the 125 GeV Higgs [34, 35] baryon
asymmetry can remain until today [11] against the washout by the electroweak
sphalerons [4]. In this scenario, the physics beyond the SM is required for
magnetogenesis but not for baryogenesis.

An advantage of this scenario is that the U(1)Y hypermagnetic fields that are
responsible for baryogenesis can persist until the present time after the
conversion into the U(1)EM magnetic fields at the EWSB as the intergalactic
magnetic fields (IGMFs), which would act as its potential probe. Indeed, the
existence of IGMFs has been suggested by the recent TeV blazar observations
with Fermi-LAT [36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. The lack of secondary
GeV photon in the blazar spectra, expected to be generated during the
propagation in the intergalactic void, can be attributed to the IGMFs so that
the lower bound of their strength and coherence length are given. If further
observation of the IGMFs can pin down their strength and coherence length as
well as their helicity, in principle we can test the scenario, which gives us
insights into the origin of the IGMFs as well as that of BAU.

Previous studies have already shown that maximally helical U(1)Y gauge fields
that can explain the observationally suggested IGMF would suffer from baryon
overproduction [11], unless the real transition from the hypermagnetic fields
to the electromagnetic fields at the electroweak crossover is much more rapid
than the one extrapolated from the result of the lattice calculation [34, 35].
One way to accommodate the baryogenesis and IGMFs indicated by the blazar
observations is to take the U(1)Y hypermagnetic fields to be partially
helical, which might be realized by, e.g., an incomplete cancellation of the
hypermagnetic helicity and the chiral asymmetry carried by the right-handed
electrons generated in axion inflation via the chiral plasma instability
[30].111Another option is to consider the second magnetogenesis after the
EWSB. In this case we do not have direct relationship between the BAU and
IGMFs. In such a case, the baryogenesis is less effective so that stronger
hyoermagnetic fields, possibly up to the ones consistent with the lower bound
of the IGMFs, are needed to explain the present BAU. The purpose of this paper
is to address further this option.

The key ingredient to investigate this situation but not to have been explored
in depth yet is the baryon isocurvature perturbation, which should be
generated at the scale corresponding to the coherence length of the
hypermagnetic fields. It is well known that the baryon isocurvature
perturbations are constrained by the observations of cosmic microwave
background (CMB) on large scales ($k\lesssim 0.1\;\mathrm{Mpc}^{-1}$; $k$ is
the comoving wavenumber) [46], which would give constraints on acausally
generated hypermagnetic fields that never entered the hydrodynamic turbulence
due to sufficiently large coherence length.222The baryon isocurvature
perturbations are degenerate with cold dark matter isocurvature perturbations.
It has been claimed that 21 cm fluctuations can potentially discriminate the
degeneracy [47], but still it is difficult to distinguish tiny isocurvature
fluctuations [48]. The hypermagnetic fields with smaller coherence length,
e.g., the ones that have once entered the hydrodynamic turbulence regime, do
not receive stringent constraints from such cosmological observations.
However, such small-scale baryon isocurvature perturbations may spoil the
successful BBN and their spectrum can be constrained by the studies of
inhomogeneous BBN [49, 50, 51]. In particular, it has recently been pointed
out that the baryon isocurvature perturbation at a scale larger than the
neutron diffusion scale at the BBN epoch, $k\lesssim 4\times
10^{8}\;\mathrm{Mpc}^{-1}$, is constrained by the deuterium overproduction due
to the second-order effect [51]. As a result, one can obtain an upper bound on
their coherence length. We find that since the baryon isocurvature
perturbation is generated even from non-helical hypermagnetic fields [6, 7],
the constraint becomes severer for less helical hypermagnetic fields, in which
case hypermagnetic fields with longer coherence length are needed for the BAU.
Consequently, together with the constraint on their properties from the
magnetohydrodynamics (MHD) [52], we find a lower bound of the helicity
fraction, $\epsilon$, which gives an upper bound of the hypermagnetic field
strength that can generate the present BAU. In terms of the present IGMF
properties, the corresponding upper bound lies below the most stringent lower
bound of the IGMF strength [44], independent of the detail of their
properties. Moreover, since the non-helical part of the hypermagnetic fields
contributes to the baryon isocurvature perturbation, it constrains their
properties even in the case the homogeneous part of the BAU is generated by
another mechanism. It turns out that regardless of their helicity properties,
any hypermagnetic fields with too large strength and coherence length are not
allowed before the EWSB. We conclude that the IGMFs suggested by the
observation of the Fermi-LAT collaboration must have been generated after the
EWSB so as not to undermine the success of the BBN, irrespective of their
helical property as well as the generation mechanism of the present BAU. The
baryogenesis from the hypermagnetic helicity decay can still be responsible
for the present BAU, but we need additional magnetogenesis or an unknown
mechanism of the magnetic field amplification after the EWSB.

The paper is organized as follows. In §2, we briefly review baryogenesis from
hypermagnetic helicity decay and derive the spectrum of baryon isocurvature
perturbation. We provide a general treatment with which we can assess whether
this scenario survives against the baryon isocurvature constraints. In §3, we
present the general formalism to constrain the scenario from baryon
isocurvature perturbation and the way to connect the constraint to the IGMF
observations, by taking into account the evolution of the magnetic fields
after the EWSB. In §4, by adopting simple forms of the magnetic field power
spectrum, we obtain constraints on the magnetic fields quantitatively in the
case they are responsible for the present BAU. In §5, we present the
constraints on the non-helical hypermagnetic fields without specifying
baryogenesis models. Finally §6 is devoted to conclusion and discussion.

##  2 Baryogenesis from hypermangetic helicity decay

We begin by reviewing the baryogenesis scenario from the hypermagnetic
helicity decay. In the previous studies [9, 10, 11] the hypermagnetic fields
are parameterized by their characteristic field strength $B_{\mathrm{p}}$ and
coherence length $\xi_{\mathrm{p}}$, assuming a delta-function-like spectrum.
For our purpose to investigate the baryon isocurvature perturbation, we here
reformulate the consequence of this scenario by taking into account the
spectrum of the hypermagnetic fields, as is also done in Ref. [6, 7].

###  2.1 Generation of the net baryon asymmetry

Let us first give the explanation how the net baryon asymmetry is generated by
the decaying hypermagnetic helicity in a realistic cosmic history following
Ref. [11]. As explained in the introduction, the change in the comoving
hypermagnetic helicity density induces the change in the BAU through the
chiral anomaly of the SM (Eq. (1.1)). Here we consider the case where the
hypermagnetic fields are generated before the EWSB with sufficiently large
coherence length and non-vanishing net helicity. Then the hypermagnetic fields
evolve according to MHD. Basically, the net hypermagnetic helicity is almost
conserved with the help of large electric conductivity [53, 54] until the
EWSB. However, the hypermagnetic helicity decay occurs in two ways in the
cosmic history. The first is that the magnetic field diffusion due to the
large but finite electric conductivity [6, 7], and the other is the
electroweak symmetry breaking where the hypermagnetic helicity is converted
into the U(1)EM magnetic helicity [11]. It turned out that the latter gives
more significant contributions on the BAU, unless the dynamics of the EWSB is
significantly different from the results of the lattice simulations [34, 35]
as well as the one-loop analytic estimate [55]. Note that after the completion
of the EWSB U(1) baryon symmetry is no longer anomalous and baryon asymmetry
is not generated from the U(1)EM magnetic helicity decay. Below, we explain in
more depth how the BAU is generated and how we can evaluate it. We here assume
that the possible cancellation of the chiral asymmetry and hypermagnetic
helicity [30] does not occur and the hypermagnetic helicity is much larger
than the chiral asymmetry in the system before the EWSB.

To evaluate the BAU generated in this mechanism, we need to take into account
the washout effects by the electroweak sphalerons [4], since this mechanism
generates only $B+L$ but not $B-L$ asymmetry. If the EWSB would have completed
earlier than the freezeout of the electroweak sphalerons, the baryon asymmetry
would not remain much (See, however, Ref. [10]). However, lattice simulations
of the EWSB in the light of the 125 GeV Higgs suggest that they occur at
almost the same time so that the electroweak sphaleron washout is incomplete.
Then the evolution of the mean baryon asymmetry can be evaluated by solving a
simplified kinetic equation for the mean baryon-to-entropy ratio,
$\overline{\eta}_{B}$ [11],

$\begin{split}\frac{d\overline{\eta}_{B}}{dx}=(\text{source})-(\text{washout}),\end{split}$
(2.1)

with the dominant contributions around the EWSB ($T\simeq 130-145$333Electron
Yukawa is dominant for the washout at $T\gtrsim 145$ GeV. GeV) being [11]

$\begin{split}(\text{source})=\frac{3}{2}(g^{2}+g^{\prime
2})\frac{d\theta_{\mathrm{w}}}{d\ln x}\sin
2\theta_{\mathrm{w}}\mathcal{S}_{\mathrm{AB}},\quad(\text{washout})=\frac{111}{34}\gamma_{\mathrm{w,sph}}\overline{\eta}_{B},\end{split}$
(2.2)

where

$\begin{split}\mathcal{S}_{\mathrm{AB}}=\frac{H}{8\pi^{2}a^{3}sT}{\overline{\bm{A}_{\mathcal{A}}\cdot\bm{B}_{\mathcal{A}}}}.\end{split}$
(2.3)

with the overline representing the volume average. Here $x=M_{0}/T$ with
$M_{0}\equiv M_{\mathrm{pl}}/\sqrt{\pi^{2}g_{*}/90}$ ($M_{\mathrm{pl}}\simeq
2.43\times 10^{18}$ GeV is the reduced Planck mass and $g_{*}$ is the
effective number of relativistic degrees of freedom during the process.) and
$T$ being the temperature of the Universe. $g$ is the gauge coupling for the
SU(2)L electroweak gauge interaction and $\theta_{\mathrm{w}}$ is the
temperature-dependent effective weak mixing angle, which gradually changes
from 0 to $\theta_{\mathrm{w}0}\equiv\tan^{-1}(g^{\prime}/g)$ at the
electroweak crossover. $\gamma_{\mathrm{w,sph}}\simeq\exp[-(146.6\pm
1.0)+(0.83\pm 0.01)T/\mathrm{GeV}]$ [34] is the transport coefficient for the
electroweak sphalerons. $H\equiv{\dot{a}}/a$ is the Hubble parameter with $a$
being the scale factor and $s=(2\pi^{2}/45)g_{*s}T^{3}$ is the entropy density
($g_{*s}$ is the relativistic degrees of freedom for entropy). Here ${\cal A}$
stands for the massless U(1) gauge field during the electroweak crossover, and
${\bm{A}}_{\cal A}$ and ${\bm{B}}_{\cal A}$ are the vector potential and
comoving magnetic fields for ${\cal A}$, respectively. They are characterized
by the effective weak mixing angle $\theta_{\mathrm{w}}$ as
${\bm{Y}}=\cos\theta_{\mathrm{w}}{\bm{A}}_{\cal A}$ and
${\bm{W}}^{3}=\sin\theta_{\mathrm{w}}{\bm{A}}_{\cal A}$, with ${\bm{Y}}$ and
${\bm{W}}^{3}$ being the vector potentials for the U(1)Y hyper gauge
interaction and the third component of the SU(2)L electroweak gauge
interaction. Here the kinetic equation is derived by taking into account the
equilibrium conditions of the rapid spectator processes such as the spin-
flipping process of chiral fermions through the Yukawa interactions other than
the one for the electrons and the strong sphalerons. Note that during the
electroweak crossover, the long-range “magnetic” field turns from that of
U(1)Y hypergauge interaction to that of the U(1)EW electromagnetic interaction
through the massless U(1) gauge field ${\cal A}$. Thus the right hand side of
the anomaly equation,
$(g^{2}/16\pi^{2})\mathrm{Tr}\left[W_{\mu\nu}\tilde{W}^{\mu\nu}\right]-(g^{\prime
2}/32\pi^{2})Y_{\mu\nu}\tilde{Y}^{\mu\nu}$, includes the terms, e.g.,
${d(\sin\theta_{\mathrm{w}}{\bm{A}}_{\cal A}})/d\ln x\cdot{\bm{B}}_{\cal
A}\sim(d\theta_{\mathrm{w}}/d\ln x){\bm{A}}_{\cal A}\cdot{\bm{B}}_{\cal
A}$.444We carry out all the computation with the Coulomb gauge
${\bm{\nabla}}\cdot{\bm{A}}_{\cal A}=0$, but the final results are gauge
independent [11]. The gradual changes of the weak mixing angle during the
electroweak crossover give significant contributions to the source term.
Around the electroweak symmetry breaking, the electroweak sphaleron becomes
less and less effective and turns to the rate-determining reaction.

When the transport coefficient of the electroweak sphalerons is larger than
the Hubble rate, the system enters the equilibrium state with
$d\overline{\eta}_{B}/dx=0$ for Eq. (2.1), which gives the mean baryon
asymmetry as

$\overline{\eta}_{B}\simeq\frac{17}{37}\frac{g^{2}+g^{\prime
2}}{\gamma_{\mathrm{w,sph}}}\frac{d\theta_{\mathrm{w}}}{d\ln x}\sin
2\theta_{\mathrm{w}}\mathcal{S}_{\mathrm{AB}},$ (2.4)

before the electroweak sphaleron freezeout. As the temperature of the Universe
decreases, the electroweak sphaleron freezes out around
$T=T_{\mathrm{fo}}\simeq 135$ GeV. (Hereafter the subscript fo represents that
the quantity is evaluated at the sphaleron freeze out, $T=T_{\mathrm{fo}}$.)
Just after the sphaleron freezeout the electroweak symmetry breaking completes
and there will be no longer significant induction of baryon asymmetry.
Numerically it is found that the resultant baryon asymmetry is evaluated the
equilibrium solution (2.4) at $T_{\mathrm{fo}}\simeq 135$ GeV [11]. Since at
the electroweak sphaleron freezeout, the transport coefficient of the
electroweak sphaleron is related to the Hubble parameter as
$\gamma_{\mathrm{w,sph}}(T_{\mathrm{fo}})\simeq
H(T_{\mathrm{fo}})/T_{\mathrm{fo}}$, the resultant baryon asymmetry is
evaluated in a simplified form as

$\begin{split}\overline{\eta}_{B}=\left.\frac{17}{296\pi^{2}}(g^{2}+g^{\prime
2})\frac{d\theta_{\mathrm{w}}}{d\ln x}\sin
2\theta_{\mathrm{w}}\frac{\overline{\bm{A}_{\mathcal{A}}\cdot\bm{B}_{\mathcal{A}}}}{a^{3}s}\right|_{T=T_{\mathrm{fo}}}.\end{split}$
(2.5)

Therefore, if we have non-vanishing
$\overline{\bm{A}_{\mathcal{A}}\cdot\bm{B}_{\mathcal{A}}}$ on average, we will
obtain non-vanishing net baryon asymmetry. For a peaky spectrum of the
hypermagnetic fields with the characteristic comoving field strength
$B_{\mathrm{c}}$ (or the physical strength
$B_{\mathrm{p}}=a^{-2}B_{\mathrm{c}}$) and the comoving coherence length
$\xi_{\mathrm{c}}$ (or the physical coherence length
$\xi_{\mathrm{p}}=a\xi_{\mathrm{c}}$) with the helicity fraction $\epsilon$,
we can write
$\overline{\bm{A}_{\mathcal{A}}\cdot\bm{B}_{\mathcal{A}}}\simeq\epsilon\xi_{\mathrm{c}}B_{\mathrm{c}}^{2}=a^{3}\epsilon\xi_{\mathrm{p}}B_{\mathrm{p}}^{2}$
and evaluate the net baryon asymmetry as

$\displaystyle\overline{\eta}_{B}$
$\displaystyle=\left.\frac{17}{296\pi^{2}}(g^{2}+g^{\prime
2})\frac{d\theta_{\mathrm{w}}}{d\ln x}\sin
2\theta_{\mathrm{w}}\frac{\epsilon\xi_{\mathrm{p}}B_{\mathrm{p}}^{2}}{s}\right|_{T=T_{\mathrm{fo}}}$
$\displaystyle\sim 10^{-10}\epsilon\left.\sin
2\theta_{\mathrm{w}}\frac{d\theta_{\mathrm{w}}}{d\ln
x}\left(\frac{\xi_{\mathrm{p}}}{10^{6}\mathrm{GeV}^{-1}}\right)\left(\frac{B_{\mathrm{p}}}{10^{-3}\mathrm{GeV}^{2}}\right)^{2}\right|_{T=T_{\mathrm{fo}}},$
(2.6)

which will be more formally derived in the next subsection in terms of the
magnetic field power spectrum.

The value of $\sin 2\theta_{\mathrm{w}}d\theta_{\mathrm{w}}/d\ln x$ at
$T=T_{\mathrm{fo}}$ has relatively large uncertainties in the analytic
expressions [55] and results of the lattice calculations [35]. On the one
hand, the time dependence of the weak mixing angle is evaluated analytically
at the one-loop level as [55]

$\begin{split}\cos^{2}\theta_{\rm w}=\cos^{2}\theta_{{\rm
w}0}\left(1+\frac{11}{12}\frac{g_{3}^{2}\sin^{2}\theta_{{\rm w}0}}{\pi
m_{W}(T)}\right),\end{split}$ (2.7)

where $\theta_{{\rm w}0}$ is the weak mixing angle at zero temperature,
$g_{3}$ is the three dimensional SU($2$) gauge coupling, and $m_{W}(T)$ is the
perturbative $W$ boson mass. By substituting

$\begin{split}g_{3}^{2}\simeq g^{2}T,\quad
m_{W}(T)=\frac{g\phi(T)}{2},\end{split}$ (2.8)

with adopting the fitting function to the numerical lattice result for the
Higgs condensate, $\phi(T)$, around the EWSB, $T\lesssim 162$ GeV [10],

$\begin{split}\phi(T)\simeq 0.23T\sqrt{162-\frac{T}{1\;{\rm
GeV}}},\end{split}$ (2.9)

we obtain

$\begin{split}\left.\frac{d\theta_{\mathrm{w}}}{d\ln x}\sin
2\theta_{\mathrm{w}}\right|_{T=T_{\mathrm{fo}}}=\left.-T\frac{d\theta_{\mathrm{w}}}{dT}\sin
2\theta_{\mathrm{w}}\right|_{T=T_{\mathrm{fo}}}\simeq 0.14,\end{split}$ (2.10)

where we have used $g=0.65$ and $\cos^{2}\theta_{{\rm w}0}=0.77.$

On the other hand, the three dimensional lattice calculation of the weak
mixing angle has been calculated in the light of the $125$ GeV Higgs, for
$140$ GeV $<T<170$ GeV in Ref. [35]. In principle the lattice calculation is
an all-orders calculation that includes even the non-perturbative effects.
However, the result has relatively large errors and agrees with the analytic
calculation only marginally. Since one cannot tell which estimate is more
reliable and the analytic estimate is applicable only limited temperature
ranges, which is not suitable for solving the kinetic equation numerically,
Ref. [11] has adopted a phenomenological fitting formula with a smoothed step
function for $\theta_{\rm w}$ as

$\begin{split}\cos^{2}\theta_{\mathrm{w}}(T)=\cos^{2}\theta_{\mathrm{w}0}+\frac{1-\cos^{2}\theta_{\mathrm{w}0}}{2}\left(1+\tanh\frac{T-T_{\mathrm{step}}}{\Delta
T}\right).\end{split}$ (2.11)

The values of the parameters $T_{\mathrm{step}}$ and $\Delta T$, chosen in
Ref. [11], which give relatively good fit for the lattice results are shown in
Table 1.555The parameterization A in Ref. [11] is omitted since it does not
give a good fit at $T<160$ GeV and Eq. (2.1) is not applicable for the
estimate of the resultant baryon asymmetry [11].

parameterizations | B | C | D | E | one-loop analytic
---|---|---|---|---|---
$T_{\mathrm{step}}/\mathrm{GeV}$ | $160$ | $160$ | $155$ | $155$ | -
$\Delta T/\mathrm{GeV}$ | $5$ | $10$ | $10$ | $20$ | -
$\left.-T\dfrac{d\theta_{\mathrm{w}}}{dT}\sin 2\theta_{\mathrm{w}}\right|_{T=T_{\mathrm{fo}}}$ | $6\times 10^{-4}$ | $4\times 10^{-2}$ | $0.1$ | $0.3$ | $0.14$
Table 1: Parameters that characterize the EWSB are shown. Parameterizations
B-E are the ones with Eq. (2.11) chosen in Ref. [11], and the last column is
the analytic estimate (Eq. (2.10)) with the the formula at the one-loop level
in Ref. [55]. We use the latter as the fiducial value of the temperature
dependence of the weak mixing angle for the estimate of the resultant baryon
asymmetry.

Once more, we do not yet have a definite answer for the estimate of the
precise time dependence of the weak mixing angle and we admit typically
$\mathcal{O}(10^{3})$ uncertainty in the estimate of $d\theta_{\rm w}/dT.$

For the practical purpose, however, we adopt Eq. (2.10) as the fiducial value
for the estimate of the baryon asymmetry, since the perturbative analytic
estimate becomes more accurate at lower temperature $T<140$ GeV, whereas the
lattice calculation gives larger errors for lower temperature666We are
grateful to Mikko Laine for pointing it out.. Combining Eqs. (2.1) and (2.10)
(and even using Tab. 1), we can see that it is possible to explain the present
BAU with appropriate properties of primordial hypermagnetic fields.

###  2.2 Magnetic field spectrum

To investigate the spatial distribution of the baryon asymmetry, it is clear
from the right hand side of Eq. (2.5) that we need to know the spatial
distribution of the magnetic fields. In this subsection, we provide the way to
parameterize the spatial distributions of magnetic fields, namely, their power
spectrum, by assuming that they act as stochastic fields, which is often
realized in cosmic history from some magnetogenesis mechanisms such as those
from axion inflation [15, 16, 17], standard inflation with a dilatonic
coupling in the kinetic function of gauge fields [56, 57, 58], or the strong
first order phase transitions [59, 60, 61]. In the next subsection, we provide
the expressions of the baryon isocurvature perturbations in terms of the
magnetic field power spectrum provided in this subsection. Here we consider
that the magnetic field power spectrum smoothly converts from the one for the
U(1)Y hypergauge field and to the one for the U(1)EM electromagnetic field
without significant decay or change of the scales since the change of the weak
mixing angle is a relatively slow process during the electroweak crossover
[35]. In what follows we do not distinguish the hypergauge field and the
electromagnetic field, as well as the intermediate massless gauge field ${\cal
A}$ and simply denote them as $A_{\mu}$ as long as the power spectrum is
concerned.

Let us now study the distribution of the gauge field $A_{i}$ in the wavenumber
space777We adopt $f(k)=\int dxe^{-ikx}f(x)$ and $f(x)=\int
dk/(2\pi)e^{ikx}f(k)$ as the Fourier transformation. by adopting the Coulomb
gauge (See the Appendix A for more explanation on the gauge fixing). The power
spectrum of the vector potential is then defined as follows,

$\begin{split}\langle
A^{*}_{i}(\bm{k},t)A_{j}(\bm{k}^{\prime},t)\rangle=(2\pi)^{3}\delta^{3}(\bm{k}-\bm{k}^{\prime})\mathcal{F}^{A}_{ij}(\bm{k},t),\end{split}$
(2.12)

with

$\begin{split}\mathcal{F}^{A}_{ij}(\bm{k},t)=P_{ij}(\hat{\bm{k}})S(k,t)+i\epsilon_{ijm}\hat{k}_{m}A(k,t),\end{split}$
(2.13)

where $\epsilon_{ijm}$ is the 3-dimensional Levi-Civita tensor with
$\epsilon_{123}=1$, and

$P_{ij}(\hat{\bm{k}})\equiv\delta_{ij}-\hat{k}_{i}\hat{k}_{j},\quad{\hat{k}}_{i}\equiv\frac{k_{i}}{k},\quad
k\equiv|{\bm{k}}|.$ (2.14)

$S(k,t)$ and $A(k,t)$ are the symmetric and anti-symmetric parts of the power
spectrum of the magnetic fields, respectively. We keep them arbitrary for the
moment to make our discussion general. Hereafter we omit the argument $t$ but
implicitly assume the time dependence of these functions, which should obey
the MHD. Assuming the Gaussianity of the magnetic field stochastic
distribution, we can characterize all the properties of the magnetic fields
solely by these two functions, $S(k)$ and $A(k)$, since the higher-order
cumulants vanish for Gaussian-distributed stochastic variables. The comoving
energy density, helicity density, and coherence length of the magnetic fields
are now formally defined as

$\displaystyle\mathcal{E}_{\mathrm{c}}$ $\displaystyle=\frac{1}{2V}\int
d^{3}x\langle{\bm{B}}({\bm{x}})^{2}\rangle$
$\displaystyle=\frac{1}{2}\int\frac{d^{3}k}{(2\pi)^{3}}k^{2}\sum_{i}\mathcal{F}^{A}_{ii}(\bm{k})=\frac{1}{2\pi^{2}}\int
dkk^{4}S(k),$ (2.15) $\displaystyle h_{\mathrm{c}}$
$\displaystyle=\frac{1}{V}\int
d^{3}x\langle{\bm{A}}({\bm{x}})\cdot{\bm{B}}({\bm{x}})\rangle$
$\displaystyle=-i\epsilon_{ijm}\int\frac{d^{3}k}{(2\pi)^{3}}k_{m}\mathcal{F}^{A}_{ij}(\bm{k})=\frac{1}{\pi^{2}}\int
dkk^{3}A(k),$ (2.16) $\displaystyle\xi_{\mathrm{c}}$ $\displaystyle=\frac{\int
dkk^{3}S(k)}{\int dkk^{4}S(k)},$ (2.17)

respectively, where $V=\int d^{3}x$ is the volume factor. The helicity
fraction $\epsilon$, which is in principle time-dependent, is defined as

$\epsilon\equiv\frac{h_{\mathrm{c}}}{2\xi_{\mathrm{c}}{\cal E}_{\mathrm{c}}}.$
(2.18)

Now from Eq. (2.5) the net mean baryon asymmetry is given in terms of the
power spectrum as

$\displaystyle{\overline{\eta}_{B}}$
$\displaystyle=\left.\frac{17}{296\pi^{2}}\frac{g^{2}+g^{\prime
2}}{a^{3}s}\frac{d\theta_{\mathrm{w}}}{d\ln x}\sin
2\theta_{\mathrm{w}}\frac{1}{\pi^{2}}\int
dkk^{3}A(k)\right|_{T=T_{\mathrm{fo}}}$ $\displaystyle\equiv\left.{\cal
C}h_{\mathrm{c}}\right|_{T=T_{\mathrm{fo}}}=\left.2{\cal
C}\epsilon\xi_{\mathrm{c}}{\cal E}_{\mathrm{c}}\right|_{T=T_{\mathrm{fo}}},$
(2.19)

where we have defined

$\displaystyle{\cal C}$
$\displaystyle\equiv\left.\frac{17}{296\pi^{2}}\frac{g^{2}+g^{\prime
2}}{a^{3}s}\frac{d\theta_{\mathrm{w}}}{d\ln x}\sin
2\theta_{\mathrm{w}}\right|_{T=T_{\mathrm{fo}}}$ $\displaystyle\simeq 2\times
10^{34}\mathrm{GeV}^{-3}\left(\dfrac{\left.\frac{d\theta_{\mathrm{w}}}{d\ln
x}\sin 2\theta_{\mathrm{w}}\right|_{T=T_{\mathrm{fo}}}}{0.14}\right)\simeq
1\times 10^{33}\;\mathrm{Mpc}^{-1}{\rm
G}^{-2}\left(\dfrac{\left.\frac{d\theta_{\mathrm{w}}}{d\ln x}\sin
2\theta_{\mathrm{w}}\right|_{T=T_{\mathrm{fo}}}}{0.14}\right).$ (2.20)

Here we have used $g^{\prime}=0.35,g=0.65,g_{*s}^{\mathrm{fo}}=106.75$, and
$a_{\mathrm{fo}}=5.8\times 10^{-16}$ (for $T_{\mathrm{fo}}=135$ GeV). We have
adopted the natural Heaviside-Lorentz units,888In this unit system,
$\hbar=c=\varepsilon_{0}=1$, while the natural Gaussian CGS units set
$\hbar=c=4\pi\varepsilon_{0}=1$. whence $1\,\textrm{G}=1.95\times
10^{-20}\,\textrm{GeV}^{2}$ and $1\,\textrm{Mpc}=1.56\times
10^{38}\,\textrm{GeV}^{-1}$.

In the case when $S(k)$ is proportional to $A(k)$, the helicity fraction is
found to be the proportional constant, $A(k)=\epsilon S(k)$. Note that the
realizability condition [62] imposes $|\epsilon|\leq 1$. In particular, by
requiring $\overline{\eta}_{B}\simeq 10^{-10}$, the following relation
applies.

$\begin{split}\left(\dfrac{\left.\frac{d\theta_{\mathrm{w}}}{d\ln x}\sin
2\theta_{\mathrm{w}}\right|_{T=T_{\mathrm{fo}}}}{0.14}\right)^{\frac{1}{2}}\left(\frac{\xi_{\rm
p,fo}}{10^{6}\;{\rm GeV}^{-1}}\right)^{\frac{1}{2}}\left(\frac{B_{\rm
p,fo}}{10^{-3}\;{\rm GeV}^{2}}\right)\simeq
5\times\epsilon_{\mathrm{fo}}^{-\frac{1}{2}}.\end{split}$ (2.21)

###  2.3 Baryon isocurvature perturbation

Now we are ready to express the baryon isocurvature perturbations in the
wavenumber space, in terms of the magnetic field power spectrum. An expression
in the position space is often useful as well. We explicitly derive it in
Appendix B.

Equation (2.5) implies that the spatial distribution of the BAU at its freeze
out, $\eta_{B,{\rm fo}}$, is proportional to ${\bm{A}}\cdot{\bm{B}}|_{T=T_{\rm
fo}}$,

$\begin{split}\eta_{B,{\rm
fo}}({\bm{x}})=\mathcal{C}\left.{\bm{A}}({\bm{x}})\cdot{\bm{B}}({\bm{x}})\right|_{T=T_{\rm
fo}}=\mathcal{C}\left.({\bm{A}}\cdot{\bm{\nabla}}\times{\bm{A}})({\bm{x}})\right|_{T=T_{\rm
fo}},\end{split}$ (2.22)

whose Fourier mode is expressed as

$\begin{split}\eta_{B,{\rm fo}}({\bm{k}})\equiv\int
d^{3}re^{-i{\bm{k}}\cdot{\bm{r}}}\eta_{B,{\rm
fo}}({\bm{r}})=i\epsilon_{ijk}\mathcal{C}\int\frac{d^{3}p}{(2\pi)^{3}}A_{i}^{*}({\bm{p}}-{\bm{k}})p_{j}A_{k}({\bm{p}}).\end{split}$
(2.23)

To investigate the distribution of the baryon-number fluctuation,
$\delta\eta_{B}({\bm{x}})\equiv\eta_{B}({\bm{x}})-{\overline{\eta}}_{B}$, we
use two point function of the relative fluctuation,
$S_{B}(\bm{x})\equiv\delta\eta_{B}({\bm{x}})/{\overline{\eta}}_{B}$, expressed
as

$\begin{split}\mathcal{G}({\bm{r}})\equiv\langle S_{B,{\rm
fo}}(\bm{x})S_{B,{\rm fo}}(\bm{x}+\bm{r})\rangle=\frac{\langle\eta_{B,{\rm
fo}}(\bm{x})\eta_{B,{\rm
fo}}(\bm{x}+\bm{r})\rangle}{{\overline{\eta}}_{B}^{2}}-1.\end{split}$ (2.24)

Then its Fourier transform or the power spectrum is given by

$\displaystyle\mathcal{G}({\bm{k}})$
$\displaystyle=\frac{1}{V}\frac{\langle\left|\eta_{B,{\rm
fo}}({\bm{k}})\right|^{2}\rangle}{{\overline{\eta}}_{B}^{2}}-(2\pi)^{3}\delta^{3}({\bm{k}})$
$\displaystyle=\epsilon_{ijk}\epsilon_{lmn}\frac{\mathcal{C}^{2}}{V{\overline{\eta}}_{B}^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}}p_{j}p^{\prime}_{m}J_{ikln}({\bm{k}}-{\bm{p}},{\bm{p}},{\bm{k}}-{\bm{p}}^{\prime},{\bm{p}}^{\prime})-(2\pi)^{3}\delta^{3}({\bm{k}}),$
(2.25)

where

$\displaystyle J_{ikln}({\bm{q}}_{1},{\bm{q}}_{2},{\bm{q}}_{3},{\bm{q}}_{4})$
$\displaystyle\equiv\langle
A_{i}^{*}({\bm{q}}_{1})A^{*}_{k}({\bm{q}}_{2})A_{l}({\bm{q}}_{3})A_{n}({\bm{q}}_{4})\rangle$
$\displaystyle=\langle
A^{*}_{i}({\bm{q}}_{1})A^{*}_{k}({\bm{q}}_{2})\rangle\langle
A_{l}({\bm{q}}_{3})A_{n}({\bm{q}}_{4})\rangle+\langle
A^{*}_{i}({\bm{q}}_{1})A_{l}({\bm{q}}_{3})\rangle\langle
A^{*}_{k}({\bm{q}}_{2})A_{n}({\bm{q}}_{4})\rangle$ $\displaystyle\hskip
160.18901pt+\langle A^{*}_{i}({\bm{q}}_{1})A_{n}({\bm{q}}_{4})\rangle\langle
A^{*}_{k}({\bm{q}}_{2})A_{l}({\bm{q}}_{3})\rangle$
$\displaystyle=(2\pi)^{6}\left[\delta^{3}({\bm{q}}_{1}+{\bm{q}}_{2})\delta^{3}({\bm{q}}_{3}+{\bm{q}}_{4})\mathcal{F}^{A}_{ik}({\bm{q}}_{1})\mathcal{F}^{A}_{ln}(-{\bm{q}}_{3})\right.$
$\displaystyle\hskip
73.97716pt+\delta^{3}({\bm{q}}_{1}-{\bm{q}}_{3})\delta^{3}({\bm{q}}_{2}-{\bm{q}}_{4})\mathcal{F}^{A}_{il}({\bm{q}}_{1})\mathcal{F}^{A}_{kn}({\bm{q}}_{2})$
$\displaystyle\hskip
73.97716pt+\left.\delta^{3}({\bm{q}}_{1}-{\bm{q}}_{4})\delta^{3}({\bm{q}}_{2}-{\bm{q}}_{3})\mathcal{F}^{A}_{in}({\bm{q}}_{1})\mathcal{F}^{A}_{kl}({\bm{q}}_{2})\right].$
(2.26)

Here we have used the reality condition of the gauge fields,
$A_{i}^{*}({\bm{k}})=A_{i}(-{\bm{k}})$. In the second equality, we have
decomposed the four-point function of the vector potential into pairs of two-
point functions, assuming the Gaussian distribution. The contribution of each
term to Eq. (2.3) is evaluated as follows. The first term in the last line of
Eq. (2.3) is calculated as

$\displaystyle\epsilon_{ijk}\epsilon_{lmn}\frac{(2\pi)^{6}\mathcal{C}^{2}}{V{\overline{\eta}}_{B}^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}}p_{j}p^{\prime}_{m}\left[\delta^{3}({\bm{k}})\right]^{2}\mathcal{F}^{A}_{ik}({\bm{k}}-{\bm{p}})\mathcal{F}^{A}_{ln}({\bm{p}}^{\prime}-{\bm{k}}$
$\displaystyle)$
$\displaystyle\quad=\frac{(2\pi)^{3}\mathcal{C}^{2}}{{\overline{\eta}}_{B}^{2}}\left[\frac{1}{\pi^{2}}\int
dpp^{3}A(p)\right]^{2}\delta^{3}({\bm{k}})=(2\pi)^{3}\delta^{3}({\bm{k}}$
$\displaystyle).$ (2.27)

This contribution is nothing but the homogeneous component of the baryon
asymmetry and cancels with the last term in Eq. (2.3). It is generated only
from the antisymmetric part of the power spectrum. On the other hand, the
remaining contributions, which generate the inhomogeneities of the baryon
asymmetry, come from both symmetric and antisymmetric parts. This can be seen
by explicitly calculating the second term together with the third term as

$\displaystyle\epsilon_{ijk}\epsilon_{lmn}$
$\displaystyle\frac{(2\pi)^{6}\mathcal{C}^{2}}{V{\overline{\eta}}_{B}^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}}p_{j}p^{\prime}_{m}\left\\{\left[\delta^{3}({\bm{p}}-{\bm{p}}^{\prime})\right]^{2}\mathcal{F}^{A}_{il}({\bm{k}}-{\bm{p}})\mathcal{F}^{A}_{kn}({\bm{p}})\right.$
$\displaystyle\hskip
142.26378pt+\left.\left[\delta^{3}({\bm{k}}-{\bm{p}}-{\bm{p}}^{\prime})\right]^{2}\mathcal{F}^{A}_{in}({\bm{k}}-{\bm{p}})\mathcal{F}^{A}_{kl}({\bm{p}})\right\\}$
$\displaystyle=$
$\displaystyle\frac{\mathcal{C}^{2}}{{\overline{\eta}}_{B}^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\left[p^{2}S(|{\bm{k}}-{\bm{p}}|)S(p)+|{\bm{k}}-{\bm{p}}|pA(|{\bm{k}}-{\bm{p}}|)A(p)\right]$
$\displaystyle\hskip
142.26378pt\times\left[1-\frac{2({\bm{k}}-{\bm{p}})\cdot{\bm{p}}}{p^{2}}+\frac{\left(({\bm{k}}-{\bm{p}})\cdot{\bm{p}}\right)^{2}}{|{\bm{k}}-{\bm{p}}|^{2}p^{2}}\right].$
(2.28)

Putting Eqs. (2.3) and (2.3) into Eq. (2.3), we obtain

$\displaystyle\mathcal{G}({\bm{q}})$
$\displaystyle=\frac{\mathcal{C}^{2}}{{\overline{\eta}}_{B}^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\left[p^{2}S(|{\bm{k}}-{\bm{p}}|)S(p)+|{\bm{k}}-{\bm{p}}|pA(|{\bm{k}}-{\bm{p}}|)A(p)\right]$
$\displaystyle\hskip
142.26378pt\times\left[1-\frac{2({\bm{k}}-{\bm{p}})\cdot{\bm{p}}}{p^{2}}+\frac{\left(({\bm{k}}-{\bm{p}})\cdot{\bm{p}}\right)^{2}}{|{\bm{k}}-{\bm{p}}|^{2}p^{2}}\right].$
(2.29)

The essential point, which has already been noticed in Ref. [7], is that the
non-helical part $S(k)$ contributes to the two-point correlation function of
the magnetic helicity. This is because $A(k)=0$ does not mean that the
helicity, ${\bm{A}}\cdot{\bm{B}}$, vanishes at every position for
$S(k)\not=0$. Thus the two-point correlation function of the magnetic helicity
yields a non-vanishing value even for $A(k)=0$, which plays crucial role in
constraining the primordial hypermagnetic fields.

##  3 Observational constraints

The baryon isocurvature perturbation, which we explored in the previous
section, imposes constraints on the baryogenesis scenario from the
hypermagnetic helicity decay, in the parameter space where the homogeneous
part of the BAU can be explained. In this section, we examine these
constraints in terms of the magnetic field properties. We also briefly
summarize the implication of the observation of the intergalactic magnetic
fields on the scenario, adopting the magnetic field evolution suggested by the
MHD.

###  3.1 Constraints from inhomogeneous Big Bang Nulceosynthesis

Let us first examine the constraint from the baryon isocurvature
perturbations. It is well-known that they are strongly constrained by the CMB
[46, 63, 64, 65, 66].999In addition, future 21 cm line observations are
expected to yield constraints on the baryon isocurvature perturbations on
smaller scales [47, 48, 67]. However, it is sensitive only to the comoving
scales larger than 10 Mpc. In the scenario of the baryogenesis from
hypermagnetic helicity decay, baryon isocurvature perturbations are produced
on much smaller scales, since they correspond to the magnetic field coherence
length as we have seen in the previous section, which depends on the
magnetogenesis mechanism. Thus the constraints from the CMB observations are
generally not applicable.

Fortunately, baryon isocurvature perturbation on much smaller scales can be
constrained by the inhomogeneous BBN [51]. Relatively small-scale baryon
number fluctuations, larger than the comoving neutron diffusion scale at the
BBN, $k_{\mathrm{d}}^{-1}\simeq 0.0025\;\mathrm{pc}$, should not be very
large; otherwise it changes the predictions of the primordial abundance of
light elements and spoils the success of the BBN. Thus we can rule out too
large baryon isocurvature perturbations at a smaller scale than the scales
constrained by the CMB observations. In contrast, a large inhomogeneity on
scales smaller than $k_{\mathrm{d}}^{-1}$ at the BBN can be smeared out by the
neutron diffusion, and the constraints get weaker.

In our scenario, the typical scale of the baryon isocurvature perturbation is
implemented in the magnetic field coherence length, characterized by
$\xi_{\mathrm{c}}$, as we have seen in the previous section. Since at the
scale $\sim\xi_{\mathrm{c}}$ the amplitude of the baryon isocurvature
perturbation peaks to reach ${\cal O}(1)$ at the EWSB, scenarios with large
coherence length, $\xi_{\mathrm{c}}\gg k_{\mathrm{d}}^{-1}$, would create
antibaryon domains in the observable Universe at the BBN and are not allowed.
Such a large coherence length is realized for the hypermagnetic fields
produced in inflationary magnetogenesis or hypermagnetic fields in the MHD
turbulence with relatively large field strength. On the contrary, for smaller
coherence length, $\xi_{\mathrm{c}}\lesssim k_{\mathrm{d}}^{-1}$, baryon
isocurvature perturbation at the BBN leads to the inhomogeneous BBN (IBBN),
explored in Refs. [68, 69, 70, 71, 72]. In this case, neutrons diffuse and
damp large baryon isocurvature perturbations at the scales smaller than the
neutron diffusion scale, and there remains tiny but non-vanishing baryon
isocurvature perturbations at the scales larger than the neutron diffusion
scale, from which one can constrain the magnetic field distributions. By
treating the baryon inhomogeneities at the BBN that survives the smearing
effect of the neutron diffusion as a linear perturbation, one can estimate the
deuterium abundance from the second order effect of perturbations to obtain a
constraint on the baryon isocurvature perturbations to avoid the deutrium
overproduction. The condition for the volume average of the baryon
isocurvature perturbation is found to be [51]

$\begin{split}\overline{S^{2}_{B,\mathrm{BBN}}({\bm{x}})}<0.016\quad(2\sigma),\end{split}$
(3.1)

where $S_{B,\mathrm{BBN}}({\bm{x}})$ is the smoothed baryon isocurvature
perturbation at the BBN.

To take into account the smearing effect of the neutron diffusion, we
introduce a Gaussian window function so that the baryon perturbation
$\delta\eta_{B}$ at the BBN is evaluated as

$\begin{split}\delta\eta_{B,\mathrm{BBN}}(\bm{r})=\int
d^{3}xW_{D}(\bm{x}-\bm{r})\delta\eta_{B,{\rm fo}}(\bm{x}),\quad
W_{D}(\bm{r})\equiv\left(\frac{D}{\pi}\right)^{\frac{3}{2}}e^{-D\bm{r}^{2}},\quad
D\equiv\frac{3}{2}k^{2}_{\rm d}.\end{split}$ (3.2)

The choice of the window function is motivated by the fact that the neutron
diffusion equation is approximated by a heat equation [49] whose (three-
dimensional) heat kernel is the Gaussian function. Thus we employ
$\eta_{B,{\rm fo}}(\bm{r})$ as the initial condition and just convolute it
with the heat kernel $W(\bm{r})$ to obtain $\eta_{B,\mathrm{BBN}}(\bm{r})$.
The heat kernel is nothing but the window function, seen in the wavenumber
space and is normalized as $\int d^{3}xW_{D}(\bm{x})=1$, or equivalently
$W_{D}(\bm{k}=0)=1$ in the wavenumber space. We set the width of the Gaussian
function so that the root mean square radius $\left(\int
d^{3}xW_{D}(\bm{x})x^{2}\right)^{1/2}=\sqrt{3/(2D)}$ coincides with
$k_{\mathrm{d}}^{-1}$, obeying the definition of the word “diffusion length”
in Ref. [49].

From Eq. (3.2), the volume average of the smoothed baryon isocurvature
perturbation is evaluated as

$\displaystyle\overline{S^{2}_{B,\mathrm{BBN}}}$
$\displaystyle=\frac{\langle\delta\eta^{2}_{B,{\rm
BBN}}({\bm{x}})\rangle}{\overline{\eta}_{B}^{2}}=\int
d^{3}xW_{D/2}({\bm{x}})\mathcal{G}({\bm{x}})$ (3.3)
$\displaystyle=\int\frac{d^{3}k}{(2\pi)^{3}}e^{-\frac{k^{2}}{2D}}\mathcal{G}({\bm{k}})\left(<0.016\right).$
(3.4)

By inserting Eq. (2.3) into the last line, we obtain a formula that directly
relates the power spectrum of the magnetic field to the baryon isocurvature
perturbations at the BBN.

$\displaystyle\overline{S^{2}_{B,\mathrm{BBN}}}=\int\frac{d^{3}k}{(2\pi)^{3}}e^{-\frac{k^{2}}{2D}}\mathcal{G}({\bm{k}})$
$\displaystyle\,=\frac{\mathcal{C}^{2}}{{\overline{\eta}}_{B}^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\int\frac{d^{3}p}{(2\pi)^{3}}e^{-\frac{k^{2}}{2D}}\left[p^{2}S(|{\bm{k}}-{\bm{p}}|)S(p)+|{\bm{k}}-{\bm{p}}|pA(|{\bm{k}}-{\bm{p}}|)A(p)\right]$
$\displaystyle\hskip
142.26378pt\times\left[1-\frac{2({\bm{k}}-{\bm{p}})\cdot{\bm{p}}}{p^{2}}+\frac{\left(({\bm{k}}-{\bm{p}})\cdot{\bm{p}}\right)^{2}}{|{\bm{k}}-{\bm{p}}|^{2}p^{2}}\right]$
$\displaystyle\,=\frac{\mathcal{C}^{2}}{4\pi^{4}{\overline{\eta}}_{B}^{2}}\int
dk_{1}dk_{2}k_{1}^{2}k_{2}^{2}\sum_{\pm}\left(\pm\left\\{\frac{(k_{1}\pm
k_{2})^{2}}{2}\left[S(k_{1})S(k_{2})\pm
A(k_{1})A(k_{2})\right]\frac{D}{k_{1}k_{2}}\left(1\mp\frac{D}{k_{1}k_{2}}\right)\right.\right.$
$\displaystyle\hskip
42.67912pt+\left.\left[\frac{k_{1}^{2}+k_{2}^{2}}{2}S(k_{1})S(k_{2})+k_{1}k_{2}A(k_{1})A(k_{2})\right]\left(\frac{D}{k_{1}k_{2}}\right)^{3}\right\\}\left.\exp\left[-\frac{(k_{1}\mp
k_{2})^{2}}{2D}\right]\right).$ (3.5)

For a given magnetic field spectra, $A(k)$ and $S(k)$, with Eqs. (3.1) and
(3.1) one can determine the constraints on their parameters, which is the main
result of the present paper. Here we do not assume that the homogeneous part
of the baryon asymmetry, $\overline{\eta}_{B}$, is produced by the
hypermagnetic helicity decay. Therefore, we can adopt Eq. (3.1) also for the
case where other baryogenesis mechanisms are responsible for the net BAU,
which will be investigated in § 5.

We also note that there is another issue at the BBN on the constraint of the
magnetic field properties. Mainly because magnetic fields contribute to the
energy density of the Universe as an additional relativistic degree of freedom
at the time of the BBN, which are constrained in terms of the effective
numbers of neutrino species, the magnetic field energy density or the field
strength is constrained as $B_{\mathrm{c,BBN}}\lesssim 10^{-6}\;\mathrm{G}$
[73] irrespective to its coherent length.

###  3.2 Magnetic field evolution until today and implications from the
intergalactic magnetic fields

The long-range (hyper)magnetic fields generated in the radiation dominated era
(including before the EWSB) will evolve according to the MHD equations and can
remain until today as the IGMFs. Thus, the constraints obtained in the way we
developed in the previous subsection can be rewritten in terms of the
parameters of the IGMFs. In this subsection, we give the formulae that connect
the properties of the magnetic fields at the EWSB and the present.

With sufficiently strong magnetic fields, the bulk velocity fields are excited
due to the MHD, and the system enters the turbulence regime once the eddy
turnover scale of the velocity fields catches up the coherence length of the
magnetic fields. Before that the magnetic fields evolve adiabatically (the
spectrum is unchanged in terms of the comoving quantities) whereas they evolve
according to the scaling law of the direct or inverse cascade process until
the recombination [52]. We assume that the magnetic fields evolve
adiabatically again after the recombination.

More concretely, we treat the magnetic field evolution as follows. At the EWSB
the coherence length of the magnetic fields should be longer than the eddy
turnover scale [52, 74],

$\xi_{\mathrm{p}}|_{T=T_{\mathrm{fo}}}>\xi_{\mathrm{p,ed}}|_{T=T_{\mathrm{fo}}}\simeq\left.v_{A}\right|_{T=T_{\mathrm{fo}}}t_{\rm
fo}=\left.\left(\frac{2{\cal
E}_{\mathrm{p}}}{\rho_{\mathrm{p}}+p_{\mathrm{p}}}\right)^{\frac{1}{2}}\right|_{T=T_{\mathrm{fo}}}t_{\rm
fo},$ (3.6)

where $\rho_{\mathrm{p}}$ and $p_{\mathrm{p}}$ are the physical energy density
and pressure of the charged plasma particles, respectively, and the subscript
p represents that the quantity is the physical one. Here we have assumed that
the fluid velocity reaches at the Alfven velocity $v_{A}=\sqrt{2{\cal
E}_{\mathrm{p}}/(\rho_{\mathrm{p}}+p_{\mathrm{p}})}=\sqrt{2{\cal
E}_{\mathrm{c}}/(\rho_{\mathrm{c}}+p_{\mathrm{c}})}$, which indicates the
equipartition between the magnetic fields and fluid velocity fields. Note that
it is convenient to investigate with physical quantities. If
$\xi_{\mathrm{p}}|_{T=T_{\mathrm{fo}}}=\xi_{\mathrm{p,ed}}|_{T=T_{\mathrm{fo}}}$,
this suggests that the magnetic fields have already started to evolve
according to the cascade process at the EWSB. If the coherence length is
longer than the eddy turnover scale at the EWSB, eventually the eddy turnover
scale catches up the coherence length of the magnetic fields at

$t=t_{\mathrm{sc}}\equiv\left.\frac{\xi_{\mathrm{p}}}{\sqrt{2{\cal
E}_{\mathrm{p}}/(\rho_{\mathrm{p}}+p_{\mathrm{p}})}}\right|_{t=t_{\mathrm{sc}}},$
(3.7)

and the cascade process is started. Note that if
$t_{\mathrm{rec}}<t_{\mathrm{sc}}$, with $t_{\rm rec}$ being the physical
cosmic time at the recombunation, the system never enters the cascade regime
and evolve always adiabatically in the entire cosmic history after
magnetogenesis.

The scaling law of the cascade process of the magnetic field evolution depends
on the complicated properties of the distributions of the magnetic fields as
well as the fluid velocities, which is often hard to determine. Instead of
identifying the scaling law with such properties, with turning back to the
comoving quantities, let us parameterize the scaling evolution of the magnetic
fields, following Ref. [75], as

${\cal
E}_{\mathrm{c}}\propto\tau^{-\frac{2(\beta+1)}{\beta+3}},\quad\xi_{\mathrm{c}}\propto\tau^{\frac{2}{\beta+3}},$
(3.8)

where $\tau$ is the conformal time, and $\beta$ is a parameter determined by
the initial conditions of the magnetic fields and the fluid velocity fields.
Note however that the magnetic field spectrum alone does not fix the value of
$\beta$. 101010$\beta$ is the parameter that quantifies the decay of the
spectral energy around the comoving wave number $k=\xi_{c}^{-1}$ [75]. It is
an independent parameter of $\alpha$ in §4.2, which parameterizes the shape of
the magnetic field power spectrum. Such a scaling law has also been considered
in Refs. [76, 77]. The eddy turnover scale evolves with
$\xi_{\mathrm{p,ed}}=t\sqrt{2{\cal
E}_{\mathrm{c}}(\tau)/(\rho_{\mathrm{c}}+p_{\mathrm{c}})}\propto\tau^{2/(\beta+3)}$
in the radiation dominated Universe and hence the relationship
$\xi\simeq\xi_{\mathrm{ed}}$ is continuously satisfied during the cascade
regime. If the magnetic fields are maximally helical, the helicity
conservation uniquely determines the scaling law of the magnetic field
evolution as $\beta=0$, which corresponds to the so-called “inverse cascade”
regime. On the other hand, if the magnetic fields are not maximally helical,
there are not apparent conserved quantities to determine $\beta$, and hence we
here take $\beta(\geq 0)$ as the parameter of the system but do not explore
its physical origin. In particular, we take $\beta=2$ as a representative
value for the non-maximally helical case, which is often realized in the
numerical simulations in Ref. [77]. We keep in mind that a smaller $\beta$ is
hardly realized. For partially helical magnetic fields with the helicity
fraction less than unity at the EWSB, $|\epsilon_{\mathrm{fo}}|<1$, the system
becomes more and more helical with the help of the helicity conservation, and
the helicity fraction $\epsilon$ grows through the cascade process [78],
$\epsilon\propto\tau^{2\beta/(\beta+3)}$. If
$\epsilon_{\mathrm{fo}}>(\tau_{\mathrm{rec}}/\tau_{\mathrm{sc}})^{-2\beta/(\beta+3)}$,
the system becomes maximally helical, $\epsilon=1$, before the recombination
at $\tau=\tau_{\mathrm{sc}}\epsilon_{\mathrm{fo}}^{-(\beta+3)/2\beta}$, and
$\beta$ turns to 0. If
$\epsilon_{\mathrm{fo}}<(\tau_{\mathrm{rec}}/\tau_{\mathrm{sc}})^{-2\beta/(\beta+3)}$,
the cascade process is described by the single $\beta$ throughout until the
recombination as long as the helicity fraction is less than 1. Assuming that
the magnetic field evolve adiabatically after the recombination until the
present, we can express the present magnetic field energy density ${\cal
E}_{0}$ and coherence length $\xi_{0}$ as

$\displaystyle{\cal E}_{0}$
$\displaystyle=\left\\{\begin{array}[]{ll}\epsilon_{\mathrm{fo}}^{\frac{2}{3}}\left(\dfrac{\tau_{\mathrm{sc}}}{\tau_{\mathrm{rec}}}\right)^{\frac{2}{3}}\mathcal{E}_{\mathrm{c}}|_{T=T_{\mathrm{fo}}}\simeq
1.1\times 10^{-8}\epsilon_{\rm
fo}^{\frac{2}{3}}\left(\dfrac{\tau_{\mathrm{sc}}}{\tau_{\mathrm{fo}}}\right)^{\frac{2}{3}}\mathcal{E}_{\mathrm{c}}|_{T=T_{\mathrm{fo}}}&\quad\text{for}\quad\epsilon_{\mathrm{fo}}>\left(\dfrac{\tau_{\mathrm{rec}}}{\tau_{\mathrm{sc}}}\right)^{-\frac{2\beta}{\beta+3}},\vspace{3mm}\\\
\left(\dfrac{\tau_{\mathrm{sc}}}{\tau_{\mathrm{rec}}}\right)^{\frac{2(\beta+1)}{\beta+3}}\mathcal{E}_{\mathrm{c}}|_{T=T_{\mathrm{fo}}}&\quad\text{for}\quad\epsilon_{\mathrm{fo}}<\left(\dfrac{\tau_{\mathrm{rec}}}{\tau_{\mathrm{sc}}}\right)^{-\frac{2\beta}{\beta+3}},\end{array}\right.$
(3.11) $\displaystyle\xi_{0}$
$\displaystyle=\left\\{\begin{array}[]{ll}\epsilon_{\mathrm{fo}}^{\frac{1}{3}}\left(\dfrac{\tau_{\mathrm{rec}}}{\tau_{\mathrm{sc}}}\right)^{\frac{2}{3}}\xi_{\mathrm{c}}|_{T=T_{\mathrm{fo}}}\simeq
9.3\times
10^{7}\epsilon_{\mathrm{fo}}^{\frac{1}{3}}\left(\dfrac{\tau_{\mathrm{fo}}}{\tau_{\mathrm{sc}}}\right)^{\frac{2}{3}}\xi_{\mathrm{c}}|_{T=T_{\mathrm{fo}}}&\quad\text{for}\quad\epsilon_{\mathrm{fo}}>\left(\dfrac{\tau_{\mathrm{rec}}}{\tau_{\mathrm{sc}}}\right)^{-\frac{2\beta}{\beta+3}},\vspace{3mm}\\\
\left(\dfrac{\tau_{\mathrm{rec}}}{\tau_{\mathrm{sc}}}\right)^{\frac{2}{\beta+3}}\xi_{\mathrm{c}}|_{T=T_{\mathrm{fo}}}&\quad\text{for}\quad\epsilon_{\mathrm{fo}}<\left(\dfrac{\tau_{\mathrm{rec}}}{\tau_{\mathrm{sc}}}\right)^{-\frac{2\beta}{\beta+3}},\end{array}\right.\quad$
(3.14)

where we have used
$\tau_{\mathrm{rec}}/\tau_{\mathrm{fo}}=(\tau_{\mathrm{rec}}/\tau_{\mathrm{eq}})\cdot(\tau_{\mathrm{eq}}/\tau_{\mathrm{fo}})\simeq
9.0\times 10^{11}$. The present helicity fraction is then given by

$\epsilon_{0}=\left\\{\begin{array}[]{ll}1&\quad\text{for}\quad\epsilon_{\mathrm{fo}}>\left(\dfrac{\tau_{\mathrm{rec}}}{\tau_{\mathrm{sc}}}\right)^{-\frac{2\beta}{\beta+3}},\vspace{3mm}\\\
\left(\dfrac{\tau_{\mathrm{rec}}}{\tau_{\mathrm{sc}}}\right)^{\frac{2\beta}{\beta+3}}\epsilon_{\mathrm{fo}}&\quad\text{for}\quad\epsilon_{\mathrm{fo}}<\left(\dfrac{\tau_{\mathrm{rec}}}{\tau_{\mathrm{sc}}}\right)^{-\frac{2\beta}{\beta+3}}\quad\text{and}\quad\tau_{\mathrm{rec}}>\tau_{\mathrm{sc}},\vspace{3mm}\\\
\epsilon_{\mathrm{fo}}&\quad\text{for}\quad\tau_{\mathrm{rec}}<\tau_{\mathrm{sc}}.\end{array}\right.$
(3.15)

We shall also use $B_{0}\equiv\sqrt{2{\cal E}_{0}}$ as the typical magnetic
field strength today.

Now we can present predictions on the magnetic field properties today and
compare them with the baryogenesis from the hypermagnetic helicity decay. From
Eqs. (2.2), (3.6), (3.11), (3.14), and (3.15), we find that the properties of
the present magnetic field, $\xi_{0}$, $B_{0}$, and $\epsilon_{0}$ determine
the baryon asymmetry produced by the hypermagnetic helicity decay regardless
of its evolution history as

$\overline{\eta}_{B}\simeq 10^{-10}\epsilon_{0}\left(\frac{{\cal
C}}{10^{33}\;\mathrm{Mpc}^{-1}\mathrm{G}^{-2}}\right)\left(\frac{\xi_{0}}{10^{-9}\;\mathrm{Mpc}}\right)\left(\frac{B_{0}}{10^{-17}\;\mathrm{G}}\right)^{2}.$
(3.16)

The fiducial value, $\mathcal{C}\sim 10^{33}\;\mathrm{Mpc}^{-1}{\rm G}^{-2}$,
is taken from Eq. (2.2). The constraints from the baryon isocurvature
perturbation in terms of the present magnetic field properties can be derived
in a similar way, but we need to specify the shape of the magnetic field
spectrum to evaluate concretely. We perform the investigation in the next
section.

Before closing this section, let us summarize the constraints on the
cosmological magnetic fields other than those from the inhomogeneous BBN.
First, once the system enters the cascade regime, the relation between the
present magnetic field strength and coherence length is determined by their
values at the recombination as

$\xi_{0}=\xi_{\mathrm{c,ed,rec}}=\left.a^{-1}\left(\frac{2{\cal
E}_{\mathrm{c}}}{\rho_{c}+p_{c}}\right)^{\frac{1}{2}}\right|_{T=T_{\mathrm{rec}}}t_{\rm
rec}\simeq 0.6\;\mathrm{pc}\left(\frac{B_{0}}{10^{-14}\mathrm{G}}\right).$
(3.17)

and hence the primordial magnetic fields generated before the recombination
should have the property, $\xi_{0}\gtrsim
0.6\;\mathrm{pc}(B_{0}/10^{-14}\;\mathrm{G})$. As we have mentioned, since the
magnetic fields contribute to additional relativistic energy density of the
Universe, the BBN gives the upper bound on the comoving magnetic field
strength when it occurs, $B_{\mathrm{c,BBN}}<10^{-6}\;\mathrm{G}$ [73],
regardless of the coherence length. Moreover, the CMB anisotropy imposes
another upper bound on the cosmological magnetic field strength whose comoving
coherence length is longer than Mpc scales as $B_{0}<10^{-9}\;\mathrm{G}$ [79,
80, 81, 82, 83, 84, 85, 86, 88, 87] (summarized in Ref. [74]), since they
would give additional contributions to the CMB anisotropy.

On the other hand, recent gamma-ray observations of blazars suggest the
existence of the intergalactic magnetic fields [36, 37, 38, 39, 40, 41, 42,
43, 44]. For example, the Fermi collaboration reported the deficit of the
secondary GeV cascade photons that should accompany the TeV photons from the
blazars, which implies the existence of the intergalactic magnetic fields. The
latest analysis suggests that the observations can be explained if the
magnetic fields are strong enough [44],

$B_{0}>\left\\{\begin{array}[]{ll}3\times
10^{-16}\;\mathrm{G}&\quad\text{for}\quad\xi_{0}>10\;\mathrm{kpc}\\\ 3\times
10^{-16}\;\mathrm{G}\times\left(\dfrac{\xi_{0}}{10\;\mathrm{kpc}}\right)^{-\frac{1}{2}}&\quad\text{for}\quad\xi_{0}<10\;\mathrm{kpc}\end{array}\right..$
(3.18)

Note that Ref. [44] gives the constraint only for the coherence length
$10^{2}\;\mathrm{pc}<\xi_{0}<10^{2}\;\mathrm{Mpc}$, and hence strictly
speaking we can use the constraint only in these ranges, but we will use the
extrapolated values as the reference. Comparing Eq. (3.18) to Eq. (3.16), the
blazar observation can be explained by the primordial magnetic fields
responsible for the present BAU only for $\epsilon_{0}\lesssim 10^{-9}$. In
particular, if the magnetic fields have entered in the cascade regime before
the recombination, from Eqs. (3.16) and (3.17) we obtain the condition for the
BAU to be explained by the hypermagnetic helicity decay in terms of the
present magnetic field properties as

$\begin{split}B_{0}\simeq 1\times 10^{-17}\;{\rm
G}\times\epsilon_{0}^{-\frac{1}{3}},\quad\xi_{0}\simeq 6\times
10^{-10}\;\mathrm{Mpc}\times\epsilon_{0}^{-\frac{1}{3}},\end{split}$ (3.19)

regardless of the shape of the spectrum.

##  4 Specific models

In the previous sections, we have presented a formalism to explain the BAU
from the hypermagnetic helicity decay in connection with the properties of the
magnetic field both at the EWSB and the present. It turned out that the
scenario is consistent with the IGMFs suggested by the blazar observations
only when the helicity fraction is very small, $\epsilon\lesssim 10^{-9}$.
From baryon isocurvature perturbations, we can already constrain the magnetic
field coherence length roughly as $\xi_{\mathrm{c,fo}}\lesssim k_{\rm d}^{-1}$
to avoid the deuterium overproduction. However, to apply the more concrete
constraint described in § 3.1, we need to specify the magnetic field spectra
at the EWSB. In this section, we specify specific forms of the power spectra,
which enable us to extract the general feature of the constraint. We also
present appropriate constraints in terms of the present magnetic field
properties.

###  4.1 Delta function

Let us first investigate the simplest choice of $S(k)$ and $A(k)=\epsilon
S(k)$ at the EWSB, a monochromatic form described by the delta function.

$\begin{split}S(k)=\pi^{2}\frac{B^{2}_{\mathrm{c,fo}}}{k_{\sigma}^{4}}\delta(k-k_{\sigma}),\quad
A(k)=\epsilon_{\mathrm{fo}}S(k),\end{split}$ (4.1)

so that the characteristic properties of the magnetic fields, as well as the
net baryon asymmetry are given by

${\cal
E}_{\mathrm{c,fo}}=\frac{1}{2}B_{\mathrm{c,fo}}^{2},\quad\xi_{\mathrm{c,fo}}=k_{\sigma}^{-1},\quad
h_{\mathrm{c,fo}}=\epsilon_{\mathrm{fo}}\xi_{\mathrm{c,fo}}B_{\mathrm{c,fo}}^{2},\quad\overline{\eta}_{B}=\epsilon_{\mathrm{fo}}{\cal
C}\xi_{\mathrm{c,fo}}B_{\mathrm{c,fo}}^{2},$ (4.2)

respectively. Note that the previous studies on the homogeneous part of the
baryon asymmetry from the hypermagnetic helicity decay [9, 11] implicitly
assumes such spectra.

The baryon isocurvature perturbation at the BBN is evaluated from Eq. (3.1) as

$\begin{split}\overline{S^{2}_{B,\mathrm{BBN}}}&=\frac{1}{4}\left(\frac{1}{\epsilon_{\mathrm{fo}}^{2}}+1\right)D\xi_{\mathrm{c,fo}}^{2}\left[2-2D\xi_{\mathrm{c,fo}}^{2}+D^{2}\xi_{\mathrm{c,fo}}^{4}\left(1-e^{-\frac{2}{D\xi_{\mathrm{c,fo}}^{2}}}\right)\right].\end{split}$
(4.3)

In the limit of large coherence length, $D\xi_{\mathrm{c,fo}}^{2}\gg 1$,

$\begin{split}\overline{S^{2}_{B,\mathrm{BBN}}}=\frac{1}{3}\left(\frac{1}{\epsilon_{\mathrm{fo}}^{2}}+1\right)+\mathcal{O}\left(\left(D\xi_{\mathrm{c,fo}}^{2}\right)^{-1}\right).\end{split}$
(4.4)

In this limit, the effect of neutron diffusion is ineffective and the
condition (3.1) is never satisfied. In the opposite limit,
$D\xi_{\mathrm{c,fo}}^{2}\ll 1$, corresponding to small-scale magnetic fields,
which is the case we are interested in so as to maintain the condition (3.1),

$\begin{split}\overline{S^{2}_{B,\mathrm{BBN}}}=\frac{1}{2}\left(\frac{1}{\epsilon_{\mathrm{fo}}^{2}}+1\right)D\xi_{\mathrm{c,fo}}^{2}+\mathcal{O}\left(\left(D\xi_{\mathrm{c,fo}}^{2}\right)^{2}\right).\end{split}$
(4.5)

A factor $\xi_{\mathrm{c,fo}}^{2}$ appears in the leading term, convincing us
that the shorter the coherence length of the fields is, the more suppressed
the baryon isocurvature perturbations. This is because neutrons are more
likely to smear out the inhomogeneity of baryon distribution. Needless to say,
the delta-function type power spectrum in the wavenumber space does not mean a
vanishing correlation on scales larger than the corresponding scale in the
position space, and hence we acquire a suppressed but finite correlation even
after smearing out by the neutron diffusion.

The condition for sufficiently small baryon isocurvature perturbation (Eq.
(3.1)), with the condition that the present BAU is explained by the
hypermagnetic helicity decay (Eq. (2.21)) being satisfied, gives the
conditions in terms of the magnetic field properties at the EWSB as well as
the present.

First we express the constraint in terms of the magnetic field properties at
the EWSB. The condition for the MHD erasure of the small-scale magnetic fields
(Eq. (3.6)) is written in terms of the physical magnetic field properties as

$B_{\mathrm{p,fo}}\lesssim 6\times
10^{-3}\;\mathrm{GeV}^{2}\times\left(\frac{\xi_{\mathrm{p,fo}}}{10^{6}\;\mathrm{GeV}^{-1}}\right),$
(4.6)

which does not depend on the helicity fraction $\epsilon_{\mathrm{fo}}$. Here
we have used the number of charged relativistic degrees of freedom
$g_{*,\mathrm{fo}}^{\mathrm{ch}}=82.75$.

The condition for the magnetic field to generate the present BAU, Eq. (2.21),
yields the lower bound of the magnetic field strength together with the
requirement $\epsilon_{\mathrm{fo}}\leq 1$, as

$B_{\mathrm{p,fo}}\gtrsim 5\times
10^{-3}\;\mathrm{GeV}^{2}\left(\frac{\xi_{\mathrm{p,fo}}}{10^{6}\;\mathrm{GeV}^{-1}}\right)^{-\frac{1}{2}}.$
(4.7)

Conversely, the condition (2.21) allow us to determine the helicity fraction
$\epsilon_{\mathrm{fo}}$ for a given magnetic field strength and coherence
length. This leads to the lower bound of the magnetic field strength from the
baryon isocurvature constraint (Eq. (3.1)) with Eq. (4.5) as

$B_{\mathrm{p}}\lesssim 3\times
10^{-3}\;\mathrm{GeV}^{2}\left(\frac{\xi_{\mathrm{p,fo}}}{10^{10}\;\mathrm{GeV}^{-1}}\right)^{-1},$
(4.8)

for $\epsilon_{\mathrm{fo}}\ll 1$, where we have used the fact that the
neutron diffusion scale at the BBN corresponds to the proper wavenumber
$k_{\mathrm{d,fo}}^{\mathrm{p}}=4.4\times 10^{-15}\;\mathrm{GeV}$ at the EWSB.

The parameter space constrained by Eqs. (4.6), (4.7), and (4.8) (with the
behavior at $\epsilon_{\mathrm{fo}}\sim 1$ being appropriately taken into
account) is shown as the green shaded region of the right panel of Fig. 1.
Here the smallest helicity fraction that satisfies all the conditions is found
to be $\epsilon_{\mathrm{fo}}\simeq 2\times 10^{-6}$.

Next, let us turn to the constraints expressed by the present IGMF properties.
First of all, large-scale magnetic fields whose coherence length is larger
than the one for the MHD turbulence (Eq. (3.17)) are ruled out. We have seen
that such magnetic fields have evolved adiabatically with the comoving
coherence length being unchanged. Then, it can be easily checked that it is
impossible to satisfy $\xi_{\mathrm{c,fo}}<k_{\mathrm{d}}^{-1}$ , which is
required by the baryon isocurvature constraints (Eq. (3.1)), simultaneously
satisfying the condition for the generation of the present BAU (Eq. (3.16)).
For the IGMF whose properties lie in the line Eq. (3.17), there remains a
possibility to satisfy the baryon isocurvature constraint since it has
experieced the cascade process and the coherence length is much shorter at the
EWSB. Since the relationship between the coherence length at the EWSB and that
at the present depends on the value of $\beta$, the baryon isocurvature
constraint today also depends on $\beta$. If $\beta>1$, the magnetic fields
that satisfy the constraint at the EWSB (green shaded region in Fig. 1) become
maximally helical until the recombination. This can be seen that in the
$B$-$\xi$ figure, the magnetic field evolves with
$B_{\mathrm{c}}\propto\xi_{\mathrm{c}}^{-(\beta+1)/2}$ whereas the baryon
isocurvature constraint yields $B_{\mathrm{c}}\propto\xi_{\mathrm{c}}^{-1}$
(Eq. (4.8)). In this case, the IGMF property is uniquely determined as

$B_{0}\simeq 1\times 10^{-17}\;\mathrm{G},\quad\xi_{0}\simeq 6\times
10^{-4}\;\mathrm{pc}.$ (4.9)

The longest coherence length is obtained for the extreme choice, $\beta=0$,
(which would be hard to realize, though), where the cascade is most effective.
In such a case, the helicity fraction does not change so that the smallest one
is $\epsilon_{0}\sim\epsilon_{\rm fo}\sim 2\times 10^{-6}$. This uniquely
determines the upper bound of the magnetic field strength and coherence length
for $0\leq\beta\leq 1$ as

$\begin{split}1\times 10^{-17}\;\mathrm{G}\lesssim B_{0}\lesssim 7\times
10^{-16}\;\mathrm{G}\times\left(\dfrac{\tau_{\rm rec}}{\tau_{\rm
fo}}\right)^{-\frac{\beta+1}{\beta+3}+\frac{1}{3}},\;6\times
10^{-4}\;\mathrm{pc}\lesssim\xi_{0}\lesssim
0.07\;\mathrm{pc}\times\left(\dfrac{\tau_{\rm rec}}{\tau_{\rm
fo}}\right)^{\frac{2}{\beta+3}-\frac{2}{3}},\end{split}$ (4.10)

with $\tau_{\mathrm{rec}}/\tau_{\mathrm{fo}}\simeq 9\times 10^{11}$ where the
lower bound is for the case $\epsilon_{0}=1$. The allowed region is depicted
as the green line in the right panel of Fig. 1. The constraint (4.10) falls
below the blazar constraint (Eq. (3.18)), $B_{0}\lesssim 3\times 10^{-14}$ G
on the line of the MHD turbulence (Eq. (3.17)), by a factor of
$\mathcal{O}(10^{2})$.

Figure 1:  Left. Constraints on the magnetic fields at the EWSB, Equations.
(4.6), (4.7), (4.8), and (4.18) are shown in terms of the physical quantities.
Right. Constraints on the present magnetic fields, Eqs. (3.19), (4.10), and
(4.20) are shown. The navy thick line, with its lower side hatched, is the
lower bounds suggested by blazar observations [44]. The navy dotted lines are
an extrapolation of Ref. [44]. In both panels, the gray shaded regions above
the eddy turnover scales are the parameter space inconsistent with the MHD
evolution. The green shaded region and line are the allowed parameter regions
for the delta-function model, while the yellow shaded region and line are the
one for the power-law model with $\alpha=0$.

###  4.2 A Power-law spectrum with an exponential cutoff

Next we study a more realistic form of the spectrum, a power-law spectrum with
an exponential cutoff,

$\begin{split}S(k)=\frac{2\pi^{2}}{\Gamma(\frac{5\;+\;\alpha}{2})}\frac{B_{\mathrm{c,fo}}^{2}}{k_{\sigma}^{5}}\left(\frac{k}{k_{\sigma}}\right)^{\alpha}\exp\left[-\left(\frac{k}{k_{\sigma}}\right)^{2}\right],\quad
A(k)=\epsilon_{\mathrm{fo}}S(k).\end{split}$ (4.11)

In this case, characteristic properties of the magnetic fields, as well as the
net baryon asymmetry produced by the fields, are given by

${\cal
E}_{\mathrm{c,fo}}=\frac{1}{2}B_{\mathrm{c,fo}}^{2},\quad\xi_{\mathrm{c,fo}}=\left(\zeta_{\alpha}k_{\sigma}\right)^{-1},\quad
h_{\mathrm{c,fo}}=\epsilon_{\mathrm{fo}}\xi_{\mathrm{c,fo}}{\cal
E}_{\mathrm{c,fo}},\quad\overline{\eta}_{B}=\epsilon_{\mathrm{fo}}{\cal
C}\xi_{\mathrm{c,fo}}B_{\mathrm{c,fo}}^{2},$ (4.12)

respectively, where
$\zeta_{\alpha}\equiv\frac{\Gamma\left(\frac{5\;+\;\alpha}{2}\right)}{\Gamma\left(2+\frac{\alpha}{2}\right)}.$
Here we restrict ourselves as $\alpha>-4$ otherwise the energy density and
coherence length are subject to infrared divergences. Such a spectrum is
introduced, for example, in Ref. [6, 7], and we expect that this single-power
law with exponential decay model captures the essential properties of a
broader class of spectrum including the one [89] that has a Batchelor spectrum
at small $k$, a Kolmogorov spectrum at intermediate $k$, and an exponential
cutoff at large $k$ because the baryon isocurvature constraints are mainly
determined by the peak and the infrared tail of the spectrum. Batchelor
spectrum cut off at large wavenumber corresponds to $\alpha=0$ in our
notation, so we are mainly interested in the parameter range around
$\alpha\sim 0$.

In this case, the baryon isocurvature perturbation at the BBN is given from
Eq. (3.1) as

$\displaystyle\overline{S^{2}_{B,\mathrm{BBN}}}=\frac{(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D}{2\Gamma\left(2+\frac{\alpha}{2}\right)^{2}}\int
dq_{1}$ $\displaystyle
dq_{2}q_{1}^{\alpha}q_{2}^{\alpha}e^{-q_{1}^{2}-q_{2}^{2}}$
$\displaystyle\sum_{\pm}$ $\displaystyle\left(e^{-\frac{(q_{1}\pm
q_{2})^{2}}{2(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D}}\left\\{\left(1\mp\frac{1}{\epsilon_{\mathrm{fo}}^{2}}\right)(q_{1}\mp
q_{2})^{2}\left[q_{1}q_{2}\pm(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\right]\right.\right.$
$\displaystyle\;\;\left.\left.\mp\left(2+\frac{1}{\epsilon_{\mathrm{fo}}^{2}}\frac{q_{1}^{2}+q_{2}^{2}}{q_{1}q_{2}}\right)\left[(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\right]^{2}\right\\}\right),$
(4.13)

where we have introduced a dimensionless wavenumber
$q_{i}=\zeta_{\alpha}\xi_{\mathrm{c,fo}}k_{i}$. In the limit of large
coherence length, for $\alpha>-3$,
$(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\gg 1$,

$\begin{split}\overline{S^{2}_{B,\mathrm{BBN}}}&=\frac{1}{3}\left[\frac{\Gamma\left(\frac{3\;+\;\alpha}{2}\right)\Gamma\left(\frac{5\;+\;\alpha}{2}\right)}{\Gamma\left(2+\frac{\alpha}{2}\right)^{2}\epsilon_{\mathrm{fo}}^{2}}+1\right]+\mathcal{O}\left(\left[(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\right]^{-1}\right),\end{split}$
(4.14)

where the terms with positive powers of
$(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D$ are canceled out by the exponential
factor. In the case $\alpha\leq-3$, since the integrand behaves as
$k_{1}^{\alpha+2}k_{2}^{\alpha+2}\times{\cal
O}(k_{1}^{2},k_{2}^{2})/(\epsilon_{\mathrm{fo}}^{2}(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D)$,
the volume average of the baryon isocurvature perturbation diverges at the
infrared as

$\overline{S^{2}_{B,\mathrm{BBN}}}\sim\frac{1}{\epsilon_{\mathrm{fo}}^{2}}(\zeta_{\alpha}\xi_{\mathrm{c,fo}}\Lambda_{\mathrm{IR}})^{3+\alpha},$
(4.15)

where $\Lambda_{\mathrm{IR}}$ is the infrared cutoff, and hence it becomes
unacceptably large for $(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\gg 1$. In
this limit, once more, the effect of neutron diffusion is ineffective and the
condition (3.1) is never satisfied.

In the other limit $(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\ll 1$, which is
the case when the condition (3.1) can be satisfied, by using the saddle point
approximation, we can approximate the volume average of the baryon
isocurvature perturbation for $\alpha>-5/2$ as

$\begin{split}\overline{S^{2}_{B,\mathrm{BBN}}}=\frac{\sqrt{\pi}2^{-2-\alpha}\Gamma\left(\frac{5}{2}+\alpha\right)}{\Gamma\left(2+\frac{\alpha}{2}\right)^{2}}\left(\frac{1}{\epsilon_{\mathrm{fo}}^{2}}+1\right)\left[(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\right]^{\frac{3}{2}}+\mathcal{O}\left(\left[(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\right]^{\frac{5}{2}}\right).\end{split}$
(4.16)

A factor $\xi_{\mathrm{c,fo}}^{-3}$ appears in the leading term, meaning that
the neutron dissipation effect is more effective than that in delta-function
model, in which case the corresponding factor is $k_{\sigma}^{-2}$. See Appx.
B for more discussion on the interpretation how these difference appears.

The apparent divergence for $-3<\alpha\leq-5/2$ is due to the breakdown of the
validity of the saddle point approximation. Since the IR behavior of the
integrand is the same as that seen in the case with
$(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\gg 1$, with
$(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{-2}D^{-1}$ being taken as the “UV”
cutoff, we can estimate as

$\overline{S^{2}_{B,\mathrm{BBN}}}\sim\frac{1}{\epsilon_{\mathrm{fo}}^{2}}\left[(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\right]^{\frac{3+\alpha}{2}}.$
(4.17)

The divergence at $\alpha<-3$ is the physical infrared divergence and the
volume average of the baryon isocurvature perturbation behaves as Eq. (4.15)
in the same way to the case with $(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\gg
1$. Hereafter we will focus on the case with $\alpha>-5/2$ for simplicity .

Now we are ready to present the constraints on the properties of the magnetic
field both at the EWSB as well as those at the present. As for the magnetic
fields at the EWSB, the constraints derived from the MHD, (4.6), and from
baryon-number generation, (4.7), apply in the same way as in the the delta-
function spectrum. Hence the baryon isocurvature constraint can be derived in
a similar way to the previous subsection. By combining Eqs. (2.21) and (4.16),
the baryon isocurvature constraint Eq. (3.3) is rewritten in a form of the
constraint on the magnetic field strength as

$B_{\mathrm{p,fo}}\lesssim 3\times 10^{-2}\;\mathrm{GeV}^{2}\times
2^{\frac{\alpha}{4}}\times\frac{\Gamma\left(2+\frac{\alpha}{2}\right)^{\frac{5}{4}}}{\Gamma\left(\frac{5}{2}+\alpha\right)^{\frac{1}{4}}\Gamma\left(\frac{5\;+\;\alpha}{2}\right)^{\frac{3}{4}}}\times\left(\frac{\xi_{\mathrm{p,fo}}}{10^{10}\;\mathrm{GeV}^{-1}}\right)^{-\frac{5}{4}},$
(4.18)

for $\epsilon_{\mathrm{fo}}\ll 1$. The parameter space for $\alpha=0$
constrained by Eqs. (4.6), (4.7), and (4.18) (with the behavior at
$\epsilon_{\mathrm{fo}}\sim 1$ being appropriately taken into account) is
shown in the yellow shaded region of the right panel of Fig. 1. The smallest
helicity fraction that satisfies all the conditions is evaluated as

$\epsilon_{\mathrm{fo}}=\epsilon_{\mathrm{fo}}^{\mathrm{min}}\equiv 1\times
10^{-8}\times
2^{-\frac{\alpha}{3}}\frac{\Gamma\left(\frac{5}{2}+\alpha\right)^{\frac{1}{3}}\Gamma\left(\frac{5\;+\;\alpha}{2}\right)}{\Gamma\left(2+\frac{\alpha}{2}\right)^{\frac{5}{3}}}.$
(4.19)

The constraints on the present magnetic field are also given in the same way
as in the delta-function spectrum. We see that most part of the parameter
region in which present magnetic fields have never entered the MHD turbulence
does not satisfy the baryon isocurvature constraints and almost only those
which lie in the line Eq. (3.17) have the possibility to satisfy them. From
the $B$ \- $\xi$ figure (Fig. 1), we can see that the magnetic fields that
satisfy the baryon isocurvature constraint at the EWSB becomes maximally
helical if they evolve more rapidly than
$B_{\mathrm{c}}\propto\xi_{\mathrm{c}}^{-5/4}$ (See also Eq. (4.18)). This
corresponds to the cascade parameter $\beta>3/2$. In such a case, again, the
IGMF property is uniquely determined as Eq. (4.9). The strongest IGMF is
obtained for $\beta=0$ and
$\epsilon_{\mathrm{fo}}\sim\epsilon_{0}=\epsilon_{\mathrm{fo}}^{\mathrm{min}}$,
which gives the parameter spaces of the IGMF properties that can explain the
BAU satisfying the baryon isocurvature constraint as

$1\times 10^{-17}\;\mathrm{G}\lesssim B_{0}\lesssim 3\times
10^{-15}\;\mathrm{G}\times\left(\dfrac{\tau_{\rm rec}}{\tau_{\rm
fo}}\right)^{-\frac{\beta+1}{\beta+3}+\frac{1}{3}},\quad 6\times
10^{-4}\;\mathrm{pc}\lesssim\xi_{0}\lesssim
0.3\;\mathrm{pc}\times\left(\dfrac{\tau_{\rm rec}}{\tau_{\rm
fo}}\right)^{\frac{2}{\beta+3}-\frac{2}{3}},$ (4.20)

with $\alpha=0$. Note that the constraints are quite insensitive to the value
of $\alpha$. These parameter space is shown in the yellow line in the right
panel of Fig. 1.

Figure 2:  The constraints on the present magnetic fields on the eddy turnover
scale, Eqs. (4.10) and (4.20), are shown as functions of the present helicity
fraction. The thick green line is the allowed parameter region for the delta-
function model, while the thick yellow line is allowed for the power-law model
with $\alpha=0$, with the evolution parameter $\beta=0$. The navy dotted line
shows the lower limit of the field strength extrapolated from the blazar
observation [44]. The green arrows show the edges of the allowed regions for
each $\beta$. For $\beta\gtrsim 1$, only around $B_{0}=1\times 10^{-17}\;$G
with $\epsilon_{0}=1$ is allowed in both models.

Figure 2 shows the $\epsilon_{0}$ dependence of the IGMF strength on the MHD
line Eq. (3.17). Yellow and green lines represent the parameter region where
the baryon isocurvature constraint is satisfied for $\beta=0$ in the case of
the power-law spectrum with an exponential cutoff and the delta-function
spectrum, respectively. For larger $\beta$, the maximum field strength becomes
smaller. In particular, for $\beta>1$, which would be often realized according
to Ref. [77], all the primordial magnetic fields that generate the correct BAU
while satisfying the baryon isocurvature constraint reaches at $B_{0}\simeq
1\times 10^{-17}\;\mathrm{G}$. Comparing to the gray shaded region, where the
IGMF cannot explain the blazar observation, we conclude that the primordial
magnetic fields that generate the correct BAU alone cannot explain the blazar
observations regardless of its initial helicity or the shape of the spectrum.
Note that the spectrum of baryon isocurvature perturbations has more power on
the smaller wavenumber modes than that of the energy density, and hence it is
less sensitive to the behavior of the ultraviolet decay. Therefore, all the
existing magnetogenesis mechanisms that explain the present BAU, such as the
axion inflation [26, 27, 28, 29, 30, 31], the chiral plasma instability [32],
or the Affleck-Dine mechanism [33] are subject to this constraint, since each
of these models produces helical hypermagnetic fields at its own
characteristic scale and they evolve according to MHD so that their infrared
spectrum is likely to become that of Batchelor spectrum. One may wonder if it
might be possible for some inflation models to generate partially helical
magnetic fields at Mpc scales today while generating the net BAU, but such a
possibility is now ruled out because of the excessive baryon isocurvature
perturbation.

##  5 Baryon isocurvature constraints on non-helical hypermagnetic fields

Thus far we have discussed the baryon isocurvature constraint on the
cosmological magnetic fields in the case they are responsible for the net BAU.
Since not only the helical part but also the non-helical part sources the
baryon isocurvature perturbation, we can universally apply what we discussed
in the previous sections to non-helical hypermagnetic fields supposing the net
BAU is generated by other mechanism, as long as magnetic fields existed before
the EWSB. In this section, we give the constraints on the non-helical
hypermagnetic fields $\epsilon_{\mathrm{fo}}=\epsilon_{0}=0$ from the baryon
isocurvature perturbation.

The volume average of the baryon isocurvature pertuerbation in this case can
be evaluated by setting $\epsilon_{\mathrm{fo}}=0$ and using the observed
value of $\overline{\eta}_{B}$ in Eq. (3.1) as

$\displaystyle\overline{S^{2}_{B,\mathrm{BBN}}}=\left\\{\begin{array}[]{ll}\dfrac{\mathcal{C}^{2}\xi_{\mathrm{c,fo}}^{4}B_{\mathrm{c,fo}}^{4}}{2\overline{\eta}_{B}^{2}}D+\mathcal{O}\left(\left(\xi_{\mathrm{c,fo}}^{2}D\right)^{2}\right)(<0.016),&\quad\text{for}\quad\xi_{\mathrm{c,fo}}^{2}D\ll
1,\\\ \\\
\dfrac{\mathcal{C}^{2}\xi_{\mathrm{c,fo}}^{2}B_{\mathrm{c,fo}}^{4}}{3\overline{\eta}_{B}^{2}}+\mathcal{O}\left(\left(\xi_{\mathrm{c,fo}}^{2}D\right)^{-1}\right)(<0.016),&\quad\text{for}\quad\xi_{\mathrm{c,fo}}^{2}D\gg
1\end{array}\right.$ (5.4)

for the delta-function power spectrum and

$\displaystyle\overline{S^{2}_{B,\mathrm{BBN}}}=\left\\{\begin{array}[]{l}\dfrac{\sqrt{\pi}2^{-\alpha}\Gamma\left(\frac{5}{2}+\alpha\right)\mathcal{C}^{2}\xi_{\mathrm{c,fo}}^{5}B^{4}_{\mathrm{c,fo}}}{4\Gamma\left(2+\frac{\alpha}{2}\right)^{2}\overline{\eta}_{B}^{2}}\left(\zeta_{\alpha}^{2}D\right)^{\frac{3}{2}}+\mathcal{O}\left(\left[(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\right]^{\frac{5}{2}}\right)(<0.016),\\\
\hskip 233.31259pt\text{for}\quad(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\ll
1,\vspace{3mm}\\\
\dfrac{\Gamma\left(\frac{3\;+\;\alpha}{2}\right)\Gamma\left(\frac{5\;+\;\alpha}{2}\right)\mathcal{C}^{2}\xi_{\mathrm{c,fo}}^{2}B^{4}_{\mathrm{c,fo}}}{3\Gamma\left(2+\frac{\alpha}{2}\right)\overline{\eta}_{B}^{2}}+\mathcal{O}\left(\left[(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\right]^{-1}\right)(<0.016),\\\
\hskip 233.31259pt\text{for}\quad(\zeta_{\alpha}\xi_{\mathrm{c,fo}})^{2}D\gg
1\end{array}\right.$ (5.9)

for the power law spectrum with an exponential cutoff. In terms of physical
quantities at the EWSB, the magnetic field strength are then constrained as

$\displaystyle B_{\mathrm{p,fo}}\lesssim\left\\{\begin{array}[]{ll}1\times
10^{-7}\ \mathrm{GeV}^{2}\left(\dfrac{\xi_{\rm p,fo}}{2\times 10^{14}\;{\rm
GeV}^{-1}}\right)^{-1},&\quad\text{for}\quad\xi_{\mathrm{p,fo}}\ll\sqrt{\dfrac{2}{3}}k_{\mathrm{d,fo}}^{\mathrm{p}\
-1},\\\ \\\ 2\times 10^{-7}\ \mathrm{GeV}^{2}\left(\dfrac{\xi_{\rm
p,fo}}{2\times 10^{14}\;{\rm
GeV}^{-1}}\right)^{-\frac{1}{2}},&\quad\text{for}\quad\xi_{\mathrm{p,fo}}\gg\sqrt{\dfrac{2}{3}}k_{\mathrm{d,fo}}^{\mathrm{p}\
-1}\end{array}\right.$ (5.13)

for the delta-function model, where
$\sqrt{\dfrac{2}{3}}k_{\mathrm{d,fo}}^{\mathrm{p}\ -1}\simeq 2\times
10^{14}\;{\rm GeV}^{-1}$. The constraints for the power-law model, Eqs. (5.9),
read

$\displaystyle\hskip
14.22636ptB_{\mathrm{p,fo}}\lesssim\left\\{\begin{array}[]{ll}1\times
10^{-7}\;\mathrm{GeV}^{2}\left(\dfrac{\xi_{\rm p,fo}}{2\times 10^{14}\;{\rm
GeV}^{-1}}\right)^{-\frac{5}{4}}\gamma_{1\alpha}^{\frac{1}{4}},&\;\text{for}\quad\xi_{\mathrm{p,fo}}\ll\sqrt{\dfrac{2}{3}}(\zeta_{\alpha}k_{\mathrm{d,fo}}^{\mathrm{p}})^{-1},\\\
\\\ 2\times 10^{-7}\;\mathrm{GeV}^{2}\left(\dfrac{\xi_{\rm p,fo}}{2\times
10^{14}\;{\rm
GeV}^{-1}}\right)^{-\frac{1}{2}}\gamma_{2\alpha}^{\frac{1}{4}},&\;\text{for}\quad\xi_{\mathrm{p,fo}}\gg\sqrt{\dfrac{2}{3}}(\zeta_{\alpha}k_{\mathrm{d,fo}}^{\mathrm{p}})^{-1},\end{array}\right.$
(5.17)

respectively, where we have defined
$\gamma_{1\alpha}\equiv\dfrac{2^{\alpha}\Gamma(2+\frac{\alpha}{2})^{5}}{\Gamma(\frac{5\;+\;\alpha}{2})^{3}\Gamma(\frac{5}{2}+\alpha)}$
and
$\gamma_{2\alpha}\equiv\dfrac{\Gamma(2+\frac{\alpha}{2})}{\Gamma(\frac{3\;+\;\alpha}{2})\Gamma(\frac{5\;+\;\alpha}{2})}$.

The parameter space satisfying these constraints is shown in the left panel of
Fig. 3 in green and yellow for the delta-function model and the power-law
model with an exponential cutoff, respectively. Combining with the MHD
turbulence condition at the EWSB (Eq. (4.6)), we find the upper bound of the
magnetic field strength at the EWSB as

$B_{\mathrm{p,fo}}^{\mathrm{max}}\simeq\left\\{\begin{array}[]{ll}0.4\;\mathrm{GeV}^{2},&\quad\text{for
delta function},\\\ \\\
2\;\mathrm{GeV}^{2}\times\gamma_{1\alpha}^{-\frac{1}{9}},&\quad\text{for power
law},\end{array}\right.$ (5.18)

respectively. From the second lines of Eqs. (5.13) and (5.17), we can
explicitly see that if the coherence length of the magnetic field at the EWSB
is larger than the neutron diffusion scale, the effect of neutron diffusion is
so small that the volume average of the inhomogeneity of baryon distribution
at the BBN is determined only by $\xi_{\mathrm{c,fo}}B_{\mathrm{c,fo}}^{2}$
practically. This is because the amplitude of spatially-dependent component of
the baryon asymmetry is still given by ${\cal
C}|{\bm{A}}|\cdot|{\bm{B}}|\sim{\cal
C}\xi_{\mathrm{c,fo}}B_{\mathrm{c,fo}}^{2}$ without the factor of the helicity
fraction $\epsilon_{\mathrm{fo}}$, as can be seen in Eq. (2.22) regardless of
the shape of the power spectrum. On the other hand, in the case the coherence
length of the magnetic fields is shorter than the neutron diffusion scale, the
constraint depends on the shape of the spectrum. See Appendix B for analysis
in the position space.

Translating these conditions to the present magnetic fields, we obtain the
following constraints on the IGMF. Large-scale magnetic fields compared with
the eddy turnover scale at the recombination, Eq. (3.17), evolve adiabatically
from EWSB until today, and hence the constraints for the magnetic fields at
the EWSB Eqs. (5.13) and (5.17) is directly translated as

$\displaystyle B_{0}\lesssim\left\\{\begin{array}[]{ll}3\times
10^{-18}\;\mathrm{G}\left(\dfrac{\xi_{0}}{0.0025\;{\rm
pc}}\right)^{-\frac{1}{2}},&\quad\text{for delta function},\\\ \\\ 4\times
10^{-18}\;\mathrm{G}\times\gamma_{2\alpha}^{\frac{1}{4}}\left(\dfrac{\xi_{0}}{0.0025\;{\rm
pc}}\right)^{-\frac{1}{2}},&\quad\text{for power law}.\end{array}\right.$
(5.22)

Note that the neutron diffusion scale accidentally coincides the scale where
the constraint line intersects with the line for the MHD turbulence scale.

On the other hand, the magnetic fields, which had a shorter coherence length
originally, gradually start to evolve according to the cascade law and reach
at the line that is determined by the eddy turnover scale at the recombination
(Eq. (3.17)). With a similar consideration as in the previous section, we find
that the strongest magnetic fields at the present is obtained in the case with
the strongest magnetic fields at the EWSB (Eq. (5.18)) for the milder decay
with smaller $\beta$ or in the case that magnetic fields enter the turbulence
regime just before the recombination for the stronger decay with larger
$\beta$. As a consequence, we obtain the upper bound of the magnetic field
today as

$\displaystyle B_{0}\lesssim\left\\{\begin{array}[]{ll}\text{max}\left[6\times
10^{-16}\;{\rm G}\times\left(\dfrac{\tau_{\rm rec}}{\tau_{\rm
fo}}\right)^{\frac{1}{3}-\frac{\beta+1}{\beta+3}},\;6\times
10^{-18}\;\mathrm{G}\right],&\;\text{for delta function},\\\ \\\
\text{max}\left[3\times 10^{-15}\;{\rm
G}\times\gamma_{1\alpha}^{-\frac{1}{9}}\times\left(\dfrac{\tau_{\rm
rec}}{\tau_{\rm fo}}\right)^{\frac{1}{3}-\frac{\beta+1}{\beta+3}},\;8\times
10^{-18}\;\mathrm{G}\times\gamma_{2\alpha}^{\frac{1}{6}}\right],&\;\text{for
power law}.\end{array}\right.$ (5.26)

The right panel of Fig. 3 shows the parameter spaces that satisfy the baryon
isocurvature constraint. Green and yellow shaded regions denote those for the
delta-function model and power law with exponential cutoff model (for
$\alpha=0$), respectively. The strongest magnetic fields are obtained at the
MHD turbulence line (Eq. (3.17)), with the cascade law with $\beta=0$.
Compared with the current observations, once more, the upper bounds are below
the blazar constraint, $B_{0}\lesssim 3\times 10^{-14}$ G, by a factor of
$\mathcal{O}(10-100)$. On the contrary, the galaxy scales, $\xi_{0}\sim
0.1-10$ kpc, the upper bound of the magnetic fields is roughly $B_{0}\lesssim
10^{-20}$ G. This would be sufficient to be the seed magnetic fields for the
magnetic fields in the galaxy and galaxy cluster [90].

Figure 3:  Constraints on the magnetic fields at the EWSB, (5.13) and (5.17)
in terms of the physical quantities (Left) and those on the present magnetic
fields, Eqs. (5.22) and (5.26) (Right) are shown. The navy line in the right
panel is the lower bounds of the IGMFs suggested by the blazar observations
[44] with an extrapolation (dotted line). In both panels, the gray shaded
regions are the parameter spaces that are inconsistent with the MHD. The green
shaded regions are the allowed parameter spaces for the delta-function model,
while the yellow shaded regions are those for the power-law model with
$\alpha=0$.

Figures 3 exhibit quite different allowed regions compared with Fig. 1 because
the net baryon asymmetry is fixed independently of the properties of the
magnetic fields. Since weaker magnetic fields that cannot generate the
observed BAU is now allowed, viable parameter spaces are widely open. However,
since the baryon isocurvature perturbation is mainly carried by the non-
helical part of the hypermagnetic fields, the constraints do not change much
for the stronger magnetic fields that can generate correct amount of the BAU,
which lie on the line that is determined by the eddy turnover scale (Eq.
(3.17)).

Thus far we have not discussed the inflationary magnetogenesis that may have
generated a relatively flat spectrum $\alpha<-4$ up to the Mpc scales today
with a comoving infrared cutoff $H_{0}<k_{\mathrm{IR}}<k_{\mathrm{d}}$ with
$H_{0}$ being the present Hubble parameter. In this case, we can write the
power spectrum as

$S(k)=\frac{(B^{\mathrm{IR}}_{\mathrm{c,fo}})^{2}}{k_{\mathrm{IR}}^{5}}\left(\frac{k}{k_{\mathrm{IR}}}\right)^{\alpha}\exp\left[-\left(\frac{k}{k_{\sigma}}\right)^{2}\right].$
(5.27)

with $\epsilon_{\mathrm{fo}}=0$. Here $B^{\mathrm{IR}}_{\mathrm{c,fo}}$
represents the magnetic field strength at the infrared cutoff. In a similar
way that is discussed in Eq. (4.15), we can evaluate the volume average of the
baryon isocurvature perturbation as

$\overline{S^{2}_{B,\mathrm{BBN}}}\sim\frac{\mathcal{C}^{2}(B^{\mathrm{IR}}_{\mathrm{c,fo}})^{4}}{\overline{\eta}_{B}^{2}k_{\mathrm{IR}}^{2}},$
(5.28)

which is almost equal to those for the delta-function like model or the power
law with exponential cutoff model. Therefore, we can conclude that even in
this case, the magnetic field strength at the IR cutoff should be tiny so that
the spectrum at the IR cut off comes below the constraint given in Eq. (5.22).
We conclude that inflationary magnetogenesis with a relatively flat spectrum
alone also cannot explain the blazar observations if they are generated before
the EWSB unless there are an amplification process.

##  6 Conclusion and Discussion

Baryogenesis from the hypermagnetic helicity decay is an attractive scenario
since it does not require any new physical degrees of freedom beyond the SM in
the mechanism itself. If this mechanism is responsible for the present BAU,
physics beyond the SM is required for the mechanism to produce the appropriate
helical hypermagnetic fields.

In this paper, we have studied constraints on the mechanism from baryon
isocurvature perturbations. Since the baryon fluctuations are produced in
response to the coherence length of the hypermagnetic fields, large scale
hypermagnetic fields are disfavored otherwise it might produce the anti-matter
region, or at least, spoils the success of the standard BBN. One important
essence is that non-helical part of the hypermagnetic fields contributes to
the baryon isocurvature perturbation while only their helical part contributes
to the homogeneous part of the net baryon asymmetry. Cosequently, baryon
isocurvature perturbations just with $\mathcal{O}(1)$ % level on scales larger
than the comoving neutron diffusion scale at BBN is ruled out because the
deuteriums are overproduced due to the second order effect.

Our main results are summarized in Fig. 1. We find that the magnetic fields
whose coherence length at the EWSB is larger than the comoving neutron
diffusion scale at BBN are completely ruled out since generation of the
correct amount of the BAU and the small enough baryon isocurvature
perturbation cannot be satisfied simultaneously. In summary, for the
primordial magnetic field to produce the BAU, the coherence length and the
strength are constrained as $10^{6}\;{\rm GeV}^{-1}<\xi_{\rm p}<10^{15}\;{\rm
GeV}^{-1}$ and $10^{-7}\;{\rm GeV}^{2}<B_{\rm p}<10\;{\rm GeV}^{2}$ in
physical quantity at the EWSB. Such magnetic fields evolve according to the
MHD cascade processes and remain as the IGMFs. For the typical cascade
parameter $\beta\simeq 2$, the present IGMF property is uniquely determined as
$\xi_{0}\simeq 10^{-3}\;\mathrm{pc}$ and $B_{0}\simeq 10^{-17}\;\mathrm{G}$
with being maximally helical today, regardless of the shape of the magnetic
field spectrum. Even allowing relatively unrealistic value of $\beta$ up to 0,
we find the present IGMF must satisfy $10^{-3}\;{\rm pc}<\xi_{0}<1\;{\rm pc}$
and $10^{-17}\;{\rm G}<B_{0}<10^{-14}\;{\rm G}$, on the line Eq. (3.17), which
is determined by the eddy turnover scale. It is found that the relationship
among the parameters that describe the magnetic fields, that is, $B_{0}$,
$\xi_{0}$), and $\epsilon_{0}$ is uniquely determined by the condition to
generate the correct amount of BAU (Eq. (3.16)) and to satisfy the MHD
evolution (Eq. (3.17)) as
$B_{0}\propto\xi_{0}\propto\epsilon_{0}^{-\frac{1}{3}}$, leaving only one
degree of freedom to the IGMFs regardless of their spectrum or the evolution
history. While this would act as the important consistency relation to test
this scenario, the upper bound from the baryon isocurvature perturbation is
(at most) marginally below the present lower bound of the IGMFs suggested by
the blazar observations [36, 37, 38, 39, 40, 41, 42, 43, 44,
45]111111Especially, if the constraint on the IGMF coherence length suggested
in Ref. [45] is correct, it is quite difficult to satisfy both the baryon
isocurvature constraints and IGMFs.

The fact that the non-helical part of the hypermagnetic fields contribute to
the baryon isocurvature perturbation enables us to constrain them at the EWSB,
assuming that another mechanism is responsible for the BAU today. In this
case, large-scale magnetic fields whose coherence length is larger than the
comoving neutron diffusion scale at BBN is permitted, but only tiny field
satisfying $B_{0}\lesssim
10^{-18}\;\mathrm{G}(\xi_{0}/0.0025\;\mathrm{pc})^{-1/2}$ today, which is well
below the lower bound of the IGMFs suggested by the blazar observations. As
for the IGMFs with a shorter wave length that have experienced the cascade
process, the constraint is almost the same as the case when the baryogenesis
from the hypermagnetic helicity decay is responsible for the present BAU. In
summary, regardless of its helicity nature, hypermagnetic fields generated
before the EWSB cannot be the origin of the IGMFs suggested by the blazar
observations. We need another mechanism to generate or amplify the magnetic
fields after EWSB. Note that the lower bound of the primordial magnetic fields
required as a source of the dynamo amplification of the galactic magnetic
fields, $B_{0}\gtrsim 10^{-30}$ G at $\xi_{0}\gtrsim 10^{2}$ pc [90], can be
satisfied by the primordial hypermagnetic fields generated before the EWSB.

There are several potential uncertainties in our analysis that could change
our results. The first is that we have assumed that the symmetric and
antisymmetric parts of the power spectrum, $S(q)$ and $A(q)$, are proportional
to each other. However, the results are essentially unchanged even if this
assumption is relaxed. We see that Eq. (3.1) can be decomposed into
contributions from the symmetric and antisymmetric parts as
$\overline{S^{2}_{B,\mathrm{BBN}}}=\overline{\eta}_{B}^{-2}[\sum(\int SS{\rm-
like\;terms})+\sum(\int AA{\rm-like\;terms})]$, where the symmetric part
should be the dominant contribution thanks to the realizability condition.
Therefore, by defining an effective helicity fraction, $\epsilon_{\rm eff}$,
by $\epsilon_{\rm eff}^{2}\equiv\sum(\int AA{\rm-like\;terms})/\sum(\int
SS{\rm-like\;terms})(\leq 1)$, the constraint is the same as that of §4.

The second is that the temperature dependence of the weak mixing angle, which
can change the efficiency factor of the baryogenesis, ${\cal C}$. In our
analysis we adopted the value inspired by the one-loop analytic estimate [55],
while the fitting formulae for the lattice simulation [35] adopted in Ref.
[11] can lead to a smaller value of ${\cal C}$. However, in a reasonable
choice of the fitting formula, the constraints on the strength and coherence
length magnetic fields can be enhanced by a factor of ${\cal O}(10)$, but it
is not enough for the upper bound from the baryon isocurvature constraints to
reach at the lower bound of the IGMFs suggested by the blazar observations,
except for the case when the magnetic fields have experienced the MHD cascade
with milder decay $\beta=0$ (but see the next discussion). Since there are no
accurate calculations on the efficiency factor ${\cal C}$ at the sphaleron
freezeout $T\simeq 135$ GeV, we cannot rule out the possibility that it has a
much smaller value in reality, but it would be unlikely. If some physics
around the EWSB changes the weak mixing angle much more rapidly while keeping
the sphaleron rate intact, it would be possible to realize a much smaller
value of ${\cal C}$ so that small enough baryon isocurvature perturbations and
the blazar observations might be simultaneously explained, which is beyond the
scope of the present paper.

Finally, the MHD cascade evolution is treated in a relatively simplified way.
We evaluate the eddy turnover scale (Eq. (3.17)) assuming the equipartition of
the magnetic and fluid kinetic energy, which leads to $v\simeq
v_{\mathrm{A}}$. If the eddy turnover scale is smaller, we might have a
stronger IGMF today but with a smaller coherence length. In order for the
remnant magnetic fields to reach at the lower bound of the IGMFs suggested by
blazars, however, we need the changes of the eddy turnover scale with more
than a few orders of magnitude, which would be unlikely. Moreover, the upper
bound of the IGMF strength in our analysis is obtained in the case with the
cascade decay parameter $\beta=0$ at very tiny helicity. Evolution with
$\beta=0$ is the case of the so-called inverse cascade with the maximally
helical magnetic fields, which is guaranteed by the helicity conservation. We
do not rule out such a possibility, but do not expect the evolution
$\beta\simeq 0$ can be realized for the non-maximally helical magnetic fields,
where there are no conserved quantity to support the evolution. In summary,
although there still remain several uncertainties in our analysis, we conclude
that they do not change our results significantly.

What we have presented in this paper gives essential constraints on the model
building of magnetogenesis. As we have shown that the strength of the IGMFs
suggested by the blazar observations is well above the baryon isocurvature
constraints, any attempts to explain it by a primordial magnetic fields before
the EWSB is ruled out. For example, a first order phase transition in the
physics beyond the SM at a higher energy scale than the EWSB does not work as
a way to explain the origin of IGMFs. Hypermagnetic fields just to explain the
BAU are still allowed, but the helicity fraction should not be too small,
$\epsilon_{\mathrm{fo}}\gtrsim 10^{-8}$. There has been a hope to test the
scenario by IGMF observations, but we conclude that main part of the
observationally suggested IGMFs should be generated or amplified after the
EWSB, while magnetic fields responsible for the BAU is subdominant
contribution of them. In order to test the scenario, we need to develop the
way to remove such “foreground” IGMFs.121212 Another option is to explain the
blazar observation by some ways other than the IGMFs such as the beam plasma
instability [91].

## Acknowledgments

The work of FU is supported by the Forefront Physics and Mathematics Program
to Drive Transformation (FoPM). This work was partially supported by JSPS
KAKENHI Grant-in-Aid for Scientific Research (C) JP19K03842 (KK), Scientific
Research (S) 20H05639(JY), Innovative Area 19H04610(KK), and 20H05248(JY).

##  Appendix A Gauge choice

Since the source term (2.3) is gauge dependent without taking a spatial
average, we have to specify a gauge to carry out our calculation. In the main
text of the present paper, we have imposed the Coulomb gauge condition,
$\bm{\nabla}\cdot\bm{A}_{\mathcal{A}}(\bm{x})=0.$ In this Appendix, we see
that we can determine the power spectrum of the vector potential,
$\bm{A}_{\mathcal{A}}$, as we do in §2.2, by choosing the Coulomb gauge.

As for the magnetic field, $\bm{B}_{\mathcal{A}}$, we can assume the
homogeneity and the isotropy, according to the cosmological principle. Because
of the homogeneity of the universe, the two-point correlation function of the
magnetic fields can be written as a function of the relative coordinate.

$\begin{split}\langle
B_{\mathcal{A}i}(\bm{x})B_{\mathcal{A}j}(\bm{x}+\bm{r})\rangle=\tilde{\mathcal{F}}^{B}_{ij}(\bm{r}).\end{split}$
(A.1)

Transforming this equation into the wavenumber space, we see that the power
spectrum of the magnetic field is given by
$\tilde{\mathcal{F}}_{ij}^{B}(\bm{k})$, the Fourier transform of
$\tilde{\mathcal{F}}_{ij}^{B}(\bm{x})$.

$\begin{split}\langle
B^{*}_{\mathcal{A}i}(\bm{k})B_{\mathcal{A}j}(\bm{k}^{\prime})\rangle=(2\pi)^{3}\delta^{3}(\bm{k}-\bm{k}^{\prime})\tilde{\mathcal{F}}_{ij}^{B}(\bm{k}).\end{split}$
(A.2)

Notice that the factor $\delta^{3}(\bm{k}-\bm{k}^{\prime})$ appearing in the
left hand side of the equation is a consequence of the homogeneity. It
immediately follows from Eq. (A.2) that
$\tilde{\mathcal{F}}_{ji}^{B*}(\bm{k})=\tilde{\mathcal{F}}_{ij}^{B}(\bm{k})$.
Because of the isotropy and the divergenceless property of the magnetic field
in addition to this, the power spectrum of the magnetic field is characterized
by two real-valued functions, $\tilde{S}^{B}(k)$ and $\tilde{A}^{B}(k)$ as

$\begin{split}\tilde{\mathcal{F}}^{B}_{ij}(\bm{k})=P_{ij}(\hat{\bm{k}})\tilde{S}^{B}(k)+i\epsilon_{ijm}\hat{k}_{m}\tilde{A}^{B}(k),\quad
P_{ij}(\hat{\bm{k}})=\delta_{ij}-\hat{k}_{i}\hat{k}_{j}.\end{split}$ (A.3)

As for the vector potential, $\bm{A}_{\mathcal{A}},$ whether $\langle
A^{*}_{\mathcal{A}i}(\bm{k})A_{\mathcal{A}j}(\bm{k}^{\prime})\rangle$ has the
factor $\delta^{3}(\bm{k}-\bm{k}^{\prime})$ depends on the choice of gauge.
The reason why a gauge transformation breaks the condition equivalent to
homogeneity is that values of the vector potential are not physical ones and
that the homogeneity of the universe does not necessarily require the
homogeneity of the vector potential field. However, with the Coulomb gauge
condition, the homogeneity is maintained, and we can define the power spectrum
of the vector potential,131313We see this fact from
$P_{ia}(\hat{\bm{k}})P_{jb}(\hat{\bm{k}}^{\prime})\epsilon_{alr}\epsilon_{bms}\langle
B^{*}_{\mathcal{A}l}(\bm{k})B_{\mathcal{A}m}(\bm{k}^{\prime})\rangle=P_{ia}(\hat{\bm{k}})P_{jb}(\hat{\bm{k}}^{\prime})k_{r}k^{\prime}_{s}\langle
A^{*}_{\mathcal{A}a}(\bm{k})A_{\mathcal{A}b}(\bm{k}^{\prime})\rangle=k_{r}k^{\prime}_{s}\langle
A^{*}_{\mathcal{A}i}(\bm{k})A_{\mathcal{A}j}(\bm{k}^{\prime})\rangle\propto\delta^{3}(\bm{k}-\bm{k}^{\prime})$,
with the second equality valid under the Coulomb gauge condition,
$P_{ij}(\hat{\bm{k}})A_{\mathcal{A}j}(\bm{k})=A_{\mathcal{A}i}(\bm{k})$.

$\begin{split}\langle
A^{*}_{\mathcal{A}i}(\bm{k})A_{\mathcal{A}j}(\bm{k}^{\prime})\rangle=:(2\pi)^{3}\delta^{3}(\bm{k}-\bm{k}^{\prime})\tilde{\mathcal{F}}^{A}_{ij}(\bm{k}).\end{split}$
(A.4)

Since $\bm{B}_{\mathcal{A}}=\bm{\nabla}\times\bm{A}_{\mathcal{A}},$ the power
spectra, $\tilde{\mathcal{F}}_{ij}^{B}$ and $\tilde{\mathcal{F}}_{ij}^{A}$,
are related via

$\begin{split}\tilde{\mathcal{F}}_{ij}^{B}(\bm{k})=\epsilon_{ilm}\epsilon_{jrs}k_{l}k_{r}\tilde{\mathcal{F}}_{ms}^{A}(\bm{k}).\end{split}$
(A.5)

Similarly to the case of magnetic field,
$\tilde{\mathcal{F}}_{ji}^{A*}(\bm{k})=\tilde{\mathcal{F}}_{ij}^{A}(\bm{k})$,
the Coulomb gauge condition $k_{i}\tilde{\mathcal{F}}_{ij}^{A}(\bm{k})=0,$ and
the isotropy of the magnetic field leaves only two real-valued functions,
$\tilde{S}(k),\;\tilde{A}(k)$, as the degrees of freedom of the power spectrum
of the vector potential,

$\begin{split}\tilde{\mathcal{F}}^{A}_{ij}(\bm{k})=P_{ij}(\hat{\bm{k}})\tilde{S}(k)+i\epsilon_{ijm}\hat{k}_{m}\tilde{A}(k).\end{split}$
(A.6)

Using Eq. (A.5), $\tilde{S}^{B},\;\tilde{A}^{B}$ and $\tilde{S},\;\tilde{A}$
are related via

$\begin{split}\tilde{S}^{B}(k)=k^{2}\tilde{S}(k),\quad\tilde{A}^{B}(k)=k^{2}\tilde{A}(k).\end{split}$
(A.7)

In addition to this, as to the one-point function, because of the Coulomb
gauge condition,

$\begin{split}\langle A_{\mathcal{A}i}(\bm{k})\rangle=0\end{split}$ (A.8)

follows from the homogeneity of the magnetic field, $\langle
B_{\mathcal{A}i}(\bm{k})\rangle=0$.141414We see this from
$-iP_{ia}(\hat{\bm{k}})\epsilon_{alr}\langle
B_{\mathcal{A}l}(\bm{k})\rangle=P_{ia}(\hat{\bm{k}})k_{r}\langle
A_{\mathcal{A}a}(\bm{k})\rangle=k_{r}\langle
A_{\mathcal{A}i}(\bm{k})\rangle=0$, with the second equality valid under the
Coulomb gauge condition.

The physical meaning of symmetric and antisymmetric parts of the power
spectrum becomes clearer if we use the popular notations

$\begin{split}E_{\mathrm{M}}(k)=\frac{k^{4}}{2\pi^{2}}\tilde{S}(k),\quad
H_{\mathrm{M}}(k)=\frac{k^{3}}{\pi^{2}}\tilde{A}(k).\end{split}$ (A.9)

An integration of $E_{\mathrm{M}}(k)$ over $k$ gives the energy density of the
fields, $\mathcal{E}=\int dkE_{\mathrm{M}}(k)$, and an integration of
$H_{\mathrm{M}}(k)$ the helicity density, $h=\int dkH_{\mathrm{M}}(k)$.
Therefore, the symmetric part of the power spectrum determines the energy of
the fields, while the antisymmetric part determines the magnetic helicity of
the fields. One can explicitly see this property from Eqs. (2.15) and (2.16)
as well.

##  Appendix B Correlation functions in the position space

In this appendix, we rederive the correlation function of the baryon
isocurvature perturbation $\mathcal{G}$, defined in Eq. (2.24), in the real
coordinate space.151515Since we already have an expression in the wavenumber
space, we can just Fourier transform it. However, here we work in the position
space from the beginning. This reformulation will be helpful to intuitively
understand how the baryon isocurvature perturbation is generated. See Ref. [7]
for the earlier work about the procedure.

#### Power spectrum in the position space

We define three functions $F_{1},F_{2},F_{3}$, that characterize the power
spectrum of the magnetic field in the position space,

$\begin{split}F_{1}(R,t)\delta_{ij}+F_{2}(R,t)R^{i}R^{j}\equiv\int\frac{d^{3}q}{(2\pi)^{3}}e^{+i\bm{q}\cdot\bm{R}}P_{ij}(\hat{\bm{q}})S(q),\end{split}$
(B.1)

or equivalently

$\displaystyle F_{1}(R)$ $\displaystyle=\frac{1}{2\pi^{2}}\int d(\ln
q)\left[\left(-\frac{1}{R^{3}}+\frac{q^{2}}{R}\right)\sin(qR)+\frac{q}{R^{2}}\cos(qR)\right]S(q),$
(B.2) $\displaystyle F_{2}(R)$ $\displaystyle=\frac{1}{2\pi^{2}}\int d(\ln
q)\left[\left(\frac{3}{R^{5}}-\frac{q^{2}}{R^{3}}\right)\sin(qR)-\frac{3q}{R^{4}}\cos(qR)\right]S(q),$
(B.3)

and

$\begin{split}F_{3}({\bm{R}},t)\equiv\int\frac{d^{3}q}{(2\pi)^{3}}e^{+i\bm{q}\cdot\bm{R}}A(q).\end{split}$
(B.4)

Here we have introduced a characteristic scale $k_{\sigma}$ that normalizes
the dimensionful parameters, such as ${\bm{q}}\equiv{\bm{k}}/k_{\sigma}$ and
${\bm{R}}\equiv k_{\sigma}{\bm{x}}$. $S(q)$ and $A(q)$ are also normalized by
another parameter $B$, which characterizes the magnetic field strength so that
the magnetic energy and helicity are evaluated as ${\cal E}\propto B^{2}$ and
$h\propto B^{2}/k_{\sigma}$, respectively. Then the power spectrum
$\mathcal{F}_{ij}^{A}$ in the position space can be written in terms of these
functions as

$\begin{split}\mathcal{F}_{ij}^{A}({\bm{R}})=F_{1}(R)\delta_{ij}+F_{2}(R)R^{i}R^{j}+\epsilon_{ijm}F_{3,m}({\bm{R}}),\end{split}$
(B.5)

where $F_{3,m}({\bm{R}})\equiv\frac{\partial}{\partial R^{m}}F_{3}({\bm{R}})$.
$F_{1}$ and $F_{2}$ characterizes the symmetric part of the two-point
correlation function of the magnetic field, and $F_{3}$ does its antisymmetric
part. We shall note that $F_{3,m}$ has the following properties. By choosing
the coordinate system so that the ${\bm{R}}$ lies in the ${\bm{e}}_{3}$
direction, or $\bm{e}_{3}\parallel\bm{R}$, for the coordinate
${\bm{e}}_{1,2}$, which are perpendicular to ${\bm{e}}_{3}$, $F_{3,m}$
vanishes because the integrand of (B.4) is odd with respect to $q_{3}$. Or we
shall write $F_{3,i}(\bm{R})=0$ for $i=1,2$. On the other hand, for the
coordinate ${\bm{e}}_{3}$, $F_{3,3}$ gives finite value as

$\begin{split}F_{3,3}(\bm{R})=\frac{1}{2\pi^{2}R^{2}}\int
dq\left[qR\cos(qR)-\sin(qR)\right]A(q)\equiv F_{3,3}(R,t),\end{split}$ (B.6)

as a function of the absolute value $R$. As long as $S(q)$ as well as $A(q)$
are localized at the scale $q\sim 1\Leftrightarrow k\sim k_{\sigma}^{-1}$,
$F_{1,2,3}(R)$ are also peaked at $R\sim 1\Leftrightarrow r\sim
k_{\sigma}^{-1}$. Note also that the net mean baryon asymmetry is given from
Eq. (2.5) in terms of $F_{3,3}$ as

${\overline{\eta}}_{B}=-6{\cal
C}k_{\sigma}^{-1}B^{2}F_{3,3}^{\prime}(0)|_{T=T_{\mathrm{fo}}},$ (B.7)

where the prime denotes the derivative with respect to $R$. Particularly in
the coordinate coordinate system with ${\bm{e}}_{i,j}$ with $i,j=1,2$ being
perpendicular to ${\bm{R}}$ whereas ${\bm{e}}_{3}$ is parallel to ${\bm{R}}$
so that ${\bm{e}}_{1},{\bm{e}}_{2}$, and ${\bm{e}}_{3}$ construct the right-
handed coordinate system (therefore $R^{i}=0$ for $i=1,2$ and $R^{3}=R$.), the
Fourier transform of the power spectrum that contains the anti-symmetric part
as

$\displaystyle\mathcal{F}^{A}_{ij}(\bm{R})$
$\displaystyle=\int\frac{d^{3}q}{(2\pi)^{3}}e^{+i{\bm{q}}{\bm{R}}}\mathcal{F}^{A}_{ij}(\bm{q})=F_{1}(R)\delta_{ij}+\epsilon_{ij}F_{3}(R),$
(B.8)

where $\epsilon_{ij}$ is the 2-dimensional Levi-Civita tensor with
$\epsilon_{12}=1$. We shall note that other components of the magnetic field
power spectrum are given as

$\displaystyle\mathcal{F}^{A}_{i3}(\bm{R})=\mathcal{F}^{A}_{3i}(\bm{R})=0,\quad\mathcal{F}^{A}_{33}(\bm{R})=F_{1}(R)+F_{2}(R)R^{2}.$
(B.9)

#### Baryon isocurvature perturbation in the position space

Let us now derive the expressions of the baryon isocurvature perturbations in
terms of $F_{i}$s. From Eq. (2.22), the two-point correlation function of the
baryon isocurvature perturbation, $\mathcal{G}({\bm{R}})$, defined in Eq.
(2.24), is proportional to a four-point function of the vector potential as

$\begin{split}\langle\left(\bm{A}\cdot\nabla\times\bm{A}\right)(\bm{x})\left(\bm{A}\cdot\nabla\times\bm{A}\right)&(\bm{x}+\bm{r})\rangle\\\
=\epsilon_{ijk}\epsilon_{lmn}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial s^{i}}\frac{\partial}{\partial
t^{l}}\langle&A_{k}(\bm{x})A_{j}(\bm{x}+\bm{s})A_{n}(\bm{x}+\bm{r})A_{m}(\bm{x}+\bm{r}+\bm{t})\rangle\\\
=\epsilon_{ijk}\epsilon_{lmn}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial s^{i}}\frac{\partial}{\partial
t^{l}}\Bigl{[}&\langle A_{k}(\bm{x})A_{j}(\bm{x}+\bm{s})\rangle\langle
A_{n}(\bm{x}+\bm{r})A_{m}(\bm{x}+\bm{r}+\bm{t})\rangle\\\ &+\langle
A_{k}(\bm{x})A_{n}(\bm{x}+\bm{r})\rangle\langle
A_{j}(\bm{x}+\bm{s})A_{m}(\bm{x}+\bm{r}+\bm{t})\rangle\\\ &+\langle
A_{k}(\bm{x})A_{m}(\bm{x}+\bm{r}+\bm{t})\rangle\langle
A_{n}(\bm{x}+\bm{r})A_{j}(\bm{x}+\bm{s})\rangle\Bigr{]}\\\
=\epsilon_{ijk}\epsilon_{lmn}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial s^{i}}\frac{\partial}{\partial
t^{l}}\Bigl{[}&\mathcal{F}^{A}_{kj}(k_{\sigma}\bm{s})\mathcal{F}^{A}_{nm}(k_{\sigma}\bm{t})+\mathcal{F}^{A}_{kn}(k_{\sigma}\bm{r})\mathcal{F}^{A}_{jm}(k_{\sigma}(\bm{r}-\bm{s}+\bm{t}))\\\
&+\mathcal{F}^{A}_{km}(k_{\sigma}(\bm{r}+\bm{t}))\mathcal{F}^{A}_{jn}(k_{\sigma}(\bm{r}-\bm{s}))\Bigr{]}\times
k_{\sigma}^{-4}B^{4},\end{split}$ (B.10)

In the second equality, we decomposed the four-point function into pairs of
two-point functions, assuming the Gaussianity of the stochasticity. In the
last equality we have used the fact that the ensemble average $\langle
A_{i}({\bm{x}})A_{j}({\bm{x}}+{\bm{r}})\rangle$ is ${\bm{x}}$-independent, and
hence $\langle A_{i}({\bm{x}})A_{j}({\bm{x}}+{\bm{r}})\rangle=(1/V)\int
d^{3}x\langle
A_{i}({\bm{x}})A_{j}({\bm{x}}+{\bm{r}})\rangle=k_{\sigma}^{-2}B^{2}\mathcal{F}^{A}_{ij}(k_{\sigma}{\bm{r}})$.
Each contribution in Eq. (B.10) is calculated in terms of the functions
$F_{i}$s as

$\displaystyle\epsilon_{ijk}\epsilon_{lmn}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial s^{i}}\frac{\partial}{\partial
t^{l}}\left(\mathcal{F}^{A}_{kj}(k_{\sigma}\bm{s})\mathcal{F}^{A}_{nm}(k_{\sigma}\bm{t})\right)=36k_{\sigma}^{2}F_{3,3}^{\prime
2}(0),$ (B.11a)
$\displaystyle\epsilon_{ijk}\epsilon_{lmn}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial s^{i}}\frac{\partial}{\partial
t^{l}}\left(\mathcal{F}^{A}_{kn}(k_{\sigma}\bm{r})\mathcal{F}^{A}_{jm}(k_{\sigma}(\bm{r}-\bm{s}+\bm{t}))\right)$
$\displaystyle\;=2k_{\sigma}^{2}\Biggl{[}-\frac{2}{R}F_{1}F_{1}^{\prime}-F_{1}F_{1}^{\prime\prime}+3F_{1}F_{2}+RF_{1}F_{2}^{\prime}-RF_{1}^{\prime}F_{2}+R^{2}F_{2}^{2}-F_{3,3}\left(F_{3,3}^{\prime\prime}+\frac{2}{R}F_{3,3}^{\prime}-\frac{2}{R^{2}}F_{3,3}\right)\Biggr{]},$
(B.11b)
$\displaystyle\epsilon_{ijk}\epsilon_{lmn}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial s^{i}}\frac{\partial}{\partial
t^{l}}\left(\mathcal{F}^{A}_{km}(k_{\sigma}(\bm{r}+\bm{t}))\mathcal{F}^{A}_{jn}(k_{\sigma}(\bm{r}-\bm{s}))\right)$
$\displaystyle\;=2k_{\sigma}^{2}\Biggl{(}F_{1}^{\prime
2}+R^{2}F_{2}^{2}-2RF_{1}^{\prime}F_{2}+F^{\prime
2}_{3}+\frac{2}{R}F_{3,3}F^{\prime}_{3,3}+\frac{3}{R^{2}}F_{3,3}^{2}\Biggr{)}.$
(B.11c)

Here we have omitted the arguments of the functions $F_{i}$ otherwise stated.
Note that the final result is independent of the choice of the coordinate
system.

The detailed calculations to obtain Eq. (B.11a), (B.11a), and (B.11c) are
summarized in what follows. We adopt the coordinate system such that the
${\bm{R}}$ lies in the ${\bm{e}}_{3}$ direction and then can use Eqs. (B.8),
(B.9). First let us compute derivatives of $F_{3,3}$.

$\displaystyle F_{3,311}(\bm{R})$
$\displaystyle=F_{3,322}(\bm{R})=\frac{F_{3,3}(R)}{R},\quad
F_{3,333}(\bm{R})=F_{3,3}^{\prime}(R),$ (B.12)
$\displaystyle\delta_{ij}F_{3,3ij}(\bm{R})$
$\displaystyle=-\int\frac{dq}{2\pi^{2}}\frac{q^{2}}{R}\sin(qR)A(q)=F_{3,3}^{\prime}(R)+\frac{2F_{3,3}(R)}{R},$
(B.13) $\displaystyle\delta_{ij}F_{3,3mij}(\bm{R})$
$\displaystyle=-\delta_{3m}\int\frac{dq}{2\pi^{2}}\frac{q^{2}}{R^{2}}(qR\cos(qR)-\sin(qR))A(q)$
$\displaystyle=\delta_{3m}\left(F_{3,3}^{\prime\prime}(R)+\frac{2F_{3,3}^{\prime}(R)}{R}-\frac{2F_{3,3}(R)}{R^{2}}\right),$
(B.14)

where subscripts denote partial derivatives, e.g.,
$F_{3,3ij}(\bm{R})\equiv\frac{\partial^{2}}{\partial R^{j}\partial
R^{i}}R_{3,3}(R)$. Now Eq. (B.11a) is calculated as

$\displaystyle\epsilon_{ijk}\epsilon_{lmn}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial s^{i}}\frac{\partial}{\partial
t^{l}}\left(\mathcal{F}^{A}_{kj}(k_{\sigma}\bm{s})\mathcal{F}^{A}_{nm}(k_{\sigma}\bm{t})\right)$
$\displaystyle=\left(\epsilon_{ijk}\lim_{{\bm{s}}\rightarrow
0}\frac{\partial}{\partial
s^{i}}\mathcal{F}^{A}_{kj}(k_{\sigma}\bm{s})\right)^{2}$
$\displaystyle=\left(-2\delta_{im}\lim_{{\bm{s}}\rightarrow
0}\frac{\partial}{\partial s^{i}}F_{3,3m}(k_{\sigma}{\bm{s}})\right)^{2}$
$\displaystyle=36k_{\sigma}^{2}F_{3,3}^{\prime 2}(0).$ (B.15)

On the calculation of Eqs. (B.11b) and (B.11c), let us note that

$\frac{\partial}{\partial
R^{j}}F_{i}(R)=\frac{R^{j}}{R}F_{i}^{\prime}(R),\quad\frac{\partial}{\partial
R^{j}}\frac{\partial}{\partial
R^{m}}F_{i}(R)=\frac{R^{j}R^{m}}{R^{2}}\left(F_{i}^{\prime\prime}(R)-\frac{F_{i}^{\prime}}{R}\right)+\frac{F_{i}^{\prime}}{R}\delta_{jm},$
(B.16)

for $i=1,2$, and hence

$\delta_{jm}\frac{\partial}{\partial R^{j}}\frac{\partial}{\partial
R^{m}}F_{i}(R)=F_{i}^{\prime\prime}(R)+\frac{2F_{i}^{\prime}(R)}{R}.$ (B.17)

The first term in Eq. (B.16) is non-vanishing only for $j=3$ so that

$\frac{\partial}{\partial
R^{j}}F_{i}(R)=F^{\prime}_{i}(R)\delta_{j3},\quad\frac{\partial}{\partial
R^{3}}\frac{\partial}{\partial R^{3}}F_{i}(R)=F^{\prime\prime}_{i}(R).$ (B.18)

By using these expressions, we reach the expression Eq. (B.11b) as

$\displaystyle\epsilon$
${}_{ijk}\epsilon_{lmn}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial s^{i}}\frac{\partial}{\partial
t^{l}}\left(\mathcal{F}^{A}_{kn}(k_{\sigma}\bm{r})\mathcal{F}^{A}_{jm}(k_{\sigma}(\bm{r}-\bm{s}+\bm{t}))\right)$
$\displaystyle=\epsilon_{ijk}\epsilon_{lmn}k_{\sigma}^{2}\biggl{(}F_{1}(R)\delta_{kn}+F_{2}(R)R_{k}R_{n}+\epsilon_{kns}F_{3,3s}(R)\biggr{)}$
$\displaystyle\quad\times\lim_{{\bm{S}},{\bm{T}}\rightarrow
0}\frac{\partial}{\partial S^{i}}\frac{\partial}{\partial
T^{l}}\biggl{[}F_{1}(|{\bm{R}}-{\bm{S}}+{\bm{T}}|)\delta_{jm}+F_{2}(|{\bm{R}}-{\bm{S}}+{\bm{T}}|)(R_{j}-S_{j}+T_{j})(R_{m}-S_{m}+T_{m})$
$\displaystyle\hskip
334.31987pt+\epsilon_{jmt}F_{3,3t}({\bm{R}}-{\bm{S}}+{\bm{T}})\biggr{]}$
$\displaystyle=2k_{\sigma}^{2}\Biggl{[}-\frac{2}{R}F_{1}F_{1}^{\prime}-F_{1}F_{1}^{\prime\prime}+3F_{1}F_{2}+RF_{1}F_{2}^{\prime}-RF_{1}^{\prime}F_{2}+R^{2}F_{2}^{2}-F_{3,3}\left(F_{3,3}^{\prime\prime}+\frac{2}{R}F_{3,3}^{\prime}-\frac{2}{R^{2}}F_{3,3}\right)\Biggr{]},$
(B.19)

where we have defined as ${\bm{S}}\equiv k_{\sigma}{\bm{s}}$ and
${\bm{T}}\equiv k_{\sigma}{\bm{t}}$. In the same way, Eq. (B.11c) can be
derived as

$\displaystyle\epsilon$
${}_{ijk}\epsilon_{lmn}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial s^{i}}\frac{\partial}{\partial
t^{l}}\left(\mathcal{F}^{A}_{km}(k_{\sigma}(\bm{r}+\bm{t}))\mathcal{F}^{A}_{jn}(k_{\sigma}(\bm{r}-\bm{s}))\right)$
$\displaystyle=\epsilon_{ijk}\epsilon_{lmn}k_{\sigma}^{2}\lim_{{\bm{s}},{\bm{t}}\rightarrow
0}\frac{\partial}{\partial
T^{l}}\biggl{[}F_{1}(|{\bm{R}}+{\bm{T}}|)\delta_{km}+F_{2}(|{\bm{R}}+{\bm{T}}|)(R_{k}+T_{k})(R_{m}+T_{m})+\epsilon_{kmw}F_{3,3w}({\bm{R}}+{\bm{T}})\biggr{]}$
$\displaystyle\hskip 93.89409pt\times\frac{\partial}{\partial
S^{i}}\biggl{[}F_{1}(|{\bm{R}}-{\bm{S}}|)\delta_{jn}+F_{2}(|{\bm{R}}-{\bm{S}}|)(R_{j}-S_{j})(R_{n}-S_{n})+\epsilon_{jnz}F_{3,3z}({\bm{R}}-{\bm{S}})\biggr{]}$
$\displaystyle=2k_{\sigma}^{2}\biggl{[}F_{1}^{\prime
2}+R^{2}F_{2}^{2}-2RF_{1}^{\prime}F_{2}+F^{\prime
2}_{3}+\frac{2}{R}F_{3,3}F^{\prime}_{3,3}+\frac{3}{R^{2}}F_{3,3}^{2}\biggr{]}.$
(B.20)

In summary, we obtain the two-point correlation function of the helicity in
terms of the power spectrum as

$\displaystyle\langle({\bm{A}}\cdot{\bm{\nabla}}\times{\bm{A}})({\bm{x}})({\bm{A}}\cdot{\bm{\nabla}}\times{\bm{A}})({\bm{x}}+{\bm{r}})\rangle-\langle({\bm{A}}\cdot{\bm{\nabla}}\times{\bm{A}})({\bm{x}})\rangle^{2}$
$\displaystyle\quad=2k_{\sigma}^{-2}B^{4}\Biggl{[}F_{1}^{\prime
2}-\frac{2}{R}F_{1}F_{1}^{\prime}-F_{1}F_{1}^{\prime\prime}+3F_{1}F_{2}+RF_{1}F_{2}^{\prime}-3RF_{1}^{\prime}F_{2}+2R^{2}F_{2}^{2}$
$\displaystyle\hskip 207.70511pt+F_{3,3}^{\prime
2}-F_{3,3}\left(F_{3,3}^{\prime\prime}-\frac{5}{R^{2}}F_{3,3}\right)\Biggr{]}.$
(B.21)

We can easily check that Eq. (B) is consistent with the one given in Ref. [7].
Since in Ref. [7] the magnetic fields are non-helical, $A(q)=0$, postulating
that another mechanism is responsible for the BAU today, we have additional
contributions from the helical part $A(q)$, which appear in terms of
$F_{3,3}$.

Now we are ready to evaluate the baryon isocurvature perturbations in terms of
the magnetic field power spectrum. From Eqs. (2.22), (B.7), (B) we obtain

$\displaystyle\mathcal{G}({\bm{R}})=\frac{{\cal
C}^{2}}{\overline{\eta}_{B}^{2}}\left.\langle({\bm{A}}\cdot{\bm{\nabla}}\times{\bm{A}})({\bm{x}})({\bm{A}}\cdot{\bm{\nabla}}\times{\bm{A}})({\bm{x}}+{\bm{r}})\rangle-\langle({\bm{A}}\cdot{\bm{\nabla}}\times{\bm{A}})({\bm{x}})\rangle^{2}\right|_{T=T_{\mathrm{fo}}}=\frac{\tilde{G}(R)}{18F^{\prime
2}_{3,3}(0)},$ (B.22)

with $R\equiv k_{\sigma}|{\bm{r}}|$ and

$\displaystyle{\tilde{G}}(R)\equiv$ $\displaystyle F_{1}^{\prime
2}(R)-\frac{2}{R}F_{1}(R)F_{1}^{\prime}(R)-F_{1}(R)F_{1}^{\prime\prime}(R)+3F_{1}(R)F_{2}(R)+RF_{1}(R)F_{2}^{\prime}(R)$
$\displaystyle-3RF_{1}^{\prime}(R)F_{2}(R)+2R^{2}F_{2}^{2}(R)+F_{3,3}^{\prime
2}(R)-F_{3,3}(R)\left(F_{3,3}^{\prime\prime}(R)-\frac{5}{R^{2}}F_{3,3}(R)\right).$
(B.23)

#### Delta function spectrum

In the case of the delta-function model, $S(q)$ and $A(q)$,

$S(q)=\delta(q-1),\quad A(q)=\epsilon\delta(q-1),$ (B.24)

$F_{i}$s become

$\displaystyle F_{1}(R)$
$\displaystyle=\frac{1}{2\pi^{2}}\left[\left(\frac{1}{R}-\frac{1}{R^{3}}\right)\sin
R+\frac{1}{R^{2}}\cos R\right],$ (B.25a) $\displaystyle F_{2}(R)$
$\displaystyle=\frac{1}{2\pi^{2}}\left[\left(-\frac{1}{R^{3}}+\frac{3}{R^{5}}\right)\sin
R-\frac{3}{R^{4}}\cos R\right],$ (B.25b) $\displaystyle F_{3,3}(R)$
$\displaystyle=\frac{\epsilon}{2\pi^{2}}\left[-\frac{1}{R^{2}}\sin
R+\frac{1}{R}\cos R\right].$ (B.25c)

Then inserting them into Eq. (B), we can compute the correlator of baryon
isocurvature perturbations as

$\begin{split}\mathcal{G}(\bm{R})&=\frac{1}{4}\left(\frac{1}{\epsilon^{2}}+1\right)\left[\frac{2}{R^{2}}+\frac{2}{R^{4}}+\frac{3}{R^{6}}-\frac{6}{R^{5}}\sin
2R+\left(\frac{4}{R^{4}}-\frac{3}{R^{6}}\right)\cos 2R\right].\end{split}$
(B.26)

#### Power-law-like spectrum

In the case of the power law with exponential cutoff model, $S(q)$ and $A(q)$,

$S(q)=q^{\alpha}\exp(-q^{2}),\quad A(q)=\epsilon q^{\alpha}\exp(-q^{2}),$
(B.27)

$F_{i}$s become

$\displaystyle F_{1}(R)$
$\displaystyle=\frac{\Gamma(\frac{3\;+\;\alpha}{2})}{60\pi^{2}}\left[10F(\frac{3+\alpha}{2},\frac{5}{2},-\frac{R^{2}}{4})-\frac{3+\alpha}{2}F(\frac{5+\alpha}{2},\frac{7}{2},-\frac{R^{2}}{4})R^{2}\right],$
(B.28a) $\displaystyle F_{2}(R)$
$\displaystyle=\frac{\Gamma(\frac{5\;+\;\alpha}{2})}{60\pi^{2}}F(\frac{5+\alpha}{2},\frac{7}{2},-\frac{R^{2}}{4}),$
(B.28b) $\displaystyle F_{3,3}(R)$
$\displaystyle=\frac{\Gamma(1+\frac{\alpha}{2})\epsilon}{4\pi^{2}}\left[F(1+\frac{\alpha}{2},\frac{1}{2},-\frac{R^{2}}{4})-F(1+\frac{\alpha}{2},\frac{3}{2},-\frac{R^{2}}{4})\right]\frac{1}{R},$
(B.28c)

where $F(a,c,z)$ is the confluent hypergeometric function. Then inserting them
into Eq. (B), we can compute the correlator of baryon isocurvature
perturbations. The asymptotic behavior in the limit $R\to 0$,
$\mathcal{G}(\bm{R})$ is

$\begin{split}\mathcal{G}(\bm{R})=\frac{1}{3}\left[\frac{\Gamma\left(\frac{3\;+\;\alpha}{2}\right)\Gamma\left(\frac{5\;+\;\alpha}{2}\right)}{\Gamma\left(2+\frac{\alpha}{2}\right)^{2}\epsilon^{2}}+1\right]-\frac{1}{18}\left[\frac{\Gamma\left(\frac{3\;+\;\alpha}{2}\right)\Gamma\left(\frac{7\;+\;\alpha}{2}\right)}{\Gamma\left(2+\frac{\alpha}{2}\right)^{2}\epsilon^{2}}+2+\frac{\alpha}{2}\right]R^{2}+\mathcal{O}(R^{4}),\end{split}$
(B.29)

while in the opposite limit $R\to\infty$,

$\begin{split}\mathcal{G}(\bm{R})=\frac{\Gamma(3+\alpha)^{2}}{\Gamma(2+\frac{\alpha}{2})^{2}}\left[\frac{1}{\epsilon^{2}}(3+\alpha)(5+\alpha)\frac{-1+\cos(\pi\alpha)}{\alpha(2+\alpha)^{2}}+(2-\alpha)\frac{1+\cos(\pi\alpha)}{(1+\alpha)^{2}}\right]R^{-8-2\alpha}\;&\\\
+\mathcal{O}(R^{-10-2\alpha})&.\end{split}$ (B.30)

Figure 4:  Comparison of the shapes of $\mathcal{G}(R)$ with the delta-
function model and the power-law model with $\alpha=0$ is shown.
$\mathcal{G}(R)$ has terms proportional to $1/\epsilon^{2}$ originated from
the non-helical part of the spectrum, and terms independent of $\epsilon$
originated from the helical part. In the delta-function model, the correlation
function is exactly proportional to $1+1/\epsilon^{2}$.

#### Comparison of models

Finally, let us compare the behaviors of the correlation function of the
baryon isocurvature perturbation for the delta-function model and the power-
law model in the position space, which enables us to understand those in the
wavenumber space presented in the main text. Since spatially averaged baryon
asymmetry (B.7) is independent of the choice of $S(q)$, no difference between
the delta-function model and the power-law model appears in the constraint
from the overproduction of baryon asymmetry. However, as to the constraint of
deuterium overproduction, Fig. 2 shows a trend of weaker constraint for the
power-law model compared to the delta-function model. The origin of this trend
is not so trivial, because the four-point function is not linear in the power
spectrum and thus Fourier modes are not independent of each other. An
intuitive interpretation is can be given as follows. The delta-function model
has a perfect periodicity and thus non-vanishing long-range correlation in the
position space. If we compare asymptotic behavior of $\mathcal{G}(R)$ in
$R\to\infty$ limit, for the delta-function model, from (B.26), we find
$\mathcal{G}\sim R^{-2}$ when $R\to\infty$. On the other hand for the power-
law model, from (B.30), we find $\mathcal{G}\sim R^{-8-2\alpha}$ when
$R\to\infty$. Since our treatment is valid only for $\alpha>-3$, the tail in
the large $R$ limit in the power-law model is more suppressed than that in the
delta-function model, as can also be seen in Fig. 4. Since the neutron
diffusion effect erases the small scale correlation, constraints on the models
with stronger large-scale tail, that is, the delta-function model, are severer
than that of the power-law model, which explains the tendency in Fig. 2.

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