# Anisotropic gravitational waves induced by hypermagnetic fields during the
electroweak phase transition epoch

Mingqiu Li<EMAIL_ADDRESS>School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing, P.R.China
100049  Qi-Shu Yan<EMAIL_ADDRESS>School of Physics Sciences, University
of Chinese Academy of Sciences, Beijing, P.R.China 100049  Center for Future
High Energy Physics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100039, China  Mei Huang<EMAIL_ADDRESS>School of
Nuclear Science and Technology, University of Chinese Academy of Sciences,
Beijing, P.R.China 100049

###### Abstract

We study the anisotropies of gravitational waves induced by weak hypermagnetic
fields which are randomly distributed and oriented during the electroweak
phase transition in the early universe. The theory setup of this study is the
standard model plus a real singlet scalar field, which can produce the needed
strongly first order electroweak phase transition. Then we investigate how the
hypermagnetic fields can convert to magnetic fields and we compute the
departure of energy difference between the symmetric phase and the broken
phase when the magnetic fields are turned on. It is found that the presence of
the hypermagnetic fields can increase the Euclidean action, thus can decrease
nucleation temperature, which can lead to a supercool plasma. We point out
that the hypermagnetic field can enhance the gravitational wave production
from a first order electroweak phase transition and the inhomogeneity of
primordial hypermagnetic field can lead to anisotropies of gravitational
waves. By examining three well-motivated distribution of hypermagnetic fields,
we calculate the corresponding angular power spectra of stochastic
gravitational wave background and find they can be significantly larger than
the contributions of the Sachs-Wolfe effects and integrated Sachs-Wolfe
effects. Our results show that the anisotropies of gravitational wave could
provide a novel probe to the primordial hypermagnetic field in the electroweak
phase transition epoch.

gravitational waves, hypermagnetic fields, electroweak phase transition

##  I  Introduction

Magnetic fields exist in various scales of the Universe, from planets, stars,
and galaxies2015ASSL407483H ; Beck_2015 , to clusters of
galaxiesBonafede2009TheCC  and remote protogalactic clouds. The absence of
blazar halos implies magnetic fields with a magnitude $B>10^{-16}$G on Mpc to
Gpc scales or $B>10^{-15}$G on kpc scalesNeronov2010EvidenceFS , while the
anisotropy and polarization of the cosmic microwave background imposes an
upper bound of $\sim 10^{-9}$G on Mpc scalesPlanck:2015zrl ; Paoletti:2022gsn
.

The origin of cosmological magnetic fields remains a mystery. There exist two
popular schools of thoughts: on the one hand, their seeds are believed to be
created in the early universe (primordial hypothesis), on the other hand, they
can be produced during the processes of large-scale structure formation and
evolution (astrophysical hypothesis).

The primordial hypothesis is attractive and cosmological magnetic fields can
be generated from cosmological phase transitionsarticleBianLigong ;
Zhang:2019vsb , turbulenceKulsrud:2007an , inflation Martin:2007ue ;
Ferreira:2013sqa ; Campanelli:2013mea , etc. After their generation, the
magnetic fields undergo inverse cascade prior to recombination and then evolve
adiabatically. Assuming nowadays the magnitude of cosmological magnetic field
is about $10^{-16}-10^{-11}\text{G}$ or so, then it is calculable that at
$T\sim 100\text{GeV}$ epoch the magnetic field strength should be about
$10^{18}-10^{23}\text{G}$, or equally $B$ field should be in the range of
$(0.14\text{GeV})^{2}-(44\text{GeV})^{2}$cite1111 . Suppose that the
primordial magnetic fields are generated by inflation in the very early
universe, it is expected that they could play an important role in the
successive evolutionary processes, like the electroweak phase transition
(EWPT) and quantum chromodynamics (QCD) phase transition. It has been found
that QCD matter under external magnetic field shows novel phenomena, including
Chiral Magnetic Effect(CME) Kharzeev:2007tn ; Kharzeev:2007jp ;
Fukushima:2008xe , Magnetic Catalysis (MC) in the vacuum Klevansky:1989vi ;
Klimenko:1990rh ; Gusynin:1995nb , Inverse Magnetic Catalysis (IMC) around the
chiral phase transition Bali:2011qj ; Bali:2012zg ; Bali:2013esa ;
Chao:2013qpa ; Yu:2014sla , and the magnetic field also dramatically affect
hadron properties Ding:2020jui ; Bali:2017ian ; Bali:2012jv ; Liu:2014uwa ;
Liu:2015pna ; Wang:2017vtn ; Mao:2018dqe ; Lin:2022ied ; Xu:2020yag . The
gravitation wave and primordial black hole induced by chirality imbalance
under magnetic field has been investigated in Ref.Shao:2022oqw .

As a working mechanism for baryogenesisMorrissey:2012db  and a physical
process producing stochastic gravitational waves (GWs)Caprini:2015zlo , the
EWPT has been studied extensively. It is well known that in the Minimal
Standard Model (SM) the EWPT is a crossoverRummukainen:1996sx , but it can be
a strongly first order phase transition when new physics are considered, like
the singlet extension of the SMVaskonen:2016yiu ; Beniwal:2017eik ;
Beniwal:2018hyi ; Alves:2018jsw , two-Higgs-doublet modelsCline:1996mga ;
Basler:2016obg ; Dorsch:2016nrg , SUSY modelsHuber:2007vva ; Huber:2015znp ;
Demidov:2017lzf , composite modelsBian:2019kmg ; Bruggisser:2018mrt , seesaw
modelsBrdar:2018num , a holographic technicolor model Chen:2017cyc  and the
minimal left-right symmetric model Li:2020eun  etc.

Before the electroweak symmetry breaking, the symmetry of the SM is
$SU(2)_{L}\times U(1)_{Y}$. Instead there is no electromagnetic field but weak
gauge field $W$ and hypercharge gauge field $Y$. In the symmetric phase, the
non-Abelian $SU(2)_{L}$ field would be screened over distances larger than the
inverse of the magnetic mass, which is nonvanishing and can be expressed as
$m_{g}\sim g_{L}^{2}T$. In contrast, the the hypercharge magnetic field
$B_{Y}$ of the Abelian hypercharge symmetry could be a long-range magnetic
force survived in early universeElmfors_1998 ; Kajantie:1996qd ;
Giovannini:1997eg . Hypermagnetic field can delay the EWPTSanchez:2006tt  or
even induce a strongly first order phase transition in the SM if the field
strength is sufficient strongElmfors_1998 ; Abedi:2019msi . An extremely
strong hypermagnetic field may even lead to a condensation of gauge
fieldChernodub:2022ywg . Therefore, in this work, we will consider weak
hypermagnetic $B_{Y}\ll m^{2}_{W}/e$ background fields (in our numerical
calculation, $B_{Y}$ is no stronger than $(40\textrm{GeV})^{2}$ ) which can be
generated before the start of the EWPT, and we simply neglect the case with a
strong hypermagnetic field $B_{Y}$ is larger than $(40\textrm{GeV})^{2}$.

During a first order phase transition, vacuum bubbles nucleate, expand,
collide, and finally occupy all the space. In the broken phase, both weak
charged vector bosons $W$ and neutral vector boson $Z$ obtain their masses and
thus are screened in the thermal media, while the electromagnetic component of
the original hypermagnetic field remains massless due to the unbroken
$U(1)_{em}$ symmetry. Outside the bubbles is the symmetric phase where
hypermagnetic fields $B_{Y}$ can exist. Inside each a bubble, it is a broken
phase and only electromagnetic fields $B$ may survive. It is interesting to
investigate how the transition from $B_{Y}$ to $B$ can affect the EWPT. As one
of the key finding in this work, it is found that the hypermagnetic field can
decrease the nucleation temperature and produce a supercool plasma.
Consequently, during the EWPT, the hypermagnetic can increase the vacuum
energy release, which can lead to a considerable enhancement the GWs emission.

The successful detection of GWs produced by binary black holes
LIGOScientific:2016aoc  demonstrate that GWs could offer a new probe to the
early universe, since the GWs can carry the information of phase transitions.
Different from the merge of massive objects which produce a chirp-like GWs in
a short period at specific positions, the first order phase transitions
produce stochastic gravitational wave background (SGWB) which spread the whole
space, fly in all the directions, and last for a long time.

During a first order phase transition, there are three major physical
processes which can produce the GWs: bubble collisionsPhysRevLett.69.2026 ,
magnetohydrodynamic turbulenceKosowsky:2001xp , and sound
wavesHindmarsh:2013xza . For a wide range of models, during the phase
transition, the main dynamics of the GW production is the sound waves, since
it may continue to produce GWs long after the bubbles have
mergedHindmarsh:2013xza . The spectrum of GWs from a first order phase
transition has been studied by numerical simulationsHindmarsh:2013xza  and
semi-analytic methodHindmarsh:2019phv .

Similar to the cosmic microwave background, interesting information may be
extracted from the anisotropies in SGWB. As far as we know, both the
inhomogeneity of GW sourcesPhysRevLett.121.201303  and gravitational lensing
effects LISACosmologyWorkingGroup:2022kbp  can lead to anisotropies of the
SGWB. In this work, we will study the anisotropies in SGWB caused by the
inhomogeneity of GW sources, which are in the form of randomly distributed and
oriented external hypermagnegtic fields.

In this paper, we will systematically study the effects of weak hypermagnetic
fields to the EWPT. To study the anisotropies in SGWB, we assume inhomogeneous
weak hypermagnetic fields as their sources. This assumption can be justified
since magnetic fields observed in the Universe have both homogeneous (or
uniform) component and inhomogneous (or non-uniform) components
Giovannini:2003yn .

It is found in this work that the hypermagnetic field can enhance the GWs
emission and the inhomogeneity of hypermagnetic background can induce
anisotropies of the SGWB. And it is also found that the magnitudes of GWs can
be strong enough to be detected by the future planned space-based GW
detectors, like TaijiGuo:2018npi , BBOCorbin:2005ny , and DECIGOMusha:2017usi
for some model parameters. The anisotropies caused by the weak hypermagnetic
field can be of the size $10^{-1}$, which is significantly larger than the
contributions of Sachs-Wolfe (SW) effects and integrated Sachs-Wolfe (ISW)
LISACosmologyWorkingGroup:2022kbp ; Li:2021iva  effects, which is typically of
the size $10^{-9}$.

The rest of the paper is organized as follows. In Section II we revisit
singlet extension of the SM. How the magnetic field influence EWPT is
investigated in Section III. In Section IV the enhancement of SGWB due to
magnetic field is study and the anisotropy of SGWB is discussed. Finally, we
summarize our result in Section V. In Appendix A we discuss the higher order
corrections from magnetic field.

##  II A singlet extension of the SM

To achieve a strongly first-order EW phase transition, we consider a singlet
extension of the SM. The model includes the SM fields and a singlet real
scalar S. By imposing a $Z_{2}$ symmetry, i.e. $S\to-S$ while the fields of SM
is invariant under such a discrete symmetry transformation, this singlet
scalar $S$ is only allowed to couple to the SM weak Higgs doublet $H$. The
Lagrangian of the SM takes the following simple form articleBeniwal

$\mathcal{L}_{SM+singlet}=\mathcal{L}_{SM}+\frac{1}{2}\partial_{\mu}S\partial^{\mu}S-\frac{\mu_{S}^{2}}{2}S^{2}-\lambda_{HS}|H|^{2}S^{2}-\frac{\lambda_{S}}{4!}S^{4},$
(1)

where $\mu_{S}^{2}$ is a mass parameter, $\lambda_{HS}$ describes the strength
of coupling between $S$ and $H$, and $\lambda_{S}$ is the self-coupling of
$S$. For simplicity, we assume that $\mu_{S}^{2}>0$ and thus $\langle
S\rangle=0$.

To study the electroweak phase transition, we should calculate the effective
potential of Higgs. The vacuum expectation values of Higgs doublet can be put
as

$\langle H\rangle=\frac{e^{i\theta}}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
v\\\ \end{array}\right),$ (2)

where $\theta$ is a phase factor and $v$ is the vacuum expectation value of
the neutral Higgs boson at finite temperature. The effective potential can be
cast into the following form PhysRevD.45.2933 ; articleCurtin1David ;
articleBeniwal :

$\displaystyle V_{eff}(v)$
$\displaystyle=-\frac{\mu^{2}}{2}v^{2}+\frac{\lambda}{4}v^{4}$ (3)
$\displaystyle+\sum_{i}\frac{n_{i}}{64\pi^{2}}\left[m_{i}^{4}(v)\left(\log\frac{m_{i}^{2}(v)}{m_{i}^{2}(v_{EW})}+\frac{3}{2}\right)+2m_{i}^{2}(v)m_{i}^{2}(v_{EW})\right]$
$\displaystyle+\sum_{i}\frac{n_{i}T}{2\pi^{2}}\int_{0}^{\infty}dkk^{2}\log\left[1\pm\exp\left(\frac{-\sqrt{k^{2}+m_{i}^{2}(v)}}{T}\right)\right]$
$\displaystyle-\sum_{i=bosons}\frac{T}{12\pi}\left[(m_{i}^{2}(v)+\Pi_{i}(T))^{3/2}-m_{i}^{3}(v)\right].$

where the first line is the tree-level potential, the second line is the one-
loop corrections at $T=0$, and the third and forth line denote the thermal
corrections. The particle index $i$ includes $i=\\{W,Z,\chi,h,S,t,\\}$ and the
corresponding degree of freedom of particles $n_{i}$ is given as
$n_{i}=\\{6,3,3,1,1,-12\\}$. We only keep the contribution of top quarks,
which have the largest Yukawa coupling to the Higgs bosons, and we have
neglected that of light fermions. It is remarkable that only the bosonic
contribution can yield to the term $v^{3}$ in the potential, which is crucial
for the first-order phase transition.

The tree-level masses of weak vector boson and scalar Higgs bosons and top
quarks as well are given below,

$\displaystyle m_{W}^{2}$ $\displaystyle=\frac{g_{L}^{2}}{4}v^{2},\quad
m_{Z}^{2}=\frac{g_{L}^{2}+g_{Y}^{2}}{4}v^{2},\quad
m_{\chi}^{2}=-\mu^{2}+\lambda v^{2}$ (4) $\displaystyle m_{h}^{2}$
$\displaystyle=-\mu^{2}+3\lambda v^{2}\quad
m_{S}^{2}=\mu_{S}^{2}+\lambda_{HS}v^{2}\quad
m_{t}^{2}=\frac{y_{t}^{2}}{2}v^{2}.$

When we consider thermal corrections, we need thermal masses of these
particles which are denoted as $\Pi_{i}(T)$ in the effective potential given
in Eq. (3). The thermal masses of vector fields are given as

$\displaystyle\Pi_{W^{\pm}}(T)$ $\displaystyle=\frac{11}{6}g_{L}^{2}T^{2},$
(5) $\displaystyle\Pi_{W^{3},Y}(T)$
$\displaystyle=\left(\begin{array}[]{cc}\frac{11}{6}g_{L}^{2}T^{2}&0\\\
0&\frac{11}{6}g_{Y}^{2}T^{2}\\\ \end{array}\right),$

which are the same as those of the SM since the scalar field $S$ does not
couple to EW gauge fields. And the thermal masses of the two scalars are given
as

$\displaystyle\Pi_{h}(T)=\Pi_{\chi}(T)=T^{2}\left(\frac{g_{Y}^{2}}{16}+\frac{3g_{L}^{2}}{16}+\frac{\lambda}{2}+\frac{y_{t}^{2}}{4}+\frac{\lambda_{S}}{12}\right),$
(6)
$\displaystyle\Pi_{S}(T)=T^{2}\left(\frac{\lambda_{HS}}{3}+\frac{\lambda_{S}}{4}\right),$

which is consistent with the relation
$\left.\frac{\partial^{2}V_{eff}}{\partial
v^{2}}\right|_{v=0}=-\mu^{2}+\Pi_{h}$.

When temperature $T=0$, we require that the effective potential should get a
minima at $v=v_{EW}\simeq 246$GeV, such a condition can constrain the coupling
$\lambda_{HS}$ and the mass parameter of $S$, as shown in Figure 1, where the
shaded area is not satisfy such a condition and is excluded.

Figure 1: Constraint on the parameter space in the plane
$200\textrm{GeV}<m_{S}<2\textrm{TeV}$ and $2<\lambda_{HS}<12$ is shown. The
blue shaded area is excluded for $V_{eff}(v_{EW})>V_{eff}(0)$. We have chosen
$\lambda_{S}=1$. The bound that the future LHC can achieved is shown as a
dashed line.

Furthermore, except the SM Higgs, the new scalar $S$ does not couple to other
particles of the SM, thus it is difficult to detect it directly by collider
data at present. Actually it can serve as a dark matter candidate. When
kinematics allows, the SM Higgs boson can decay into a pair of $S$ and can
contribute to the invisible Higgs decay. When its mass is above $m_{H}/2$, its
detection becomes challenging.

Nevertheless, its effects can be measured via its radiative corrections to the
self-couplings of the SM like Higgs boson. For example, the triple Higgs
coupling can be an available probe. At one-loop order, the correction to the
triple Higgs coupling of $h^{3}$ can be modified as

$\displaystyle\lambda_{3}/\lambda_{3}^{SM}=\frac{1}{6\lambda_{3}^{SM}}\left.\frac{\partial^{3}V_{eff}(v)}{\partial
v^{3}}\right|_{v=v_{EW},T=0}\simeq
1+0.198\lambda_{HS}^{3}\left(\frac{100\text{GeV}}{m_{S}}\right)^{2}.$ (7)

From the di-Higgs boson searches at the LHC Gouzevitch:2020rwo , values
outside the interval $-1.8<\lambda_{3}/\lambda_{3}^{SM}<9.2$ are excluded at
95% confidence level according to the recent experimental results
Degrassi:2021uik . The future LHC 14 TeV with an integrated luminosity of 3
$\text{ab}^{-1}$ can provide a more severe constraint
$\lambda_{3}/\lambda_{3}^{SM}\in(0.7,1.3)$Barger:2013jfa , which will be able
to detect most of parameter space we interested as Figure 1 shows.

##  III Electroweak phase transition in an external magnetic field background

To examine how a homogenous large scale magnetic field can affect the
electroweak phase transition, we add a source term for the hyper field tensor
to the LagrangianElmfors_1998 (appendix A),

$\mathcal{L}=\mathcal{L}_{SM+singlet}+\frac{1}{2}Y_{\mu\nu}J^{\mu\nu}_{ext}\,,$
(8)

where $Y_{\mu\nu}$ is the field strength of the gauge field of $U(1)_{Y}$
symmetry, which is defined as

$Y_{\mu\nu}=\partial_{\mu}Y_{\nu}-\partial_{\nu}Y_{\mu}\,.$ (9)

After electroweak symmetry breaking, by using Eq. (2), the Lagrangian of Higgs
sector and gauge kinetic terms can be rewritten as

$\displaystyle L_{eff}=$
$\displaystyle\frac{1}{2}\partial_{\mu}v\partial^{\mu}v+\frac{v^{2}}{2}\left(\partial_{\mu}\theta+\frac{g_{L}}{2}W^{3}_{\mu}-\frac{g_{Y}}{2}Y_{\mu}\right)^{2}$
(10)
$\displaystyle-\frac{1}{4}Y_{\mu\nu}Y^{\mu\nu}+\frac{1}{2}Y_{\mu\nu}J^{\mu\nu}_{ext}-\frac{1}{4}(W^{3})_{\mu\nu}(W^{3})^{\mu\nu}-V_{eff}(v)$

where we assume vacuum expectation values of charge gauge fields are zero,
i.e. $\langle W^{1}\rangle=\langle W^{2}\rangle=0$, since they are charged
under the unbroken $U(1)_{em}$ symmetry.

Then the expectation value of neutral Higgs boson $v$ and the gauge fields
$A_{\mu}$ and $Z_{\mu}$ should be extreme point of the Lagrangian and satisfy
the following conditions

$\displaystyle
v^{2}\partial^{\mu}\left(\partial_{\mu}\theta+\frac{g_{Z}}{2}Z_{\mu}\right)$
$\displaystyle=0$ (11a) $\displaystyle\nabla^{2}v-\frac{\partial
V_{eff}}{\partial
v}+v\left(\partial_{\mu}\theta+\frac{g_{Z}}{2}Z_{\mu}\right)^{2}$
$\displaystyle=0$ (11b)
$\displaystyle\partial_{\nu}Z^{\mu\nu}+\sin\theta_{W}\partial_{\nu}J_{\text{ext
}}^{\mu\nu}+g_{Z}v^{2}\left(\partial^{\mu}\theta+\frac{g_{Z}}{2}Z^{\mu}\right)$
$\displaystyle=0$ (11c)
$\displaystyle\partial_{\nu}A^{\mu\nu}-\cos\theta_{W}\partial_{\nu}J_{\mathrm{ext}}^{\mu\nu}$
$\displaystyle=0$ (11d)

where $g_{Z}=\sqrt{g_{L}^{2}+g_{Y}^{2}}$, $\cos\theta_{W}=g_{L}/g_{Z}$ and
$\sin\theta_{W}=g_{Y}/g_{Z}$ and $\theta_{W}$ is the Weinberg angle.
Meanwhile, in Eqs.(11a)-(11d), after electroweak symmetry breaking, we
redefine the gauge fields from $Y_{\mu}$ and $W_{\mu}^{3}$ to physical vector
fields $A_{\mu}$ and $Z_{\mu}$ (vice versa) as

$\displaystyle\left.\begin{array}[]{l}Z_{\mu}\equiv\cos\theta_{W}W_{\mu}^{3}-\sin\theta_{W}Y_{\mu}\\\
A_{\mu}\equiv\sin\theta_{W}W_{\mu}^{3}+\cos\theta_{W}Y_{\mu}\end{array}\right\\}\Leftrightarrow\left\\{\begin{array}[]{c}Y_{\mu}=\cos\theta_{W}A_{\mu}-\sin\theta_{W}Z_{\mu}\\\
W_{\mu}^{3}=\sin\theta_{W}A_{\mu}+\cos\theta_{W}Z_{\mu}\end{array}\right..$
(12)

For the symmetric phase, we have $v=0$, and the solutions of these equations
should respect the following conditions

$\displaystyle\left\\{\begin{array}[]{l}Z^{\mu\nu}=-\sin\theta_{W}J_{ext}^{\mu\nu}\\\
A^{\mu\nu}=\cos\theta_{W}J_{ext}^{\mu\nu}\end{array}\right.\Leftrightarrow\left\\{\begin{array}[]{c}Y^{\mu\nu}=J^{\mu\nu}_{ext}\\\
(W^{3})^{\mu\nu}=0\end{array}\right..$ (13)

For the symmetry-broken phase, we have $v>0$, and from Eqs.(11a)-(11d) the
solutions should satisfy the following relations

$\displaystyle\left\\{\begin{array}[]{l}Z^{\mu\nu}=0\\\
A^{\mu\nu}=\cos\theta_{W}J_{ext}^{\mu\nu}\end{array}\right.\Leftrightarrow\left\\{\begin{array}[]{c}Y^{\mu\nu}=\cos^{2}\theta_{w}J^{\mu\nu}_{ext}\\\
(W^{3})^{\mu\nu}=\cos\theta_{w}\sin\theta J^{\mu\nu}_{ext}\end{array}\right..$
(14)

It is obvious that in either phase, the field strength of electromagnetic
field $A_{\mu\nu}$ remains constant, while the field strength of $Z$ field
$Z_{\mu\nu}$ depends upon broken or symmetric phase.

Assuming there exists a homogeneous magnetic field background, thus the Gibbs
free energy of the system in the broken and symmetric phases can be put as
Elmfors_1998

$\Omega=-L=\left\\{\begin{array}[]{lr}V_{eff}(0)-\frac{1}{2}B_{ext}^{2}&\text{symmetric
phase}\\\ V_{eff}(v)-\frac{1}{2}\cos^{2}\theta_{w}B_{ext}^{2}&\text{broken
phase}\end{array}\right.,$ (15)

where $B_{ext}$ denotes the space components of $J_{ext}$ (i.e.
$B_{ext}^{i}=\epsilon^{ijk}J^{jk}_{ext}$ and i,j,k only take space indices and
$\epsilon^{ijk}$ is the antisymmetric rank-3 tensor), and the Maxwell magnetic
field is related to it by
$B^{i}=\epsilon^{ijk}A^{jk}=\cos\theta_{W}B^{i}_{ext}$. In the following
section, instead of using hypermagnetic fields, we use magnetic fields as our
model input, since these two types of fields are simply related.

The electroweak phase transition of the universe occurs via the nucleation,
expansion and merging of EW vacuum bubbles. The number of bubbles nucleated
per time per volume can be computed and given by the following
formulaLINDE1983421 ; PhysRevD.16.1762 :

$\displaystyle\Gamma\simeq T^{4}\exp{\left(-\frac{S_{3}}{T}\right)}.$ (16)

Here $S_{3}$ is the Euclidean action of the field configuration which is a
saddle point, i.e., solution of Eqs. (11a)-(11d). Assuming the background
magnetic field is parallel to the z-axis (i.e. $\vec{B}_{ext}=B^{3}_{ext}$),
and using a cylindrical coordinate frame and the unitary gauge (i.e.
$\theta=0$), we can put Eq. (11b) and Eq. (11c) into the following forms

$\displaystyle\frac{\partial^{2}v}{\partial
r^{2}}+\frac{\partial^{2}v}{\partial z^{2}}+\frac{1}{r}\frac{\partial
v}{\partial r}-\frac{\partial V}{\partial
v}-\frac{1}{g_{Z}^{2}\,v^{3}}(\frac{\partial Z_{12}}{\partial r})^{2}$
$\displaystyle=0,$ (17) $\displaystyle v\frac{\partial^{2}Z_{12}}{\partial
r^{2}}+(\frac{v}{r}-2\frac{\partial v}{\partial r})\frac{\partial
Z_{12}}{\partial r}-\frac{g_{Z}^{2}}{2}v^{3}Z_{12}$ $\displaystyle=0.$

Meanwhile, Eq. (11a) simply reduces as the covariant Lorentz gauge. To solve
these two partial differential equations, we specify the boundary conditions,
which are given as

$\displaystyle\left.\frac{\partial v}{\partial
x_{i}}\right|_{\vec{x}=0}=0,\quad v(+\infty)$ $\displaystyle=0,$ (18)
$\displaystyle\left.\frac{\partial Z_{12}}{\partial r}\right|_{r=0}=0,\quad
Z_{12}(+\infty)$ $\displaystyle=-\sin\theta_{W}B^{3}_{ext}.$

In order to evaluate the probability of nucleation of the corresponding
bubbles, the solutions of these two differential equations (bounce solutions)
can be substituted into the Euclidean action $S_{3}$,

$S_{3}=\int dx^{3}\left[\frac{1}{2}(\nabla
v)^{2}+V_{eff}(v)+\frac{1}{2}Z_{12}^{2}+\sin\theta_{W}Z_{12}B_{ext}^{3}+\frac{1}{2}\sin\theta_{W}^{2}(B_{ext}^{3})^{2}\right].$
(19)

The last constant term in the integral is added to guarantee the Lagrangian is
zero at symmetric phase.

The nucleation temperature $T_{n}$ is defined such that at this temperature
there is at least one bubble produced within a Hubble volume. For the
electroweak transition, $T_{n}$ can be well approximated the following
equation Apreda2002GravitationalWF

$\displaystyle\left.\frac{S_{3}(T)}{T}\right|_{T=T_{n}}\simeq 140$ (20)

In the left plot of Figure 2, we show the values of $S_{3}/T$ varying of $T$
with $B=(40\textrm{GeV})^{2}$, $B=(30\textrm{GeV})^{2}$, and $B=0$,
respectively. In the right plot, we show the $\Delta S_{3}/T$ when magnetic
field is turned on. We have used $B=\cos\theta_{W}B_{ext}$ to determine the
corresponding Maxwell magnetic field. The $\Delta S_{3}/T$ is defined as

$\left.\frac{\Delta S_{3}}{T}\right.=\left.\frac{S_{3}}{T}\right|_{B\neq
0}-\left.\frac{S_{3}}{T}\right|_{B=0}\,.$ (21)

From the left plot of Figure (2), it is apparent that the background magnetic
field can affect the Euclidean action and thus affect the nucleation process.
The decrease of nucleation temperature due to the effects of the external
magnetic field is shown in Figure (3). Although the decrease is less than one
percent for $B<1000\text{GeV}^{2}$, it can have significant effect to the key
parameters of phase transition, like $\alpha$ and $\beta$.

Figure 2: The left plot shows the value of $S_{3}/T$ at different temperature.
The right plot shows the change of $S_{3}/T$, i.e. $\Delta
S_{3}/T=(S_{3}/T)_{B\neq 0}-(S_{3}/T)_{B=0}$ at different temperature. To show
these results, we have fixed other parameters as
$\lambda_{S}=1,\lambda_{HS}=3.67,m_{S}=500\text{GeV}$. Figure 3: Decrease of
nucleation temperature at different background magnetic field. We have fixed
other parameters as $\lambda_{S}=1,m_{S}=500\text{GeV}$. When magnetic field
is absent, $T_{n}$ is 95.496GeV for $\lambda_{HS}=3.6$ and $T_{n}=72.871$GeV
for $\lambda_{HS}=3.67$.

Bounce solution is the critical bubble, of which the surface tension is equal
to the pressure difference on the bubble wall. Thus it is the most likely
initial configuration. To find the bounce solution when a constant
hypermagnetic field exists, in this work we assume that the external
hypermagnetic field is weak. In the first step, we use a bounce solution of
$v^{0}$ when external field vanishes as a trial solution, then we substitute
the $v^{0}$ into the second equation of Eq. (17) to find the corresponding
$Z_{12}^{0}$. In the second step, we substitute the solution of $Z_{12}^{0}$
back into the first equation of Eq. (17) to find the corrected solution of
$\delta v$. Then we repeat the first and second steps until the solution
converges to specified precision and the procedure stops. In principle, we
should use the total $v^{t}=v^{0}+\delta v^{1}+\delta v^{2}+...$ and
$Z_{12}^{0}+Z_{12}^{1}+...$ to compute the $S_{3}$. In practice, it is found
that $\text{max}[\delta v^{1}]$ ($\text{max}[Z_{12}^{1}]$) is less than
$4\%\times v^{0}$ ($Z_{12}^{0}$) when the external magnetic field is less than
$(40\textrm{GeV})^{2}$, therefore, we only use the $v^{t}=v^{0}+\delta v^{1}$
and $Z_{12}^{0}$ to compute $S_{3}$.

One of the solution of $Z_{12}$ are present as Figure 4. Inside the bubble
$Z_{12}$ decrease to zero and outside the bubble $Z_{12}$ approaches to its
value in the false vacuum. This is consistent with the fact that in true
vacuum $Z$ boson fields obtain mass while in false vacuum $Y$ is massless.
After EW bubbles crossing over all space the EWPT ends, and the $Z$ field
vanishes everywhere and only magnetic field $A_{ij}$ survives.

Figure 4: The distribution of $Z_{12}$ around a critical bubble are shown. In
the left plot, the values of $Z_{12}$ along $z$ and $r$ directions are shown.
In the right plot, 3D values of $Z_{12}$ along the $z$ and $r$ directions are
shown where the color bar denotes the values of $Z_{12}$. To show the results,
we choose a set of parameters: $\lambda_{S}=1$, $\lambda_{HS}=3.67$,
$m_{S}=500\text{GeV}$, and $T=72.9\text{GeV}$. Figure 5: Higgs condensation
alone r direction is shown for two cases $Z_{12}=0$ and
$Z_{12}=(40\textrm{GeV})^{2}$, respectively. Other model parameters are the
same as given in Figure 4.

In Figure 5, we show the Higgs expectation value $v(r,0)$. From the first
equation of Eq. (17), we can see that the last term can be neglected when
$r\rightarrow 0$, thus $v(0,z)$ is not affected by the magnetic field and it
should equal to $v(r,0)$ when the magnetic field is zero. As the hypermagnetic
field is turned on, to balance the pressure induced by a Z field, more
pressure or less surface tension due to Higgs are needed, and thus $v(r,0)$
should change accordingly.

##  IV Magnitudes and anisotropicity of Gravitational waves

According to ref Hindmarsh2017gnf , the dominant source of GWs are sound waves
created after bubbles collision. The energy density of GWs rescaled by the
critical energy of universe is evaluated as Hindmarsh:2015qta ; Guo:2020grp

$\displaystyle h^{2}\Omega_{sw}(f)\simeq 2.65\times
10^{-6}\left(\frac{H_{n}}{\beta}\right)\left(\frac{\kappa_{v}\alpha}{1+\alpha}\right)^{2}\left(\frac{100}{g_{*}}\right)^{1/3}v_{w}\left(\frac{f}{f_{sw}}\right)^{3}\left[\frac{7}{4+3\left(\frac{f}{f_{sw}}\right)^{2}}\right]^{7/2},$
(22)

where the peak frequency $f_{sw}$ is approximated by

$\displaystyle f_{sw}$ $\displaystyle\ \simeq\ $ $\displaystyle 1.9\times
10^{-2}\frac{1}{v_{w}}\left(\frac{\beta}{H_{n}}\right)\left(\frac{T_{*}}{100\text{GeV}}\right)\left(\frac{g_{*}}{100}\right)^{1/6}\text{mHz},$
(23)

and

$\kappa_{v}\simeq\frac{\alpha}{0.73+0.083\sqrt{\alpha}+\alpha},$ (24)

is the fraction of vacuum energy that is converted to bulk motion when bubble
wall velocity $v_{w}\sim 1$ Espinosa:2010hh ; Giese:2020rtr . In order to
determine magnitudes of the stochastic gravitational waves produced by the
EWPT, we should determine two key parameters, $\alpha$ and $\beta$.

The parameter $\alpha$ describes the fraction of free energy released during
the EW phase transition transits to the energy of radiation. As it is known
that part of the radiation energy becomes gravitational waves. When the
external background magnetic field is taken into account, the parameter
$\alpha$ can be put as

$\displaystyle\alpha$
$\displaystyle=\frac{30}{g_{*}\pi^{2}T^{4}}\left(-\frac{T}{4}\frac{d}{dT}\Delta\Omega+\Delta\Omega\right)$
(25)

where $\Delta\Omega$ is the free energy released during the EW phase
transition and can be expressed as

$\Delta\Omega=V_{eff}(0)-V_{eff}(v)-\frac{1}{2}\tan^{2}\theta_{W}B^{2}\,.$
(26)

To examine how the last term affect the parameter $\alpha$, how magnetic field
evolve as temperature decrease is needed. Prior to the recombination epoch,
the magnetic field $B^{i}=|B|e^{i}=\epsilon^{ijk}A_{jk}$ ( here $|B|$ denotes
the strength of magnetic field and $e^{i}$ denotes the orientation of the
magnetic field) decays in the turbulent plasma. After recombination, the
magnetic field evolves adiabatically. Thus for maximally helical magnetic
field we can assume that the strength of magnetic field today $B_{0}$ is
related to the field strength in early universe by cite1111

$\displaystyle
B(T)=\left(\frac{a}{a_{0}}\right)^{-2}\left(\frac{\tau}{\tau_{rec}}\right)^{-1/2}B_{0}.$
(27)

Here $a$ is the scale factor and $\tau$ is the conformal time. for $B_{0}$ in
$10^{-15}$G to $10^{-11}$G, at nucleation temperature $T_{n}\simeq 72.9$GeV
the corresponding magnetic field $B(T)$ is about
$(0.31\text{GeV})^{2}\sim(31\text{GeV})^{2}$. Using Eq. (27), how the last
term in Eq. (26) affect $\alpha$ is showed in Figure 6. In Figure 7 we show
the increase of $\alpha$ at nucleation temperature. Comparing to the
$\tan^{2}\theta_{W}B^{2}$ term in Eq. (26), the decrease of nucleation
temperature has more significant effect.

Figure 6: The dependence of $\alpha$ parameter upon magnetic field $B$ is
shown. To show the results, we choose a set of parameters:
$\lambda_{S}=1,\lambda_{HS}=3.67,m_{S}=500\text{GeV},T=72.9\text{GeV}$.

The inverse duration of phase transition $\beta$ is a crucial parameter to
describe the rate of phase transition, which is given by

$\displaystyle\frac{\beta}{H_{n}}=T\left.\frac{d(S_{3}/T)}{dT}\right|_{T=T_{n}}$
(28)

where $H_{n}$ is Hubble parameter at the nucleation temperature $T_{n}$. It is
found that the external field can affect the value of $\beta/H_{n}$, As the
right plot in Figure 7 shows.

Figure 7: Left plot shows the increase of $\alpha$ at nucleation temperature
versus magnetic field. When magnetic field is absent, $\alpha$ is $0.0136$ for
$\lambda_{HS}=3.6$ and $\alpha=0.0412$ for $\lambda_{HS}=3.67$. In right plot,
$\Delta\beta/H_{n}:=\left.\beta/H_{n}\right|_{B=0}-\left.\beta/H_{n}\right|_{B\neq
0}$. When magnetic field is absent, $\beta/H_{n}$ are 671 or 133 for
$\lambda_{HS}=3.6$ and $\lambda_{HS}=3.67$, respectively. Other parameter are
the same as figure 3.

In Figures 8, we show the spectra of GWs when $B=0$ and the effects of
external magnetic fields. It is observed that the magnitude is quite dependent
upon the model parameter $\lambda_{HS}$, the maximum magnitude with
$\lambda_{HS}=3.67$ can be two orders larger than that of $\lambda_{HS}=3.60$,
as shown in the left plot. From the right plot, it is clear that when the
model parameter $\lambda_{HS}$ is fixed, the larger the strength of external
magnetic field, the larger is the magnitude of energy density of GWs. For
example, when $\lambda_{HS}=3.67$ and $B=800/\textrm{GeV}^{2}$, the magnitude
can be enhanced by $10\%$. Such an enhancement can be attributed to the
increase of $\alpha$ and $\beta/H_{n}$ shown in Figure 7.

Figure 8: Left plot is to show the energy densities of GWs with $B=0$ with two
cases $\lambda_{HS}=3.67$ and $\lambda_{HS}=3.6$. Right plot is to show the
effect of magnetic field. To show the results, we choose a set of parameters:
$\lambda_{S}=1$ and $m_{S}=500\text{GeV}$.

In the numerical analysis above, we have assumed a homogeneous magnetic field
background. But in the cosmological scale, the magnetic fields are
inhomogeneous Vachaspati:2020blt . Our study is valid only if $\lambda_{B}\gg
R_{b}$, where $\lambda_{B}$ denotes the peak length of magnetic field spectrum
and $R_{b}$ is the average distance between bubbles which can be estimated as
$(8\pi)^{1/3}v_{w}/\beta$. Below we investigate the consequence of the
inhomogeneous magnetic fields to the GWs. For the sake of convenience, we will
fix the model parameter $\lambda_{HS}=3.67$ and $m_{S}=500\textrm{GeV}$.

As it is well-known, the inhomogeneity of hypermagnetic fields necessarily
leads to the inhomogeneity of quantity $S_{3}$. If we assume that inside the
plasma everywhere has the same termperature $T$, then a place A with a
stronger hypermagnetic field can have a larger $S_{3}(A)$. In contrast, a
place B with a weaker hypermagnetic field can have a smaller $S_{3}(B)$, i.e.
$S_{3}(B)/T<S_{3}(A)/T$. Then the EWPT at the place B can occur earlier than
it does at the place A, thus the anisotropies of GWs can be generated. Similar
to its counterpart in the cosmic microwave background, anisotropies of GWs can
be decomposed in spherical harmonics:

$\delta_{\mathrm{GW}}\left(\hat{n}\right)=\frac{\Omega(\hat{n})-\bar{\Omega}}{\bar{\Omega}}=\sum_{\ell}\sum_{m=-\ell}^{\ell}\delta_{\mathrm{GW},\ell\mathrm{m}}Y_{\ell
m}(\hat{n}).$ (29)

Under the assumption of statistical isotropy, the multipole coefficients can
be defined as

$\left\langle\delta_{\mathrm{GW},\ell\mathrm{m}}\delta_{\mathrm{GW},\ell^{\prime}\mathrm{m}^{\prime}}^{*}\right\rangle=C_{\ell}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}.$
(30)

In their nature, the anisotropies of GWs come from perturbations of spacetime
and the inhomogeneous emission of GWsLISACosmologyWorkingGroup:2022kbp ;
Bartolo:2019yeu . The latter contribution to $C_{l}$ is

$C_{\ell}=4\pi\int\frac{dk}{k}\left[j_{\ell}\left(k\left(\eta_{0}-\eta_{\mathrm{in}}\right)\right)\right]^{2}P_{\delta}(k)\,,$
(31)

where $j_{l}$ is spherical Bessel function. $\eta_{in}$ is the time when GWs
produced and $\eta_{0}$ is the age of universe. The initial power spectrum of
$\delta_{GW}$ is defined by

$\left\langle\delta_{GW}\left(\eta_{\text{in
}},\vec{k}\right)\delta_{GW}^{*}\left(\eta_{\text{in
}},\vec{k}^{\prime}\right)\right\rangle=\frac{2\pi^{2}}{k^{3}}P_{\delta}(k)(2\pi)^{3}\delta\left(\vec{k}-\vec{k}^{\prime}\right)\,,$
(32)

and $\delta_{GW}(\vec{k})$ is the Fourier transformation of
$\delta_{GW}(\vec{x})$. The spectrum $P_{\delta}$ is depended on magnetic
fields. Cosmological magnetic fields are often characterized by power spectrum
and helicity spectrum, which are defined by the correlator:

$\langle
B^{i}_{\bf{k}}B^{*j}_{{\bf{k}}^{\prime}}\rangle=\left[{\frac{E_{M}(k)}{4\pi
k^{2}}}p_{ij}+i\epsilon_{ijl}k^{l}{\frac{H_{M}(k)}{8\pi
k^{2}}}\right](2\pi)^{6}\delta({\vec{k}}-{\vec{k}}^{\prime})\,,$ (33)

where $p_{ij}=\delta_{ij}-\hat{k}_{i}\hat{k}_{j}.$ $E_{M}(k)$ describe the
energy density $\rho_{B}$ of magnetic fields in k space and $H_{M}(k)$ is
related to the helicity $h$. The quantities $\rho_{B}$ and $h$ are

$\rho_{B}=\int dk\,E_{M}(k),\quad h=\int dk\,H_{M}(k).$ (34)

The two spectra obey the relationship $E_{M}(k)\geq k|H_{M}(k)|/2$ and
magnetic fields are called maximally helical fields if the equal sign is
achieved. For weak magnetic fields, we can model the anisotropies of GWs as
$\delta_{GW}(x)=CB^{2}(x)$. As shown in Figure 8, for the red line we take
$C=4.06\times 10^{-8}\text{GeV}^{-4}$, and for the blue line we take
$C=1.61\times 10^{-7}\text{GeV}^{-4}$. In Fourier space, the perturbation of
energy caused by magnetic fields can be expressed as

$\displaystyle\delta_{GW}(\vec{k})$ $\displaystyle=C\int
d^{3}xB^{2}(x)e^{i\vec{k}\cdot\vec{x}}$ (35)
$\displaystyle=C\int{d^{3}x\frac{d^{3}k_{1}}{(2\pi)^{3}}}\frac{d^{3}k_{2}}{(2\pi)^{3}}B^{i}_{\bf{k}_{1}}B^{i}_{\bf{k}_{2}}e^{-i\left(\bf{k}_{1}+{k}_{2}-{k}\right)\cdot\bf{x}}$
$\displaystyle=C\int{\frac{d^{3}k_{1}}{(2\pi)^{3}}}B^{i}_{\bf{k}_{1}}B^{i}_{\bf{k}-{k}_{1}}\,.$

Thus, the correlator of $\delta_{GW}$ can be put as

$\displaystyle\langle\delta_{GW}(\vec{k})\delta_{GW}^{*}(\vec{k^{{}^{\prime}}})\rangle$
$\displaystyle=C^{2}\int\frac{d^{3}k_{1}}{(2\pi)^{3}}\frac{d^{3}k_{2}}{(2\pi)^{3}}\langle
B^{i}_{\bf{k}_{1}}B^{i}_{\bf{k}-{k}_{1}}B^{*j}_{\bf{k}_{2}}B^{*j}_{\bf{k^{{}^{\prime}}}-{k}_{2}}\rangle$
(36)
$\displaystyle=C^{2}\int\frac{d^{3}k_{1}}{(2\pi)^{3}}\frac{d^{3}k_{2}}{(2\pi)^{3}}\left(\langle
B^{i}_{\bf{k}_{1}}B^{i}_{\bf{k}-{k}_{1}}\rangle\langle
B^{*j}_{\bf{k}_{2}}B^{*j}_{\bf{k^{{}^{\prime}}}-{k}_{2}}\rangle\right.$
$\displaystyle+\langle B^{i}_{\bf{k}_{1}}B^{*j}_{\bf{k}_{2}}\rangle\langle
B^{i}_{\bf{k}-{k}_{1}}B^{*j}_{\bf{k^{{}^{\prime}}}-{k}_{2}}\rangle+\left.\langle
B^{i}_{\bf{k}_{1}}B^{*j}_{\bf{k^{{}^{\prime}}}-{k}_{2}}\rangle\langle
B^{i}_{\bf{k}-{k}_{1}}B^{*j}_{\bf{k}_{2}}\rangle\right)$
$\displaystyle=C^{2}(2\pi)^{4}\delta({\vec{k}}-{\vec{k}}^{\prime})$
$\displaystyle\times\int
d^{3}k_{1}\left[\frac{E_{M}(k)}{k^{2}}\frac{E_{M}\left(\left|\vec{k}-\vec{k}_{1}\right|\right)}{\left|\vec{k}-\vec{k}_{1}\right|^{2}}-\frac{H_{M}(k)}{2}\frac{H_{M}\left(\left|\vec{k}-\vec{k}_{1}\right|\right)}{2\left|\vec{k}-\vec{k}_{1}\right|^{2}}\right]$
$\displaystyle=2C^{2}(2\pi)^{5}\delta({\vec{k}}-{\vec{k}}^{\prime})\left[\frac{E_{M}(k)}{k^{2}}\rho_{B}-\frac{H_{M}(k)}{4}h\right]\,.$

We have used Eq. (33) and Eq. (34) in the last two steps. Comparing with the
definition of $P_{\delta}$ given in Eq. (32), we arrive at the following
relation

$\displaystyle P_{\delta}(k)=4C^{2}kE_{M}(k)\rho_{B}-C^{2}k^{3}H_{M}(k)h.$
(37)

As shown in Figure 8, when $\lambda_{HS}=3.67$,
$\delta_{\mathrm{GW}}(\hat{n}):=\delta_{GW}(\hat{x}(\eta_{0}-\eta_{in}))$ can
change in the range $[0,0.15]$. Approximately, the power spectra of GWs
$P_{\delta}$ can be estimated as $\delta^{2}_{GW}(\vec{x})$. The detail of
$P_{\delta}$ is dependent upon the spectra of hypermagnetic field. Below, we
examine the anisotropies of GWs by considering the following three power
spectra for magnetic fields:

  * •

Scenario 1: $P_{\delta}=A^{2}$

  * •

Scenario 2: $P_{\delta}(k)=A^{2}\exp[-(k\lambda_{B}-2\pi)^{2}]$

  * •

Scenario 3: $P_{\delta}(k)=4C^{2}kE_{M}(k)\rho_{B},$

where
$E_{M}(k)=\left\\{\begin{array}[]{cc}\frac{10\rho_{B}}{17k_{B}}\left(\frac{k}{k_{B}}\right)^{4}&k<k_{B}\\\
\frac{10\rho_{B}}{17k_{B}}\left(\frac{k}{k_{B}}\right)^{-5/3}&k>k_{B}\\\
\end{array}\right.\,,\,\rho_{B}C=\frac{17}{20}A$.

Figure 9: The angular power spectra of the stochastic gravitational wave
background are shown. The blue, red and green lines correspond to scenario 1,
scenario 2, and scenario 3, respectively. To make the plot, we choose the
following parameters $A=0.1,\lambda_{B}=2\pi/k_{B}=100\text{Mpc}$.

For all three scenarios, we choose the parameters
$A=0.1,\lambda_{B}=2\pi/k_{B}=100\text{Mpc}$. In Figure 9, we show the angular
power spectra of GWs in these scenarios.

The first scenario corresponds a constant spectra of magnetic fields. If
magnetic fields were produced on large coherence scales during
inflationCampanelli:2013mea ; Ferreira:2013sqa ; Martin:2007ue , the spectra
of magnetic fields can be scale invariant, which can lead to a constant power
spectra $P_{\delta}$ in the GWs.

The second scenario corresponds to a magnetic spectrum having a peak at length
scale $\lambda_{B}$, which leads a peak of GWs at the same scale.

Due to the effects of inverse cascade, for a broad range of initial
conditions, the spectrum evolves to $E_{M}\propto k^{4}$ at small
kReppin:2017uud ; Brandenburg:2017neh . Thus, in scenario 3, we assume a a
Batchelor spectrum $E_{M}\propto k^{4}$ for large scale and a Kolmogorov
spectrum $E_{M}\propto k^{-5/3}$ for small scale. Typically, the helicity
$H_{M}$ can be either positive or negative. For the sake of simplicity, we
assume $h=0$.

For both scenario 2 and scenario 3, the angular spectrum $C_{l}$ has a peak at
$l\sim 2\pi\eta_{0}/\lambda_{B}$ or so.

##  V Discussions and conclusion

In this work, we studied the anisotropies of GWs caused by the inhomogenious
weak external hypermagnetic fields. The theory model is the SM+singlet model,
where a real scalar field is added as an extension of the SM. Such a new
physics model is necessary, in order to offer the strongly first order
electroweak phase transition, which can produce stochastic gravitational
waves. We have examined the effects of external hypermagnetic fields in the
following three steps.

At the first step, We examined the effects of the weak external hypermagnetic
field to the bounce solutions and to the electroweak phase transition. It is
found that the magnetic field can enhance the value of the 3-D Euclidean
action and consequently can decrease the nucleation temperature.

At the second step, we evaluate the GWs spectra in the random magnetic field
background. It is found that the GWs can be detected by Taiji, BBO, DECIGO for
some parameters of the theory model. Even with the assumption that the
external magnetic field is weak, it is found that the hypermagnetic field can
further enhance the GWs energy density by about ten percent if we choose
$B\sim 800\text{GeV}^{2}$ at EWPT epoch or $B_{0}\sim 8\times
10^{-12}\text{G}$ today. The main underlying reason which leads to a
significant enhancement of GWs production is due to the decrease of nucleation
temperature caused by external hypermagnetic field, which can enlarge the
parameters $\alpha$ and $H_{n}/\beta$, thus can increase $\Omega$.

At the third step, we analyzed anisotropies of GWs caused by the magnetic
field inhomogeneity. We relate the angular power spectrum of SGWB to the
spectrum of magnetic field. Assuming the enhancement of GWs energy density is
$\delta_{GW}\sim 0.1$, from the numerical comparison, we observe that the
anisotropies can be much larger than the SW and ISW effects come from the
primordial density perturbations which are about
$l(l+1)C_{l}^{SW+ISW}/2\pi\sim 10^{-9}$Li:2021iva . We also find that for
scale invariant magnetic fields, the angular power spectrum $l(l+1)C_{l}\sim
0.06$ is a constant; for Gaussian-like and random distributed magnetic fields,
the angular power spectrum $l(l+1)C_{l}$ reaches its maximum at $l\sim
2\pi\eta_{0}/\lambda_{B}$ with $l(l+1)C_{l}<0.06$. In our model, $C_{l}\propto
B^{4}$, thus it can be secondary if the primordial magnetic fields are
extremely weak. For helical hypermagnetic fields, the helicity decay can lead
to baryon asymmetry through chiral anomaly. To avoid large amplitude of baryon
isocurvature perturbations, magnetic fields at EWPT epoch should less than
$1\text{GeV}^{2}$ as Ref.Kamada:2020bmb  recommended. If we adopt the
constraint $B<1\text{GeV}^{2}$, $l(l+1)C_{l}<1.5\times 10^{-13}$, which is
negligible. Our results demonstrate that the detection of the anisotropies of
GWs may offer a handle to explore primordial magnetic fields of the universe
at the EWPT epoch.

In our study, we have assumed weak external fields in order to find the bounce
solutions. It is possible to extend our study to include the case with strong
external fields, which can be explored in our future works.

##  Appendix A Magnetic corrections to effective potential

In this work, we have neglected the higher order corrections from magnetic
field due to the weak field strength assumption. we will demonstrate the
higher order corrections are negligible in this section. Terms linear to
hypermagnetic field in the Lagrangian are

$\displaystyle L$ $\displaystyle\supset-Y_{\mu}J^{\mu}$ (38) $\displaystyle
J^{\mu}$
$\displaystyle=\frac{g_{Y}}{2}\left(i\partial^{\mu}H^{\dagger}H-iH^{\dagger}\partial^{\mu}H-\sum_{i}q_{i}\bar{\psi}_{i}\gamma^{\mu}\psi_{i}\right)$

To induce a hypermagnetic field, we should add a nontrivial external source
$J^{\mu}$, which means that the plasma in universe are not isotropic.
Motivated by the Maxwell equations $J^{\mu}=\partial_{\nu}F^{\nu\mu}$, we
rewrite the source term as $J^{\mu}_{ext}=\partial_{\nu}J_{ext}^{\nu\mu}$,
then

$\displaystyle L\supset-Y_{\mu}J_{ext}^{\mu}$
$\displaystyle=-Y_{\mu}\partial_{\nu}J^{\nu\mu}_{ext}$ (39)
$\displaystyle=-\partial_{\nu}\left(Y_{\mu}J_{ext}^{\nu\mu}\right)+\partial_{\nu}Y_{\mu}J_{ext}^{\nu\mu}$
$\displaystyle=-\partial_{\nu}\left(Y_{\mu}J_{ext}^{\nu\mu}\right)+\frac{1}{2}\left(\partial_{\nu}Y_{\mu}J^{\nu\mu}_{ext}-\partial_{\mu}Y_{\nu}J^{\nu\mu}_{ext}\right)$
$\displaystyle=-\partial_{\nu}\left(Y_{\mu}J_{ext}^{\nu\mu}\right)+\frac{1}{2}Y_{\nu\mu}J^{\nu\mu}_{ext}.$

Neglecting the first term which is a surface term, we use the second term to
induce hypermagnetic field in Eq. (8). Assuming the hypermagnetic field is
along the Z axis, we can simply set $J^{12}_{ext}=-J^{21}_{ext}=B_{ext}$, and
other components are zero.

The next order hypermagnetic field contributions come from the one loop
corrections for the effective potential. In the symmetric phase, only fermions
and Higgs couple to the external field $B_{ext}$. The propagator for scalar
isSanchez:2006tt

$\displaystyle iD_{H}(k)$ $\displaystyle=\int_{0}^{\infty}\frac{ds}{\cos
qB_{ext}s}\exp\left[is\left(k_{\|}^{2}-k_{\perp}^{2}\frac{\tan
qB_{ext}s}{qB_{ext}s}-m^{2}\right)\right]$ (40)
$\displaystyle=\frac{1}{k^{2}-m^{2}}\left(1-\frac{(qB_{ext})^{2}}{\left(k^{2}-m^{2}\right)^{2}}-\frac{2(qB_{ext})^{2}k_{\perp}^{2}}{\left(k^{2}-m^{2}\right)^{3}}+\mathcal{O}(q^{2}B_{ext}^{2})\right).$

The contribution to effective potential from scalars can be computed as

$\displaystyle V_{H}^{1}$
$\displaystyle=\frac{T}{2}\sum_{n,i}\int\frac{d^{3}k}{(2\pi)^{3}}\log
D^{-1}_{H}(\omega_{n},\vec{k};m_{i})$ (41)
$\displaystyle\simeq\frac{T}{2}\sum_{n,i}\int\frac{d^{3}k}{(2\pi)^{3}}\log(\omega_{n}^{2}+k^{2}+m_{i}^{2})$
$\displaystyle+(qB_{ext})^{2}\frac{T}{2}\sum_{n,i}\int\frac{d^{3}k}{(2\pi)^{3}}\left[\frac{1}{(\omega_{n}^{2}+k^{2}+m_{i}^{2})^{2}}-\frac{2k_{\perp}^{2}}{(\omega_{n}^{2}+k^{2}+m_{i}^{2})^{3}}\right]$
$\displaystyle=\frac{T}{2}\sum_{n,i}\int\frac{d^{3}k}{(2\pi)^{3}}\log(\omega_{n}^{2}+k^{2}+m_{i}^{2})+(qB_{ext})^{2}\frac{T}{2}\sum_{i}\int\frac{1}{3}\frac{d^{3}k}{(2\pi)^{3}}\left[\nabla_{\vec{k}}\frac{\vec{k}}{(\omega_{n}^{2}+k^{2}+m_{i}^{2})^{2}}\right]\,.$

After the renormalization to remove the UV divergence, the first term is what
we used in Eq. (3). The second term vanishes as demonstrated in Reference
Ayala:2004dx . Similarly, the contributions from fermions are given
belowSanchez:2006tt

$\displaystyle{}^{2}V_{f}^{(1)}$
$\displaystyle=2(qB_{ext})^{2}T\sum_{n}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{\omega_{n}^{2}+k_{3}^{2}+m_{i}^{2}}{\left(\omega_{n}^{2}+k^{2}+m_{i}^{2}\right)^{3}}$
(42)
$\displaystyle=\frac{2(qB_{ext})^{2}}{32\pi^{2}}\left[\frac{1}{\epsilon}-\frac{7\xi(3)}{2\pi^{2}}\frac{m_{i}^{2}}{T^{2}}+\ln\left(\frac{4\mu^{2}}{\pi
T^{2}}\right)+\gamma_{E}+\frac{2}{3}\right].$

We have used the Mellin summation methodBedingham:2000ct  to sum the Matsubara
frequencies. After the renormalization procedure, it is found that the
contributions of fermions can be put as $f\times(B_{ext})^{2}$, where the
factor $f$ is defined as $f=\frac{q^{2}}{16\pi^{2}}$, which is of size
$10^{-2}$ and can be safely omitted when compared with the term given in Eq.
(15).

###### Acknowledgements.

We thank Y.D. Chen, A.P. Huang, F.P.Huang, J. D. Shao, and D.W. Wang for
helpful discussions. This work is supported in part by the Fundamental
Research Funds for the Central Universities, National Natural Science
Foundation of China (NSFC) Grant Nos. 12235016, 12221005, 11725523 and
11735007, the Strategic Priority Research Program of Chinese Academy of
Sciences under Grant Nos XDB34030000 and XDPB15, the start-up funding from
University of Chinese Academy of Sciences (UCAS). The work of Q.S. Yan is
supported by NSFC Grant Nos. 11475180 and No. 11875260.

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