#  On some fractal sets only containing irrationals

Kan Jiang  Department of Mathematics, Ningbo University, People’s Republic of
China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove that many fractal sets generated by the associated
dynamical systems only contain irrationals. We give two applications. Firstly,
we explicitly construct some overlapping self-similar sets which only consist
of irrationals. Secondly, we prove that some scaled univoque sets only contain
irrationals.

###### 2010 Mathematics Subject Classification:

Primary: 28A80, Secondary:28A78

##  1\. Introduction

Kurt Mahler [16] posed the following famous conjecture:

######  Conjecture 1.1.

Any number from the ternary Cantor set is either a rational or a
transcendental number.

This conjecture is elegant. It is a classical problem in Diophantine
approximation. To the best of our knowledge, this conjecture is still far from
being solved. However, it motivates us to consider the numbers in fractal
sets. For instance, it is natural to ask when some fractal sets only consist
of irrationals or transcendental numbers. We introduce some related work on
this topic. Jia, Li and Jiang [11] proved the following result:

######  Proposition 1.2.

Let $E\subset\mathbb{R}$ be a set with zero Lebesgue measure. Then for
Lebesgue almost every $t$, we have that

$E+t=\\{x+t:x\in E\\}$

is a set only containing irrationals or transcendental numbers.

Moreover, they considered some special Cantor sets, and constructed explicitly
$t$’s such that $E+t$ only contains irrationals. In this paper, we shall
analyze some more general fractal sets, including invariant sets of some
dynamical systems, self-similar sets with overlaps and so forth. We prove that
these fractal sets only contain irrational numbers. Before introducing the
main results of this paper, we give some definitions and notation. Let
$m\in\mathbb{N}_{\geq 2}$ and $t\in(0,1)$. We say that $t$ has an $m$-adic
expansion if there exists some $(t_{k})\in\\{0,1,2,\cdots,m-1\\}^{\mathbb{N}}$
such that

$t=\sum_{k=1}^{\infty}\dfrac{t_{k}}{m^{k}}.$

Given any non-empty word $(x_{1}x_{2}\cdots
x_{k})\in\\{0,1,2,\cdots,m-1\\}^{k},k\geq 1.$ If there exists some
$k_{0}\in\mathbb{N}$ such that

$t_{k_{0}+1}t_{k_{0}+2}\cdots t_{k_{0}+k}=x_{1}x_{2}\cdots x_{k},$

then we call $(t_{k})$ a universal expansion of $t$.

We say that an expansion $(t_{k})$ is normal if for any finite nonempty word a
taken from

$\\{0,1,2,\cdots,m-1\\}^{*}=\cup_{k=1}^{\infty}\\{0,1,2,\cdots,m-1\\}^{k}$

we have that the limiting frequency of the appearances of a as a subword of
$(t_{k})$ is

$\dfrac{1}{m^{|\textbf{a}|}},$

where $|\textbf{a}|$ denotes the length of $\textbf{a}.$ If $x$ has a normal
expansion, then we call $x$ a normal number. We define the following sets.

$A\pm t=\\{x\pm t:x\in A\\},tA=\\{tx:x\in
A\setminus\\{0\\}\\},t^{-1}A=\\{t^{-1}x:x\in A\setminus\\{0\\}\\}.$

In what follows, we will use similar notation. Now we state the main result of
this paper.

######  Theorem 1.3.

Let $m\in\mathbb{N}_{\geq 2}.$ We define

$T:[0,1)\to[0,1)$

by

$T(x)=mx\mod 1.$

Let $A$ be any nowhere dense invariant set of $T$. For any $t\in(0,1)$ such
that $t$ has a universal $m$-adic expansion, then we simultaneously have

$A-t\subset\mathbb{Q}^{c},A+t\subset\mathbb{Q}^{c},\dfrac{A}{t}\subset\mathbb{Q}^{c}.$

Moreover, if $t\in(1,\infty)$ and $(1/t)$ has a universal $m$-adic expansion,
then

$tA\subset\mathbb{Q}^{c}.$

In particular, for Lebesgue almost every $t\in(0,1)$, we have the above
inclusions.

######  Remark 1.4.

A set $A$ is called an invariant set of $T$ if $T(A)\subset A.$ There are many
invariant sets with respect to $T$. Therefore, our result gives a uniform $t$
such that $A\pm t,A/t$ and $tA$ only contain irrationals. Dynamically, for any
$t\in(0,1)$ with an $m$-adic universal expansion, its orbit

$\\{T^{n}(t):n\geq 0\\}$

is dense in $[0,1]$. In fact, our condition can be weaken. We only need to
assume that $(t_{k})$ is “local” universal, i.e. it suffices to assume that

$(t_{k})=x_{1}x_{2}\cdots x_{i}s_{1}s_{2}s_{3}\cdots,$

where $(s_{j})$ is a universal expansion and $x_{1}x_{2}\cdots x_{i}$ is any
word from

$\\{0,1,2,\cdots,m-1\\}^{i}.$

Here $i$ can be any positive integer. Roughly speaking, “local” universal
means the dynamical orbit of $t$ is locally dense in some subset of $[0,1]$.
The last statement is trivial as it is well-known that for Lebesgue almost
every $t\in(0,1)$, $t$ has an $m$-adic universal expansion. This is due to the
famous Borel’s normal number theorem.

If $A$ is a special set, for instance, the survival set of some open dynamical
system [9], then we still have the above result. For simplicity, we only
consider one hole. For multiple holes, our result is still correct.

######  Corollary 1.5.

Given a hole $(a,b)\subset[0,1)$. Let $A$ be the survival set of the following
open dynamics, i.e.

$A=\\{x\in[0,1):T^{n}(x)\notin(a,b)\mbox{ for any }n\geq 0\\}.$

Then for any $t\in(0,1)$ such that $t$ has a universal $m$-adic expansion, we
have

$A-t\subset\mathbb{Q}^{c},A+t\subset\mathbb{Q}^{c},\dfrac{A}{t}\subset\mathbb{Q}^{c}.$

Moreover, if $t\in(1,\infty)$ and $(1/t)$ has a universal $m$-adic expansion,
then

$tA\subset\mathbb{Q}^{c}.$

With similar discussion, we are allowed to prove the following result.

######  Corollary 1.6.

Given any $m\in\mathbb{N}_{\geq 2}$. Let $\Sigma$ be a subshift of finite type
[14] over $\\{0,1,2,\cdots,m-1\\}^{\mathbb{N}}$. Let

$A=\left\\{\sum_{i=1}^{\infty}\dfrac{x_{i}}{m^{i}}:(x_{i})\in\Sigma\right\\}.$

If $A$ is nowhere dense in $[0,1]$, then for any $t\in(0,1)$ such that $t$ has
a universal $m$-adic expansion, we have

$A-t\subset\mathbb{Q}^{c},A+t\subset\mathbb{Q}^{c},\dfrac{A}{t}\subset\mathbb{Q}^{c}.$

Moreover, if $t\in(1,\infty)$ and $(1/t)$ has a universal $m$-adic expansion,
then

$tA\subset\mathbb{Q}^{c}.$

In Theorem 1.3, if we choose $t$ with stronger property, then we are able to
prove the following result. We still use the notation defined in Theorem 1.3.

######  Theorem 1.7.

Let $A$ be any nowhere dense invariant set of $T$. Then for any normal number
$t\in(0,1)$,

$A+it\subset\mathbb{Q}^{c},\dfrac{A}{it}\subset\mathbb{Q}^{c},i\in\mathbb{Z}\setminus\\{0\\}.$

In the above theorem, we let

$A+it=\\{x+it:x\in A\\},\dfrac{A}{it}=\left\\{\dfrac{x}{it}:x\in
A\setminus\\{0\\}\right\\},i\neq 0.$

As an application of the idea of Theorem 1.3, we explicitly construct some
overlapping self-similar sets which only consist of irrationals.

######  Theorem 1.8.

Let $K$ be the attractor of the IFS

$\left\\{f_{i}(x)=\dfrac{x+a_{i}}{q}\right\\}_{i=1}^{n},$

where

$q\in\mathbb{N}_{\geq 3},a_{i}\in\mathbb{Q}\cap[0,q-1].$

Suppose that $K$ is nowhere dense in $[0,1]$. Then for any $t\in(0,1)$ such
that $t$ has a universal $q$-adic expansion, we have

$K-t\subset\mathbb{Q}^{c},K+t\subset\mathbb{Q}^{c},\dfrac{K}{t}\subset\mathbb{Q}^{c}.$

Moreover, for any $t\in(1,+\infty)$ such that $1/t$ has a universal $q$-adic
expansion in $[0,1]$, then we have

$tK\subset\mathbb{Q}^{c}.$

The main idea of Theorem 1.3 is still useful for some non-integer expanding
maps. We give another application to the univoque set. Let $\beta\in(1,2)$.
Given any $x\in[0,1/(\beta-1)]$. Then there exists an expansion
$(x_{n})\in\\{0,1\\}^{\mathbb{N}}$ such that

$x=\sum_{n=1}^{\infty}\dfrac{x_{n}}{\beta^{n}}.$

If $(x_{n})$ is unique, then we call $x$ a univoque point. Usually, for a
generic point, it has uncountably many different expansions [2, 3, 4, 8]. We
denote by $U_{\beta}$ all the univoque points in base $\beta.$ It is well-
known that for any $\beta\in(\beta_{*},2)$, $U_{\beta}$ is a uncountable set,
where $\beta_{*}$ is the Komornik-Loreti constant [1, 8, 12]. Now, we state
the final result of this paper.

######  Theorem 1.9.

Let $1<\beta<2.$ Let $G:[0,1)\to[0,1)$ be a map defined by $G(x)=\beta x\mod
1.$ Suppose that $A$ is a nowhere dense invariant set of $G$. Then for any
$t\in(0,1)$ such that

$\\{\beta^{k}t\mod 1:k\geq 0\\}$

is dense in $[0,1)$, we have

$tA\subset\mathbb{Q}^{c}.$

Moreover, for any $t\in(1,+\infty)$ such that

$\\{\beta^{k}(1/t)\mod 1:k\geq 0\\}$

is dense in $[0,1)$, then $(1/t)A\subset\mathbb{Q}^{c}.$

In particular, if $A=U_{\beta}\cap\left[\dfrac{2-\beta}{\beta-1},1\right]$ and
$\beta_{*}<\beta<2$, then for some appropriate $t$’s,
$tA\subset\mathbb{Q}^{c}\mbox{ or }(1/t)A\subset\mathbb{Q}^{c}.$

This paper is arranged as follows. In Section 2, we give the proofs of main
results. In Section 3, we give some problems.

##  2\. Proofs of main results

###### Proof of Theorem 1.3.

We only prove $A+t,t^{-1}A\subset\mathbb{Q}^{c}$. The other two inclusions can
be proved analogously. First, we show $A+t\subset\mathbb{Q}^{c}$. Suppose that
there exists some $x\in A$ such that

$x+t=r\in\mathbb{Q}.$

Then we consider

$\\{m^{k}(r-x)\mod 1:k\geq 0\\}=\\{m^{k}t\mod 1:k\geq 0\\}.$

Note that the right side of the above equation is dense in $[0,1]$. This is
because $t$ has a universal expansion. For the left side, it is not dense.
Therefore, we obtain a contradiction and finish the proof.

Now, we prove the left side of the above equation is not dense. We let

$r=\kappa_{1}/\kappa_{2},\kappa_{i}\in\mathbb{Z}.$

Therefore,

$m^{k}(r-x)\mod 1=(m^{k}r-m^{k}x)\mod 1.$

Since $x\in A$, there exists an $m$-adic expansion

$(x_{i})\in\\{0,1,2,\cdots,m-1\\}^{\mathbb{N}}$

such that

$x=\sum_{i=1}^{\infty}\dfrac{x_{i}}{m^{i}}.$

Since $A$ is invariant, it follows that $T(x)\in A.$ Therefore,

$x^{\prime}:=m^{k}x\mod
1=T^{k}(x)=\sum_{i=1}^{\infty}\dfrac{x_{k+i}}{m^{i}}\in A.$

Now, we have

$m^{k}(r-x)\mod 1=(m^{k}r-m^{k}x)\mod
1=m^{k}\dfrac{\kappa_{1}}{\kappa_{2}}-x^{\prime}\mod 1.$

Therefore,

$\\{m^{k}(r-x)\mod 1:k\geq
0\\}\subset\cup_{j=-1}^{1}\cup_{i=0}^{2\kappa_{2}}\left(\dfrac{i}{\kappa_{2}}+j-A\right)\mod
1.$

Here $a-F=\\{a-x:x\in F\\},a\in\mathbb{R}.$

It is well-known that any finite union of nowhere dense sets is still nowhere
dense. Hence, $\\{m^{k}(r-x)\mod 1:k\geq 0\\}$ is not dense in $[0,1)$. We
finish the proof of $A+t\subset\mathbb{Q}^{c}.$

Next, we prove that $t^{-1}A\subset\mathbb{Q}^{c}$. We still use some notation
when we prove $A+t\subset\mathbb{Q}^{c}.$ Suppose that there exists some $x\in
A$ such that

$t^{-1}x=r\in\mathbb{Q}.$

We let

$r=\dfrac{\kappa_{3}}{\kappa_{4}},$

where $\kappa_{i}\in\mathbb{Z}.$ Note that

$\\{m^{k}(r^{-1}x)\mod 1:k\geq 0\\}=\\{m^{k}(t)\mod 1:k\geq 0\\}.$

The right side of the above equation is dense. Now, we prove that the left
side is not dense. Notice that

$\displaystyle m^{k}(r^{-1}x)\mod 1$ $\displaystyle=$ $\displaystyle
r^{-1}m^{k}x\mod 1=r^{-1}\left(m^{k}x\right)\mod 1$ $\displaystyle=$
$\displaystyle r^{-1}\left(\sum_{i=1}^{k}m^{k-i}x_{i}+T^{k}(x)\right)\mod 1$
$\displaystyle=$ $\displaystyle
r^{-1}\left(\sum_{i=1}^{k}m^{k-i}x_{i}+x^{\prime}\right)\mod 1$
$\displaystyle\subset$
$\displaystyle\left(r^{-1}A+\dfrac{\kappa_{4}}{\kappa_{3}}M_{k}\right)\mod 1,$

where $M_{k}:=\sum_{i=1}^{k}m^{k-i}x_{i}\in\mathbb{N}^{+}$. Note that

$\left\\{\left(r^{-1}A+\dfrac{\kappa_{4}}{\kappa_{3}}M_{k}\right)\mod 1:k\geq
0\right\\}\subset\left(\left\\{r^{-1}A+\dfrac{i}{\kappa_{3}}\mod
1\right\\}_{i=0}^{\kappa_{3}}\right).$

We observe that for each $i$,

$r^{-1}A+\dfrac{i}{\kappa_{3}}\mod 1$

is a nowhere dense set as $A$ is nowhere dense. Therefore,

$\\{m^{k}(r^{-1}x)\mod 1:k\geq 0\\}$

is contained in a union of some nowhere dense sets. Hence, we have proved that

$\\{m^{k}(r^{-1}x)\mod 1:k\geq 0\\}$

is not dense in $[0,1]$. ∎

###### Proof of Corollary 1.5.

We need to check that the survival set is invariant. By the definition of
survival set, we have that for any $x\in A$, $T(x)\in A$. Therefore,
$T(A)\subset A.$ By the definition of open dynamics, the survival set $A$ is
closed. Therefore, to prove $A$ is nowhere dense, it suffices to prove that
$A$ does not have interior. Otherwise, if $A$ contains some interval, we
denote it by $(\delta_{1},\delta_{2})$. It is well-known by the ergodic
theorem that for Lebesgue almost every $x\in(\delta_{1},\delta_{2})$, the
orbit of $x$ under the map $T$ is dense in $[0,1]$. This is clearly a
contradiction as the the orbits of points from the survival set never hit the
hole $(a,b)$. We finish the proof. ∎

###### Proof of Corollary 1.6.

The proof is similar to that of Theorem 1.3. We leave it to the reader. ∎

###### Proof of Theorem 1.7.

We recall two well-known results on the sequences of uniform distribution
[13].

Claim1: Let $q\in\mathbb{N}_{\geq 2}$, $t\in(0,1)$. Then

$\\{(q^{k}t):k\geq 0\\}$

is of uniform distribution in $[0,1]$ if and only if $t$ is a normal number in
base $q$, where $(q^{k}t)$ denotes the fractional part of $q^{k}t.$

Claim2: If

$\\{(q^{k}t):k\geq 0\\}$

is of uniform distribution in $[0,1]$, then for any
$s\in\mathbb{Z}\setminus\\{0\\}$, we have

$\\{(q^{k}ts):k\geq 0\\}$

is of uniform distribution in $[0,1]$.

We only prove

$A+it\subset\mathbb{Q}^{c},i\neq 0.$

If there is some $x\in A$ such that

$x+it=r\in\mathbb{Q}.$

Then we consider

(2.1) $m^{k}(r-x)\mod 1=m^{k}it\mod 1,k\geq 0.$

Since $t$ is normal, it follows by Claim 1 and Claim 2 that

$\\{m^{k}it:k\geq 0\\}$

is uniformly distributed modulo $1$. However, the left side of equation (2.1)
is not dense in $[0,1]$ (the reader may refer to the proof of Theorem 1.3). We
find a contradiction. ∎

###### Proof of Theorem 1.8.

We only prove $K+t\subset\mathbb{Q}^{c}$. The other two cases can be proved in
a similar way. Suppose on contrary that there exists some $x\in K$ such that

$x+t=r\in\mathbb{Q}.$

Then we have

$q^{k}(r-x)\mod 1=q^{k}t\mod 1.$

Since $x\in K$, it follows that there exists a greedy expansion (roughly
speaking, a greedy expansion is largest in the sense of lexicographical order,
the reader may refer to [3] for more information)
$(b_{i})\in\\{a_{1},a_{2},\cdots,a_{n}\\}^{\mathbb{N}}$ such that

$x=\sum_{k=1}^{\infty}\dfrac{b_{k}}{q^{k}}.$

Therefore,

$\\{q^{k}(r-x)\mod 1:k\geq 0\\}\subset(F-K(\mod 1))\cap[0,1],$

where

$F=\left(\bigcup_{t_{i},k\in\mathbb{N},0\leq i\leq
n}(q^{k}r-(t_{1}a_{1}+t_{2}a_{2}+\cdots t_{n}a_{n}))\right),$

and $F-K(\mod 1)=\\{y-x\mod 1:x\in K,y\in F\\}.$ However, we can only take
finite values from $F(\mod 1)=\\{x\mod 1:x\in F\\}$. This is because
$a_{i},r\in\mathbb{Q},1\leq i\leq n$. We may assume

$r=\dfrac{C}{D},a_{i}=\dfrac{B_{i}}{A_{i}},$

where

$C,D,A_{i},B,B_{i}\in\mathbb{Z}^{+}.$

Note that $K\subset[0,1]$. Hence, if we take some $y\in F$ which is bigger
than $2$, then

$y-K(\mod 1)=y-1-K(\mod 1).$

In other words, when we consider $F-K(\mod 1)$, we can take only finitely many
values from $F$. More precisely, we may take at most

$2D(\Pi_{i=1}^{n}A_{i})+1$

numbers from $F$.

Hence, $F-K(\mod 1)$ is contained in a union of finitely many nowhere dense
sets. Therefore, $\\{q^{k}(r-x)\mod 1:k\geq 0\\}$ is also nowhere dense. This
contradicts with the fact

$\\{q^{k}t\mod 1:k\geq 0\\}$

is dense in $[0,1].$ ∎

###### Proof of Theorem 1.9.

We only prove $t^{-1}A\subset\mathbb{Q}^{c}$. The other one can be proved in a
similar way. Suppose that there exists some $x\in A$ such that

$t^{-1}x=r\in\mathbb{Q}.$

We let

$r=\dfrac{\kappa_{5}}{\kappa_{6}},$

where $\kappa_{i}\in\mathbb{Z}.$ Note that

$\\{\beta^{k}(r^{-1}x)\mod 1:k\geq 0\\}=\\{\beta^{k}t\mod 1:k\geq 0\\}.$

The right side of the above equation is dense. The left side, however, is not.
Now, we prove that the left side is not dense. Notice that

$\displaystyle\beta^{k}(r^{-1}x)\mod 1$ $\displaystyle=$ $\displaystyle
r^{-1}\beta^{k}x\mod 1=r^{-1}\left(\beta^{k}x\right)\mod 1$ $\displaystyle=$
$\displaystyle r^{-1}\left(\sum_{i=1}^{k}\beta^{k-i}x_{i}+G^{k}(x)\right)\mod
1$ $\displaystyle=$ $\displaystyle
r^{-1}\left(\sum_{i=1}^{k}\beta^{k-i}x_{i}+x^{\prime}\right)\mod 1$
$\displaystyle\subset$
$\displaystyle\left(r^{-1}A+\dfrac{\kappa_{6}}{\kappa_{5}}M_{k}\right)\mod 1,$

where $x=\sum_{i=1}^{\infty}\dfrac{x_{i}}{\beta^{i}}$ has a greedy expansion
$(x_{i})$ in base $\beta$ ([3]), and

$M_{k}:=\sum_{i=1}^{k}m^{k-i}x_{i}\in\mathbb{N}^{+}.$

Note that

$\left\\{\left(r^{-1}A+\dfrac{\kappa_{6}}{\kappa_{5}}M_{k}\right)\mod 1:k\geq
0\right\\}\subset\left(\left\\{r^{-1}A+\dfrac{i}{\kappa_{5}}\mod
1\right\\}_{i=0}^{\kappa_{5}}\right).$

We observe that for each $i$,

$r^{-1}A+\dfrac{i}{\kappa_{5}}\mod 1$

is a nowhere dense set as $A$ is nowhere dense. Therefore,

$\\{\beta^{k}(r^{-1}x)\mod 1:k\geq 0\\}$

is contained in a union of some nowhere dense sets. Hence, we have proved that

$\\{\beta^{k}(r^{-1}x)\mod 1:k\geq 0\\}$

is not dense in $[0,1]$.

For the univoque set, we have the following simple properties. First, we
always have

$G\left(U_{\beta}\cap\left[\dfrac{2-\beta}{\beta-1},1\right]\right)\subset
U_{\beta}\cap\left[\dfrac{2-\beta}{\beta-1},1\right].$

Next, for any $x\in U_{\beta}$, there exists some $k\in\mathbb{N}$ such that
for any $i\geq k$

$G^{i}(x)\in\left[\dfrac{2-\beta}{\beta-1},1\right].$

In terms of the above two properties, we have the desired results for the
univoque sets. ∎

##  3\. Some questions

We pose some problems.

  * (1)

How can we find a $t$ such that $C+t$ only consists of transcendental numbers,
where $C$ is the ternary Cantor set.

  * (2)

For any two non-trivial invariant sets, denoted by $A$ and $B$, under the maps
$T_{2}$ and $T_{3}$, respectively, then how can we find a concrete uniform $t$
such that

$A+t,B+t\subset\mathbb{Q}^{c},$

where $T_{2}(x)=2x\mod 1\mbox{ and }T_{3}(x)=3x\mod 1$ are defined on $[0,1)$.

  * (3)

In Theorem 1.3, we do not know whether for any $t\in(0,1)$ such that $t$ has a
universal $m$-adic expansion, we always have

$tA=\\{ta:a\in A\setminus\\{0\\}\\}\subset\mathbb{Q}^{c}.$

For the last question, the only thing we know is the following fact from
uniform distribution modulo one. Let $q\in\mathbb{N}_{\geq 2}$. Then for
Lebesgue almost every $a$,

$\\{q^{k}a\mod 1:k\geq 0\\}$

is of uniform distribution in the unit interval. Therefore, for Lebesgue
almost every $a$, we simultaneously have that

$\\{q^{k}a\mod 1:k\geq 0\\}$

and

$\\{q^{k}a^{-1}\mod 1:k\geq 0\\}$

is of uniform distribution in $[0,1]$. By the above analysis, we claim that
for Lebesgue almost every $t\in(0,1)$,

$A-t\subset\mathbb{Q}^{c},A+t\subset\mathbb{Q}^{c},tA\subset\mathbb{Q}^{c},\dfrac{A}{t}\subset\mathbb{Q}^{c}.$

Throughout the paper, we essentially analyze the following problem. Let

$\beta\in(1,\infty)$

be a real, and $r$ be a rational number. Then when do we have that the
following set

$\\{\beta^{k}r\mod 1:k\geq 0\\}$

takes only finitely many values. More generally, how can we find the non-
trivial lower and upper bounds of the above set. This is related to the theory
of sequences modulo one, see [5, 7] and references therein. For the above
problem, it is natural to consider some Pisot numbers and rational numbers. In
[5], Dubickas proved that when $\beta$ is a Pisot number (an algebraic number
is called a Pisot number if all of whose other Galois conjugates have modulus
strictly less than $1$) and $r$ is rational, then

$\\{\beta^{k}r\mod 1:k\geq 0\\}$

has finitely many limit points.

In [7], Flatto, Lagarias, and Pollington considered when $\beta$ is a rational
number, they give the lower bound of the length of

$\\{\beta^{k}r\mod 1:k\geq 0\\}.$

All these results are useful to our analysis on the translations of the
invariant sets and self-similar sets.

## Acknowledgements

This work is supported by K.C. Wong Magna Fund in Ningbo University. This work
is also supported by Zhejiang Provincial Natural Science Foundation of China
with No.LY20A010009. The author would like to thank Simon Baker and Derong
Kong for some discussions on the uniform distribution of sequences.

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# Relic Gravitational Waves from the Chiral Plasma Instability
in the Standard Cosmological Model

Axel Brandenburg<EMAIL_ADDRESS>Nordita, KTH Royal Institute of
Technology and Stockholm University, 10691 Stockholm, Sweden  Department of
Astronomy, AlbaNova University Center, Stockholm University, 10691 Stockholm,
Sweden  School of Natural Sciences and Medicine, Ilia State University, 0194
Tbilisi, Georgia  McWilliams Center for Cosmology and Department of Physics,
Carnegie Mellon University, Pittsburgh, PA 15213, USA  Emma Clarke
111Corresponding author; the authors are listed alphabetically.
<EMAIL_ADDRESS>McWilliams Center for Cosmology and Department of
Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA  Tina
Kahniashvili<EMAIL_ADDRESS>McWilliams Center for Cosmology and
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
School of Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi,
Georgia  Abastumani Astrophysical Observatory, Tbilisi, GE-0179, Georgia
Andrew J. Long<EMAIL_ADDRESS>Department of Physics and Astronomy, Rice
University, 6100 Main St., Houston, TX 77005, USA  Guotong Sun
<EMAIL_ADDRESS>McWilliams Center for Cosmology and Department of
Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA

###### Abstract

In the primordial plasma, at temperatures above the scale of electroweak
symmetry breaking, the presence of chiral asymmetries is expected to induce
the development of helical hypermagnetic fields through the phenomenon of
chiral plasma instability. It results in magnetohydrodynamic turbulence due to
the high conductivity and low viscosity and sources gravitational waves that
survive in the universe today as a stochastic polarized gravitational wave
background. In this article, we show that this scenario only relies on
Standard Model physics, and therefore the observable signatures, namely the
relic magnetic field and gravitational background, are linked to a single
parameter controlling the initial chiral asymmetry. We estimate the magnetic
field and gravitational wave spectra, and validate these estimates with 3D
numerical simulations.

††preprint: NORDITA-2023-034

##  I Introduction

The excess of matter over antimatter on cosmological scales in the universe
today is well measured but its origin is not yet established. In studies of
early universe cosmology, it is typically assumed that the matter-antimatter
asymmetry arose dynamically in the first fractions of a second after the Big
Bang through a process called baryogenesis Kolb:1979qa . In addition to
creating the baryon asymmetry, e.g., the excess of nuclei over antinuclei,
baryogenesis may have created other (possibly unstable) particle asymmetries
as well; a few examples include lepton asymmetry Fukugita:1986hr , Higgs
asymmetry Servant:2013uwa , neutrino asymmetry Dick:1999je ; Murayama:2002je ,
and right-chiral electron asymmetry Campbell:1990fa . Some of these are
examples of chiral asymmetries, $n_{5}=n_{R}-n_{L}$, namely an excess (or
deficit) of right-chiral particles and antiparticles over their left-chiral
partners. A particular linear combination of various particle asymmetries,
which we call the hypercharge-weighted chiral asymmetry, has attracted
interest because of its connections with primordial magnetogenesis
Joyce:1997uy  through a phenomenon known as the chiral plasma instability
Akamatsu:2013pjd .

The primordial magnetic field may survive in the universe today as an
intergalactic magnetic field, thereby opening a pathway to test this scenario
Neronov:2010gir ; Vachaspati:2020blt . In addition, the primordial magnetic
field and its interaction with the turbulent plasma are expected to source
gravitational radiation; see Ref. Deryagin:1986qq  for pioneering work and
Ref. Brandenburg:2021aln  for numerical simulations of the gravitational waves
induced by the primordial magnetic field originating from the chiral plasma
instability. The production of magnetic fields (possibly dark fields) and
gravitational wave radiation has also been extensively explored in a different
class of theories where the role of the chemical potential is played by axions
or axion-like particles Anber:2009ua ; Barnaby:2012xt ; Domcke:2016bkh ;
Machado:2018nqk . In our work, we investigate the gravitational wave
signatures of a primordial hypercharge-weighted chiral asymmetry via the
chiral plasma instability.

Contrary to earlier numerical simulations, we study here a parameter regime
that is more realistic in various respects. The resulting gravitational wave
energy from our simulations confirm the scaling with the sixth power of the
chiral chemical potential and the fifth power of the inverse square root of
the chiral dilution parameter, which can be combined into a single parameter,
as already found previously Brandenburg:2021aln .

##  II Description of the model

We consider the primordial Standard Model plasma at temperatures $T\gtrsim
100\ \mathrm{TeV}$ in the phase of unbroken electroweak symmetry. We remain
agnostic as to the physics of baryogenesis, but assume that a nonzero
hypercharge-weighted chiral asymmetry is present in the plasma initially. We
study the growth of an initially vanishingly small hypermagnetic field via the
chiral plasma instability and calculate the resulting gravitational wave
radiation. The hypermagnetic field generated by the chiral plasma instability
is always maximally helical and therefore also leads to the production of
maximally circularly polarized gravitational waves. The present work is
conceptually different from that of Refs. Brandenburg:2023rul ;
Brandenburg:2023aco , where a helical magnetic field was present initially
such that the net chirality of the system was balanced to zero by a fermion
chirality of opposite sign.

One appealing aspect of our approach is its minimalism: we only assume
Standard Model particle physics and the standard cosmological model after
reheating. Our only free parameter is the initial hypercharge-weighted chiral
asymmetry, which presumably arises from physics beyond the Standard Model. We
work in the Lorentz-Heaviside unit system with $\hbar=c=k_{B}=1$. We account
for the cosmological expansion using an Friedmann-Lemaître-Robertson-Walker
metric with dimensionless scale factor $a(t)$ and set $a_{0}=1$ today. Unless
otherwise specified, all dimensionful variables are comoving; this includes
conformal time $\mathrm{d}t=\mathrm{d}t_{\mathrm{phys}}/a$, comoving magnetic
field ${\bm{B}}=a^{2}{\bm{B}}_{\mathrm{phys}}$, comoving magnetic correlation
length $\xi_{\rm M}=\xi_{\mathrm{M,phys}}/a$, comoving temperature
$T=aT_{\mathrm{phys}}$, comoving wave number $k=ak_{\mathrm{phys}}$, comoving
Hubble parameter $H=aH_{\mathrm{phys}}$ (with $H\equiv(da/dt)/a$), and
comoving energy density of any relativistic component (including frozen-in
magnetic fields, gravitational waves, etc)
$\mathcal{E}=\mathcal{E}_{\mathrm{phys}}a^{4}$. We denote Newton’s
gravitational constant by $G$, the Planck mass by
$M_{\mathrm{Pl}}=1/\sqrt{G}=1.2\times 10^{19}\,\mathrm{GeV}$, the physical
Hubble constant by
$H_{\mathrm{phys},0}=100h_{0}\,\mathrm{km}/\mathrm{sec}/\mathrm{Mpc}$, and the
critical energy density today by $\mathcal{E}_{\mathrm{cr}}=3H_{0}^{2}/(8\pi
G)$. We use the subscript “CPI” to denote the time when the chiral plasma
instability (CPI) develops. Assuming that the plasma’s entropy density is
conserved between the CPI epoch and today leads to the relation $g_{\ast
S,{\text{\sc cpi}}}a_{\text{\sc cpi}}^{3}T_{\mathrm{phys},{\text{\sc
cpi}}}^{3}=g_{\ast S,0}a_{0}^{3}T_{\mathrm{phys},0}^{3}$. Taking $g_{\ast
S,0}=3.91$ and $T_{\mathrm{phys},0}=0.234\,\mathrm{meV}$ gives

$\frac{a_{\text{\sc cpi}}}{a_{0}}=(8\times 10^{-19})\bigg{(}\frac{g_{\ast
S,{\text{\sc
cpi}}}}{106.75}\bigg{)}^{-1/3}\bigg{(}\frac{T_{\mathrm{phys},{\text{\sc
cpi}}}}{100\,{\rm TeV}}\bigg{)}^{-1}\;.$ (1)

We fiducialize the effective number of relativistic degrees of freedom during
the CPI epoch to $g_{\ast S,{\text{\sc cpi}}}=106.75$, which is the expected
value for Standard Model cosmology at temperatures above $100\,\mathrm{GeV}$.
We fiducialize the physical plasma temperature at the CPI epoch to
$T_{\mathrm{phys},{\text{\sc cpi}}}=100\,\mathrm{TeV}$.

### Chiral magnetic effect.

The chiral plasma instability and chiral magnetic effect (CME) were first
studied in the context of a relativistic electron-positron plasma described by
quantum electrodynamics (QED). Although chirality is conserved at the
classical level for massless electrons, chirality is broken in the quantum
theory and this is expressed by the Adler-Bell-Jackiw axial anomaly
Bell:1969ts ; Adler:1969gk . A manifestation of the anomalous chiral symmetry
is the CME Vilenkin:1980fu : in a QED plasma that possesses a chiral
asymmetry, a magnetic field induces a proportional current. The CME
corresponds to an anomalous contribution to the electric current density
${\bm{J}}({\bm{x}},t)=\mu_{5}(t){\bm{B}}({\bm{x}},t)$ where
$\mu_{5}=2\alpha\tilde{\mu}_{5}/\pi$ is proportional to the chiral chemical
potential $\tilde{\mu}_{5}$, $\alpha=e^{2}/4\pi\approx 1/137$ is the
electromagnetic fine structure constant, and ${\bm{B}}$ is the magnetic field.
Implications of the CME for a turbulent QED plasma have been studied
extensively with a combination of analytical techniques and numerical
simulations Boyarsky:2011uy ; Boyarsky:2012ex ; Boyarsky:2015faa ;
Brandenburg:2017rcb ; Brandenburg:2021aln ; Brandenburg:2023aco ; see also
Ref. Kamada:2022nyt  for a recent review article.

### Adaptation to hypercharge.

The formalism used to study the CME in QED is easily adapted to the
hypercharge sector of the Standard Model for a plasma in the phase of unbroken
electroweak symmetry at temperatures $T_{\mathrm{phys}}\gtrsim 100\
\mathrm{GeV}$. The quantity of interest is the hypercharge-weighted chiral
chemical potential $\tilde{\mu}_{Y,5}(t)$, which is given by
$\tilde{\mu}_{Y,5}(t)=\sum_{i}\varepsilon_{i}g_{i}Y_{i}^{2}\tilde{\mu}_{i}(t)$,
where the sum runs over all Standard Model particle species (indexed by $i$),
$\varepsilon_{i}=\pm 1$ for right/left-chiral particles (and $0$ otherwise),
$g_{i}$ is a multiplicity factor (counting color, spin, etc), $Y_{i}$ is the
hypercharge of species $i$, and $\tilde{\mu}_{i}$ is the chemical potential
that parameterizes the asymmetry (excess of particles over antiparticle
partners) in species $i$ via $n_{i}\propto\tilde{\mu}_{i}T^{2}$; see Refs.
Kamada:2016eeb ; Kamada:2016cnb  for additional details.

### Chiral plasma instability.

In the presence of a chiral asymmetry, the equations of magnetohydrodynamics
(MHD) are modified due to the CME, and the new equations exhibit a tachyonic
instability toward the growth of long-wavelength modes of the magnetic field,
which is known as the chiral plasma instability Akamatsu:2013pjd . To
illustrate the instability in the hypercharge sector of the primordial plasma,
we present the evolution equation for the hyper-magnetic field assuming
negligible plasma velocity:
$\dot{\bm{B}}_{Y}=\eta_{Y}\nabla^{2}{\bm{B}}_{Y}+(2\alpha_{Y}\tilde{\mu}_{Y,5}/\pi)\eta_{Y}{\bm{\nabla}}\times{\bm{B}}_{Y}$.
Here and below, dots represent partial derivatives with respect to conformal
time, ${\bm{B}}_{Y}({\bm{x}},t)$ is the hypermagnetic field,
$\eta_{Y}=1/\sigma_{Y}$ is the hypermagnetic diffusivity, $\sigma_{Y}$ is the
hypercharge conductivity, and $\alpha_{Y}=g^{\prime 2}/4\pi\approx 0.01$ is
the hypercharge fine structure constant. Long-wavelength modes of the
hypermagnetic field with wave number $k<k_{\text{\sc
cpi}}=2\alpha_{Y}|\tilde{\mu}_{Y,5}(t)|/\pi$ experience a tachyonic
instability in one of the two circular polarization modes, and their amplitude
increases exponentially $\propto\mathrm{exp}(t/t_{\text{\sc cpi}})$. The
fastest growing modes have $k=k_{\text{\sc cpi}}/2$, and for these modes
$t_{\text{\sc cpi}}=4/\eta_{Y}k_{\text{\sc
cpi}}^{2}=\pi^{2}/\eta_{Y}\alpha_{Y}^{2}|\tilde{\mu}_{Y,5}|^{2}$. Assuming a
radiation-dominated cosmology with $g_{\ast}=106.75$, the physical plasma
temperature at this time is

$T_{\mathrm{phys},{\text{\sc cpi}}}=\bigl{(}70\
\mathrm{TeV}\bigr{)}\biggl{(}\frac{\eta_{Y}}{0.01T^{-1}}\biggr{)}\biggl{(}\frac{|\tilde{\mu}_{Y,5}|/T}{10^{-3}}\biggr{)}^{2}\;.$
(2)

In other words, although the chiral asymmetry may be present in the plasma
from a very early time, its effect on the hypermagnetic field does not develop
until (possibly much) later when the age of the universe is comparable to
$t_{\text{\sc cpi}}$ and the plasma has cooled to temperature
$T_{\mathrm{phys},{\text{\sc cpi}}}$. Reducing the magnitude of the chiral
asymmetry, i.e., assuming a smaller $|\tilde{\mu}_{Y,5}|/T$ initially, delays
the onset of the chiral plasma instability.

### Chiral asymmetry erasure.

In a relativistic electron-positron plasma described by the theory of QED, the
electromagnetic charge is exactly conserved and the chiral charge is
approximately conserved. The violation of chiral charge conservation derives
from both the chiral anomaly, which leads to the phenomenon of chiral plasma
instability discussed above, as well as explicit breaking induced by the
nonzero electron mass. The chiral charge changes in a scattering that converts
right-chiral particles into left-chiral particles, or vice versa, and the rate
for such ‘spin-flip’ scatterings is proportional the squared electron mass
$(m_{e}/T)^{2}$. Although the chiral charge is not exactly conserved, it is
important to recognize that it is approximately conserved on time scales that
are small compared to the inverse spin-flip rate. Similarly, the hypercharge-
weighted chiral asymmetry is eventually driven to zero by scatterings
involving the Yukawa couplings; the most relevant processes are Higgs decays
and inverse decays with right-chiral electrons. The rate for these chirality-
changing reactions is $\Gamma_{\mathrm{f}}\approx 10^{-2}y_{e}^{2}T$, with
$y_{e}$ electron Yukawa coupling $y_{e}=\sqrt{2}m_{e}/v\simeq 3\times 10^{-6}$
and assuming a standard radiation-dominated cosmology, these reactions come
into equilibrium when the plasma cools to a physical temperature of
$T_{\mathrm{phys},\mathrm{f}}\simeq 80\ \mathrm{TeV}$ Bodeker:2019ajh . To
ensure that the chiral plasma instability develops before the hypercharge-
weighted chiral asymmetry is erased by Higgs decays and inverse decays, it is
necessary to have $|\tilde{\mu}_{Y,5}|/T>10^{-3}$. For reference, the observed
baryon asymmetry of the universe today corresponds to a much smaller chemical
potential of $\tilde{\mu}_{\text{\sf B}}/T\approx 10^{-8}$, but it is not
unusual for large chemical potentials to be generated during the course of
baryogenesis. New physics such as a matter-dominated phase or an injection of
$e_{R}$ asymmetry can change the temperature of chiral asymmetry erasure; for
an example, see Ref. Chen:2019wnk .

### Magnetogenesis.

As the chiral plasma instability develops, the growing helical hypermagnetic
field is accompanied by a depletion of the hypercharge-weighted chiral
asymmetry. This is because the hypercharge-weighted chiral number density
$n_{Y,5}=\tilde{\mu}_{Y,5}T^{2}/6$ and the hypermagnetic helicity
$\mathcal{H}_{B}$ are linked by the chiral anomaly, which imposes
$\dot{n}_{Y,5}\propto-\alpha_{Y}\dot{\mathcal{H}}_{\rm M}/\pi$ Joyce:1997uy .
If the chiral plasma instability shuts off after the hypercharge-weighted
chiral asymmetry depletes by an order one factor, the hypermagnetic helicity
can be estimated as $\mathcal{H}_{{\rm M},{\text{\sc
cpi}}}\sim\pi|\tilde{\mu}_{Y,5}|T_{\text{\sc cpi}}^{2}/6\alpha_{Y}$. The
coherence length and field strength are estimated as $\xi_{{\rm M},{\text{\sc
cpi}}}\approx 2\pi/(k_{\text{\sc cpi}}/2)$ and $B_{\text{\sc
cpi}}\approx\sqrt{\mathcal{H}_{\rm M}/\xi_{{\rm M},{\text{\sc cpi}}}}$, which
gives $\xi_{{\rm M},{\text{\sc cpi}}}\approx(5\times 10^{5}\
\mathrm{cm})(|\tilde{\mu}_{Y,5}|/10^{-3}T)^{-1}$ and $B_{\text{\sc
cpi}}\approx(5\times 10^{-11}\ \mathrm{G})\,(|\tilde{\mu}_{Y,5}|/10^{-3}T)$.
If the magnetic field evolves according to the inverse cascade scaling (in the
fully helical case), $\xi_{\rm M}\propto t^{2/3}$ and $B\propto t^{-1/3}$
Hat84 , until recombination, then the physical coherence length and field
strength today (assuming a frozen-in magnetic field and neglecting MHD
dynamics at late epochs, after re-ionization) are expected to be on the order
of

$\begin{split}\xi_{\mathrm{M},\mathrm{phys},0}&=\bigl{(}9\times 10^{-4}\
\mathrm{pc}\bigr{)}\biggl{(}\frac{\eta_{Y}}{0.01T^{-1}}\biggr{)}^{2/3}\biggl{(}\frac{|\tilde{\mu}_{Y,5}|/T}{10^{-3}}\biggr{)}^{1/3}\;,\\\
B_{\mathrm{phys},0}&=\bigl{(}7\times 10^{-16}\
\mathrm{G}\bigr{)}\biggl{(}\frac{\eta_{Y}}{0.01T^{-1}}\biggr{)}^{-1/3}\biggl{(}\frac{|\tilde{\mu}_{Y,5}|/T}{10^{-3}}\biggr{)}^{1/3}\;.\end{split}$
(3)

A larger chiral asymmetry leads to a stronger magnetic field on larger length
scales today.

### Gravitational wave generation.

The time-varying quadrupole moment of the growing hypermagnetic field provides
a source of gravitational wave radiation Deryagin:1986qq . As the chiral
plasma instability develops, most of the magnetic energy is carried by the
modes with coherence length $\xi_{{\rm M},{\text{\sc cpi}}}$ (i.e., the
magnetic energy is characterized by a spectrum that peaks at wave number
$k_{I}\simeq 2\pi/\xi_{{\rm M},{\text{\sc cpi}}}$). As long as the magnetic
field is still growing, however, the induced gravitational wave spectrum peaks
at the characteristic wave number $k=2/t_{\text{\sc cpi}}=\eta_{Y}k_{\text{\sc
cpi}}^{2}/2$ Brandenburg:2021aln . For $\eta_{Y}k_{\text{\sc cpi}}/2<1$, this
wave number is below the cutoff wave number for gravitational waves,
$k_{\text{\sc cpi}}$. Above this wave number, very little gravitational wave
energy is produced by the chiral plasma instability Brandenburg:2021aln . The
gravitational wave cutoff frequency is $f_{\rm GW}\simeq 2k_{I}/(2\pi)\simeq
2/\xi_{{\rm M},{\text{\sc cpi}}}$ (the factor “2” is due to the quadratic
nature of the source). Since the gravitational waves’ comoving frequency
remains constant, the physical frequency today corresponds to
$f_{\mathrm{GW},0}=2/\xi_{{\rm M},{\text{\sc cpi}}}$. Once the CPI stops and
$\tilde{\mu}_{Y,5}$ becomes depleted, the low wave number part of the
gravitational wave spectrum becomes shallower and the peak moves toward
smaller wave numbers.

The energy density carried by the gravitational waves is estimated as
$\mathcal{E}_{\mathrm{GW}}\sim(G/2\pi)a_{\text{\sc cpi}}^{-2}\xi_{{\rm
M},{\text{\sc cpi}}}^{2}B_{\text{\sc cpi}}^{4}$. This estimate follows from
deriving the energy density $\mathcal{E}_{\mathrm{GW}}({\bm{x}},t)$ in the
standard way222In physical space, we have
${\mathcal{E}}_{\mathrm{GW}}^{\rm(phys)}({\bm{x}}_{\rm phys},t_{\rm
phys})=\langle\partial_{t_{\rm phys}}{h}^{\rm(phys)}_{ij}({\bm{x}}_{\rm
phys},t_{\rm phys})\partial_{t_{\rm phys}}{h}^{\rm(phys)}_{ij}({\bm{x}}_{{\rm
phys}},t_{\rm phys})\rangle/(32\pi G)$ where $h_{ij}^{(\rm
phys)}({\bm{x}}_{\rm phys},t_{\rm phys)})$ is the transverse and traceless
tensor mode of the metric perturbations, and using the gravitational wave
equation $\partial_{t_{\rm phys}}^{2}h_{ij}^{(\rm phys)}-\nabla^{2}_{\rm
phys}h_{ij}^{(\rm phys)}=16\pi GT_{ij}^{(\rm phys)}$ where $T_{ij}^{({\rm
phys})}\sim B_{i}^{\rm(phys)}B_{j}^{\rm(phys)}$ is the transverse and
traceless part of the anisotropic part of the magnetic field stress-energy
tensor, to estimate the field amplitude Gogoberidze:2007an ; RoperPol:2018sap
. Maggiore:1999vm . Next we define
$\Omega_{\mathrm{GW}}=\mathcal{E}_{\mathrm{GW}}/\mathcal{E}_{\mathrm{cr}}$ to
be the gravitational wave energy fraction today.

Numerical estimates give

$\begin{split}f_{\mathrm{GW},0}&=\bigl{(}1\times 10^{5}\
\mathrm{Hz}\bigr{)}\biggl{(}\frac{|\tilde{\mu}_{Y,5}|/T}{10^{-3}}\biggr{)},\\\
\Omega_{\mathrm{GW}}h_{0}^{2}&=\bigl{(}7\times
10^{-39}\bigr{)}\biggl{(}\frac{\eta_{Y}}{0.01T^{-1}}\biggr{)}^{2}\biggl{(}\frac{|\tilde{\mu}_{Y,5}|/T}{10^{-3}}\biggr{)}^{6}\;.\end{split}$
(4)

A larger hypercharge-weighted chiral asymmetry moves the peak of the
gravitational wave spectrum to higher frequencies (since the chiral plasma
instability develops earlier) and increases the gravitational wave strength.
For reference, the LIGO-Virgo-KAGRA gravitational wave interferometer array is
sensitive to a stochastic gravitational wave background at the level of $\sim
10^{-7}$ for frequencies of $\sim 10$–$100\ \mathrm{Hz}$
LIGOScientific:2016jlg . The future space-based detectors such as the Laser
Interferometer Space Antenna (LISA) will push this sensitivity down to $\sim
10^{-12}$ at frequencies of $\sim 1$–$10\ \mathrm{mHz}$ Bartolo:2018qqn ;
Baker:2019nia ; Caprini:2019pxz . At still lower frequencies of $\sim
10^{-9}$–$10^{-7}\ \mathrm{Hz}$, pulsar timing arrays (PTAs), such as Parkes
PTA (PPTA) Reardon:2023gzh , European PTA (EPTA) Antoniadis:2023ott , North
American Nanohertz Observatory for Gravitational Waves (NANOGrav)
NANOGrav:2023hvm , Chinese PTA (CPTA) Xu:2023wog , Indian PTA (InPTA)
ChandraJoshi:2022etw , and MeerKAT Pulsar Timing Array (MPTA) Miles:2022lkg
are sensitive to a stochastic gravitational wave background at the level of
$\Omega_{\mathrm{GW}}h_{0}^{2}\sim 10^{-10}$. Various strategies for probing
higher-frequency gravitational waves, even up to the $\mathrm{GHz}$ band, have
been explored in recent years; see Ref. Aggarwal:2020olq  for a review of
these activities. Nevertheless, a detection of gravitational wave radiation at
the level expected here, even for $|\tilde{\mu}_{Y,5}|/T\approx 1$, seems far
out of reach.

### Baryon number overproduction.

The presence of a helical hypermagnetic field in the early universe is
expected to give rise to a baryon asymmetry Fujita:2016igl ; Kamada:2016eeb ;
Kamada:2016cnb . This is because time-varying hypermagnetic helicity sources
baryon and lepton number through the electroweak anomaly Giovannini:1997eg .
Specifically, the conversion of a hypermagnetic field into an electromagnetic
field at the electroweak epoch at $T_{\mathrm{phys}}\approx 100\ \mathrm{GeV}$
sources baryon number after the electroweak sphaleron has gone out of
equilibrium, leading to a boost in the baryon asymmetry Kamada:2016cnb .

The baryon number can easily be over-produced if the magnetic field strength
is too large. Avoidance of this baryon-number overproduction imposes an upper
bound of $|\tilde{\mu}_{Y,5}|/T\lesssim 10^{-2}$ Domcke:2022uue . This bound
is somewhat uncertain as the baryon production calculation depends on a
detailed modeling of magnetic field evolution at the Standard Model
electroweak crossover Kamada:2016cnb , which is not well understood.

##  III Numerical simulations

In order to validate the preceding estimates, we have performed three-
dimensional numerical simulations using the Pencil Code PC . These simulations
allow us to study the growth and evolution of the magnetic field during the
chiral plasma instability and to evaluate the spectrum of the resulting
gravitational wave radiation.

We model the Standard Model matter and radiation as a single component plasma
of charged particles interacting with the hypermagnetic field. Several
properties of the plasma are relevant to the evolution: the magnetic
diffusivity (for simplicity here and below we suppress the subscript “$Y$”)
$\eta(t)=1/\sigma(t)$, the kinematic viscosity $\nu(t)$, the chiral diffusion
coefficient $D_{5}(t)$, the chiral depletion parameter $\lambda(t)$, and the
chiral chemical potential
$\mu_{50}\equiv\mu_{5}(\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}{},0)=2\alpha\tilde{\mu}_{5}/\pi$
that enters as an initial condition. One can calculate $\sigma$, $\eta$,
$\nu$, and $D_{5}$ from first principles using Standard Model particle
physics. The hypercharge conductivity is predicted to be $\sigma\sim
T/\alpha\approx 100\,T$ Arnold:2000dr  implying $\eta\approx 0.01T^{-1}$, and
we assume for simplicity $\eta=\nu=D_{5}$. The chiral depletion parameter
$\lambda$ arises from the Standard Model chiral anomalies, and past studies
have obtained the prediction $\lambda=192\,\alpha^{2}/T^{2}\simeq
0.02\,T^{-2}$ Rogachevskii:2017uyc ; Brandenburg:2017rcb . The initial chiral
chemical potential can be written as $\mu_{50}\approx(6\times
10^{-6}\,T)(\tilde{\mu}_{5}/T/10^{-3})$ by fiducializing to
$\tilde{\mu}_{5}/T=10^{-3}$.

Given the limited dynamic range of numerical simulations, it is not possible
to set the parameters, $\eta$, $\nu$, $D_{5}$, $\lambda$, and $\mu_{50}$,
equal to the Standard Model predictions. Instead we consider sets of
simulations with different parameters. They can be distinguished by the
relative ordering of the characteristic quantities
$v_{\lambda}=\mu_{50}/(\mathcal{E}_{\mathrm{cr}}\lambda)^{1/2}$ and
$v_{\mu}=\mu_{50}\eta$. We consider runs in regimes I (where
$v_{\lambda}>v_{\mu}$) and II (where $v_{\lambda}<v_{\mu}$).

The simulations solve a coupled system of partial differential equations that
account for MHD and the CME Brandenburg:2017rcb  to determine the evolution of
the magnetic field ${\bm{B}}({\bm{x}},t)$, the energy density of the plasma
$\rho({\bm{x}},t)$, the plasma velocity ${\bm{u}}({\bm{x}},t)$, and the chiral
chemical potential $\mu_{5}({\bm{x}},t)$. In the following, we solve the
following set of equations Rogachevskii:2017uyc

$\displaystyle{\partial\mathchoice{\mbox{\boldmath$\displaystyle
A$}}{\mbox{\boldmath$\textstyle A$}}{\mbox{\boldmath$\scriptstyle
A$}}{\mbox{\boldmath$\scriptscriptstyle A$}}{}\over\partial t}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}{}\times\mathchoice{\mbox{\boldmath$\displaystyle
B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle
B$}}{\mbox{\boldmath$\scriptscriptstyle
B$}}{}+\eta(\mu_{5}\mathchoice{\mbox{\boldmath$\displaystyle
B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle
B$}}{\mbox{\boldmath$\scriptscriptstyle
B$}}{}-\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}{}),$ (5)
$\displaystyle{\partial\mu_{5}\over\partial t}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\mathchoice{\mbox{\boldmath$\displaystyle\nabla$}}{\mbox{\boldmath$\textstyle\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}{}\cdot(\mu_{5}\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}{})-\lambda\eta(\mu_{5}\mathchoice{\mbox{\boldmath$\displaystyle
B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle
B$}}{\mbox{\boldmath$\scriptscriptstyle
B$}}{}-\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}{})\cdot\mathchoice{\mbox{\boldmath$\displaystyle
B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle
B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}{}+D_{5}\nabla^{2}\mu_{5},\;\;$
(6) $\displaystyle{{\rm D}{}\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle u$}}{}\over{\rm D}{}t}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle{2\over\rho}\mathchoice{\mbox{\boldmath$\displaystyle\nabla$}}{\mbox{\boldmath$\textstyle\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}{}\cdot\left(\rho\nu\bm{\mathsf{S}}\right)-{1\over
4}\mathchoice{\mbox{\boldmath$\displaystyle\nabla$}}{\mbox{\boldmath$\textstyle\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}{}\ln\rho+{\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle u$}}{}\over
3}\left(\mathchoice{\mbox{\boldmath$\displaystyle\nabla$}}{\mbox{\boldmath$\textstyle\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}{}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}{}+\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}{}\cdot\mathchoice{\mbox{\boldmath$\displaystyle\nabla$}}{\mbox{\boldmath$\textstyle\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}{}\ln\rho\right)$
(7) $\displaystyle-{\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}{}\over\rho}\left[\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}{}\cdot(\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}{}\times\mathchoice{\mbox{\boldmath$\displaystyle
B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle
B$}}{\mbox{\boldmath$\scriptscriptstyle
B$}}{})+\eta\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}{}^{2}\right]+{3\over
4\rho}\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}{}\times\mathchoice{\mbox{\boldmath$\displaystyle
B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle
B$}}{\mbox{\boldmath$\scriptscriptstyle B$}}{},$
$\displaystyle{\partial\ln\rho\over\partial t}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{4}{3}\left(\mathchoice{\mbox{\boldmath$\displaystyle\nabla$}}{\mbox{\boldmath$\textstyle\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}{}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}{}+\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}{}\cdot\mathchoice{\mbox{\boldmath$\displaystyle\nabla$}}{\mbox{\boldmath$\textstyle\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}{}\ln\rho\right)$
(8)
$\displaystyle+\frac{1}{\rho}\left[\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}{}\cdot(\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle
J$}}{}\times\mathchoice{\mbox{\boldmath$\displaystyle
B$}}{\mbox{\boldmath$\textstyle B$}}{\mbox{\boldmath$\scriptstyle
B$}}{\mbox{\boldmath$\scriptscriptstyle
B$}}{})+\eta\mathchoice{\mbox{\boldmath$\displaystyle
J$}}{\mbox{\boldmath$\textstyle J$}}{\mbox{\boldmath$\scriptstyle
J$}}{\mbox{\boldmath$\scriptscriptstyle J$}}{}^{2}\right],$

where ${\sf
S}_{ij}=(\partial_{j}u_{i}+\partial_{i}u_{j})/2-\delta_{ij}\mathchoice{\mbox{\boldmath$\displaystyle\nabla$}}{\mbox{\boldmath$\textstyle\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}{}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
u$}}{\mbox{\boldmath$\textstyle u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle u$}}{}/3$ are the components of the
rate-of-strain tensor. We solve Eqs (5)–(8) using the Pencil Code
PencilCode:2020eyn , which is a massively parallel MHD code using sixth-order
finite differences and a third-order time stepping scheme.

For discussion of the simulation results, we employ “code units.” Times are
measured in units of $t_{\ast}=1/H_{\ast}$, lengths in units of
$l_{\ast}=c/H_{\ast}$, and energies in units of
$E_{\ast}=\mathcal{E}_{\mathrm{cr},\ast}l_{\ast}^{3}$. Setting
$\hbar=c=k_{B}=1$ we define $H_{\ast}$ by the relation $H_{\ast}=\sqrt{(8\pi
G/3)(\pi^{2}/30)(g_{\ast}T_{\ast}^{4}/a_{\ast}^{2})}$ where $g_{\ast}=100$,
$T_{\mathrm{phys},\ast}=T_{\ast}/a_{\ast}=100\,\mathrm{TeV}$, and
$T_{\ast}=(g_{\ast S,0}/g_{\ast S,\ast})^{1/3}T_{0}\simeq 8\times
10^{-5}\,\mathrm{eV}$. If the CPI develops at a physical plasma temperature of
$100\,\mathrm{TeV}$ then the age of the universe is $\sim 1/H_{\ast}$ and
Hubble-scale Fourier modes have $k\sim H_{\ast}$. The linearized gravitational
wave equations are solved in wave number space RoperPol:2018sap ,

$\frac{\partial^{2}}{\partial
t^{2}}\tilde{h}_{+/\times}(\mathchoice{\mbox{\boldmath$\displaystyle
k$}}{\mbox{\boldmath$\textstyle k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}{},t)+k^{2}\tilde{h}_{+/\times}(\mathchoice{\mbox{\boldmath$\displaystyle
k$}}{\mbox{\boldmath$\textstyle k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}}{},t)={6\,H_{\ast}\over
t\,\mathcal{E}_{\mathrm{cr}}}\tilde{T}_{+/\times}(\mathchoice{\mbox{\boldmath$\displaystyle
k$}}{\mbox{\boldmath$\textstyle k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}}{},t),$ (9)

where $\tilde{h}_{+/\times}={\sf e}_{ij}^{+/\times}({\sf P}_{il}{\sf
P}_{jm}-{\textstyle{1\over 2}}{\sf P}_{ij}{\sf P}_{lm})\,\tilde{{\sf
h}}_{lm}(\mathchoice{\mbox{\boldmath$\displaystyle
k$}}{\mbox{\boldmath$\textstyle k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}}{},t)$ are the Fourier-transformed
$+$ and $\times$ modes of $\textstyle h$, with ${\sf
e}^{+}_{ij}(\mathchoice{\mbox{\boldmath$\displaystyle
k$}}{\mbox{\boldmath$\textstyle k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}{})\,\,=\,e_{i}^{1}e_{j}^{1}-e_{i}^{2}e_{j}^{2}$ and ${\sf
e}^{\times}_{ij}(\mathchoice{\mbox{\boldmath$\displaystyle
k$}}{\mbox{\boldmath$\textstyle k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}{})\,=\,e_{i}^{1}e_{j}^{2}+e_{i}^{2}e_{j}^{1}$ being the linear
polarization basis, $\mathchoice{\mbox{\boldmath$\displaystyle
e$}}{\mbox{\boldmath$\textstyle e$}}{\mbox{\boldmath$\scriptstyle
e$}}{\mbox{\boldmath$\scriptscriptstyle e$}}{}^{1}$ and
$\mathchoice{\mbox{\boldmath$\displaystyle e$}}{\mbox{\boldmath$\textstyle
e$}}{\mbox{\boldmath$\scriptstyle e$}}{\mbox{\boldmath$\scriptscriptstyle
e$}}{}^{2}$ are unit vectors perpendicular to $\textstyle k$ and perpendicular
to each other, and ${\sf P}_{ij}(\mathchoice{\mbox{\boldmath$\displaystyle
k$}}{\mbox{\boldmath$\textstyle k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}}{})=\delta_{ij}-k_{i}k_{j}$ is the
projection operator. $\tilde{T}_{+/\times}$ are defined analogously. We solve
Eq. (9) accurate to second order in the time step and use $1024^{3}$ mesh
points in all of our calculations. Our initial conditions have a weak seed
magnetic field and vanishing plasma velocities, and the chiral chemical
potential is homogeneous and equal to the value given above. At each time
step, we calculate the spectrum of gravitational wave radiation by solving the
gravitational wave equation sourced by the stress-energy of the plasma and
magnetic field; see Ref. RoperPol:2018sap  for details regarding our
computational approach.

Table 1:  Summary of Runs discussed in this paper. Runs B1, B10, A1, and A12
of Ref. Brandenburg:2021aln  are included for comparison. In the last row,
theoretically expected values are listed where
$\eta_{2}=\eta_{Y}/(0.01T^{-1})$ and $\mu_{3}=\tilde{\mu}_{Y,5}/10^{-3}T$.

Run $\eta H_{\ast}$ $(\mathcal{E}_{\mathrm{cr}}\lambda)^{1/2}/H_{\ast}$
$\mu_{50}/H_{\ast}$ $v_{\mu}$ $v_{\lambda}$ $\eta\mu_{50}^{2}/H_{\ast}$
$k_{1}/H_{\ast}$ ${\cal E}_{\rm M}^{\max}/{\mathcal{E}}_{\rm cr}$ ${\cal
E}_{\rm GW}^{\rm sat}/{\mathcal{E}}_{\rm cr}$ $q$ B1 $1\times 10^{-6}$
$2\times 10^{4}$ $10^{4}$ $1\times 10^{-2}$ $5\times 10^{-1}$ $1\times 10^{2}$
$1\times 10^{2}$ $1.6\times 10^{-2}$ $4.7\times 10^{-12}$ $0.027$ B10 $1\times
10^{-3}$ $2\times 10^{4}$ $10^{4}$ $1\times 10^{1}$ $5\times 10^{-1}$ $1\times
10^{5}$ $1\times 10^{2}$ $6.0\times 10^{-2}$ $6.0\times 10^{-9}$ $12$ A1
$1\times 10^{-6}$ $5\times 10^{4}$ $10^{4}$ $1\times 10^{-2}$ $2\times
10^{-1}$ $1\times 10^{2}$ $1\times 10^{2}$ $4.6\times 10^{-3}$ $8.9\times
10^{-14}$ $0.032$ A12 $5\times 10^{-3}$ $5\times 10^{4}$ $10^{4}$ $5\times
10^{1}$ $2\times 10^{-1}$ $5\times 10^{5}$ $5\times 10^{1}$ $9.2\times
10^{-3}$ $3.0\times 10^{-10}$ $18$ X1 $5\times 10^{-8}$ $10^{10}$ $10^{6}$
$5\times 10^{-2}$ $1\times 10^{-4}$ $5\times 10^{4}$ $5\times 10^{3}$
$2.4\times 10^{-9}$ $8.8\times 10^{-31}$ $0.39$ X2 $5\times 10^{-9}$ $10^{10}$
$10^{6}$ $5\times 10^{-3}$ $1\times 10^{-4}$ $5\times 10^{3}$ $5\times 10^{3}$
$2.4\times 10^{-9}$ $1.6\times 10^{-30}$ $0.53$ X3 $5\times 10^{-10}$
$10^{10}$ $10^{6}$ $5\times 10^{-4}$ $1\times 10^{-4}$ $5\times 10^{2}$
$5\times 10^{3}$ $2.4\times 10^{-9}$ $1.1\times 10^{-30}$ $0.44$ X4 $5\times
10^{-11}$ $10^{10}$ $10^{6}$ $5\times 10^{-5}$ $1\times 10^{-4}$ $5\times
10^{1}$ $5\times 10^{3}$ $2.3\times 10^{-9}$ $3.1\times 10^{-31}$ $0.12$ Y1
$5\times 10^{-8}$ $7\times 10^{11}$ $10^{6}$ $5\times 10^{-2}$ $1\times
10^{-6}$ $5\times 10^{4}$ $5\times 10^{3}$ $4.9\times 10^{-13}$ $3.6\times
10^{-38}$ $0.39$ Y2 $5\times 10^{-8}$ $7\times 10^{11}$ $10^{6}$ $5\times
10^{-2}$ $1\times 10^{-6}$ $5\times 10^{4}$ $2\times 10^{3}$ $4.4\times
10^{-13}$ $3.2\times 10^{-37}$ $1.3$ Y3 $5\times 10^{-8}$ $7\times 10^{11}$
$10^{6}$ $5\times 10^{-2}$ $1\times 10^{-6}$ $5\times 10^{4}$ $1\times 10^{3}$
$3.3\times 10^{-13}$ $6.9\times 10^{-37}$ $2.5$ expected $10^{-15}\eta_{2}$
$6\times 10^{12}$ $5\times 10^{7}\mu_{3}$ $6\times 10^{-8}\eta_{2}\mu_{3}$
$8\times 10^{-6}\mu_{3}$ $3\eta_{2}\mu_{3}^{2}$ — $6\times
10^{-15}\mu_{3}^{2}$ $7\times 10^{-39}\eta_{2}^{2}\mu_{3}^{6}$ —

In Table LABEL:Tsummary, we summarize the parameters used in our simulations,
including the smallest wavenumber $k_{1}$, and the key results. We consider
two series of runs that we refer to as X and Y. We also compare with two pairs
of runs, A1 and A12, as well as B1 and B10, both from Ref. Brandenburg:2021aln
. where $\mu/v_{\lambda}$ increases from 0.02 and 0.05 to 20 and 250,
respectively. Also the efficiency parameters increases from 0.03 to 12 and 18,
respectively. The runs of series X and Y are subdivided further into Runs
X1–X4 and Runs Y1–Y3. Our runs of series X have increasing values of $v_{\mu}$
and cross from regime II (for Run X1) into regime I (for Run X4). For the runs
of series X, we take $\eta=\nu=D_{5}=5\times 10^{-11}/H_{\ast}$,
$\mathcal{E}_{\mathrm{cr}}\lambda=10^{20}\,H_{\ast}^{2}$, and
$\mu_{50}=10^{6}\,H_{\ast}$. We also give the efficiency of gravitational wave
production,

$q=(k_{\rm peak}/H_{\ast})\left.\sqrt{{\cal E}_{\rm GW}^{\rm
sat}{\mathcal{E}}_{\rm cr}}\right/{\cal E}_{\rm M}^{\max},$ (10)

where we estimate $k_{\rm peak}=k_{\mu}\min(1,v_{\mu}/v_{\lambda})$ where
$k_{\mu}=\mu_{50}/2$. This means that $k_{\rm peak}=k_{\mu}$ when
$v_{\mu}>v_{\lambda}$ (regime II) and $k_{\rm peak}=k_{\lambda}/4$ when
$v_{\mu}<v_{\lambda}$ (regime I); see Ref. Brandenburg:2017rcb .

In the last row of Table LABEL:Tsummary, we compare with the theoretically
expected values. Obviously, our values of $\eta$ are about seven orders of
magnitude too large. This reflects the fact that our simulations are unable to
capture a sufficiently large range of length scales. Consequently, also our
values of $v_{\mu}$ are by about six orders of magnitude too large.
Furthermore, $\eta\mu_{50}^{2}/H_{\ast}$ is by about four orders of magnitude
too large. Most of the other simulation values in the table are not so far
from the theoretically expected values.

The evolution of the magnetic and gravitational wave energy spectra for Run X4
is shown in Fig. 1. The energy densities may be written as
$\mathcal{E}=\int_{0}^{\infty}\\!\mathrm{d}k\,E(k)$ where $k$ is the wave
number and $E(k)$ is the energy spectrum. For the parameters of Run X4, the
instability length scale corresponds to a wavenumber of $k_{\text{\sc
cpi}}=10^{6}H_{\ast}$, which agrees with the wave number above which $E_{\rm
GW}(k)$ drops sharply. The instability time scale normalized to the Hubble
time is $t_{\text{\sc cpi}}H_{\ast}=0.08$, which is about 100 times longer
than the time step. The magnetic energy spectrum grows initially for modes
with $k\approx k_{\text{\sc cpi}}/2=5\times 10^{5}H_{\ast}$ (see the upper set
of dotted lines in Fig. 1). Later, the peak evolves to smaller $k$ with an
inverse cascade scaling, which is consistent with earlier simulations
Brandenburg:2017rcb . The generated magnetic field is then maximally helical;
see Fig. 8(b) of Ref. Brandenburg:2021bfx .

The gravitational wave energy spectra grow in time as long as the magnetic
energy has not yet reached its maximum. In this phase, as discussed above, the
gravitational wave spectrum is expected to peak at the characteristic wave
number $k=2/t_{\text{\sc cpi}}=\eta k_{\text{\sc cpi}}^{2}/2=25\,H_{\ast}$,
which is here much smaller than $k_{\text{\sc cpi}}=10^{6}H_{\ast}$, but
larger than the horizon wave number, $k=H_{\ast}$. When the magnetic energy
density has reached its maximum value, the gravitational wave spectrum has
nearly saturated and is then approximately independent of $k$ for
$k<k_{\text{\sc cpi}}/2$. In principle, it is possible to have a declining
$k^{-2}$ spectrum in the range $\eta k_{\text{\sc cpi}}^{2}/2\leq k\leq
k_{\text{\sc cpi}}$, but this is only seen in our models with larger
diffusivity. The absence of a $k^{-2}$ subrange in the gravitational wave
spectrum could also be an artifact of insufficient numerical resolution. In
any case, once the gravitational wave spectrum saturates, we would expect the
development of a flat ($E_{\rm GW}\propto k^{0}$) spectrum. Such a flat
spectrum is expected to extend all the way to the horizon wave number
$k=H_{\text{\sc cpi}}$ RoperPol:2019wvy ; RoperPol:2022iel ; Sharma:2022ysf .
Therefore, the total gravitational wave energy is expected to be proportional
to $k_{\text{\sc cpi}}/2-H_{\ast}$. However, since $k_{\text{\sc cpi}}/2$ is
already much larger than our lower cutoff value $k_{1}$, the error in our
estimate of ${\cal E}_{\rm GW}\propto k_{\text{\sc cpi}}/2-k_{1}$ is
negligible.

Figure 1:  Spectra (per linear wave number interval) of magnetic energy
$E_{\rm M}(k,t)$ (upper curves) and gravitational wave energy $E_{\rm
GW}(k,t)$ (lower curves) from the chiral plasma instability and turbulent MHD
evolution for Run X4, where $\mu_{5}=10^{6}\,H_{\ast}$,
$\mathcal{E}_{\mathrm{cr}}\lambda=10^{20}/H_{\ast}$, $\eta=5\times
10^{-11}\,H_{\ast}$ which implies $v_{\lambda}=10^{-4}$ and $v_{\mu}=5\times
10^{-5}$ (corresponding to regime I). The solid curves are for
$tH_{\ast}=2.98$, when ${\cal E}_{\rm M}$ is maximum. The dotted curves are
for $tH_{\ast}=2.41$ (black), $2.56$ (red), and $2.71$ (orange), before ${\cal
E}_{\rm M}$ is maximum, while the dashed curves are for $tH_{\ast}=3.66$
(blue) and $tH_{\ast}=5.37$ (green), when ${\cal E}_{\rm M}$ is decaying.
Figure 2:  Comparison of $h_{0}^{2}\Omega_{\rm GW}(f)$ versus $f$ for runs
with $k_{1}/H_{\ast}=10^{3}$ (Run Y3, blue), $2\times 10^{3}$ (Run Y2,
orange), and $5\times 10^{3}$ (Run Y1, red), with $\lambda=49\times
10^{22}H_{\ast}^{2}/\mathcal{E}_{\mathrm{cr},\ast}$ and $\eta=5\times
10^{-8}H_{\ast}^{-1}$ with $H_{0}=100\,h_{0}\,{\rm km}\,{\rm s}^{-1}\,{\rm
Mpc}^{-1}$. Run X4 with with
$\lambda=10^{20}H_{\ast}^{2}/\mathcal{E}_{\mathrm{cr},\ast}$ and again
$\eta=5\times 10^{-8}H_{\ast}^{-1}$ is shown as the black line for reference.

In Fig. 2, we show gravitational wave spectra for a few runs with smaller
values of the minimum wave number in the simulations. We see that the spectra
remain nearly flat, but the spectra are also becoming more irregular at large
wave numbers. This is likely an artifact of insufficient numerical resolution.
We also see that most of the gravitational wave energy is at frequencies below
about $1\,\,{\rm kHz}$, but this value would increase with increasing values
of $\mu_{50}$, beyond the value of $10^{6}H_{\ast}$ adopted here. The fiducial
value of $\tilde{\mu}_{Y,5}=10^{-3}T$ in Eq. (4) corresponds to
$\mu_{50}\approx 5\times 10^{7}\,H_{\ast}$, and since the simulations
presented in Fig. 2 have $\mu_{50}=10^{6}H_{\ast}$, the gravitational wave
frequencies are proportionally smaller.

Earlier work showed that ${\cal E}_{\rm GW}^{\rm sat}$ grows approximately
linearly with $\eta$ and was proportional to
$(\mathcal{E}_{\mathrm{cr}}\lambda)^{-5/2}$, which leads to the combined
dependence Brandenburg:2021aln

${\cal E}_{\rm GW}^{\rm sat}/{\mathcal{E}}_{\rm cr}\approx 6\times
10^{-8}\,v_{\lambda}^{5}v_{\mu},$ (11)

which implies that ${\cal E}_{\rm GW}^{\rm sat}\propto\mu_{50}^{6}$. In Fig.
3, we plot ${\cal E}_{\rm GW}^{\rm sat}$ versus $v_{\lambda}^{5}v_{\mu}$ for
Runs X1–X4 and Y1–Y3. We see that Eq. (11) agrees reasonably well with our
numerical data. Compared with the runs of Ref. Brandenburg:2021aln , the new
one in Fig. 1 has much smaller values of $v_{\lambda}$ (here
$v_{\lambda}=10^{-4}$ instead of 0.5 for the old runs of Series B) and
$v_{\mu}$ (here $v_{\mu}=5\times 10^{-5}$ instead of $10^{-2}$, which was
their smallest value). This has been achieved by having $k_{\text{\sc cpi}}$
much larger (here $10^{6}$ instead of $10^{4}$, for example). This also means
that we have to choose a correspondingly larger value of the minimum wave
number, $k_{1}$.

Figure 3:  Dependence of ${\cal E}_{\rm GW}^{\rm sat}$ on
$v_{\lambda}^{5}v_{\mu}$ for our runs of Series X (red) and Y (blue), as well
as Series A (black) and B (orange) of Ref. Brandenburg:2021aln .  Figure 4:
Summary of the viable parameter space for the hypercharge-weighted chiral
chemical potential $\tilde{\mu}_{Y,5}$ and predictions for the relic magnetic
field and gravitational wave radiation.

##  IV Conclusions

Our estimates of the key variables are summarized in Fig. 4. Since we assume
Standard Model particles and interactions, as well as a standard cosmology
with radiation domination at temperatures $T>100\ \mathrm{TeV}$, the
observables depend only on the single dimensionless parameter
$|\tilde{\mu}_{Y,5}|/T$, which controls the size of the initial hypercharge-
weighted chiral asymmetry. To ensure that the instability develops before the
chiral asymmetry is washed out by reactions such as Higgs decays and inverse
decays, we need $|\tilde{\mu}_{Y,5}|/T\gtrsim 10^{-3}$. On the other hand, to
avoid over-producing the baryon asymmetry we need
$|\tilde{\mu}_{Y,5}|/T\lesssim 10^{-2}$. This leaves an approximately one-
decade wide window of viable parameter space. The predicted magnetic field
strength today, assuming inverse cascade scaling from production until
recombination, is at the level of $10^{-15}\ \mathrm{Gauss}$. An intergalactic
magnetic field at this level is strong enough to explain observations of
distant TeV blazars, which provide evidence for a nonzero intergalactic
magnetic field at the level $\gtrsim 10^{-16}\ \mathrm{Gauss}$ Neronov:2010gir
, although the predicted coherence length is too small. Alternative models
with CPI-like equations of motion and late-time dynamics Zhitnitsky:2019ijg ;
Basilakos:2019acj  may lead to stronger large-scale fields. The same magnetic
field may help to explain the origin of galactic magnetic fields by providing
a seed for the galactic dynamo. The strength of the gravitational wave signal
is expected to depend strongly on the value of $|\tilde{\mu}_{Y,5}|/T$, going
as its sixth power. The typical frequency of this signal is expected to fall
near $\sim\mathrm{GHz}$, putting it into a frequency band that is being
targeted by several recently proposed probes of high-frequency gravitational
wave radiation. However, within the viable window, the gravitational wave
signal is likely far too weak for detection.

Data availability—The source code used for the simulations of this study, the
Pencil Code, is freely available from Ref. PC . The simulation setups and the
corresponding data are freely available from Ref. DATA .

Acknowledgements—We thank Kohei Kamada, Sayan Mandal, and Jonathan Stepp for
useful discussions and comments. A.J.L. is grateful to the Mainz Institute for
Theoretical Physics (MITP) of the DFG Cluster of Excellence PRISMA+ (Project
ID 39083149) as well as to the Aspen Center for Physics, which is supported by
National Science Foundation grant PHY-2210452, for their hospitality during
the completion of this work. Support through the NASA ATP award 80NSSC22K0825,
the Swedish Research Council, grant 2019-04234, and Shota Rustaveli GNSF
(grant FR/18-1462) are gratefully acknowledged. We acknowledge the allocation
of computing resources provided by the Swedish National Allocations Committee
at the Center for Parallel Computers at the Royal Institute of Technology in
Stockholm. E.C. acknowledges the Pake Fellowship, and G.S. acknowledges
support from the Undergraduate Research Office in the form of a Summer
Undergraduate Research Fellowships (SURF) at Carnegie Mellon University.

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