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# A topological embedding of $T_{3}$ into $\mathbb{Z}^{2}$

Samuel Kelly

(November 2023)

###### Abstract

We prove that for any finite tree $T$ with $n$ vertices and maximal degree
$3$, there is a topological embedding of $T$ into the integer grid $Z^{2}$
which maps vertices to vertices and whose image meets at most $\frac{7}{3}n$
vertices. This resolves Question 7.7 of [BH21], and furthermore gives the
first example of a pair of graphs $X,Y$ such that there is no regular map
$X\to Y$ but the coarse wiring profile of $X$ into $Y$ grows linearly.

##  1 Introduction

Coarse wiring profiles were introduced by Barratt-Hume as a coarse geometric
variant of thick topological embeddings previously considered by Kolmogorov-
Barzdin and Gromov-Guth [BH21, Bar93, GG11]. We first recall the key
definitions from [BH21]:

######  Definition 1.0.1 (Wiring).

Let $\Gamma,\Gamma^{\prime}$ be graphs. A wiring of $\Gamma$ into
$\Gamma^{\prime}$ is a continuous map $f:\Gamma\to\Gamma^{\prime}$ which maps
vertices to vertices and edges to unions of edges. A wiring $f$ is a coarse
$k$-wiring if

  1. 1.

the restriction of $f$ to $V\Gamma$ is $\leq\text{$k$-to-$1$}$ i.e $|\\{v\in
V\Gamma|f(v)=w)|\leq k$ for all $w\in V\Gamma^{\prime}$.

  2. 2.

each edge $e\in E\Gamma^{\prime}$ is contained in at most $k$ of the image
paths of $f$.

The volume of a wiring $\mathrm{vol}(f)$ is the number of vertices in its
image.

######  Definition 1.0.2 (Wiring Profile).

Let $\Gamma$ be a finite graph and $Y$ be a graph. We denote by
$\text{wir}^{k}(\Gamma\to Y)$ the minimal volume of a coarse $k$-wiring of
$\Gamma$ into $Y$. If no such coarse $k$-wiring exists, we say
$\text{wir}^{k}(\Gamma\to Y)=\infty$. Let $X$ and $Y$ be graphs. The
$k$-wiring profile of $X$ into $Y$ is the function

$\mathrm{wir}_{X\to Y}^{k}(n)=\max\Big{\\{}\mathrm{wir}^{k}(\Gamma\to
Y)\mid\Gamma\leq X\text{ , }|\Gamma|\leq n\Big{\\}}$

Coarse wiring profiles are typically considered with respect to the following
partial order. Given two functions $f,g:\mathbb{N}\to\mathbb{N}$, we write
$f\lesssim g$ if there is a constant $C$ such that $f(n)\leq Cg(Cn)+C$ for all
$n$, and we write $f\simeq g$ if $f\lesssim g$ and $g\lesssim f$. The goal of
the paper is the following:

######  Theorem 1.0.3.

Let $T_{3}$ denote the infinite $3$-regular tree, and let $\mathbb{Z}^{2}$
denote the 2-dimensional integer lattice. We have

$\mathrm{wir}_{T_{3}\to\mathbb{Z}^{2}}^{k}(n)\simeq n.$

To achieve this, we show that the $1$-wiring profile from $T_{3}$ to
$\mathbb{Z}^{2}$ is linear. Further, applying the permanence properties of
wiring profiles proved in [BH21] results in the following corollary.

######  Corollary 1.0.3.1.

Let $Y$ be a Cayley graph of a finitely generated group $G$ which is not
virtually cyclic. If, for all $k$,

$\mathrm{wir}_{T_{3}\to Y}^{k}(n)\not\simeq n$

then $G$ is amenable but not elementary amenable.

###### Proof.

When $G$ is virtually nilpotent but not virtually cyclic, $G$ admits a
$\mathbb{Z}^{2}$ subgroup. To see this, note that $G$ admits a finite-index
torsion-free subgroup $G^{\prime}$ with non-trivial centre, let $x$ be such a
non-trivial element. Since $G$ is not virtually cyclic, there is some element
$y\in G^{\prime}$ such that no power of $y$ is equal to a power of $x$. It
follows that $x,y$ generate a $\mathbb{Z}^{2}$ subgroup of $G^{\prime}$, and
therefore of $G$. Therefore, $\mathrm{wir}_{\mathbb{Z}^{2}\to Y}^{k}(n)\simeq
n$ for all sufficiently large $k$. Applying Theorem 1.0.3 and [BH21, Lemma
3.2], we see that $\mathrm{wir}_{T_{3}\to Y}^{k}(n)\simeq n$ for all
sufficiently large $k$.

When $G$ is elementary amenable but not virtually nilpotent, it admits a non-
abelian free subsemigroup [Cho80], so there is a regular map $T_{3}\to Y$, and
$\mathrm{wir}_{T_{3}\to Y}^{k}(n)\simeq n$ for all sufficiently large $k$.
Alternatively, if $G$ is not amenable, then it admits a quasi-isometric
embedding of $T_{3}$ [BS97], so again $\mathrm{wir}_{T_{3}\to Y}^{k}(n)\simeq
n$ for all sufficiently large $k$. ∎

It is very natural to ask whether the following stronger result holds.

######  Question 1.0.4.

Let $Y$ be a Cayley graph of a finitely generated group $G$. If, for some $k$,

$\mathrm{wir}_{T_{3}\to Y}^{k}(n)\simeq n$

is $G$ necessarily virtually cyclic?

## Acknowledgements

The author was funded by a University of Bristol Student Summership Bursary
throughout this work and would like to thank David Hume for his kind
supervision.

##  2 The $1$-wiring profile $T_{3}\to\mathbb{Z}^{2}$ is linear

To prove Theorem 1.0.3, we provide a linear upper bound for the $1$-wiring
profile via an algorithm whose input is a subgraph $\Gamma\subset T_{3}$ and
whose output is a $1$-wiring of $\Gamma$ into $\mathbb{Z}^{2}$ of volume less
than $\frac{7}{3}|\Gamma|$. A $1$-wiring $f:\Gamma\to\mathbb{Z}^{2}$ is
injective on the vertex set and maps each edge to a unique path in
$\mathbb{Z}^{2}$. The amount of vertices in the image of a path is equal to
the length of the image path in the taxicab metric on $\mathbb{Z}^{2}$, thus

$\mathrm{vol}(f)=1+\sum_{vw\in E\Gamma}d_{\mathbb{Z}^{2}}(f(v),f(w))$

The algorithm is defined recursively. We assign some vertex $v\in\Gamma$ as a
root to obtain a parent-child hierarchy. At each stage in the algorithm, we
suppose we have wirings for the subtrees stemming from the children. We place
our root at $(0,0)$ and copy over the wiring for the first child, placing its
root on the $y$-axis, just high enough so that the entire wiring is in the
upper half plane. If there is a second child, we again copy its wiring but
this time we rotate it $90^{\circ}$ degrees clockwise before placing its root
on the positive $x$-axis, just far away enough so that every image point has a
positive $x$-coordinate.

Formally, the construction is given by $S(\Gamma,v)$, which takes a binary
tree $\Gamma$ and some root vertex $v\in\Gamma$ as parameters and outputs a
wiring for $\Gamma$ in $\mathbb{Z}^{2}$ rooted at $v$. Denoting the children
of $v$ by $v_{1}$ and $v_{2}$, the child subtree, denoted $\Gamma_{v_{i}}$ of
$v_{i}$ is the connected component of $\Gamma$ containing $v_{i}$ but not $v$.
Given a wiring $f$ for $\Gamma$, we define the connector, denoted
$\mathrm{conn}(f)$, to be $\mathrm{conn}(f)=|\min\\{y\mid(x,y)\in
f(\Gamma)\\}|$; this is the extra length needed to connect a child to its
root. We define $S:(\Gamma,v)\mapsto f$ as follows, where $f$ denotes the
wiring $f:\Gamma\to\mathbb{Z}^{2}$ we return.

  1. 1.

Set $f(v)=(0,0)$.

  2. 2.

If $v$ has children.

    1. (a)

If $v$ has only one child $w$. Calculate the wiring for $\Gamma_{w}$ and
denote it $f_{w}$. We map the $u\in\Gamma_{w}\subset\Gamma$ to
$f(u)=f_{w}(u)+(0,\mathrm{conn}(f_{w})+1)$.

    2. (b)

If $v$ has two children $w_{1},w_{2}$, order them so that
$|T_{w_{1}}|\geq|T_{w_{2}}|$. Calculate their wirings and denote them $f_{1}$
and $f_{2}$ respectively. For the largest child, we do the same as in $(a)$.
We map the $u\in\Gamma_{w_{1}}\subset\Gamma$ to
$f(u)=f_{1}(u)+(0,\mathrm{conn}(f_{1})+1)$. For the second wiring, we first
rotate the image vertices $90^{\circ}$ degrees clockwise about the origin,
denoting this $f_{2}^{*}$. We then map the $u\in\Gamma_{w_{2}}\subset\Gamma$
to $f(u)=f_{2}^{*}(u)+(\mathrm{conn}(f_{2})+1,0)$.

  3. 3.

Return $f$.

To see that $f$ is a $1$-wiring, observe that at each step, the first child
lies in the North-East quadrant and the second child lies in the South-East
quadrant. Let $B_{n}$ be the $n$-th rooted perfect binary tree with an extra
degree $1$ vertex placed above the traditional root. We have illustrated the
image of $B_{4}$ in Figure 1. Note also that $B_{3}$’s image is illustrated in
blue and $B_{2}$’s is illustrated in red (after being rotated clockwise
$90^{\circ}$). The colouring scheme is to supplement a future argument.

$(0,0)$ Figure 1: The image of $B_{4}$

We now turn our attention towards bounding the volume of this wiring. To
motivate our approach, consider subdividing any edge in $B_{4}$ and the
resultant change in volume of Figure 1. The more rotations the edge we
subdivide has undergone, the greater the volume added per subdivision.
However, add too many and one changes the size ordering which results in the
algorithm outputting a new wiring of lesser volume. Thus, maximising the
volume of our tree is tantamount to maximising the amount of subdivisions on
rotated edges whist adhering to the constraints of the size ordering. To prove
Theorem 1.0.3, we simplify the problem by considering only one such size-
ordering at a time and study how we can maximise the volume by distributing
subdivisions according to this ordering. As an example, to subdivide the black
(only) edge that has been rotated $4$ times and keep the same size ordering,
we must apply at least $15$ other subdivisions. For the following argument, we
suppose for any graph that the root vertex $v$ on which we initiated the
algorithm was of order $1$. We use this root to give us the parent-child
hierarchy. We write $S(\Gamma,v)=f_{\Gamma}$.

Let’s standardise the meaning of our ”size-ordering”. Given our rooted tree
$\Gamma$, we assign a left-to-right order for each pair of children by size so
that the left child’s subtree has more vertices than the right’s. We define
the reduction of $\Gamma$, denoted $R(\Gamma)$, to be the series-reduced tree
homeomorphic to $\Gamma$ with the same parent-child hierarchy and left-right
ordering as $\Gamma$ imposed on its vertices. Note that it is the right
children that get rotated. The behaviour of the volume of our wiring
$S(\Gamma,v)$ is entirely determined by the reduction of $\Gamma$. We write
$\Gamma_{1}\sim\Gamma_{2}$ if $R(\Gamma_{1})=R(\Gamma_{2})$. A reduction $R$
is the result of reducing a tree; it is a finite, ordered, rooted, full binary
tree. For a reduction $R$, let $V_{R}$ denote the maximal volume of a graph
$\Gamma\sim R$.

$V_{R}=\sup\bigg{\\{}\frac{\mathrm{vol}(f_{\Gamma})}{|\Gamma|}\text{
}\bigg{\rvert}\text{ }\Gamma\sim R\bigg{\\}}$

This $V_{R}$ is attained as a limit of graphs that are subdivisions of $R$
adhering to the ordering. To prove Theorem 1.0.3, we show that for every
reduction $R$, $V_{R}\leq\frac{7}{3}$. Fortunately, we need only show this is
true for the $R=B_{n}$.

######  Lemma 2.0.1.

Let $R$ be a reduction, there exists $n$ such that $V_{R}\leq V_{B_{n}}$.

###### Proof.

We first show a stronger result. Suppose that a reduction $R_{1}$ is a subtree
of another reduction $R_{2}$ and that the parent-child hierarchy and left-
right orderings for their vertices in common are equivalent, then
$V_{R_{1}}\leq V_{R_{2}}$.

Let $(\Gamma_{m})_{m=1}^{\infty}$ be a sequence of graphs with
$\lim_{m\to\infty}|\Gamma_{m}|=\infty$ that attains the supremum $V_{R_{1}}$,
i.e. $\Gamma_{m}\sim R_{1}$ for all $m$, and

$\lim_{m\to\infty}\frac{f_{\Gamma_{m}}}{|\Gamma_{m}|}=V_{R_{1}}$

We now create a new sequence $(\Gamma^{\prime}_{m})_{m=1}^{\infty}$ by taking
the vertices in $R_{2}\setminus R_{1}$ and the graphs $\Gamma_{m}$, and
appending the $R_{2}\setminus R_{1}$ onto the leaves of each $\Gamma_{m}$ so
that $\Gamma^{\prime}_{m}\sim R_{2}$. We never subdivide any edges between the
elements of $R_{2}\setminus R_{1}$. Thus,

$\lim_{m\to\infty}\frac{f_{\Gamma^{\prime}_{m}}+|R_{2}\setminus
R_{1}|}{|\Gamma_{m}|+|R_{2}\setminus
R_{1}|}=\lim_{m\to\infty}\frac{f_{\Gamma_{m}}}{|\Gamma_{m}|}=V_{R_{1}}$

and thus $V_{R_{1}}\leq V_{R_{2}}$.

Let $L(R)$ denote the maximal quantity of two-child parents in a path from
root to leaf in $R$. The lemma follows as all reductions $R$ with $L(R)=n$ are
subgraphs of $B_{n}$. ∎

The remainder of the paper is dedicated to showing that $V(n)=V_{B_{n}}$ is
uniformly bounded. Firstly, let’s define a method for calculating $V_{R}$.

######  Lemma 2.0.2.

Enumerating the leaves of a reduction $R$ from left to right in order by
$1,\dots,k$, let $v_{i}$ denote the volume per subdivision of leaf $i$ and
$s_{i}\in[0,1]$ denote the proportion of subdivisions we apply to leaf $i$.
Then

$V_{R}=\max\\{v_{1}s_{1}+\dots+v_{k}s_{k}\mid s_{1}+\dots+s_{k}=1\text{ and
the $s_{i}$ satisfy the size-ordering}\\}$

###### Proof.

Let $R$ be a reduction. We first show that we need only subdivide the edges
connecting to leaves.

Let $(\Gamma_{n})_{n=1}^{\infty}$ be a sequence of graphs with $\Gamma_{n}\sim
R$ attaining $V_{R}$. Every $\Gamma_{n}$ is a subdivision of $R$ satisfying
the left-right ordering. A subdivision of an edge close to the root gives less
volume per vertex added than a subdivision of a deeper edge. To produce our
desired sequence of graphs $\Gamma^{\prime}_{n}$, simply move all of the
subdivisions of non-leaf edges to their left-most leaf child. The volume of
this new sequence is greater than or equal to the volume of the original and,
thus, also attains the supremum.

Following this approach of only subdividing leaf-edges, after applying $N$
subdivisions to $R$ the volume over the order is given by

$\frac{v_{1}(s_{1}N)+\dots+v_{k}(s_{k}N)+\mathrm{vol}(f_{R})}{N+|R|}$

whose limit as $N$ grows large yields the desired result. ∎

To bound $V_{B_{n}}$, we exploit the self-similar nature of our image of
$B_{n}$. Recall the illustration of $B_{4}$ in Figure 1. We see that $B_{n}$
is comprised of $B_{n-1}$ (in blue), $B_{n-2}$ (in red) and a spiral shape (in
black) which we call $S_{n}$. We define $S_{n}$ to be the reduction that
outputs the structure of the black nodes in Figure 1, but such that the blue
and red sections are straight paths (i.e. single leaves with many
subdivisions) of the same quantity of nodes as in $B_{n-1}$ and $B_{n-2}$
respectively. Once we have calculated the value of $V_{S_{n}}$, we obtain the
value of $V_{B_{n}}$ by replacing the two straight paths with copies of
$B_{n-1}$ and $B_{n-2}$. As the edges in the spiral have undergone the most
rotations, we place as many subdivisions there as possible. This leaves a
quarter of the subdivisions left for the red $B_{n-2}$ and a half for the blue
$B_{n-1}$. We have, note we must remove the straight paths from $S_{n}$,

$V_{B_{n}}=\frac{1}{2}(V_{B_{n-1}}-1)+\frac{1}{4}(V_{B_{n-2}}-1)+V_{S_{n}}$

We will shortly prove that $V_{S_{n}}\leq\frac{4}{3}$ for all $n$. Thus,

$V(n)\leq\frac{7}{12}+\frac{1}{2}V(n-1)+\frac{1}{4}V(n-2)$

with initial conditions $V(0)=1$ and $V(1)=1$. This is soluble, $V_{n}$ is
monotone increasing and bounded above by its limit of $\frac{7}{3}$. Thus
$V(n)\leq\frac{7}{3}$ for all $n$. We finish with the proof that $S_{n}$ is
bounded above by $\frac{4}{3}$.

######  Lemma 2.0.3.

For all $n$, $V_{S_{n}}\leq\frac{4}{3}$.

###### Proof.

We must distribute the subdivisions of the leaves to maximise the volume. The
left-most two leaves (the straight paths in place of the blue and red
sections) provide a volume per subdivision of $1$. Thus, we assign the minimum
amount of subdivisions to them, i.e. $\frac{1}{2}$ and $\frac{1}{4}$.
Enumerate all of the other leaves from left to right by the numbers $0,\dots
2^{n-2}-1$. Again, let $s_{l}$ denote the proportion of subdivisions we assign
to leaf $l$ and $v_{l}$ denote the volume per subdivision of leaf $l$. Let
$B(x)$ denote the amount of $1$s in the binary expansion of $x$. The amount of
rotations a leaf $l$ undergoes is given by $B(l)+2$ and $v_{l}$ is bounded by
$2+\lfloor B(l)/2\rfloor$. In practise, very few of the leaves attain this
bound. This is due to what we call shadowing. In Figure 1, see that if
$s_{2}>s_{1}$, then $v_{1}=1$ and vice versa. Thus, if we have assigned any
subdivisions to leaf $1$, we gain nothing from assigning them to leaf $2$.
However, this is not the case for leaf $3$. The most rotated leaf produces no
shadows. Thus, for $S_{4}$, our maximum is attained by assigning
$s_{3}=s_{2}=\frac{1}{4}$, $s_{1}=0$ and $s_{0}=\frac{1}{2}$. This gives
(recall the first two leaves),
$V_{S_{4}}=(1)\frac{1}{2}+(1)\frac{1}{4}+(2s_{0}+2s_{2}+3s_{3})\frac{1}{4}=\frac{21}{16}$.

For the general case, recall the more rotations a leaf has undergone, the more
volume per subdivision. The more shadowed leaves we still extend, the less the
volume. Thus, at any given parent $v$, we assign half the possible
subdivisions to its left-most leaf in the subtree $\Gamma_{v}$ and half of
them to be distributed by its right child who does the same. Now, we have no
shadows interfering with each other and the depth of all the leaves we assign
to is maximised. The leaves we assigned subdivisions to were the leaves at
positions $0$ and $p_{j}$ with $p_{j}=\sum_{i=n-2-j}^{n-3}2^{i}$. The leaf $0$
has $v_{0}=2$ and $s_{0}=\frac{1}{2}$. The leaves $p_{j}$ have
$v_{p_{j}}=2+\lfloor B(p_{j})/2\rfloor=2+\lfloor j/2\rfloor$. Assigning the
maximum amount of subdivisions to the deepest vertex gives $s_{0}=\frac{1}{2}$
and $s_{p_{j}}=\frac{1}{2^{j+1}}$ for $1\leq j\leq n-3$ and
$s_{p_{n-2}}=\frac{1}{2^{n-2}}$. Thus,

$\displaystyle V_{S_{n}}$
$\displaystyle=\frac{1}{2}+\frac{1}{4}+\frac{1}{4}\bigg{(}s_{0}v_{0}+s_{p_{n-2}}v_{p_{n-2}}+\sum_{j=1}^{n-3}s_{p_{j}}v_{p_{j}}\bigg{)}$
$\displaystyle V_{S_{n}}$
$\displaystyle=1+\frac{2+\lfloor(n-2)/2\rfloor}{2^{n}}+\sum_{j=1}^{n-3}\frac{2+\lfloor
j/2\rfloor}{2^{j+3}}$

which simplifies to

$V_{S_{n}}=\frac{4}{3}-\frac{1}{2^{n}}\bigg{(}2+\left\lfloor\dfrac{n-1}{2}\right\rfloor-\left\lfloor\dfrac{n}{2}\right\rfloor\bigg{)}\leq\frac{4}{3}$

∎

## References

  * [Bar93] Ya. M. Barzdin.  On the Realization of Networks in Three-Dimensional Space, pages 194–202.  Springer Netherlands, Dordrecht, 1993.
  * [BH21] Benjamin Barrett and David Hume.  Thick embeddings of graphs into symmetric spaces via coarse geometry, 2021.
  * [BS97] Itai Benjamini and Oded Schramm.  Every Graph with a Positive Cheeger Constant Contains a Tree with a Positive Cheeger Constant.  GAFA, Geom. funct. anal., (7):403–419, 1997.
  * [Cho80] Ching Chou.  Elementary amenable groups.  Illinois Journal of Mathematics, 24(3):396 – 407, 1980.
  * [GG11] Mikhail Gromov and Larry Guth.  Generalizations of the kolmogorov-barzdin embedding estimates.  Duke Mathematical Journal, 161, 03 2011.


# The effect of ionizing background fluctuations on the spatial correlations
of high redshift Ly$\alpha$-emitting galaxies

Avery Meiksin1 & Teresita Suarez 1
1SUPA, The Royal Observatory, Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK
Contact e-mail<EMAIL_ADDRESS>Universities Physics Alliance

(Accepted 8 August 2022. Received 5 August 2022; in original form 30 May
2022.)

###### Abstract

We investigate the possible influence of fluctuations in the metagalactic
photoionizing ultra-violet background (UVBG) on the clustering of
Ly$\alpha$-emitting galaxies through the modulation of the ionization level of
the gas surrounding the systems. At redshifts $z>5$, even when assuming the
reionization of the intergalactic medium has completed, the fluctuations are
sufficiently large that they may non-negligibly enhance, and possibly even
dominate, the angular correlation function on scales up to a few hundred
arcsecs. Whilst a comparison with observations at $z\simeq 5.7$ is
statistically consistent with no influence of UVBG fluctuations, allowing for
the fluctuations opens up the range of acceptable models to include those with
relatively low bias factors for the Ly$\alpha$-emitting galaxies. In this
case, the evolution in the bias factor of Ly$\alpha$-emitters over the
approximate redshift range $3<z<7$ corresponds to a nearly constant halo mass
for Ly$\alpha$-emitting galaxies of $\sim 10^{10.5}{M_{\odot}}$.

††pubyear: 2022††pagerange: The effect of ionizing background fluctuations on
the spatial correlations of high redshift Ly$\alpha$-emitting galaxies–A

##  1 Introduction

The discovery of high redshift Ly$\alpha$-emitting galaxies (Hu & McMahon,
1996; Pascarelle et al., 1996) has led to a series of Ly$\alpha$ emitter (LAE)
surveys of increasing depth (Ouchi et al., 2020). Surveys at $z>7$ likely
probe the Epoch of Reionisation (EoR), when the Universe was transformed from
neutral to ionized by the first galaxies and Quasi-Stellar Objects (QSOs),
over a characteristic redshift range $6\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10$ (Planck Collaboration, 2020).
LAE surveys therefore provide not only information about some of the
properties of the earliest galaxies in the Universe, but also about the
evolving nature of the Intergalactic Medium (IGM).

The evolution in the luminosity distribution of LAEs has been used to
constrain the ionization state of the IGM during the EoR. The decrease in the
LAE Ly$\alpha$ luminosity function compared with the ultra-violet (UV)
continuum luminosity function of Lyman-break galaxies (LBGs) agrees with an
increasing neutral hydrogen fraction of the IGM with redshift. The mean
neutral hydrogen fraction is estimated to be $\bar{x}_{\textrm{HI}}>0.1$ at
$z>6.6$ (Konno et al., 2018), and it possibly exceeds
$\bar{x}_{\textrm{HI}}>0.2$ at $z>7$ (eg Itoh et al., 2018; Goto et al.,
2021). The evolving rate of detection of Ly$\alpha$ emission in LBGs was used
by Mason et al. (2018) to suggest $\bar{x}_{\textrm{HI}}=0.59^{+0.11}_{-0.15}$
at $z\sim 7$.

The effects of incomplete reionization on the properties of LAEs are less able
to constrain the ionization state of the IGM at $z<6$. The strongest
constraints come from the Ly$\alpha$ forest measured in bright, background
QSOs. The distributions of the measured Ly$\alpha$ effective optical depths
$\tau_{\alpha}$ at $z>5$ are unexpectedly broad compared with the predictions
of numerical simulations of the IGM. The breadths of the distributions
increase with redshift, with patches of high attenuation extending over 100
$h^{-1}$ comoving Mpc appearing along some lines of sight at $z>6$ (Becker et
al., 2015, 2018; Bosman et al., 2018; Yang et al., 2020; Bosman et al., 2022).
It has been suggested that late reionization may account for the breadths of
the Ly$\alpha$ effective optical depth distributions, with the EoR ending as
late as $z\simeq 5.2-5.5$ (Kulkarni et al., 2019a; Keating et al., 2020; Nasir
& D’Aloisio, 2020). In this scenario, extended regions of high optical depth
originate in residual patches of neutral hydrogen.

An alternative explanation suggested for the breadths of the distributions is
the “dappling” of the IGM by a fluctuating metagalactic UVBG (Meiksin, 2020).
Fluctuations in the UVBG arise from both spatial correlations between the
sources and from the shot noise due to their discreteness (Gontcho A Gontcho
et al., 2014; Pontzen, 2014; Suarez & Pontzen, 2017; Meiksin & McQuinn, 2019).
Whilst the fractional contribution of QSO sources to the UV background is
below half at $z>3$ and diminishes with increasing redshift, the QSOs continue
to dominate the shot noise contribution to the background. The large scale
UVBG fluctuations resulting from the source shot noise reproduce the measured
breadths in the distributions of the Ly$\alpha$ effective opacity at $3<z<6$.

LAEs have been measured to cluster spatially. The strength of the clustering
has been used to infer the nature of LAEs from the estimated bias factors.
Comparison with the expected clustering of dark matter shows an increase in
the LAE bias from $b_{L}\sim 1.5$ at $z<4$ to as high as $b_{L}\sim 6$ at
$z>5$, with inferred dark matter halo masses of $10^{10.3}-10^{11.3}\,{\rm
M}_{\odot}$ and a duty cycle, representing the fraction of haloes in a LAE
phase, of less than 1% (Ouchi et al., 2018). From the evolution of the bias
factor, Ouchi et al. (2018) infer LAEs at z > 2 may be the progenitors of
massive elliptical galaxies today. The estimates of the LAE bias factors,
however, show considerable scatter at the higher redshifts, with estimates for
$b_{L}$ at $z=5.7$ ranging between $4.1\pm 0.2$ and $6.1\pm 0.7$, and at
$z=6.6$ between $b_{L}=3.6\pm 0.7$ and $4.5\pm 0.6$, depending on the field(s)
analysed (Ouchi et al., 2010, 2018). The origin of the discrepant values,
especially at $z=5.7$, is unclear. It has been suggested that results for some
of the smaller fields may be affected by cosmic variance (Kusakabe et al.,
2018), with the clustering in one field at $z=5.7$ dominated by a protocluster
(Ouchi et al., 2010). Firmly establishing the evolution of the LAE bias
factors will help elucidate the nature of the host systems at high redshifts
and of their counterparts today.

The interpretation of the clustering signal is also complicated by the
possible effects of Ly$\alpha$ photon scattering by the intervening IGM. The
clustering of LAEs at $z>5$ may depend on the physical character of the EoR
through the ionization structure of the IGM. The patchwork of ionized and
still-neutral regions during the EoR will be imprinted on the spatial
correlations of LAEs because of the relative difference in attenuation of the
Ly$\alpha$ emission line of the galaxies in different patches (Furlanetto et
al., 2006; McQuinn et al., 2007). Comparison with observations suggests a mean
IGM neutral hydrogen fraction below 30% at $z=6.6$ (Ouchi et al., 2018).

The clustering of LAEs may also depend on radiative transfer effects on the
sample selection, since the Ly$\alpha$ emission line may be modulated by
scattering through the intervening IGM, which traces the same dark matter
density field as do the galaxies (Wyithe & Dijkstra, 2011; Zheng et al., 2011;
Gurung-López et al., 2020). Local non-linear redshift space distortions may
further confound interpretation of the clustering strength (Byrohl et al.,
2019).

In this paper, we suggest that another factor governing the clustering of LAE
systems are fluctuations in the UVBG after the end of cosmic reionization. The
gas surrounding a LAE may give rise to a non-negligible Ly$\alpha$ optical
depth redward of the Ly$\alpha$ emission line (eg Verhamme et al., 2006;
Laursen et al., 2011). UVBG fluctuations will then modify the detectability of
the systems and imprint their spatial correlations on the spatial correlation
function of the LAEs. We assess the possible contribution of UVBG fluctuations
to the spatial correlations of LAEs, and constrain the attenuating properties
of the gas surrounding the LAEs. We shall show that allowing for UVBG
fluctuations increases the uncertainty in the LAE bias factors inferred from
the angular correlation function.

Modelling the effect of the UVBG on the spatial correlations of LAEs, however,
is hampered by two difficulties: (1) the rarity of the QSOs at high redshift
requires either very large simulation volumes, on the order of 800$h^{-1}$ Mpc
(comoving) or very many statistical realisations in smaller volumes, to
capture the shot-noise contribution (Meiksin, 2020), and (2) the origin and
subsequent radiative transfer of the Ly$\alpha$ emission line from LAEs is
unknown. We overcome the first difficulty by using a perturbative approach to
model the UVBG fluctuations (Meiksin & McQuinn, 2019), as the underlying
density fluctuations are small over the physical scales for which we estimate
the LAE spatial correlations. The latter difficulty includes uncertainties in
the amount of obscuration of the escaping Ly$\alpha$ photons from the unknown
clumpiness and flow pattern of the surrounding circumgalactic and
intergalactic media. We adopt a simple statistical model described below to
assess the amount of obscuration of the Ly$\alpha$ emission line by the gas
surrounding LAEs. The combined model is designed to capture the range of
possible impact the UVBG fluctuations may have on the LAE spatial
correlations.

The paper is organised as follows. In the next section, we describe our model
for assessing the possible effect of UVBG fluctuations on the measured angular
correlation function of LAEs. In Sec. 3, comparisons between the model
predictions and measurements are presented. We discuss these in Sec. 4, and
provide a summary of our conclusions in the last section. A flat cosmology is
assumed with $\Omega_{m}=0.31$, $\sigma_{8}=0.81$ and $n=0.97$ (Planck
Collaboration, 2020).

##  2 The model

###  2.1 UV background model

The mean photoionizing UVBG is modelled following Meiksin & McQuinn (2019) and
Meiksin (2020), with some recent updates. In brief, both galactic and QSO
sources are included in the mean metagalactic emissivity. The evolving
Schechter luminosity function fit from Bouwens et al. (2015) is adopted for
the galactic contribution. We use Model 3 of Kulkarni et al. (2019b) for the
QSO luminosity function, confined to sources with absolute AB magnitudes at
1450 A of $-30<M_{1450}<-21$. The mean free path of ionizing photons at the
Lyman photoelectric edge is taken from Worseck et al. (2014) for $z<5$ and
Becker et al. (2021) for $z\geq 5$. The mean metagalactic emissivity at the
Lyman edge is adjusted to reproduce the median effective IGM Ly$\alpha$
optical depth measurements listed in Meiksin (2009) for $z<5$, and from Bosman
et al. (2018) for $5\leq z<5.4$ and Yang et al. (2020) for $z\geq 5.4$.

The fluctuations in the UVBG arise from fluctuations in the Lyman continuum
opacity of the IGM, and from the spatial clustering of the sources and the
source shot noise. We assume a galaxy bias factor $b_{G}=3$ (e.g. Bielby et
al., 2013), noting it may be higher at $z>5$. An evolving bias factor is
adopted for the QSOs of $b_{Q}=0.278(1+z)^{2}+0.57$ (Laurent et al., 2017). At
$z>5$, the fluctuations are dominated by the shot noise in the QSO counts over
the scales of interest. For a short source lifetime compared with the age of
the Universe, the shot noise contribution is proportional to the source
lifetime. For a long lifetime, the shot noise is given by the steady-state
limit.

Following Meiksin & McQuinn (2019), we define the following quantities: for an
UVBG intensity $I_{\nu}$, emissivity $j_{\nu}$ and photoelectric cross section
$\sigma_{\nu}$, $f=\int\,d\nu(I_{\nu}/h_{\mathrm{P}}\nu)\sigma_{\nu}$,
$j=\int\,d\nu(j_{\nu}/h_{\mathrm{P}}\nu)\sigma_{\nu}$ and $\zeta=\langle
I_{L}\rangle\sigma_{L}/h_{\mathrm{P}}\langle f\rangle$, where $h_{\mathrm{P}}$
is Planck’s constant and $I_{L}$ and $\sigma_{L}$ are the values of the
intensity and cross section at the Lyman edge, respectively. Here,
$\langle\dots\rangle$ denotes a spatial average. A dimensionless opacity
$\chi=(c/H)\langle\alpha_{\mathrm{eff}}\rangle$ is also defined, where
$\alpha_{\mathrm{eff}}$ is the frequency-averaged opacity weighted by
$I_{\nu}\sigma_{\nu}/h_{\mathrm{P}}\nu$, and $H$ is the Hubble parameter. The
dimensionless ratio $\phi=c\langle j\rangle/[H(\chi+\zeta)\langle f\rangle]$
describes the evolution of the radiation field. For a non-evolving radiation
field, $\phi=1$. The mean metagalactic photoionization rate is related to $f$
by $\Gamma=4\pi\langle f\rangle$. The Fourier component for comoving
wavenumber $k$ of the perturbation to the photoionization rate is given in the
steady-state limit by

$\tilde{\delta}_{\Gamma,{\rm
SS}}(\kappa)=\frac{\phi(\chi+\zeta)\tilde{\delta}_{j}(\kappa)-b_{\chi,\delta}\chi\tilde{\delta}(\kappa)}{{\frac{\kappa}{a}\left[{\rm
atan}{\left(\frac{\kappa}{a\phi(\chi+\zeta)}\right)}\right]^{-1}}+b_{\chi,\Gamma}\chi},$
(1)

where $\tilde{\delta}_{j}$ and $\tilde{\delta}$ are the perturbations in the
source density and gas density respectively, $\kappa=(c/H)k$ is a
dimensionless comoving wavenumber, and $a=1/(1+z)$. Here,
$\tilde{\delta}\log\chi=b_{\chi,\delta}\tilde{\delta}+b_{\chi,\Gamma}\tilde{\delta}_{\Gamma}$.
Eq. (1) is similar to the findings of Gontcho A Gontcho et al. (2014) and
Pontzen (2014), who both additionally assume $\phi=1$. The power spectrum of
the light fluctuations in a comoving volume $V_{u}$ is
$P_{\Gamma}^{\mathrm{SS}}(k)=V_{u}\langle\tilde{\delta}_{\Gamma}^{\dagger}(k)\tilde{\delta}_{\Gamma}(k)\rangle^{\mathrm{SS}}$.
In addition to a component proportional to the dark matter power spectrum and
dependent on the bias factors of the sources, a source shot noise term also
contributes additively to the full power spectrum. The shot noise contribution
is given by

$V_{u}\langle\tilde{\delta}^{*}_{\Gamma}(k)\tilde{\delta}_{\Gamma}(k)\rangle^{\mathrm{SS}}_{\mathrm{shot}}\sim\left[\frac{\pi}{2}\phi\left(\chi+\zeta\right)\frac{a}{\kappa}\right]^{2}\frac{1}{n_{\rm
eff}},$ (2)

in the limit $\kappa\gg 1$, corresponding to modes with wavelengths short
compared with the cosmological horizon. The effective number density of the
sources is given by $n_{\mathrm{eff}}=\langle L\rangle_{\Phi}^{2}/\langle
L^{2}\rangle_{\Phi}$, where the averages $\langle\dots\rangle_{\Phi}$ are
carried out over the luminosity density $\Phi(L)$ of the sources.

For sources with finite lifetimes $\tau_{S}$, the shot noise contribution may
be substantially reduced. We adopt an approximate form which accurately
interpolates between the $\kappa\ll 1$ and $\kappa\gg 1$ limits,

$V_{u}\langle\tilde{\delta}^{*}_{\Gamma}(k)\tilde{\delta}_{\Gamma}\rangle_{\rm
shot}(k)\simeq\frac{1}{A}\frac{\left[\phi(\chi+\zeta)\right]^{2}}{1+[\kappa/\kappa_{*}^{\rm
SN}(\kappa)]^{2}}\frac{1}{n_{\rm eff}},$ (3)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{2}{H\tau_{S}}\left[(\phi+b_{\chi,\Gamma})\chi+\phi\zeta+\frac{3}{4}-\frac{1}{2}\alpha_{n}\right]$
$\displaystyle-\left[\frac{4}{\pi^{2}}\left(\phi(\chi+\zeta)-1/2\right)+b_{\chi,\Gamma}\chi\right]^{2},$

for the approximation $n_{\mathrm{eff}}\sim(1+z)^{-\alpha_{n}}$, and
$\kappa_{*}^{\rm SN}(\kappa)=2\pi[c/H(z)]/[(1+z)\lambda_{*}^{\rm SN}(\kappa)]$
with

$\displaystyle\lambda_{*}^{\rm SN}$ $\displaystyle=$ $\displaystyle
4\frac{c}{H}A^{-1/2}$
$\displaystyle\times\left\\{1+\frac{\pi}{2}\left[\frac{4}{\pi^{2}}\left(\phi(\chi+\zeta)-1/2\right)+b_{\chi,\Gamma}\chi\right]\frac{a}{\kappa(a)}\right\\},$

for $H\tau_{S}\ll 1$ (Meiksin & McQuinn, 2019). When the distance light
travels during the source lifetime exceeds the mean free path of the ionizing
radiation, the steady-state value is a good approximation. On scales small
compared with the light travel distance over the source lifetime, $k\gg
1/(c\tau_{S})$, Eq. (3) goes over to Eq. (2). Additional suppression of the
power spectrum of UVBG fluctuations may arise if QSOs emit in tight beams
(Suarez & Pontzen, 2017); this effect is not included here.

When including the UVBG fluctuations, we adjust the escape fraction of
ionizing photons from galaxies to match the median optical depth measurements.
In practice this has only a small effect on the shot noise contribution
because it is dominated by the QSOs.

###  2.2 Observed Ly$\alpha$ emission line

Modelling the Ly$\alpha$ emission line is more problematic. The observed line
depends on the emission line mechanism and the subsequent Ly$\alpha$ radiative
transfer both within and near the Ly$\alpha$-emitting galaxy and through the
IGM. The emission line may originate in a compact region or from several
regions extending across the galaxy, which will affect the line profile
generated. This will also affect the Ly$\alpha$ radiative transfer through the
interstellar medium of the galaxy, which may be static or moving, possibly in
a wind. The profile will be further modified by the gas in the circumgalactic
medium (CGM) of the galaxy, and whether the gas is expanding in a wind or
infalling, and possibly part of an H $\scriptstyle\rm II\ $ region produced by
the LAE. The possible presence of dust absorption is still another
complicating factor. All of these factors may vary with time over the lifetime
of an individual LAE, and between different LAEs, possibly dependent on
properties of the host galaxy such as its mass.

The astrophysical effects are clearly well beyond definitive detailed
modelling given the current understanding of the physical properties of LAEs
and their immediate environments. Several detailed Ly$\alpha$ radiative
transfer computations have been performed of the emission line leaving the
galaxy and its CGM under varying assumptions. Whilst the radiative transfer
through an optically thick slab results in a double-horned emission line
profile (Harrington, 1973), radiative transfer through a wind or accretion
flow generally suppresses either the blue side in a wind (Verhamme et al.,
2006), or the red side in an accretion flow (Dijkstra et al., 2006; Verhamme
et al., 2006), producing emission that is redward dominated in the case of
outflow or blueward dominated in the case of infall. Backscattering off the
far-side of the gas relative to the observer results in peaks displaced by as
much as twice the outflow or infall velocities, possibly achieving offsets as
great as $\pm 400\,{\rm{km\,s^{-1}}}$ (Verhamme et al., 2006).

The details of the emission line profile are sensitive to the origin of the
emission (central point source or extended emitting region), the total H
$\scriptstyle\rm I\ $ column density, the gas temperature and the expansion or
infall velocity, as well as broadening by any turbulence that may be present.
For emission through an expanding shell, and depending on the H
$\scriptstyle\rm I\ $ column density, for a sufficiently large expansion
velocity the redward lobe splits into two, one peaking at about twice the wind
velocity redward of line centre and the other just shortward of line centre
(Verhamme et al., 2006). The profile also may depend on the orientation of the
LAE. For photons generated within a gaseous disk, when viewed face-on the
profile may be dominated by a single lobe, either blueward of line centre if
the source is at the centre of an accretion flow, or redward if at the centre
of a wind. When viewed edge-on, however, the profile may split into two lobes
around line centre for a sufficiently high internal H $\scriptstyle\rm I\ $
column density (eg Verhamme et al., 2012).

In reionization simulations, various approaches have been taken to model the
Ly$\alpha$ profiles including the effect of IGM attenuation. These typically
assume complete suppression of any feature blueward of the systemic Ly$\alpha$
wavelength of a LAE, and partial attenuation of the redward feature by the
damping wing from neutral patches in the IGM (Furlanetto et al., 2006; McQuinn
et al., 2007; Iliev et al., 2008; Dijkstra et al., 2011). As we are
considering the effects of IGM attenuation after reionization has completed,
such a simpified treatment is inappropriate. Instead the emission line is
attenuated using the mean IGM transmission $e^{-\tau_{\alpha}}$ at the
cosmological redshift of the LAE. How much of the profile is attenuated,
however, depends on several factors: the centroid of the emission line
compared with the systemic velocity of the LAE, the peculiar velocity of the
LAE, the size of a possible H $\scriptstyle\rm II\ $ region produced by the
LAE, gaseous outflow produced by winds from the hosting galaxy, and
cosmological infall around the LAE. The latter, in particular, can result in
attenuation even when the emission line is redward of the systemic velocity of
a comoving LAE (Laursen et al., 2011; Weinberger et al., 2018). The detailed
study of Laursen et al. (2011) shows IGM attenuation on both the blue and red
sides of the local cosmological restframe Ly$\alpha$ wavelength, extending
redward by more than $200\,{\rm{km\,s^{-1}}}$ at $z=5.8$ in an early
reionization scenario (with characteristic reionization redshift
$z_{\mathrm{ri}}=10$) and to over $600\,{\rm{km\,s^{-1}}}$ for late
reionization ($z_{\mathrm{ri}}=6$) (their figures 3 and 4). Typically three-
quarters of the Ly$\alpha$ radiation emergent from the galaxy is scattered out
of the line of sight by the large-scale IGM for the early reionization
scenario. The measured Ly$\alpha$ luminosity is highly sensitive to galaxy
mass and the profile of the emission line emerging from the galaxy, whilst the
amount further attenuated by the IGM varies widely, depending on the line of
sight (their figure 8).

For simplicity we model the emission line as a single gaussian profile of
width $\sigma_{l}$. Although the profile should split into two lobes for a LAE
viewed edge-on, we treat it as unresolved, as it will make no difference to
the total Ly$\alpha$ luminosity of a symmetric feature: the blueward half of
the emission line will be suppressed by roughly the same factor. We allow the
centroid to be displaced relative to the restframe Ly$\alpha$ attenuation edge
of the IGM. In the event LAEs photoionize their surroundings, the displacement
is relative to the near (lower redshift) edge of the H $\scriptstyle\rm II\ $
bubble.

The effect of the IGM attenuation is to alter the intrinsic luminosity $L$ of
the Ly$\alpha$ emission line, giving the observed value

$L_{\mathrm{obs}}=\frac{L}{1+f}\left(1+fe^{-\tau_{\alpha}}\right).$ (6)

For a gaussian emission line profile with velocity centroid $v$ relative to
the IGM attenuation edge and width $\sigma_{l}$, the factor $f$ is given by

$f(v)=\frac{1-\mathrm{erf}\left(\frac{v}{2^{1/2}\sigma_{l}}\right)}{1+\mathrm{erf}\left(\frac{v}{2^{1/2}\sigma_{l}}\right)}.$
(7)

In the limit $v\gg\sigma_{l}$ (eg, wind-dominated emission), $f\rightarrow 0$
and $L_{\mathrm{obs}}\rightarrow L$, while for $-v\gg\sigma_{l}$ (eg, infall-
dominated emission), $f\rightarrow\infty$ and $L_{\mathrm{obs}}\rightarrow
Le^{-\tau_{\alpha}}$. In the limit of a small velocity offset
$|v|\ll\sigma_{l}$, $f\rightarrow 1$ (eg, as for a disk seen edge-on), and
$L_{\mathrm{obs}}\rightarrow L(1+e^{-\tau_{\alpha}})/2$.

We show below that only in the presence of substantial IGM attenuation will
UVBG fluctuations have much of an effect on the angular correlation function
of the LAEs. At moderate redshifts ($z\sim 3$), little effect is expected; but
the effects may be strong at $z>5$. In accordance with simulations of the IGM
attenuation discussed above, the IGM attenuation edge may be displaced
substantially redward of the cosmological restframe Ly$\alpha$ wavelength at
these redshifts. The characteristic LAE emission line FWHM at $z=5.7$ is
$265\pm 37\,{\rm{km\,s^{-1}}}$ (Ouchi et al., 2010), corresponding to
$\sigma_{l}\simeq 110\,{\rm{km\,s^{-1}}}$. The combined displacement redward
of the LAE emission line may be by as much as four times this value, allowing
also for infall. Whilst there is evidence that the Ly$\alpha$ emission line
itself is redshifted relative to the systemic velocity of the
Ly$\alpha$-emitting galaxy (eg McLinden et al., 2011; Verhamme et al., 2018),
the observations are largely at $z\sim 3-4$, when winds may be driven by
higher rates of star formation in more massive systems. Many of the systems
also show blueward emission, although at a level consistent with outflows. It
is unknown whether the Ly$\alpha$ emission is redshifted compared to the
systemic velocity of most LAEs at $z>5$, but even if so, it is unknown whether
the gaseous surroundings at these redshifts are outflowing, or dominated by
cosmic accretion. If outflows dominate, then no strong contribution of UVBG
fluctuations to the clustering of the LAEs is expected. Observations suggest
the impact of intervening IGM scattering on the measured Ly$\alpha$ profile of
LAEs increases with redshift (Hayes et al., 2021).

###  2.3 Spatial correlations of Ly$\alpha$ emitters

The effect of fluctuations in the density field and the UVBG on the number
counts of LAEs may be quantified using the LAE luminosity distribution. This
is well fit by a Schechter function (Ouchi et al., 2020). Since the luminosity
detection threshold111The detection threshold is in practice based on
equivalent width. We simplify the analysis by formulating the detection in
terms of a luminosity threshold. is small compared with the exponential cut-
off in the Schechter function, the total number of detected systems may be
approximated as a power-law $d{\cal N}/dL=n_{*}L^{\alpha}$ for LAE Ly$\alpha$
luminosity $L$. These systems will dominate the spatial correlation signal
unless the high luminosity systems cluster much more strongly than the low.
There is some evidence for a statistical correlation between clustering
strength and LAE luminosity at $z<5$ (Ouchi et al., 2003, 2005), but this may
arise from AGN, which are not in as great abundance among LAEs at higher
redshifts (Ouchi et al., 2008).

To allow for the line-of-sight variation in the amount of IGM attenuation, we
model the relative offset between the LAE emission line and IGM attenuation
edge as a gaussian random process with centroid velocity offset $v_{0}$ and
dispersion $\sigma_{v}$. The observed distribution is then given by

$\frac{d{\cal
N}}{dL_{\mathrm{obs}}}=\frac{1}{(2\pi\sigma^{2}_{v})^{1/2}}\int_{-\infty}^{\infty}dv\,e^{-(v-v_{0})^{2}/2\sigma^{2}_{v}}\frac{d{\cal
N}}{dL}|_{L(v|L_{\mathrm{obs}})}\frac{dL}{dL_{\mathrm{obs}}}(v),$ (8)

where $L(v|L_{\mathrm{obs}})$ is the intrinsic Ly$\alpha$ luminosity $L$
required to obtain an observed luminosity $L_{\mathrm{obs}}$ given a velocity
offset $v$, according to Eqs. (6) and (7). The number of systems above a
particular observed value $L_{0}$ becomes (for $\alpha<-1$),

${\cal
N}(L_{\mathrm{obs}}>L_{0})\simeq-\frac{\bar{n}_{*}}{\alpha+1}L_{0}^{\alpha+1},$
(9)

where, using Eq. (8),

$\bar{n}_{*}=\frac{n_{*}}{(2\pi\sigma^{2}_{v})^{1/2}}\int_{-\infty}^{\infty}dv\,e^{-(v-v_{0})^{2}/2\sigma^{2}_{v}}\left[\frac{1+f(v)}{1+f(v)e^{-\tau_{\alpha}}}\right]^{\alpha+1}.$
(10)

Good fits to the observed distribution of LAE luminosities are provided by
$\alpha=-1.8$ for $z\leq 5.7$ and $\alpha=-2.5$ for $z\geq 5.7$ (Ouchi et al.,
2020).

Assuming $n*$ is proportional to the local cosmological mass density, the
perturbation in the number of detected LAEs, allowing for both mass density
and UVBG fluctuations, is then given by

$\frac{\hat{\delta}{\cal N}}{\cal
N}(\mathbf{k},z)=\left[b_{L}+f(\Omega_{m})\mu_{k}^{2}\right]\hat{\delta}(\mathbf{k},z)+(\alpha+1)g(v,\tau_{\alpha})\hat{\delta}\tau_{\alpha}(\mathbf{k},z),$
(11)

where $b_{L}$ is the cosmological density bias factor for the LAEs and

$g(v_{0},\tau_{\alpha})=\frac{\int_{-\infty}^{\infty}dv\,e^{-\frac{(v-v_{0})^{2}}{2\sigma^{2}_{v}}}\left[\frac{1+f(v)}{1+f(v)e^{-\tau_{\alpha}}}\right]^{\alpha+1}\frac{f(v)e^{-\tau_{\alpha}}}{1+f(v)e^{-\tau_{\alpha}}}}{\int_{-\infty}^{\infty}dv\,e^{-\frac{(v-v_{0})^{2}}{2\sigma^{2}_{v}}}\left[\frac{1+f(v)}{1+f(v)e^{-\tau_{\alpha}}}\right]^{\alpha+1}}.$
(12)

This provides a convenient parametrisation of the influence of UVBG
fluctuations on the LAE correlation function in that $g$ is confined to the
range $0\leq g\leq 1$. For simplicity, unless stated otherwise, we take the
same width for the velocity offset gaussian as for the line profile
($\sigma_{v}=\sigma_{l}$). This avoids a partial degeneracy between the offset
of the centroid and the width of the distribution. Although we use the full
integration in Eq. (12) for all results presented below, we note that at
$z=5.7$ and for $\alpha=-1.8$, $g$ is well-approximated over
$-4\sigma_{l}<v_{0}<0$ by

$g(v_{0},z=5.7)\simeq\frac{1}{2}\left[1-\tanh\left(\frac{3+v_{0}/\sigma_{l}}{2^{1/2}}\right)\right].$
(13)

The additional term $f(\Omega_{m})\mu_{k}^{2}$ in Eq. (11) accounts for
redshift space distortions (Kaiser, 1987), where $f(\Omega_{m})$ is the
dimensionless linear growth rate of the density fluctuation growing mode (a
flat Universe is assumed) and $\mu_{k}=\mathbf{\hat{n}}\cdot\mathbf{\hat{k}}$
for line-of-sight direction $\mathbf{\hat{n}}$.

The LAE power spectrum, allowing for both density and UVBG fluctuations and
for redshift space distortions, is then

$\displaystyle P_{L}(k,\mu_{k},z)$ $\displaystyle=$ $\displaystyle
V_{u}\left\langle\frac{\delta\hat{\cal N}^{*}}{\cal N}({\bf
k},z)\frac{\delta\hat{\cal N}}{\cal N}({\bf k},z)\right\rangle$ (14)
$\displaystyle=$ $\displaystyle
b_{\mathrm{eff}}^{2}V_{u}\left\langle\hat{\delta}^{*}(\mathbf{k},z)\hat{\delta}(\mathbf{k},z)\right\rangle\left[1+\frac{f(\Omega_{m})}{b_{\mathrm{eff}}}\mu^{2}_{k}\right]^{2}$
$\displaystyle+$ $\displaystyle
b_{\alpha,\Gamma}(\alpha+1)g\tau_{\alpha}b_{\mathrm{eff}}V_{u}\Biggl{[}\left\langle\hat{\delta}^{*}_{\Gamma}(\mathbf{k},z)\hat{\delta}(\mathbf{k},z)\right\rangle\phantom{\Biggr{]}}$
$\displaystyle\phantom{\Biggl{[}}+\left\langle\hat{\delta}_{\Gamma}(\mathbf{k},z)\hat{\delta}^{*}(\mathbf{k},z)\right\rangle\Biggr{]}\left[1+\frac{f(\Omega_{m})}{b_{\mathrm{eff}}}\mu^{2}_{k}\right]$
$\displaystyle+$
$\displaystyle\left[b_{\alpha,\Gamma}(\alpha+1)g\tau_{\alpha}\right]^{2}V_{u}\left\langle\hat{\delta}^{*}_{\Gamma}(\mathbf{k},z)\hat{\delta}_{\Gamma}(\mathbf{k},z)\right\rangle,$

where

$b_{\mathrm{eff}}=b_{L}+b_{\alpha,\delta}(\alpha+1)g\tau_{\alpha},$ (15)

and we have introduced the bias factors
$b_{\alpha,\delta}=d\log\tau_{\alpha}/d\log\rho_{b}$ and
$b_{\alpha,\Gamma}=d\log\tau_{\alpha}/d\log\Gamma$, where $\rho_{b}$ is the
baryon density. The factor $b_{\textrm{eff}}$ includes the radiative transfer
selection effect on the LAE sample discussed by Zheng et al. (2011) and Wyithe
& Dijkstra (2011), as an enhancement in the IGM density reduces the
detectability of a LAE. For $\alpha<-1$, this results in a suppression of the
LAE clustering signal. Because fluctuations in the UVBG will also affect
$\tau_{\alpha}$, additional terms proportional to $b_{\alpha,\Gamma}$
contribute. These arise from fluctuations in the mean free path of Lyman
continuum photons and from fluctuations in the spatial density of the
radiation sources, here assumed to trace the underyling dark matter density
field. The additional contributions are proportional to
$\hat{\delta}^{*}_{\Gamma}\hat{\delta}$ (and its complex conjugate). Another
contribution arises from the shot noise of the sources, which is included in
the term proportional to $\hat{\delta}^{*}_{\Gamma}\hat{\delta}_{\Gamma}$.
(See Meiksin & McQuinn, 2019, for details.)

The values for $b_{\alpha,\delta}$ and $b_{\alpha,\Gamma}$ are computed for
the log-normal approximation model for the Ly$\alpha$ forest described in
Meiksin (2020), for which the baryon fluctuations are the same as the dark
matter fluctuations over the length scales of interest (large compared with
the Jeans length of the IGM). The redshift dependences of the coefficients are
well fit over $2<z<6$ by

$b_{\alpha,\delta}\simeq-0.07834(1+z)+1.12334$

and

$b_{\alpha,\Gamma}\simeq 0.07645(1+z)-0.71815.$

The computation of the predicted angular correlation function is facilitated
by decomposing the power spectrum, including redshift space distortions, into
its Legendre components

$P_{\it l}(k,z)=\frac{2{\it
l}+1}{2}\int_{-1}^{1}\,d\mu_{k}\,P_{L}(k,\mu_{k},z)L_{\it l}(\mu_{k}),$ (16)

(Hamilton, 1998), where $L_{\it l}(\mu_{k})$ is a Legendre polynomial of order
${\it l}$. In the linear density approximation (and assuming a flat sky) only
the ${\it l}=0,2$ and 4 components are non-vanishing. The Legendre components
of the redshift-space correlation function corresponding to the power spectrum
components are given by

$\xi_{\it l}(r,z)=\frac{i^{\it l}}{2\pi^{2}}\int_{0}^{\infty}\,dk\,k^{2}j_{\it
l}(kr)P_{\it l}(k,z).$ (17)

The angular dependence may be recovered through

$\xi(r,\mu,z)=\sum_{i=0}^{2}L_{2i}(\mu)\xi_{2i}(r,z),$ (18)

where $\mu=\mathbf{\hat{n}}\cdot\mathbf{\hat{r}}$ for two points separated by
$\mathbf{r}$. The angular correlation function at redshift $z$ for objects
separated by a ray with projected separation $b_{\perp}\simeq D_{A}\theta$ (in
the small angle approximation) relative to the line of sight from the observer
is then

$\omega(\theta,\mu)\simeq\sum_{i=0}^{2}\xi_{2i}(D_{A}[1+z]\theta/[1-\mu^{2}]^{1/2})L_{2i}(\mu),$
(19)

where $D_{A}$ is the angular-diameter distance to redshift $z$ and $\mu$ is
the cosine of the angle between the ray connecting the objects and the line of
sight. The redshift factor $[1+z]$ multiplying $D_{A}$ is introduced since $k$
and $r$ are assumed to be in comoving units in Eq. (17).

Measurements of the angular correlation function of LAEs are carried out for
systems confined to redshift shells defined by the narrow-band filter used for
measuring the Ly$\alpha$ emission line. A shell of width $\Delta z$ in
redshift corresponds to a comoving width $\Delta{\sc l}=(c/H)\Delta z$. To
compute the angular correlation function in a shell of width $\Delta z$, the
spatial correlations must be averaged along the line of sight through the
shell:

$\omega(\theta)\simeq\frac{2}{\Delta{\sc l}}\sum_{i=0}^{2}\int_{0}^{\Delta{\sc
l}/2}d{\sc l}\,\xi_{2i}\left({\sc l}/\mu\right)L_{2i}(\mu),$ (20)

where $\mu={\sc l}/[{\sc l}^{2}+(D_{A}[1+z]\theta)^{2}]^{1/2}$.

##  3 Results

Figure 1: LAE angular correlation function at z = 5.7 for LAE bias factor
$b_{\mathrm{LAE}}=4.1$ and allowing for the maximum effect of UVBG
fluctuations ($g=1$). Results are shown in the steady-state limit with the
contribution from source shot noise (long-dashed blue curve) and without
(dotted cyan curve), and for time varying UVBG fluctuations with a QSO
lifetime $\tau_{Q}=10$ Myr (dot-dashed green curve). Also shown is the result
for no UVBG fluctuations, (solid black curves; the lighter curve is without
redshift space distortions). The data points are taken from Ouchi et al.
(2020), based on the samples analysed in Ouchi et al. (2010) (circles) and
Ouchi et al. (2018) (squares). (At $z=5.7$, 100 arcsecs corresponds to a
comoving separation of $2.7h^{-1}$ Mpc.)

We first consider the limiting case $g=1$ for the expected angular correlation
function of LAEs, corresponding to $-v_{0}\gg\sigma_{l}$ and
$f\rightarrow\infty$, so that the entire emission line is attenuated, giving
the maximal signature of UVBG fluctuations on the spatial correlations of the
LAEs. For $z<5$, we find the contribution from UVBG fluctuations is negligible
(less than 2%), but the contribution grows rapidly with redshift as a result
of the rapidly rising intergalactic effective Ly$\alpha$ optical depth.

In Fig. 1, we show the effect of UVBG fluctuations with $g=1$ on the angular
correlation function at $z\simeq 5.7$, matching a redshift value for LAEs
measured in the SILVERRUSH Subaru survey (Ouchi et al., 2018). We average the
spatial correlations over a redshift shell $\Delta z\simeq 0.092$ wide,
corresponding to the width of the narrow-band filter used in the survey. Two
models for the UVBG fluctuations are illustrated, one for the steady-state
limit and a second time-dependent model with a QSO source lifetime of
$\tau_{Q}=10$ Myr. (The lifetime of the galaxies is taken to be 100 Myr, but
the finite value has a negligible effect on the UVBG fluctuations because of
the high abundance of the galaxies.) Also shown (black solid curves) is a
model with no UVBG fluctuations using the best estimate LAE bias factor from
Ouchi et al. (2018) of $b_{L}=4.1$. The light curve does not include redshift
space distortions, which enhance the angular correlations on large angular
scales.222This unintuitive result perhaps deserves some explanation. For small
angular separations, the displacement between most pairs lies nearly along the
line of sight, so that $\mu\approx 1$ and
$\xi(r,\mu,z)\rightarrow\xi(r,1,z)=\xi_{0}(r,z)+\xi_{2}(r,z)+\xi_{4}(r,z)$.
For large angular separations, the displacements between pairs are mainly
orthogonal to the line of sight, so that $\mu\approx 0$ and
$\xi(r,\mu,z)\rightarrow\xi(r,0,z)=\xi_{0}(r,z)-\xi_{2}(r,z)/2+3\xi_{4}(r,z)/8$.
Without redshift space distortions $\xi_{2}=\xi_{4}=0$, and $\xi=\xi_{0}$.
Whilst allowing for redshift space distortions boosts $\xi_{0}$, the boost is
largely cancelled by $\xi_{2}(<0)$ in $\xi(r,1,z)$ on the scales of interest.
By contrast, $\xi_{2}$ boosts $\xi(r,0,z)$.

The UVBG fluctuations greatly boost the correlations on angular scales
$\theta<1000$ arcsec. The enhancement arises primarily from the source shot
noise contribution to the UVBG fluctuations. For a QSO lifetime of
$\tau_{Q}=10$ Myr, the resulting angular correlation function is slightly
smaller than for the steady-state limit. Decreasing the QSO lifetime further
reduces the strength of the correlations on large angular scales, with the
angle beyond which the correlations are weakened decreasing with decreasing
source lifetime (not shown). Reducing the LAE bias factor $b_{L}$ diminishes
the strength of the correlations on large angular scales, with little change
at small angles, where the UVBG fluctuation contribution dominates. It is
noteworthy that the UVBG fluctuations excluding the source shot noise
contribution results in a reduction in the strength of the correlations on
scales smaller than a few hundred arcsecs compared to the case with no UVBG
fluctuations (dotted cyan curve). This is because the effective bias factor
$b_{\mathrm{eff}}$ in Eq. (15) is reduced below $b_{L}$ by the UVBG
fluctuations for $\alpha<-1$: overdense regions give rise to an excess in IGM
attenuation that suppresses the detection of LAEs.

Figure 2: Best fit models to the measured LAE angular correlation functions at
$z=5.7$. Upper panel: Comparison with the data from Ouchi et al. (2010).
Models are shown without UVBG fluctuations ($g=0$; solid black curve), the
steady-state model (dashed blue curve), and allowing for UVBG fluctuations for
a QSO lifetime $\tau_{Q}=10$ Myr (dotted green curve). Lower panel: Comparison
with the data from Ouchi et al. (2018). Models are shown without UVBG
fluctuations ($g=0$; solid black curve) and the best-fitting steady-state
model for the most likely value of $g$, marginalised over $b_{L}$,
corresponding to $b_{L}=0.23$ and $g=0.59$ (dashed blue curve).

The models are fit to the data by minimising
$\chi^{2}(b_{L},g)=\Sigma_{i}[\omega_{m}(\theta_{i})-\omega_{d}(\theta_{i})]^{2}/\sigma_{d,i}^{2}$,
where $\omega_{m}$ is the model prediction for the angular correlation
function and $\omega_{d}$ and $\sigma_{d}$ are the measured value and its
error (Ouchi et al., 2018, 2020). Marginalised probability distributions for
$b_{L}$ or $g$ are computed by integrating the extraneous variable over the
likelihood ${\cal L}=\exp[-\chi^{2}(b_{L},g)/2]$.

Figure 3: The break down into component contributions to $\omega(\theta)$ for the steady-state model with $b_{L}=2.3$ and $g=0.59$. Shown are the total $\omega$ (solid black curve), the total $\omega_{\mathrm{nsn}}$ excluding the source shot noise contribution to the UVBG fluctuations (short-dashed black curve) and the separate contributions from the LAE system density correlations $\omega_{LL}$ (dotted green curve), the cross-correlation $\omega_{L\Gamma}$ between the LAE system density fluctuations and the UVBG fluctuations (dot-dashed magenta curve), the UVBG fluctuation correlations $\omega_{\Gamma\Gamma}$ (long-dashed blue curve), and the UVBG fluctuation correlations excluding the source shot noise contribution $\omega_{\Gamma\Gamma,\mathrm{nsn}}$ (dotted-long-dashed cyan curve).  $\tau_{Q}$ | sample | $b_{L}$ | $g$ | $\chi^{2}$
---|---|---|---|---
(Myr) |  |  |  |
– | O18 | $3.5^{+0.3}_{-0.4}\,\left({}^{+0.6}_{-0.8}\right)$ | $0$ | 1.40
– | O10 | $5.1\pm 0.3\,\left({}^{+0.6}_{-0.7}\right)$ | $0$ | 10.6
10 | O18 | $3.4^{+0.4}_{-0.6}\,\left({}^{+0.8}_{-2.0}\right)$ | $0.40^{+0.13}_{-0.35}\,\left(<0.69\right)$ | 1.40
10 | O10 | $5.1^{+0.3}_{-0.4}\,\left({}^{+0.6}_{-0.8}\right)$ | $<0.29\,\left(<0.49\right)$ | 10.1
$\infty$ | O18 | $3.4^{+0.5}_{-1.0}\,\left({}^{+0.7}_{-2.7}\right)$ | $0.59^{+0.09}_{-0.39}\left(<0.70\right)$ | 1.40
$\infty$ | O10 | $5.0^{+0.4}_{-0.3}\,\left({}^{+0.7}_{-0.8}\right)$ | $0.18\pm 0.17\left(<0.54\right)$ | 10.6
Table 1: Marginalised estimates for LAE bias $b_{L}$ and attenuation factor
$g$, without UVBG fluctuations (top section; $g=0$), for a QSO lifetime
$\tau_{Q}=$10 Myr (middle section) and the steady state model ($\infty$)
(bottom section). Estimates based on $w(\theta)$ are computed using the data
of Ouchi et al. (2010) (O10) and Ouchi et al. (2018) (O18). Ranges are given
for 68% (95%) confidence intervals or upper limits. The last column gives
$\chi^{2}$ for the best-fitting model.

Best fits to the measured correlations at $z=5.7$ (including the integral
correction for the finite survey areas), are shown in Fig. 2, using the
measured values from Ouchi et al. (2010) (O10) and Ouchi et al. (2018) (O18).
The point from O10 with lowest angular separation is excluded since the dark
matter density fluctuations become non-linear on this scale (Ouchi et al.,
2018). The best-fit models to the O10 result are found to nearly coincide with
the best-fitting model with no UVBG fluctuations ($g=0$). The best-fitting
model to the O18 result has $g=0$ for both the steady state and $\tau_{Q}=10$
Myr cases. Also shown is the best-fitting model fixing $g=0.59$, corresponding
to the most likely value for $g$ after marginalising over $b_{L}$ in the
steady-state model, as given in Table 1. For $g=0.59$, $\chi^{2}$ is minimum
at $b_{L}=2.3$. The difference from the overall best model curves shows that
much of the weight of the fits comes from the data at small angle separations.
This is illustrated in Fig. 3 for the model with $b_{L}=2.3$ and $g=0.59$. The
curve for $\omega_{\Gamma\Gamma}$ shows that the correlations become
increasingly dominated by the UVBG fluctuations on scales below 400 arcsecs. A
comparison between $\omega_{\mathrm{nsn}}$ and the full correlation function
$\omega$ shows the fluctuations are dominated by the contribution to the UVBG
fluctuations from the source shot noise on scales below 100 arcsecs, given by
the difference between $\omega_{\Gamma\Gamma}$ and
$\omega_{\Gamma\Gamma,\mathrm{nsn}}$. The density-dependent radiative transfer
selection effect is subdominant on these scales for this model.

(a)  (b)

(c)  (d)

Figure 4: Likelihood contour levels in $b_{L}$ and $g$ for fits to the
measured angular correlation functions at $z=5.7$. Results shown for UVBG
fluctuation models with $\tau_{Q}=10$ Myr to the data from a) O10 and b) O18,
and for the steady-state limit for the data from c) O10 and d) O18. (The
contour levels correspond to 1$\sigma$, 2$\sigma$ and 3$\sigma$ enclosed
probabilities.)

Likelihood contours for the fits are presented in Fig. 4. Allowing for the
UVBG fluctuations opens up solutions with low LAE bias $b_{L}$ and high $g$,
particularly for the O18 data. For both the O10 and O18 data, low $b_{L}$ and
high $g$ solutions become relatively more preferable for the steady state
model compared with the $\tau_{Q}=10$ Myr model.

The estimates for the LAE bias $b_{L}$ (marginalised over $g$) and for $g$
(marginalised over $b_{L}$) are displayed in Table 1 for a QSO lifetime
$\tau_{Q}=10$ Myr and in the steady-state limit. A bias of $b_{L}\approx 3.4$
is found for the O18 data, and $b_{L}\approx 5$ from the smaller O10 sample,
with agreement at the 68% confidence level regardless of the assumed QSO
lifetime, or excluding UVBG fluctuations ($g=0$). At the 95% confidence
limits, however, the solutions open up to allowing substantially reduced
values for $b_{L}$ with $g$ as high as $\sim 0.5$ for the O10 sample and $\sim
0.7$ for the O18 sample.

Because much of the weight for the increased range in $g$ arises from the
correlations at small angles, and the best-fitting models lie systematically
above the data points at small angles (see Fig. 2), we also provide results
using an alternative fitting approach based on allowing for possible
underestimates in the error bars following Hogg et al. (2010). For the steady-
state model fit to the O10 sample, the model parameter expectation values and
68% (95%) confidence intervals obtained are
$b_{L}=5.0^{+0.4}_{-0.6}\left({}^{+0.5}_{-1.2}\right)$ and
$g=0.32^{+0.24}_{-0.21}\left({}^{+0.32}_{-0.27}\right)$, while for the O18
sample, the expectation values obtained are
$b_{L}=3.2^{+0.6}_{-1.3}\left({}^{+0.8}_{-2.4}\right)$ and $g=0.46\pm
0.28\,\left({}^{+0.32}_{-0.41}\right)$. These are comparable to the values
found from the more direct maximum likelihood approach above, although with
somewhat extended error bars allowing for a larger contribution from UVBG
fluctuations. See the Appendix for further details.

##  4 Discussion

Whilst UVBG fluctuations are too small to affect the clustering of LAEs at
redshifts $z<5$, allowing for UVBG fluctuations at higher redshifts is found
to open up the LAE bias factors inferred from LAE clustering to a broader
range, especially at the 95% confidence interval (CI) level. Assuming no UVBG
fluctuations ($g=0$) at $z=5.7$, we find $b_{L}\simeq 3.5\pm 0.3$
(${}^{+0.6}_{-0.8}$, 95% CI) using the O18 LAE sample, and $b_{L}\simeq 5.1\pm
0.3$ (${}^{+0.6}_{-0.7}$, 95% CI) using the O10 sample. Allowing for UVBG
fluctuations broadens the error ranges to $b_{L}\simeq 3.4^{+0.5}_{-0.8}$
(${}^{+0.7}_{-2.3}$, 95% CI) for the O18 sample, and $b_{L}\simeq 5.0\pm 0.3$
($\pm 0.7$, 95% CI) for the O10 sample. At the 95% confidence level, the
angular correlations at angular separations smaller than 400 arcsecs may be
dominated by UVBG fluctuations, allowing for solutions with much reduced LAE
bias factors, especially for the larger O18 sample.

The evolution in the bias factors has been used to infer the nature of the
haloes in which LAE systems reside. For a fixed halo mass, the bias factor
increases rapidly with redshift. On this interpretation, the evolution in bias
factors suggests LAEs occupy haloes in the mass range
$10^{10}-10^{12}\,M_{\odot}$ at $z>2$ and may be the progenitors of present-
day massive elliptical galaxies (Ouchi et al., 2020). There is, however,
considerable scatter of unclear origin in the redshift trend of the bias
factors. Whilst contamination by randomly distributed sources would reduce the
inferred bias factors, it has also been suggested that the small volumes of
some of the surveys may result in a large scatter from cosmic variance,
particularly if a protocluster is in the field (Ouchi et al., 2010; Kusakabe
et al., 2018). A protocluster containing luminous photoionizing sources may
also increase the Ly$\alpha$ transmission of nearby LAEs, enhancing their
clustering. For the O18 samples at $z=5.7$, the findings here suggest low bias
factors $b_{L}\sim 2$ may be consistent with the angular correlation strengths
in the presence of a large UVBG fluctuation contribution. In this case, the
evolution of the bias factor of LAEs is consistent with a constant halo mass
of about $10^{10.5}{M_{\odot}}$ (see fig.16 of Ouchi et al., 2020).

The statistical limits on $g$ are very broad. The 68% confidence levels nearly
include $g=0$. For the O18 sample, the overall best-fitting models are for
$g=0$. Nonetheless, values as high as $g<0.7$ are acceptable at the 95%
confidence level for the O18 sample, and $g<0.5$ for the O10 sample.

The upper limits on $g$ of 0.5–0.7 correspond to upper limits on the blueward
velocity offset of $3-3.4\sigma_{l}$, where $\sigma_{l}$ is the line width of
the emission line, for the LAE emission compared with the Ly$\alpha$
attenuation edge of the surrounding gas. This corresponds to the higher values
found in the simulations of Laursen et al. (2011) for $\sigma_{l}\approx
100-150\,{\rm{km\,s^{-1}}}$ in an early reionization model
($z_{\mathrm{ri}}=10$), but are typical for late cosmic reionization
($z_{\mathrm{ri}}=6$). Values of $g\simeq 0.05-0.1$ correspond to blueward
displacements of the LAE emission line of $0.9-1.5\sigma_{l}$, typical for
early reionization.

Figure 5: Dependence of the angular correlation function on width of the
redshift shell. Upper panel: Angular correlation function without ($b=3.5$,
$g=0$) and with ($b=2.3$, $g=0.6$) UVBG fluctuations in the steady-state
limit, for shell widths of 28, 14 and 7 $h^{-1}\,$Mpc (comoving). The data
points are from Ouchi et al. (2018). Lower panel: The ratio of the angular
correlation functions for shell widths of 14 and 7 $h^{-1}\,$Mpc, to the
correlation functions for 28 $h^{-1}\,$Mpc.

The angular correlation function on small scales is suppressed by the finite
width of the redshift shell imposed by the narrow-band filter centred on the
Ly$\alpha$ wavelength for a given redshift. The width of the redshift shell at
$z=5.7$ corresponds to an averaging of the spatial correlation function up to
separations of nearly 30$h^{-1}$ Mpc (comoving). As a possible means of
distingishing between a correlation function dominated by the (biassed)
underlying dark matter density fluctuation field and by the UVBG fluctuations,
we show in Fig. 5 the effect of choosing narrower filters corresponding to a
half and a quarter of the actual filter width. Using the best-fit models
without and with UVBG fluctuations for the steady state model, the resulting
trends are virtually identical for the O10 data and exactly identical for the
O18 data (since the best-fit model allowing UVBG fluctuations has $g=0$). For
this reason, we show instead the trend for the best-fit model for the most
likely value of $g$, corresponding to $b_{L}=2.3$ and $g=0.59$, for the O18
data. For this model, UVBG fluctuations dominate the angular correlations on
angular scales smaller than 400 arcsecs, with the shot noise contribution
dominating on scales smaller than 100 arcsecs. Whilst reducing the thickness
of the redshift shell increases the strength of the correlations (upper
panel), the trends are nearly identical whether or not UVBG fluctuations
dominate the correlations (lower panel). A small difference may be
distinuished for the quarter-width filter case, but this would require
measuring the correlations on these scales to better than 10% accuracy.

On the other hand, the result shows that the rise in the strength of the
angular correlations is robust when sub-samples analysed are confined to
increasingly narrow redshift ranges. This provides a test of the possible
contribution of contaminating sources to the angular correlations. Although
spectroscopic follow-up suggests a contamination rate of only $\sim 8$% by
foreground objects (Ouchi et al., 2018), if they cluster more strongly than
the LAE systems, they would alter the expected scaling of the angular
correlation function with filter width.333If the foreground objects are weakly
or un-clustered, however, they only dilute the measured correlations and the
scaling would remain unaffected.

(a)  (b)

Figure 6: Contour levels of the spatial correlation function
$\xi(r_{||},r_{\perp})$, as a function of the separations $r_{||}$ along the
line of sight and $r_{\perp}$ transverse to the line of sight (in comoving
Mpc/ h). Left panel: The best fit model to the O18 data for no UVBG
fluctuations ($b_{L}=3.5$). Right panel: The steady-state UVBG fluctuation
model with $b_{L}=2.3$ and $g=0.59$. (The outermost contour level is 0.1, and
the levels increase inward by multiplicative steps of 1.89.)

The availability of a large spectroscopic sample would offer the possiblity of
measuring the spatial correlation function $\xi(r_{||},r_{\perp})$ as a
function of line-of-sight $r_{||}$ and transverse $r_{\perp}$ separations.
Fig. 6 shows $\xi(r_{||},r_{\perp})$ for the best-fit model without UVBG
fluctuations to the O18 data, with $b_{L}=3.5$ (left panel) and the steady-
state UVBG model with $b_{L}=2.3$ and $g=0.59$ (right panel). The angular
correlation functions (Fig. 2), are nearly identical. The UVBG fluctuations
tend to isotropise $\xi(r_{||},r_{\perp})$ compared with the expectation for a
biassed halo model. Distinguishing the two models, however, would require high
precision redshift determinations.

##  5 Conclusions

We summarise our principal conclusions:

1\. Allowing for UVBG fluctuations opens up a broader range of acceptable
models for the bias factor of LAEs. For the expected range in redshifted
foreground IGM Ly$\alpha$ photon attenuation as may arise from cosmic infall
around the LAE systems in a reionized IGM, UVBG fluctuations may non-
negligibly contribute to the clustering of LAEs at $z>5$ as measured by the
angular correlation function.

2\. Comparison with the measured angular correlation function of LAEs in the
SILVERRUSH Subaru survey at $z\simeq 5.7$ shows that, whilst the clustering is
statistically consistent with no contribution from UVBG fluctuations, the
measured clustering also supports models with a substantial contribution from
UVBG fluctuations. These include models in which the UVBG fluctuations
dominate the clustering signal over angular separations smaller than a few
hundred arcseconds, corresponding to projected separations smaller than
$15h^{-1}$ Mpc (comoving). On scales smaller than 100 arcsecs (about $3h^{-1}$
Mpc), the shot noise contribution to the UVBG fluctuations dominates the
correlations in these models. The scales over which shot noise dominates is
smaller for QSO lifetimes shorter than 10 Myr.

3\. It may be possible to distinguish models with and without a substantial
contribution from UVBG fluctuations to the clustering of LAEs if the angular
correlation function may be measured to 10% accuracy on angular scales 10–100
arcsec at $z\simeq 5.7$. Another approach may be to measure the spatial
correlations as a function of line-of-sight and transverse separations, as the
UVBG fluctuations tend to isotropise the correlations.

There is much room for improving the constraint on the contribution of UVBG
fluctuations to the clustering of LAE systems at high redshift. The estimate
provided here made several simplifying assumptions concerning the luminosity
distribution of the LAEs, their emission line widths and the velocity offset
of the emission region relative to the Ly$\alpha$ attenuation edge of the
surrounding gas. Our main conclusion is that UVBG fluctuations are
sufficiently large that they may comprise a significant contribution to the
overall LAE clustering strength at $z>5$ even after the IGM has been
reionized. UVBG fluctuations may continue to contribute to the clustering of
LAEs into the reionization epoch as well, requiring an extension to models of
the role of reionization on the LAE correlation function. More precise
measurements of the LAE correlation function would better constrain the UVBG
fluctuation contribution both during and after reionization completes, which
in turn may help clarify the properties of the gaseous environment of LAE
systems, their feedback on it, and of the nature of the LAE systems
themselves.

## Acknowledgements

We thank Joanne Cohn and Martin White for helpful discussions, and the
anonymous referee for comments that improved the presentation. TS gratefully
acknowledges support from the UK Science and Technology Facilities Council,
Consolidated Grant ST/R000972/1.

This research made use of the python packages emcee (Foreman-Mackey et al.,
2013) and corner.py (Foreman-Mackey, 2016).

## Data Availability

All data used in this paper are publicly available from the cited references.

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##  Appendix A Alternative error estimator

Following Hogg et al. (2010), we define the likelihood as

$\ln
p(\\{\omega_{d,n}\\}|\\{\theta_{n}\\},b_{L},g,f)=-\frac{1}{2}\sum_{n}\left[\frac{(\omega_{d,n}-\omega_{m,n})^{2}}{s_{n}^{2}}+\ln(2\pi
s_{n}^{2})\right]\textrm{,}$ (21)

where

$s_{n}^{2}=\sigma^{2}_{n}+f^{2}\omega_{m,n}$ (22)

for model correlation $\omega_{m,n}$ at angle $\theta_{n}$, measured
correlations $\omega_{d,n}$ with error $\sigma_{n}$, and $f$ is a factor
giving the fractional variance added in the noise model. Due to the dynamical
range of the data, adding a single value to the variance would overestimate
the error on the data points with small correlation strengths. Instead the
variance at any give point is increased by a fraction of the correlation
strength at the same point as predicted by the model. We use a uniform prior
on the logarithm of $f$ instead of $f$ itself to force $f$ to be non-negative.

Figure 7: MCMC parameters fits for the O10 sample. Lower left panel:
Likelihood function for $b_{L}$ and $g$, marginalised over $f$. Upper panel:
Probability density for $b_{L}$, marginalised over $g$ and $f$. Lower right
panel: Probability density for $g$, marginalised over $b_{L}$ and $f$. The
dashed lines in the probability distribution plots indicate the expectation
values and 68% confidence intervals.

Figure 8: As for Fig. 7, but for the O18 sample.

We compute the marginalised expected values for $b_{L}$ and $g$ using the
Markov Chain Monte Carlo (MCMC) python module emcee444
https://emcee.readthedocs.io/en/develop/ . The resulting joint contour plots
for $b_{L}$ and $g$, marginalised over $f$, and the marginalised distributions
for $b_{L}$ and $g$ are shown in Figs. 7 and 8 for the O10 and O18 samples,
respectively. The plots were produced using the python module corner.py555
https://github.com/dfm/corner.py .