# The Dual Degree Cech Bifiltration

Morten Brun1 1 Department of Mathematics, University of Bergen, Norway
<EMAIL_ADDRESS>
In topological data analysis (TDA), a longstanding challenge is to recognize
underlying geometric structures in noisy data. One motivating examples is the
shape of a point cloud in Euclidean space given by image patches [9]. Carlsson
et al. [5] proposed a method to detect topological features in point clouds by
first filtering by density and then applying persistent homology. Later more
refined methods have been developed, such as the degree Rips complex [10] of
Lesnick and Wright and the multicover bifiltration [2, 6, 7, 12]. In this
paper we introduce the dual Degree Cech bifiltration, a Prohorov stable
bicomplex of a point cloud in a metric space with the point cloud itself as
vertex set. It is of the same homotopy type as the Measure Dowker bifiltration
[8, 14] but it has a different vertex set.

The dual Degree Cech bifiltration can be constructed both in an ambient and an
intrinsic way. The intrinsic dual Degree Cech bifiltration is
$(1,2)$-intereaved with the ambient dual Degree Cech bifiltration in the
distance parameter. This interleaving can be used to leverage a stability
result for the intrinsically defined dual Degree Cech bifiltration. Such a
stability result recently occured in work by Hellmer and Spaliński [8, Theorem
4.2].

##  1\. Introduction

Let $\mu$ be a measure with support $S(\mu)$ on the Borel $\sigma$-algebra of
a metric space $M$ with metric $d\colon M\times M\to[0,\infty]$. The dual
degree Cech bifiltration of $d$ and $\mu$ is the family
$\\{DC(d,\mu)_{m,r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}$ of simplicial
complexes where

$DC(d,\mu)_{m,r}=\\{\tau\in\Delta(S(\mu))\mid\exists\,x\in M\text{ with
}\tau\subseteq B_{d}(x,r)\text{ and }\mu(B_{d}(x,r))\geq m\\}.$

Here $\Delta(S(\mu))$ is the set of finite non-empty subsets of $S(\mu)$ and
$B_{d}(x,r)$ is the closed ball centered at $x$ with radius $r$, that is,
$B_{d}(x,r)=\\{y\in M\mid d(x,y)\leq r\\}$. The vertex sets of these
simplicial complexes is the support $S(\mu)$ of $\mu$.

Observe that if $\mu$ is the counting measure of a finite subset $S$ of $M$
then $DC(d,\mu)_{1,r}$ is the ordinary Cech complex of the finite subset
$S=S(\mu)$ of the metric space $M$ at filtration degree $r$.

Let us refer to Definition 2.10 for the concept of homotopy weak interleaving
distance and to Definition 6.1 for the concept of Prohorov distance appearing
in the next Theorem. The concept of homotopy weak interleaving distance is a
variation on the usual concept of interleaving distance for bifiltrations. One
justification for the homotopy weak interleaving distance is that interleaving
distance between homology persistence modules is bounded above by both
homotopy weak interleaving distance and interleaving distance.

The following result states that the dual Degree Cech bifiltration is Prohorov
stable.

######  Theorem 1.1.

If $\mu_{0}$ and $\mu_{1}$ are two Borel measures on a separable metric space
$M$, then the homotopy weak interleaving distance between the bifiltered
simplicial complexes $DC(d,\mu_{0})$ and $DC(d,\mu_{1})$ is bounded above by
the Prohorov distance between $\mu_{0}$ and $\mu_{1}$.

The dual degree Cech bifiltration is closely related to the measure
bifiltration
$\\{\mathcal{M}(d,\mu)_{m,r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}$
associated to a metric $d$ on a set $M$ and a Borel measure $\mu$ on $M$. The
space $\mathcal{M}(d,\mu)_{m,r}$ consists of the points $x$ in $M$ satisfying
that the ball $B_{d}(x,r)$ has measure at least $m$. That is,

$\mathcal{M}(d,\mu)_{m,r}=\\{x\in M\mid\mu(B_{d}(x,r))\geq m\\}.$

This definition appears several places in the literature, see for example [8,
Definition 2.6] or [2, Definition 2.6] where it is denoted
$\mathcal{B}(\mu)_{m,r}$ and $\mathcal{B}(M)_{m,r}$ respectively. By [2,
Theorem 3.1], given Borel measures $\mu$ and $\eta$ on a metric space $M$ with
metric $d$, the interleaving distance of the bifiltered topological spaces
$\mathcal{M}(d,\mu)$ and $\mathcal{M}(d,\eta)$ is bounded above by the
Prohorov distance between $\mu$ and $\eta$. That is, the measure bifiltration
is Prohorov stable.

######  Definition 1.2.

Given a subset $X$ of a metric space $M$ with metric $d$ and $r\geq 0$, the
$r$-offset $X^{r}$ is the union of all closed balls $B_{d}(x,r)$ of radius $r$
centered at a point $x$ in $X$.

######  Theorem 1.3.

Let $M$ be a separable metric space with metric $d$ and with a Borel measure
$\mu$. The nerve of the cover of $\mathcal{M}(d,\mu)_{m,r}$, consisting of the
closed subsets $B_{d}(x,r)\cap\mathcal{M}(d,\mu)_{m,r}$ given by intersections
with balls of radius $r$ centered at points $x$ in the $r$-offset
$\mathcal{M}(d,\mu)^{r}_{m,r}$, is homotopy equivalent to the dual degree Cech
bifiltration $DC(d,\mu)_{m,r}$. Such homotopy equivalences can be chosen to be
compatible with the bifiltration.

Unfortunately Theorem 1.3 contains no information about the homotopy type of
the dual degree Cech bifiltration.

Restricting $d$ and $\mu$ to the support of $\mu$ we obtain an intrinsic
version of the dual degree Cech bifiltration. Like the usual intrinsic Cech
filtration is multiplicatively interleaved with the (ambient) Cech filtration,
the intrinsic degree Cech bifiltration is multiplicatively interleaved with
the (ambient) dual degree Cech bifiltration.

######  Theorem 1.4.

Let $M$ be a separable metric space with metric $d$ and let $\mu$ be a Borel
measure on $M$ with support $S$. Let $d|_{S\times S}$ be the restriction of
$d$ to $S$ and let $\mu|_{S}$ be the restriction of $\mu$ to $S$. For all
$(m,r)\in\mathbb{R}\times[0,\infty]$, there are inclusions

$DC(d|_{S\times S},\mu|_{S})_{m,r}\subseteq DC(d,\mu)_{m,r}\subseteq
DC(d|_{S\times S},\mu|_{S})_{m,2r}.$

We refer to Definition 2.8 for the concept of a weak interleaving of
bifiltered simplicial complexes and to Definition 2.7 for the concept of a
forward shift map occurring below. Combining Theorems 1.1 and 1.4 we obtain
the following result that is closely related to [8, Theorem 4.2].

######  Theorem 1.5.

If $\mu$ and $\eta$ are two Borel measures with compact supports on a
separable metric space $M$ of Prohorov distance at less than $\varepsilon$,
then there are maps $\pi_{1}\colon S(\mu)\to S(\eta)$ and $\pi_{0}\colon
S(\eta)\to S(\mu)$ which for all $(m,r)\in\mathbb{R}\times[0,\infty]$ induce
maps

$\pi_{1}\colon DC(d_{S(\mu)\times S(\mu)},\mu|_{S(\mu)})_{m,r}\to
DC(d_{S(\eta)\times
S(\eta)},\eta|_{S(\eta)})_{m-\varepsilon,2(r+\varepsilon)}$

and

$\pi_{0}\colon DC(d_{S(\eta)\times S(\eta)},\eta|_{S(\eta)})_{m,r}\to
DC(d_{S(\mu)\times S(\mu)},\mu|_{S(\mu)})_{m-\varepsilon,2(r+\varepsilon)}.$

These maps form a weak interleaving of the intrinsic dual degree Cech
bifiltrations $DC(d_{S(\mu)\times S(\mu)},\mu|_{S(\mu)})$ and
$DC(d_{S(\eta)\times S(\eta)},\eta|_{S(\eta)})$ with respect to the forward
shift map $(m,r)\mapsto(m-\varepsilon,2(r+\varepsilon))$.

The rest of this paper is organized as follows. In Section 2 we introduce
variations on the concepts of bifiltrations and interleavings and state some
results about them. In Section 3 we extend results about Dowker duality to our
context. In Section 4 we prove that the passage from set bifiltrations to
bifiltered simplicial complexes is stable with respect to our notion of
interleaving distance. In Section 5 we use Dowker duality to prove that a
restriction of the vertex set of the dual Dowker nerve to the support of a
measure preserves homotopy type. In Section 6 we show that Prohorov distance
and interleaving distance are related. Finally, in Section 7 we prove Theorem
1.5.

##  2\. Density Dependent Bifiltration

In this section we introduce notation around bifiltrations and we state some
results about them.

### Bifiltered Simplicial Complexes

Given a set $X$, we let $\Delta(X)$ be the collection of all finite non-empty
subsets of $X$, considered as a partially ordered set under inclusion.

######  Definition 2.1.

A simplicial complex on the vertex set $X$ is a subset $K$ of $\Delta(X)$ with
the property that if $\sigma\in K$ and $\tau\subseteq\sigma$ is a subset of
the set $\sigma$, then $\tau\in K$.

We allow simplicial complexes to have ghost vertices, that is, we do not
require that the singleton set $\\{x\\}$ is a member of $K$ for every $x\in
X$. Note that $\Delta(X)$ itself is a simplicial complex, referred to as the
simplex with the vertex set $X$. Let us stress that for us, a simplicial
complex is a collection of non-empty sets.

######  Definition 2.2.

A bifiltered simplicial complex on the vertex set $X$ is a collection

$K=\\{K_{m,r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}$

of simplicial complexes with vertex set $X$ such that $K_{m,r}\subseteq
K_{m^{\prime},r^{\prime}}$ whenever $m^{\prime}\leq m$ and $r\leq r^{\prime}$.

In the terminology of categories and functors, a bifiltered simplicial complex
$K$ is a functor

$K\colon\mathbb{R}^{\mathrm{op}}\times[0,\infty]\to\operatorname{sCx},\quad(m,r)\mapsto
Nf_{m,r}.$

Here $\mathbb{R}^{\mathrm{op}}$ is the opposite of the category $\mathbb{R}$,
where $\mathbb{R}$ is the category with the elements of $\mathbb{R}$ as
objects and with exactly one morphism $s\to t$ for each pair
$(s,t)\in\mathbb{R}\times\mathbb{R}$ with $s\leq t$. Similarly the category
$[0,\infty]$ has one object for each element of $[0,\infty]$.

The objects of the category $\operatorname{sCx}$ are simplicial complexes, and
a morphism $\phi\colon A\to B$ in $\operatorname{sCx}$ from a simplicial
complex $A$ on the vertex set $X$ to a simplicial complex $B$ on the vertex
set $Y$ consists of a function $\phi\colon X\to Y$ with the property that if
$\sigma\subseteq X$ is a member of $A$, then its image
$\phi(\sigma)=\\{\phi(x)\,|\,x\in\sigma\\}$ under $\phi$ is a member of $B$.
Composition of morphisms of simplicial complexes is given by composition of
functions.

######  Definition 2.3.

Let $K^{0}$ and $K^{1}$ be bifiltered simplicial complexes on the vertex sets
$X_{0}$ and $X_{1}$ respectively. A weak $\varepsilon$-interleaving of $K^{0}$
and $K^{1}$ is a pair maps $\pi_{1}\colon X_{0}\to X_{1}$ and $\pi_{0}\colon
X_{1}\to X_{0}$ satisfying the following conditions for every
$(m,r)\in\mathbb{R}\times[0,\infty]$:

  1. (1)

The map $\pi_{1}$ induces a map $\pi_{1}\colon K^{0}_{m,r}\to
K^{1}_{m-\varepsilon,r+\varepsilon}$.

  2. (2)

The map $\pi_{0}$ induces a map $\pi_{0}\colon K^{1}_{m,r}\to
K^{0}_{m-\varepsilon,r+\varepsilon}$.

  3. (3)

The map $\pi_{1}\pi_{0}\colon K^{1}_{m,r}\to
K^{1}_{m-2\varepsilon,r+2\varepsilon}$ is homotopic to the inclusion map.

  4. (4)

The map $\pi_{0}\pi_{1}\colon K^{0}_{m,r}\to
K^{0}_{m-2\varepsilon,r+2\varepsilon}$ is homotopic to the inclusion map.

If there exists a weak $\varepsilon$-interleaving of $K^{0}$ and $K^{1}$, we
say that are weakly $\varepsilon$-interleaved.

Let us emphasize that the difference between a weak interleaving and an
interleaving as it appear in the literature is that the maps
$\pi_{1}\circ\pi_{0}$ and $\pi_{0}\circ\pi_{1}$ are not required to be
inclusion maps. We have introduced weak interleavings because they naturally
occur in our context.

######  Definition 2.4.

The weak interleaving distance $d_{WI}(K^{0},K^{1})$ between two bifiltered
simplicial complexes $K^{0}$ and $K^{1}$ is the infimum of all
$\varepsilon\geq 0$ such that $K^{0}$ and $K^{1}$ are $\varepsilon$-weakly
interleaved.

Since homotopic maps induce identical maps on homology groups, after applying
homology the difference between weak interleavings and interleavings
disappears. In particular, homology is non-increasing under with respect to
weak interleaving distance of bifiltrations and interleaving distance of
persistence modules.

A bifiltered topological space $X$ consists of a collection
$\\{X_{m,r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}$ of topological spaces
such that $X_{m,r}\subseteq X_{m^{\prime},r^{\prime}}$ whenever
$m^{\prime}\leq m$ and $r\leq r^{\prime}$. A morphism $f\colon X\to Y$ of
bifiltered topological spaces consists of continuous maps $f_{m,r}\colon
X_{m,r}\to Y_{m,r}$ such that for all $m^{\prime}\leq m$ and $r\leq
r^{\prime}$, the following diagram commutes:

${X_{m,r}}$${Y_{m,r}}$${X_{m^{\prime},r^{\prime}}}$${Y_{m^{\prime},r^{\prime}}.}$$\scriptstyle{f_{m,r}}$$\scriptstyle{f_{m^{\prime},r^{\prime}}}$

######  Definition 2.5.

A morphism $f\colon K\to L$ is a homotopy equivalence if for each
$(m,r)\in\mathbb{R}\times[0,\infty]$, the induced map
$|f_{m,r}|\colon|K_{m,r}|\to|L_{m,r}|$ of geometric realizations is a homotopy
equivalence. If a (possibly infinite) zig-zag chain of homotopy equivalences
between $K$ and $L$ exists, then $K$ and $L$ are homotopy equivalent.

######  Remark 2.6.

A homotopy equivalence in the above sense is often called an objectwise
homotopy equivalence. We refrain from going into details about this and refer
the interested reader to [2, Definition 2.35].

######  Definition 2.7.

A forward shift map is an order preserving map of the form
$\alpha\colon\mathbb{R}^{\mathrm{op}}\times[0,\infty]\to\mathbb{R}^{\mathrm{op}}\times[0,\infty]$
with the property that $\alpha(m,r)\geq(m,r)$ for all
$(m,r)\in\mathbb{R}^{\mathrm{op}}\times[0,\infty]$.

Notice that if $\alpha(m,r)=(\alpha_{0}(m,r),\alpha_{1}(m,r))$ then
$(m,r)\leq\alpha(m,r)$ means that $m\geq\alpha_{0}(m,r)$ and
$r\leq\alpha_{1}(m,r)$.

######  Definition 2.8.

Let $K=\\{K_{m,r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}$ and
$L=\\{L_{m,r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}$ be bifiltrations and
let
$\alpha,\beta\colon\mathbb{R}^{\mathrm{op}}\times[0,\infty]\to\mathbb{R}^{\mathrm{op}}\times[0,\infty]$
be forward shift maps. A weak $(\alpha,\beta)$-interleaving between $K$ and
$L$ consists of maps $\pi_{m,r}\colon K_{m,r}\to L_{\alpha(m,r)}$ and
$\psi_{m,r}\colon L_{m,r}\to K_{\beta(m,r)}$ of simplicial complexes for every
$(m,r)\in\mathbb{R}\times[0,\infty]$ such that
$\psi_{\alpha(m,r)}\circ\pi_{m,r}$ and $\pi_{\beta(m,r)}\circ\psi_{m,r}$ are
homotopic to the inclusions $K_{m,r}\to K_{\beta\circ\alpha(m,r)}$ and
$L_{m,r}\to L_{\alpha\circ\beta(m,r)}$ respectively.

If a weak $(\alpha,\beta)$-interleaving exists, the bifiltrations $K$ and $L$
are $(\alpha,\beta)$-weakly interleaved.

If $\alpha=\beta$ are given by $\alpha(m,r)=(m-\varepsilon,r+\varepsilon)$ for
some $\varepsilon\geq 0$, then we say that $K$ and $L$ are
$\varepsilon$-weakly interleaved.

######  Definition 2.9.

Let $K$ and $L$ be bifiltrations and let $\alpha,\beta$ be forward shift maps.
We say that $K$ and $L$ are homotopy $(\alpha,\beta)$-weakly interleaved if
there exists bifiltrations $K^{\prime}$ and $L^{\prime}$ that are homotopy
equivalent to $K$ and $L$ respectively and a weak
$(\alpha,\beta)$-interleaving between $K^{\prime}$ and $L^{\prime}$.

If $\alpha=\beta$ are given by $\alpha(m,r)=(m-\varepsilon,r+\varepsilon)$ for
some $\varepsilon\geq 0$, then we say that $K$ and $L$ are homotopy
$\varepsilon$-weakly interleaved.

######  Definition 2.10.

The homotopy weak interleaving distance between two bifiltrations $K$ and $L$
is the infimum of all $\varepsilon\geq 0$ such that $K$ and $L$ are homotopy
$\varepsilon$-weakly interleaved.

Note that homology is also non-increasing under with respect to homotopy weak
interleaving distance of bifiltrations and interleaving distance of
persistence modules.

### Set Bifiltrations

Given a set $X$, let $\Delta(X)^{\mathrm{op}}$ be the opposite poset of the
poset $\Delta(X)$ of finite subsets of $X$ ordered by inclusion. In this
convention a function
$f\colon\Delta(X)^{\mathrm{op}}\times[0,\infty]\to[0,\infty]$ is order
preserving if and only if $f(\sigma,r)\leq f(\sigma^{\prime},r^{\prime})$
whenever $\sigma^{\prime}\subseteq\sigma$ in $\Delta(X)$ and $r\leq
r^{\prime}$ in $[0,\infty]$.

######  Definition 2.11.

A set bifiltration on a set $X$ is an order preserving function of the form
$f\colon\Delta(X)^{\mathrm{op}}\times[0,\infty]\to[0,\infty]$.

######  Definition 2.12.

Let $f_{0}$ and $f_{1}$ be set bifiltrations on the sets $X_{0}$ and $X_{1}$
respectively. A weak $\varepsilon$-interleaving of $f_{0}$ and $f_{1}$ is a
pair of maps $\pi_{1}\colon X_{0}\to X_{1}$ and $\pi_{0}\colon X_{1}\to X_{0}$
satisfying the following conditions for every
$(\sigma,r)\in\Delta(X_{0})^{\mathrm{op}}\times[0,\infty]$:

  1. (1)

$f_{0}(\sigma,r)\leq f_{1}(\pi_{1}(\sigma),r+\varepsilon)+\varepsilon$ for
every $(\sigma,r)\in\Delta(X_{0})^{\mathrm{op}}\times[0,\infty]$.

  2. (2)

$f_{1}(\sigma,r)\leq f_{0}(\pi_{0}(\sigma),r+\varepsilon)+\varepsilon$ for
every $(\sigma,r)\in\Delta(X_{1})^{\mathrm{op}}\times[0,\infty]$.

  3. (3)

$f_{0}(\sigma,r)\leq
f_{0}(\sigma\cup\pi_{0}\pi_{1}(\sigma),r+2\varepsilon)+2\varepsilon$ for every
$(\sigma,r)\in\Delta(X_{0})^{\mathrm{op}}\times[0,\infty]$.

  4. (4)

$f_{1}(\sigma,r)\leq
f_{1}(\sigma\cup\pi_{1}\pi_{0}(\sigma),r+2\varepsilon)+2\varepsilon$ for every
$(\sigma,r)\in\Delta(X_{1})^{\mathrm{op}}\times[0,\infty]$.

If a weak $\varepsilon$-interleaving of $f_{0}$ and $f_{1}$ exists, we say
that are $\varepsilon$-weakly interleaved.

######  Definition 2.13.

The interleaving distance $d_{WI}(f_{0},f_{1})$ between two set bifiltrations
$f_{0}$ and $f_{1}$ is the infimum of all $\varepsilon\geq 0$ such that
$f_{0}$ and $f_{1}$ are $\varepsilon$-weakly interleaved.

######  Definition 2.14.

The bifiltered simplicial complex associated to a set bifiltration $f$ on a
set $X$ is the bifiltration of $\Delta(X)$ given by the collection
$Nf=\\{Nf_{m,r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}$ of simplicial
complexes where a finite subset $\sigma$ of $X$ is in $Nf_{m,r}$ if and only
if $f(\sigma,r)\geq m$.

Notice that in Definition 2.14, we have $Nf_{0,r}=\Delta(X)$ for every
$r\in[0,\infty]$. In section 4 we prove the following result stating that the
interleaving distance is non-increasing under the associated bifiltered
complex construction:

######  Proposition 2.15.

The interleaving distance is non-increasing under the associated bifiltered
simplicial complex construction, that is if $f_{0}$ and $f_{1}$ are set
bifiltrations, then

$d_{WI}(Nf_{0},Nf_{1})\leq d_{WI}(f_{0},f_{1}).$

We are going to use this result throughout the paper when we consider the
interleaving distance of bifiltered simplicial complexes associated to set
bifiltrations. In particular for set bifiltrations arising from measures on
metric spaces.

### Dowker Dual Bifiltrations

######  Definition 2.16.

A Dowker dissimilarity $\Lambda$ consists of two sets $X$ and $Y$ and a
function

$\Lambda\colon X\times Y\to[0,\infty].$

######  Definition 2.17.

Given a Dowker dissimilarity $\Lambda\colon X\times Y\to[0,\infty]$ and $x\in
X$ and $r\in[0,\infty]$, the $\Lambda$-ball centered at $x$ with radius $r$ is
the set

$B_{\Lambda}(x,r)=\\{y\in Y\mid\Lambda(x,y)\leq r\\}.$

Given a finite subset $\sigma$ of $X$ and $r\in[0,\infty]$, the $\Lambda$-ball
centered at $\sigma$ with radius $r$ is the intersection

$B_{\Lambda}(\sigma,r)=\bigcap_{x\in\sigma}B_{\Lambda}(x,r)$

of all $\Lambda$-balls centered at points in $\sigma$.

######  Definition 2.18.

Let $\Lambda\colon X\times Y\to[0,\infty]$ be a Dowker dissimilarity and let
$f$ be a set bifiltration on $X$. The $\Lambda$-Dowker dual bifiltration
$D_{\Lambda}Nf$ of the bifiltered simplicial complex $Nf$ associated to $f$ is
the collection
$\\{(D_{\Lambda}Nf)_{m,r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}$ of
simplicial complexes with $Y$ as vertex set defined as follows: a finite
subset $\tau$ of $Y$ is in $(D_{\Lambda}Nf)_{m,r}$ if and only there exists
$x\in X$ with $f(\\{x\\},r)\geq m$ and $\tau\subseteq B_{\Lambda}(x,r)$.

######  Remark 2.19.

If $\Lambda^{T}\colon Y\times X\to[0,\infty]$ is the transpose of $\Lambda$,
given by $\Lambda^{T}(y,x)=\Lambda(x,y)$, then $\tau\subseteq
B_{\Lambda}(x,r)$ if and only if $x\in B_{\Lambda^{T}}(\tau,r)$.

######  Definition 2.20.

A Dowker bifiltration $(\Lambda,f)$ is a pair of a Dowker dissimilarity
$\Lambda\colon X\times Y\to[0,\infty]$ and a set bifiltration $f$ on $X$ with
the property that for every finite subset $\sigma$ of $X$ and $r\in[0,\infty]$
with $f(\sigma,r)>0$ we have that $B_{\Lambda}(\sigma,r)$ is non-empty.

The following result is a direct consequence of Proposition 3.2.

######  Proposition 2.21.

For every Dowker bifiltration $(\Lambda,f)$, the $\Lambda$-dual bifiltration
$D_{\Lambda}Nf$, of the bifiltered simplicial complex $Nf$ associated to $f$,
is homotopy equivalent to $Nf$.

##  3\. Rectangle Bifiltration

In this section we extend Dowker duality from Dowker dissimilarities to Dowker
bifiltrations. The main ingredient in this extension is the concept of a
rectangle bifiltration.

######  Definition 3.1.

Let $\Lambda\colon X\times Y\to[0,\infty]$ be a Dowker dissimilarity and let
$f$ be a set bifiltration on $X$. The rectangle bifiltration of $(\Lambda,f)$
is the collection $\\{E(\Lambda,f)_{m,r}\\}_{(m,r)}$ of simplicial complexes
with $X\times Y$ as vertex set where a finite subset $U$ of $X\times Y$ is in
$E(\Lambda,f)_{m,r}$ if and only both the projection of $U$ to $X$ is in
$Nf_{m,r}$, the projection of $U$ to $Y$ is in $D_{\Lambda}Nf_{m,r}$ and
$\Lambda(x,y)\leq r$ for every $(x,y)\in U$.

######  Proposition 3.2.

Let $(\Lambda,f)$ be a Dowker bifiltration. If $\Lambda$ is of the form
$\Lambda\colon X\times Y\to[0,\infty]$ then, for all
$(m,r)\in\mathbb{R}\times[0,\infty]$, the projections $\pi_{X}\colon X\times
Y\to X$ and $\pi_{Y}\colon X\times Y\to Y$ induce homotopy equivalences
$\pi_{X}\colon E(\Lambda,f)_{m,r}\to Nf_{m,r}$ and $\pi_{Y}\colon
E(\Lambda,f)_{m,r}\to D_{\Lambda}Nf_{m,r}$.

The proof we present of this proposition is a slight modification of the proof
of [4, Theorem 4.3]. In order to present the proof we repeat some background
and notation from [4].

If $K$ is a simplicial complex, then $K$ is a partially ordered set under
inclusion, and its order complex is the barycentric subdivision of $K$. It is
well-known that the barycentric subdivision of $K$ is homeomorphic to $K$
itself [13, 3.3.9].

Let $\pi:(K,V)\to(K^{\prime},V^{\prime})$ be a simplicial map. The fiber of
$\pi$ over $\sigma\in K^{\prime}$ is the simplicial complex
$\pi/\sigma=\\{\tau\in K\,|\,\pi(\tau)\subseteq\sigma\\}$.

######  Theorem 3.3 ([1, Thm. 10.5], [11, Prop. 1.6]).

Let $\pi:(K,V)\to(K^{\prime},V^{\prime})$ be a simplicial map. If the fiber
$\pi/\sigma$ is a contractible simplicial complex for every $\sigma\in
K^{\prime}$, then $\pi$ induces a homotopy equivalence
$|\pi|:|K|\to|K^{\prime}|$.

The nerve of a collection $\mathcal{U}$ of sets is the simplicial complex with
vertex set $\mathcal{U}$ and simplices given by the finite subsets of
$\mathcal{U}$ with non-empty intersection.

######  Theorem 3.4 (Borsuk Nerve Theorem [3, Cor. 9.3]).

Let $\mathcal{U}$ be a cover of a simplicial complex $K$ by simplicial
subcomplexes. If every finite intersection of elements in $\mathcal{U}$ is
contractible, then the geometric realization of the nerve of $\mathcal{U}$ is
homotopy equivalent to the geometric realization of $K$

###### Proof of Proposition 3.2.

Let $\pi_{X}\colon E(\Lambda,f)_{m,r}\to Nf_{m,r}$ be the map of simplicial
complexes induced by the projection $X\times Y\to X$. We first prove that
$\pi_{X}$ is a homotopy equivalence. Given a finite subset $\sigma$ of $X$, we
let

$Y(\sigma)=B_{\Lambda}(\sigma,r)=\\{y\in Y\,\mid\,\Lambda(x,y)\leq r\text{ for
all }x\in\sigma\\},$

and we let $F(\sigma)=\Delta(\sigma\times Y(\sigma))$ be the set of all finite
non-empty subsets of $\sigma\times Y(\sigma)$. We first prove that the fibres
of the map $\pi_{X}\colon E(\Lambda,f)_{m,r}\to Nf_{m,r}$ are contractible.

Let $\sigma\in Nf_{m,r}$. By assumption $Y(\sigma)$ and $F(\sigma)$ are non-
empty. The fibre $\pi_{X}/\sigma$ is the union

$\pi_{X}/\sigma=\bigcup_{\tau\subseteq\sigma}F(\tau).$

We will show that the nerve of the cover
$\\{F(\tau)\,\mid\,\tau\subseteq\sigma\\}$ of $\pi_{X}/\sigma$ is both
homotopy equivalent to $\pi_{X}/\sigma$ and contractible. In order to see that
the nerve of the cover is homotopy equivalent to $\pi_{X}/\sigma$, it suffices
to show that the cover is a good cover.

In order to see that the cover is good, we note that

$\bigcap_{i=1}^{k}F(\tau_{i})=\Delta(\bigcap_{i=1}^{k}\tau_{i}\times
Y(\tau_{i})).$

This is a simplex, and simplices are contractible. Thus,
$\\{F(\tau)\,\mid\,\tau\subseteq\sigma\\}$ is a good cover.

In order to show that the nerve of the cover
$\\{F(\tau)\,\mid\,\tau\subseteq\sigma\\}$ is contractible we show that it has
the vertex $F(\sigma)$ as a cone point. Let $\gamma$ be a simplex in the nerve
of the cover. Then $\gamma$ is of the form
$\gamma=\\{F(\tau_{1}),\dots,F(\tau_{k})\\}$ with
$\bigcap_{i=1}^{k}\tau_{i}\times Y(\tau_{i})$ non-empty. In particular
$\bigcap_{i=1}^{k}\tau_{i}$ is non-empty. Let $x\in\bigcap_{i=1}^{k}\tau_{i}$
and let $y\in Y(\sigma)$. Since $\tau_{i}\subseteq\sigma$ for every $i$, we
have that $y\in\bigcap_{i=1}^{k}Y(\tau_{i})$. We conclude that
$(x,y)\in(\sigma\times Y(\sigma))\cap\bigcap_{i=1}^{k}\tau_{i}\times
Y(\tau_{i})$ so that $\gamma\cup\\{\sigma\times Y(\sigma)\\}$ is a simplex in
the nerve of the cover. Thus, the nerve of the cover is contractible. We use
Theorem 3.3 to conclude that $\pi_{X}\colon E(\Lambda,f)_{m,r}\to Nf_{m,r}$ is
a homotopy equivalence.

We next prove that the map $\pi_{Y}\colon E(\Lambda,f)_{m,r}\to
D_{\Lambda}Nf_{m,r}$ is a homotopy equivalence. Given a finite subset $\tau$
of $Y$, we let

$X(\tau)=B_{\Lambda^{T}}(\tau,r)=\\{x\in Y\,\mid\,\Lambda(x,y)\leq r\text{ for
all }y\in\tau\\},$

and we let $G(\tau)$ be the set of all subsets of $X(\tau)\times\tau$. We
first prove that the fibres of the map $\pi_{Y}\colon E(\Lambda,f)_{m,r}\to
D_{\Lambda}Nf_{m,r}$ are contractible. Let $\tau\in D_{\Lambda}Nf_{m,r}$ and
let $x\in B$ with $f(\\{x\\},r)\geq m$ and $\tau\subseteq B_{\Lambda}(x,r)$.
Then $x\in X(\\{\tau\\})$ so $G(\tau)$ is non-empty. The fibre $\pi_{Y}/\tau$
is the union

$\pi_{Y}/\tau=\bigcup_{\sigma\subseteq\tau}G(\sigma).$

We will show that the nerve of the cover
$\\{G(\sigma)\,\mid\,\sigma\subseteq\tau\\}$ is both homotopy equivalent to
the fibre $\pi_{Y}/\tau$ and contractible. In order to see that the nerve of
the cover is homotopy equivalent to the fibre $\pi_{Y}/\tau$, it suffices to
show that the cover is a good cover.

In order to see that the cover is good, we note that

$\bigcap_{i=1}^{k}G(\sigma_{i})=\Delta(\bigcap_{i=1}^{k}X(\sigma_{i})\times\sigma_{i}).$

This is a simplex, and simplices are contractible. Thus, the cover is good.

In order to show that the nerve of the cover is contractible we show that it
has the vertex $G(\tau)$ as a cone point. Let $\tau$ be a simplex in the nerve
of the cover. Then $\tau$ is of the form
$\tau=\\{G(\sigma_{1}),\dots,G(\sigma_{k})\\}$ with
$\bigcap_{i=1}^{k}X(\sigma_{i})\times\sigma_{i}$ non-empty. In particular
$\bigcap_{i=1}^{k}\sigma_{i}$ is non-empty. Let
$y\in\bigcap_{i=1}^{k}\sigma_{i}$ and let $x\in X(\tau)$. Since
$\sigma_{i}\subseteq\tau$ for every $i$, we have that
$x\in\bigcap_{i=1}^{k}X(\sigma_{i})$. We conclude that
$(x,y)\in(X(\tau)\times\tau)\cap\bigcap_{i=1}^{k}X(\sigma_{i})\times\sigma_{i}$
so that $\gamma\cup\\{X(\tau)\times\tau\\}$ is a simplex in the nerve of the
cover. Thus, the nerve of the cover is contractible.

Finally, we use Theorem 3.3 to conclude that $\pi_{Y}\colon
E(\Lambda,f)_{m,r}\to D_{\Lambda}Nf_{m,r}$ is a homotopy equivalence. ∎

##  4\. Set bifiltration Distance

In this section we pursue the concept of interleaving of set bifiltrations and
prove Theorem 2.15 stating that the transition to the associated bifiltered
simplicial complexes is $1$-Lipschitz with respect to the interleaving
distance. We also consider interleavings of set bifiltrations with respect to
forward shift maps.

The following lemma illustrates the main idea of this section.

######  Lemma 4.1.

Let $f_{0}$ and $f_{1}$ be set bifiltrations on the sets $X_{0}$ and $X_{1}$
respectively and let $\varepsilon,\delta\in[0,\infty]$. If $\pi_{0}\colon
X_{1}\to X_{0}$ is a function such that

$f_{1}(\sigma,r)\leq f_{0}(\pi_{0}\sigma,r+\delta)+\varepsilon$

for all $\sigma\in\Delta(X_{1})$ and $r\in[0,\infty]$, then $\pi_{0}\colon
X_{1}\to X_{0}$ induces a map

$\pi_{0}\colon N(f_{1})_{m,r}\to N(f_{0})_{m-\varepsilon,r+\delta}.$

If $\pi^{\prime}_{0}\colon X_{1}\to X_{0}$ is another function such that
$f_{1}(\sigma,r)\leq
f_{0}(\pi^{\prime}_{0}\sigma\cup\pi_{0}\sigma,r+\delta)+\varepsilon$ for all
$\sigma\in\Delta(X)$ and $r\in[0,\infty]$ then $\pi_{0}^{\prime}$ induces a
map

$\pi^{\prime}_{0}\colon N(f_{1})_{m,r}\to N(f_{0})_{m-\varepsilon,r+\delta}$

that is contiguous to $\pi_{0}$.

###### Proof.

Let $\sigma\in N(f_{1})_{m,r}$. Then

$m\leq f_{1}(\sigma,r)\leq f_{0}(\pi_{0}\sigma,r+\delta)+\varepsilon.$

Rearranging the inequality we get $f_{0}(\pi_{0}\sigma,r+\delta)\geq
m-\varepsilon$, so $\pi_{0}\sigma\in N(f_{0})_{m-\varepsilon,r+\delta}$. A
similar argument shows if $\pi^{\prime}_{0}$ is a described in the statement,
then $\pi^{\prime}_{0}\sigma\cup\pi_{0}\sigma\in
N(f_{0})_{m-\varepsilon,r+\delta}$, so $\pi^{\prime}_{0}$ induces a map
$\pi^{\prime}_{0}\colon N(f_{1})_{m,r}\to
N(f_{0})_{m-\varepsilon,r+\varepsilon}$ that is contiguous to $\pi_{0}$. ∎

######  Corollary 4.2.

Let $f$ and $g$ be set bifiltrations on the same set $X$. Given
$\varepsilon>0$ such that $f(\sigma,r)\leq
g(\sigma,r+\varepsilon)+\varepsilon$ for every $\sigma\in\Delta(X)$ and
$r\in[0,\infty]$, we have that $Nf_{m,r}\subseteq
N\Lambda^{g}_{m-\varepsilon,r+\varepsilon}$.

###### Proof of Proposition 2.15.

It suffices to show that if $f_{0}$ and $f_{1}$ are $\varepsilon$-weakly
interleaved, then $Nf_{0}$ and $Nf_{1}$ are $\varepsilon$-weakly interleaved.
Let $X_{0}$ and $X_{1}$ be the sets underlying $f_{0}$ and $f_{1}$
respectively. Let $\pi_{0}\colon X_{1}\to X_{0}$ and $\pi_{1}\colon X_{0}\to
X_{1}$ be functions such that for every
$(\sigma,r)\in\Delta(X_{0})^{\mathrm{op}}\times[0,\infty]$ we have:

  1. (1)

$f_{0}(\sigma,r)\leq f_{1}(\pi_{1}(\sigma),r+\varepsilon)+\varepsilon$ for
every $(\sigma,r)\in\Delta(X_{0})^{\mathrm{op}}\times[0,\infty]$.

  2. (2)

$f_{1}(\sigma,r)\leq f_{0}(\pi_{0}(\sigma),r+\varepsilon)+\varepsilon$ for
every $(\sigma,r)\in\Delta(X_{1})^{\mathrm{op}}\times[0,\infty]$.

  3. (3)

$f_{0}(\sigma,r)\leq
f_{0}(\sigma\cup\pi_{0}\pi_{1}(\sigma),r+2\varepsilon)+2\varepsilon$ for every
$(\sigma,r)\in\Delta(X_{0})^{\mathrm{op}}\times[0,\infty]$.

  4. (4)

$f_{1}(\sigma,r)\leq
f_{1}(\sigma\cup\pi_{1}\pi_{0}(\sigma),r+2\varepsilon)+2\varepsilon$ for every
$(\sigma,r)\in\Delta(X_{1})^{\mathrm{op}}\times[0,\infty]$.

By $(2)$ and Lemma 4.1 we have that $\pi_{0}$ induces a simplicial map
$\pi_{0}\colon(Nf_{1})_{m,r}\to(Nf_{0})_{m-\varepsilon,r+\varepsilon}$. By
$(4)$ and Lemma 4.1 we have that $\pi_{1}\pi_{0}$ induces a simplicial map
$\pi_{1}\pi_{0}(\colon
Nf_{1})_{m,r}\to(Nf_{1})_{m-2\varepsilon,r+2\varepsilon}$ that is contiguous
to the inclusion map. By $(1)$ and Lemma 4.1 we have that $\pi_{1}$ induces a
map $\pi_{1}\colon(Nf_{0})_{m,r}\to(Nf_{1})_{m-\varepsilon,r+\varepsilon}$. By
$(3)$ and Lemma 4.1 we have that $\pi_{0}\pi_{1}$ induces a map
$\pi_{0}\pi_{1}\colon(Nf_{0})_{m,r}\to(Nf_{0})_{m-2\varepsilon,r+2\varepsilon}$
that is contiguous to the identity map.

We conclude that $Nf_{0}$ and $Nf_{1}$ are $\varepsilon$-weakly interleaved. ∎

Applying Proposition 3.2 and Proposition 2.15 we get:

######  Corollary 4.3.

Let $(\Lambda_{0},f_{0})$ and $(\Lambda_{1},f_{1})$ be Dowker bifiltrations.
If the set bifiltration distance of $f_{0}$ and $f_{1}$ is less than
$\varepsilon$ then the bifiltered complexes $D_{\Lambda_{0}}Nf_{0}$ and
$D_{\Lambda_{1}}Nf_{1}$ are homotopy $\varepsilon$-weakly interleaved.

######  Definition 4.4.

Let $f_{0}$ and $f_{1}$ be set bifiltrations on the sets $X_{0}$ and $X_{1}$
respectively and let $\alpha$ and $\beta$ be forward shift maps with
components $\alpha=(\alpha_{0},\alpha_{1})$ and $\beta=(\beta_{0},\beta_{1})$.
A weak $(\alpha,\beta)$ interleaving between $f_{0}$ and $f_{1}$ consists of
functions $\pi_{0}\colon X_{1}\to X_{0}$ and $\pi_{1}\colon X_{0}\to X_{1}$
satisfying the following conditions:

  1. (1)

$\alpha_{0}(f_{1}(\sigma,r),r)\leq
f_{0}(\pi_{0}\sigma,\alpha_{1}(f_{1}(\sigma,r),r))$ for all
$\sigma\in\Delta(X_{1})$, and $r\in[0,\infty]$.

  2. (2)

$\beta_{0}(f_{0}(\tau,r),r)\leq f_{1}(\pi_{1}\tau,\beta_{1}(f_{0}(\tau,r),r))$
for all $\tau\in\Delta(X_{0})$ and $r\in[0,\infty]$.

  3. (3)

$\beta_{0}(\alpha(f_{1}(\sigma,r),r))\leq
f_{1}(\sigma\cup\pi_{1}\pi_{0}\sigma,\beta_{1}(\alpha(f_{1}(\sigma,r),r)))$
for all $\sigma\in\Delta(X_{1})$ and $r\in[0,\infty]$.

  4. (4)

$\alpha_{0}(\beta(f_{0}(\tau,r),r))\leq
f_{0}(\tau\cup\pi_{0}\pi_{1}\tau,\alpha_{1}(\beta(f_{0}(\tau,r),r)))$ for all
$\tau\in\Delta(X_{0})$, and $r\in[0,\infty]$.

If an $(\alpha,\beta)$-interleaving exists, the filtered Dowker
dissimilarities are $(\alpha,\beta)$-weakly interleaved.

If $\alpha=\beta$ we say that $(\pi_{0},\pi_{1})$ is an $\alpha$-interleaving
and that the set bifiltrations are $\alpha$-weakly interleaved.

If $\alpha=\beta$ and $\alpha(m,r)=(m-\varepsilon,r+\varepsilon)$ we say that
$(\pi_{0},\pi_{1})$ is an $\varepsilon$-interleaving and that the set
bifiltrations are $\varepsilon$-weakly interleaved.

######  Proposition 4.5.

Let $f_{0}$ and $f_{1}$ be set bifiltrations on the sets $X_{0}$ and $X_{1}$
respectively and let $\alpha$ and $\beta$ be forward shift maps with
components $\alpha=(\alpha_{0},\alpha_{1})$ and $\beta=(\beta_{0},\beta_{1})$.
Let $\pi_{0}\colon X_{1}\to X_{0}$ and $\pi_{1}\colon X_{0}\to X_{1}$ be a
weak $(\alpha,\beta)$ interleaving of the set bifiltrations $f_{0}$ and
$f_{1}$. Then $\pi_{0}$ and $\pi_{1}$ induce a weak $(\alpha,\beta)$
interleaving between $Nf_{0}$ and $Nf_{1}$.

###### Proof.

Let $m\in\mathbb{R}$ and $r\in[0,\infty]$.

Let $\sigma\in N(f_{1})_{m,r}$. By part $(1)$ of Definition 4.4 we have
$\alpha_{0}(f_{1}(\sigma,r),r)\leq
f_{0}(\pi_{0}\sigma,\alpha_{1}(f_{1}(\sigma,r),r))$. Since $m\leq
f_{1}(\sigma,r)$ we have that
$\alpha_{0}(m,r)\leq\alpha_{0}(f_{1}(\sigma,r),r)$ and that
$\alpha_{1}(m,r)\geq\alpha_{1}(f_{1}(\sigma,r),r)$. Collecting these
inequalities we get

$\alpha_{0}(m,r)\leq\alpha_{0}(f_{1}(\sigma,r),r)\leq
f_{0}(\pi_{0}\sigma,\alpha_{1}(f_{1}(\sigma,r),r))\leq
f_{0}(\pi_{0}\sigma,\alpha_{1}(m,r)).$

This means that $\pi_{0}\sigma\in N(f_{0})_{\alpha(m,r)}$ and that $\pi_{0}$
induces a map

$\pi_{0}\colon N(f_{1})_{m,r}\to N(f_{0})_{\alpha(m,r)}.$

Let $\tau\in N(f_{0})_{m,r}$. By part $(2)$ of Definition 4.4 we have
$\beta_{0}(f_{0}(\tau,r),r)\leq
f_{1}(\pi_{1}\tau,\beta_{1}(f_{0}(\tau,r),r))$. Since $m\leq f_{0}(\tau,r)$ we
have that $\beta_{0}(m,r)\leq\beta_{0}(f_{0}(\tau,r),r)$ and that
$\beta_{1}(m,r)\geq\beta_{1}(f_{0}(\tau,r),r)$. Collecting these inequalities
we get

$\beta_{0}(m,r)\leq\beta_{0}(f_{0}(\tau,r),r)\leq
f_{1}(\pi_{0}\tau,\beta_{1}(f_{0}(\tau,r),r))\leq
f_{1}(\pi_{0}\tau,\beta_{1}(m,r)).$

This means that $\pi_{1}\tau\in N(f_{1})_{\beta(m,r)}$ and that $\pi_{1}$
induces a map

$\pi_{1}\colon N(f_{0})_{m,r}\to N(f_{1})_{\beta(m,r)}.$

Let $\sigma\in N(f_{1})_{m,r}$. By part $(3)$ of Definition 4.4 we have
$\beta_{0}(\alpha(f_{1}(\sigma,r),r))\leq
f_{1}(\sigma\cup\pi_{0}\pi_{1}\sigma,\beta_{1}(\alpha(f_{1}(\sigma,r),r)))$.
Since $m\leq f_{1}(\sigma,r)$ we have that
$\beta_{0}\alpha(m,r)\leq\beta_{0}\alpha(f_{1}(\sigma,r),r)$ and that
$\beta_{1}\alpha(m,r)\geq\beta_{1}\alpha(f_{1}(\sigma,r),r)$. Collecting these
inequalities we get

$\displaystyle\beta_{0}\alpha(m,r)$ $\displaystyle\leq$
$\displaystyle\beta_{0}\alpha(f_{1}(\sigma,r),r)$ $\displaystyle\leq$
$\displaystyle
f_{1}(\sigma\cup\pi_{1}\pi_{0}\sigma,\beta_{1}\alpha(f_{1}(\sigma,r),r))$
$\displaystyle\leq$ $\displaystyle
f_{1}(\sigma\cup\pi_{1}\pi_{0}\sigma,\beta_{1}\alpha(m,r)).$

This means that $\sigma\cup\pi_{1}\pi_{0}\sigma\in
N(f_{1})_{\alpha\beta(m,r)}$ and that $\pi_{0}$ induces a map

$\pi_{0}\colon N(f_{1})_{m,r}\to N(f_{1})_{\alpha\beta(m,r)}$

that is homotopic to the inclusion map.

The proof that $\pi_{1}$ induces a map

$\pi_{1}\colon N(f_{0})_{m,r}\to N(f_{0})_{\beta\alpha(m,r)}$

that is homotopic to the inclusion map is similar. ∎

##  5\. Restriction to the Support

Recall that a measurable space $(X,\Sigma)$ consists of a set $X$ and a
$\sigma$-algebra $\Sigma$ of subsets of $X$. That $\Sigma$ is a
$\sigma$-algebra means that it is closed under complements and countable
unions and countable intersections. Also recall that a measure $\mu$ on the
$\sigma$-algebra $\Sigma$ is a function $\mu\colon\Sigma\to[0,\infty]$ that is
countably additive, in that sense that $\mu(\emptyset)=0$ and that for
disjoint sets $B_{1},B_{2},\ldots$ in $\Sigma$ we have
$\mu(\cup_{i=1}^{\infty}B_{i})=\sum_{i=1}^{\infty}\mu(B_{i})$. A measure space
is a triple $(X,\Sigma,\mu)$ where $(X,\Sigma)$ measurable space and $\mu$ is
a measure on $\Sigma$.

######  Definition 5.1.

The $\sigma$-algebra $\Sigma_{\Lambda}$ associated to a Dowker dissimilarity

$\Lambda\colon X\times Y\to[0,\infty]$

is the smallest $\sigma$-algebra on $Y$ containing the sets

$B_{\Lambda}(x,r)=\\{y\in Y\,\mid\,\Lambda(x,y)\leq r\\}$

and

$\operatorname{int}(B_{\Lambda}(x,r))=\\{y\in Y\,\mid\,\Lambda(x,y)<r\\}$

for every $x\in X$ and $r\in[0,\infty]$.

######  Definition 5.2.

Let $\Lambda\colon X\times Y\to[0,\infty]$ be a Dowker dissimilarity and let
$\mu$ be a measure on the $\sigma$-algebra $\Sigma_{\Lambda}$. The support
$S(\mu)$ of $\mu$ consists of the elements $y\in Y$ with the property that if
$y\in B_{\Lambda}(x,r)$ for any $x\in X$ and $r>0$ then
$\mu(B_{\Lambda}(x,r))>0$.

######  Definition 5.3.

A Dowker measure space is a pair $(\Lambda,\mu)$ of a Dowker dissimilarity
$\Lambda\colon X\times Y\to[0,\infty]$ and a measure $\mu$ on the
$\sigma$-algebra $\Sigma_{\Lambda}$ such that both $S(\mu)\in\Sigma_{\Lambda}$
and $\mu(Y\setminus S(\mu))=0$.

######  Example 5.4.

Let $M$ be a separable metric space with metric $d$ and let $\mu$ be a Borel
measure on $M$. Write $\mu|_{\Sigma_{d}}$ for the restriction of $\mu$ to the
subset $\Sigma_{d}$ of the Borel $\sigma$-algebra of $M$. We claim that
$(d,\mu|_{\Sigma_{d}})$ is a Dowker measure space. Firstly, the complement
$M\setminus S(\mu)$ is open. Secondly, $M$ has a countable basis
$\\{B_{1},B_{2},\dots,\\}$ consisting of balls of the form
$\operatorname{int}(B_{d}(x,r))$. Thus, $M\setminus S(\mu)$ is a countable
union of sets in $\Sigma_{d}$, and hence $M\setminus S(\mu)$ is itself in
$\Sigma_{d}$. Moreover, for every $x\in M\setminus S(\mu)$ we can pick a basis
open $B_{i}$ with $x\in B_{i}$ and $\mu(B_{i})=0$. Thus, $M\setminus S(\mu)$
is a countable union of sets of measure $0$, and hence $\mu(M\setminus
S(\mu))=0$. Finally, since $M\setminus S(\mu)$ is in $\Sigma_{d}$, also
$S(\mu)$ is in $\Sigma_{d}$.

######  Example 5.5 (Distance to Measure Dowker Bifiltration).

Let $(\Lambda,\mu)$ be a Dowker measure space.

Given $p>0$ we define a Dowker bifiltration $(\Lambda,f_{p})$, where $f_{p}$
is the function
$f_{p}\colon\Delta(X)^{\mathrm{op}}\times[0,\infty]\to[0,\infty]$ given by

$f_{p}(\sigma,r)=\left(\int_{y\in Y}\min\\{\Lambda(x,y)^{p}\colon
x\in\sigma\\}\cdot\mathbbm{1}_{B_{\Lambda}(\sigma,r)}d\mu(y)\right)^{1/p}.$

Note that if $\mu$ is a Borel measure on $Y$, then, as $p\to 0$, the value
$f_{p}(\sigma,r)$ converges to $\mu(B_{\Lambda}(\sigma,r))$.

Given a simplicial complex $K$ with vertex set $V$ and a subset $S$ of $V$, we
let $K|_{S}$ be the simplicial complex with vertex set $S$ and simplices of
the form $\sigma\cap S$ for $\sigma\in K$.

######  Proposition 5.6.

Let $(\Lambda,\mu)$ be a Dowker measure space and let

$f\colon\Delta(X)^{\mathrm{op}}\times[0,\infty]\to[0,\infty]$

be the set bifiltration $f(\sigma,r)=\mu(B_{\Lambda}(\sigma,r))$. The
inclusion

$DC(\Lambda,\mu)_{m,r}=(D_{\Lambda}Nf_{m,r})|_{S(\mu)}\subseteq
D_{\Lambda}Nf_{m,r}$

is a homotopy equivalence.

###### Proof.

Let $\Lambda$ be of the form $\Lambda\colon X\times Y\to[0,\infty]$ and let
$\tau\in Nf_{m,r}$. Then there exists $x\in X$ with $\mu(B_{\Lambda}(x,r))\geq
m$ and $\tau\subseteq B_{\Lambda}(x,r)$. If $m=0$, then both
$(D_{\Lambda}Nf_{m,r})|_{S(\mu)}$ and $D_{\Lambda}Nf_{m,r}$ are contractible,
and the inclusion is a $(D_{\Lambda}Nf_{m,r})|_{S(\mu)}\subseteq
D_{\Lambda}Nf_{m,r}$ homotopy equivalence. Now, assume $m>0$. In the rest of
this proof we write $S$ for $S(\mu)$ and $\Lambda|_{X\times S}$ for the
restriction of $\Lambda$ to $X\times S$. We have a commutative diagram

${Nf_{m,r}}$${E(\Lambda|_{X\times S},f)_{m,r}}$${D_{\Lambda|_{X\times
S}}Nf_{m,r}}$${Nf_{m,r}}$${E(\Lambda,f)_{m,r}}$${D_{\Lambda}Nf_{m,r}}$$\scriptstyle{id}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\pi_{Y}}$$\scriptstyle{j}$$\scriptstyle{i}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\pi_{Y}}$

where, by Proposition 3.2, the horizontal maps are homotopy equivalences. The
map $i$ is the inclusion

$D_{\Lambda|_{X\times S}}Nf_{m,r}=(D_{\Lambda}Nf_{m,r})|_{S}\subseteq
D_{\Lambda}Nf_{m,r}$

and $j$ is the inclusion

$E(\Lambda|_{X\times S},f)_{m,r}=(E(\Lambda,f)_{m,r})|_{X\times S}\subseteq
E(\Lambda,f)_{m,r}.$

We conclude that the inclusion $i$ is a homotopy equivalence, as desired. ∎

###### Proof of theorem 1.3.

Let $f\colon\Delta(M)^{\mathrm{op}}\times[0,\infty]\to[0,\infty]$ be the set
bifiltration $f(\sigma,r)=\mu(B_{d}(\sigma,r))$. By definition we have
$DC(d,\mu)=(D_{d}Nf)|_{S(\mu)}$. By Proposition 5.6 this bifiltered simplicial
complex is homotopy equivalent to the dual measure Dowker bifiltration
$D_{d}Nf$. Let $\mathcal{U}$ be the cover of $\mathcal{M}(d,\mu)_{m,r}$
consisting of the sets $B_{d}(y,r)\cap\mathcal{M}(d,\mu)_{m,r}$ for
$y\in\mathcal{M}(d,\mu)_{m,r}^{r}$.

If $\tau\in N\mathcal{U}$ is an element of the nerve of this cover, then the
intersection $\bigcap_{y\in\tau}B_{d}(y,r)\cap\mathcal{M}(d,\mu)_{m,r}$ is
non-empty. Pick
$x\in\bigcap_{y\in\tau}B_{d}(y,r)\cap\mathcal{M}(d,\mu)_{m,r}$. Then
$f(\\{x\\},r)=\mu(B_{d}(x,r))\geq m$ and $\tau\subseteq B_{d}(x,r)$, so
$\tau\in D_{d}Nf_{m,r}$.

Conversely, if $\tau\in D_{d}Nf_{m,r}$, then we can pick $x\in M$ such that
$f(\\{x\\},\sigma)=\mu(B_{d}(x,r))\geq m$ and $\tau\subseteq
B_{d}(x,r)\subseteq\mathcal{M}(d,\mu)_{m,r}^{r}$. This means that $x\in
B_{d}(\tau,r)\cap\mathcal{M}(d,\mu)_{m,r}$, so $\tau\in N\mathcal{U}$. ∎

##  6\. Prohorov Distance

In this section we relate interleaving distance to Prohorov distance and
extract results for the Prohorov distance from the results in Section 4.

Let $M$ be a metric space with metric $d\colon M\times M\to[0,\infty]$ and let
$\Omega$ be the Borel $\sigma$-algebra on $M$. Recall from Definition 1.2 that
given $B\in\Omega$ and $\varepsilon>0$, the $\varepsilon$-offset
$B^{\varepsilon}_{d}$ of $B$ is the union of all closed $\varepsilon$-balls
with centers in $B$. Observe that, by the triangle inequality, for $x\in M$
and $r\geq 0$, $B_{d}(x,r)^{\varepsilon}$ is a subset of
$B_{d}(x,r+\varepsilon)$.

######  Definition 6.1.

The Prohorov distance between two Borel probability measures $\mu_{0}$ and
$\mu_{1}$ on $M$ is the infimum of all $\varepsilon\geq 0$ such that

$\mu_{0}(B)\leq\mu_{1}(B^{\varepsilon})+\varepsilon$

and

$\mu_{1}(B)\leq\mu_{0}(B^{\varepsilon})+\varepsilon$

for all $B\in\Omega$.

###### Proof of theorem 1.1.

Let $\mu_{0}$ and $\mu_{1}$ be two measures on the separable metric space $M$.
Define $f_{0}\colon\Delta(M)^{\mathrm{op}}\times[0,\infty]\to[0,\infty]$ and
$f_{1}\colon\Delta(M)^{\mathrm{op}}\times[0,\infty]\to[0,\infty]$ by
$f_{0}(\sigma,r)=\mu_{0}(B_{d}(\sigma,r))$ and
$f_{1}(\sigma,r)=\mu_{1}(B_{d}(\sigma,r))$. Since $\mu_{0}$ and $\mu_{1}$ are
of Prohorov distance less than $\varepsilon$, the identity on $M$ gives a
homotopy weak $\varepsilon$-interleaving between the set bifiltrations $f_{0}$
and $f_{1}$. By Corollary 4.3 the bifiltered complexes $D_{d}Nf_{0}$ and
$D_{d}Nf_{1}$ are homotopy weakly $\varepsilon$-interleaved. By Proposition
5.6 also the bifiltered complexes $DC(d,\mu_{0})=D_{d}Nf_{0}|_{S(\mu_{0})}$
and $DC(d,\mu_{1})=D_{d}Nf_{1}|_{S(\mu_{1})}$ are homotopy weakly
$\varepsilon$-interleaved. ∎

Note that this proof of Proposition 1.1 does not involve applying the Nerve
Lemma to a cover of $M$.

##  7\. Gromov-Prohorov Distance

######  Definition 7.1.

A metric measure space is a triple $(X,d,\mu)$ consisting of a metric space
$(X,d)$ and a Borel measure $\mu$ on $X$.

######  Definition 7.2.

Let $f\colon X\to Y$ be a measurable function between measurable spaces
$(X,\Sigma_{X})$ and $(Y,\Sigma_{Y})$. Given a measure $\mu$ on $\Sigma_{X}$
the pushforward measure $f_{*}\mu$ is the measure on $\Sigma_{Y}$ defined by
$f_{*}\mu(B)=\mu(f^{-1}(B))$ for all $B\in\Sigma_{Y}$.

######  Definition 7.3.

Let $(X_{0},d_{0},\mu_{0})$ and $(X_{1},d_{1},\mu_{1})$ be metric measure
spaces. Their Gromov-Prohorov distance is the infimum over all common distance
preserving embeddings $\iota_{0}\colon X_{0}\to M$ and $\iota_{1}\colon
X_{1}\to M$ into a common metric space $M$ of the Prohorov distance between
$\iota_{0*}\mu_{0}$ and $\iota_{1*}\mu_{1}$.

######  Definition 7.4.

Let $\iota\colon X\to M$ be a distance preserving embedding of a metric space
$X$ into a metric space $M$. A nearest neighbor projection $p\colon M\to X$ is
a function taking $x\in M$ to an element $z=p(x)\in X$ minimizing the distance
$d(x,\iota(z))$.

Note that in general a nearest neighbor projection is not unique, and that it
may happen that no nearest neighbor projection exists. If $X$ is compact, then
a nearest neighbor projection always exists.

######  Lemma 7.5.

Let $\iota_{0}\colon X_{0}\to M$ and $\iota_{1}\colon X_{1}\to M$ be
embeddings of two compact metric spaces $(X_{0},d_{0})$ and $(X_{1},d_{1})$
into a common metric space $(M,d)$ with a nearest neighbor projection
$p_{0}\colon M\to X_{0}$.

Given $x\in X_{1}$ and $y\in X_{0}$ we have

$d(\iota_{0}p_{0}\iota_{1}x,\iota_{0}y)\leq 2d(\iota_{1}x,\iota_{0}y).$

###### Proof.

By symmetry and the definition of $p_{0}$,

$d(\iota_{0}p_{0}\iota_{1}x,\iota_{1}x)\leq
d(\iota_{0}y,\iota_{1}x)=d(\iota_{1}x,\iota_{0}y).$

By the triangle inequality we have

$d(\iota_{0}p_{0}\iota_{1}x,\iota_{0}y)\leq
d(\iota_{0}p_{0}\iota_{1}x,\iota_{1}x)+d(\iota_{1}x,\iota_{0}y).$

However, we have just seen that

$d(\iota_{0}p_{0}\iota_{1}x,\iota_{1}x)\leq d(\iota_{1}x,\iota_{0}y),$

so

$d(\iota_{0}p_{0}\iota_{1}x,\iota_{0}y)\leq 2d(\iota_{1}x,\iota_{0}y).$

∎

Recall that given $\varepsilon>0$, the forward shift map $\beta^{\varepsilon}$
is defined by $\beta^{\varepsilon}(m,r)=(m-\varepsilon,2(r+\varepsilon))$. The
following is Theorem 1.5 stated in terms of interleavings.

######  Proposition 7.6.

Let $(X_{0},d_{0},\mu_{0})$ and $(X_{1},d_{1},\mu_{1})$ be compact metric
measure spaces. If $\varepsilon$ strictly greater than the Gromov-Prohorov
distance between $(X_{0},d_{0},\mu_{0})$ and $(X_{1},d_{1},\mu_{1})$ then the
bifiltered simplicial complexes $DC(d_{0},\mu_{0})$ and $DC(d_{1},\mu_{1})$
are $\beta^{\varepsilon}$-weakly interleaved.

###### Proof.

Let $\iota_{0}\colon X_{0}\to M$ and $\iota_{1}\colon X_{1}\to M$ be
embeddings of the metric spaces $(X_{0},d_{0})$ and $(X_{1},d_{1})$ into a
common metric space $(M,d)$ such that the Prohorov distance between
$\iota_{0*}\mu_{0}$ and $\iota_{1*}\mu_{1}$ is less than $\varepsilon$.

Let $f_{0}$ and $f_{1}$ be the set bifiltrations

$f_{1}\colon\Delta(X_{1})^{\mathrm{op}}\times[0,\infty]\to[0,\infty],\quad
f_{1}(\sigma,r)=\mu_{1}(B_{d_{1}}(\sigma,r))$

and

$f_{0}\colon\Delta(X_{0})^{\mathrm{op}}\times[0,\infty]\to[0,\infty],\quad
f_{0}(\sigma,r)=\mu_{0}(B_{d_{0}}(\sigma,r)).$

Then $DC(d_{0},\mu_{0})=(D_{d_{0}}Nf_{0})|_{S(\mu_{0})}$ and
$DC(d_{1},\mu_{1})=(D_{d_{1}}Nf_{1})|_{S(\mu_{1})}$.

Let $p_{1}\colon M\to X_{1}$ and $p_{0}\colon M\to X_{0}$ be nearest neighbor
projections. We let $\pi_{0}=p_{0}\circ\iota_{1}$ and
$\pi_{1}=p_{1}\circ\iota_{0}$. Since $X_{0}$ and $X_{1}$ are compact, they are
separable, so as explained in Example 5.4 $(d_{0},\mu_{0})$ and
$(d_{1},\mu_{1})$ are Dowker measure spaces. By Proposition 4.3, Proposition
5.6 and Proposition 4.5 it suffices to show that the conditions of Definition
4.4 are satisfied by the maps $\pi_{0}$ and $\pi_{1}$ and the set
bifiltrations $f_{0}$ and $f_{1}$.

In the current context, the first condition of Definition 4.4 is that

$f_{1}(\sigma,r)-\varepsilon\leq f_{0}(\pi_{0}\sigma,2(r+\varepsilon)),$

that is,

$\mu_{1}(B_{d_{1}}(\sigma,r))-\varepsilon\leq\mu_{0}(B_{d_{0}}(\pi_{0}\sigma,2(r+\varepsilon)))$

for all $\sigma\in\Delta(X_{1})$, and $r\in[0,\infty]$. Let us first note that
since the Prohorov distance between $\iota_{1*}\mu_{1}$ and
$\iota_{0*}\mu_{0}$ is less than $\varepsilon$, for $\sigma\in\Delta(X_{1})$
and $r\in[0,\infty]$ we have

$\mu_{1}(B_{d_{1}}(\sigma,r))={\iota_{1*}}\mu_{1}(B_{d}(\iota_{1}(\sigma),r))\leq{\iota_{0*}}\mu_{0}(B_{d}(\iota_{1}(\sigma),r+\varepsilon))+\varepsilon.$

In order to verify the first condition of Definition 4.4 it suffices to show
that

${\iota_{0*}}\mu_{0}(B_{d}(\iota_{1}(\sigma),r+\varepsilon))\leq\iota_{0*}\mu_{0}(B_{d}(\iota_{0}(\pi_{0}\sigma),2(r+\varepsilon))=\mu_{0}(B_{d_{0}}(\pi_{0}\sigma,2(r+\varepsilon)).$

For this, it suffices to show that

$B_{d}(\iota_{1}(\sigma),r+\varepsilon)\cap\iota_{0}(X_{0})\subseteq
B_{d}(\iota_{0}(\pi_{0}\sigma),2(r+\varepsilon)).$

For this in turn, it suffices to show that, for every $x\in X_{1}$, we have

$B_{d}(\iota_{1}(x),r+\varepsilon)\cap\iota_{0}(X_{0})\subseteq
B_{d}(\iota_{0}(\pi_{0}x),2(r+\varepsilon)).$

Now, let $y\in X_{0}$ with $\iota_{0}y\in B_{d}(\iota_{1}(x),r+\varepsilon)$.
By Lemma 7.5 we have

$d(\iota_{0}\pi_{0}x,\iota_{0}y)\leq 2d(\iota_{1}x,\iota_{0}y)\leq
2(r+\varepsilon).$

We conclude that $\iota_{0}y\in B_{d}(\iota_{0}(\pi_{0}(x)),2(r+\varepsilon))$
as desired.

The same argument with the roles of $X_{0}$ and $X_{1}$ interchanged shows
that

$f_{0}(\sigma,r)-\varepsilon\leq f_{1}(\pi_{1}\sigma,2(r+\varepsilon)),$

that is,

$\mu_{0}(B_{d_{0}}(\sigma,r))-\varepsilon\leq\mu_{1}(B_{d_{1}}(\pi_{1}\sigma,2(r+\varepsilon)))$

for all $\sigma\in\Delta(X_{0})$, so condition $(2)$ of Definition 4.4 is
satisfied.

In the current context, the third condition of Definition 4.4 is

$f_{1}(\sigma,r)-2\varepsilon\leq
f_{1}(\sigma\cup\pi_{1}\pi_{0}\sigma,2(2(r+\varepsilon)+\varepsilon)),$

that is,

$\mu_{1}(B_{d_{1}}(\sigma,r))-2\varepsilon\leq\mu_{1}(B_{d_{1}}(\sigma\cup\pi_{1}\pi_{0}\sigma,4r+6\varepsilon))$

for all $\sigma\in\Delta(X_{1})$, and $r\in[0,\infty]$.

Since condition (1) of Definition 4.4 is satisfied we have

$\mu_{1}(B_{d_{1}}(\sigma,r))\leq\mu_{0}(B_{d_{0}}(\pi_{0}\sigma,2(r+\varepsilon)))+\varepsilon$

Prohorov stability implies that

$\mu_{0}(B_{d_{0}}(\pi_{0}\sigma,2(r+\varepsilon)))\leq\iota_{1*}\mu_{1}(B_{d}(\iota_{0}\pi_{0}\sigma,2(r+\varepsilon)))+\varepsilon.$

Assume that $y\in X_{1}$ with $\iota_{1}y\in
B_{d}(\iota_{0}\pi_{0}\sigma,2(r+\varepsilon))$. By the definition of the
nearest neighbor projection $p_{0}$, for $x\in\sigma$ we have that

$d(\iota_{1}y,\iota_{1}x)\leq
d(\iota_{1}y,\iota_{0}p_{0}\iota_{1}x)+d(\iota_{0}p_{0}\iota_{1}x,\iota_{1}x)\leq
2d(\iota_{1}y,\iota_{0}p_{0}\iota_{1}x)=2d(\iota_{1}y,\iota_{0}\pi_{0}x).$

On the other hand, by Lemma 7.5 with the roles of $X_{0}$ and $X_{1}$
interchanged we have

$d(\iota_{1}y,\iota_{1}\pi_{1}\pi_{0}x)\leq 2d(\iota_{1}y,\iota_{0}\pi_{0}x).$

The above discussion shows that

$B_{d}(\iota_{0}\pi_{0}\sigma,2(r+\varepsilon))\cap\iota_{1}(X_{1})\subseteq
B_{d}(\iota_{1}\sigma\cup\iota_{1}\pi_{1}\pi_{0}\sigma,4(r+\varepsilon)).$

Since $\iota_{1*}\mu_{1}$ is order preserving, we have

$\iota_{1*}\mu_{1}(B_{d}(\iota_{0}\pi_{0}\sigma,2(r+\varepsilon)))\leq\iota_{1*}\mu_{1}(B_{d}(\iota_{1}\sigma\cup\iota_{1}\pi_{1}\pi_{0}\sigma,4(r+\varepsilon))).$

Note that

$\iota_{1*}\mu_{1}(B_{d}(\iota_{1}\sigma\cup\iota_{1}\pi_{1}\pi_{0}\sigma,4(r+\varepsilon)))=\mu_{1}(B_{d_{1}}(\sigma\cup\pi_{1}\pi_{0}\sigma,4(r+\varepsilon)))$

and that since $\mu_{1}$ is order preserving we have

$\mu_{1}(B_{d_{1}}(\sigma\cup\pi_{1}\pi_{0}\sigma,4(r+\varepsilon)))\leq\mu_{1}(B_{d_{1}}(\sigma\cup\pi_{1}\pi_{0}\sigma,4r+6\varepsilon)).$

Collecting the above inequalities we have

$\mu_{1}(B_{d_{1}}(\sigma,r))-2\varepsilon\leq\mu_{1}(B_{d_{1}}(\sigma\cup\pi_{1}\pi_{0}\sigma,4r+6\varepsilon))$

as desired.

The fourth condition of Definition 4.4 is verified by a similar argument. ∎

######  Remark 7.7 (Measure Dowker bifiltration).

Let $(X,\Lambda,\mu)$ be a metric measure space and let $f$ be the degree
bifiltration associated to $\mu$. Then

$\\{\mathcal{MD}(X,(Y,\Omega,\mu),\Lambda)_{m,r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}=\\{Nf_{m,2r}\\}_{(m,r)\in\mathbb{R}\times[0,\infty]}$

is the measure Dowker bifiltration of [8, Definition 3.6]. Thus Proposition
7.6 is a slight extension of [8, Theorem 4.2].

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Reionization history
as a probe of primordial fluctuations

Teppei Minoda$\,{}^{a}$, Shintaro Yoshiura$\,{}^{b}$, Tomo Takahashi$\,{}^{c}$

aThe University of Melbourne, School of Physics, Parkville, VIC 3010,
Australia
bMizusawa VLBI Observatory, National Astronomical Observatory Japan, 2-21-1
Osawa, Mitaka, Tokyo 181-8588, Japan
cDepartment of Physics, Saga University, Saga 840-8502, Japan

We argue that observations of the reionization history such as the luminosity
function of the Lyman-$\alpha$ emitters can be used as a probe of primordial
density fluctuations, particularly on small scales. Although the primordial
curvature perturbations are well constrained from measurements of cosmic
microwave background (CMB) anisotropies and large-scale structure, these
observational data probe the curvature perturbations only on large scales, and
hence its information on smaller scales will give us further insight on
primordial fluctuations. Since the formation of early galaxies is sensitive to
the amplitude of small-scale perturbations, and then, in turn, gives an impact
on the reionization history, one can probe the primordial power spectrum on
small scales through observations of reionization. In this work, we focus on
the running spectral indices of the primordial power spectrum to characterize
the small-scale perturbations, and investigate their constraints from
observations of the luminosity function of the Lyman-$\alpha$ emitters. We
show that the reionization, in combination with large-scale observations such
as CMB, would be a useful tool to investigate primordial density fluctuations.

##  1 Introduction

The statistical property of primordial perturbations is well understood by
measurements of cosmic microwave background (CMB) [1] and large-scale
structure (LSS) [2, 3, 4, 5, 6, 7] in this decade. The perturbations are
almost adiabatic and Gaussian. The power spectrum of the primordial
perturbations is nearly scale-invariant, but slightly red-tilted. These
characteristics of the primordial perturbations provide strong constraints on
inflation and their generation mechanism.

The scale dependence of primordial power spectrum ${\cal P}_{\zeta}$ is
usually characterized by a constant spectral index $n_{s}$ with which the
power spectrum is given by

${\cal P}_{\zeta}\propto k^{n_{s}-1}\,.$ (1.1)

However, in usual inflation models, $n_{s}$ is not exactly constant, but has
some scale dependence, and in some cases, it can be strongly scale-dependent.
Such a scale dependence is commonly characterized by the so-called “runnings”
in which the power spectrum is represented by

${\cal P}_{\zeta}\propto
k^{n_{s}-1+\frac{1}{2}\alpha_{s}\ln(k/k_{0})+\frac{1}{6}\beta_{s}[\ln(k/k_{0})]^{2}}\,,$
(1.2)

with $\alpha_{s}$ and $\beta_{s}$ being denoted as running parameters at the
pivot scale $k_{0}$. Because observations such as CMB and LSS probe the
primordial fluctuations only on large scales as
$10^{-3}~{}\mathrm{Mpc}^{-1}\lesssim k\lesssim 1~{}\mathrm{Mpc}^{-1}$ [1], we
need to investigate the fluctuations on different scales in order to study the
scale dependence of the primordial fluctuations further. Indeed, there have
been many attempts to investigate fluctuations on small scales $k\gtrsim
1~{}\mathrm{Mpc}^{-1}$ to probe primordial fluctuations using observations
such as the 21-cm global signal [8, 9], 21-cm power spectrum [10, 11], 21-cm
forest [12], Lyman-$\alpha$ forest [13, 14], minihalos [15], primordial black
holes (PBH) formation [16], supernovae lensing [17], spectral distortions of
CMB [18, 19, 20, 21], galaxy UV luminosity function [22], the substructure of
dark matter halo [23] and so on.

Our approach in this paper is to study the impact of the primordial power
spectrum on the reionization history. Since fluctuations on small scales
$k\gtrsim 1~{}\mathrm{Mpc}^{-1}$ are connected to the halos with mass
$M\lesssim 10^{11}M_{\odot}$, which are expected to be formed around the
cosmic dawn and the epoch of reionization, and thereby we can utilize
observations of the reionization history as a probe of such small-scale
primordial fluctuations. One famous cosmological constraint on the
reionization history is the Thomson optical depth of CMB, which corresponds to
the column density of the free electron. Therefore, it provides the redshift-
integrated information on the reionization history. We can also probe the
reionization physics from observations of the Lyman-$\alpha$ emitter (LAE)
luminosity function which gives the redshift-dependent information on the
reionization history. We focus on the latter observation to probe the
primordial fluctuations from the reionization history. In particular, we use
the SILVERRUSH survey data with the Subaru telescope [24], from which we can
constrain the running of the spectral index $\alpha_{s}$ and $\beta_{s}$.

In the following, we first introduce observational constraints on the
reionization history in Section 2.1, and our method to calculate the evolution
of the free electron fraction in Section 2.2. Then in Section 2.3, we provide
the constraints on the runnings of the primordial power spectrum from
observations of the reionization. We conclude and summarize this paper in
Section 3.

##  2 Reionization history as a probe of primordial fluctuations

###  2.1 Observational constraint on the reionization history

In this work, we put a constraint on the running parameters of the primordial
power spectrum from observations of the Ly-$\alpha$ luminosity function with
SILVERRUSH by calculating the reionization history [24]. Indeed one can also
use the measurement of the CMB optical depth from Planck [25] to obtain the
constraint. However, as we show in the following, the constraint on the
optical depth from the Ly-$\alpha$ luminosity function with SILVERRUSH is
tighter than that from CMB, and hence we focus on the constraint from the
Ly-$\alpha$ luminosity function. Firstly, by using a simple $\tanh$-type
reionization model, we demonstrate how the reionization history is constrained
by the latest Planck and SILVERRUSH data. The free electron fraction at a
given redshift $z$ is often modeled by the hyperbolic tangent function as [26]

$\displaystyle
x_{\mathrm{e}}(z,z_{\mathrm{re}})=\cfrac{1}{2}\left[1+\tanh{\left(\cfrac{y(z_{\mathrm{re}})-y(z)}{\Delta
y}\right)}\right],$ (2.1)

where $z_{\mathrm{re}}$ is the model parameter that represents the central
redshift of reionization, and $y(z)=(1+z)^{2/3},~{}\Delta
y=3/2(1+z_{\mathrm{re}})^{1/2}\Delta z$, with $\Delta z=0.5$. We note that
although the redshift width of reionization $\Delta z$ is in general
considered to be a free parameter as assumed in the original reference [26],
we fix it as $\Delta z=0.5$ since this width fits the SILVERRUSH results well
and the recent Planck analysis also assumes $\Delta z=0.5$ [25].

Figure 1 shows the $\tanh$-type reionization history for different
$z_{\mathrm{re}}$, together with the constraint on the free electron fraction
from the SILVERRUSH results of the Ly-$\alpha$ luminosity function. Three data
points from the SILVERRUSH results are depicted in the figure. The most
stringent constraint among them is $0.49\leq x_{\mathrm{e}}\leq 0.69$ at
$z=7.3$. This constraint can be translated into that on the central redshift
of the reionization $z_{\mathrm{re}}$ as $7.29\leq z_{\mathrm{re}}\leq 7.50$
for the modeling of $x_{\mathrm{e}}$ given in Eq. (2.1). The corresponding
range for $x_{\mathrm{e}}$ is shown with the pink-shaded region in Figure 1.
We note that, even when we change the redshift width of the reionization as
$\Delta z=1.0$, the constraint on the central redshift of reionization is
given by $7.42\leq z_{\mathrm{re}}\leq 7.70$, which is consistent with that
for the case of $\Delta z=0.5$. Therefore the change of $\Delta z$ does not
give a significant impact on our conclusion.

Other SILVERRUSH data points, $x_{\mathrm{e}}\geq 0.69$ at $z=7.0$ and
$x_{\mathrm{e}}\geq 0.69$ at $z=6.6$, can be translated into
$z_{\mathrm{re}}\geq 7.20$ and $z_{\mathrm{re}}\geq 6.80$, respectively. These
constraints are weaker than that from the data point at $z=7.3$, so we do not
use two data points at $z=7.0$ and 6.6 in the following analysis.

Figure 1:  Redshift evolution of the free electron fraction with
$z_{\mathrm{re}}=7.0$ (solid), $7.4$ (dashed), and $7.8$ (dotted). Three red
points at $z=6.6,7.0,$ and $7.3$ are the SILVERRUSH results, given in Table 2
of Ref. [24]. The pink-shaded region corresponds to the range of
$x_{\mathrm{e}}$ from the constraint of the SILVERRUSH results for the
$\tanh$-type reionization model with $7.29\leq z_{\mathrm{re}}\leq 7.50$.

To compare the constraint from SILVERRUSH with that from the Planck optical
depth, we calculate the Thomson optical depth, which is obtained by
integrating the $\tanh$-type reionization history, via

$\displaystyle\tau=n_{\mathrm{H}}c\sigma_{\mathrm{T}}\int_{0}^{z_{\max}}dz~{}x_{\mathrm{e}}(z,z_{\mathrm{re}})\frac{(1+z)^{2}}{H(z)},$
(2.2)

with the number density of hydrogen $n_{\mathrm{H}}$, the speed of light $c$,
the Thomson scattering cross-section $\sigma_{\mathrm{T}}$, and the Hubble
parameter $H(z)$. As referred in [25], we take the maximum redshift in the
integration as $z_{\max}=50$. We show the Thomson optical depth from the
$\tanh$-type reionization with different $z_{\mathrm{re}}$ in Figure 2. The
SILVERRUSH constraint on the central redshift of the reionization is $7.29\leq
z_{\mathrm{re}}\leq 7.50$, and plotted with the dot-hatched region in Figure
2. Assuming the $\tanh$-type reionization history, this constraint on
$z_{\mathrm{re}}$ can be translated to the constraint on the Thomson optical
depth, as $0.0456\leq\tau\leq 0.0475$, which is obtained from the overlapping
range between the blue thick line and the magenta dot-hatched region. This
constraint on $\tau$ is plotted with the horizontal thick-shaded region. For
comparison, we also show the Planck constraint on the optical depth,
$0.0440\leq\tau\leq 0.0549$ given in Eq. (86a) in Ref. [25] with the
horizontal green cross-hatched region. It is clear that the SILVERRUSH
constraint is stronger than the Planck one when we assume the $\tanh$-type
reionization.

Figure 2: Thomson optical depth obtained by Eq. (2.2) with the $\tanh$-type
reionization history, as a function of $z_{\mathrm{re}}$ (the blue thick
line). The green cross-hatched horizontal region is the constraint on the
optical depth from Planck 2018 results. The magenta dot-hatched vertical one
is obtained by the $\tanh$-type reionization history with the SILVERRUSH
results, as $7.29\leq z_{\mathrm{re}}\leq 7.50$. The horizontal thick-shaded
region corresponds to the range of $\tau$ suggested from the SILVERRUSH
constraints on $z_{\mathrm{re}}$ for the $\tanh$-type reionization model.

###  2.2 Reionization calculation with the 21cmFAST

We adopt the publicly available semi-numerical simulation code 21cmFAST [27]
to calculate the reionization history. We modify the code to extend the
cosmological model in 21cmFAST, to include the “running” and “running of
running” of the primordial curvature power spectrum, $\alpha_{s}$ and
$\beta_{s}$ as

$\mathcal{P}_{\zeta}(k)=A_{s}\left(\cfrac{k}{k_{0}}\right)^{n_{s}-1+\frac{1}{2}\alpha_{s}\ln\left(\frac{k}{k_{0}}\right)+\frac{1}{6}\beta_{s}\left[\ln\left(\frac{k}{k_{0}}\right)\right]^{2}},$
(2.3)

where $A_{s}$ is the amplitude at the pivot scale $k_{0}$, and $n_{s}$ is the
spectral index. We fix the pivot scale as $k_{0}=0.05~{}\mathrm{Mpc}^{-1}$ as
the Planck collaboration adopted [25]. To simplify the following analysis, we
fix the $\Lambda$CDM parameters with the Planck 2018 best-fitted values [25]
although we vary several astrophysical parameters as we describe below since
the reionization history is more affected by astrophysical processes. Figure 3
shows the matter power spectra with varying running parameters. Here, the
larger $\alpha_{s}$ and $\beta_{s}$ enhance the small-scale power spectrum.
However, on large scales, the larger $\alpha_{s}$ enhances the power spectrum
as well, but the larger $\beta_{s}$ reduces the large-scale matter power
spectrum. Although cosmological observations such as CMB and LSS can probe
large-scale fluctuations, constraints from such observations show the
degeneracy between $\alpha_{s}$ and $\beta_{s}$. However, the combined
analysis with other observations especially probing the small-scale
fluctuations is expected to break such a degeneracy. In this paper, we focus
on the reionization history as such a cosmological probe for the running
spectral indices. Figure 4 shows the time evolution of the free electron
fraction with different running parameters. Increasing $\alpha_{s}$ and
$\beta_{s}$ enhances the amplitude of small-scale matter fluctuations, and
thereby high-redshift halo formation is promoted and the reionization gets
earlier.

Figure 3: Effects of the running spectral indices $\alpha_{s}$ and $\beta_{s}$
on the matter power spectra. Figure 4: Effects of the running spectral indices
$\alpha_{s}$ and $\beta_{s}$ on the reionization history. Astrophysical
parameters are assumed to be the best-fitted values from HERA constraints
[28]. Although the HERA best-fitted model slightly deviates from the
SILVERRUSH constraints, taking account of the uncertainties, the HERA results
are consistent with the SILVERRUSH (see Fig. 5).

When calculating the reionization history, we use the astrophysical parameters
adopted in the 21cmFAST version 2 [27]#1#1#1The latest version of 21cmFAST is
3.2.1 (updated September 13, 2023), which has improved calculation speed and
numerical stability. However, version 2 is slightly easier in extension and
does not affect the calculation results. Therefore we use version 2 in this
paper.. In particular, three parameters $f_{*,10}$, $f_{\mathrm{esc,10}}$, and
$M_{\mathrm{turn}}$ give a significant impact on the reionizaiton history.
These model parameters are defined by the stellar mass-to-baryon mass fraction
in halos

$\displaystyle
f_{*}(M_{\mathrm{halo}})=f_{*,10}\left(\cfrac{M_{\mathrm{halo}}}{10^{10}M_{\odot}}\right)^{\alpha_{*}},$
(2.4)

the halo mass-dependent escape fraction of the ionizing photons

$\displaystyle
f_{\mathrm{esc}}(M_{\mathrm{halo}})=f_{\mathrm{esc,10}}\left(\cfrac{M_{\mathrm{halo}}}{10^{10}M_{\odot}}\right)^{\alpha_{\mathrm{esc}}},$
(2.5)

and the duty cycle of the star-forming galaxy

$\displaystyle
f_{\mathrm{duty}}(M_{\mathrm{halo}})=\exp\left(-\cfrac{M_{\mathrm{turn}}}{M_{\mathrm{halo}}}\right)\,,$
(2.6)

where $\alpha_{*}$ and $\alpha_{\mathrm{esc}}$ represent their power-law
dependence. Figure 5 shows the time evolution of the free electron fraction
with different astrophysical parameters, $f_{*,10}$, $f_{\mathrm{esc,10}}$,
and $M_{\mathrm{turn}}$. Other astrophysical parameters are set to the best-
fitted values of HERA results [28] as $\alpha_{*}=0.50$,
$\alpha_{\mathrm{esc}}=0.02$, $t_{\mathrm{*}}=0.60$,
$L_{\mathrm{X}}/\mathrm{SFR}=10^{40.64}~{}\mathrm{erg/s/M_{\odot}~{}yr}$.
These parameters can also affect the reionization history. The effects on the
free electron fraction at $z=7.3$ are $\Delta x_{\mathrm{e}}\lesssim 0.01$ for
varying $t_{\mathrm{*}}$, $\Delta x_{\mathrm{e}}\lesssim 0.05$ for varying
$L_{\mathrm{X}}/\mathrm{SFR}$. Thus, these effects are smaller than the above
three, $f_{*,10}$, $f_{\mathrm{esc,10}}$, and $M_{\mathrm{turn}}$. On the
other hand, $\Delta x_{\mathrm{e}}$ can be larger than 0.1 for varying
$\alpha_{*}$ and $\alpha_{\mathrm{esc}}$. However, we only use the free
electron fraction at $z=7.3$ and it is difficult to constrain the $\alpha_{*}$
and $\alpha_{\mathrm{esc}}$ independently of $f_{*,10}$ and
$f_{\mathrm{esc,10}}$, respectively. Thus, we also fix $\alpha_{*}$ and
$\alpha_{\mathrm{esc}}$. For the simulation parameters in 21cmFAST, we take
such that the box size is (400 Mpc)3, spatial resolution is $800^{3}$ for
high-resolution calculation, and $200^{3}$ for low-resolution.

Figure 5: Reionization history calculated by 21cmFAST with varying
astrophysical parameters, $f_{*,10}$, $f_{\mathrm{esc,10}}$, and
$M_{\mathrm{turn}}$, from top to bottom. The values for these parameters
assumed here (shown inside the figure) correspond to the mean, 1$\sigma$ upper
and lower values from the HERA results [28].

Our extended 21cmFAST enables us to calculate the reionization history varying
the primordial running spectral indices and astrophysical parameters. However,
the parameter inference with MCMC analysis, which is widely used to obtain
constraints in cosmology and reionization [29, 30, 28, 31], requires
computational resources so much#2#2#2In Ref. [32], using 21CMMC[29], they
performed an MCMC analysis with varying two astrophysical parameters and two
parameters related to the bump-like features of the primordial power spectrum.
They only considered the bumps at large scales ($k<1\rm Mpc^{-1}$) and did not
use the constraint from the reionization history.. To save the calculation
time, we construct the fitting function for the free electron fraction at
redshift $z=7.3$, based on our extended 21cmFAST results. Here, we use the
free electron fraction only at $z=7.3$, at which the most stringent constraint
is given by the SILVERRUSH data. The fitting function that we obtained is

$\displaystyle x_{\mathrm{e}}(z=7.3)=\min(1.0,\tilde{x}_{\mathrm{e,7.3}}),$
(2.7)

where

$\displaystyle\tilde{x}_{\mathrm{e,7.3}}=$
$\displaystyle\left\\{0.421\mathrm{e}^{A}+B\left[\left(\cfrac{f_{*,10}}{0.058}\right)^{C}-1.0\right]+D\log\left(\cfrac{M_{\mathrm{turn}}}{3.8\times
10^{8}M_{\odot}}\right)\right\\}\left(\cfrac{f_{\mathrm{esc,10}}}{0.078}\right)^{E},$
(2.8)

with

$\displaystyle A$ $\displaystyle=8.43\alpha_{s}+11.41\beta_{s},$
$\displaystyle B$ $\displaystyle=4.00\alpha_{s}+7.75\beta_{s}+0.38,$
$\displaystyle C$ $\displaystyle=-5.24\alpha_{s}-12.04\beta_{s}+1.34,$
$\displaystyle D$ $\displaystyle=-0.7\alpha_{s}-1.20\beta_{s}-0.07,$
$\displaystyle E$ $\displaystyle=-3.11\beta_{s}+1.08.$

In order to check the validity of our fitting function, we performed test
calculations with the random parameter sets
($\alpha_{s},\beta_{s},f_{\mathrm{esc,10}},f_{*,10},M_{\mathrm{turn}}$) in the
ranges of
$\alpha_{s}=(-0.05,0.05),\beta_{s}=(-0.05,0.05),f_{\mathrm{esc,10}}=(0.03,0.3),f_{*,10}=(0.024,0.091),$
and $M_{\mathrm{turn}}/M_{\odot}=(1.6\times 10^{8},1.5\times 10^{9})$. We show
the comparison between the extended 21cmFAST results and the fitting function
in Table 1. The absolute value of the average difference of the free electron
fraction between that from 21cmFAST and the fitting function $\Delta
x_{\mathrm{e}}$ is about 0.02, which is well smaller than the error in the
SILVERRUSH data. Therefore we can safely use our fitting function to analyze
the constraint. Our fitting function (2.8) is obtained from the simulation
data from 21cmFAST with
$\alpha_{s}=(-0.015,0.02),\beta_{s}=(-0.015,0.02),f_{\mathrm{esc,10}}=(0.03,0.3),f_{*,10}=(0.024,0.091),M_{\mathrm{turn}}/M_{\odot}=(1.6\times
10^{8},1.5\times 10^{9})$. However, as seen in the results of Table 1, the
fitting function can also be applied outside the range of parameters that we
used. Although one may expect that we can also use other parameters such as
$n_{s},\alpha_{*},\alpha_{\mathrm{esc}}$ to improve the accuracy of our
fitting function, it turns out that the inclusion of more parameters in the
fitting does not help much. Therefore, we decided to fix other parameters to
obtain the fitting function.

Table 1: Comparison between the free electron fraction calculated with the extended 21cmFAST code and our fitting function Eq. (2.8). The average value of the absolute difference between the 21cmFAST and Eq. (2.8) is about 0.02 among the parameter sets we used. $\alpha_{s}$ | $\beta_{s}$ | $f_{\mathrm{esc,10}}$ | $f_{*,10}$ | $M_{\mathrm{turn}}/10^{8}M_{\odot}$ | $x_{\mathrm{e}}^{\mathrm{sim}}(z=7.3)$ | $x_{\mathrm{e}}^{\mathrm{fit}}(z=7.3)$ | $\Delta x_{\mathrm{e}}$
---|---|---|---|---|---|---|---
0.002 | 0.031 | 0.074 | 0.055 | 10.240 | 0.442 | 0.451 | -0.009
-0.038 | 0.015 | 0.113 | 0.067 | 3.857 | 0.688 | 0.650 | 0.038
-0.005 | -0.013 | 0.124 | 0.067 | 3.788 | 0.748 | 0.705 | 0.042
-0.034 | -0.000 | 0.064 | 0.087 | 6.155 | 0.383 | 0.409 | -0.025
0.017 | -0.030 | 0.100 | 0.079 | 4.691 | 0.688 | 0.645 | 0.043
0.032 | 0.039 | 0.050 | 0.068 | 2.139 | 0.684 | 0.676 | 0.008
0.007 | -0.002 | 0.086 | 0.074 | 2.164 | 0.718 | 0.704 | 0.014
0.005 | 0.019 | 0.166 | 0.033 | 2.845 | 0.730 | 0.724 | 0.007
0.011 | 0.011 | 0.149 | 0.042 | 5.242 | 0.729 | 0.666 | 0.063
0.009 | -0.018 | 0.100 | 0.055 | 5.932 | 0.447 | 0.437 | 0.011

###  2.3 Constraints from the SILVERRUSH observation

In this study, we perform MCMC analysis, using the public code emcee [33], by
comparing the SILVERRUSH results with Eq. (2.8) using a Gaussian likelihood.
We use two priors on the running parameters: the flat one with
$\alpha_{s}=(-0.1,0.1),\beta_{s}=(-0.1,0.1)$, and the Gaussian Planck prior.
The Planck prior assumes the 2-dimensional Gaussian distribution function with
the mean $\alpha_{s}=0.0012,\beta_{s}=0.0119$ and the covariance matrix from
the Planck Legacy Archive [34]. Three astrophysical parameters are varied with
flat priors as $f_{\mathrm{esc,10}}=(0.03,0.3),f_{*,10}=(0.024,0.091),$ and
$\log_{10}(M_{\mathrm{turn}}/M_{\odot})=(8.2,9.2)$, which correspond to the
1$\sigma$ uncertainty range from the HERA result [28].

First, we show constraints on $\alpha_{s}$ and $\beta_{s}$ for the case that
the astrophysical parameters are fixed as
$f_{\mathrm{esc,10}}=0.078,f_{*,10}=0.058,$ and
$\log_{10}(M_{\mathrm{turn}}/M_{\odot})=8.58$, which are taken from the recent
HERA best-fitted values in Figure 6. When imposing the flat priors for
$\alpha_{s}$ and $\beta_{s}$, a strong degeneracy between $\alpha_{s}$ and
$\beta_{s}$ appears. This degeneracy shows a negative correlation because the
reionization redshift is considered to be determined by the small-scale matter
power spectrum where effects of $\alpha_{s}$ and $\beta_{s}$ are canceled. On
the other hand, CMB anisotropies probe large-scale perturbations, and
therefore the direction of the degeneracy is positive since the response of
large-scale fluctuations to the change of $\alpha_{s}$ and $\beta_{s}$ is
opposite to that of small-scale as seen from Figure 3. Figure 6 shows that
combining CMB and reionization observations can break these degeneracies and
bring stringent constraints on $\alpha_{s}$ and $\beta_{s}$.

Figure 6: Triangle-plot of constraints on $(\alpha_{s},\beta_{s})$ from
observations of the reionization history. The constraint with the flat prior
on $(\alpha_{s},\beta_{s})$ is labeled as “SILVERRUSH only” (orange), and one
with the Gaussian Planck prior is labeled as “Planck+SILVERRUSH” (blue). For
comparison, the constraint obtained from Planck results is also shown as
“Planck only” (green) [1, 34].

When one discusses the constraints on the running spectral indices from the
reionization history, the degeneracies between the running and astrophysical
parameters should be considered. Figure 7 shows the results of MCMC analysis
when all parameters are sampled. When we assume a flat prior on $\alpha_{s}$
and $\beta_{s}$, although one can see a weak correlation between $\alpha_{s}$
and $\beta_{s}$, most parameters are poorly constrained and the 68% C.L.
errors on the running spectral indices are
$\Delta\alpha_{s}\sim\Delta\beta_{s}\sim{\cal O}(0.1)$. On the other hand,
when we impose the Planck prior for $\alpha_{s}$ and $\beta_{s}$, the
constraints become significantly improved as
$\Delta\alpha_{s}\sim\Delta\beta_{s}\sim 0.01$ while constraints on
astrophysical parameters are not changed much between those for the two
different priors. Furthermore, we found that $f_{\mathrm{esc,10}}$ and
$f_{*,10}$ are slightly more sensitive to the reionization history, compared
with $M_{\mathrm{turn}}$ as expected from Figure 5. In table 2, we summarize
the best-fit values and 68% C.L. errors for two prior cases with and without
fixing astrophysical parameters.

Figure 7: Triangle-plot of constraint on $(\alpha_{s},\beta_{s},f_{\mathrm{*,10}},f_{\mathrm{esc,10}},M_{\mathrm{turn}})$ from the SLVERRUSH observations with the flat prior on $(\alpha_{\mathrm{s}},\beta_{\mathrm{s}})$ (orange), and that with the Gaussian Planck prior [34] (blue).  Table 2: The best-fit values and 68% C.L. errors for the running and astrophysical parameters from SLIVERRUSH observations.  | w/o astrophysical parameters | w. astrophysical parameters
---|---|---
| flat prior | Planck prior | flat prior | Planck prior
$\alpha_{s}$ | — | $0.006^{+0.007}_{-0.007}$ | $-0.048_{-0.103}^{+0.149}$ | $0.001_{-0.010}^{+0.010}$
$\beta_{s}$ | — | $0.019^{+0.008}_{-0.009}$ | $0.048_{-0.118}^{+0.110}$ | $0.012_{-0.013}^{+0.013}$
$f_{\mathrm{esc,10}}$ |  |  | $0.087_{-0.064}^{+0.147}$ | $0.112_{-0.041}^{+0.055}$
$f_{*,10}$ |  |  | $0.104_{-0.072}^{+0.182}$ | $0.064_{-0.024}^{+0.022}$
$\log_{10}(M_{\mathrm{turn}}/M_{\odot})$ |  |  | $9.148_{-1.370}^{+0.937}$ | $9.053_{-0.737}^{+0.912}$

##  3 Conclusion

In this work, we have argued that the reionization history can be used as a
novel probe of primordial fluctuations. To this end, we have investigated the
impact of the small-scale primordial curvature perturbations on the
reionization history. We modified the semi-numerical simulation code 21cmFAST
to examine the reionization history incorporating the running of the spectral
indices of the primordial power spectrum up to the second order. Based on
calculations with 21cmFAST, we constructed the fitting function for the free
electron fraction at $z=7.3$. We also performed the MCMC analysis by using the
fitting function and observational data of the reionization history from the
SILVERRUSH LAEs to obtain the constraints on the running spectral indices. In
case the astrophysical uncertainties are neglected, the running spectral
indices can be strongly constrained by combining the observation of the
reionization history and that of CMB. On the other hand, when considering
uncertainties of the astrophysical parameters as inferred in the recent 21-cm
observations [28], we found that the current observational constraints on the
reionization history cannot severely constrain the running spectral indices.
However, in near future, the measurements of astrophysical parameters can
become more precise and, in such a case, the reionization history would be a
useful probe of primordial fluctuations and can augment other methods to
severely constrain them.

Finally, we comment on the other observations of the reionization history. In
this work, we mainly used the free electron fraction derived from the
luminosity function of the LAEs observed in the SILVERRUSH survey. Our choice
of observational data is just to demonstrate the power of constraining
primordial perturbations using the reionization history. Actually, many
observational constraints on the reionization history are reported, such as
the Ly-$\alpha$ equivalent width of the Lyman break galaxies (LBGs) [35, 36,
37, 38], the number fraction of LBGs emitting Ly-$\alpha$ in all observed LBGs
[39], gamma-ray burst and the QSO damping wings [40, 41, 42, 43, 44, 45], the
Ly-$\alpha$ and Ly-$\beta$ forest dark gaps of QSO spectra [46, 47], and the
Gunn-Peterson trough of QSOs [48, 49]. These observational data are
independent but some of the above results show inconsistencies (for examples,
see Figure 11 of Ref. [24]). Among them, the reionization history obtained by
the LAE luminosity function with the Subaru telescope that we used in this
work provides constraints at different redshifts, and they are consistent with
each other [50, 51, 52, 53, 24]. It is worth noting that measurements of the
LAE luminosity function using other telescopes provide different constraints
on the reionization history [54, 55], and therefore the best-fit values in our
analysis may depend on the observational data choice.

The current observational constraint errors on the free electron fractions are
$\Delta x_{\mathrm{e}}\sim\mathcal{O}(0.1)$. However, we would expect that
more precise measurements of the reionization history can decrease the
uncertainties on cosmological and astrophysical parameters. However, it may be
difficult to resolve degeneracies among the parameters in our analysis, i.e.,
$\alpha_{s},\beta_{s},f_{\mathrm{*,10}},f_{\mathrm{esc,10}},$ and
$M_{\mathrm{turn}}$ by using observations of the reionization history only.
Further improvement on the constraints on the primordial curvature
perturbations is expected to be obtained by combining other types of
observations, such as the recent high-redshift galaxy luminosity function
obtained by the James Webb Space Telescope [56, 57, 58], which is worth
investigating. We leave this issue to future work.

## Acknowledgements

TM is supported by JSPS Overseas Research Fellowship. SY is supported by JSPS
Research Fellowships for Young Scientists. This work was supported in part by
JSPS KAKENHI Grant Nos. 21J00416 (SY), 22KJ3092 (SY) and 19K03874 (TT).
Numerical computations were carried out on Cray XC50 at Center for
Computational Astrophysics, National Astronomical Observatory of Japan.

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