# On submaximal curves in fake projective planes

Piotr Pokora and Halszka Tutaj-Gasińska  Department of Mathematics,
Pedagogical University of Cracow, Podchora̧żych 2, PL-30-084 Kraków, Poland.
<EMAIL_ADDRESS><EMAIL_ADDRESS>Jagiellonian Univeristy, Faculty
of Mathematics and Computer Science, Łojasiewicza 6, PL-30-348 Cracow Poland
<EMAIL_ADDRESS>
###### Abstract.

The main purpose of the note is to exclude the existence of certain submaximal
curves in fake projective planes. This will lead to lower bounds on multipoint
Seshadri constants of ‘fake’ $\mathcal{O}(1)$ on fake projective planes.
Keywords: fake projective planes, submaximal curves, Seshadri constants, ample
line bundles
AMS2010: 14J29, 14C20

##  1\. Introduction

In the note, we study the existence of certain submaximal curves in the
context of the Seshadri constants for ample line bundles on fake projective
planes. Let us recall that by a fake projective plane we understand a smooth
complex projective surface of general type having the same Betti numbers as
the complex projective plane. The existence of such fake projective planes was
proved by Mumford [8], and now we know that there are exactly $50$ pairs of
fake projective planes, [3]. We would like to focus on the case of multipoint
Seshadri constants for fake projective planes. In the case of the single point
Seshadri constants, L. Di Cerbo [4] proved that these constants coincide with
the single point Seshadri constants of the complex projective plane. The key
advantage of this results is that it provides (probably) the first sharp
result on single point Seshadri constants for surfaces of general type. Here
we want to follow this path and look at the multipoint Seshadri constants for
ample line bundles on fake projective planes. Before we present our main
result, let us recall some basic facts.

######  Definition 1.1.

Let $X$ be a complex smooth projective surface and $L$ an ample line bundle.
For a point $x\in X$ the Seshadri constant of $L$ at $x$ is defined as

$\varepsilon(X,L;x)=\displaystyle{{\rm inf}_{C\subset X}}\frac{C.L}{{\rm
mult}_{x}C},$

where the infimum is taken over all irreducible and reduced curves $C$ passing
through $x\in X$.

######  Definition 1.2.

Let $X$ be a complex smooth projective surface and $L$ an ample line bundle at
$r$ points $x_{1},...,x_{r}$, then the multipoint Seshadri constant of $L$ is
defined as

$\varepsilon(X,L;x_{1},...,x_{r})=\inf\bigg{\\{}\frac{C.L}{\sum_{i=1}^{r}\text{mult}_{x_{i}}C}\bigg{\\}},$

where the infimum is taken over all irreducible and reduced curves such that
$C\cap\\{x_{1},...,x_{r}\\}\neq\emptyset$. If the points $x_{1},...,x_{r}$ are
very general, then we use the abbreviation $\varepsilon(X,L;r)$.

Let $X$ be a fake projective plane, i.e., a surface with
$b_{1}(X)=0,b_{2}(X)=1$ and $X$ is not isomorphic to
$\mathbb{P}^{2}=\mathbb{P}^{2}_{\mathbb{C}}$. Then its canonical bundle is
ample and $X$ is of general type. So a fake projective plane is nothing but a
surface of general type with $p_{g}=0$ and $c_{1}^{2}(X)=3c_{2}(X)=9$.
Furthermore, its fundamental group $\pi_{1}(X)$ is infinite. Moreover, $X$ has
Picard number $1$, and we can find an ample generator $L_{1}$ (which turns out
to be not uniquely determined) such that $L_{1}^{2}=1$.

There are not so many results on multipoint Seshadri constants for surfaces of
general type and ample line bundles (in some places the assumption on the
ampleness is weakened to big and nef line bundles). In [14], T. Szemberg
showed that if $X$ is a smooth complex projective surface of Picard number $1$
and $L$ is the ample generator of the Néron-Severi group of $X$, then

$\varepsilon(X,L;r)\geq\bigg{\lfloor}\sqrt{\frac{L^{2}}{r}}\bigg{\rfloor}.$

In the same context, one can show that if $X$ is a smooth complex projective
surface of Picard number $1$, $L$ is the ample generator of the Néron-Severi
group, and the number of points $r=k^{2}$ for some $k\in\mathbb{Z}_{\geq 1}$,
then

$\varepsilon(X,L;r)=\frac{\sqrt{L^{2}}}{k},$

so in the present paper _we are going to focus on the non-trivial case when
$r\neq k^{2}$ for some $k$_.

As it was observed by J. Roé [10], there is an interesting link between multi–
and single point Seshadri constants, so let us phrase [10, Theorem 3] in the
setting of our paper and adequate for the scope of our work. If $X$ is a fake
projective plane, $p\in X$ any point, and $r\geq 1$, then one has

$\varepsilon(X,L_{1};r)\geq\varepsilon(X,L_{1};p)\cdot\varepsilon(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1),r).$

By a result due to L. Di Cerbo [4] we know that $\varepsilon(X,L_{1};p)=1$, so
we obtain the following inequality

$\varepsilon(X,L_{1};r)\geq\varepsilon(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1),r).$

The main result of the paper, Theorem 3.1, gives us bounds for Seshadri
constants on fake projective planes. It tells us, for instance, that for
$r\in\\{2,3,5,6,7\\}$ we have strict inequality above. This stands to the
opposite to the case of single point Seshadri constants where
$\varepsilon(X,kL_{1};p)=\varepsilon(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(k);p^{\prime})$.
Our strategy to show the strict inequality is based on the fact that we are
able to exclude the existence of certain _submaximal_ curves, i.e., those
irreducible and reduced curves in fake projective planes breaking the so-
called Nagata bound. We explain in detail these notions in the next section.

The main reason for our studies on multipoint Seshadri constants for fake
projective planes is the Nagata-Biran-Szemberg Conjecture, see [2]. It claims
that the multipoint Seshadri constant of an ample line bundle $L$ at a very
general set of points is maximal when $r\geq k^{2}_{0}L^{2}$, where $k_{0}$ is
the smallest integer such that the linear system $|k_{0}L|$ contains a smooth
non-rational curve. In the case of fake projective planes we know that there
are no rational and elliptic curves on them, which means that multipoint
Seshadri constants in this case must be maximal unconditionally.

##  2\. Preliminaries

Let $x_{1},\dots,x_{r}$ be (very) general points in a surface $X$ (which is
always smooth, projective and complex). Let us recall that by Oguiso’s result,
see [9], the maximum of the multipoint Seshadri constants is achieved for a
generic set of $r$ points. That is why we use the abbreviation
$\varepsilon(X,L;r)$ for the $r$-point Seshadri constant with $r$ points in
very general position. Let $C$ be a curve on $X$ and denote
$m_{i}=\text{mult}_{x_{i}}C$. We recall some fundamental results about curves
by G. Xu [15, Lemma 1] and M. Roth [11].

######  Lemma 2.1 (Xu).

Let $X$ be a smooth complex projective surface and
$C_{U}=\\{(C_{u},x_{u})|u\in U\\}$ be a non-trivial family of pointed
irreducible and reduced curves in X such that ${\rm mult}_{x_{u}}C_{u}\geq m$
for some integer $m\geq 2$, then

$C^{2}_{u}\geq m(m-1)+1.$

######  Remark 2.2.

In our consideration, we will use the result of A. Knutsen, T. Szemberg, W.
Syzdek and also by F. Bastianelli, see [7] and [1], extending the result of
Xu, saying that for a smooth projective surface $S$ and a smooth variety $U$,
and a nontrivial family $\\{(C_{u},x_{u})\\}_{u\in U}$ where $x_{u}$ is a very
general point of $S$ and $C_{u}$ is a curve satisfying the condition
$\text{mult}_{x_{u}}C_{u}\geq m$ for every $u\in U$ and for some integer
$m\geq 2$, we have for a general curve $C$ of this family

$C^{2}\geq m(m-1)+\text{gon}(\widetilde{C}),$

where $\widetilde{C}$ denotes the normalization of $C$. As mentioned above, on
fake projective planes there are no rational or elliptic curves, so
$\text{gon}(\widetilde{C})\geq 2$, and we will use the inequality

(1) $C^{2}\geq m(m-1)+2.$

To state next useful results we need the following definition.

######  Definition 2.3.

Let $X$ be a smooth projective surface and $L$ an ample line bundle. The
Seshadri constant is said to be submaximal if the inequality holds

$\varepsilon(X,L;r)<\sqrt{\frac{L^{2}}{r}}.$

In that case a curve $C\subset X$ which computes the ratio
$\frac{C.L}{\sum_{i}{\rm mult}_{x_{i}}C}$ is called _submaximal_.

We have the following important results for submaximal curves which are proven
by M. Roth [11] (manuscript soon on arXiv).

######  Theorem 2.4.

Let $X$ be a smooth complex projective surface and $C\subset X$ a submaximal
curve with respect to a fixed $L$ and very general points
$\\{x_{1},...,x_{r}\\}$ with $r\geq 1$. Then we have the following properties:

  1. a)

At least $(r-1)$ of the $m_{i}$’s are equal.

  2. b)

If $m_{i}$ and $m_{j}$ are distinct multiplicities, then

$-D^{2}\leq(m_{i}-m_{j})^{2}<-\frac{r}{r-1}D^{2},$

where $D$ is the proper transform of $C$ under the blowing up of $X$ at
$\\{x_{1},...,x_{r}\\}$.

  3. c)

If one of the multiplicities is equal to zero, then others non-zero
multiplicities are equal to $1$ and $C^{2}=-1$.

  4. d)

If none of the multiplicities is zero and $(r,m_{r})\neq(1,1)$, then
$\sum_{i=1}^{r}m_{i}=\lceil\sqrt{rC^{2}}\rceil$.

  5. e)

In the case when all multiplicities are non-zero and additionally equal, say
to $m\geq 1$, then

$m=\bigg{\lceil}\sqrt{\frac{C^{2}}{r}}\bigg{\rceil},\quad
m-\sqrt{\frac{C^{2}}{r}}\leq\frac{1}{r},\quad\text{ and }\quad
rm^{2}-C^{2}\leq m.$

  6. f)

If none of the multiplicities are zero and one of the multiplicities is
different from the others, then

$\sum_{i=1}^{r}m_{i}-\sqrt{rC^{2}}<\frac{1}{r}.$

######  Remark 2.5.

It is worth pointing out that Theorem 2.4 a) is a nice generalization of
Corollary 4.6 from Szemberg’s Habilitation Thesis [13] where he was
considering polarized surfaces $(X,L)$ having Picard number $\rho(X)=1$.

######  Remark 2.6.

It is worth emphasizing that Theorem 2.4 a) was also proved by Harbourne and
Roé in [6, Corrolary 2.2.2].

Now we focus on fake projective planes.

######  Remark 2.7.

For a fake projective plane $X$, we have $Pic(X)\simeq H^{2}(X;\mathbb{Z})$.
Moreover, the torsion part of $H^{2}(X;\mathbb{Z})$ is equal to
$H_{1}(X;\mathbb{Z})$ which is never vanishing.

######  Remark 2.8.

For a fake projective plane $X$, we denote by $L_{1}$ any ample generator of
$Pic(X)$ modulo torsion. Let us emphasize that $L_{1}$ is not uniquely
determined.

Here we are going to find lower bounds on the multipoint Seshadri constant for
fake projective planes and a specially chosen line bundle, namely $L_{1}$. Let
us recall that $L_{1}$, being ample, has $L_{1}^{2}=1$, but it is an open
problem to determine whether in general one has $h^{0}(X,L_{1})>0$. Here we
are not going to approach this question, but it seems to be very interesting
to check whether $L_{1}$ might be effective due to important implications –
please consult [5, Remark 3.9].

##  3\. Main Result

The bound on the Seshadri constants in $r$ general points on fake projective
plane is given by the following theorem.

######  Theorem 3.1.

Let $X$ be a fake projective plane and denote by $L_{1}$ an ample generator of
the Néron-Severi group and assume that $r$ is not a square. Then

$\varepsilon(X,L_{1};r)\geq\frac{1}{\sqrt{r}+\delta(r)},$

where

$\delta(2)=0.031,\quad\delta(3)=0.018,\quad\delta(5)=0.014,$
$\delta(6)=0.022,\quad\delta(7)=0.011,\quad\delta(8)=0.012,$

and

$\delta(r)=0.013\text{ for }r\geq 10.$

If $r=s^{2}$ for $s\in\mathbb{Z}_{>0}$, then

$\varepsilon(X,L_{1};r)=\frac{1}{s}.$

Before we present our proof of the Theorem, let us make a few remarks.

######  Remark 3.2.

  * 1)

For $\mathbb{P}^{2}$ we have the following lower bound [12, Theorem 2.3]:

$\varepsilon(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1);r)\geq\frac{\sqrt{49r+8}}{7r+1}$

provided that $r\geq 10$. Thus, combining [10, Theorem 3] with the above lower
bound we get

$\varepsilon({\rm FPP},L_{1};r)\geq\frac{\sqrt{49r+8}}{7r+1},$

provided that $r\geq 10$.

  * 2)

It is easy to check that

$\frac{1}{\sqrt{r}+0.013}>\frac{\sqrt{49r+8}}{7r+1}$

for $10\leq r\leq 22,$ so our bound is better in this range.

  * 3)

Analysing the proof of our theorem, the reader may notice that changing the
range of $r$ to improve the bound, e.g. for $23\leq r$ we have

$\varepsilon({\rm FPP},L_{1};r)\geq\frac{1}{\sqrt{r}+0.010},$

so having $r$ big enough (and a strong enough computer) it may be possible to
find really good bounds for $\varepsilon({\rm FPP},L_{1};r)$.

######  Remark 3.3.

Here we would like to compare multipoint Seshadri constants for fake
projective planes and the projective plane.

$r$ | $\varepsilon(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1);r)$ | $\varepsilon({\rm FPP},L_{1};r)$
---|---|---
| Exact value |  Bound from Thm 3.1
$2$ | $1/2$ | $>0.69$
$3$ | $1/2$ | $\geq 0.5701$
$4$ | $1/2$ | $1/2$
$5$ | $2/5$ | $\geq 0.44$
$6$ | $2/5$ | $\geq 0.4046$
$7$ | $\frac{3}{8}=0.375$ | $\geq 0.3763$
$8$ | $\frac{6}{17}\approx 0.3529$ | $\geq 0.3391$
$9$ | $1/3$ | $1/3$
|  Bound from Remark 3.2 |  Bound from Thm 3.1
$10$ | $\geq 0.3143$ | $\geq 0.3149$
$11$ | $\geq 0.2998$ | $\geq 0.3003$
$12$ | $\geq 0.2872$ | $\geq 0.2876$
$13$ | $\geq 0.2760$ | $\geq 0.2763$
$14$ | $\geq 0.2661$ | $\geq 0.2663$
$15$ | $\geq 0.2571$ | $\geq 0.2573$
$16$ | $1/4$ | $1/4$

Now we present the proof of Theorem 3.1.

###### Proof.

The idea of the proof goes as follows. Suppose we have a submaximal curve
passing in fake projective plane $X$ through $r$ points with multiplicities
$m$ (in $r-1$ points) and $M$ (in one point) – it is also possible that $M=m$.
Using Theorem 2.4 we check that the inequality
$\frac{k}{(r-1)m+M}\geq\frac{1}{\sqrt{r}+\delta(r)}$ is satisfied for $k$ big
enough. The remaining cases (small $k$) are excluded with the help of the Xu-
type statement (1).

Let us now go into details. Suppose there exists a submaximal curve
$C\in|kL_{1}|$ passing through general points $x_{1}\dots x_{r}$ with
multiplicities $m$ (in $r-1$ points) and $M$ (in one point). Using M. Roth’s
result, i.e., Theorem 2.4 we have that every submaximal curve must satisfy

$\frac{C.L_{1}}{(r-1)m+M}\geq\frac{k}{k\sqrt{r}+\frac{1}{r}}$

provided that $M\neq m$, and if $m=M$ we have

$\frac{C.L_{1}}{rm}\geq\frac{k}{k\sqrt{r}+\frac{1}{2}}.$

In both cases we have

$\frac{C.L_{1}}{(r-1)m+M}\geq\frac{k}{k\sqrt{r}+\frac{1}{2}}.$

Next, we check when

$\frac{k}{k\sqrt{r}+\frac{1}{2}}\geq\frac{1}{\sqrt{r}+\delta(r)}.$

This may be checked of course only for the smallest considered $\delta$, so we
need to check when

$\frac{k}{k\sqrt{r}+\frac{1}{2}}\geq\frac{1}{\sqrt{r}+0.01}.$

The inequality is satisfied for $k\geq 50$, so this means that all submaximal
curves violating the bound $\frac{1}{\sqrt{r}+\delta(r)}$, if they exist, must
be in $|kL_{1}|$ with $k\leq 49$.

Firstly, observe that a submaximal curve must have at least one multiplicity
greater than $1$. Indeed, if all points have multiplicity one, the conditions
imposed by them are independent, thus computing $h^{0}(kL_{1})$ and imposing
$r$ conditions we get that

$k\geq\frac{3+\sqrt{1+8r}}{2}$

whereas the submaximality gives

$\frac{k}{r}<\frac{1}{\sqrt{r}},$

a contradiction. Thus, at least one multiplicity of a submaximal $C$ is
greater or equal $2$. So, from Inequality (1), every submaximal curve $C$ must
satisfy one of the following inequalities:

$f_{1}(k,r,m):=rm^{2}-m+2-k^{2}\leq 0\text{ if only $M=m$,}$

or

$f_{2}(k,r,m,M):=(r-1)m^{2}+M^{2}-M+2-k^{2}\leq 0\text{ if only $1<M<m$,}$

or

$f_{3}(k,r,m,M):=(r-1)m^{2}+M^{2}-m+2-k^{2}\leq 0\text{ if only $1<m<M$,}$

or

$f_{4}(k,r,m):=(r-1)m^{2}+1-m+2-k^{2}\leq 0\text{ if $m_{r}=1,m_{1}=\dots
m_{r-1}=m>1$,}$

or

$f_{5}(k,r,m):=(r-1)+m^{2}-m+2-k^{2}\leq 0\text{ if $m_{1}>1,m_{2}=\dots
m_{r}=1$.}$

We only have to check that if $r\geq 2$ and $k\leq 49$ there are no positive
integer solutions (for a given $k$, $m,M$, and $r$) for the systems of
inequalities given below. If $m=M$, then

$f_{1}\leq 0,\ \frac{k}{rm}<\frac{1}{\sqrt{r}+\delta(r)}.$

If $M<m$, then

$f_{2}\leq 0,\ \frac{k}{(r-1)m+M}<\frac{1}{\sqrt{r}+\delta(r)}.$

In the case when $m<M$:

$f_{3}\leq 0,\ \frac{k}{(r-1)m+M}<\frac{1}{\sqrt{r}+\delta(r)}.$

Finally, when one multiplicity is $1$ and the others are greater than $1$:

$f_{4}\leq 0,\ \frac{k}{(r-1)m+1}<\frac{1}{\sqrt{r}+\delta(r)}$

and when all but one multiplicities are $1$:

$f_{5}\leq 0,\ \frac{k}{(r-1)+m}<\frac{1}{\sqrt{r}+\delta(r)}$

This may be done by hand, or, preferably with help of a computer program, e.g.
MATHEMATICA. ∎

######  Remark 3.4.

In order to get a better bound on $\varepsilon(X,L_{1};2)$, namely

$\varepsilon(X,L_{1};2)>0.7$

we would have to exclude, for example, the existence of $C\in|7L_{1}|$ having
exactly two singular points of multiplicity $m=5$, what seems to be not
obvious.

## Acknowledgments

We would like to thank L. Borisov, L. Di Cerbo, and M. Stover for discussions
on fake projective planes, and to Joaquim Roé for useful suggestions.
Additionally, we would like to warmly thank Mike Roth for sharing with us his
manuscript and very useful discussion on the subject of the present note. The
first author is partially supported by the National Science Center (Poland)
Sonata Grant Nr 2018/31/D/ST1/00177.

## References

  * [1] F. Bastianelli, Remarks on the nef cone on symmetric products of curves. Manuscripta Math. 130(1): 113 – 120 (2009).
  * [2] Th. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. L. Knutsen, and W. Syzdek, A primer on Seshadri constants. _Contemporary Mathematics_ 496: 33 – 70 (2009).
  * [3] D. I. Cartwright and T. Steger, Enumeration of the 50 fake projective planes. C. R., Math., Acad. Sci. Paris 348, No. 1-2: 11 – 13 (2010).
  * [4] L. Di Cerbo, The Toledo invariant, and Seshadri constants of fake projective planes. J. Math. Soc. Japan 69(4): 1601 – 1610 (2017).
  * [5] G. Di Brino and L. Di Cerbo, Exceptional collections and the bicanonical map of Keum’s fake projective planes. Commun. Contemp. Math. 20(1): #1650066 (2018).
  * [6] B. Harbourne and J. Roé, Discrete behavior of Seshadri constants on surfaces. J. Pure Appl. Algebra 212(3): 616 – 627 (2008).
  * [7] A. L. Knutsen, W. Syzdek, T. Szemberg, Moving curves and Seshadri constants. Math. Res. Lett. 16 no. 4: 711 – 719 (2009).
  * [8] D. Mumford, An algebraic surface with $K$ ample, $(K^{2})=9$, $p_{g}=q=0$. Amer. J. Math. 101: 233 – 244 (1979).
  * [9] K. Oguiso, Seshadri constants in a family of surfaces. Math. Ann. 323(4): 625 – 631 (2002).
  * [10] J. Roé, A relation between one-point and multi-point Seshadri constant. J. Algebra 274: 643 – 651 (2004).
  * [11] M. Roth, Numerical restrictions on curves classes breaking the Nagata bound. Unpublished manuscript.
  * [12] B. Strycharz-Szemberg and T. Szemberg, Remarks on the Nagata conjecture. Serdica Math. J. 30, No. 2-3: 405 – 430 (2004).
  * [13] T. Szemberg, Global and local positivity of line bundles. Habilitationsschrift, Universität Duisburg-Essen, 2001.
  * [14] T. Szemberg, An effective and sharp lower bound on Seshadri constants on surfaces with Picard number $1$. J. Algebra 319: 3112 – 3119 (2008).
  * [15] G. Xu, Ample line bundles on smooth surfaces. J. Reine Angew. Math. 469: 199 – 209 (1995).
  * [16] Wolfram Research, Inc., Mathematica, Version 12.0, Champaign, IL (2019).


# Gravitational waves generation in turbulent hypermagnetic fields before the
electroweak phase transition

Maxim Dvornikov

###### Abstract

We study the production of relic gravitational waves (GWs) in turbulent
hypermagnetic fields (HMFs) in the symmetric phase of the early universe
before the electroweak phase transition (EWPT). The noise of HMFs is modeled
by the analog of the magnetic hydrodynamics turbulence. The evolution of HMFs
is driven by the analogs of the chiral magnetic effect and the Adler anomalies
in the presence of the nonzero asymmetries of leptons and Higgs bosons. We
track the evolution of the energy density of GWs from $10\,\text{TeV}$ down to
EWPT and analyze its dependence on the parameters of the system. We also
discuss the possibility to observe the predicted GW background by the current
GW detectors.

##  1 Introduction

The observation of gravitational waves (GWs) by the LIGO-Virgo collaborations
[1] was one of the direct confirmations of the validity of the General
Relativity. Since then, the records of the numerous sources of GWs have been
collected [2]. There are vast plans [3] to build new GW detectors in addition
to the currently operating ones.

Besides the GW signal emitted by certain astrophysical objects, there should
be stochastic GWs filling all the universe [4]. The detection of stochastic
GWs is quite challenging because of the rather low frequencies of the signal.
Some of the detection methods are reviewed in ref. [5].

There are numerous sources of the stochastic GW background. They can be of the
astrophysical origin [6] or stem from the early universe [7]. Even for the
cosmological GW background, one can point out on multiple sources. Relic GWs
can be produced in the inflation epoch, during various phase transitions, by
cosmic strings etc. Many of the mechanisms are reviewed in ref. [7].

We are interested in the generation of primordial GWs by turbulent magnetic
fields (see, e.g., ref. [8]). However, instead of dealing with a magnetic
field, we shall treat the production of GWs before the electroweak phase
transition (EWPT). In this case, instead of a photon $A^{\mu}$, one has the
massless hypercharge field
$Y^{\mu}=\sin\theta_{\mathrm{W}}Z^{\mu}+\cos\theta_{\mathrm{W}}A^{\mu}$, where
$Z^{\mu}$ is the $Z$-boson and $\theta_{\mathrm{W}}$ is the Weinberg angle.
Thus, we replace the electromagnetic field tensor with
$F_{\mu\nu}\to(\mathbf{E}_{\mathrm{Y}},\mathbf{B}_{\mathrm{Y}})$, where
$\mathbf{E}_{\mathrm{Y}}$ and $\mathbf{B}_{\mathrm{Y}}$ are hyperelectric and
hypermagnetic fields (HMF).

All leptons can be treated as massless particles before EWPT. Moreover, one
can have nonzero lepton asymmetries $n_{f}\neq n_{\bar{f}}$ in this epoch.
Thus, we can apply the analog of the chiral magnetic effect (the CME) [9] for
the description of the HMF evolution. The CME consists in the excitation of an
electric current of massless particles, forming plasma with a nonzero chiral
imbalance, along an external magnetic field. The magnetic field in this
situation becomes unstable. There is a backreaction from the helical magnetic
field to the particle asymmetry resulting from the Adler anomaly (see, e.g.,
ref. [10]). It leads to the decay of a helical magnetic field and the
production a nonzero asymmetry of massless fermions. For the first time, this
scenario was used for the leptogenesis in the early universe in ref. [11].

The mechanism, which involves the CME and the Adler anomaly, was used mainly
in connection to the lepto- and baryogenesis in (hyper-)magnetic fields in the
early universe (see, e.g., refs. [12, 13]). However, we can treat evolving
HMFs as a source of primordial GWs. The production of GWs driven by unstable
magnetic fields accounting for the chiral anomalies was considered recently in
refs. [14, 15].

In ref. [16], we studied the influence of the HMF turbulence on the lepto- and
baryogenesis in the early universe before EWPT. The noise of HMFs was modeled
by the hypermagnetic hydrodynamics [(H)MHD] turbulence. In this approach, the
plasma motion in the Navier-Stokes equation was driven mainly by the Lorentz
force [17]. To calculate the baryon asymmetry of the universe (BAU) we were
interested in the evolution of the lepton asymmetries. Now, if we apply the
model in ref. [16] to describe the GWs production by the HMF turbulence, we
will deal with the evolution of the spectra of the HMF energy density and
helicity.

This work is organized in the following way. In section 2, we rederive the
energy density of GWs and its spectrum in turbulent HMFs in the symmetric
phase of the early universe before EWPT. We summarize the dynamics of HMFs, as
well as the lepton and Higgs boson asymmetries in section 3. The results of
the numerical simulations are present in section 4. The possibility to observe
predicted relic GWs with the modern GWs detectors is considered in section
4.1. We conclude in section 5. The main kinetic equations are rewritten in the
form convenient for numerical simulations in appendix A.

##  2 Production of GWs

We consider the tensor perturbation $h_{\mu\nu}$ of the background
Friedmann–Robertson–Walker metric
$\bar{g}_{\mu\nu}=\text{diag}(1,-a^{2},-a^{2},-a^{2})$, where $a=a(t)$ is the
scale factor. The interval reads now

$\mathrm{d}s^{2}=g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu},\quad
g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu},$ (2.1)

where we use the physical coordinates $x^{\mu}=(t,\mathbf{x})$. We can express
the spatial part of $h_{\mu\nu}$ as $h_{ij}=a^{2}D_{ij}$, where the matrix
$D_{ij}$ is supposed to vary harmonically $D_{ij}(\mathbf{x},t)\sim
e^{\mathrm{i}\mathbf{kx}}D_{ij}(\mathbf{k},t)$. Here
$\mathbf{k}=a\mathbf{k}_{\mathrm{phys}}$ is the conformal momentum. Using the
transverse-traceless gauge, in which $D_{ii}=0$ and $k_{i}D_{ij}=0$, one gets
that $D_{ij}$ obeys the equation [18],

$-a^{2}\ddot{D}_{ij}-3a\dot{a}\dot{D}_{ij}+\nabla^{2}D_{ij}=16\pi
G\pi_{ij}^{(\mathrm{T})},$ (2.2)

where $\pi_{ij}^{(\mathrm{T})}=P_{ikjl}T_{kl}$, $T_{kl}$ is the energy-
momentum tensor, $P_{ikjl}=P_{ik}P_{jl}-\tfrac{1}{2}P_{ij}P_{kl}$,
$P_{ij}=\delta_{ij}-\hat{k}_{i}\hat{k}_{j}$, $\hat{\mathbf{k}}=\mathbf{k}/k$
is the unit vector, $G=M_{\mathrm{Pl}}^{-2}$ is the Newton constant, and
$M_{\mathrm{Pl}}=1.2\times 10^{19}\,\text{GeV}$ is the Planck mass. We can see
that $k_{i}\pi_{ij}^{(\mathrm{T})}=0$ and $\pi_{ii}^{(\mathrm{T})}=0$.

We suppose that metric perturbations are caused by the noise of HMF in the
symmetric phase before EWPT. In this case, $T_{\mu\nu}$ reads

$T_{\mu\nu}=\frac{1}{4}\bar{g}_{\mu\nu}F_{\alpha\beta}^{(\mathrm{Y})}F_{\mathrm{Y}}^{\alpha\beta}-F_{\mu\alpha}^{(\mathrm{Y})}F_{\nu\beta}^{(\mathrm{Y})}\bar{g}^{\alpha\beta},$
(2.3)

where $F_{\mu\nu}^{(\mathrm{Y})}=\partial_{\mu}Y_{\nu}-\partial_{\nu}Y_{\mu}$
is the tensor of the hyperfield potential $Y^{\mu}$. Considering length scales
greater than the Debye radius, we get that the hyperelectric field is screened
effectively. The energy-momentum tensor takes the form,

$T_{ij}=-\frac{1}{a^{2}}\left(B_{\mathrm{Y}i}^{(c)}B_{\mathrm{Y}j}^{(c)}-\frac{1}{2}\delta_{ij}B_{\mathrm{Y}}^{(c)2}\right),$
(2.4)

where $\mathbf{B}_{\mathrm{Y}}^{(c)}=a^{2}\mathbf{B}_{\mathrm{Y}}$ is the
conformal HMF.

Using eq. (2.4), we rewrite eq. (2.2) in the form,

$D_{ij}^{\prime\prime}+2HD_{ij}^{\prime}+k^{2}D_{ij}=f_{ij},\quad
f_{ij}=\frac{16\pi
G}{a^{2}}P_{ikjl}\left(B_{\mathrm{Y}k}^{(c)}B_{\mathrm{Y}l}^{(c)}-\frac{1}{2}\delta_{kl}B_{\mathrm{Y}}^{(c)2}\right),$
(2.5)

where $H=a^{\prime}/a$ is the Hubble constant and the prime means the
derivative with respect to the conformal time $\eta$ defined by
$\mathrm{d}\eta=\mathrm{d}t/a$. Since we are at the radiation dominated
universe, we can choose $a(t)=\sqrt{t/t_{\text{Univ}}}$, where
$t_{\text{Univ}}=1.4\times 10^{10}\,\text{yr}$ is the age of the
Universe.111Strictly speaking, the choice of the scale factor is not exact
since there is a matter dominated universe presently. The current scale factor
reads $a_{\mathrm{matt}}\propto t^{2/3}$ which modifies the normalization
constant in $a_{\mathrm{rad}}\propto\sqrt{t}$. It corresponds to the scale
factor $a_{\text{now}}=a(t=t_{\text{Univ}})=1$ nowadays. The conformal time is
chosen such that $\eta(t)=2\sqrt{t_{\text{Univ}}}(\sqrt{t}-\sqrt{t_{*}})$,
where $t_{*}$ is the initial time when perturbations start growing. Note that
$\eta(t_{*})=0$. Equation (2.5), rewritten using new variables, takes the
form,

$D_{ij}^{\prime\prime}+\frac{2}{\eta+\eta_{0}}D_{ij}^{\prime}+k^{2}D_{ij}=f_{ij}(\mathbf{k},\eta),$
(2.6)

where $\eta_{0}=2\sqrt{t_{\text{Univ}}t_{*}}$. We supply eq. (2.6) with the
initial condition $D_{ij}(\mathbf{k},0)=D^{\prime}_{ij}(\mathbf{k},0)=0$. Note
that $\eta+\eta_{0}=2t_{\text{Univ}}a(\eta)$. The general solution, satisfying
the chosen initial condition, reads

$D_{ij}(\eta)=\frac{1}{ka(\eta)}\int_{0}^{\eta}\frac{\sin(k[\eta-\eta_{1}])}{a(\eta_{1})}f_{ij}^{(c)}(\mathbf{k},\eta_{1})\mathrm{d}\eta_{1},$
(2.7)

where $f_{ij}^{(c)}=a^{2}f_{ij}$.

Equation (2.7) is used to compute the energy density of GWs
$\rho_{\mathrm{GW}}$ which is the time component $\rho_{\mathrm{GW}}=t_{00}$
of the effective energy-momentum tensor of GW [19], $t_{\mu\nu}=\frac{1}{32\pi
G}\left\langle\partial_{\mu}h_{\alpha\beta}\partial_{\nu}h^{\alpha\beta}\right\rangle$.
The conformal energy density reads

$\rho_{\mathrm{GW}}^{(c)}=a^{4}\rho_{\mathrm{GW}}=\frac{a^{2}}{32\pi
G}\left\langle\frac{\mathrm{d}}{\mathrm{d}\eta}(a^{2}D_{ij})\cdot\frac{\mathrm{d}}{\mathrm{d}\eta}(a^{-2}D_{ij})\right\rangle.$
(2.8)

Assuming that the time scale of a perturbation is shorter than the expansion
rate of the universe, we can write down the spectral density of GWs as

$\displaystyle\rho_{\mathrm{GW}}^{(c)}(\mathbf{k},\eta)=$
$\displaystyle\delta^{3}(\mathbf{k}-\mathbf{k}^{\prime})\frac{1}{8\pi
G}\int_{0}^{\eta}\int_{0}^{\eta}\frac{\mathrm{d}\eta_{1}\mathrm{d}\eta_{2}}{a(\eta_{1})a(\eta_{2})}$
$\displaystyle\times\left\langle\cos(k[\eta-\eta_{1}])\cos(k[\eta-\eta_{2}])f_{ij}^{(c)}(\mathbf{k},\eta_{1})f_{ij}^{(c)*}(\mathbf{k}^{\prime},\eta_{2})\right\rangle.$
(2.9)

We change the variables in the double integral in eq. (2) to
$\eta_{1}=\xi+\xi^{\prime}$ and $\eta_{2}=\xi$. We also separate the averaging
between cosines and $f_{ij}^{(c)}$. The integral in eq. (2) transforms to

$\int_{0}^{\eta}d\xi\int_{-\xi}^{\eta-\xi}\frac{\mathrm{d}\xi^{\prime}}{a(\xi+\xi^{\prime})a(\xi)}\left\langle\cos(k[\eta-\xi-\xi^{\prime}])\cos(k[\eta-\xi])\right\rangle\\\
\times\left\langle
f_{ij}^{(c)}(\mathbf{k},\xi+\xi^{\prime})f_{ij}^{(c)*}(\mathbf{k}^{\prime},\xi)\right\rangle.$
(2.10)

Unlike ref. [8], where the generation of GWs was studied during a relatively
short time after a phase transition, we neglect the variable $\xi^{\prime}$ in
the inner integrand in eq. (2.10) since we study the production of GWs in a
long time interval. The average of the cosines factor is
$\left\langle\cos^{2}(k[\eta-\xi])\right\rangle=1/2$. Eventually, the spectral
density of GWs takes the form,

$\rho_{\mathrm{GW}}^{(c)}(\mathbf{k},\eta)=\delta^{3}(\mathbf{k}-\mathbf{k}^{\prime})\frac{\eta}{16\pi
G}\int_{0}^{\eta}\frac{\mathrm{d}\xi}{a^{2}(\xi)}\left\langle
f_{ij}^{(c)}(\mathbf{k},\xi)f_{ij}^{(c)*}(\mathbf{k}^{\prime},\xi)\right\rangle.$
(2.11)

Note that $\rho_{\mathrm{GW}}^{(c)}(\mathbf{k},\eta)$ is proportional to
$\eta$ as in ref. [8].

The function $f_{ij}^{(c)}$ is the binary combination of the hypermagnetic
components. To compute $f_{ij}^{(c)}$, we should, first, determine
$\left\langle
B_{\mathrm{Y}i}^{(c)}(\mathbf{k},\eta)B_{\mathrm{Y}j}^{(c)*}(\mathbf{k}^{\prime},\eta)\right\rangle$.
We take that [20, 21]

$\left\langle
B_{\mathrm{Y}i}^{(c)}(\mathbf{k},\eta)B_{\mathrm{Y}j}^{(c)*}(\mathbf{k}^{\prime},\eta)\right\rangle=(2\pi)^{3}\delta^{3}(\mathbf{k}-\mathbf{k}^{\prime})\frac{1}{2}\left[(\delta_{ij}-\hat{k}_{i}\hat{k}_{j})S_{\mathrm{Y}}(k,\eta)-\mathrm{i}\varepsilon_{ijn}\hat{k}_{n}A_{\mathrm{Y}}(k,\eta)\right],$
(2.12)

where
$S_{\mathrm{Y}}(k,\eta)=(2\pi)^{2}\rho_{\mathrm{Y}}^{(c)}(k,\eta)/k^{2}$,
$A_{\mathrm{Y}}(k,\xi)=2\pi{}^{2}h_{\mathrm{Y}}^{(c)}(k,\eta)/k$,
$\rho_{\mathrm{Y}}^{(c)}(k,\eta)$ is the spectrum of the hypermagnetic energy
density, and $h_{\mathrm{Y}}^{(c)}(k,\eta)$ is the spectrum of hypermagnetic
helicity density. Both $\rho_{\mathrm{Y}}^{(c)}(k,\eta)$ and
$h_{\mathrm{Y}}^{(c)}(k,\eta)$ are defined in conformal variables. The total
hypermagnetic energy and the hypermagnetic helicity are given by
$\rho_{\mathrm{Y}}^{(c)}(\eta)\equiv
B_{\mathrm{Y}}^{(c)2}/2=\smallint\mathrm{d}k\rho_{\mathrm{Y}}^{(c)}(k,\eta)$
and
$h_{\mathrm{Y}}^{(c)}(\eta)\equiv\tfrac{1}{V}\smallint\mathrm{d}^{3}x(\mathbf{Y}^{(c)}\cdot\mathbf{B}_{\mathrm{Y}}^{(c)})=\smallint\mathrm{d}kh_{\mathrm{Y}}^{(c)}(k,\eta)$.

Using eq. (2.12), after lengthy but straightforward calculations, we get that

$\displaystyle\left\langle
f_{ij}^{(c)}(\mathbf{k},\xi)f_{ij}^{(c)*}(\mathbf{k}^{\prime},\xi)\right\rangle=$
$\displaystyle\delta(\mathbf{k}-\mathbf{k}^{\prime})\frac{32G^{2}}{\pi}$
$\displaystyle\times\int\frac{\mathrm{d}^{3}q}{(2\pi)^{3}}\big{[}(1+\cos^{2}\alpha)(1+\cos^{2}\beta)S_{\mathrm{Y}}(|\mathbf{q}|,\xi)S_{\mathrm{Y}}(|\mathbf{k}-\mathbf{q}|,\xi)$
$\displaystyle+4\cos\alpha\cos\beta
A_{\mathrm{Y}}(|\mathbf{q}|,\xi)A_{\mathrm{Y}}(|\mathbf{k}-\mathbf{q}|,\xi)\big{]},$
(2.13)

where $\cos\alpha=(\hat{k}\cdot\hat{q})$ and
$\cos\beta=(\hat{k}\cdot\widehat{k-q})$. Finally, with help of eqs. (2) and
(2), we obtain the expression for the spectral density of GWs in the form,

$\displaystyle\rho_{\mathrm{GW}}^{(c)}(k,\eta)=$
$\displaystyle\frac{t_{\text{Univ}}^{2}G}{4k^{3}\pi^{2}}\eta\int_{0}^{\eta}\frac{\mathrm{d}\xi}{(\eta_{0}+\xi)^{2}}\int_{0}^{\infty}\frac{\mathrm{d}q}{q^{3}}\int_{|k-q|}^{k+q}\frac{\mathrm{d}p}{p^{3}}$
$\displaystyle\times\big{\\{}[4k^{2}q^{2}+(k^{2}+q^{2}-p^{2})^{2}][4k^{2}p^{2}+(k^{2}-q^{2}+p^{2})^{2}]\rho_{\mathrm{Y}}^{(c)}(q,\xi)\rho_{\mathrm{Y}}^{(c)}(p,\xi)$
$\displaystyle+4k^{2}q^{2}p^{2}(k^{2}+q^{2}-p^{2})(k^{2}-q^{2}+p^{2})h_{\mathrm{Y}}^{(c)}(q,\xi)h_{\mathrm{Y}}^{(c)}(p,\xi)\big{\\}}.$
(2.14)

The total conformal energy density can be obtained using eq. (2) as

$\rho_{\mathrm{GW}}^{(c)}(\eta)=\int_{0}^{\infty}\rho_{\mathrm{GW}}^{(c)}(k,\eta)\mathrm{d}k.$
(2.15)

We use the assumption of the spectra isotropy to derive eqs. (2) and (2.15).

To produce relic GWs, we use random HMFs with the spectrum in the range
$k_{\mathrm{min}}<k<k_{\mathrm{max}}$ (see section 3 below). In this case,
basing on eq. (2), one gets that spectrum of GWs produced is in the interval
$0<k<2k_{\mathrm{max}}$.

##  3 HMFs evolution

According to eq. (2), the GWs generation is owing to the presence of the
nonzero $\rho_{\mathrm{Y}}^{(c)}(k,\xi)$ and $h_{\mathrm{Y}}^{(c)}(k,\xi)$. We
have to choose the appropriate model for the evolution of HMFs.

In the symmetric phase of the universe evolution before EWPT, the behavior of
HMFs can be driven by the analog of the CME in the presence of nonzero lepton
asymmetries. This model was studied in details in ref. [16] in connection to
the BAU problem. The full set of the kinetic equations reads,

$\displaystyle\frac{\partial\tilde{\mathcal{E}}_{{\rm
B_{\mathrm{Y}}}}}{\partial\tilde{\eta}}$
$\displaystyle=-2\tilde{k}^{2}\eta_{\mathrm{eff}}\tilde{\mathcal{E}}_{{\rm
B_{\mathrm{Y}}}}+\alpha_{+}\tilde{k}^{2}\tilde{\mathcal{H}}_{{\rm
B_{\mathrm{Y}}}},$ $\displaystyle\frac{\partial\tilde{\mathcal{H}}_{{\rm
B_{\mathrm{Y}}}}}{\partial\tilde{\eta}}$
$\displaystyle=-2\tilde{k}^{2}\eta_{\mathrm{eff}}\tilde{\mathcal{H}}_{{\rm
B_{\mathrm{Y}}}}+4\alpha_{-}\tilde{\mathcal{E}}_{{\rm B_{\mathrm{Y}}}},$
$\displaystyle\frac{\mathrm{d}\xi_{e\mathrm{R}}}{\mathrm{d}\tilde{\eta}}$
$\displaystyle=-\frac{3\alpha^{\prime}}{\pi}\int_{\tilde{k}_{\mathrm{min}}}^{\tilde{k}_{\mathrm{max}}}\mathrm{d}\tilde{k}\frac{\partial\tilde{\mathcal{H}}_{{\rm
B_{\mathrm{Y}}}}}{\partial\eta}-\Gamma(\xi_{e\mathrm{R}}-\xi_{e\mathrm{L}}+\xi_{0}),$
$\displaystyle\frac{\mathrm{d}\xi_{e\mathrm{L}}}{\mathrm{d}\tilde{\eta}}$
$\displaystyle=\frac{3\alpha^{\prime}}{4\pi}\int_{k_{\mathrm{min}}}^{\tilde{k}_{\mathrm{max}}}\mathrm{d}\tilde{k}\frac{\partial\tilde{\mathcal{H}}_{{\rm
B_{\mathrm{Y}}}}}{\partial\eta}-\frac{\Gamma}{2}(\xi_{e\mathrm{L}}-\xi_{e\mathrm{R}}-\xi_{0})-\frac{\Gamma_{\mathrm{sph}}}{2}\xi_{e\mathrm{L}},$
$\displaystyle\frac{\mathrm{d}\xi_{0}}{\mathrm{d}\tilde{\eta}}$
$\displaystyle=-\frac{\Gamma}{2}(\xi_{e\mathrm{R}}+\xi_{0}-\xi_{e\mathrm{L}}),$
(3.1)

where $\tilde{\mathcal{E}}_{{\rm B_{\mathrm{Y}}}}(\tilde{k},\tilde{\eta})$ and
$\tilde{\mathcal{H}}_{{\rm B_{\mathrm{Y}}}}(\tilde{k},\tilde{\eta})$ are the
dimensionless spectral densities of the HMF energy and helicity. They are
related to $\rho_{\mathrm{Y}}^{(c)}(q,\xi)$ and $h_{\mathrm{Y}}^{(c)}(k,\xi)$
by $\rho_{\mathrm{Y}}^{(c)}(k,\xi)=\tilde{\mathcal{E}}_{{\rm
B_{\mathrm{Y}}}}(\tilde{k},\tilde{\eta})T_{0}^{3}$ and
$h_{\mathrm{Y}}^{(c)}(k,\xi)=\tilde{\mathcal{H}}_{{\rm
B_{\mathrm{Y}}}}(\tilde{k},\tilde{\eta})T_{0}^{2}$, where
$T_{0}=2.7\,\text{K}$ is the temperature of the cosmic microwave background
radiation. The dimensionless conformal time $\tilde{\eta}$ and the conformal
momentum $\tilde{k}$ in eq. (3) are
$\eta=(2t_{\mathrm{Univ}}T_{0}/\tilde{M}_{\mathrm{Pl}})\tilde{\eta}$ and
$k=T_{0}\tilde{k}$. Here
$\tilde{M}_{\mathrm{Pl}}=M_{\mathrm{Pl}}/1.66\sqrt{g_{*}}$ is the effective
Planck mass, $g_{*}=106.75$ is the number of the relativistic degrees of
freedom in the hot plasma before EWPT.

The asymmetries of right and left fermions $\xi_{e\mathrm{R,L}}$, as well as
that of Higgs bosons $\xi_{0}$, are
$\xi_{e\mathrm{R,L}}=6(n_{e\mathrm{R,L}}-n_{\bar{e}\mathrm{R,L}})/T^{3}$and
$\xi_{0}=3(n_{\varphi_{0}}-n_{\bar{\varphi}_{0}})/T^{3}$, where
$n_{(e,\bar{e})(\mathrm{R,L)}}$ are the number densities of right electrons,
left fermions, their antiparticles, $n_{\varphi_{0},\bar{\varphi}_{0}}$ are
the number densities of Higgs bosons and antibosons, and $T$ is the plasma
temperature. We account for only the lightest lepton generation since other
leptons are out of equilibrium sooner because of their greater Yukawa coupling
constants [22]. The contribution of left fermions is taken into account to
make the system in eq. (3) self consistent [12].

The spin-flip rate $\Gamma$ because of the interaction of fermions with Higgs
bosons reads [22]

$\Gamma(\eta)=\frac{242}{\tilde{\eta}_{\mathrm{EW}}}\left[1-\frac{\tilde{\eta}^{2}}{\tilde{\eta}_{\mathrm{EW}}^{2}}\right],\quad\tilde{\eta}_{\mathrm{EW}}=\frac{\tilde{M}_{\mathrm{Pl}}}{T_{\mathrm{EW}}}=7\times
10^{15},$ (3.2)

where $T_{\mathrm{EW}}=10^{2}\,\text{GeV}$ is the temperature of EWPT. The
dimensionless sphaleron transitions rate is $\Gamma_{\mathrm{sph}}=8\times
10^{-7}$ [23].

We choose the initial time of the HMFs evolution as $t_{*}=t_{\mathrm{RL}}$
which corresponds to $T=T_{\mathrm{RL}}=10\,\text{TeV}.$ Note that both $\eta$
and $\tilde{\eta}$ are vanishing at $T=T_{\mathrm{RL}}$. Below this
temperature, Higgs decays become faster than universe expansion. Thus, left
fermions start to be produced. The minimal wave vector
$\tilde{k}_{\mathrm{min}}$, which corresponds to the maximal length scale, is
chosen as the inverse conformal horizon size at $T=T_{\mathrm{RL}}$,
$\tilde{k}_{\mathrm{min}}\sim T_{\mathrm{RL}}/\tilde{M}_{\mathrm{Pl}}\approx
10^{-14}$. The maximal wave vector $\tilde{k}_{\mathrm{max}}$, which is
related to the minimal length scale, is the free parameter in our model. The
minimal scale is chosen to be greater than the Debye length to satisfy the
plasma electroneutrality. We shall vary $\tilde{k}_{\mathrm{max}}$ in the
range $10^{-5}<\tilde{k}_{\mathrm{max}}<10^{-2}$. In this situation, the
minimal length scale of HMFs is still greater than the conformal Debye length
$\tilde{r}_{\mathrm{D}}$.

The effective magnetic diffusion coefficient $\eta_{\mathrm{eff}}$ and the
effective $\alpha$-dynamo parameters $\alpha_{\pm}$ account for the analogs of
both the CME and the (H)MHD turbulence for HMFs. They are [21]

$\displaystyle\eta_{\mathrm{eff}}$
$\displaystyle=\sigma_{c}^{-1}+\frac{4}{3}\frac{(\alpha^{\prime})^{-2}}{\tilde{\rho}+\tilde{p}}\int_{\tilde{k}_{\mathrm{min}}}^{\tilde{k}_{\mathrm{max}}}\mathrm{d}\tilde{k}\mathcal{\tilde{E}}_{{\rm
B_{\mathrm{Y}}}},\quad\alpha^{\prime}=\frac{g^{\prime 2}}{4\pi},$
$\displaystyle\alpha_{\pm}$
$\displaystyle=\alpha_{\mathrm{Y}}(\tilde{\eta})\mp\frac{2}{3}\frac{(\alpha^{\prime})^{-2}}{\tilde{\rho}+\tilde{p}}\int_{\tilde{k}_{\mathrm{min}}}^{\tilde{k}_{\mathrm{max}}}\mathrm{d}\tilde{k}\tilde{k}^{2}\mathcal{\tilde{H}}_{{\rm
B_{\mathrm{Y}}}},$ (3.3)

where $\sigma_{c}\approx 10^{2}$ is the conformal conductivity of relativistic
plasma, $\tilde{\rho}$ and $\tilde{p}$ are the plasma density and pressure
expressed in conformal variables, $g^{\prime}=e/\cos\theta_{\mathrm{W}}$ is
the hypercharge, and

$\alpha_{\mathrm{Y}}(\tilde{\eta})=\frac{\alpha^{\prime}}{\pi\sigma_{c}}\left[\xi_{e\mathrm{R}}(\tilde{\eta})-\frac{\xi_{e\mathrm{L}}(\tilde{\eta})}{2}\right],$
(3.4)

is the $\alpha$-dynamo parameter when only the analog of the CME for HMFs is
accounted for. We choose $p=\rho/3$ for the ultrarelativistic plasma. The
value of $\alpha^{\prime}=9.5\times 10^{-3}.$ Note that $\alpha_{\mathrm{Y}}$
in eq. (3.4) is analogous to the $\alpha$-parameter, responsible for the
instability of Maxwell magnetic fields in the early universe after EWPT owing
to the CME, which is used in ref. [24].

HMF can be unstable if $\alpha_{\mathrm{Y}}$ is nonzero. It is the consequence
of the analog of the CME in which the (hyper-)electric current is induced
along the (hyper-)magnetic field. Besides the CME, one has the chiral vortical
effect (the CVE) in the system of ultrarelativistic fermions. The CVE is the
generation of a current
$\mathbf{j}_{\mathrm{CVE}}=\tfrac{\mu_{5}\mu}{\pi^{2}}\bm{\omega}$ [25] along
the plasma vorticity $\bm{\omega}=\tfrac{1}{2}(\nabla\times\mathbf{v})$.
Taking into account that $\mathbf{v}\propto(\mathbf{j}\times\mathbf{B})$ in
the chosen model for (H)MHD turbulence [17], as well as
$(\nabla\times\mathbf{B})=\mathbf{j}+\mathbf{j}_{\mathrm{CVE}}$, one gets that
the CVE contribution to the induction equation is quadratic in HMF. Thus, such
a contribution is cubic in HMF in the equations for the HMF energy density and
the helicity, which are the binary combinations of HMFs. Supposing that random
HMFs are Gaussian, i.e. all odd correlators are vanishing, one gets that the
CVE does not contribute to eq. (3).

The initial condition for eq. (3) is chosen as $\xi_{e\mathrm{L}}=\xi_{0}=0$
and $\xi_{e\mathrm{R}}=10^{-10}$. Such values give the appropriate value of
the observed $\text{BAU}\sim 10^{-10}$ if this model is applied for this
problem [16]. The initial spectra are $\mathcal{\tilde{E}}_{{\rm
B_{\mathrm{Y}}}}^{(0)}(\tilde{k})=\mathcal{C}_{\mathrm{Y}}\tilde{k}^{n_{\mathrm{Y}}}$
and $\mathcal{\tilde{H}}_{{\rm
B_{\mathrm{Y}}}}^{(0)}(\tilde{k})=2q\mathcal{\tilde{E}}_{{\rm
B_{\mathrm{Y}}}}^{(0)}(\tilde{k})/\tilde{k}$, where $0<q<1$ is the
phenomenological parameter fixing the helicity of a seed HMF. The constant
$\mathcal{C}_{\mathrm{Y}}$ is chosen such that

$\tilde{B}_{\mathrm{Y}}^{(0)}=\left[2\int_{\tilde{k}_{\mathrm{min}}}^{\tilde{k}_{\mathrm{max}}}\mathrm{d}\tilde{k}\mathcal{\tilde{E}}_{{\rm
B_{\mathrm{Y}}}}^{(0)}(\tilde{k})\right]^{1/2}$ (3.5)

is the dimensionless strength of a seed HMF. We vary
$\tilde{B}_{\mathrm{Y}}^{(0)}$ in the range $1.4\times
10^{-2}<\tilde{B}_{\mathrm{Y}}^{(0)}<1.4\times 10^{-1}$. Such HMFs contribute
to neither the universe expansion nor the primordial nucleosynthesis [16].

We take the initial Kolmogorov spectrum with $n_{\mathrm{Y}}=-5/3$. The
Kolmogorov spectrum has the singularity at $\tilde{k}=0$ which has to be
regularized by setting the lower bound $\tilde{k}>\tilde{k}_{\mathrm{min}}$,
which corresponds to the maximal length scale, $k_{\mathrm{min}}\propto
L_{\mathrm{max}}^{-1}$. As we mentioned above, we choose $L_{\mathrm{max}}$
comparable with the horizon size at $T=T_{\mathrm{RL}}$ (see also ref. [16]).
A more realistic choice of the seed spectrum was used in ref. [26]. It
consists in the Batchelor spectrum $\mathcal{\tilde{E}}_{{\rm
B_{\mathrm{Y}}}}^{(0)}(\tilde{k})\propto\tilde{k}^{4}$ for small $\tilde{k}$
below a certain value and the Kolmogorov one above it.

The energy of plasma dissipates into heat because of viscous processes if the
typical length scale is less than the Kolmogorov one
$L_{\mathrm{visc}}\propto\nu^{3/4}$ [27], where $\nu$ is the kinematic
viscosity coefficient. In our approach, we keep only the Lorentz force in the
Navier-Stokes equation and neglect other terms including the viscous one [17].
Therefore we can take that $L_{\mathrm{visc}}$ is vanishing. It means that the
only relevant lower bound for the minimal length scale is determined by the
Debye length,
$\tilde{k}_{\mathrm{max}}^{-1}=\tilde{L}_{\mathrm{min}}\gg\tilde{r}_{\mathrm{D}}$.

For the derivation of eqs. (3)-(3.4) and the discussion of their applicability
for the description of the HMFs evolution the reader is referred to refs. [12,
16, 21]. Note that the coefficients analogous to that in eq. (3) were also
studied in ref. [20]. We can also rewrite eqs. (2)-(3) in the new variables
convenient for numerical simulations. The details are provided in appendix A.

##  4 Results

In this section, we show the behavior of the energy density of GWs based on
the numerical solution of eq. (A) with the parameters and the initial
condition adopted in section 3.

In figure 1, we present the evolution of the system from $T=T_{\mathrm{RL}}$
down to EWPT for the fixed $\tilde{k}_{\mathrm{max}}=10^{-3}$ and different
$\tilde{B}_{\mathrm{Y}}^{(0)}$ and $q$. Comparing figures 1 and 1, where the
plots of HMF are shown, we can see that there is a very small dependence of
the results on $q$.

Figure 1: The evolution of the system at different
$\tilde{B}_{\mathrm{Y}}^{(0)}$ and $q$ for the fixed
$\tilde{k}_{\mathrm{max}}=10^{-3}$ based on the numerical solution of eq. (A).
The initial condition is $\xi_{e\mathrm{R}}^{(0)}=10^{-10}$,
$\xi_{e\mathrm{L}}^{(0)}=\xi_{0}^{(0)}=0$, and $\rho_{\mathrm{GW}}^{(c)}=0$ at
$T_{\mathrm{RL}}=10\,\text{TeV}$. We account for the (H)MHD turbulence in
solid lines, whereas the dashed ones are plotted without taking the turbulence
into account. Red lines correspond to $\tilde{B}_{\mathrm{Y}}^{(0)}=1.4\times
10^{-1}$ and blue lines to $\tilde{B}_{\mathrm{Y}}^{(0)}=1.4\times 10^{-2}$.
Panels (a) and (b): the evolution of the HMF; panels (c) and (d): the
evolution of the energy density of GWs; cf. eqs. (2.15) and (A.8); panels (e)
and (f): the evolution of the $\alpha$-dynamo parameter
$\sim\xi_{e\mathrm{R}}-\xi_{e\mathrm{L}}/2$. In panels (a), (c), and (e), we
set $q=1$; and panels (b), (d), and (f) correspond to $q=0.1$.

The HMF evolution depends on whether we account for the (H)MHD turbulence or
not. The HMF evolution, without taking the (H)MHD turbulence into account,
corresponds to $\eta_{\mathrm{eff}}=\sigma_{c}^{-1}$ and
$\alpha_{\pm}=\alpha_{\mathrm{Y}}$ in eq. (3). The main contribution of the
HMFs noise was mentioned in ref. [16] to result from $\eta_{\mathrm{eff}}$.
Thus, the greater $\tilde{B}_{\mathrm{Y}}^{(0)}$ is, the faster HMF decays.
This feature can be seen in figures 1 and 1. Red and blue dashed lines,
plotted without the (H)MHD turbulence, overlap. It should be noted that,
despite red solid lines are below blue ones, the absolute value of HMFs, shown
in red, is almost one order of magnitude greater than of these depicted in
blue in figures 1 and 1.

The evolution of the energy density of GWs $\rho_{\mathrm{GW}}^{(c)}$ in the
cooling universe is shown in figures 1 and 1. First, greater
$\tilde{B}_{\mathrm{Y}}^{(0)}$ leads to greater $\rho_{\mathrm{GW}}^{(c)}$;
cf. red and blue solid lines in figures 1 and 1. Second, greater
$\tilde{B}_{\mathrm{Y}}^{(0)}$ result in the faster decay of HMFs. Thus the
energy density of GWs, which depends on $\tilde{B}_{\mathrm{Y}}$, will evolve
slower for greater $\tilde{B}_{\mathrm{Y}}^{(0)}$. This feature can be also
observed figures 1 and 1: the gap between solid and dashed lines in red is
wider than between these in blue. Third, $\rho_{\mathrm{GW}}^{(c)}$ grows
while the universe cools down and HMFs decays; cf. figures 1 and 1. It is a
consequence of the cumulative effect of the integration over the conformal
time in eq. (2).

The behavior of the $\alpha$-dynamo parameter
$\alpha_{\mathrm{Y}}\sim\xi_{e\mathrm{R}}-\xi_{e\mathrm{L}}/2$ is depicted in
figures 1 and 1. In fact, $\alpha_{\mathrm{Y}}$ is almost completely
determined by the evolution of $\xi_{e\mathrm{R}}$ since
$|\xi_{e\mathrm{L}}|\ll\xi_{e\mathrm{R}}$, as found in ref. [16].

One can see in figures 1 and 1 that $\alpha_{\mathrm{Y}}$ has a spike at
$T\gtrsim T_{\mathrm{RL}}$. It can be explained by the backreaction of the
hypermagnetic helicity on the asymmetries. Indeed, this spike results from the
term $I_{\mathrm{H}}$ in eq. (A) for the asymmetry of right electrons
$M_{\mathrm{R}}$. The level of the spike in higher for the greater initial
helicity. This feature is seen in figures 1 and 1. The asymmetries fall down
at greater evolution times because of the Higgs decays and especially the
sphaleron process which acts on the asymmetry of left fermions. Taking into
account the Higgs decays and the sphaleron process results in the
nonconservation of the sum of the chiral imbalance and the (hyper-)magnetic
helicity, which is usually conserved if only the Adler anomaly is considered.
These processes are analogous to the spin-flip of fermions in electromagnetic
plasma [28, 29]. The evolution of HMF in figures 1 and 1 does not reveal the
spikes at $T\gtrsim T_{\mathrm{RL}}$ since it is dominated by the diffusion
terms $\propto-2\tilde{k}^{2}\sigma_{c}^{-1}\tilde{\mathcal{E}}_{{\rm
B_{\mathrm{Y}}}}$ in eq. (3).

The obtained behavior of HMFs can be explained if we calculate the typical
scale of the HMFs instability. Using the results of ref. [30], one gets that

$\tilde{k}_{\star}=\frac{\pi\sigma_{c}}{2\alpha^{\prime}}\left(\int_{0}^{\tilde{\eta}}\mathrm{d}\tilde{\eta}\xi_{e\mathrm{R}}\right)^{-1},$
(4.1)

where we keep only the right asymmetry for simplicity. Equation (4.1) can be
obtained, e.g., from the induction equation for HMF,
$\partial_{\eta}\tilde{\mathbf{B}}_{\mathrm{Y}}=\alpha_{\mathrm{Y}}(\nabla\times\tilde{\mathbf{B}}_{\mathrm{Y}})+\dotsb$,
where we keep only the instability term. Assuming the Chern-Simons wave
distribution and using eq. (3.4), one gets that
$\tilde{B}_{\mathrm{Y}}\propto\exp(\tilde{k}/\tilde{k}_{\star})$. A more
careful derivation of $\tilde{k}_{\star}$ is provided in ref. [30].

One expects the HMF amplification if $\tilde{k}>\tilde{k}_{\star}$. The fast
growth of $\xi_{e\mathrm{R}}$ to the value $\sim 10^{-8}$ takes place in the
temperature range $\Delta T=T_{\mathrm{RL}}-T=10^{-1}\,\text{GeV}$ (see, e.g.,
figure 1 and ref. [16]). Thus we can estimate $\tilde{k}_{\star}\sim 10^{3}$,
which is much greater than $\tilde{k}_{\mathrm{max}}=10^{-3}$ used in figure
1. It means that the diffusion is dominant in the HMF evolution. To trigger a
sizable instability of HMFs in the considered system, one has to consider,
e.g., a greater value of $\tilde{k}_{\mathrm{max}}\sim\tilde{k}_{\star}$.
However, the condition of the plasma electroneutrality is violated even for
$\tilde{k}_{\mathrm{max}}=0.1\sim\tilde{r}_{\mathrm{D}}^{-1}$.

The spectra of the hypermagnetic energy $R$ and the hypermagnetic helicity $H$
(see eq. (A)) at different plasma temperatures are depicted in figure 2. We
show the initial spectra at $T=T_{\mathrm{RL}}$, which are Kolmogorov and
depicted with dashed lines, and the spectra at EWPT. The spectra of seed
fields evolve to the solid lines very rapidly. Therefore we do not show them
for any intermediate temperatures.

The spectra at $T=T_{\mathrm{EW}}$ coincide with the seed ones, shown with
dashed lines, for $\kappa\lesssim 10^{-5}$. For greater $\kappa$, the spectra
start to decrease. This behavior is accounted for by the diffusion terms
$\propto-\kappa^{2}$ in eq. (A). Thus, modes with greater $\kappa$ are damped
more efficiently. This evolution of the HMFs is promising from the
observational point of view since HMFs with large scales survive. Moreover, we
can see in figure 2 that the spectra corresponding to a stronger seed field,
shown with red color, start to decrease at smaller $\kappa$ compared to blue
ones. It happens owing to the turbulence contribution to the effective
diffusion coefficient. The above features justify our suggestion that the
evolution of HMFs in this model is dominated by the diffusion.

The spectra at $T=T_{\mathrm{EW}}$ demonstrate the drastic change in their
behavior when they drop below $\sim 10^{-20}$. This feature is related to the
limited accuracy of numerical simulations. Thus, we can rely only on the part
of the curves in figure 2 which are above $\sim 10^{-20}$.

Figure 2: The spectra of the hypermagnetic energy $R$ and the hypermagnetic
helicity $H$ versus the dimensioless wave vector $\kappa$ (see eq. (A)) for
different $\tilde{B}_{\mathrm{Y}}^{(0)}$ and $q$, as well as the fixed
$\tilde{k}_{\mathrm{max}}=10^{-3}$. The initial condition is the same as in
figure 1. Here, we account for the (H)MHD turbulence. Dashed lines show the
spectra of seed HMFs at $T=T_{\mathrm{RL}}$. Solid lines represent the
situation when the universe cools down to EWPT. Red lines correspond to
$\tilde{B}_{\mathrm{Y}}^{(0)}=1.4\times 10^{-1}$ and blue lines to
$\tilde{B}_{\mathrm{Y}}^{(0)}=1.4\times 10^{-2}$. Panels (a) and (b): the
spectra of the energy density; panels (c) and (d): the spectra of the
hypermagnetic helicity; In panels (a) and (c), we set $q=1$; and panels (b)
and (d) correspond to $q=0.1$.

The influence of the HMFs scale on the production of GWs is shown in figure 3,
where we fix the strength of the seed HMF by
$\tilde{B}_{\mathrm{Y}}^{(0)}=1.4\times 10^{-1}$ and vary
$\tilde{k}_{\mathrm{max}}$. The effect of the (H)MHD turbulence on the
evolution of HMFs was mentioned in refs. [16, 21] to be more sizable for small
scale fields, i.e. for great $\tilde{k}_{\mathrm{max}}$. Thus HMFs,
corresponding to greater $\tilde{k}_{\mathrm{max}}$ decay faster. It results
in the slower enhancement of $\rho_{\mathrm{GW}}^{(c)}$ for such fields. Such
a behavior is seen in figure 3 where red lines, with
$\tilde{k}_{\mathrm{max}}=10^{-2}$, are below blue ones, with
$\tilde{k}_{\mathrm{max}}=10^{-3}$. We can also see the negligible dependence
on the initial hypermagnetic helicity; cf. figures 3 and 3.

Figure 3: The evolution of the energy density of GWs at different
$\tilde{k}_{\mathrm{max}}$ and $q$ for the fixed
$\tilde{B}_{\mathrm{Y}}^{(0)}=1.4\times 10^{-1}$ based on the numerical
solution of eq. (A). The initial condition is the same as in figure 1. We
account for the (H)MHD turbulence in solid lines, wheres the dashed ones are
plotted without taking the turbulence into account. Red lines correspond to
$\tilde{k}_{\mathrm{max}}=10^{-2}$ and blue lines to
$\tilde{k}_{\mathrm{max}}=10^{-3}$. (a) $q=1$; and (b) $q=0.1$.

The evolution of the system, shown in figures 1 and 3, qualitatively resembles
the results obtained in ref. [14], where the generation of primordial GWs is
driven by the CME. However, unlike ref. [14], where the turbulence was modeled
fully numerically, we use a semi-analytical approach, which allows one to
analyze the influence of different factors, like the strength of a seed HMF
and its scale, on the production of GWs. Moreover, we study more realistic
situation when the generation of relic GWs is driven by the lepton and Higgs
asymmetries before EWPT. For this purpose, we utilize the analogs of the CME
for the HMFs and of the Adler anomalies for the asymmetries.

###  4.1 Observability of relic GWs

Let us consider the possibility for the predicted GW signal to be observed
with the current experimental techniques. For this purpose we take that
$\tilde{B}_{\mathrm{Y}}^{(0)}=1.4\times 10^{-1}$,
$\tilde{k}_{\mathrm{max}}=10^{-5}$, and $q=1$. The chosen
$B_{\mathrm{Y}}^{(0)}=7\times 10^{26}\,\text{G}$ still does not violate the
Big Bang nucleosynthesis constraint $B_{\mathrm{BBN}}=10^{11}\text{G}$ at
$T_{\mathrm{BBN}}=0.1\,\text{MeV}$ [31]. The value of
$\tilde{k}_{\mathrm{max}}$ is taken so that the HMFs noise is still noticeable
but its influence is small (see solid and dashed lines in figure 4). The rest
of the parameters is the same as in figure 1.

Figure 4: The energy density of GWs versus the plasma temperature for
$\tilde{B}_{\mathrm{Y}}^{(0)}=1.4\times 10^{-1}$
($B_{\mathrm{Y}}^{(0)}=7\times 10^{26}\,\text{G}$),
$\tilde{k}_{\mathrm{max}}=10^{-5}$, and $q=1$. In the solid line, we account
for the (H)MHD turbulence, whereas, in dashed one, not. The rest of the
parameters and the initial condition are the same as in figure 1. The
horizontal dash-dotted line is the averaged lower bound for the GW energy
density, which can be experimentally probed presently.

One can see in the end of the solid line at $T=10^{2}\,\text{GeV}$ in figure 4
that the conformal energy density of GWs at $T_{\mathrm{EW}}$ is
$\rho_{\mathrm{GW}}^{(c)}(T_{\mathrm{EW}})=5\times
10^{15}\rho_{\mathrm{GW}}^{(0)}=7.5\times
10^{-8}\,\text{eV}\cdot\text{cm}^{-3}$, where $\rho_{\mathrm{GW}}^{(0)}$ is
given in eq. (A.8). We also assume that no other sources of GWs are after
EWPT. Thus the current energy density of GWs is
$\rho_{\mathrm{GW}}^{(\mathrm{now})}=\rho_{\mathrm{GW}}^{(c)}/a_{\mathrm{now}}^{3}=\rho_{\mathrm{GW}}^{(c)}(T_{\mathrm{EW}})$.

Some modern GWs detectors (see, e.g., ref. [32]) can potentially detect
stochastic GWs with $\Omega\sim 10^{-10}$, where
$\Omega=\tfrac{f\rho_{\mathrm{GW}}(f)}{\rho_{\mathrm{crit}}}$, $f=k/2\pi$ is
the frequency of GWs in Hz, $\rho_{\mathrm{GW}}(f)$ is the spectrum of the
energy density with respect to $f$, and $\rho_{\mathrm{crit}}=0.5\times
10^{-5}\,\text{GeV}\cdot\text{cm}^{-3}$ is the critical energy density of the
universe. We can evaluate the total observable energy density as
$\rho_{\mathrm{GW}}^{(\mathrm{obs})}\sim 3.1\times
10^{-6}\,\text{eV}\cdot\text{cm}^{-3}$, which is only $40$ times greater than
the predicted value $\rho_{\mathrm{GW}}^{(\mathrm{now})}=7.5\times
10^{-8}\,\text{eV}\cdot\text{cm}^{-3}$. Thus, the GW background, described
within our model, is potentially detectable using current GWs detectors after
some enhancement of their sensitivity.

We can see in figure 4 that both the turbulent GW energy density, shown by the
solid line, and the nonturbulent one, depicted by the dashed line, are below
the observations threshold, represented by the dash-dotted line. The value of
$(\rho_{\mathrm{GW}}^{(c)}/\rho_{\mathrm{GW}}^{(0)})_{\mathrm{obs}}=2.1\times
10^{17}$ corresponds to the sensitivity of, e.g., the NANOGrav experiment
[32]. It corresponds to $\Omega\sim 10^{-10}.$

##  5 Conclusion

In this work, we have studied the production of relic GWs in turbulent plasma
in the symmetric phase before the EWPT. Random HMFs, with the (H)MHD
turbulence, are the driver for the GWs generation. HMFs are taken to couple to
the gravity through their energy-momentum tensor. The evolution of HMFs in
primordial plasma is governed by the analogs of the CME and the Adler
anomalies in the presence of nonzero asymmetries of right and left electrons,
left neutrinos, their antiparticles, as well as Higgs bosons.

We have used the realistic parameters and the initial condition of HMFs for
the production of GWs. The same parameters, while utilized in the problem of
the BAU generation (see, e.g., ref. [16]), result in BAU close to the observed
value, $(\text{BAU})_{\mathrm{obs}}\sim 10^{-10}$. It is the advantage of our
work compared to ref. [14], where the analogous problem, i.e. the generation
of relic GWs driven by the CME, was studied. Moreover, we use the semi-
analytical model, which is based on the (H)MHD turbulence [17, 20, 21]. It
allows one to guess the dependence of the final results on the parameters of
the system, in contrast to ref. [14], where purely numerical simulations were
used.

In section 2, we have rederived eq. (2.5) for the tensor perturbations of the
metric in the presence of HMFs. Then, using its formal solution in eq. (2.7),
we have obtained the expression for the spectrum of the energy density of GWs
in eq. (2) represented in conformal variables. After averaging and using the
binary combinations of HMFs, such as their energy and the helicity, we get the
final expressions for the energy density of GWs and its spectrum in eqs.
(2.15) and (2).

Section 3 is devoted to the formulation of the dynamics of HMFs and the
initial condition for them. Basically, it is similar to that in ref. [16],
where we studied the BAU generation driven by turbulent HMFs. Such HMFs evolve
owing to the analog of the CME in the presence of nonzero lepton asymmetries.
We have also accounted for the backreaction of helical HMFs to the lepton
asymmetries evolution because of the analog of the Adler anomalies for HMFs.
The noise of HMFs is modeled by the analog of the (H)MHD turbulence. The main
kinetic equations for the binary combinations of HMFs and the particle
asymmetries are summarized in eqs. (3)-(3.4).

In section 4, we have presented the results of the numerical solution of the
evolution eq. (3) and the energy density of GWs in eq. (2.15) basing on this
solution. In figures 1 and 3, we have shown the strength of HMF
$\tilde{B}_{\mathrm{Y}}$, the energy density of GWs
$\rho_{\mathrm{GW}}^{(c)}$, and the $\alpha$-dynamo parameter
$\sim\xi_{e\mathrm{R}}-\xi_{e\mathrm{L}}/2$ in the temperature range from
$T=T_{\mathrm{RL}}=10\,\text{TeV}$ down to
$T=T_{\mathrm{EW}}=10^{2}\,\text{GeV}$. We have analyzed the dependence of
$\rho_{\mathrm{GW}}^{(c)}$ on the strength of the seed HMF
$\tilde{B}_{\mathrm{Y}}^{(0)}$ for the fixed minimal scale
$\tilde{k^{-1}}_{\mathrm{max}}$ (see figure 1), as well as on the minimal
scale for the fixed strength of the seed HMF (see figure 3). There is a
negligible dependence of $\rho_{\mathrm{GW}}^{(c)}$ on the initial helicity of
HMFs. There results qualitatively resemble the findings of ref. [14].

We have briefly discussed the possibility to observe the predicted GWs
background in section 4.1. Using the following realistic initial conditions:
$B_{\mathrm{Y}}^{(0)}=7\times 10^{26}\,\text{G}$,
$\tilde{k}_{\mathrm{max}}=10^{-5}$, $q=1$, $\xi_{e\mathrm{R}}^{(0)}=10^{-10}$,
and $\xi_{e\mathrm{L}}^{(0)}=\xi_{0}^{(0)}=0$, we have obtained that the
current energy density of GWs produced is only $\sim 40$ times below the
threshold of the modern GWs detectors. Thus such a signal can be potentially
observable in the nearest future.

In appendix A, we have rewritten the main expressions in the form adapted for
numerical simulations.

## Acknowledgments

I am thankful to A. Yu. Smirnov and V. B. Semikoz for useful discussions.

##  Appendix A New variables

Following ref. [16], it is convenient to use the new variables in eq. (3),

$\displaystyle\tilde{\mathcal{E}}_{{\rm
B_{\mathrm{Y}}}}(\tilde{k},\tilde{\eta})$
$\displaystyle=\frac{\tilde{k}_{\mathrm{max}}\pi^{2}}{6\alpha^{\prime
2}}R(\kappa,\tau),\quad\tilde{\mathcal{H}}_{{\rm
B_{\mathrm{Y}}}}(\tilde{k},\tilde{\eta})=\frac{\pi^{2}}{3\alpha^{\prime
2}}H(\kappa,\tau),$ $\displaystyle\xi_{\mathrm{R,L,0}}(\tilde{\eta})$
$\displaystyle=\frac{\pi\tilde{k}_{\mathrm{max}}}{\alpha^{\prime}}M_{\mathrm{R,L,0}}(\tau),\quad\tau=\frac{2\tilde{k}_{\mathrm{max}}^{2}}{\sigma_{c}}\tilde{\eta},\quad\tilde{k}=\tilde{k}_{\mathrm{max}}\kappa,$
(A.1)

where $\kappa_{m}<\kappa<1$,
$\kappa_{m}=\tilde{k}_{\mathrm{min}}/\tilde{k}_{\mathrm{max}}$, and $\tau\geq
0$ is the new dimensionless time. Using eq. (A), we rewrite eq. (3) in the
form [16],

$\displaystyle\frac{\partial R}{\partial\tau}=$
$\displaystyle-\kappa^{2}\left(1+\lambda_{t}I_{\mathrm{R}}\right)R+\kappa^{2}\left(M_{\mathrm{R}}-\frac{M_{\mathrm{L}}}{2}-\lambda_{t}I_{\mathrm{H}}\right)H,$
$\displaystyle\frac{\partial H}{\partial\tau}=$
$\displaystyle-\kappa^{2}\left(1+\lambda_{t}I_{\mathrm{R}}\right)H+\left(M_{\mathrm{R}}-\frac{M_{\mathrm{L}}}{2}+\lambda_{t}I_{\mathrm{H}}\right)R,$
$\displaystyle\frac{\mathrm{d}M_{\mathrm{R}}}{\mathrm{d}\tau}=$ $\displaystyle
I_{\mathrm{H}}-\left(M_{\mathrm{R}}-\frac{M_{\mathrm{L}}}{2}\right)I_{\mathrm{R}}-\Gamma^{\prime}(M_{\mathrm{R}}-M_{\mathrm{L}}+M_{0}),$
$\displaystyle\frac{\mathrm{d}M_{\mathrm{L}}}{\mathrm{d}\tau}=$
$\displaystyle-\frac{1}{4}I_{\mathrm{H}}+\frac{1}{4}\left(M_{\mathrm{R}}-\frac{M_{\mathrm{L}}}{2}\right)I_{\mathrm{R}}-\Gamma^{\prime}(M_{\mathrm{L}}-M_{\mathrm{R}}-M_{0})/2-\frac{\Gamma^{\prime}_{s}}{2}M_{\mathrm{L}},$
$\displaystyle\frac{\mathrm{d}M_{0}}{\mathrm{d}\tau}=$
$\displaystyle-\Gamma^{\prime}(M_{\mathrm{R}}+M_{0}-M_{\mathrm{L}})/2,$ (A.2)

where

$\displaystyle I_{\mathrm{R}}(\tau)=$
$\displaystyle\int_{\kappa_{m}}^{1}\mathrm{d}\kappa^{\prime}R(\kappa^{\prime},\tau),\quad
I_{\mathrm{H}}(\tau)=\int_{\kappa_{m}}^{1}\mathrm{d}\kappa^{\prime}\kappa^{\prime
2}H(\kappa^{\prime},\tau),\quad\lambda_{t}=\frac{2\sigma_{c}\tilde{k}_{\mathrm{max}}^{2}\pi^{2}}{9\alpha^{\prime
4}(\tilde{p}+\tilde{\rho})},$ $\displaystyle\Gamma^{\prime}(\tau)=$
$\displaystyle\frac{121\sigma_{c}}{\tilde{\eta}_{\mathrm{EW}}\tilde{k}_{\mathrm{max}}^{2}}\left[1-\frac{T_{\mathrm{EW}}^{2}}{T_{\mathrm{RL}}^{2}}\left(1+\frac{T_{\mathrm{RL}}}{M_{0}}\frac{\sigma_{c}}{2\tilde{k}_{\mathrm{max}}^{2}}\tau\right)^{2}\right],$
(A.3)

and
$\Gamma^{\prime}_{s}=\sigma_{c}\Gamma_{\mathrm{sph}}/2\tilde{k}_{\mathrm{max}}^{2}$.

Equation (A) should be completed with the initial condition, which has the
form,

$R(\kappa,0)=C_{\mathrm{Y}}\kappa^{n_{\mathrm{Y}}},\quad
H(\kappa,0)=q\frac{R(\kappa,0)}{\kappa},$ (A.4)

where

$C_{\mathrm{Y}}=\frac{3\alpha^{\prime
2}(1+n_{\mathrm{Y}})[\tilde{B}_{\mathrm{Y}}^{(0)}]^{2}}{\pi^{2}\tilde{k}_{\mathrm{max}}^{2}(1-\kappa_{m}^{1+n_{\mathrm{Y}}})}.$
(A.5)

Equations (A)-(A.5) can be solved numerically.

The spectral density of GWs and their energy density in eqs. (2) and (2.15)
should be also adapted to the new variables. The spectrum of the energy
density is

$\displaystyle\rho_{\mathrm{GW}}^{(c)}(\kappa,\tau)=$
$\displaystyle\frac{\sigma_{c}^{2}\pi^{2}t_{\text{Univ}}^{2}T_{0}^{7}T_{\mathrm{RL}}^{2}G}{576\alpha^{\prime
4}\tilde{M}_{\mathrm{Pl}}^{2}\tilde{k}_{\mathrm{max}}}\frac{\tau}{\kappa^{3}}\int_{0}^{\tau}\frac{\mathrm{d}\tau^{\prime}}{(1+7.1\times
10^{-13}\tau^{\prime}/\tilde{k}_{\mathrm{max}})^{2}}$
$\displaystyle\times\iint_{S(\kappa)}\frac{\mathrm{d}\upsilon\mathrm{d}\varpi}{\upsilon{}^{3}\varpi^{3}}$
$\displaystyle\times\big{\\{}[4\kappa^{2}\upsilon^{2}+(\kappa^{2}+\upsilon^{2}-\varpi^{2})^{2}][4\kappa^{2}\varpi^{2}+(\kappa^{2}-\upsilon^{2}+\varpi^{2})^{2}]R(\upsilon,\tau^{\prime})R(\varpi,\tau^{\prime})$
$\displaystyle+16\kappa^{2}\upsilon^{2}\varpi^{2}(\kappa^{2}+\upsilon^{2}-\varpi^{2})(\kappa^{2}-\upsilon^{2}+\varpi^{2})H(\upsilon,\tau^{\prime})H(\varpi,\tau^{\prime})\big{\\}},$
(A.6)

where $0<\kappa<2$ and $S(\kappa)$ is the 2D integration domain in the
$(\upsilon,\varpi)$-plane. The energy density reads

$\displaystyle\rho_{\mathrm{GW}}^{(c)}(\tau)=$
$\displaystyle\int_{0}^{2k_{\mathrm{max}}}\rho_{\mathrm{GW}}^{(c)}(k,\tau)\mathrm{d}k=\rho_{\mathrm{GW}}^{(0)}\tau\int_{0}^{\tau}\frac{\mathrm{d}\tau^{\prime}}{(1+7.1\times
10^{-13}\tau^{\prime}/\tilde{k}_{\mathrm{max}})^{2}}$
$\displaystyle\times\int_{0}^{2}\frac{\mathrm{d}\kappa}{\kappa^{3}}\iint_{S(\kappa)}\frac{\mathrm{d}\upsilon\mathrm{d}\varpi}{\upsilon{}^{3}\varpi^{3}}$
$\displaystyle\times\big{\\{}[4\kappa^{2}\upsilon^{2}+(\kappa^{2}+\upsilon^{2}-\varpi^{2})^{2}][4\kappa^{2}\varpi^{2}+(\kappa^{2}-\upsilon^{2}+\varpi^{2})^{2}]R(\upsilon,\tau^{\prime})R(\varpi,\tau^{\prime})$
$\displaystyle+16\kappa^{2}\upsilon^{2}\varpi^{2}(\kappa^{2}+\upsilon^{2}-\varpi^{2})(\kappa^{2}-\upsilon^{2}+\varpi^{2})H(\upsilon,\tau^{\prime})H(\varpi,\tau^{\prime})\big{\\}},$
(A.7)

where

$\rho_{\mathrm{GW}}^{(0)}=\frac{\sigma_{c}^{2}\pi^{2}t_{\text{Univ}}^{2}T_{0}^{8}T_{\mathrm{RL}}^{2}G}{576\alpha^{\prime
4}\tilde{M}_{\mathrm{Pl}}^{2}}=1.5\times
10^{-23}\,\text{eV}\cdot\text{cm}^{-3}.$ (A.8)

Equation (A) is used to plot figures 1, 1, 3, and 4 basing on the numerical
solution, $R(\kappa,\tau)$ and $H(\kappa,\tau)$, of eq. (A).

## References

  * [1] LIGO Scientific collaboration and Virgo collaboration, B.P. Abbott et al., Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837].
  * [2] LIGO Scientific collaboration and Virgo collaboration, R. Abbott et al., GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo during the First Half of the Third Observing Run, Phys. Rev. X 11 (2021) 021053 [arXiv:2010.14527].
  * [3] M. Bailes et al., Gravitational-wave physics and astronomy in the 2020s and 2030s, Nat. Rev. Phys. 3 (2021) 344.
  * [4] N. Christensen, Stochastic gravitational wave backgrounds, Rep. Prog. Phys. 82 (2019) 016903 [arXiv:1811.08797].
  * [5] J.D. Romano and N.J. Cornish, Detection methods for stochastic gravitational-wave backgrounds: A unified treatment, Living Rev. Relativity 20 (2017) 2 [arXiv:1608.06889].
  * [6] T. Regimbau, The astrophysical gravitational wave stochastic background, Res. Astron. Astrophys. 11 (2011) 369 [arXiv:1101.2762].
  * [7] C. Caprini and D.G. Figueroa, Cosmological backgrounds of gravitational waves, Class. Quantum Grav. 35 (2018) 163001 [arXiv:1801.04268].
  * [8] A. Kosowsky, A. Mack and T. Kahniashvili, Gravitational radiation from cosmological turbulence, Phys. Rev. D 66 (2002) 024030 [astro-ph/0111483].
  * [9] K. Fukushima, D.E. Kharzeev and H.J. Warringa, The Chiral Magnetic Effect, Phys. Rev. D 78 (2008) 074033 [arXiv:0808.3382].
  * [10] M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Perseus Books, Reading, U.S.A. (1995), pp. 651–688.
  * [11] M. Joyce and M. Shaposhnikov, Primordial magnetic fields, right-handed electrons, and the Abelian anomaly, Phys. Rev. Lett. 79 (1997) 1193 [astro-ph/9703005].
  * [12] M. Dvornikov and V. B. Semikoz, Lepton asymmetry growth in the symmetric phase of an electroweak plasma with hypermagnetic fields versus its washing out by sphalerons, Phys. Rev. D 87 (2013) 025023 [arXiv:1212.1416].
  * [13] K. Kamada and A. Long, Baryogenesis from decaying magnetic helicity, Phys. Rev. D 94 (2016) 063501 [arXiv:1606.08891].
  * [14] A. Brandenburg, Y. He, T. Kahniashvili, M. Rheinhardt and J. Schober, Relic gravitational waves from the chiral magnetic effect, Astrophys. J. 911 (2021) 110 [arXiv:2101.08178].
  * [15] A.K. Pandey, Gravitational waves in neutrino plasma and NANOGrav signal, Eur. Phys. J. C 81 (2021) 399 [arXiv:2011.05821].
  * [16] M. Dvornikov and V.B. Semikoz, Influence of the hypermagnetic field noise on the baryon asymmetry generation in the symmetric phase of the early universe, Eur. Phys. J. C 81 (2021) 1001 [arXiv:2110.01071].
  * [17] G. Sigl, Cosmological magnetic fields from primordial helical seeds, Phys. Rev. D 66 (2002) 123002 [astro-ph/0202424].
  * [18] S. Weinberg, Cosmology, Oxford University Press, New York, U.S.A. (2020), pp. 219–235.
  * [19] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, seventh edition, Nauka, Moscow, U.S.S.R. (1988), pp. 444–447.
  * [20] L. Campanelli, Evolution of magnetic fields in freely decaying magnetohydrodynamic turbulence, Phys. Rev. Lett. 98 (2007) 251302 [arXiv:0705.2308].
  * [21] M. Dvornikov and V.B. Semikoz, Influence of the turbulent motion on the chiral magnetic effect in the early Universe, Phys. Rev. D 95 (2017) 043538 [arXiv:1612.05897].
  * [22] B.A. Campbell, S. Davidson, J. Ellis and K.A. Olive, On the baryon, lepton-flavor and right-handed electron asymmetries of the universe, Phys. Lett. B 297 (1992) 118 [hep-ph/9302221].
  * [23] D.S. Gorbunov, V.A. Rubakov, Introduction to the Theory of the Early Universe: Hot Big Bang Theory, World Scientific, Singapore (2011), p. 250.
  * [24] I. Rogachevskii, O. Ruchayskiy, A. Boyarsky, J. Fröhlich, N. Kleeorin, A. Brandenburg and J. Schober, Laminar and turbulent dynamos in chiral magnetohydrodynamics—I: Theory, Astrophys. J. 846 (2017) 153 [arXiv:1705.00378].
  * [25] D.E. Kharzeev, J. Liao, S.A. Voloshin and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions—A status report, Prog. Part. Nucl. Phys. 88 (2016) 1 [arxiv:1511.04050].
  * [26] A. Roper Pol, S. Mandal, A. Brandenburg, T. Kahniashvili and A. Kosowsky, Numerical simulations of gravitational waves from early-universe turbulence, Phys. Rev. D 102 (2020) 083512 [arXiv:1903.08585].
  * [27] P.A. Davidson, Turbulence: An Introduction for Scientists and Engineers, second edition, Oxford University Press, Oxford, UK (2015), p. 25.
  * [28] M. Dvornikov, Relaxation of the chiral imbalance and the generation of magnetic fields in magnetars, J. Exp. Theor. Phys. 123 (2016) 967 [arXiv:1510.06228].
  * [29] A. Boyarsky, V. Cheianov, O. Ruchayskiy and O. Sobol, Equilibration of the chiral asymmetry due to finite electron mass in electron-positron plasma, Phys. Rev. D 103 (2021) 013003 [arXiv:2008.00360].
  * [30] V.B. Semikoz, A.Yu. Smirnov and D.D. Sokoloff, Generation of hypermagnetic helicity and leptogenesis in the early Universe, Phys. Rev. D 93 (2016) 103003 [arXiv:1604.02273].
  * [31] B. Cheng, D.N. Schramm and J.W. Truran, Constraints on the strength of a primordial magnetic field from Big Bang Nucleosynthesis, Phys. Rev. D 49 (1994) 5006 [astro-ph/9308041].
  * [32] NANOGrav collaboration, Z. Arzoumanian et al., The NANOGrav 12.5-year Data Set: Search For An Isotropic Stochastic Gravitational-Wave Background, Astrophys. J. Lett. 905 (2020) L34 [arXiv:2009.04496].