# Schur rings over infinite dihedral group

Gang chen  School of Mathematics and Statistics, Central China Normal
University, Wuhan 430079, China<EMAIL_ADDRESS>,  Jiawei He
School of Mathematics and Information Sciences, Nanchang HangKong University,
Nanchang 330063, China<EMAIL_ADDRESS>and  Zhiman Wu  Fengcheng
No.9 Middle School, Yichun 331100, China<EMAIL_ADDRESS>
###### Abstract.

Schur rings over the infinite dihedral group
$\mathcal{Z}\rtimes\mathcal{Z}_{2}$ are studied according to properties of
Schur rings over infinite groups and the classification of Schur rings over
infinite cyclic groups. Schur rings over $\mathcal{Z}\rtimes{\mathcal{Z}}_{2}$
are classified under the assumption that $\mathcal{Z}$ is an
$\mathcal{A}$-subgroup. Those Schur rings are proved to be traditional.

##  1\. Introduction

Schur rings are a type of subrings of group rings which are determined by a
partition of the underline group, and they were originally developed by Schur
[8] and Wielandt [9] in order to study permutation groups as an alternative to
character theory. In addition, theory of Schur rings has been used in many
other applications. For the applications of Schur rings to association
schemes, see for example, subsections 2.4 and 4.4 of [2].

Classification of Schur rings over a given group $G$ has been a subject of
much research interest. Most papers have studied Schur rings over finite
groups. Leung and Man have classified Schur rings over finite cyclic groups.
There are four types of Schur rings over finite cyclic groups which are:
partitions induced by direct product decomposition of groups (Schur rings of
direct produt type); partitions induced by orbits of automorphism subgroups
(orbit Schur rings); partitions induced by cosets of normal subgroups (Schur
rings of wedge product type); trivial Schur ring. We say that a Schur ring
over a group is a traditional Schur ring if it is one of these four types.

Some scholars have extended some properties of Schur rings over finite groups
to Schur rings over infinite groups and have classified Schur rings over
infinite cyclic groups, the group $\mathcal{Z}\times{\mathcal{Z}_{2}}$
$(\mathcal{Z}$ is an infinite cyclic group, $\mathcal{Z}_{2}$ is a cyclic
group of order 2) and torsion-free locally cyclic groups. In particular, the
classification of Schur rings over infinite cyclic groups and the main results
of Leung and Man tell us that all Schur rings over cyclic groups are
traditional.

Although general structure theorems are available for Schur rings over finite
cyclic groups, the problem of classification of Schur rings over infinite
groups is proved to be a difficult problem. Some special cases have been
considered, for example, Bastian et al. [1] study the Schur rings over some
types of infinite groups. According to their results, there are nontraditional
Schur rings over free groups and free products. Recently, Chen et al. [3]
classify Schur rings over $\mathcal{Z}\times\mathcal{Z}_{3}$. In particular,
all these Schur rings are proved to be traditional. In this paper we consider
the classification of Schur rings over $\mathcal{Z}\rtimes{\mathcal{Z}_{2}}$.

Throughout let $G$ denote an arbitrary group with identity element $1$, and
$\mathbb{F}$ an arbitrary field of characteristic $0$ and we assume that it
contains $\mathbb{Z}$ as a subring. The group algebra over $G$ with
coefficients in $\mathbb{F}$ is denoted $\mathbb{F}[G]$. The infinite dihedral
group $\langle z,s~{}:~{}s^{2}=1,s^{-1}zs=z^{-1}\rangle$ is denoted by
${\mathcal{D}}_{\infty}$.

As we shall see, the main results of the paper tells us that when $\langle
z\rangle$ is an $\mathcal{A}$-subgroup for a Schur ring over
${\mathcal{D}}_{\infty}$. Then can get the exact classification of
$\mathcal{A}$, which are prove to be traditional. The following results
implies that under some condition, $\langle z\rangle$ is really an
$\mathcal{A}$-subgroup.

######  Proposition 1.1.

Let $\mathcal{A}$ be a Schur ring over ${\mathcal{D}}_{\infty}$,
${\mathcal{D}}$ the basic set of $\mathcal{A}$, and $C$ the basic set
containsing $z$. Then the following statements hold:

  1. (i)

If $C$ is contained in $\mathcal{Z}$, then $\langle z\rangle$ is an
$\mathcal{A}$-subgroup.

  2. (ii)

Suppose that $C$ contains elements of the form $z^{i}s$. Then there exist at
least two elements $z^{i}s$ and $z^{j}s$ with $i\neq j$ in $C$.

  3. (iii)

Set

$l:=max_{D\in{\mathcal{D}}}\\{\>i-j\>:\>z^{i}s,z^{j}s\in
D\>\\}~{}~{}and~{}~{}t:=min_{D\in{\mathcal{D}}}\\{\>|i-j|\>:\>z^{i}s,z^{j}s\in
D\>\\}.$

Suppose that $2\nmid l$ and $t>2$, then $\mathcal{Z}$ is an
$\mathcal{A}$-subgroup.

The following are the main results of the present paper.

######  Theorem 1.1.

Let $\mathcal{A}$ be a Schur ring over ${\mathcal{D}}_{\infty}$. Assume that
$\mathcal{Z}=\langle z\rangle$ is an $\mathcal{A}$-subgroup. Then
$\mathcal{A}$ is isomorphic to one of the following:

  1. (i)

$\mathcal{A}=\mathbb{F}[\mathcal{Z}\rtimes{\mathcal{Z}}_{2}];$

  2. (ii)

$\mathcal{A}=\mathrm{Span}_{\mathbb{F}}\left\\{\>{\underline{C}\>:\>C\in{\mathcal{D}}\>}\right\\}$,
where
${\mathcal{D}}=\\{\\{z^{j}s,z^{i-j}s\\}:\>j\in\mathbb{Z}\>\\}\cup\\{\\{z^{j},z^{-j}\\}:j\in\mathbb{Z}\\}$
for some $i\in\mathbb{Z}\backslash\\{0\\}$.

In addition the Schur ring in $(ii)$ is an orbit Schur ring.

######  Theorem 1.2.

Let

$\mathcal{A}=\mathrm{Span}_{\mathbb{F}}\\{~{}\underline{C}~{}:~{}C\in\mathcal{D}~{}\\},$

where

${\mathcal{D}}=\\{\\{z^{i},z^{-i},z^{i}s,z^{-i}s\\}:~{}2\nmid
i,i\in\mathbb{Z}~{}\\}\cup\\{\\{z^{i}s^{j},z^{-i}s^{j}\\}:2\mid
i,~{}i\in\mathbb{Z},j\in\\{0,1\\}\\}.$

Then $\mathcal{A}$ is a Schur ring over ${\mathcal{D}}_{\infty}$ such that
$\langle z\rangle$ is not an $\mathcal{A}$-subgroup and furthermore
$\mathcal{A}$ is not traditional.

##  2\. Preliminary

For any finite non-empty subset $C\subseteq G$, denote
$\underline{C}=\sum_{g\in C}g$ and call $\underline{C}$ a simple quantity. In
addition, we define $C^{*}=\\{g^{-1}:g\in C\\}$. A partition ${\mathcal{D}}$
of $G$ is said to be finite support provided any $C\in{\mathcal{D}}$ is a
finite subset of $G$. Our notation for Schur ring is taken from [7].

###  2.1. Schur Rings

######  Definition 2.1.

Let ${\mathcal{D}}$ be a finite support partition of $G$ and $\mathcal{A}$ a
subspace of $\mathbb{F}[G]$ spanned by
$\\{\underline{C}:C\in{\mathcal{D}}\\}$. We say that $\mathcal{A}$ is a Schur
ring (or $S$-ring) over $G$ if

  1. (1)

$\\{1\\}\in{\mathcal{D}}$,

  2. (2)

for any $C\in{\mathcal{D}}$, $C^{*}\in{\mathcal{D}}$,

  3. (3)

for all $C,D\in{\mathcal{D}}$,
$\underline{C}\cdot\underline{D}=\sum_{E\in{\mathcal{D}}}\lambda_{CDE}\underline{E}$,
where all but finitely many $\lambda_{CDE}$ equal $0$.

For a Schur ring $\mathcal{A}$ over $G$, the associated partition
${\mathcal{D}}$ is denoted ${\mathcal{D}}(\mathcal{A})$ and each element in
${\mathcal{D}}(\mathcal{A})$ is called a basic set of $\mathcal{A}$. Each
$\lambda_{CDE}$ is called a structure constant. We say that a subset $C$ of
$G$ is an $\mathcal{A}$-set if $C$ is a union of some basic sets of
$\mathcal{A}$. An $\mathcal{A}$-subgroup is simultaneously an
$\mathcal{A}$-set and a subgroup.

Suppose $C=\sum_{g\in G}\alpha_{g}g$ and $D=\sum_{g\in
G}\beta_{g}g\in\mathbb{F}[G]$ (note that here only finitely many nonzero
coefficients $\alpha_{g}$ and $\beta_{g}$). Then define $C^{*}=\sum_{g\in
G}\alpha_{g}g^{-1}$ and $\operatorname{supp}(C)=\\{g:\alpha_{g}\neq 0\\}$.
Additionally, the Hadamard product is defined as $C\circ D=\sum_{g\in
G}\alpha_{g}\beta_{g}g$.

######  Lemma 2.1.

Let $\mathcal{A}$ be a Schur ring over a group $G$. Suppose $C=\\{g\\}$ is a
basic set. Then for any basic set $D$, $gD$ is a basic set.

The following theorem was proved by Wielandt in [10] when $G$ is finite. The
general case appeared as [1, Corollary 2.12].

######  Theorem 2.1.

Let $G$ be a group and $\mathcal{A}$ a subring of $\mathbb{F}[G]$ which is
spanned by $\\{\underline{C}:C\in{\mathcal{D}}(\mathcal{A})\\}$ as an
$\mathbb{F}$-vector space. Then $\mathcal{A}$ is a Schur ring over $G$ with
basic set ${\mathcal{D}}(\mathcal{A})$ if and only if $\mathcal{A}$ is closed
under $\circ$ and $*$, $1\in\mathcal{D}(\mathcal{A})$, and for all $g\in G$
there exists some $C\in\mathcal{D}(\mathcal{A})$ such that
$g\in\operatorname{supp}(C)$.

Let $f:\mathbb{F}\rightarrow\mathbb{F}$ be a function such that $f(0)=0$. For
any $\alpha=\sum_{g\in G}\alpha_{g}g$, we set

$f(\alpha)=\sum_{g\in G}f(\alpha_{g})g.$

The following proposition was proved by Wielandt in [10, Proposition 22.3].
The general case appeared as [1, Propostion 2.4].

######  Proposition 2.1.

Let $\mathcal{A}$ be a Schur ring over a group $G$ and the function be as
above. Then $f(\alpha)\in\mathcal{A}$ whenever $\alpha\in\mathcal{A}$.

Let $f$ take value $1$ at one non-zero number and $0$ at other numbers. We get
the following statement, which is known as the Schur-Wielandt principle; see
[7, Corollary 1.10].

######  Corollary 2.1.

Let $\mathcal{A}$ be a Schur ring over $G$. For any $\alpha\in\mathcal{A}$,
suppose $g\in\operatorname{supp}({\alpha})$ with $\alpha_{g}=c\neq 0$. The set

$\\{g\in G:\alpha_{g}=c\\}$

is an $\mathcal{A}$-set.

The following two propositions, which were first proved by Wielandt in [10]
and were generalized in [1], tell us how to generate $\mathcal{A}$-subgroups
in any Schur ring $\mathcal{A}$.

######  Proposition 2.2.

Let $\mathcal{A}$ be a Schur ring over $G$. Let $\alpha\in\mathcal{A}$ and

${\rm Stab}(\alpha)=\\{g\in G:g\alpha=\alpha\\}.$

Then ${\rm Stab}(\alpha)$ is an $\mathcal{A}$-subgroup.

######  Corollary 2.2.

Let $\mathcal{A}$ be a Schur ring over an infinite group $G$. Then for any
basic set $C$

${\rm Stab}(C)=\\{g\in G:gC=C\\}$

is a finite $\mathcal{A}$-subgroup.

Proof. Observe that $gC=C$ if and only if $g\underline{C}=\underline{C}$. It
then follows from Proposition 2.3 that ${\rm Stab}(C)$ is an
$\mathcal{A}$-subgroup. Since we are assuming that $C$ is a finite set, ${\rm
Stab}(C)$ is a finite subgroup of $G$. $\square$

######  Proposition 2.3.

Let $\mathcal{A}$ be a Schur ring over $G$. Let $\alpha\in\mathcal{A}$ and
$H=\langle\operatorname{supp}(\alpha)\rangle$. Then $H$ is an
$\mathcal{A}$-subgroup.

Suppose that $H$ is an $\mathcal{A}$-subgroup, then
$\\{C:C\in{\mathcal{D}}(\mathcal{A}),C\subseteq H\\}$ consists of a basic set
of a Schur ring over $H$. It is denoted $\mathcal{A}_{H}$.

When we have normal $\mathcal{A}$-subgroup, we can construct Schur ring over
the factor group as shown in the following lemma.

######  Lemma 2.2.

([4, Lemma 1.2]) Let $\varphi:G\rightarrow H$ be a group homomorphism with
${\rm Ker}(\varphi)=K$ and $\mathcal{A}$ be a Schur ring over $G$. Suppose
that $K$ is an $\mathcal{A}$-subgroup. Then the image $\varphi(\mathcal{A})$
is a Schur ring over $\varphi(G)$ where
${\mathcal{D}}(\varphi(\mathcal{A}))=\\{\varphi(C):C\in{\mathcal{D}}(\mathcal{A})\\}$.

In particular, if $H=G/K$, the factor group of $G$ over $K$, and $\varphi$ is
the natural homomorphism, the corresponding $S$-ring is denoted
$\mathcal{A}_{G/K}$.

######  Theorem 2.2.

([1, Theorem 3.3]) The only Schur rings over the infinite cyclic group
$\langle z\rangle$ are either discrete or symmetric Schur ring. Here each
basic set of the discrete Schur ring is a singleton set and each basic set of
the the symmetric Schur rings has the form $\\{z^{i},z^{-i}\\}$ for some
nonnegative integer $i$.

###  2.2. Traditional Schur rings

Discrete and trivial Schur rings. The group ring $\mathbb{F}G$ itself forms a
Schur ring over $G$, which is called the discrete Schur ring over $G$. When
$G$ is a finite group, the trivial partition $\\{1,G\\}$ produces a Schur ring
which is known as the trivial Schur ring over $G$.

Tensor products. Assume that $G=H\times K$, $\mathcal{A}_{H}$ and
$\mathcal{A}_{K}$ are Schur rings over $H$ and $K$, respectively. Moreover,
the set

$\\{CD:C\in\mathcal{D}(\mathcal{A}_{H}),D\in\mathcal{D}(\mathcal{A}_{H})\\}$

consists of the basic sets of a Schur ring over $G$, which is first proved by
Wielandt in [9]. The corresponding Schur ring is called the tensor product of
$\mathcal{A}_{H}$ and $\mathcal{A}_{K}$.

Orbit Schur rings. If $\mathcal{K}$ is a finite subgroup of
$\operatorname{Aut}(G)$, the set of elements of $\mathbb{F}[G]$ fixed by
$\mathcal{K}$ is a Schur ring over G, denoted $\mathbb{F}[G]^{\mathcal{K}}$
and called the orbit Schur ring associated with $\mathcal{K}$. Obviously, the
group ring $\mathbb{F}G$ itself also forms an orbit Schur ring.

Wedge products. A Schur ring $\mathcal{A}$ over $G$ is a wedge product if
there exist nontrivial proper $\mathcal{A}$-subgroups $H,K$ such that $K\leq
H,K\unlhd G$, and every basic set outside $H$ is a union of $K$-cosets. In
this case, the series

$1<K\leq H<G$

is called a wedge-decomposition of $\mathcal{A}$. Furthermore, we write
$\mathcal{A}=\mathcal{A}_{1}\wedge\mathcal{A}_{2}$, where
$\mathcal{A}_{1}=\mathcal{A}_{H}$ and $\mathcal{A}_{2}=\mathcal{A}_{G/K}$.

A Schur ring over a group $G$ is called traditional if it is either a trivial
Schur ring (when $G$ is finite), or an orbit Schur ring, or a tensor product
of Schur rings over smaller subgroups, or a wedge product. A group $G$ is said
to be traditional if each Schur ring over it is traditional.

##  3\. Proof of the main Results

Proof of Proposition 1.1 If $z^{i}s\notin C$ for all $i\in\mathbb{Z}$, then
$\mathcal{Z}=\langle C\rangle$ is an $\mathcal{A}$-subgroup. Then statement
$(i)$ follows.

Next, assume that $z^{i}s\in C$ for some $i\in\mathbb{Z}$. If there is only a
unique $0\neq i\in\mathbb{Z}$ such that $z^{i}s$ is contained in $C$. Observe
that $z^{i}s\in C\cap C^{*}$, and we conclude that $C=C^{*}$. This yields that

$\underline{C\backslash\\{z^{i}s\\}}\cdot
z^{i}s=z^{i}s\cdot\underline{C\backslash\\{z^{i}s\\}}.$

Also, for any odd prime number $p$,

$\begin{split}\underline{C^{p}}&={(\underline{C\backslash\\{z^{i}s\\}}+z^{i}s)}{(\underline{C\backslash\\{z^{i}s\\}}+z^{i}s)}...{(\underline{C\backslash\\{z^{i}s\\}}+z^{i}s)}\\\
&\equiv({\underline{C\backslash\\{z^{i}s\\}}})^{p}+(z^{i}s)^{p}\\\
&\equiv({\underline{C\backslash\\{z^{i}s\\}}})^{(p)}+(z^{i}s)^{p}\\\
&={\underline{C}^{(p)}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(mod~{}~{}p),\end{split}$

and so $C^{(p)}$ is an $\mathcal{A}$-set. This means that $C\subseteq C^{(p)}$
as ${(z^{i}s)}^{p}=z^{i}s$, and hence $C=C^{(p)}$ because

$|C|\leq|C^{(p)}|\leq|C|<\infty.$

This implies that $z^{p}\in C$ for any odd prime number $p$, which contradicts
the finiteness of $C$. Thus there exists a positive integer $k$ greater than
or equal to 2 such that

$z^{m_{1}}s,z^{m_{2}}s,...,z^{m_{k}}s\in C,$

where $m_{1},m_{2},...,m_{k}\in\mathbb{Z}$ and $m_{1}<m_{2}<...<m_{k}.$ Thus,
statements $(ii)$ follows.

Now, we will prove conclusion $(iii)$. Note that

$\begin{split}&\underline{C\backslash\\{z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s\\}}\cdot(z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s)\\\
=&(z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s)\cdot\underline{C\backslash\\{z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s\\}}.\end{split}$

Thus,

$\begin{split}{\underline{C}}^{2}&=\\{\underline{C\backslash\\{z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s\\}}+(z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s)\\}^{2}\\\
&\equiv(\underline{C\backslash\\{z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s\\}})^{2}+(z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s)^{2}\\\
&\equiv({\underline{C\backslash\\{z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s\\}}})^{(2)}+(z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s)^{2}~{}~{}~{}~{}~{}~{}~{}(mod~{}~{}~{}2).\end{split}$

Let $K$ denote the set of all elements of ${\underline{C}}^{2}$ with odd
coefficients. Clearly, $K\subseteq\mathcal{Z}$ and is an $\mathcal{A}$-set.
Observe that

$\begin{split}&(z^{m_{1}}s+z^{m_{2}}s+...+z^{m_{k}}s)^{2}\\\
=&(1+z^{m_{1}-m_{2}}+z^{m_{1}-m_{3}}+...+z^{m_{1}-m_{k}})+\\\
&(1+z^{m_{2}-m_{1}}+z^{m_{2}-m_{3}}+...+z^{m_{2}-m_{k}})+...+\\\
&(1+z^{m_{k}-m_{1}}+z^{m_{k}-m_{2}}+...+z^{m_{k}-m_{k-1}}),\end{split}$

by the assumption in statement $(iii)$, one can see that $m_{k}-m_{1}$ is odd
and

$m_{i+1}-m_{i}>2,i=1,\ldots,k-1.$

These facts yield that

$\\{z^{m_{1}-m_{k}},z^{m_{k}-m_{1}}\\}\subseteq
K,~{}\\{z^{2},z^{-2}\\}\subseteq K.$

Therefore $\mathcal{Z}=\langle K\rangle$ is an $\mathcal{A}$-set, and hence we
complete the proof of Proposition 1.1. $\square$

Proof of Theorem 1.1 In the sequel, we fix the following notation:

$G=\mathcal{Z}\rtimes\mathcal{Z}_{2}=\langle z\rangle\rtimes\langle s\rangle,$

and $\mathcal{A}$ is a Schur ring over $G$ such that $\mathcal{Z}$ is an
$\mathcal{A}$-subgroup.

By Theorem 2.2, we consider the following two:

Case 1. Suppose $\mathcal{A}_{\mathcal{Z}}=\mathbb{F}[\mathcal{Z}]$ is a
discrete Schur ring.

We claim that each basic set in ${\mathcal{D}}(\mathcal{A})$ is a singleton
set. Otherwise there exists $i\neq j\in\mathbb{Z}$ such that $z^{i}s,z^{j}s\in
C$ for some basic set $C$. It then follows that $z^{i-j}C$ is a basic set by
Lemma 2.1. Observe that

$z^{i}s=z^{i-j}\cdot z^{j}s\in z^{i-j}C,$

and so $z^{i-j}C=C$. This yields that $z^{i-j}$ belongs to ${\rm Stab}(C)$ and
which would contain the infinite subgroup $\langle z^{i-j}\rangle$. This is a
contradiction to Lemma 2.2.

As a consequence, $\mathcal{A}=\mathbb{F}[G]$.

Case 2. Suppose $\mathcal{A}_{\mathcal{Z}}={\mathbb{F}[\mathcal{Z}]}^{\pm}$ is
the symmetric Schur ring. Let $D$ be the basic set containing $s$.

Subcase 2.1. Suppose $|D|=1$. It then follows that

${\mathcal{D}}(\mathcal{A})=\\{\\{z^{j}s,z^{-j}s\\}:j\in\mathbb{Z}\\}\cup\\{\\{z^{j},z^{-j}\\}|j\in\mathbb{Z}\\}.$

Thus case (ii) in Theorem 1.1 occurs with $i=0$.

Subcase 2.2. Suppose $|D|=2$. Then there exists
$i\in\mathbb{Z}\backslash\\{0\\}$ such that $D=\\{s,z^{i}s\\}$.

Claim 1. There exists a basic set of the form $\\{z^{i_{0}}s\\}$ for some
$i_{0}\in\mathbb{Z}\backslash\\{0\\}$ if and only if $i$ is even. Furthermore,
if this happens then $\\{z^{\frac{i}{2}}s\\}$ is the unique basic set of size
one and

${\mathcal{D}}=\\{\\{z^{j+\frac{i}{2}}s,z^{\frac{i}{2}-j}s\\}:j\in\mathbb{Z}\\}\cup\\{\\{z^{j},z^{-j}\\}:j\in\mathbb{Z}\\}.$

Proof. If $\\{z^{i_{0}}s\\}$ is a basic set for some
$i_{0}\in\mathbb{Z}\backslash\\{0\\}$, applying Lemma 2.1, we obtain that

$\\{s,z^{2i_{0}}s\\}=\\{z^{i_{0}}s\\}\\{z^{-i_{0}},z^{i_{0}}\\}$

is a basic set. This yields that

$D=\\{s,z^{2i_{0}}s\\},$

and hence $2i_{0}=i$. This yields that $i$ must be even and
$\\{z^{\frac{i}{2}}s\\}$ is the unique basic set of size one consisting of
nonidentity.

Conversely, assume that $i$ is even. Then

$(s+z^{i}s)(z^{\frac{i}{2}}+z^{-\frac{i}{2}})=z^{\frac{3i}{2}}s+z^{\frac{i}{2}}s+2z^{\frac{i}{2}}s$

belongs to $\mathcal{A}$. Since $i\neq 0$, we obtain that
$\\{z^{\frac{i}{2}}s\\}$ is a basic set by Corollary 2.1. Applying Lemma 2.1,
one can see in this case that

${\mathcal{D}}=\\{\\{z^{j+\frac{i}{2}}s,z^{\frac{i}{2}-j}s\\}:j\in\mathbb{Z}\\}\\}\cup\\{\\{z^{j},z^{-j}\\}:j\in\mathbb{Z}\\}\\}.$

This completes the proof of Claim 1. $\square$

Now we may assume that $i$ is odd. Then every basic set of nonidentity
elements has size at least $2$ by Claim 1. For any
$j\in\mathbb{Z}\backslash\\{0\\}$, then both

$\\{z^{j}s,z^{i+j}s,z^{-j}s,z^{i-j}s\\}=\\{z^{j},z^{-j}\\}\\{s,z^{i}s\\},$

and

$\\{z^{-i-j}s,z^{i+j}s,z^{-j}s,z^{2i+j}s\\}=\\{s,z^{i}s\\}\\{z^{i+j},z^{-i-j}\\}$

are $\mathcal{A}$-sets. It follows that

$\\{z^{i+j}s,z^{-j}s\\}=\\{z^{j}s,z^{i+j}s,z^{-j}s,z^{i-j}s\\}\cap\\{z^{-i-j}s,z^{i+j}s,z^{-j}s,z^{2i+j}\\}$

is a basic set. As a consequence,

$\\{z^{i+j}s,z^{-j}s\\}\quad\text{and}\quad\\{z^{j}s,z^{i-j}s\\}$

are two basic sets. Thus,

${\mathcal{D}}=\\{\\{z^{j}s,z^{i-j}s\\}:j\in\mathbb{Z}\\}\cup\\{\\{z^{j},z^{-j}\\}:j\in\mathbb{Z}\\}.$

Subcase 2.3. Suppose $|D|\geq 3$. Then there are $i\neq
j\in\mathbb{Z}\backslash\\{0\\}$ such that $z^{i}s$ and $z^{j}s$ belong to
$D$. Observe that $C=\\{z^{i},z^{-i}\\}$ is a basic set by the assumption.
Since $s\in CD\cap D$, one can see that $D\subseteq CD$.

Claim 2. The structure constant $\lambda_{CDD}=1$.

Proof. Otherwise, suppose the structure constant

$\lambda_{CDD}=|\\{(x,y)\in C\times D:xy=z^{i}s\\}|\geq 2,$

then $z^{2i}s\in D$. Observe that

$\lambda_{CDD}=|\\{(x,y)\in C\times D:xy=z^{2i}s\\}|\geq 2,$

which implies that $z^{3i}s\in D$. It then follows that that $z^{ni}s\in D$
for all nonnegative integer $n$, which is a contradiction to the assumption
that $D$ has finite support. The claim then follows. $\square$

Claim 3. The integers $i$ and $j$ satisfy $j\neq 2i,i\neq 2j$.

Proof. Towards a contradiction, suppose that $j=2i$. Observe that

$z^{i}\cdot s=z^{i}s,~{}z^{-i}\cdot z^{j}s=z^{i}s,$

which yields that $\lambda_{CDD}\geq 2$, which is a contradiction to Claim 2.
So, $j\neq 2i$. Similarly, we can conclude that $i\neq 2j$ and the claim
follows.

$\square$

Since $D\subseteq CD$ and $z^{j}s\in D$, one can see that

$z^{j-i}s\in D~{}~{}\mathrm{or}~{}~{}z^{j+i}s\in D.$ (3.1)

Let $E=\\{z^{j},z^{-j}\\}$. Observe that $s\in D\cap ED$, and hence
$D\subseteq ED$. Also $z^{i}s\in D$. This implies that

$z^{i-j}s\in D~{}~{}\mathrm{or}~{}~{}z^{j+i}s\in D.$ (3.2)

If $z^{i+j}s\in D$, one can see that

$z^{i+j}s,s\in FD,$

where $F=\\{z^{i+j},z^{-(i+j)}\\}$. So $D\subseteq FD$.

It follows that one of the following

$z^{-i}s,z^{-j}s,z^{i+2j}s,z^{2i+j}s$

belongs to $D$.

If $z^{-i}s\in D$, then

$z^{i}\cdot z^{-i}s=s,z^{-i}\cdot z^{i}s=s$

and thus

$\lambda_{CDD}=|\\{(x,y)\in C\times D:xy=s\\}|\geq 2,$

which is a contradiction to Claim 2. So, $z^{-i}s$ does not belong to $D$.
Similarly, $z^{-j}s$ does not belong to $D$.

Now, suppose that $z^{2i+j}s$ lies in $D$. Then

$z^{i}\cdot z^{j}s=z^{i+j}s,~{}z^{-i}\cdot z^{2i+j}s=z^{i+j}s.$

This yields that

$\lambda_{CDD}=|\\{(x,y)\in C\times D:xy=z^{i+j}s\\}|\geq 2.$

This is a contradiction to Claim 2. This means that $z^{2i+j}s$ does not lie
in $D$. Similarly $z^{i+2j}s$ does not lie in $D$. As a consequence,
$z^{i+j}s$ does not lie in $D$.

Now, by statements 3.1 and 3.2 we conclude that $z^{i-j}s$ and $z^{j-i}s$
belong to $D$. Let $K=\\{z^{i-j},z^{j-i}\\}$. Then

$z^{i-j}\cdot z^{j-i}s=s,~{}z^{j-i}\cdot z^{i-j}s=s,$

and hence

$\lambda_{KDD}=|\\{(x,y)\in K\times D:xy=s\\}|\geq 2.$

Observe that

$|\\{(x,y)\in K\times D:xy=z^{i-j}s\\}|=\lambda_{KDD}\geq 2,$

which implies that $z^{2(i-j)}\in D$. As in the proof of Claim 2, one can see
that $z^{n(i-j)}s\in D$ for all nonnegative integers $n$. This is a
contradiction to $|D|<\infty$. Therefore neither $z^{i-j}s$ nor $z^{j-i}s$
lies in $D$. Consequently $|D|\leq 2$, i.e., subcase 2.3. does not occur.

$\square$

Now

${\mathcal{D}}=\\{\\{z^{j}s,z^{i-j}s\\}:j\in\mathbb{Z}\\}\cup\\{\\{z^{j},z^{-j}\\}:j\in\mathbb{Z}\\},$

where $D=\\{s,z^{i}s\\}$ is the basic set containing $s$. Let

$\varphi:~{}G\longrightarrow G,~{}s\mapsto z^{i}s,~{}z\mapsto z^{-1}.$

It is easy to see that $\varphi\in\mathrm{Aut}(G)$. Moreover,
$\mathcal{A}={\mathbb{F}[\mathcal{Z}\rtimes{\mathcal{Z}}_{2}]}^{\langle\varphi\rangle}$.

$\square$

Proof of Theorem 1.2 Now let
$\mathcal{A}=\mathrm{Span}_{\mathbb{F}}\\{~{}\underline{C}:C\in\mathcal{D}~{}\\},$
where

${\mathcal{D}}=\\{\\{z^{i},z^{-i},z^{i}s,z^{-i}s\\}:2\nmid
i,i\in\mathbb{Z}~{}\\}\cup\\{\\{z^{i}s^{j},z^{-i}s^{j}\\}:2\mid
i,~{}i\in\mathbb{Z},j\in\\{0,1\\}~{}\\}.$

Clearly, $\\{e\\}\in\mathcal{D}$ and $\mathcal{D}$ is closed to the Schur
multiplication.

Let $i_{1},i_{2}\in\mathbb{Z}$ with $2\nmid i_{1},2\nmid i_{2}$, then
$2|(i_{1}-i_{2})$. So,

$\begin{split}&(z^{i_{1}}+z^{-i_{1}}+z^{i_{1}}s+z^{-i_{1}}s)(z^{i_{2}}+z^{-i_{2}}+z^{i_{2}}s+z^{-i_{2}}s)\\\
=&2(z^{i_{1}+i_{2}}+z^{-i_{1}-i_{2}})+2(z^{i_{1}+i_{2}}s+z^{-i_{1}-i_{2}}s)+\\\
&2(z^{i_{1}-i_{2}}+z^{i_{2}-i_{1}})+2(z^{i_{1}-i_{2}}s+z^{i_{2}-i_{1}}s)\in\mathcal{A}.\end{split}$

Choose $i_{3},i_{4}\in\mathbb{Z}$ with $2\nmid i_{3},2\mid i_{4}$, then
$2\nmid(i_{3}-i_{4})$. One can easily see that

$\begin{split}&(z^{i_{3}}+z^{-i_{3}}+z^{i_{3}}s+z^{-i_{3}}s)(z^{i_{4}}+z^{-i_{4}})\\\
=&(z^{i_{3}+i_{4}}+z^{-i_{3}-i_{4}}+z^{i_{3}+i_{4}}s+z^{-i_{3}-i_{4}}s)+(z^{i_{3}-i_{4}}+z^{i_{4}-i_{3}}+z^{i_{3}-i_{4}}s+z^{i_{4}-i_{3}}s)\in\mathcal{A},\end{split}$
$\begin{split}&(z^{i_{4}}+z^{-i_{4}})(z^{i_{3}}+z^{-i_{3}}+z^{i_{3}}s+z^{-i_{3}}s)\\\
=&(z^{i_{3}+i_{4}}+z^{-i_{3}-i_{4}}+z^{i_{3}+i_{4}}s+z^{-i_{3}-i_{4}}s)+(z^{i_{3}-i_{4}}+z^{i_{4}-i_{3}}+z^{i_{3}-i_{4}}s+z^{i_{4}-i_{3}}s)\in\mathcal{A}.\end{split}$

Moreover,

$\begin{split}&(z^{i_{3}}+z^{-i_{3}}+z^{i_{3}}s+z^{-i_{3}}s)(z^{i_{4}}s+z^{-i_{4}}s)\\\
=&(z^{i_{3}+i_{4}}+z^{-i_{3}-i_{4}}+z^{i_{3}+i_{4}}s+z^{-i_{3}-i_{4}}s)+(z^{i_{3}-i_{4}}+z^{i_{4}-i_{3}}+z^{i_{3}-i_{4}}s+z^{i_{4}-i_{3}}s)\in\mathcal{A},\end{split}$
$\begin{split}&(z^{i_{4}}s+z^{-i_{4}}s)(z^{i_{3}}+z^{-i_{3}}+z^{i_{3}}s+z^{-i_{3}}s)\\\
=&(z^{i_{3}+i_{4}}+z^{-i_{3}-i_{4}}+z^{i_{3}+i_{4}}s+z^{-i_{3}-i_{4}}s)+(z^{i_{3}-i_{4}}+z^{i_{4}-i_{3}}+z^{i_{3}-i_{4}}s+z^{i_{4}-i_{3}}s)\in\mathcal{A}.\end{split}$

Let $i_{5},i_{6}\in\mathbb{Z}$ and $2\mid i_{5},2\mid i_{6}$,
$2\mid(i_{5}-i_{6})$. Thus,

$\begin{split}&(z^{i_{5}}+z^{-i_{5}})(z^{i_{6}}+z^{-i_{6}})=(z^{i_{5}+i_{6}}+z^{-i_{5}-i_{6}})+(z^{i_{5}-i_{6}}+z^{i_{6}-i_{5}})\in\mathcal{A},\\\
&(z^{i_{5}}+z^{-i_{5}})(z^{i_{6}}s+z^{-i_{6}}s)=(z^{i_{5}+i_{6}}s+z^{-i_{5}-i_{6}}s)+(z^{i_{5}-i_{6}}s+z^{i_{6}-i_{5}}s)\in\mathcal{A},\\\
&(z^{i_{5}}s+z^{-i_{5}}s)(z^{i_{6}}+z^{-i_{6}})=(z^{i_{5}+i_{6}}s+z^{-i_{5}-i_{6}}s)+(z^{i_{5}-i_{6}}s+z^{i_{6}-i_{5}}s)\in\mathcal{A},\\\
&(z^{i_{5}}s+z^{-i_{5}}s)(z^{i_{6}}s+z^{-i_{6}}s)=(z^{i_{5}+i_{6}}+z^{-i_{5}-i_{6}})+(z^{i_{5}-i_{6}}+z^{i_{6}-i_{5}})\in\mathcal{A}.\end{split}$

Hence, $\mathcal{A}$ is a Schur ring over ${\mathcal{D}}_{\infty}$.

Since there is no finite normal subgroup over ${\mathcal{D}}_{\infty}$, this
means that $\mathcal{A}$ can not be the Schur rings of wedge product type.
Obviously, there is no $\eta\in\mathrm{Aut}({\mathcal{D}}_{\infty})$ such that
$\eta(z^{i})=z^{i}s$ for all $2\nmid i\in\mathbb{Z}$, which yields that
$\mathcal{A}$ is not an orbit Schur ring. So, $\mathcal{A}$ is not
traditional.

##  4\. Acknowledgment

The work of the first author is supported by Natural Science Foundation of
China (No. 11971189, No. 12161035)

## References

  * [1] N. Bastian, J. Brewer, M. Humphries, A. Misseldine, C. Thompson, On Schur rings over infinite groups. Algebr. Represent. Theor. 23(3):493-511 (2020).
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  * [3] G. Chen, J. W. He, Schur rings over $\mathcal{Z}\times\mathcal{Z}_{3}$. Communications in algebra, 49(1): 4434-4446 (2021).
  * [4] K. H. Leung, S. L. Ma, The structure of Schur rings over cyclic groups. J. Pure Appl. Algebra 66(3): 287-302 (1990).
  * [5] K. H. Leung, S. H. Man, On Schur rings over cyclic groups II. J. Algebra 183(2):273-285 (1996).
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  * [7] I. Ponomarenko, _Schur rings and algebraic combinatorics_. Lecture Notes. Wuhan (2015).
  * [8] I. Schur, _Zur Theorie Der Einfach Transitiven Permutationsgruppen_ , Vol. 118. Berlin: Sitzungsber. Preuss. Akad. Wiss. Phy-Math Klasse, pp. 309-310 (1933).
  * [9] H. Wielandt, _Zur theorie der einfach transitiven permutationsgruppen II_. Math. Z. 52(1): 384-393 (1950).
  * [10] H. Wielandt, _Finite Permutation Groups_. New York, NY: Academic Press (1964).


# Constraints on compact dark matter from the non-observation of
gravitational-wave strong lensing

A. Barsode1 S. J. Kapadia2,1 P. Ajith1,3 1 International Centre for
Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore
560089, India  2 Inter-University Centre for Astronomy and Astrophysics, Pune
411007, India  3 Canadian Institute for Advanced Research, CIFAR Azrieli
Global Scholar, MaRS Centre, West Tower, 661 University Ave., Suite 505,
Toronto, ON M5G 1M1, Canada

###### Abstract

We use the non-observation of strong lensing of gravitational waves (GWs) in
the first three observation runs of LIGO-Virgo detectors to constrain the
fraction of dark matter in the form of compact objects in the mass range
$10^{6}-10^{9}~{}{{\mathrm{M}_{\odot}}}$. Using a Bayesian formalism
supplemented by astrophysical simulations of strong lensing of GWs, we
constrain the compact dark matter fraction to $\lesssim 0.4-0.6$ with
currently available data and show that they may get significantly tighter in
the future. We find that multiple lensing — i.e. GWs getting deflected by
multiple compact objects on their way to us — is possible. By ignoring this,
we underestimate the constraints by a few percent.

###### Subject headings:

strong lensing, gravitational waves, dark matter, compact objects, multiple
lensing

##  1\. Introduction

Gravitational wave (GW) astronomy has emerged as a powerful cosmological probe
that complements electromagnetic (EM) observations. The weak interaction of
GWs with matter renders them exceptionally well-suited for delving into the
dark universe. So far, the ground-based interferometric detectors LIGO and
Virgo have detected over 90 GW events in their first three observing runs
(Abbott et al., 2019, 2021a, 2023a, 2024a). Most of these have been identified
as binary black hole (BBH) coalescences, with a few as binary neutron star
(BNS) and neutron star-black hole (NSBH) coalescences.

Because the coupling between matter and gravity is weak, GWs undergo minimal
attenuation on their way through the cosmos. However, like light, their paths
may get distorted due to the presence of intervening matter, through the
effect of gravitational lensing. Galaxies and galaxy clusters are expected to
be the most promising lenses. Since the characteristic length scale ($\sim$
Schwarzschild radius, $G{M_{\ell}}/c^{2}\sim 10^{12}$ km) of these objects is
much larger than the wavelength of GWs ($\lambda_{\mathrm{GW}}\sim 10^{3}$
km), their lensing can be treated in the geometric optics approximation.
Theoretical studies of lensing of GWs due to galaxies have suggested that
$\sim 0.1-1\%$ of all GW signals observable by the first three observation
runs of LIGO-Virgo detectors may be strongly lensed, producing multiple images
(Ng et al., 2018; Li et al., 2018; Oguri, 2018). The expected lensing
probability due to clusters is likely an order of magnitude smaller (Smith et
al., 2017).

Compact objects (COs) like black holes (BHs) could also produce lensing
effects on GWs. If the gravitational radius of the lensing CO is comparable to
the GW wavelength ($G{M_{\ell}}/c^{2}\sim\lambda_{\mathrm{GW}}$), then
geometric optics approximation becomes inaccurate and lensing will involve
wave optics effects (diffraction, interference, etc). In the frequency band of
ground-based GW observatories ($f_{\mathrm{GW}}\sim 10-10^{3}~{}\mathrm{Hz}$),
CO lenses in the mass range $\sim 10^{2}-10^{5}~{}M_{\odot}$ are expected to
produce wave optics effects (called, GW microlensing) (Takahashi & Nakamura,
2003), while more massive CO lenses (${M_{\ell}}\gtrsim 10^{5}~{}M_{\odot}$)
can produce multiple images (strong lensing), similar to galaxies. The
probability of observing such effects depends on the abundance of such COs in
the universe. The expected abundance of astrophysical BHs (produced from the
collapse of baryonic matter) is not large enough to produce observable lensing
signatures in the near future. However, if a significant fraction of dark
matter is in the form of primordial BHs or other exotic COs, their lensing
effects will be observable soon.

Several searches performed in LIGO-Virgo data failed to obtain significant
evidence of wave optics effects or multiple images in the $\sim 90$ GW events
observed so far (Hannuksela et al., 2019; McIsaac et al., 2020; Abbott et al.,
2023b, 2021b; Janquart et al., 2023; Goyal et al., 2024). While this non-
observation is so far not in conflict with the expectations for standard lens
populations, it can be used to rule out some more exotic proposals. For
example, the non-observation of microlensing-induced distortions in the GW
signals has been used to constrain the fraction of dark matter in the form of
COs in the mass range $10^{2}-10^{5}~{}{\mathrm{M}_{\odot}}$ (Basak et al.,
2022). While their constraints are modest, it is nevertheless a novel and
independent demonstration of using GWs to probe the nature of dark matter.

In this paper, we constrain the fraction
${f_{\mathrm{DM}}}\equiv\Omega_{\mathrm{CO}}/\Omega_{\mathrm{DM}}$ of dark
matter in the form of COs in the mass range
$10^{6}-10^{9}~{}{\mathrm{M}_{\odot}}$ from the non-observation of strong
lensing of GWs. Here, $\Omega_{\mathrm{DM}}$ is the energy density of the dark
matter and $\Omega_{\mathrm{CO}}$ is that in the form of COs. While the
condition that GWs be strongly lensed imposes the lower limit of $\sim
10^{5}~{}{\mathrm{M}_{\odot}}$, there is no upper bound on the masses of
objects that may be constrained. Yet, it may not make sense to go beyond the
masses of the smallest dwarf galaxies ($\sim 10^{9}~{}{\mathrm{M}_{\odot}}$).

Constraints have already been put on this range of CO masses (Carr et al.,
2021); the tightest ones arising from the non-observation of distortions in
CMB spectrum (${f_{\mathrm{DM}}}<10^{-6}$ for
${M_{\ell}}>10~{}{\mathrm{M}_{\odot}}$). However, these are model dependent
and may be evaded if the COs are presumed to be born lighter and allowed to
accreate matter (Carr & Kühnel, 2022). The most robust constraints come from
the non-observation of dynamical effects these COs would have on various
structures in the universe (${f_{\mathrm{DM}}}<10^{-4}$ for ${M_{\ell}}\sim
10^{7}~{}{\mathrm{M}_{\odot}}$) (Carr & Sakellariadou, 1999). Non-observation
of strong lensing of EM sources has also been used for constraining
${f_{\mathrm{DM}}}\lesssim 10^{-2}$ (Wilkinson et al., 2001; Zhou et al.,
2022; Nemiroff et al., 2001; Ji et al., 2018; Leung et al., 2022;
Zumalacárregui & Seljak, 2018).

In comparison, we constrain ${f_{\mathrm{DM}}}$ to be less than $0.4-0.6$ at
90% credible interval using currently available data. While these constraints
are quite modest, this is the first demonstration of a new probe of compact
dark matter in this mass range using GWs. Further, we shall show that these
constraints will get significantly better in the next few years as the number
of GW detections grows rapidly.

##  2\. Strong lensing by compact dark matter

###  2.1. The strong lensing optical depth

In order to constrain the fraction ${f_{\mathrm{DM}}}$ of dark matter in the
form of COs, we need to compute the relationship of ${f_{\mathrm{DM}}}$ with
the detectable fraction of GW events lensed by such objects. The primary
quantity bridging these two is the strong lensing optical depth $\tau(z_{s})$
— the average number of lenses whose area of influence (described by the
Einstein radius) intersects the line of sight to a source at redshift $z_{s}$:

$\tau(z_{s})=\int_{0}^{z_{s}}\dfrac{d\tau}{d{z_{\ell}}}\left({z_{\ell}},z_{s}\right),$
(1)

where ${d\tau}/{d{z_{\ell}}}$ is the differential optical depth contributed by
lenses at redshift ${z_{\ell}}$. If we assume that these COs are uniformly
distributed in comoving volume, the differential optical depth is given by
(see, e.g., Jung & Shin (2019))

$\dfrac{d\tau}{d{z_{\ell}}}\left({z_{\ell}},z_{s}\right)={f_{\mathrm{DM}}}\Omega_{\mathrm{DM}}\dfrac{3H_{0}^{2}}{2c}\dfrac{\left(1+{z_{\ell}}\right)^{2}}{H({z_{\ell}})}\dfrac{D({z_{\ell}})D({z_{\ell}},z_{s})}{D(z_{s})}.$
(2)

Here, $D({z_{\ell}}),D(z_{s})$ and $D({z_{\ell}},z_{s})$ denote the angular
diameter distance to the lens, to the source and between the lens and the
source, $H(z)$ is the Hubble parameter with $H_{0}$ being its current value,
$\Omega_{\mathrm{DM}}$ is the energy density of dark matter in units of the
critical energy density of the universe, and $c$ is the speed of light.

If one assumes that dark matter is made up of COs, the propagation of waves no
longer follows the null geodesics of the smooth universe. A simple way to
understand this is by considering the waves propagating through a light cone
to us from a source. If the source is not lensed, then, by definition, there
is no CO inside this light cone, and therefore, no dark matter. Waves would
travel differently in such an “empty cone” than if it had been filled with a
smooth distribution of dark matter. Detailed calculations presented in
Nikolaev & Chervon (2015) and Nikolaev & Chervon (2016) show that the angular
diameter distance in an empty cone cosmology can be larger than the standard
“filled cone” cosmology. As a result, the optical depth is smaller in an empty
cone (lower panel of Fig. 1).

The empty cone cosmology refers only to the case ${f_{\mathrm{DM}}}=1$. For
$0<{f_{\mathrm{DM}}}<1$, the cone may be “partially” filled. In this case, the
optical depth is between that of the empty and completely filled case.
However, it is difficult to incorporate this effect in our simulations, so we
resort to using the empty cone assumption for all ${f_{\mathrm{DM}}}$. This
would underestimate the optical depth, and therefore the lensing fraction, and
consequently make our ${f_{\mathrm{DM}}}$ constraints more conservative. For
the background cosmology, we use the Planck18 (Aghanim et al., 2020) parameter
values: $h=0.674,\ \Omega_{m}=0.315$.

###  2.2. The case for multiple lensing

The optical depth definition in Eqs. (1–2) considers sources falling inside
the Einstein radius $r_{E}$ of the lenses ($y\leq 1$, where $y$ is the impact
parameter in units of $r_{E}$). However, for a single point lens, two images
will be formed even when the sources are outside the Einstein radius (see,
e.g., Meneghetti (2021)), though magnification of the second image rapidly
decreases when $y\gtrsim 1$ (Fig. 7). The average number of lenses having an
impact parameter less than $y$ is given by $y^{2}\,\tau(z_{s})$.

Usually, a fixed upper cutoff on $y$ is chosen, with the expectation that all
the sufficiently magnified lensed events will be taken into consideration
because any lensed event with an impact parameter greater than $y_{0}$, though
more likely to occur, will not be magnified enough to be detectable.

Figure 1.— Upper panel: The intrinsic distribution of the redshifts of BBHs,
as well as the distribution of detected BBHs (both unlensed and lensed – these
are scaled to reflect their abundance relative to the intrinsic population).
Please refer to Appendix B for details of our simulations. Lower panel: Strong
lensing optical depth for the smooth (filled cone) and inhomogenous (empty
cone) cosmologies.

The important caveat is that the above expectation is valid only when the
optical depth is small. If it isn’t, there is a good chance that GWs may
encounter multiple lenses along the way ($y_{0}^{2}\tau(z_{s})>1$). Almost all
events will be lensed (the lensing probability is
$P_{\ell}(z_{s})=1-e^{-y_{0}^{2}\tau(z_{s})}$), and their magnifications may
no longer follow the same behavior as for the single lens case. As the upper
panel of Fig. 1 shows, the horizon for detecting strongly lensed events in
LIGO-Virgo detectors is deep enough that we must include all the dark matter
up to $z=7$ as potential lenses, leading to very high optical depths. Thus, it
would appear that we must include the effects of multiple lensing in our
analysis of strong lensing by compact dark matter.

Dealing with multiple lensing is complicated due to the intrinsic non-
linearity of the lens equation (see, e.g., Schneider et al. (1992)). The
simplest way to deal with multiple lensing is to consider only the effect of
the strongest lens among the lenses encountered by a source (Peacock, 1986).
We show in Appendix A that this will result in a more conservative estimation
(by $<2.5\%$) of the fraction of lensed events from our simulations. Keeping
in mind, therefore, that our constraints will be more conservative, we proceed
with using the strongest lens approximation.

##  3\. Has LIGO-Virgo detected multiple images due to compact dark matter
lenses?

Using the strongest lens approximation, we can simulate lensing of GWs and
obtain a relationship between ${f_{\mathrm{DM}}}$ and the detectable lensing
fraction $u({f_{\mathrm{DM}}},{M_{\ell}})$, for COs of mass ${M_{\ell}}$. If
the expected lensing fraction is higher and if we observe no lensed events,
constraints on ${f_{\mathrm{DM}}}$ will be tighter. In this section, we argue
that though we cannot be absolutely certain about the non-detection of strong
lensing in LIGO-Virgo data, we can be reasonably sure that we have not
detected strong lensing due to COs in the mass range
$10^{6}-10^{9}{\mathrm{M}_{\odot}}$.

Figure 2.— The solid lines show the distribution of time delays between
detectable strongly lensed image pairs as obtained from our simulation (scale
relative to the “O3, $10^{9}{\mathrm{M}_{\odot}}$” case). The assumed
observing run and the mass of the lensing CO are shown in the legend. The
dashed line shows the distribution of time delays between pairs of events
observed in O3 (scaled to peak at 1). Even if all of the dark matter was made
of $10^{9}~{}{\mathrm{M}_{\odot}}$ COs, the lensing time delay would not
exceed $\sim 10^{5}$ s (gray shaded region). We have used a GW source redshift
distribution from Madau & Dickinson (2014) for this simulation.

Figure 2 shows the _expected_ distribution of the time delays between lensed
images obtained from simulations (see Appendix B for details). We find that a
multiply imaged event pair that is strongly lensed by COs of mass
$10^{6}-10^{9}~{}{\mathrm{M}_{\odot}}$ cannot have a time delay $\gtrsim
10^{5}$ s. On the other hand, all the events in the first two observing runs
(O1 and O2) of LIGO-Virgo have time delays of at least a few days (Abbott et
al., 2019, 2021a), and therefore, none of them could be pairs of images lensed
by COs. In the third observing run (O3), all except 23 pairs have a time delay
$\gtrsim 10^{5}$ s (Abbott et al., 2023a, 2024a). Therefore, these pairs could
not have been lensed by COs. Similarly, we rule out the top 5 candidates of
the subthreshold lensing search by Abbott et al. (2023b) since all of these
have time delays greater than a day.

The 23 pairs which do have time delay smaller than $\sim 10^{5}$ s have little
consistency in their sky locations and other inferred source parameters. Their
strong lensing likelihood ratios as calculated using the _posterior overlap
analysis_ (Haris et al., 2018) are $\leq 1.8\times 10^{-2}$ (Abbott et al.,
2024b). This likelihood ratio (Bayes factor) quantifies the overlap integral
between the posterior distributions of the parameters corresponding to these
events. It is highly improbable for truly lensed pairs to have Bayes factors
as small as $\sim 10^{-2}$ (Haris et al., 2018). The 24th pair in increasing
order of time delays, GW200219_094415 - GW200220_124850, has a Bayes factor of
$21.9$ (Abbott et al., 2024b), which is perhaps too high to be dismissed as
unlensed. In fact, we chose $10^{9}{\mathrm{M}_{\odot}}$ as the upper limit of
CO mass range so that the time delays caused by them would not exceed this
pair’s time delay (97475 s) and we can be certain that we have not detected
any lensing due to COs of this mass.

Before proceeding, we briefly mention how we could still put constraints on
${f_{\mathrm{DM}}}$ if we observe a pair of events having a time delay smaller
than what a CO lens might give. In such a scenario, the probability mass
contained below this observed time delay in the simulated time delay
distribution is still not observed. However, the probability mass contained
above the observed time delay has to be down-weighted because of our inability
to rule out lensing. The weight factor is simply given by the probability of
getting a lensed pair of events from CO lenses having a Bayes factor smaller
than the Bayes factor of the observed pair.

##  4\. Constraining ${f_{\mathrm{DM}}}$ from non-observation of strong
lensing

To use the non-detection of strongly lensed GWs to constrain the fraction of
compact dark matter, ${f_{\mathrm{DM}}}$, we use the same Bayesian formalism
as that used in Basak et al. (2022). We assume that the total number of
detected GW events and the number of lensed events follow Poisson
distributions with means $\Lambda$ and $\Lambda_{\ell}$, respectively111The
point mass lenses will create two images. We call an event as lensed only when
both of these are detectable. Note, also, that the lensed events may come from
a larger distance as compared to their unlensed counterparts, owing to the
lensing magnification (see the top panel of Fig. 1). We account for this in
our astrophysical simulations.. The detectable lensing fraction is defined as
a ratio of the Poisson means of the numbers of lensed and total events
detected, $u\equiv\Lambda_{\ell}/\Lambda$. A posterior on $u$ can be obtained
using the fact that we have observed 90 events so far in O1, O2, and O3, and
none of them were lensed (please refer to §3 of Basak et al. (2022) for a full
derivation).

$P(u\mid
N,{N_{\ell}}=0)\propto\int\limits_{0}^{\infty}d\Lambda\,\Lambda^{N+1}e^{-\Lambda(u+1)}\,\mathcal{P}_{\ell}(u\Lambda)\,\mathcal{P}(\Lambda)$
(3)

Here, $N(=90)$ is the total number of detected GW signals, ${N_{\ell}}(=0)$ is
the number of strongly lensed events among them, $\mathcal{P}_{\ell}$ and
$\mathcal{P}$ are the assumed priors for the lensed and total event rates
respectively. The proportionality constant is fixed by normalizing $P(u\mid
N,{N_{\ell}}=0)$.

Next, we simulate strong lensing by compact dark matter to estimate the
relationship between the expected lensing fraction $u$ versus the fraction of
dark matter ${f_{\mathrm{DM}}}$, for a given mass ${M_{\ell}}$ of the COs. The
details of our simulation are given in Appendix B. The final result of these
simulations, a map of $u$ as a function of ${f_{\mathrm{DM}}}$ and
${M_{\ell}}$, is shown in Fig. 3 for two choices of GW source redshift
distribution.

Figure 3.— Expected fraction $u$ of detectable strongly lensed events if a
fraction ${f_{\mathrm{DM}}}$ of dark matter is made up of COs of mass
${M_{\ell}}$.

Overall, the lensing fractions are higher for Belczynski et al. (2016)’s
redshift distribution, which has a higher merger rate at higher redshifts than
Madau & Dickinson (2014)’s. This is a direct consequence of the fact that the
optical depth is higher at higher redshifts, leading to higher chances of
lensing. Since the probability of encountering lenses increases with
${f_{\mathrm{DM}}}$, it is natural to expect $u$ to increase monotonically
with ${f_{\mathrm{DM}}}$.

More interesting is the behavior of $u$ with ${M_{\ell}}$: for small
${M_{\ell}}$, there are very few detectable events with time delays larger
than the signal duration so that they may appear as distinct strongly lensed
images. Consequently, the lensing fraction is nearly zero for
${M_{\ell}}\lesssim 10^{6}~{}{\mathrm{M}_{\odot}}$. For a point lens, the
lensing time delay is directly proportional to ${M_{\ell}}$, leading to a
steady growth in the number of events having time delay greater than the
signal duration. Eventually, the fraction saturates near ${M_{\ell}}\sim
10^{7}~{}{\mathrm{M}_{\odot}}$ as all the detectable events have sufficiently
large time delays.

With the relationship between $u$ and ${f_{\mathrm{DM}}}$ thus known, we take
its numerical derivative to estimate the Jacobian required to transform the
posterior on $u$ (Eq. 3) to that on ${f_{\mathrm{DM}}}$ (Basak et al., 2022).

$P({f_{\mathrm{DM}}}\mid N,{N_{\ell}}=0,{M_{\ell}})=P(u\mid
N,{N_{\ell}}=0)\,\dfrac{du\left({f_{\mathrm{DM}}},{M_{\ell}}\right)}{d{f_{\mathrm{DM}}}}$
(4)

Figures 4 and 5, respectively, show the posteriors and 90% upper bounds on
${f_{\mathrm{DM}}}$ for different choices of prior and source redshift
distributions. The posteriors show higher support for lower values of
${f_{\mathrm{DM}}}$ and are better constrained for higher lens masses, being
most constraining for ${M_{\ell}}\sim 10^{9}~{}{\mathrm{M}_{\odot}}$. The
latter is a direct consequence of the lensing fraction being higher for higher
${M_{\ell}}$. We can constrain ${f_{\mathrm{DM}}}\lesssim 0.4-0.6$ with
current LIGO-Virgo data. We remark that most of the uncertainty arises due to
our choice of prior, with little dependence on the assumed redshift
distribution.

Figure 4.— Posterior distribution of the fraction of dark matter
${f_{\mathrm{DM}}}$ in the form of COs of mass ${M_{\ell}}$. Solid (dotted)
lines and filled squares (circles) respectively show the posteriors and 90%
upper bounds on ${f_{\mathrm{DM}}}$ assuming flat (Jeffreys) prior in Eq. (3).
We have assumed that the BBH mergers have component masses are distributed
according to the “power law + peak” model from Abbott et al. (2023c), and are
distributed in redshift following Madau & Dickinson (2014) (thick lines) or
Belczynski et al. (2016) (thin lines). Figure 5.— 90% upper bounds on the
fraction of dark matter ${f_{\mathrm{DM}}}$ in the form of COs of mass
${M_{\ell}}$. Results are shown for two different source redshift
distributions (shaded regions vs hatches) and two different priors in Eq. (3),
flat (blue shades), and Jeffreys (red shades).

If LIGO-Virgo detectors continue to detect GW signals without detecting the
presence of strong lensing, or if such detections are confidently associated
with non-CO lenses, our constraints will become much tighter. Figure 6 shows
the projected 90% upper bounds on ${f_{\mathrm{DM}}}$ for future observing
runs. After $N\sim\mathcal{O}(1/u)$ GW signals have been detected, the upper
bounds rapidly drop with further detections, raising our hopes of competing
with alternative probes of compact dark matter in the near future.

Figure 6.— Current and expected 90% upper bounds on the fraction of dark
matter ${f_{\mathrm{DM}}}$ in the form of COs of mass
${M_{\ell}}=10^{7}~{}{\mathrm{M}_{\odot}}$, as a function of the number of BBH
observed mergers with no detected lensed events. The black and grey lines
correspond to flat and Jeffreys prior, respectively. We have used a GW source
redshift distribution from Madau & Dickinson (2014) for this simulation.

We remind the reader that these results are based on a lower bound obtained on
the lensing fraction using the strongest lens approximation (see Appendix A)
and an empty cone optical depth (see Appendix 2.1). Consequently, the
${f_{\mathrm{DM}}}$ constraints are a conservative upper bound, and may, in
fact, be slightly tighter.

##  5\. Conclusion

Various searches have found no evidence of strong lensing in the $\sim 90$ GW
events observed by LIGO and Virgo so far. While this non-detection is curbed
by uncertainties, a powerful argument based on time delays between observed
events can be used to conclusively rule out the presence of lensing by COs of
mass $10^{6}-10^{9}~{}{\mathrm{M}_{\odot}}$. We have used this non-observation
to constrain the fraction of dark matter in the form of such COs.

We used a Bayesian formalism coupled with simulations of strong lensing of GW
sources in an inhomogeneous cosmology. If dark matter had been in the form of
COs of mass $10^{9}~{}{\mathrm{M}_{\odot}}$, $\sim$3.5% of all detected GW
events would have been strongly lensed. Non-observation of such lensed signals
allows us to rule out $40-60\%$ of dark matter in the form of COs of mass
$10^{6}-10^{9}~{}{\mathrm{M}_{\odot}}$. While these are quite modest when
compared to those in the literature (Carr et al., 2021), they will only
improve with the availability of more data, provided we can conclusively rule
out strong lensing from observed data.

The latter expectation may seem too optimistic; current estimates of lensing
rates (Ng et al., 2018; Oguri, 2018) predict that a detection of strongly
lensed event is imminent in the upcoming runs of LIGO-Virgo. In case such a
detection is made, do our future predictions on ${f_{\mathrm{DM}}}$
constraints become void entirely? We think that there is still hope of ruling
out compact dark matter lenses in such a case and outline a procedure for
doing so.

Firstly, a point lens is known to produce exactly two images separated by a
morse phase difference of exactly $\pi/2$. Therefore, if more than two lensed
counterparts of the same event are detected, or if only two lensed images are
detected, but having a morse phase difference of 0 or $\pi$, they could not
have been lensed by a point lens. Our simulations show that galaxy lenses have
roughly equal chances of producing exactly 2 or more than 2 images. However,
our precise ability to rule out CO lensing from multiple images produced by
galaxy/cluster lenses needs to be characterized. This is something that we are
investigating now.

Our constraints on ${f_{\mathrm{DM}}}$ are conservative, and they could be
tighter. Part of this can be attributed to the fact that we have used an empty
cone optical depth for all ${f_{\mathrm{DM}}}$, even though the optical depth
may be slightly larger in partially filled cones. In addition, our result is
conservative because we have neglected multiple lensing. However, through
simulations of double lensing across different source redshifts, we find that
the error would be smaller than 2.5%, and ignoring multiple lensing would make
our constraints only slightly weaker.

## Acknowledgments

We thank Anuj Mishra for his careful review of the manuscript and Rajaram
Nityananda for guidance regarding inhomogeneous cosmologies. We are also
grateful to Srashti Goyal, Loganayagam R., Soummyadip Basak, as well as
members of the ICTS Astrophysical Relativity group and LIGO-Virgo-KAGRA
collaboration’s lensing group for helpful discussions and suggestions. Our
research is supported by the Department of Atomic Energy, Government of India,
under Project No. RTI4001. The numerical calculations reported in the paper
were performed on the Alice computing cluster at ICTS-TIFR.

##  Appendix A Multiple gravitational lensing

Figure 7.— Magnification of images due to a point mass lens as a function of
the impact parameters $y$, see Eq. (A1).

We treat multiple lensing as a perturbative phenomenon where the second
strongest lens perturbs the solution corresponding to the strongest lens; the
third strongest lens perturbs the solution corresponding to the strongest two
lenses; and so on. The formula for magnification by a point lens (e.g.,
Meneghetti (2021)) gives a natural criterion for deciding the lenses’ strength

$\mu_{\pm}(y)=\dfrac{1}{2}\pm\dfrac{y^{2}+2}{2y\sqrt{y^{2}+4}}.$ (A1)

As shown in Fig. 7, $\mu_{\pm}$ decrease monotonically with $y$; hence a lens
with a smaller $y$ is relatively “stronger” than one with a larger $y$.

One may interpret the optical depth $\tau(z_{s})/\pi$ as a surface density of
lenses projected onto the source plane, where separations are measured in
units of the dimensionless impact parameter $y$. If lenses are uniformly
distributed on the source plane, the probability distribution of the smallest
impact parameter $y_{1}$ can be derived using elementary considerations of the
Poisson distribution.

The probability $P(y_{1})\delta y_{1}$ of finding the closest lens between
$[y_{1},y_{1}+\delta y_{1}]$ is given by the product of probability of finding
no lenses in a disk of radius $y_{1}$, and the probability of finding exactly
one lens in an annulus of inner radius $y_{1}$ and thickness $\delta y_{1}$.
These probabilities are given by

$P(n\mid\lambda)=\dfrac{\lambda^{n}e^{-\lambda}}{n!}$

with $\\{n,~{}\lambda\\}=\\{0,~{}\tau(z_{s})\,y_{1}^{2}\\}$ and
$\\{1,~{}2\tau(z_{s})\,y_{1}\delta y_{1}\\}$ respectively. After taking the
small $\delta y_{1}$ limit, we get

$\displaystyle P(y_{1}\mid
z_{s})=2\tau(z_{s})\,y_{1}\,e^{-\tau(z_{s})y_{1}^{2}}.$ (A2)

Fleury & García-Bellido (2020) obtained the same expression using slightly
different arguments. We plot this distribution for three values of $\tau$ in
Fig. 8. While the impact parameters are linearly distributed for small
$y_{1}$, the distribution is highly non-linear at large values of $y_{1}$,
especially for large $\tau$. It is clear that for large $\tau$, one should not
assume $P(y_{1})\propto y$, as is commonly done in the recent literature on GW
lensing.

Figure 8.— Probability distribution of the strongest lens’s impact parameter,
Eq.(A2). The dotted lines show the linear approximation $P(y_{1})\propto
y_{1}$. Note that, for large $\tau$, the distribution deviates from the linear
approximation even at small values of $y_{1}$.

One can obtain similar distributions for the second smallest, third smallest,
etc., impact parameters. For example, the probability distribution function of
the second smallest impact parameter $y_{2}$, conditional on $y_{1}$, is given
by the product of finding no lenses in an annulus of inner and outer radii
$y_{1},y_{2}$ respectively, and exactly one lens in a differential annulus of
radius $y_{2}$

$\displaystyle P(y_{2}\mid
y_{1},z_{s})=2\tau(z_{s})\,y_{2}\,e^{-\tau(z_{s})\,(y_{2}^{2}-y_{1}^{2})}.$
(A3)

If more than one lens is present, no exact analytical formula exists for the
magnifications of multiple images. However, several approximation schemes
exist in the literature (Fleury & García-Bellido, 2020). The strongest lens
approximation, first derived by Peacock (1986), simply ignores the
contribution to magnification from lenses other than the strongest. Therefore,
the magnifications are given by $\mu(y_{1})$ where $y_{1}$ follows the
distribution given in Eq.(A2).

Because the optical depth is rather large (Fig. 1), the validity of strongest
lens approximation needs to be checked. The most straightforward way is to
simulate many random configurations of two compact dark matter lenses and
check whether the distributions of magnification $\mu_{1,2}$ and time delays
$\Delta T$ — the only lensing observables relevant to us — remain similar. We
randomly draw the configuration of the first lens by following the same
procedure as that stated in Appendix B, supplement it with Eq. (A3) for
drawing the second lens, and compute $\mu_{1,2},\Delta T$ using a general
purpose numerical solver lenstronomy (Birrer & Amara, 2018; Birrer et al.,
2021).

Figure 9.— (Reversed) cumulative distributions of magnifications and time
delays of the loudest two images from one (lighter solid lines) and two
(darker dashed lines) lenses. While the distributions of $\mu_{2}$ and $\Delta
T$ remain more or less unchanged, the distribution of $\mu_{1}$ is altered
slightly due to the presence of a second lens. Figure 10.— Distribution of
ratio of $\mu_{1}$ calculating assuming double or single lensing. Vertical
dashed lines indicate the 10th and 90th percentile bounds.

Figure 9 shows the (reversed) cumulative distributions of $\mu_{1,2}$ and
$\Delta T$ as obtained from the simulation. It is apparent that though The
distributions of $\mu_{2}$ and $\Delta T$ remain more or less unchanged,
double lensing shifts the distribution of $\mu_{1}$ towards higher values. We
also plot the histogram of the ratio of $\mu_{1}$ calculated with and without
the second lens (Fig. 10). We find that the presence of a second lens can
cause the magnification to increase (on average) by $\sim 1\%$, though it may
fluctuate by $\sim 7-13\%$ on an individual basis.

Our final goal of this exercise is to estimate the detectable fraction of
lensed events. Fluctuations due to double lensing could affect it in two ways:
they might change in magnification, which may make some marginal events
detectable/not detectable. Because these fluctuations are asymmetric around
unity, there will be an overall excess magnification due to double lensing. In
the worst case when $z_{s}=7$, the peak of the magnification ratio in Fig. 10
shifts by at most 1% (i.e. to 1.01). This will have a similar effect as
reducing the signal-to-noise (S/N) threshold for detection by $\sqrt{1.01}$.
Our simulations show that with this change, the lensing fraction increases at
most by 2.5% (i.e. by a factor of 1.025). Since the optical depth isn’t
stupendously large, the effect of three or more lenses is expected to be even
more subdominant. It is clear, then, that though multiple lensing is possible,
the error incurred by ignoring it is negligible, and would only make our
estimates slightly conservative.

##  Appendix B Astrophysical simulations of strongly lensed mergers

Here we sketch our methodology for estimating the detectable fraction of
strongly lensed events. This requires simulating a population of BBH mergers,
computing the effects of strong lensing due to point lenses constituting a
fraction ${f_{\mathrm{DM}}}$ of dark matter, and checking their detectability.

###  B.1. Simulating astrophysical sources of GWs

Since we want to calculate the fraction of lensed events that LIGO-Virgo might
have observed in the first three observing runs, the most relevant sources are
BBH mergers. Moreover, they are also the most likely sources to be lensed,
thanks to the larger detection horizon. We generate a population of $10^{8}$
BBHs following physically motivated distribution functions for various
parameters. We assume that their masses are distributed according to the
“power law + peak” model inferred from the population of BBHs detected by
LIGO-Virgo (Abbott et al., 2023c).

While Abbott et al. (2023c) also provide a redshift distribution for the
binaries, its range is limited to $z\lesssim 1.5$. Since strong lensing can
magnify events that would otherwise be out of our horizon, we must simulate a
source population far deeper than our estimated horizon redshift for unlensed
events (Wierda et al., 2021). Hence, instead of using the Abbott et al.
(2023c) redshift distribution, we assume that the redshift distribution of BBH
mergers is consistent with the cosmic star formation history given in Madau &
Dickinson (2014). We find that a maximum redshift of 7 is sufficient to ensure
that we don’t miss out on potential lensed events from beyond the horizon
(Fig. 1).

We assume the BBH to be in quasi-circular orbits with uniformly distributed
component spins aligned with the direction of angular momentum. We further
assume that they are uniformly distributed in binary phase and polarization,
and isotropic in inclination and sky angles. Their arrival times are assigned
uniformly within the duration of the observing run.

###  B.2. Simulating strong lensing by compact dark matter

As justified in Appendix A, we use the strongest lens approximation outlined
in Peacock (1986) and Fleury & García-Bellido (2020) to simulate the effects
of strong lensing. For each BBH source in our simulated population, we draw a
lens redshift ${z_{\ell}}$ according to the differential optical depth (Eq. 2)

$P({z_{\ell}}\mid z_{s})\propto\dfrac{d\tau}{d{z_{\ell}}}({z_{\ell}},z_{s}),$
(B1)

with the normalization fixed using numerical integration. We use an
inhomogeneous “empty cone” cosmology for all our optical depth calculations
irrespective of the value of ${f_{\mathrm{DM}}}$ (see the discussion in
Sec.2.1). This causes an underestimation of the lensing probability for small
${f_{\mathrm{DM}}}$, leading to slightly conservative constraints.

The dimensionless impact parameter $y$ corresponding to the strongest lens is
chosen according to Eq. (A2). With ${z_{\ell}}$ and $y$ known, the
magnifications of the two images are given by Eq. (A1), and the time delay
between them is given by Takahashi & Nakamura (2003)

${\Delta
T_{\ell}}=4{M_{\ell}}(1+{z_{\ell}})\left[\dfrac{y\sqrt{y^{2}+4}}{2}+\log{\dfrac{\sqrt{y^{2}+4}+y}{\sqrt{y^{2}+4}-y}}\right],$
(B2)

where ${M_{\ell}}$ is the mass of the point lens.

###  B.3. Checking the detectability of unlensed and lensed GWs

In principle, almost every source in our simulation is lensed (i.e. has a lens
in front of it at redshift $0<{z_{\ell}}<z_{s}$, with some impact parameter
$y$). However, most of these sources will not be intrinsically loud enough, or
may not be magnified enough, to be observable by our GW detectors. To utilize
the non-observation of strong lensing, we must compute the fraction of
detectable lensed events in different observing scenarios.

To keep the analysis simple, we use the optimal S/N that a source would
produce at our detectors as our detection statistic. We use the IMRPhenomD
(Pürrer et al., 2023) waveform model to generate BBH signals and compute the
optimal S/N in the frequency range 20–1024 Hz using the pycbc package (Nitz et
al., 2024). We use the representative/anticipated PSDs from LIGO-Virgo
Collaboration (2015a, b, 2018, 2020).

We magnify each source’s S/N by multiplying it with ${{|\mu|^{1/2}}}$. Then,
following similar detection criteria as Goyal et al. (2024), a source is
deemed detectable as a “lensed” event if the magnified network S/N of the
loudest image crosses a threshold of 8, and that for the fainter image crosses
7. If only the former criterion is satisfied, the event is deemed “unlensed”.
If neither criterion is satisfied, the event is deemed undetectable.

Finally, we count the number of sources satisfying the lensing detection
criterion to obtain a Monte-Carlo estimate of the detectable lensing fraction
$u\equiv\Lambda_{\ell}/\Lambda\simeq{N}^{\mathrm{sim}}_{\ell}/{N}^{\mathrm{sim}}$,
where ${N}^{\mathrm{sim}}$ is the detected number of simulated events
(lensed+unlensed) corresponding to each observing run, and
${N}^{\mathrm{sim}}_{\ell}$ is the detected number of lensed events from that
simulation. The total lensing fraction weighted over different observing runs
given by

$u=\dfrac{1}{N}\sum_{R}{N}_{R}u_{R}.$ (B3)

where $N_{R}$ is the number of detections in run $R$, and $N$ is the total
number of detections considered. We take $N_{R}=3,8,79$ respectively for the
first three observing runs (Abbott et al., 2019, 2021a, 2023a, 2024a). For
future projections, we take $N_{\mathrm{O4}}=81\times 2=162$ for the fourth
observing run by extrapolating from O4a (LIGO-Virgo Collaboration, 2024).

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