# Coarse selectors of groups

Igor Protasov  I.Protasov: Taras Shevchenko National University of Kyiv,
Department of Computer Science and Cybernetics, Academic Glushkov pr. 4d,
03680 Kyiv, Ukraine<EMAIL_ADDRESS>
###### Abstract.

For a group $G$, $\mathcal{F}_{G}$ denotes the set of all non-empty finite
subsets of $G$. We extend the finitary coarse structure of $G$ from $G\times
G$ to $\mathcal{F}_{G}\times\mathcal{F}_{G}$ and say that a macro-uniform
mapping $f:\mathcal{F}_{G}\rightarrow\mathcal{F}_{G}$ (resp.
$f:[G]^{2}\rightarrow G$) is a finitary selector (resp. 2-selector) of $G$ if
$f(A)\in A$ for each $A\in\mathcal{F}_{G}$ (resp. $A\in[G]^{2}$). We prove
that a group $G$ admits a finitary selector iff $G$ admits a 2-selector and
iff $G$ is a finite extension of an infinite cyclic subgroup or $G$ is
countable and locally finite. We use this result to characterize groups
admitting linear orders compatible with finitary coarse structures.

1991 MSC: 20F69, 54C65.

Keywords: finitary coarse structure, Cayley graph, selector.

##  1\.  Introduction and results

The notions of selectors went from Topology. Let $X$ be a topological space,
$exp\ X$ denotes the set of all non-empty closed subsets of $X$ endowed with
some (initially, the Vietoris) topology, $\mathcal{F}$ be a non-empty subset
of $exp\ X$. A continuous mapping $f:\mathcal{F}\rightarrow X$ is called an
$\mathcal{F}$-selector of $X$ if $f(A)\in A$ for each $A\in\mathcal{F}$. The
question on selectors of topological spaces was studied in a plenty of papers,
we mention only [1], [4], [9], [10].

Formally, coarse spaces, introduced independently and simultaniously in [17]
and [13], can be considered as asymptotic counterparts of uniform topological
spaces. But actually, this notion is rooted in Geometry, Geometric Group
Theory and Combinatorics, see [17, Chapter 1], [6, Chapter 4] and [13]. Every
group $G$ admits the natural finitary coarse structure which, in the case of
finitely generated $G$, can be viewed as the metric structure of a Cayley
graph of $G$. At this point, we need some basic definitions.

Given a set $X$, a family $\mathcal{E}$ of subsets of $X\times X$ is called a
coarse structure on $X$ if

  * •

each $E\in\mathcal{E}$ contains the diagonal
$\bigtriangleup_{X}:=\\{(x,x):x\in X\\}$ of $X$;

  * •

if $E$, $E^{\prime}\in\mathcal{E}$ then $E\circ E^{\prime}\in\mathcal{E}$ and
$E^{-1}\in\mathcal{E}$, where $E\circ E^{\prime}=\\{(x,y):\exists
z\;\;((x,z)\in E,\ (z,y)\in E^{\prime})\\}$, $E^{-1}=\\{(y,x):(x,y)\in E\\}$;

  * •

if $E\in\mathcal{E}$ and $\bigtriangleup_{X}\subseteq E^{\prime}\subseteq E$
then $E^{\prime}\in\mathcal{E}$.

Elements $E\in\mathcal{E}$ of the coarse structure are called entourages on
$X$.

For $x\in X$ and $E\in\mathcal{E}$ the set $E[x]:=\\{y\in
X:(x,y)\in\mathcal{E}\\}$ is called the ball of radius $E$ centered at $x$.
Since $E=\bigcup_{x\in X}(\\{x\\}\times E[x])$, the entourage $E$ is uniquely
determined by the family of balls $\\{E[x]:x\in X\\}$. A subfamily
${\mathcal{E}}^{\prime}\subseteq\mathcal{E}$ is called a base of the coarse
structure $\mathcal{E}$ if each set $E\in\mathcal{E}$ is contained in some
$E^{\prime}\in\mathcal{E}^{\prime}$.

The pair $(X,\mathcal{E})$ is called a coarse space [17] or a ballean [13],
[16].

A coarse space $(X,\mathcal{E})$ is called connected if, for any $x,y\in X$,
there exists $E\in\mathcal{E}$ such that $y\in E[x]$.

A subset $Y\subseteq X$ is called bounded if $Y\subseteq E[x]$ for some
$E\in\mathcal{E}$ and $x\in X$. If $(X,\mathcal{E})$ is connected then the
family $\mathcal{B}_{X}$ of all bounded subsets of $X$ is a bornology on $X$.
We recall that a family $\mathcal{B}$ of subsets of a set $X$ is a bornology
if $\mathcal{B}$ contains the family $[X]^{<\omega}$ of all finite subsets of
$X$ and $\mathcal{B}$ is closed under finite unions and taking subsets. A
bornology $\mathcal{B}$ on a set $X$ is called unbounded if
$X\notin\mathcal{B}$. A subfamily $\mathcal{B}^{\prime}$ of $\mathcal{B}$ is
called a base for $\mathcal{B}$ if, for each $B\in\mathcal{B}$, there exists
$B^{\prime}\in\mathcal{B}^{\prime}$ such that $B\subseteq B^{\prime}$.

Each subset $Y\subseteq X$ defines a subspace $(Y,\mathcal{E}|_{Y})$ of
$(X,\mathcal{E})$, where $\mathcal{E}|_{Y}=\\{E\cap(Y\times
Y):E\in\mathcal{E}\\}$. A subspace $(Y,\mathcal{E}|_{Y})$ is called large if
there exists $E\in\mathcal{E}$ such that $X=E[Y]$, where $E[Y]=\bigcup_{y\in
Y}E[y]$.

Let $(X,\mathcal{E})$, $(X^{\prime},\mathcal{E}^{\prime})$ be coarse spaces. A
mapping $f:X\to X^{\prime}$ is called macro-uniform  if for every
$E\in\mathcal{E}$ there exists $E^{\prime}\in\mathcal{E}^{\prime}$ such that
$f(E(x))\subseteq E^{\prime}(f(x))$ for each $x\in X$. If $f$ is a bijection
such that $f$ and $f^{-1}$ are macro-uniform, then $f$ is called an
asymorphism. If $(X,\mathcal{E})$ and $(X^{\prime},\mathcal{E}^{\prime})$
contain large asymorphic subspaces, then they are called coarsely equivalent.

Given a coarse spaces $(X,\mathcal{E})$, we denote by $exp\ X$ the set of all
non-empty subsets of $X$ and endow $exp\ X$ with the coarse structure $exp\
\mathcal{E}$ with the base $\\{exp\ E:E\in\mathcal{E}\\}$, where

$(A,B)\in exp\ E\Leftrightarrow A\subseteq E[B],\ \ B\subseteq E[A].$

The coarse space $(exp\ X,exp\ \mathcal{E})$ is called the hyperballean of
$(X,\mathcal{E})$, for hyperballeans see [2], [3], [14], [15].

Now we are ready to the key definition. Let $(X,\mathcal{E})$ be coarse space,
$\mathcal{F}$ be a non-empty subspace of $exp\ X$. A macro-uniform mapping
$f:\mathcal{F}\longrightarrow X$ is called an $\mathcal{F}$-selector of
$(X,\mathcal{E})$ if $f(A)\in A$ for each $A\in\mathcal{F}$. In the case
$\mathcal{F}=exp\ X$, $\mathcal{F}=\mathcal{B}\setminus\\{0\\}$,
$\mathcal{F}=[X]^{2}$ we get a global selector, a bornologous selector and a
2-selector respectively. The investigation of selectors of coarse was
initiated in [11], [12].

Every group $G$ with the identity $e$ can be considered as the coarse spaces
$(G,\mathcal{E})$, where $\mathcal{E}$ is the (right) finitary coarse
structure with the base

$\\{\\{(x,y):x\in Fy\\}:F\in[G]^{<\omega},\ e\in F\\}.$

We note that the bornology of $(G,\mathcal{E})$ coincides with
$\mathcal{F}_{G}$ and use the name finitary selector in place the bornologous
selector.

Every metric $d$ on a set $X$ defines the coarse structure $\mathcal{E}_{d}$
on $X$ with the base $\\{\\{(x,y):d(x,y)\leq r\\}:r>0\\}$. Given a connected
graph $\Gamma$, $\Gamma=\Gamma[V]$, we denote by $d$ the path metric on the
set $V$ of vertices of $\Gamma$ and consider $\Gamma$ as the coarse space
$(V,\mathcal{E}_{d})$. We recall that $\Gamma$ is locally finite if the set
$\\{y:d(x,y)\leq 1\\}$ if finite for each $x\in V$.

Our goal is to prove the following theorem.

Theorem 1.  For a group $G$, the following statements are equivalent

$(i)$ $G$ admits a finitary selector;

$(ii)$ $G$ admits a 2-selector;

$(iii)$ $G$ is a finite extension of an infinite cyclic subgroup or $G$ is
countable and locally finite (i.e. every finite subset of $G$ generates a
finite subgroup).

In the prof of Theorem 1 we use the following characterization of locally
finite graphs admitting selectors. By $\mathbb{N}$ and $\mathbb{Z}$, we denote
graphs, on the sets of natural and integer numbers in which two vertices $a,b$
are incident if and only if $|a-b|=1$. We note also that two graphs are
coarsely equivalent if and only if they are quasi-isometric, see [6, Chapter
4] for quasi-isometric spaces.

Theorem 2.  For a locally finite graph $\Gamma$, the following statements are
equivalent:

$(i)$ $\Gamma$ admits a finitary selector;

$(ii)$ $\Gamma$ admits a 2-selector;

$(iii)$ $\Gamma$ is either finite or coarsely equivalent to $\mathbb{N}$ and
$\mathbb{Z}$.

We prove Theorem 2 in Section 2 and Theorem 1 in Section 3. In Section 4, we
apply Theorem 1 to characterize groups admitting linear orders compatible with
finitary coarse structures.

##  2\.  Proof of Theorem 2

The implication $(i)\Rightarrow(ii)$ is evident. To prove
$(ii)\Rightarrow(iii)$, we choose a 2-selector $f$ of $\Gamma[V]$ and get
$(iii)$ at the end of some chain of elementary observations.

We define a binary relation $\prec$ on $V$ as follows: $a\prec b$ iff $a\neq
b$ and $f(\\{a,b\\})=a$.

We use also the Hausdorff metric on the set of all non-empty finite subsets of
$V$ defined by $d_{H}(A,B)=max\\{d(a,B)$, $\ d(b,A):a\in A,b\in B\\}$,
$d(a,B)=min\\{d(a,b):b\in B\\}$. We note that the coarse structure on
$[V]^{2}$ is defined by $d_{H}$. Since $f$ is macro-uniform, there exists the
minimal natural number $r$ such that if $A,B\in[V]^{2}$ and $d_{H}(A,B)\leq 1$
then $d(f(A),f(B))\leq r$. We fix and use this $r$.

We recall that a sequence of vertices $a_{0},\dots,a_{m}$ is a geodesic path
if $d(a_{0},a_{m})=m$ and $d(a_{i},a_{i+1})=1$ for each
$i\in\\{0,\dots,m-1\\}$.

Claim 1. Let $a_{0},\dots,a_{m}$ be a geodesic path in $V$ and $m\geq r$. If
$a_{0}\prec a_{r}$ (resp. $a_{r}\prec a_{0}$) then $a_{i}\prec a_{j}$ (resp.
$a_{j}\prec a_{i}$) for all $i,j$ such that $j-i\geq r$.

Let $a_{0}\prec a_{r}$. By the choice of $r$, we have $a_{0}\prec
a_{r+1},\dots a_{0}\prec a_{j}$ and $a_{1}\prec a_{j},\dots a_{i}\prec a_{j}$.

Claim 2. Let $v\in V$, $B(v,r)=\\{x\in V:d(x,v)\leq r\\}$ and $U$ be a subset
of $V\setminus B(v,r)$ such that the graph $\Gamma[U]$ is connected. Then
either $v\prec u$ for each $u\in U$ or $u\prec v$ for each $u\in U$.

We take arbitrary $u,u^{\prime}\in U$ and choose $a_{0},\dots,a_{k}$ in $U$
such that $a_{0}=u$, $a_{k}=u^{\prime}$ and $d(a_{i},a_{i+1})=1$ for each
$i\in\\{0,\dots,k-1\\}$. Let $a_{0}\prec v$. By the choice of $r$, we have
$a_{1}\prec v$, $\dots,a_{k}\prec v$.

Claim 3. Let $u,v,v^{\prime}\in V$, $d(v,v^{\prime})=n$ and $d(u,v)>n+r$. If
$u\prec v$ (resp. $v\prec u$) then $u\prec v^{\prime}$ (resp. $v^{\prime}\prec
u$).

We choose a geodesic path $a_{0},\dots,a_{m}$ from $v$ to $v^{\prime}$. Let
$u\prec v$. By the choice of $r$, $u\prec a_{0}$, $u\prec a_{1},\dots,u\prec
a_{n}$.

Claim 4. Let $a_{0},\dots,a_{m}$ be a geodesic path in $V$, $v\in V$,
$d(v,\\{a_{0},\dots,a_{m}\\})=d(v,a_{k})$, $k>2r+1$, $m-k>2r+1$. Then
$d(v,a_{k})\leq r$.

We take the first alternative given by Claim 1, the second is analogical. Then
$a_{0}\prec a_{k},a_{k}\prec a_{m}$. Assuming that $d(v,a_{k})>r$, we can
replace $v$ to some point on a geodesic path from $v$ to $a_{k}$ and get
$d(v,a_{k})=r+1$. We take the first alternative given by Claim 2, the second
is analogical. Then $v\prec a_{0}$, $v\prec a_{m}$. But $v\prec a_{0}$ and
$a_{0}\prec a_{k}$ contradict Claim 3.

We recall that a sequence $(a_{n})_{n<\omega}$ in $V$ is a ray  if
$d(a_{i},a_{j})=j-i$ for all $i<j$. Evidently, $\Gamma[\\{a_{n}:n<\omega\\}]$
is asymorphic to $\mathbb{N}$.

Claim 5. Let $(a_{n})_{n<\omega}$ , $(c_{n})_{n<\omega}$ be rays in $V$,
$A=\\{a_{n}:n<\omega\\},$ $C=\\{c_{n}:n<\omega\\}$ and $A\cap C=\emptyset$.
Let $t_{0},\dots,t_{k}$ be a geodesic path from $a_{0}$ to $c_{0}$,
$T=\\{t_{0},\dots,t_{k}\\}.$ Assume that $T\cap\\{A\\}=\\{a_{0}\\}$, $T\cap
C=\\{c_{0}\\}$. If there exists a finite subset $H$ of $V$ such that every
geodesic path from a vertex $a\in A$ to a vertex $c\in C$ meets $H$ then
$(A\cup C\cup T,d)$ is asymorphic to $\mathbb{Z}$.

We define a bijection $f:A\cup C\cup T\rightarrow\mathbb{Z}$ by

$f(c_{i})=-i-1,\ \ f(t_{i})=i,\ \ f(a_{i})=i+k+1$

and show that $f$ is an asymorphism.

If $x,y\in A\cup C\cup T$ then $|f(x)-f(y)|\leq d(x,y)$. Hence, $f^{-1}$ is
macro-uniform.

We denote by $p=max\\{d(a_{0},h),d(b_{0},h):h\in H\\}$. Then the restriction
of $f$ to $C\cup T\cup\\{a_{0},\dots,a_{p}\\}$ is an asymorphism and the
restriction of $f$ to $A\cup T\cup\\{c_{0},\dots,c_{p}\\}$ is an asymorphism.
Let $n>p$, $m>p$. Since a geodesic path from $c_{n}$ to $a_{m}$ meets $H$, we
have

$d(a_{m},c_{n})\leq n-p+m-p=|f(a_{m})-f(c_{n})|-k-2p,$

so $f$ is macro-uniform and the claim is proven.

We suppose that $V$ is infinite. Since $\Gamma[V]$ is locally finite, there
exists a ray $(a_{n})_{n<\omega}$ in $V$. We put $A=\\{a_{n}:n<\omega\\}$. If
$V\setminus B(A,r)$ is finite then $\Gamma[V]$ is coarsely equivalent to
$\mathbb{N}$.

We suppose $V\setminus B(A,r)$ is infinite, take $u\in V\setminus B(A,r)$ and
show that every path $P$ from $u$ to a point from $B(A,r)$ meets
$B(\\{a_{0},\dots,a_{2r+1}\\},r+1)$. We take a point $v\in P$ such that
$d(v,A)=r+1$ and take $k$ such that $d(v,a_{k})=r+1$. By Claim 4, $k\leq
2r+1$, so $v\in B(\\{a_{0},\dots,a_{2r+1}\\},r+1)$. We choose a ray
$(c_{n})_{n<\omega}$ in $V\setminus B(A,r)$ and put $C=(c_{n})_{n<\omega}$. We
delete (if necessary) a finite number of points from $A$ so that $A,C$ and $T$
satisfy the assumptions of Claim 5 with $F=B(\\{a_{0},\dots,a_{2r+1}\\},r+1)$.
Then $(B(A\cup C\cup T),d)$ is coarsely equivalent to $\mathbb{Z}$.

We show that $V\setminus B(A\cup C,r)$ is finite, so $\Gamma[V]$ is coasly
equivalent to $\mathbb{Z}$. We suppose the contrary and choose a ray
$(x_{n})_{n<\omega}$ in $V\setminus B(A\cup C,r)$. Applying arguments from
above paragraph, we can construct a subset $X$ of $V$ such that $(X,d)$ is
coarsely equivalent to a tree $T$ which is a union of three rays with common
beginning. Since $(X,d)$ has a 2-selector, by Proposition 5 from [12], $T$
also admits a 2-selector. On the other hand, Claim 4 states that $T$ does not
admit a 2-selector and we get a contradiction.

It remains to prove $(iii)\Rightarrow(i)$. This is evident if $\Gamma$ is
finite. By [12, Proposition 5], it suffices to show that $\mathbb{N}$ and
$\mathbb{Z}$ admit finitary selectors. In both cases, a mapping $f$ defined by
$f(A)=max\ A$ is finitary selector.

##  3\.  Proof of Theorem 1

Let $G$ be a group with the finite system $S$ of generators, $S=S^{-1}$. We
recall that the Cayley graph $Cay(G,S)$ is a graph with the set of vertices
$G$ and the set of edges $\\{(x,y):x\neq y,\ xy^{-1}\in S\\}$. We note that
the finitary coarse space of $G$ is asymorphic to the coarse space of
$Cay(G,S)$.

Now let $G$ be an arbitrary group. The implication $(i)\Rightarrow(ii)$ is
evident.

We prove $(ii)\Rightarrow(iii)$. By [12, Theorem 4], $G$ is countable. Let $f$
be a 2-selector of $G$. We use the binary relation $\prec$ on $G$, defined in
Section 2, and consider two cases.

Case 1. $G$ has an element $a$ of infinite order. We denote by $A$ the
subgroup of $G$, generated by $a$, and show that $|G:A|$ is finite.

On the contrary, let $|G:A|$ is infinite. We put $S=\\{e,a,a^{-1}\\}$, denote
by $\Gamma[A]$ the graph $Cay(A,S)$ and choose a natural number $r$ such that
if $B,C\in[A]^{2}$ and $d_{H}(B,C)\leq 1$ then $d(f(A),f(B))\leq r$. By Claim
1, either $a^{m}\prec a^{n}$ for all $m,n\in\mathbb{Z}$ such that $n-m\geq r$
or $a^{n}\prec a^{m}$ for all $m,n\in\mathbb{Z}$ such that $n-m\geq r$.

Since $f:[G]^{2}\rightarrow G$ is macro-uniform, there exists a finite subset
$F$ of $G$ such that $F=F^{-1}$, $e\in F$ and if $B,C\in[G]^{2}$ and
$A\subseteq SB$, $B\subseteq SA$ then $f(A)\in Ff(B)$. Since $|G:A|$ is
infinite, we can choose $h\in G\setminus FA$, so $Fh\cap A=\emptyset$. Then
either $a^{n}\prec h$ for each $n\in\mathbb{Z}$ or $h\prec a^{n}$ for each
$n\in\mathbb{Z}$. We consider the first alternative, the second is analogical.

Since $f$ is macro-uniform, we can choose $m\in\mathbb{N}$, $m\geq r$ such
that $e\prec a^{m}$ and $h\prec a^{m}$, but $h\prec a^{m}$ contradicts above
paragraph.

Case 2. $G$ is a torsion group. We suppose that $G$ is not locally finite,
choose a finite subset $S$ of $G$ such that the subgroup $H$, generated by
$S$, is infinite. We denote $\Gamma[H]=Cay(H,S)$. By Theorem 2, $\Gamma[H]$ is
coarsely equivalent to $\mathbb{N}$ or $\mathbb{Z}$.

We take $v\in\Gamma[H]$ and denote $S(v,n)=\\{u\in H:d(v,u)=n\\}$,
$n\in\mathbb{N}$. By [8, Theorem 1] or [16, Theorem 5.4.1], there exists a
natural number $k$ such that $|S(v,n)|\leq k$ for each $n\in\mathbb{N}$.
Hence, $H$ is of linear growth. Applying either [5] or [7], we conclude that
$H$ has an element of infinite order, a contradiction with the choice of $G$.

It remains to verify $(iii)\Rightarrow(i)$. If $G$ is a finite extension of an
infinite cyclic subgroup then we apply Theorem 2. If $G$ is countable and
locally finite, one can refer to Theorem 5 in [12], but we give the following
direct proof to use in the proof of Theorem 3.

We write $G$ as the union of an increasing chain $\\{G_{n}:n<\omega\\}$,
$G_{0}=\\{e\\}$ of finite subgroup. For each $n$, we choose some system
$R_{n}$, $e\in R_{n}$ of representatives of right cosets of $G_{n+1}$ by
$G_{n}$, so $G_{n+1}=G_{n}R_{n}$. We denote

$X=\\{(x_{n})_{n<\omega}:x_{n}\in R_{n}\ and\ x_{n}=e\ for\ all\ but\
finitery\ many\ n\\}$

and define a bijection $h:G\rightarrow X$ as follows.

We put $h(e)=(x_{n})_{n<\omega},\ x_{n}=e$. Let $g\in G$, $g\neq e$. We choose
$n_{0}$ such that $g\in G_{n_{0}+1}\setminus G_{n_{0}}$ and write
$g=g_{0}r_{n_{0}}$, $g_{0}\in G_{g_{0}},\ r_{n_{0}}\in R_{n_{0}}$. If
$g_{0}\neq e$ then we find $n_{1}$, $g_{1}\in G_{n_{1}}$, $r_{n_{1}}\in
R_{n_{1}}$ such that $g_{0}=g_{1}r_{n_{1}}$. After a finite number $k$ of
steps, we get $g=r_{n_{k}}\dots r_{n_{1}}\ r_{n_{0}}$. We put $h(g)=(y_{n})$,
where $y_{n}=r_{n}$ if $n\in\\{n_{k},\dots,n_{0}\\}$, otherwise, $y_{n}=e$.

Now we define a linear order $\leq$ on $X$. For each $n<\omega$, we choose
some linear order $\leq_{n}$ on $R_{n}$ with the minimal element $e$. If
$(x_{n})_{n<\omega}\ \neq\ (y_{n})_{n<\omega}$ then we choose the minimal $k$
such that $x_{n}=y_{n}$ for each $n>k$. If $x_{k}<_{k}\ y_{k}$ then we put
$(x_{n})_{n<\omega}\ <\ (y_{n})_{n<\omega}$.

We note that $(X,\leq)$ is well-ordered, so every non-empty subset of $X$ has
the minimal element. To define a global selector $f:exp\ G\rightarrow G$, we
take an arbitrary $A\in exp\ G$ and put $f(A)=min\ h(A)$.

##  4\.  Linear orders

Let $(X,\mathcal{E})$ be a coarse space. We say that a linear order $\leq$ on
$X$ is compatible with the coarse structure $\mathcal{E}$ if, for every
$E\in\mathcal{E}$, there exists $F\in\mathcal{E}$ such that $E\subseteq F$ and
if $\\{x,y\\}\in[X]^{2}$, $x<y$ $(y<x)$ and $y\in X\setminus F[x]$ then
$x^{\prime}<y$ $(y<x^{\prime})$ for each $x^{\prime}\in E[x]$.

Let $(X,\mathcal{E})$ be a coarse space, $\leq$ be a linear order on $X$. We
say that an entourage $E\in\mathcal{E}$ is interval (with respect to $\leq$)
if, for each $x\in X$, there exist $a_{x},b_{x}\in X$ such that $a_{x}\leq
x\leq b_{x}$ and $E[x]=[a_{x},b_{x}]$. We say that $\mathcal{E}$ is an
interval coarse structure if there is a base of $\mathcal{E}$ consisting of
interval entourages. Clearly, if $\mathcal{E}$ is interval then $\leq$ is
compatible with $\mathcal{E}$.

Theorem 3.  Let $G$ be a group, $\mathcal{E}$ denotes the finitary coarse
structure on $G$. Then the following statements are equivalent

$(i)$ there exists a linear order $\leq$ on $G$ such that $\mathcal{E}$ is
interval with respect to $\mathcal{E}$;

$(ii)$ there exists a linear order $\leq$ on $G$ compatible with
$\mathcal{E}$;

$(iii)$ $G$ admits a 2-selector.

Proof. The imlication $(i)\Rightarrow(ii)$ is evident, $(ii)\Rightarrow(iii)$
follows from Proposition 2 in [12]. To prove $(iii)\Rightarrow(i)$, we use
Theorem 1 and consider two cases.

Case 1. $G$ is a finite extension of an infinite cyclic group $A$. We can
suppose that $A$ is a normal subgroup. Let $A=\\{a^{n}:n\in\mathbb{Z}\\}$,
$\\{f_{0},\dots,f_{m}\\}$ be a set of representatives of cosets of $G$ by $A$,
$f_{0}=e$, $F=\\{f_{0},\dots,f_{m}\\}$. We set
$F_{n}=F\\{a^{-n},\dots,a^{n}\\}$, $E_{n}=\\{(x,y):xy^{-1}\in F_{n}\\}$ and
note that $\\{E_{n}:n\in\omega\\}$ is a base for $\mathcal{E}$.

We endow $G$ with a linear order $\leq$ defined by the rule:
$f_{i}a^{k}<f_{j}a^{n}$ if and only if either $k<n$ or $k=n$ and $i<j$.

We choose the minimal natural number $d$ such that $f_{j}f_{j}\in
F\\{a^{-d},\dots,a^{d}\\}$ for all $i,j\in\\{0,\dots,m\\}$. Since
$f_{i}^{-1}af_{j}\in\\{a,a^{-1}\\}$, we have

$E_{n}[f_{i}a^{k}]\ \subseteq\ [f_{0}a^{k-n-d},\ f_{m}a^{k+n+d}].$

On the other hand,

$[f_{0}\ a^{k-n},\ f_{m}\ a^{k+n}]\ \subseteq\ F_{n}\ a^{k}\ \subseteq\ F_{n}\
f_{i}^{-1}\ (f_{i}\ a^{k}).$

Hence, $\mathcal{E}$ has an interval base with respect to $\leq$.

Case 2. $G$ is countable and locally finite. Then $\mathcal{E}$ is interval
with respect to the linear order $\leq$ defined in the proof of Theorem 1. $\
\ \ \Box$

Let $\leq$ be a linear order on $G$ compatible with $\mathcal{E}$. Does there
exist a global selector of $G$? The following theorem gives the negative
answer.

Theorem 4.  The group $\mathbb{Z}$ does not admit a global selector.

Proof. We suppose the contrary and let $f$ be a global selector. Since $f$ is
macro-uniform, there exists a natural number $n$ such that if $X,Y\in exp\ G$
and

$X\subseteq[-1,1]+Y,\ Y\subseteq[-1,1]+X$

then $f(Y)\in[f(X)-n,\ f(x)+n].$

We put $A=(n+1)\mathbb{Z}$, $a=f(A)$, $A^{\prime}=A\setminus\\{a\\}$. Then

$f(A^{\prime}\cup\\{a-1\\})\in\\{a-1,a-(n+1)\\}$,
$f(A^{\prime}\cup\\{a-2\\})\in$ $\\{a-2,a-(n+1)\\},\dots$,
$f(A^{\prime})=a-(n+1)$,

$f(A^{\prime}\cup\\{a+1\\})\in\\{a+1,a+n+1\\}$,
$f(A^{\prime}\cup\\{a+2\\})\in\\{a+2,a+n+1\\}$, $\dots,f(A^{\prime})=a+n+1$,

and we get a contradiction. $\ \ \ \Box$

## References

  * [1] G. Artico, U. Marconi, J. Pelant, L. Rotter, M. Tkachenko, Selections and suborderability, Fundam. Math. 175 (2002), 1-33.
  * [2] D. Dikranjan, I. Protasov, K. Protasova, N. Zava, Balleans, hyperballeans and ideals, Appl. Gen. Topology 2 (2019), 431-447.
  * [3] D. Dikranjan, I. Protasov, N. Zava, Hyperballeans of groups, Topology Appl. 263 (2019), 172-198.
  * [4] R. Engelking, R. W. Health, E. Michael, Topological well ordering and continuous selections, Invent. Math. 6(1968), 150-158.
  * [5] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. I. H. E. S. 53(1981), 53-73.
  * [6] P. de la Harpe,Topics in Geometrical Group Theory, University Chicago Press, 2000.
  * [7] W. Imrich, N. Seifter,A bounded for groups of linear growth, Archiv. Math. 48(1987), 100-104.
  * [8] A. Kuchaiev, I.V. Protasov, Coarse rays, Ukr. Math. Bull. 2 (2008), 185-192.
  * [9] J. van Mill, E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), 601-605.
  * [10] J. van Mill, J. Pelant, R. Pol, Selections that characterize topological completeness, Fundam. Math. 149 (1996), 127-141.
  * [11] I. Protasov, Selectors of discrete coarse spaces, Comment. Math. Univ. Carolin (to appear), preprint arXiv: 2101.07199.
  * [12] I. Protasov, Selectors and orderings of coarse spaces, Ukr. Math. Bull. (to appear), preprint arXiv: 2102.02053.
  * [13] I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs, Math. Stud. Monogr. Ser., vol. 11, VNTL, Lviv, 2003.
  * [14] I. Protasov, K. Protasova, On hyperballeans of bounded geometry, Europ. J. Math. 4 (2018), 1515-1520.
  * [15] I. Protasov, K. Protasova, The normality of macrocubes and hiperballeans, Europ. J. Math. https://doi.org/10.1007/s40879-020-00400-x.
  * [16] I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., vol. 12, VNTL, Lviv, 2007.
  * [17] J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, American Mathematical Society, Providence RI , 2003.


# UV Cooling via O VI Emission in the Superwind of M82 Observed with the Far
Ultraviolet Spectroscopic Explorer (FUSE)

Jin-Ah Kim University of Arizona, Steward Observatory, 933 N. Cherry Ave.,
Tucson, AZ 85721, USA  Haeun Chung University of Arizona, Steward Observatory,
933 N. Cherry Ave., Tucson, AZ 85721, USA  Carlos J. Vargas University of
Arizona, Steward Observatory, 933 N. Cherry Ave., Tucson, AZ 85721, USA  Erika
Hamden University of Arizona, Steward Observatory, 933 N. Cherry Ave., Tucson,
AZ 85721, USA

###### Abstract

We examined archival Far Ultraviolet Spectroscopic Explorer data to search for
far-ultraviolet emission lines in the starburst galaxy M82. The observations
were made in an outflow region that extends beyond the galactic disk. We found
the O VI $\lambda\lambda$ 1032, 1038 emission lines from the galaxy’s southern
outflow region. The O VI lines suggest that the outflowing warm-hot gas is
undergoing radiative cooling. We measured a radial velocity of $\sim$420 km
s-1 from the O VI lines, which is faster than the velocity seen in H$\alpha$
observations. The O VI $\lambda$1038 emission line seems to be blended with
the C II $\lambda$1037 emission line, which has a radial velocity of $\sim$300
km s-1, similar to what is observed in H$\alpha$ observations. The outflow
medium of M82 appears to be composed of gas in multiple phases with varying
temperatures and kinematics. Future spectroscopic observations in high energy
regimes covering a wider spatial area are necessary to understand better the
properties of the warm-hot gas medium in the outflow.

Ultraviolet spectroscopy, Galaxies, Circumgalactic medium, Starburst galaxies

††facilities: MAST (FUSE, GALEX)††software: Astropy (Astropy Collaboration et
al., 2013, 2018, 2022), emcee (Foreman-Mackey et al., 2013), Matplotlib
(Hunter, 2007), NumPy (Harris et al., 2020), regions (Bradley et al., 2022),
SciPy (Virtanen et al., 2020)

##  1 Introduction

The circumgalactic medium (CGM) is the gas surrounding a galactic disk and
filling a galactic halo. The CGM and the interstellar medium are constantly
interacting, transforming into each other through mechanisms such as gas
accretion and galactic outflows (e.g., Oppenheimer & Davé, 2008; Hafen et al.,
2019). This interaction can change the gas available to form stars and metal
mixing in galaxies. Understanding the CGM is crucial to comprehending star
formation and metal composition in galaxies.

The presence of the CGM was first suggested based on the detection of
absorption lines formed while light passes through extended galactic halos
(see Tumlinson et al., 2017, for review). Many CGM studies still rely on
detecting absorption lines by observing background sources such as quasars or
halo stars (e.g., Bahcall & Spitzer, 1969; Gupta et al., 2012; Lehner et al.,
2015). Another method to investigate the CGM is analyzing the continuum or
line emission emitted by the CGM directly (e.g., Anderson et al., 2013; Hayes
et al., 2016). These emission line studies are rare and challenging due to the
low surface brightness of the emission. Nevertheless, emission line studies
have the advantage of allowing us to explore areas without bright background
sources.

The CGM is a complex system consisting of multiple phases of matter, each with
varying temperature and velocities (see Figures 4 and 5 for details in
Tumlinson et al., 2017). Various emission and absorption lines emerge from
different ionized energy states as illustrated in Figure 6 of Tumlinson et al.
(2017). Lines associated with warm-hot CGM gas are primarily detected using
X-ray or ultraviolet (UV) wavelengths. For example, highly ionized metal lines
like O VIII and Mg X are mostly induced in a hot phase ($\textrm{T}\sim
10^{6}~{}\textrm{K}$), while O VI line is induced in a warm-hot phase
($\textrm{T}\sim 10^{5.5}~{}\textrm{K}$). C II and N II lines occur in a
colder phase ($\textrm{T}\sim 10^{4.5}~{}\textrm{K}$).

M82 is a starburst galaxy, 3.61 kpc away from us (Karachentsev et al., 2018).
A strong outflow from the galaxy has been known through observations at
multiple wavelengths (e.g., Lynds & Sandage, 1963; Shopbell & Bland-Hawthorn,
1998; Strickland & Heckman, 2009). Multiwavelength observations, from X-ray,
far-UV (FUV), and visible, through radio wavelengths, show diffuse emission
extending beyond the galactic disk. The extended morphologies in these
multiple observations are similar to each other and indicate a powerful
galactic wind (Leroy et al., 2015). Therefore, M82 is an excellent laboratory
to study how a strong galactic wind can affect the CGM.

We analyze Far Ultraviolet Spectroscopic Explorer (FUSE) observational data to
examine FUV emission lines around M82. In this paper, we present O VI emission
lines observed in M82’s outflow. In section 2, we describe the archival data
and the data reduction we used. The O VI lines we found are shown in section 3
and discussed in section 4.

##  2 Data

FUSE observations were made around the M82 halo under program C131 in 2002.
The observational data used in this study may be accessed from the MAST
archive at http://dx.doi.org/10.17909/y0wa-4b44 (catalog
doi:10.17909/y0wa-4b44). FUSE observed in four regions near M82, shown in
Figure 1. Three of them are located on the southern side of the galaxy, and
the other region is on the northern side. The regions A, B, and C in Figure 1
are 1.2, 1.8, and 2.4 kpc far, respectively, from the galactic center. Region
D is around 1.9 kpc offset from the galactic plane.

FUSE observations were made within areas of strong X-ray emission (see Figure
1 in Hoopes et al., 2003). The X-ray emission observed in M82 has a bipolar
outflow shape (Strickland et al., 2004). The bipolar outflow feature is
similar as seen with the GALEX FUV contours in Figure 1 and also found by
observations at various wavelengths (e.g., Leroy et al., 2015). Note that,
although some line emission could be emitted by the warm-hot CGM, the FUV
diffuse emission observed by GALEX would be mainly attributed to light
originating in the galactic disk and scattering by the extended dust halo
(Hoopes et al., 2005; Coker et al., 2013).

Figure 1: Four LWRS observed regions over an optical image around M82. The
aperture size of LWRS is 30″× 30″. A color map shows the DSS POSS-II red
optical image. Contour lines are at the GALEX FUV intensity of 0.5, 1, and
2$\times 10^{-18}$ erg cm-2 s-1 Å-1 arcsec-2.

The FUSE satellite (Moos et al., 2000; Sahnow et al., 2000) was launched in
1999 and operated until 2007. FUSE was designed to target FUV wavelengths
ranging from around 900 to 1200 Å, with a spectral resolving power of roughly
20,000. We specifically analyze data obtained through the low-resolution
(LWRS) aperture (30″× 30″), the largest aperture of FUSE. Our analysis is
based on data collected in LiF1A and LiF2B channels, which cover O VI emission
lines at 1032 and 1038 Å. The LiF1A channel covers from 987 to 1082 Å, and the
LiF2B channel covers from 979 to 1075 Å. Although the SiC1A and SiC2B channels
also cover these wavelengths, we did not incorporate the data from the SiC
channels in our study. The presence of solar-scattered O VI emission lines in
the SiC channels can interfere with data analysis. The LiF channels are
located on the instrument’s side, facing away from the Sun to minimize
emission lines from solar-scattered light. The effective area is also far
lower in the SiC1A and SiC2B channels than in the LiF1A and LiF2B channels,
resulting in a lower signal-to-noise ratio (S/N) in the SiC1A and SiC2B
channels. Due to these two effects, the LiF1A and LiF2B channels are better
suited for studying O VI lines than the SiC channels.

We processed the FUSE observational data through CalFUSE version 3.2.3, a data
reduction pipeline for the FUSE data. CalFUSE is described in Dixon et al.
(2007). We reduced each RAW data file and generated intermediate data files
(IDF files). We then combined and calibrated the multiple IDF files using the
data analysis tools within the CalFUSE pipeline package, specifically
idf_combine and cf_extract_spectra. We used CalFUSE’s default parameters but
set the initial wavelength for LiF Channels (LIF_W0) to 987.1 Å. This was done
to ensure that the spectra from the LiF1A and LiF2B channels started at the
same wavelength. We rebinned the calibrated data using cf_arith to have a
wavelength bin size of 0.104 Å (16 detector pixels). The CalFUSE calibrated
files provide fluxes, flux errors, raw photon counts, and estimated background
counts as a heliocentric wavelength spectrum. The reduced data products are
available in http://dx.doi.org/10.5281/zenodo.10680806 (catalog
doi:10.5281/zenodo.10680806).

Our main interest in this study is O VI $\lambda$1031.9
($1s^{2}~{}2s~{}^{2}S_{1/2}-1s^{2}~{}2p~{}^{2}P_{3/2}$) and $\lambda$1037.6
($1s^{2}~{}2s~{}^{2}S_{1/2}-1s^{2}~{}2p~{}^{2}P_{1/2}$) emission lines, which
are primarily emitted from the warm-hot CGM at $\textrm{T}\sim
10^{5.5}~{}\textrm{K}$. Hoopes et al. (2003) also studied FUSE observations
for M82 but found no detectable O VI $\lambda$1032 emission lines. They used
data reduced by an older version of CalFUSE (2.0.5). Several parameters and
functions have been updated since version 2.0.5. (see Dixon et al., 2007, for
details). One noticeable change is the range of acceptable pulse height
values. The pulse height used in Hoopes et al. (2003) is between 4 and 14. The
default pulse height limits in CalFUSE version 3.2.3. are from 2 to 25 in the
LiF1A channel and 3 to 24 in the LiF2B channel. Dixon et al. (2007) point out
that narrower pulse height limits than the default values can cause
significant flux losses. Chung et al. (2021) also show that the updated pulse
height limit helps detect emission lines invisible with the old pulse height
limit. Here we revisit the FUSE observational data using the up-to-date
CalFUSE pipeline and see if any spectral lines have been missed.

##  3 Results

Figure 2 displays the individual FUSE spectra of four regions in M82. We
plotted in the spectra the dead-time corrected photon count values (referred
to as WEIGHTS in CalFUSE) from CalFUSE calibrated files created using the
LiF2B channel data. From the data on the southern side (Regions A, B, and C),
we found potential emission peaks at around 1031 and 1037 Å. The two peaks are
expected to correspond to blue-shifted O VI $\lambda\lambda$ 1032, 1038
emission lines. The locations of two potential lines are marked as gray
shades, and the lines are shifted as much as the V${}_{\tiny\textrm{LOS}}$
value from Table 1. A S/N estimated in each shaded region is less than 3,
while the longer wavelength line in Regions A and C has a S/N higher than 2.5.
The average of S/Ns is approximately 2. Despite the low S/N, the line feature
consistently seen in three regions makes the presence of the O VI emission
lines promising. The spectra from the LiF1A channel data have similar peaks,
but their S/Ns are lower than those in the LiF2B spectra.

The spectrum of the northern side (Region D) does not have apparent emission
line features. In Region D of Figure 2, S/Ns at the shaded wavelength regimes
are less than 1. In particular, the LiF1A spectrum at Region D is dominated by
background noise. This is likely due to the fact that the outflow on the
northern side is positioned behind the northwest side of the galaxy disk,
which is closer to us (Lynds & Sandage, 1963). As the emission from the
outflow on the northern side should pass through the galactic disk, it is
subject to greater extinction than the emission on the southern side. As a
result, the emission lines on the northern side may be obscured and may not be
visible.

Figure 2: FUSE spectra from the LiF2B channel. Gray shaded regions indicate
potential locations of the O VI $\lambda\lambda$ 1032, 1038 emission lines.
The width of the shades is 0.53 Å. A width of 0.53 Å covers $\sim$90% of the
emission flux when an intrinsic line FWHM is $\sim$81 km s-1. Horizontal solid
gray lines depict background values. Vertical dotted lines are positioned at
the geocoronal emission lines. On the top-right corner of each spectrum, the
number is a total exposure time, including both day and night exposures.

To enhance the S/N in a resultant spectrum, we combined the spectra from three
regions A, B, and C using idf_combine. As all the observations were made
between 2002-02-02 and 2002-02-03, there was no significant change in
effective area of the telescope during this short period. Outflow velocity
measured in M82 with optical observations changes slowly beyond a distance of
around 30″ from the center (Shopbell & Bland-Hawthorn, 1998). Because of the
opposite outflow directions on the northern and southern sides of the galaxy,
we combined only the data from the southern side.

Figure 3: FUSE spectra combined from the observations on the southern side.
The top and bottom panels are the spectrum from the LiF 1A and 2B channels,
respectively. Gray shades are the same as in Figure 2. Horizontal gray lines
are background values. The dashed lines are the background values calculated
through CalFUSE, and the solid lines are the adjusted values given our MCMC
fitting result.

Figure 3 shows the count spectrum combined from observational data for Regions
A, B, and C, separately in the LiF 1A and 2B channels. Both channels show a
clear emission peak near 1031.2 Å. The other peak is located near 1037.9 Å,
although the peak locations in LiF 1A and 2B channels are slightly offset.
Because of the low S/N, the line position and flux estimated in each combined
spectrum may not be precise. For more accurate results, we incorporate the
spectra from both channels to estimate the physical properties of the emission
lines.

We used a Markov chain Monte Carlo (MCMC) approach to derive properties of O
VI $\lambda\lambda$ 1032, 1038 emission lines. We created an intrinsic model
spectrum containing the O VI doublet lines, which are parameterized with a
heliocentric radial velocity (V${}_{\tiny\textrm{LOS}}$), a full width at half
maximum (FWHM) of lines, and individual line intensities
(I${}_{\textrm{\footnotesize{{O~{}VI}}},\lambda 1032}$ and
I${}_{\textrm{\footnotesize{{O~{}VI}}},\lambda 1038}$). The model spectrum was
convolved with a Gaussian profile (0.021 Å) corresponding to the FUSE line
spread function and a top-hat function with a 0.362 Å width accounting for the
filling effect on the LWRS aperture. We converted the values in the model
spectrum from flux to photon counts and added background values. The added
background values are derived from multiplying the background count spectrum,
provided by CalFUSE, and a scaling factor, which is one of fitting parameters.
The scaling factors correct the overestimated background shown in the LiF1A
spectrum of Figure 3 and account for the continuum in the wavelength regime.
The adjusted background values according to the fitting result are shown as
solid horizontal lines in Figure 4. Lastly, using a continuous Poisson
probability function,
$P(N_{obs})\propto(N_{model})^{N_{obs}}~{}exp(-N_{model})/\Gamma(N_{obs}+1)$,
we estimated the likelihood of the observational data given the model spectrum
separately for the LiF 1A and 2B channels. We performed MCMC fitting, assuming
the product of individual likelihoods in both channels as the likelihood of
observational data given model spectra. We estimated the likelihood separately
in the LiF 1A and 2B channels because of the different background values and
the different flux conversion factors from different effective areas. Table 1
presents the MCMC results including the median value and the range of the 68%
confidence level of parameter probability distributions.

Figure 4 compares the model spectra and the observational spectrum. The model
spectrum plotted in black was created using the median values for each model
parameter in Table 1. The observational spectrum is generated from the flux
spectra in the CalFUSE outputs of the LiF 1A and 2B channels. We combined the
flux spectra from the two channels by subtracting the flux, corresponding to
the background adjusting values from the MCMC result, and weighting by the
inverse square of the flux errors. The deviation between the observational
spectrum and the median model spectrum is shown in a lower panel of Figure 4.

Based on our fitting result, it appears that
I${}_{\textrm{\footnotesize{{O~{}VI}}},\lambda 1032}$ is lower than
I${}_{\textrm{\footnotesize{{O~{}VI}}},\lambda 1038}$. This is unexpected,
because these lines are resonance doublets, therefore O VI ${\lambda 1032}$
line should be two times brighter than O VI ${\lambda 1038}$ line. One
possible explanation for this unexpected result is that another emission line
may be blended with the O VI $\lambda$ 1038 line, leading to in an incorrect
estimation of I${}_{\textrm{\footnotesize{{O~{}VI}}},\lambda 1038}$. We will
discuss this further in Section 4.2.

Note that the line FWHM may not accurately represent properties of the medium
emitting the O VI lines. The top-hat function with a 0.362 Å width is
estimated when emission sources extensively fill an observing aperture. If the
emission is locally concentrated in the aperture, the intrinsic line FWHM may
be underestimated in our method. Also, since the spectrum is combined from
three different regions, the velocity variation in these regions may appear as
a larger line FWHM.

Figure 4: Model spectra plotted over the observations. The observation
spectrum is combined from the LiF 1A and 2B spectra. The thick black spectrum
is modeled with the median values in Table 1. Gray lines depict sample spectra
drawn from the MCMC chains. Red arrows in the top panel indicate the possible
locations of C II $\lambda\lambda$ 1036, 1037. Solid red arrows are with the
radial velocity in Table 1. Dashed red arrows are shifted with a radial
velocity of -100 km s-1, taken from H$\alpha$ observations (Shopbell & Bland-
Hawthorn, 1998). The bottom panel displays the deviation between the
observation and the median model spectrum. Red error bars represent a
intensity uncertainty in the spectrum’s wavelength range.  Table 1: MCMC
results including O VI $\lambda\lambda$1032, 1038 emission lines.

V${}_{\tiny\textrm{LOS}}$ \- V${}_{\tiny\textrm{sys,{M82}}}$ | FWHM | I${}_{\textrm{\footnotesize{{O~{}VI}}},\lambda 1032}$ | I${}_{\textrm{\footnotesize{{O~{}VI}}},\lambda 1038}$
---|---|---|---
[km s-1] | [km s-1] | [erg cm-2 s-1 arcsec-2] | [erg cm-2 s-1 arcsec-2]
-415.67 ${}^{+42.69}_{-12.65}$ | 81.42 ${}^{+216.17}_{-53.13}$ | 1.78 ${}^{+0.66}_{-0.52}$ $\times 10^{-18}$ | 2.05 ${}^{+1.64}_{-0.66}$ $\times 10^{-18}$

Note. — Median values and 68% confidence ranges from parameter probability
distributions. The velocities are heliocentric. The velocity of M82
(V${}_{\textrm{\tiny sys, {M82}}}$) is 203 km s-1 (de Vaucouleurs et al.,
1995). Intensities are uncorrected for extinction.

Additionally, we repeated the MCMC fitting process, focusing solely on the O
VI $\lambda$1032 emission line. The results in Table 1 are estimated assuming
that the FWHM is consistent for both O VI $\lambda$1032 and $\lambda$1038
lines. However, if the O VI ${\lambda 1038}$ line is blended with other lines,
as mentioned previously, the line FWHM and intensity observed at O VI
$\lambda$1038 would not be exclusively attributed to the O VI ${\lambda 1038}$
emission line. Consequently, it is unnecessary for the O VI $\lambda$1032 and
$\lambda$1038 lines to have the same line FWHM. It is also challenging to
determine the properties of the O VI ${\lambda 1038}$ emission line. The
fitting results only of O VI $\lambda$1032 emission line are presented in
Figure 5 and Table 2. The radial velocity and intensity of the line remain
unchanged from the previous fitting of the O VI $\lambda\lambda$1032, 1038
doublet. However, the line FWHM is noticeably narrower when derived from the
single line fitting. This is also a potential evidence that another emission
line may overlap with the O VI $\lambda$1038 emission line, causing it to
appear broader in the spectrum.

Figure 5: The same as Figure 4, but only including fitting for O VI
$\lambda$1032 emission line. The model parameters can be found in Table 2.
Table 2: MCMC results only for O VI $\lambda$1032 emission line and an upper
limit for the C II $\lambda$1037 emission line.

O VI $\lambda$1032 | C II $\lambda$1037
---|---
V${}_{\tiny\textrm{LOS}}$ \- V${}_{\tiny\textrm{sys,{M82}}}$ | FWHM | I${}_{\textrm{\footnotesize{{O~{}VI}}},\lambda 1032}$ | I${}_{\textrm{\footnotesize{{C~{}II}}},\lambda 1037}$
[km s-1] | [km s-1] | [erg cm-2 s-1 arcsec-2] | [erg cm-2 s-1 arcsec-2]
-418.16 ${}^{+0.10}_{-0.10}$ | 30.25 ${}^{+33.91}_{-19.89}$ | 1.73 ${}^{+0.46}_{-0.45}$ $\times 10^{-18}$ |  $<$ 1.20 $\times 10^{-18}$

Note. — The same as Table 1 but the fitting results only for O VI
$\lambda$1032 emission line. I${}_{\textrm{\footnotesize{{C~{}II}}},\lambda
1037}$ is estimated as a 2-$\sigma$ upper limit based on the intensity
uncertainty and a wavelength range of 0.37 Å.

##  4 Discussion

###  4.1 Comparison with other wavelength observations

The O VI emission lines we observed are most likely emitted from the outflow
of M82. Based on our observations with the O VI lines, the warm-hot outflow
velocity on the southern side of M82 is approximately -420 km s-1. While it is
conceivable that high-velocity clouds within the Milky Way may produce these
emission lines, the lack of detected emission lines on the northern region
leads us to consider that M82 is the more likely source. Also, according to
Westmeier (2018), the radial velocity in the local standard of rest of high-
velocity clouds around the line of sight toward M82 is around 180 km s-1.

The spectral lines in the outflow on the southern side of M82 have been found
with multiple observations. Shopbell & Bland-Hawthorn (1998) discovered
H$\alpha$ lines at a velocity of about -30 and -300 km s-1 and another optical
emission lines, N II and O III, with a similar velocity trend. CO lines
observed by Salak et al. (2013) are shifted by around -30 and -100 km s-1
relative to the galactic systemic velocity. Note that the outflow velocities
mentioned here are line-of-sight velocities, so they are lower than the actual
outflow velocity due to the inclination effect.

Our result shows that the radial velocity of O VI emission lines is higher
than that of optical and CO emission lines. The O VI emission lines are
primarily emitted from a hotter medium ($\textrm{T}\sim
10^{5.5}~{}\textrm{K}$) than optical and CO emission lines are. Therefore, our
finding indicates that a hotter phase medium is being expelled faster than a
colder medium. Numerical simulations also demonstrated that different phases
could have varying velocities. For example, models such as the M82’s outflow
find that hotter gas moves faster than colder gas (Schneider et al., 2020).
Bow shock simulations, although simulating a supersonic motion of galaxies,
also exhibit reduced velocities in regions with lower temperatures (Schulreich
& Breitschwerdt, 2011). Another galactic outflow simulation proposed that
H$\alpha$ emission may come from gas cloudlets, broken away from the gas cloud
being expelled from a galactic disk, while O VI emission occurs around the gas
cloudlets (Cooper et al., 2008, 2009). To better understand a multiphase gas
outflow’s complex structure, more observations and numerical analysis would be
required.

Another noticeable difference between our findings and optical or radio line
observations is line splitting; that is, each emission line is shifted by two
different velocities. The line splitting seen in optical and CO observations
is explained as emission from the surface of a biconical outflow (see Figure 8
in Salak et al., 2013). On the other hand, the line splitting is not seen in
our FUSE spectrum. However, our observation of a single O VI emission line
does not contradict the bi-conical model. The gas medium filling the cones
could obstruct the detection of another O VI line emitted from the rear side
of the cone due to the high extinction in UV wavelengths. With assumptions
that RV of 3.1 and H$\alpha$/H$\beta$ of 4 (Heckman et al., 1990) and using
equations in Cardelli et al. (1989) and Xiao et al. (2012), the optical depth
of the total medium in M82’s outflow is approximately 4 at 1035 Å. If the
front-side emission is radiated in front of the outflow medium and the rear-
side emission passes through the total medium, the ratio between the front-
and rear-side emission flux is roughly 50. With the low S/N of the observation
data we used, it is likely that the fainter emission peaks would be
undetected. It is also possible that the front and rear sides of the region
emitting O VI are not clearly distinct, unlike the colder phase medium. To
better understand the structural difference between the region emitting O VI
and the colder gas medium, we would require additional observations that are
spatially resolved across a wide field at multiple wavelengths.

###  4.2 Flux at O VI $\lambda$1038

In the results in Table 1, the line at O VI $\lambda$1038 is brighter than
that at O VI $\lambda$1032\. The emission line at O VI $\lambda$1038 also
looks broader than that at O VI $\lambda$1032 in Figure 4. Theoretically, when
electrons are mainly excited by collisions, the flux ratio of the doublet
lines is 2:1, with O VI $\lambda$1032 being brighter. The variation of a flux
ratio of doublet lines, similar to O VI doublet lines, also has been observed
in star-forming galaxies (e.g., Henry et al., 2018; Xu et al., 2023). However,
the flux ratio lower than 1 is uncommon.

The observed flux ratio of doublet lines can vary depending on optical depths
due to self-absorption (e.g., Shelton et al., 2001; Chisholm et al., 2020).
Seon (2023) performs simulations for Mg II doublet lines, similar to O VI
doublet lines, that are impacted by self-absorption and scattering effects.
However, the flux ratio of the simulated doublets in emission spectra is not
found to be less than 1.2. In cases with a very high expansion velocity and
continuum sources, the flux ratio of the simulations goes lower than 1.
Nevertheless, our observations do not show noticeably enhanced continuum
levels to support the cases. Additionally, the contribution of ions excited by
photoionization could decrease the flux ratio close to 1 (Shelton et al.,
2001), but not less than 1. Therefore, it is difficult to explain our observed
flux ratio, less than 1, with a simple model.

Another plausible explanation for brighter flux at our O VI $\lambda$1038 is
that the line is overlapped with another emission lines, most likley C II
$\lambda$1036.3 ($2s~{}2p^{2}~{}^{2}S_{1/2}-2s^{2}~{}2p~{}^{2}P_{1/2}$) and C
II $\lambda$1037.0 ($2s~{}2p^{2}~{}^{2}S_{1/2}-2s^{2}~{}2p~{}^{2}P_{3/2}$). C
II $\lambda$1037 emission line has been observed in the Galactic halo (Shelton
et al., 2001; Shelton, 2002). Dixon et al. (2006) also mention that the O VI
$\lambda$1038 line is often possible to be blended with the C II $\lambda$1037
line. In Figure 4, we illustrate possible locations of C II emission lines.
The C II lines, which are shifted in the same way as the observed O VI lines
(solid arrows), are positioned apart from the O VI lines. However, the
plausible C II $\lambda$1037 line shifted same as the H$\alpha$ velocity
(dashed arrows), V${}_{\textrm{\tiny LOS}}\sim-100\textrm{ km s}^{-1}$, is
close to the O VI $\lambda$1038 line. C II emission lines are effectively
emitted from the regions of T$\sim 10^{4.5}$ K (Figure 6 in Tumlinson et al.,
2017) which is close to the temperature for emitting H$\alpha$ emission.
Therefore, C II emitters may have different velocities than O VI emitters but
similar to H$\alpha$ emitters. Other UV emission lines, which are primarily
emitted at a similar temperature with C II lines, like C III $\lambda$977.02
and N III $\lambda$989.79 lines, are hard to be detected due to their
proximity to geocoronal emission lines.

###  4.3 O VI emitters

In the FUSE spectrum, we are unable to detect C II $\lambda$1037 emission
lines at the wavelength where the emission were shifted by the same amount as
the observed O VI lines. Instead, we estimate an upper intensity limit in the
regime where the C II emission line at the same radial velocity as the
observed O VI lines would be expected. This limit has been calculated based on
intensity uncertainties, within a wavelength range of 0.37 Å. The width of the
range covers more than 90% of the total flux of a line with a FWHM of 30 km
s-1. The estimated upper limit is noted in Table 2. Accordingly, a lower limit
of the brightness ratio between O VI $\lambda$1032 and C II $\lambda$1037
lines is 1.4. We have compared this intensity ratio with various models to
speculate about the physical properties of the O VI emitting regions.

By comparing with models in Slavin et al. (1993), we estimate that the
temperature is higher than $10^{5.3}$ K if the O VI lines are emitted by a
turbulent mixing layer. We also compare the brightness ratio with the shock
models in the Mexican Million Models database (shock part; 3MdBs),
specifically the model results from Allen et al. (2008) and Alarie & Morisset
(2019). As per the model, a shock velocity lower than 100 km/s is insufficient
to produce the lower limit of the brightness ratio. A shock velocity higher
than approximately 130 km s-1 is more likely to account for the brightness
ratio. On the other hand, infrared observations suggested that the warm H2
outflow may have lower velocity shock, which is lower than 40 km s-1 (Beirão
et al., 2015). As shown in the different outflowing velocities between O VI
and molecular observations, the O VI emitters and molecular outflow region are
likely to be separated.

##  5 Conclusion

We explore the outflow in M82 using FUSE observations. We found O VI
$\lambda\lambda$1032, 1038 emission lines in the southern side of M82 using
FUSE observations. These UV emission lines indicate that the outflowing medium
in the warm gas phase is undergoing radiative cooling. The O VI emission lines
are shifted by higher radial velocity than H$\alpha$ or CO observations. The
higher flux at O VI $\lambda$1038 than one at O VI $\lambda$1032 could be
explained by the presence of C II emission lines at different line-of-sight
velocity from O VI lines. It is likely that the outflowing medium in the
outflow of M82 has a complex velocity phase in different temperature phases.
Therefore, in addition to the previous studies mostly focused on the outflow
medium in the colder phase observed at visible or longer wavelengths, more
advanced observations at higher energy regimes are required to definitively
determine the behavior of outflowing gas.

This work is supported by NASA under award No. 80NSSC23K1129. Some of the data
presented in this paper were obtained from the Mikulski Archive for Space
Telescopes (MAST) at the Space Telescope Science Institute. The specific
observations analyzed can be accessed via http://dx.doi.org/10.17909/y0wa-4b44
(catalog doi:10.17909/y0wa-4b44). STScI is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS5–26555.
Support to MAST for these data is provided by the NASA Office of Space Science
via grant NAG5–7584 and by other grants and contracts. This work is based on
observations made with the NASA-CNES-CSA Far Ultraviolet Spectroscopic
Explorer. FUSE is operated for NASA by the Johns Hopkins University under NASA
contract NAS5-32985. This work includes observations made with the NASA Galaxy
Evolution Explorer. GALEX is operated for NASA by the California Institute of
Technology under NASA contract NAS5-98034. The Second Palomar Observatory Sky
Survey (POSS-II) was made by the California Institute of Technology with funds
from the National Science Foundation, the National Aeronautics and Space
Administration, the National Geographic Society, the Sloan Foundation, the
Samuel Oschin Foundation, and the Eastman Kodak Corporation.

## References

  * Alarie & Morisset (2019) Alarie, A., & Morisset, C. 2019, Rev. Mexicana Astron. Astrofis., 55, 377, doi: 10.22201/ia.01851101p.2019.55.02.21
  * Allen et al. (2008) Allen, M. G., Groves, B. A., Dopita, M. A., Sutherland, R. S., & Kewley, L. J. 2008, ApJS, 178, 20, doi: 10.1086/589652
  * Anderson et al. (2013) Anderson, M. E., Bregman, J. N., & Dai, X. 2013, ApJ, 762, 106, doi: 10.1088/0004-637X/762/2/106
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
  * Bahcall & Spitzer (1969) Bahcall, J. N., & Spitzer, Lyman, J. 1969, ApJ, 156, L63, doi: 10.1086/180350
  * Beirão et al. (2015) Beirão, P., Armus, L., Lehnert, M. D., et al. 2015, MNRAS, 451, 2640, doi: 10.1093/mnras/stv1101
  * Bradley et al. (2022) Bradley, L., Deil, C., Ginsburg, A., et al. 2022, astropy/regions: v0.7, v0.7, Zenodo, doi: 10.5281/zenodo.7259631
  * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245, doi: 10.1086/167900
  * Chisholm et al. (2020) Chisholm, J., Prochaska, J. X., Schaerer, D., Gazagnes, S., & Henry, A. 2020, MNRAS, 498, 2554, doi: 10.1093/mnras/staa2470
  * Chung et al. (2021) Chung, H., Vargas, C. J., & Hamden, E. 2021, ApJ, 916, 7, doi: 10.3847/1538-4357/ac04af
  * Coker et al. (2013) Coker, C. T., Thompson, T. A., & Martini, P. 2013, ApJ, 778, 79, doi: 10.1088/0004-637X/778/1/79
  * Cooper et al. (2008) Cooper, J. L., Bicknell, G. V., Sutherland, R. S., & Bland-Hawthorn, J. 2008, ApJ, 674, 157, doi: 10.1086/524918
  * Cooper et al. (2009) —. 2009, ApJ, 703, 330, doi: 10.1088/0004-637X/703/1/330
  * de Vaucouleurs et al. (1995) de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., et al. 1995, VizieR Online Data Catalog, VII/155
  * Dixon et al. (2007) Dixon, W. V., Sahnow, D. J., Barrett, P. E., et al. 2007, PASP, 119, 527, doi: 10.1086/518617
  * Dixon et al. (2006) Dixon, W. V. D., Sankrit, R., & Otte, B. 2006, ApJ, 647, 328, doi: 10.1086/505168
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Gupta et al. (2012) Gupta, A., Mathur, S., Krongold, Y., Nicastro, F., & Galeazzi, M. 2012, ApJ, 756, L8, doi: 10.1088/2041-8205/756/1/L8
  * Hafen et al. (2019) Hafen, Z., Faucher-Giguère, C.-A., Anglés-Alcázar, D., et al. 2019, MNRAS, 488, 1248, doi: 10.1093/mnras/stz1773
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hayes et al. (2016) Hayes, M., Melinder, J., Östlin, G., et al. 2016, ApJ, 828, 49, doi: 10.3847/0004-637X/828/1/49
  * Heckman et al. (1990) Heckman, T. M., Armus, L., & Miley, G. K. 1990, ApJS, 74, 833, doi: 10.1086/191522
  * Henry et al. (2018) Henry, A., Berg, D. A., Scarlata, C., Verhamme, A., & Erb, D. 2018, ApJ, 855, 96, doi: 10.3847/1538-4357/aab099
  * Hoopes et al. (2003) Hoopes, C. G., Heckman, T. M., Strickland, D. K., & Howk, J. C. 2003, ApJ, 596, L175, doi: 10.1086/379533
  * Hoopes et al. (2005) Hoopes, C. G., Heckman, T. M., Strickland, D. K., et al. 2005, ApJ, 619, L99, doi: 10.1086/423032
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Karachentsev et al. (2018) Karachentsev, I. D., Kaisina, E. I., & Makarov, D. I. 2018, MNRAS, 479, 4136, doi: 10.1093/mnras/sty1774
  * Lehner et al. (2015) Lehner, N., Howk, J. C., & Wakker, B. P. 2015, ApJ, 804, 79, doi: 10.1088/0004-637X/804/2/79
  * Leroy et al. (2015) Leroy, A. K., Walter, F., Martini, P., et al. 2015, ApJ, 814, 83, doi: 10.1088/0004-637X/814/2/83
  * Lynds & Sandage (1963) Lynds, C. R., & Sandage, A. R. 1963, ApJ, 137, 1005, doi: 10.1086/147579
  * Moos et al. (2000) Moos, H. W., Cash, W. C., Cowie, L. L., et al. 2000, ApJ, 538, L1, doi: 10.1086/312795
  * Oppenheimer & Davé (2008) Oppenheimer, B. D., & Davé, R. 2008, MNRAS, 387, 577, doi: 10.1111/j.1365-2966.2008.13280.x
  * Sahnow et al. (2000) Sahnow, D. J., Moos, H. W., Ake, T. B., et al. 2000, ApJ, 538, L7, doi: 10.1086/312794
  * Salak et al. (2013) Salak, D., Nakai, N., Miyamoto, Y., Yamauchi, A., & Tsuru, T. G. 2013, PASJ, 65, 66, doi: 10.1093/pasj/65.3.66
  * Schneider et al. (2020) Schneider, E. E., Ostriker, E. C., Robertson, B. E., & Thompson, T. A. 2020, ApJ, 895, 43, doi: 10.3847/1538-4357/ab8ae8
  * Schulreich & Breitschwerdt (2011) Schulreich, M. M., & Breitschwerdt, D. 2011, A&A, 531, A13, doi: 10.1051/0004-6361/200912436
  * Seon (2023) Seon, K.-i. 2023, arXiv e-prints, arXiv:2310.17908, doi: 10.48550/arXiv.2310.17908
  * Shelton (2002) Shelton, R. L. 2002, ApJ, 569, 758, doi: 10.1086/339263
  * Shelton et al. (2001) Shelton, R. L., Kruk, J. W., Murphy, E. M., et al. 2001, ApJ, 560, 730, doi: 10.1086/322478
  * Shopbell & Bland-Hawthorn (1998) Shopbell, P. L., & Bland-Hawthorn, J. 1998, ApJ, 493, 129, doi: 10.1086/305108
  * Slavin et al. (1993) Slavin, J. D., Shull, J. M., & Begelman, M. C. 1993, ApJ, 407, 83, doi: 10.1086/172494
  * Strickland & Heckman (2009) Strickland, D. K., & Heckman, T. M. 2009, ApJ, 697, 2030, doi: 10.1088/0004-637X/697/2/2030
  * Strickland et al. (2004) Strickland, D. K., Heckman, T. M., Colbert, E. J. M., Hoopes, C. G., & Weaver, K. A. 2004, ApJS, 151, 193, doi: 10.1086/382214
  * Tumlinson et al. (2017) Tumlinson, J., Peeples, M. S., & Werk, J. K. 2017, ARA&A, 55, 389, doi: 10.1146/annurev-astro-091916-055240
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Westmeier (2018) Westmeier, T. 2018, MNRAS, 474, 289, doi: 10.1093/mnras/stx2757
  * Xiao et al. (2012) Xiao, T., Wang, T., Wang, H., et al. 2012, MNRAS, 421, 486, doi: 10.1111/j.1365-2966.2011.20327.x
  * Xu et al. (2023) Xu, X., Henry, A., Heckman, T., et al. 2023, ApJ, 943, 94, doi: 10.3847/1538-4357/aca89a


# Thin accretion disk images of rotating hairy Horndeski black holes

Mohaddese Heydari-Fard1 , Malihe Heydari-Fard2 and Nematollah Riazi1
_1 Department of Physics, Shahid Beheshti University, Evin 19839, Tehran,
Iran_
_2 Department of Physics, The University of Qom, 3716146611, Qom, Iran_
Electronic address<EMAIL_ADDRESS>address:
<EMAIL_ADDRESS>address<EMAIL_ADDRESS>
###### Abstract

By considering the steady-state Novikov-Thorne model, we study thin accretion
disk processes for rotating hairy black holes in the framework of the
Horndeski gravity. We obtain the electromagnetic properties of accretion disk
around such black holes and investigate the effects of the hair parameter $h$
on them. We find that by decreasing the hair parameter from the Kerr limit,
$h\rightarrow 0$, the radius of the innermost stable circular orbit decreases
which makes thin accretion disks around rotating hairy black holes in
Horndeski gravity more efficient than that for the Kerr black hole in general
relativity. Furthermore, using the numerical ray-tracing method, we plot thin
accretion disk images around these black holes and investigate the effects of
hair parameter on the central shadow area of accretion disk.
PACS numbers: 97.10.Gz, 04.70.–s, 04.50.Kd
Keywords: Accretion and accretion disks, Physics of black holes, Modified
theories of gravity

##  1 Introduction

Black holes (BHs) are solutions to Einstein field equations of general
relativity (GR). Despite the historical debates on the BH’s existence, there
have been several observational evidences in recent decades which support the
existence of BHs. For instance, the discovery of gravitational waves from
binary BH mergers [1], the detection of stochastic gravitational wave
background by the Pulsar Timing Array [2]–[3]and images of the shadow of BHs
in the center of M87 galaxy and Milky Way [4]–[5], are some of them. Moreover,
the observed electromagnetic spectra of accretion disks around compact objects
can also confirm the existence of BHs [6]–[7].

It is well known that the astrophysical BHs gain their mass through the
accretion process. In this process, the accreting matter accumulates into a BH
by its gravitational attraction and releases gravitational energy in the form
of radiation. The emission spectra of such radiation is associated to nearly
geodesic motion of test particles and the structure of central BH. Therefore,
the electromagnetic spectrum of accretion disk around a BH can be used to
explore additional imprints of modified theory of gravity that we are
interested in testing. Such studies including the investigation of accretion
disks in $f(R)$ gravity, scalar-vector-tensor-gravity, brane-world scenario,
Einstein-Maxwell-dilaton and Einstein-Gauss-Bonnet gravity have been done in
[8]–[18]. The effects of dark matter, non-linear electrodynamics and naked
singularities on the disk properties have also been studied in [19]–[26],
respectively. Moreover, exploring the image of the accretion disk around a BH
is another interesting topic that has been the focus of attention in recent
years [27]–[33].

Although the theory of GR has been well tested in weak and strong field
regimes, it needs to be further modified to address its shortcomings, such as
an inflationary era in the early universe, the nature of the dark matter and
dark energy, the cosmological constant and the Hubble tension.

Among modifications of GR are the scalar-tensor theories in which one or more
scalar fields couple to curvature and interfere with the gravitational
interaction [34]. The most general scalar-tensor theory in four-dimensions is
the Horndeski gravity that leads to second-order field equations and it is
free of Ostrogradski instabilities [35]. Besides the cosmological studies in
this theory [36]–[41], the BH solutions of Horndeski gravity have been
extensively studied [42]–[46]. Also, by considering the quartic scalar field
model, the hairy BH solution of Horndeski gravity has been obtained in [47].
The extension of this solution to the case of rotating hairy Horndeski BHs is
done in [48]. The authors have also investigated the gravitational deflection
of light in such a space-time and by modelling of Sgr A* BH with hairy
Horndeski BH showed that the correction in deflection angle can be measured
with the current observational facilities. The strong lensing observables of
static hairy Horndeski BHs, including deflection angle, separation and
magnification have been discussed in [49]. Testing Horndeski gravity from the
observational data of the BH shadow has been considered in [50]–[51] and some
constraints on the hair parameter have been obtained. The motion of test
particles and their harmonic oscillations in the spacetime of static hairy BHs
in Hordeski gravity, as well as the astrophysical applications of
quasiperiodic oscillations has been studied in [53]. Furthermore, superradiant
energy extraction [54], perturbations of massless external fields [55], and
the effects of tilted thin accretion disks on the optical appearance of hairy
Horndeki BHs [56] were studied.

The image of static hairy Horndeski BHs illuminated by thin disk and thin
spherical accretion flows, has been investigated in [57]. Since the
astrophysical BHs are expected to be rotating due to the accretion effects, in
the present work we consider the rotating hairy Horndeski BHs and study thin
accretion disk luminosity and its image around these BHs. We employ the
steady-state Novikov-Thorne model [58] to study thin accretion disk processes
of rotating BHs in Horndeski gravity, and investigate the effects of hair
parameter on both the disk properties and its image.

The structure of the present paper is as follows. In section 2, we present the
test particles motion and thin accretion disk model in a general stationary
axisymmetric space-time. A brief review of rotating hairy Horndeski BHs is
given in section 3. Then, the disk properties around such BHs are obtained in
section 4 and the effects of the hair parameter are discussed. In section 5,
we plot the ray-traced redshifted image and intensity and polarization profile
of an accretion disk around a rotating hairy Horndeski BH. The paper ends with
drawing our conclusions.

##  2 Thin accretion disk model

In order to study the electromagnetic properties of accretion disk, we need to
obtain the equations of motion of test particles in the space-time of the
central compact object. Therefore, we take into account the Lagrangian ${\cal
L}$ of a test particle in the vicinity of the central BH as follows

${\cal L}=\frac{1}{2}g_{\mu\nu}\dot{x^{\mu}}\dot{x^{\nu}},$ (1)

where a dot denotes derivative with respect to the affine parameter and
$g_{\mu\nu}$ is the line element of a generic stationary and axisymmetric
space-time

$ds^{2}=g_{tt}dt^{2}+2g_{t\phi}dtd\phi+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2},$
(2)

for which the metric components $g_{tt},g_{rr},g_{\theta\theta}$,
$g_{\phi\phi}$ and $g_{t\phi}$ are assumed to be only functions of $r$ and
$\theta$ coordinates. Thus, there are two constants of motion, the energy and
the angular momentum per unit rest-mass, $\tilde{E}$ and $\tilde{L}$, which
are given by

$g_{tt}\dot{t}+g_{t\phi}\dot{\phi}=-\tilde{E},$ (3)
$g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}=\tilde{L}.$ (4)

Using above equations and the normalization condition for massive particles,
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$, we obtain

$g_{rr}\dot{r}^{2}+g_{\theta\theta}\dot{\theta}^{2}=V_{\rm eff}(r,\theta),$
(5)

where the effective potential is

$V_{\rm
eff}(r,\theta)=-1+\frac{\tilde{E}^{2}g_{\phi\phi}+2\tilde{E}\tilde{L}g_{t\phi}+\tilde{L}^{2}g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}.$
(6)

Then, from the radial component of the geodesic equation with conditions
$\dot{r}=\dot{\theta}=\ddot{r}=0$ for equatorial circular orbits, we find the
following relation for the angular velocity, $\Omega=\dot{\phi}/\dot{t}$, [59]

$\Omega_{\pm}=\frac{-g_{t\phi,r}\pm\sqrt{(g_{t\phi,r})^{2}-g_{tt,r}g_{\phi\phi,r}}}{g_{\phi\phi,r}},$
(7)

where the upper and lower signs refer to co-rotating and counter-rotating
orbits, respectively. Now, using equations (3)-(4) and from condition
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$, the specific energy ${\tilde{E}}$
and the specific angular momentum ${\tilde{L}}$ for a particle on a circular
orbit can be expresed as

${\tilde{E}}=-\frac{g_{tt}+g_{t\phi}\Omega}{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}},$ (8)
${\tilde{L}}=\frac{g_{t\phi}+g_{\phi\phi}\Omega}{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}.$ (9)

The innermost stable circular orbit, $r_{\rm isco}$, can be obtained from the
condition $V_{\rm eff,rr}=0$, which leads to the following equation

$V_{\rm eff,rr}\mid_{r=r_{\rm
isco}}=\frac{1}{g_{t\phi}^{2}-g_{tt}g_{t\phi}}\left[{\tilde{E}^{2}g_{\phi\phi,rr}}+2\tilde{E}\tilde{L}g_{t\phi,rr}+{\tilde{L}^{2}g_{tt,rr}}-\left(g_{t\phi}^{2}-g_{tt}g_{\phi\phi}\right)_{,rr}\right]\mid_{r=r_{\rm
isco}}=0.$ (10)

The standard framework to describe thin accretion disk processes is the
Novikov-Thorne model [58], which is a generalization of the Newtonian approach
of Shakura-Sunyaev [60]. The model is based on some typical assumptions such
as the vertical size of disk, $H$, is negligible compared to its horizontal
size, $H\ll r$. Also, it is assumed that the disk is in the equatorial plane
of the central compact object and its inner edge is located at the ISCO
radius. The disk is considered in a steady-state which means that the mass
accretion rate, $\dot{M}$, is constant in time. In this steady-state model the
accreting gas is assumed to be in thermodynamical equilibrium and thus the
disk temperature is related to the energy flux via the equation

$F(r)=\sigma_{\rm SB}T^{4}(r),$ (11)

where $\sigma_{\rm SB}=5.67\times 10^{-5}\rm erg$ $\rm s^{-1}cm^{-2}K^{-4}$ is
the Stefan-Boltzmann constant and the energy flux emitted from the disk
surface is given by [58, 61]

$F(r)=-\frac{\dot{M}_{0}\Omega_{,r}}{4\pi\sqrt{-g}\left(\tilde{E}-\Omega\tilde{L}\right)^{2}}\int^{r}_{r_{\rm
isco}}\left(\tilde{E}-\Omega\tilde{L}\right)\tilde{L}_{,r}dr.$ (12)

Moreover, the luminosity of disk can be found as [62]

$L(\nu)=4\pi d^{2}I(\nu)=\frac{16\pi^{2}\hbar\cos\gamma}{c^{2}}\int_{r_{\rm
in}}^{r_{\rm out}}\int_{0}^{2\pi}\frac{\nu_{\rm
e}^{3}rdrd\phi}{\exp{[\frac{h\nu_{\rm e}}{k_{\rm B}T}]}-1},$ (13)

where $d$ is the distance from the disk center, $\gamma$ is the disk
inclination angle, and $r_{\rm in}$ and $r_{\rm out}$ denote the inner and
outer radii of the edge of disk, respectively. Also, $\hbar$ and $k_{\rm B}$
are respectively the reduced Planck constant and Boltzmann constant, while
$\nu_{\rm e}=\nu(1+z)$ is the emitted frequency with the redshift factor $z$
given by

$1+z=\frac{1+\Omega r\sin\phi\sin\gamma}{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}.$ (14)

Finally, we define the accretion efficiency as

$\epsilon=1-\tilde{E}_{\rm isco}.$ (15)

which denotes the capability of central massive object to convert rest mass
into radiation.

##  3 Rotating BHs in Horndeski gravity

The action of Horndeski gravity is described by

${\cal S}=\int
d^{4}x\sqrt{-g}\left[Q_{2}(\chi)+Q_{3}(\chi)\Box\phi+Q_{4}(\chi)R+Q_{4,\chi}\left((\Box\phi)^{2}-(\nabla^{\mu}\nabla^{\nu}\phi)(\nabla_{\mu}\nabla_{\nu}\phi)\right)\right],$
(16)

where $g$, $R$ and $\phi$ are the metric determinant, Ricci scalar and a
scalar field, respectively. Also, $Q_{i=2,..4}(\chi)$ are arbitrary functions
of the kinetic term $\chi=-1/2\partial^{\mu}\phi\partial_{\mu}\phi$. Note the
above action is a quartic Horndeski scalar field model in which the terms
containing $Q_{5}$ are absent [47]. From the definition of the four-current

$j^{\nu}=\frac{1}{\sqrt{-g}}\frac{\delta{\cal S}}{\delta(\phi_{,\nu})},$ (17)

it follows that

$\displaystyle j^{\nu}$ $\displaystyle=$ $\displaystyle-
Q_{2,\chi}\phi^{,\nu}-Q_{3,\chi}(\phi^{,\nu}\Box\phi+\chi^{,\nu})-Q_{4,\chi}(\phi^{,\nu}R-2R^{\nu\sigma}\phi_{,\sigma})$
(18) $\displaystyle-$ $\displaystyle
Q_{4,\chi,\chi}[\phi^{,\nu}((\Box\phi)^{2}-(\nabla_{\alpha}\nabla_{\beta}\phi)(\nabla^{\alpha}\nabla^{\beta}\phi))+2(\chi^{,\nu}\Box\phi-\chi_{,\mu}\nabla^{\mu}\nabla^{\nu}\phi)],$

where we have used the usual convention of the Riemann tensor

$\nabla_{\rho}\nabla_{\beta}\nabla_{\alpha}\phi-\nabla{\beta}\nabla{\rho}\nabla_{\alpha}\phi=-R^{\sigma}_{\alpha\rho\beta}\nabla_{\sigma}\phi.$
(19)

Varying action (16) with respect to metric $g^{\mu\nu}$ we find the field
equations

$Q_{4}G_{\mu\nu}=T_{\mu\nu},$ (20)

where $T_{\mu\nu}$ is given by

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(Q_{2,\chi}\phi_{,\mu}\phi_{,\nu}+Q_{2}g_{\mu\nu})+\frac{1}{2}Q_{3,\chi}(\phi_{,\mu}\phi_{,\nu}\Box\phi-
g_{\mu\nu}\chi_{,\alpha}\phi^{,\alpha}+\chi_{,\mu}\phi_{\nu}+\chi_{,\nu}\phi_{\mu})$
(21) $\displaystyle-$ $\displaystyle
Q_{4,\chi}\left(\frac{1}{2}g_{\mu\nu}[(\Box\phi)^{2}-(\nabla_{\alpha}\nabla_{\beta}\phi)(\nabla^{\alpha}\nabla^{\beta}\phi)-2R_{\sigma\gamma}\phi^{,\sigma}\phi^{,\gamma}]-\nabla_{\mu}\nabla_{\nu}\phi\Box\phi\right.$
$\displaystyle+$
$\displaystyle\left.\nabla_{\gamma}\nabla_{\mu}\phi\nabla^{\gamma}\nabla_{\nu}\phi-\frac{1}{2}\phi_{,\mu}\phi_{,\nu}R+R_{\sigma\mu}\phi^{,\sigma}\phi_{,\nu}+R_{\sigma\nu}\phi^{,\sigma}\phi_{,\mu}+R_{\sigma\nu\gamma\mu}\phi^{,\sigma}\phi^{,\gamma}\right)$
$\displaystyle-$ $\displaystyle
Q_{4,\chi,\chi}\left(g_{\mu\nu}(\chi_{\alpha}\phi^{,\alpha}\Box\phi+\chi_{\alpha}\chi^{,\alpha})+\frac{1}{2}\phi_{,\mu}\phi_{,\nu}(\nabla_{\alpha}\nabla_{\beta}\phi\nabla^{\alpha}\nabla^{\beta}\phi-(\Box\phi)^{2})\right.$
$\displaystyle-$
$\displaystyle\left.\chi_{,\mu}\chi_{,\nu}-\Box\phi(\chi_{,\mu}\phi_{,\nu}+\chi_{,\nu}\phi_{,\mu})-\chi_{,\gamma}[\phi^{,\gamma}\nabla_{\mu}\nabla_{\nu}\phi-(\nabla^{\gamma}\nabla_{\mu}\phi)\phi_{,\nu}-(\nabla^{\gamma}\nabla_{\nu}\phi)\phi_{,\mu}]\right).$

By considering the canonical action for the scalar field $\phi=\phi(r)$, and
also imposing conditions of the finite energy of $\phi$, i.e.,
$E=\int_{V}\sqrt{-g}T^{0}_{0}d^{3}x$ and a vanishing radial four-current at
infinity $j^{r}=0$, one can solve the field equations (20) and obtain the
static and spherically symmetric solution as

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (22)

with

$f(r)=1-\frac{2M}{r}+\frac{h}{r}\ln\left(\frac{r}{2M}\right),$ (23)

where $M$ is the BH mass and $h$ is referred to as the hair parameter with the
dimension of length. When $h/M>0$ there is only one horizon at $r=2M$, while
for the values of $h$ in the interval $-2<h/M<0$, there are two horizons; one
is the outer event horizon at $r_{+}=2M$ and other is the inner Cauchy
horizon, $r_{-}$. These two horizons coincide at $r=2M$ and we get an extremal
BH [47]. Also, when $h\rightarrow 0$ the metric (23) reduces to the
Schwarzschild solution, and it is asymptotically flat. Thus, here we restrict
our analysis to the case of $-2<h/M<0$ as considered in [48]–[57]. Using the
revised Newman-Janis algorithm, Walia et al., obtained the rotating BH in
Horndeski gravity which in the Boyer–Lindqist coordinates is given by [48]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta}{dr^{2}}+\Sigma
d\theta^{2}$ (24) $\displaystyle+$
$\displaystyle\frac{2a\sin^{2}\theta}{\Sigma}\left(\Delta-(r^{2}+a^{2})\right)dtd\varphi$
$\displaystyle+$
$\displaystyle\frac{\sin^{2}\theta}{\Sigma}\left[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]d\varphi^{2},$

with

$\Delta=r^{2}+a^{2}-2Mr+hr\ln\left(\frac{r}{2M}\right),\quad\Sigma=r^{2}+a^{2}\cos^{2}\theta.$
(25)

In the absence of hair, $h\rightarrow 0$, the above metric reduces to the Kerr
metric and for $a=0$ it reduces to the line element of a static hairy
Horndeski BH in equation (22). From the condition $g^{rr}=\Delta=0$ one can
obtain the location of horizons. We have plotted $\Delta(r)$ as a function of
radial coordinate for different values of $h$, and for $a=0.6M$ and $a=0.9M$
in Fig. 1. As is clear, in each panel for some values of the hair parameter
$h$, there are two horizons, namely the inner Cauchy horizon, $r_{-}$, and the
outer event horizon, $r_{+}$, so that $r_{-}<r_{+}$. However for a critical
value of $h=h_{\rm c}$, the two horizons coincide, $r_{-}=r_{+}$,
corresponding to an extremal BH. Also, for $h<h_{\rm c}$ there are no BH
space-time [50]. Also, we see that the radius of the event horizon decrease
with increasing $|h|$.

Figure 1: The behavior of horizons as a function of the radial coordinate $r$
for different values of the hair parameter $h$. The rotation parameter is set
to $a=0.6M$ (left panel) and $a=0.9M$ (right panel), respectively. In each
panel the solid curve corresponds to the Kerr BH.

##  4 Thin accretion disk properties of rotating hairy Horndeski BHs

Let us now study thin accretion disk properties around rotating hairy BHs in
Horndeski gravity and compare the results with that of the Kerr BH in GR.
First, using equations (7)-(9) we obtain the angular velocity, specific energy
and specific angular momentum of test particles moving in circular orbits
around a rotating hairy Horndeski BH as follows

$\Omega=\frac{1}{a+\sqrt{2}r/u(r)},$ (26)
${\tilde{E}}=\frac{\sqrt{2}(r-2M+h\ln(\frac{r}{2M}))+au(r)}{\sqrt{r[-6M+2r-h(1-3\ln(\frac{r}{2M}))+2\sqrt{2}au(r)]}},$
(27)
${\tilde{L}}=\frac{-\sqrt{2}a(2M-h\ln(\frac{r}{2M}))+(r^{2}+a^{2})u(r)}{\sqrt{r[-6M+2r-h(1-3\ln(\frac{r}{2M}))+2\sqrt{2}au(r)]}},$
(28)

with

$u(r)=\sqrt{\frac{2M+h(1-\ln(\frac{r}{2M}))}{r}}.$ (29)

The innermost stable circular orbit (ISCO) equation (10) also reads

$\displaystyle 2r(6M^{2}-Mr+4Mh+h^{2})-4\sqrt{2}ar(2M+h)u(r)$ (30)
$\displaystyle+$ $\displaystyle
3rh^{2}\ln(\frac{r}{2M})^{2}+a^{2}(6M+4h)+h\ln(\frac{r}{2M})$
$\displaystyle\times$
$\displaystyle\left(-3a^{2}+r(r-12M-4h)+4\sqrt{2}aru(r)\right)=0,$

which can not be solved analytically. Therefore, we have numerically obtained
the ISCO radius for different values of $a$ and $h$ parameters. Also, in the
absence of hair, $h=0$, with $a=0$ the above equation reduces to $r_{\rm
isco}=6M$, which is the ISCO radius of the Schwarzschild BH. In Fig. 2, we
have shown the effects of $a$ and $h$ parameters on the ISCO radius, showing
that with increasing both the spin parameter and the absolute value of the
hair parameter the ISCO radius decreases.

Now, using equations (26)-(28) for the space-time of rotating hairy Horndeski
BHs, we are ready to investigate the effects of hair as well as the spin
parameter on the electromagnetic properties of thin disk around these BHs. The
behavior of the energy flux for $a=0.6M$ and $a=0.9M$ are shown in the left
and right panels of Fig. 3, respectively. As we see from Fig. 1, the critical
value of hair parameter for $a=0.6M$ is $h_{\rm c}=-0.9833M$ and in the case
of $a=0.9M$, is $h_{\rm c}=-0.2771M$. So, in each panel, the $h$ values were
chosen in such a way that are larger than the critical value $h_{\rm c}$ and
thus we have a BH solution. We see that for a fixed $a$, with increasing
$|h|$, the radiant energy flux increases too, so that the Kerr BH with $h=0$
has the smallest energy flux. Also, an enhancement in the rotation parameter
increases the energy flux. The disk temperature is plotted in Fig. 4, where a
similar behavior can be seen.

In Fig. 5,we have displayed the effect of the hair parameter on the disk
spectra for rotating Horndeski BHs. Similar to the energy flux and the disk
temperature, deviation of $L(\nu)$ for Horndeski BHs from that of the Kerr BH
becomes more pronounced with increasing $|h|$. Moreover, the cut-off
frequencies, in which the maximum of luminosity occur, shift to higher
frequencies with increase of $|h|$.

Figure 2: The behaviour of the ISCO radius as a function of Horndeski
parameter $h$ and the spin parameter $a$.

Figure 3: The energy flux $F(r)$ from a disk around a rotating hairy Horndeski
BH with the mass accretion rate $\dot{M}=2\times 10^{-6}M_{\odot}\rm yr^{-1}$,
for different values of the hair parameter $h$. The rotation parameter is set
to $a=0.6M$ (left panel) and $a=0.9M$ (right panel), respectively. In each
panel the solid curve corresponds to the Kerr BH.

Figure 4: The disk temperature $T(r)$ for a rotating hairy Horndeski BH with
mass accretion rate $\dot{M}=2\times 10^{-6}M_{\odot}\rm yr^{-1}$, for
different values of the hair parameter $h$. The rotation parameter is set to
$a=0.6M$ (left panel) and $a=0.9M$ (right panel), respectively. In each panel
the solid curve corresponds to the Kerr BH.

Figure 5: The emission spectrum $\nu L(\nu)$ of the accretion disk around a
rotating hairy Horndeski BH with mass accretion rate $\dot{M}=2\times
10^{-6}M_{\odot}yr^{-1}$ and inclination $\gamma=0^{\circ}$ for different
values of the hair parameter $h$, as a function of frequency $\nu$. The
rotation parameter is set to $a=0.6M$ (left panel) and $a=0.9M$ (right panel),
respectively. In each panel the solid curve corresponds to the Kerr BH.

The dependence of the accretion efficiency $\epsilon$ on both the spin and
hair parameter is plotted in Fig. 6. It shows that the radiative efficiency of
Horndeski BHs increases with increasing of $a$ and $|h|$. It means that the
accretion of matter on hairy Horndeski BHs is more efficient than that on the
Kerr BH with $h=0$. Therefore, the rotating hairy Horndeski BHs can provide a
more efficient processes for converting the energy of accreting matter into
electromagnetic radiation than that the Kerr BH in the absence of hair.

Figure 6: The accretion efficiency $\epsilon$ of the Horndeski BHs as a
function of the hair parameter $h$ and the spin parameter $a$.

##  5 Image of disk around a rotating hairy Horndeski BH

In this section, we use the numerical ray tracing method proposed by Chen et
al [63] to construct the image of a thin accretion disk around a rotating
hairy Horndeski BH. The coordinate system of the BH and its surrounding disk
and also the coordinate system of the distant observer is shown in Fig. 7 and
has been discussed in details in Ref. [64].

The BH surrounded by an accretion disk is described with Cartesian coordinates
system $(x,y,z)$, while the observer at a distant $D$ from the BH with
inclination angle $i$ is described with Cartesian coordinates system
$(X,Y,Z)$. Here, the image plane lies in the plane $Z=0$, meaning that the
$Z$-axis is perpendicular to the image plane. The relation between two
coordinates is given by

$\displaystyle x=D\sin i-Y\cos i+Z\sin i,$ $\displaystyle y=X,$ $\displaystyle
z=D\cos i+Y\sin i+Z\cos i.$ (31)

Figure 7: Cartesian coordinates (x,y,z) are centred at the BH, and the
Cartesian coordinates (X,Y,Z) are for the image plane of distant observer,
located at the distant $D$ from the BH with inclination angle $i$. Figure from
[64].

###  5.1 Photon initial conditions

For the first step we write the photon initial conditions in the image plane
of the distant observer. For this purpose we consider the photon places at the
image plane at the position $(X_{0},Y_{0},0)$ with 4-momentum
${k}^{\alpha}=(k_{0},0,0,-k_{0})$ so that the photon 3-momentum
$\mathbf{k_{0}}=-k_{0}\hat{Z}$ is perpendicular to the image plane. The
initial conditions for the photon position are given as

$\displaystyle t_{0}=0,$ $\displaystyle
r_{0}=\sqrt{D^{2}+X_{0}^{2}+Y_{0}^{2}},$
$\displaystyle\theta_{0}=\arccos\frac{D\cos i+Y_{0}\sin i}{r_{0}},$
$\displaystyle\phi_{0}=\arctan\frac{X_{0}}{D\sin i-Y_{0}\cos i}.$ (32)

Using the relation $k^{\mu}=(\partial
x^{\mu}/\partial\tilde{x}^{\alpha})\tilde{k}^{\alpha}$, where
$\tilde{k}^{\mu}$ is the photon 4-momentum in Cartesian coordinates, we obtain
the photon 4-momentum in the spherical coordinates system

$\displaystyle k_{0}^{r}=-\frac{D}{r_{0}}|\mathbf{k_{0}}|,$ $\displaystyle
k_{0}^{\theta}=\frac{\cos i-(Y_{0}\sin i+D\cos
i)\frac{D}{r_{0}^{2}}}{\sqrt{X_{0}^{2}+(D\sin i-Y_{0}\cos
i)^{2}}}|\mathbf{k_{0}}|,$ $\displaystyle k_{0}^{\phi}=\frac{X_{0}\sin
i}{X_{0}^{2}+(D\sin i-Y_{0}\cos i)^{2}}|\mathbf{k_{0}}|,$ (33)

where $k_{0}^{t}$ cab be obtained from the condition
$g_{\mu\nu}k^{\mu}k^{\nu}=0$ as follows

$\displaystyle
k_{0}^{t}=\sqrt{(k_{0}^{r})^{2}+r_{0}^{2}(k_{0}^{\theta})^{2}+r_{0}^{2}\sin^{2}\theta_{0}(k_{0}^{\phi})^{2}}.$
(34)

###  5.2 Photon trajectory

Now by using the initial conditions (32)-(34) we can integrate the following
geodesic equation

$\frac{d^{2}x^{\mu}}{d\lambda^{2}}+\Gamma^{\mu}_{\nu\rho}\frac{dx^{\nu}}{d\lambda}\frac{dx^{\rho}}{d\lambda}=0,$
(35)

backward in time from detection point $(X_{0},Y_{0},0)$ at the image plane of
the distant observer to the emission point. In equation (35), $\lambda$ is the
affine parameter. In a similar way with Ref. [63] we first solve the geodesic
equations using the Runge-Kutta method in MATLAB software.

###  5.3 Strong lensing by hairy Horndeski BH

It is believed that the large luminosity of active galactic nuclei (AGN) and
quasars is the result of gas accreted by supersessive BH. The quasar accretion
disk emits X-rays. This emission is lensed by the central BH before it arrives
at distant observer. The radiation intensity profile emitted from the disk
surface has been observed to follow a power law. In this work, following [63],
we consider the radiation intensity profile emitted from the disk surface as

$\displaystyle
I(\nu,\mu,r)\propto\frac{1}{r^{n}}\frac{\omega(\mu)}{\nu^{\Gamma-1}},$ (36)

where $\nu$ is the frequency of photon in the rest frame and $\mu$ is the
cosine between photon 4-momentum and the normal vector of disk measured by the
comoving observer. $\omega(\mu)$ denotes the angular-dependence of intensity
profile and we take it from Chandrasekhar’s book [65]. Also, $\Gamma$ and $n$
are the photon index and the radial steepness of the profile, respectively.
For the source profile (36) the following flux integral

$\displaystyle F_{\nu_{\rm o}}\equiv\int g^{3}I^{\rm source}_{\nu_{\rm
e}}(e^{a},p_{b})d\Omega_{0},$ (37)

can be easily computed analytically or numerically through backward ray-
tracing. Here ${\nu_{\rm e}}$ is the source frequency and
$g\equiv\frac{{\nu_{\rm o}}}{{\nu_{\rm e}}}$ is the redshift factor.

In Ref. [63] authors investigated the polarization and gravitational Faraday
rotation for rotating Kerr BH in details. For further study, the interested
reader is referred to [66]–[69]. Note that for a rotating BH, the photon
polarization plane rotates due to gravitational Faraday rotation [70], which
makes the disk radiation being partially polarized due to the magnetic field
of the disk atmosphere. Also, due to the Thomson scattering of photons by free
electrons in the dense atmosphere of accretion disk, the degree of
polarization varies depending on the change in the disk’s inclination angle,
so that as it increases the degree of polarization increases too.

Both relativistic effects including the gravitational redshift due to the
existence of the BH and the relativistic Doppler effect due to the rotation of
the disk lead to the changes of the observed image, as shown in the Fig.
8-Fig. 10. The figures show the Doppler blueshift in the left half-part of the
plane which can exceed the overall gravitational redshift (left panels). Also,
the intensity distribution is concentrated on a small region near the BH on
the left side of accretion disk (right panels).

In Fig. 8 and Fig. 9, we plot the ray-traced redshifted image (left panel) and
also the intensity and polarization profile of an accretion disk (right panel)
around a Kerr BH with $h=0$ and a rotating hairy Horndeski BH for $h=-0.8M$,
respectively. We set the spin parameter $a=0.6M$, $M=1$, and the inclination
angle $i=80^{\circ}$. Fig. 10 is similar to Fig. 9 but presented at the
arbitrary inclination angle $i=50^{\circ}$. The color bar shows the degree of
the redshift of radiant light emitted from the accretion disk in ray-traced
redshifted images, and the intensity of radiation on the accretion disk for
intensity and polarization profiles. Also, we note that the size and the
direction of polarization are shown by the length and direction of arrow in
the right panel of figures.

The ray-traced redshift images are dependent on the hair parameter $h$ and the
inclination angle of the accretion disk $i$. Comparing Fig. 8 and Fig. 9 shows
that the region of the central shadow increases with the decrease of $|h|$.
Also, the size and orientation of polarization vectors in the right panels of
the figures decrease with increasing $|h|$, so that the Kerr space-time with
$h=0$ has the most change of polarization vector in a small region on the
accretion disk in the vicinity of BH. On the other hand, comparing Fig. 9 and
Fig. 10 shows that the inclination angle has a notable effect on both the ray-
traced redshifted image and the intensity and polarization profile of
accretion disk, so that for the high inclination angles the blueshift in the
left half of the plate increases.

Figure 8: Ray-traced redshifted image (left panel) and intensity and
polarization profile (right panel) of a lensed accretion disk around Kerr BH.
The inclination angle is set to $i=80^{\circ}$ and spin parameter is $a=0.6M$
with $M=1$. The inner edge of disk is at $r_{\rm in}=r_{\rm isco}$ and the
outer edge of disk is at $r_{\rm out}=20M$.

Figure 9: Ray-traced redshifted image (left panel) and intensity and
polarization profile (right panel) of a lensed accretion disk around rotating
hairy Horndeski BH. The inclination angle is set to $i=80^{\circ}$, spin
parameter is $a=0.6M$ and hair parameter is $h=-0.8M$ with $M=1$. The inner
edge of disk is at $r_{\rm in}=r_{\rm isco}$ and the outer edge of disk is at
$r_{\rm out}=20M$.

Figure 10: Ray-traced redshifted image (left panel) and intensity and
polarization profile (right panel) of a lensed accretion disk around rotating
hairy Horndeski BH. The inclination angle is set to $i=50^{\circ}$, spin
parameter is $a=0.6M$ and hair parameter is $h=-0.8M$ with $M=1$. The inner
edge of disk is at $r_{\rm in}=r_{\rm isco}$ and the outer edge of disk is at
$r_{\rm out}=20M$.

##  6 Conclusions

In the present work, we considered thin accretion disks around rotating hairy
BHs in Horndeski gravity. First, we studied the geodesic motion of test
particles in the space-time of these BHs and obtained the relevant quantities,
namely the specific angular momentum $\tilde{L}$, specific energy $\tilde{E}$,
angular velocity $\Omega$ and the ISCO radius for particles moving in circular
orbits. We numerically obtained the ISCO radius for both spin, $a$, and the
hair parameter, $h$. It is shown that by increasing both the spin parameter
and the absolute value of the hair parameter, the ISCO radius decreases. Then,
using the steady-state Novikov-Thorne model, we obtained the physical
properties of accretion disks including the energy flux $F(r)$, temperature
distribution $T(r)$, and the emission spectra $L(\nu)$ for some values of the
hair parameter and plotted their profiles. The decrease of ISCO radius shows
that the presence of scalar filed weakens the strength of gravitational field
and thus the inner edge of disk shifts to lower and lower radii. Therefore,
the disks around rotating BHs in Horndeski gravity are hotter and more
luminous than the Kerr BH in the absence of hair.

Furthermore, we investigated the images of thin accretion disks around
rotating hairy Horndeski BHs using a ray-tracing technique. We made the ray-
traced redshifted images, intensity and polarization profiles of disks for
both hairy Horndeski BH and the Kerr BH in the absence of hair. It is shown
that with increasing the absolute value of the hair parameter $h$, the central
shadow area decreases and the size and orientation of polarization vectors
also decrease. Moreover, the results show that the inclination angle has a
pronounced effect on the disk images.

## Acknowledgments

The work of Mohaddese Heydari-Fard is supported by the Iran National Science
Foundation (INSF) and the Research Council of Shahid Beheshti University under
research project No. 4016024.

## References

  * [1] B. P. Abbott et al., (LIGO Scientific, Virgo), Phys. Rev. Lett 116 (2016) 061102.
  * [2] H. Xu, S. Y. Chen, Y. J. Guo, et al., Res. Astron. Astrophys. 23 (2023) 075024.
  * [3] G. Agazie, A. Anumarlapudi, A. M. Archibald, et al., Astrophys. J. Lett. 951 (2023) L8.
  * [4] K. Akiyama et al., (Event Horizon Telescope), Astrophys. J 875 (2019) L1.
  * [5] K. Akiyama et al., (Event Horizon Telescope), Astrophys. J 930 (2022) L12.
  * [6] F. Yuan and R. Narayan, Ann. Rev. Astron. Astrophys 52 (2014) 529.
  * [7] S. Nampalliwar and C. Bambi, arXiv:1810.07041 [astro-ph.HE].
  * [8] C. S. J. Pun, Z. Kovács and T. Harko, Phys. Rev. D 78 (2008) 024043.
  * [9] K. V. Staykov, D. D. Doneva and S. S. Yazadjiev, JCAP 2016 (2016) 061.
  * [10] P.I. Dyadina and N.A. Avdeev, arXiv:2308.02375 [gr-qc].
  * [11] D. Perez, F. G. L. Armengol and G. E. Romero, Phys. Rev. D 95 (2017) 104047.
  * [12] M. Heydari-Fard, Class. Quant. Grav 27 (2010) 235004.
  * [13] R. Kh. Karimov, R. N. Izmailov, A. Bhattacharya and K. K. Nandi, Eur. Phys. J. C 78 (2018) 788.
  * [14] M. Heydari-Fard, M. Heydari-Fard and H. R. Sepangi, Eur. Phys. J. C 80 (2020) 351.
  * [15] M. Heydari-Fard and H. R. Sepangi, Phys. Lett. B 816 (2021) 136276.
  * [16] C. Liu, T. Zhu and Q. Wu, Chin. Phys. C 45 (2021) 015105.
  * [17] M. Heydari-Fard M. Heydari-Fard and H. R. Sepangi, Eur. Phys. J. C 81 (2021) 473.
  * [18] O. Donmez, F. Dogan and T. Sahin, Universe 8 (2022) 458.
  * [19] K. Boshkayev, A. Idrissov, O. Luongo and D. Malafarina, Mon. Not. Roy. Astron. Soc 496 (2020) 1115.
  * [20] E. Kurmanov, K. Boshkayev, R. Giambò, T. Konysbayev, O. Luongo, D. Malafarina and H. Quevedo, Astrophys. J 925 (2022) 210.
  * [21] K. Boshkayev, T. Konysbayev, Y. Kurmanov, O. Luongo and D. Malafarina, Astrophys. J 936 (2022) 96.
  * [22] M. Heydari-Fard, S. G. Honarvar and M. Heydari-Fard, Mon. Not. Roy. Astron. Soc 521 (2023) 708.
  * [23] G. Abbas and H. Rehman, Fortsch. Phys 71 (2023) 2200205.
  * [24] A. Uniyal, R. C. Pantig and A. Övgün, Phys. Dark. Univ 40 (2023) 101178.
  * [25] A. R. Akbarieh, M. Khoshragbaf and M. Atazadeh, arXiv:2302.02784 [gr-qc].
  * [26] R. K. Karimov, R. N. Izmailov, A. A. Potapov and K. K. Nandi, Eur. Phys. J. C 82 (2022) 239.
  * [27] J. P. Luminet, Astron. Astrophys 75 (1979) 228.
  * [28] G. Gyulchev, P. Nedkova, T. Vetsov and S. Yazadjiev, Eur. Phys. J. C 81 (2021) 885.
  * [29] C. Liu, L. Tang and J. Jing, Int. J. Mod. Phys. D 31 (2022) 2250041.
  * [30] Y. Hou, Z. Zhang, H. Yan, M. Guo and B. Chen, Phys. Rev. D 106 (2022) 064058.
  * [31] Y. X. Huang, S. Guo, Y. H. Cui, Q. Q. Jiang and K. Lin, Phys. Rev. D 107 (2023) 123009.
  * [32] J. L. Rosa, Phys. Rev. D 107 (2023) 084048.
  * [33] G. J. Olmo, J. L. Rosa, D. Rubiera-Garcia and D. Saez-Chillon Gomez, Class. Quant. Grav 40 (2023) 174002.
  * [34] C. Brans and R. H. Dicke, Phys. Rev 124 (1961) 925; R. H. Dicke, ibid 125 (1962) 2163.
  * [35] G. W. Horndeski, Int. J. Theor. Phys 10 (1974) 363.
  * [36] R. Kase and S. Tsujikawa, Int. J. Mod. Phys. D 28 (2019) 1942005.
  * [37] J. L. Ripley, Int. J. Mod. Phys. D 31 (2022) 2230017.
  * [38] M. Petronikolou, S. Basilakos and E. N. Saridakis, Phys. Rev. D 106 (2022) 124051.
  * [39] K. i. Kubota, S. Arai and S. Mukohyama, Phys. Rev. D 107 (2023) 064002.
  * [40] T. Kobayashi, Rep. Prog. Phys 82 (2019) 086901.
  * [41] R. Kase and S. Tsujikawa , Int. J. Mod. Phys. D 28 (2019) 1942005.
  * [42] M. Rinaldi, Phys. Rev. D 86 (2012) 084048.
  * [43] C. Charmousis, T. Kolyvaris, E. Papantonopoulos and M. Tsoukalas, JHEP 07 (2014) 085.
  * [44] A. Cisterna and C. Erices, Phys. Rev. D 89 (2014) 084038.
  * [45] A. Anabalon, A. Cisterna and J. Oliva, Phys. Rev. D 89 (2014) 084050.
  * [46] M. Bravo-Gaete and M. M. Stetsko, Phys. Rev. D 105 (2022) 024038.
  * [47] S. E. P. Bergliaffa, R. Maier and N. d. Silvano, arXiv:2107.07839 [gr-qc].
  * [48] R. K. Walia, S. D. Maharaj and S. G. Ghosh, Eur. Phys. J. C 82 (2022) 547.
  * [49] J. Kumar, S. U. Islam and S. G. Ghosh, Eur. Phys. J. C 82 (2022) 443.
  * [50] M. Afrin and S. G. Ghosh, Astrophys. J. 932 (2022) 51.
  * [51] S. G. Ghosh and M. Afrin, Astrophys. J. 944 (2023) 174.
  * [52] S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, A. Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh and P. S. Joshi, et al. Class. Quant. Grav 40 (2023) 165007.
  * [53] J. Rayimbaev, K. F. Dialektopoulos, F. Sarikulov and A. Abdujabbarov, Eur. Phys. J. C 83 (2023) 572.
  * [54] S. K. Jha, M. Khodadi, A. Rahaman and A. Sheykhi, Phys. Rev. D 107 (2023) 084052.
  * [55] Z. H. Yang, Y. H. Lei, X. M. Kuang and J. P. Wu, arXiv:2309.03565 [gr-qc].
  * [56] S. Hu, D. Li, C. Deng, X. Wu and E. Liang, arXiv:2309.10557 [gr-qc].
  * [57] X. J. Wang, X. M. Kuang, Y. Meng, B. Wang and J. P. Wu, Phys. Rev. D 107 (2023) 124052.
  * [58] I. D. Novikov and K. S. Thorne, Astrophysics and black holes, in Black Holes, edited by C. De Witt and B. De Witt (Gordon and Breach, New York, 1973).
  * [59] C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity, Springer, Singapore (2017).
  * [60] N. I. Shakura and R. A. Sunyaev, Astron. Astrophys 24 (1973) 33.
  * [61] D. N. Page and K. S. Thorne, Astrophys. J 191 (1974) 499.
  * [62] D. F. Torres, Nucl. Phys. B 626 (2002) 377.
  * [63] B. Chen, R. Kantowski, X. Dai, E. Baron and P. Maddumage, Astrophys. J. Suppl 218 (2015) 4.
  * [64] C. Bambi, A. Cardenas-Avendano, T. Dauser, J. A. Garcia and S. Nampalliwar, Astrophys. J 842 (2017) 76.
  * [65] S. Chandrasekhar, Radiative Transfer, New York, Dover (1960).
  * [66] N. L. Balazs, Phys. Rev 110 (1958) 236.
  * [67] J. Plebanski, Phys. Rev 118 (1960) 1396.
  * [68] L. X. Li, R. Narayan and J. E. McClintock, Astrophys. J 691 (2009) 847.
  * [69] J. D. Schnittman and J. H. Krolik, Astrophys. J 701 (2009) 1175.
  * [70] P. A. Connors and R. F. Stark, Nature 269 (1977) 128.