# 2-Colored Rogers-Ramanujan Partition Identities

MOHAMMAD ZADEHDABBAGH,

###### Abstract.

In this paper, we combined two types of partitions and introduced 2-colored
Rogers-Ramanujan partitions. By finding some functional equations and using a
constructive method, some identities have been found. Some Overpartition
identities coincide with our findings. A correspondence between colored
partitions and overpartitions is provided.
Key words: Colored Partitions, Rogers-Ramanujan, Overpartition

Department of Mathematics, Faculty of Engineering and Natural Sciences,
Sabanci University, Istanbul, Turkey

E-mail<EMAIL_ADDRESS>
##  1\. Introduction

Agarwal and Andrews [1] gave some identities for Rogers-Ramanujan type
partitions with n copies of n, which correspond to colored partitions. Sandon
and Zanello [6] proved identities on the colored case of some partition types.
This gave us the idea of combining colored partitions and Roger-Ramanujan type
partitions and apply a constructive method to find identities on those
partitions.

Rogers-Ramanujan identities can be interpreted using partitions in which the
difference between every two consecutive parts is at least 2. We combined
these types of partitions and colored ones to have the following partition
type,

######  Definition 1.0.1.

A 2-colored Rogers-Ramanujan partition of $n$ consists of two separated parts,
each of the same color, and the difference between every two consecutive parts
of the same color is at least two, moreover, parts in different colors do not
intersect.

As an example, all 2-colored Rogers-Ramanujan partitions of 6 are

$6,\color[rgb]{1,0,0}6\color[rgb]{0,0,0},5+1,\color[rgb]{1,0,0}5\color[rgb]{0,0,0}+1,5+\color[rgb]{1,0,0}1\color[rgb]{0,0,0},\color[rgb]{1,0,0}5\color[rgb]{0,0,0}+\color[rgb]{1,0,0}1\color[rgb]{0,0,0},4+2,\color[rgb]{1,0,0}4\color[rgb]{0,0,0}+2,4+\color[rgb]{1,0,0}2\color[rgb]{0,0,0},\color[rgb]{1,0,0}4\color[rgb]{0,0,0}+\color[rgb]{1,0,0}2\color[rgb]{0,0,0},3+\color[rgb]{1,0,0}2\color[rgb]{0,0,0}+1,\color[rgb]{1,0,0}3\color[rgb]{0,0,0}+2+\color[rgb]{1,0,0}1$

A constructive way from Kurşungöz’s papers [3] and [4] applied on the
generating functions for these types of partitions to find identities on
2-colored Rogers-Ramanujan partitions.

For this purpose, functional equations relating to generating functions for
given partition types have been found. By these functional equations, in two
main steps, we can find the generating function as an infinite sum. Finally,
by some transformation such as Jacobi’s triple products [2], we can find the
divisibility part of identities.

######  Theorem 1.1 (Jacobi’s triple product).

For $z\neq 0$, $|q|<1$,

$\sum_{n=-\infty}^{\infty}z^{n}q^{n^{2}}=\prod_{n=0}^{\infty}(1-q^{2n+2})(1+zq^{2n+1})(1+z^{-1}q^{2n+1}).$

By the method and the transformation mentioned above we reached the following
identity

######  Theorem 1.2.

Let $R(n)$ denote the number of 2-colored Rogers-Ramanujan partitions of $n$,
then

$R(n)=\frac{(-q)_{\infty}(q^{2},q^{2},q^{4};q^{4})_{\infty}}{(q)_{\infty}}.$

We observed that our results are identical to identities for special cases of
overpartitions in [5], that identity is

######  Theorem 1.3.

For $k\geq a\geq 1$, let $D_{k,a}(n)$ denote the number of overpartitions of
$n$ of the form $d_{1}+d_{2}+\cdots+d_{s}$, such that $1$ can occur as a non-
overlined part at most $a-1$ times, and $d_{j}-d_{j+k-1}\geq 1$ if $d_{j}$ is
overlined and $d_{j}-d_{j+k-1}\geq 2$ otherwise. For $k>i\geq 1$, let
$C_{k,i}(n)$ denote the number of overpartitions of $n$ whose non-overlined
parts are not congruent to $0,\pm i$ modulo $2k$ and let $C_{k,k}(n)$ denote
the number of overpartitions of $n$ with parts not divisible by $k$. Then
$C_{k,i}(n)=D_{k,i}(n)$.

In the first section, we will go over the 2-colored Rogers-Ramanujan partition
type, accordingly, we will find two functional equations, and then
constructively, mentioned above, we will find a partition identity. In the
third section, a correspondence between our identities and one for
overpartitions is given. Finally, possible future works are introduced in the
last section.

##  2\. Colored Rogers-Ramanujan Partitions

Throughout this paper, for $|q|<1$,

$(a;q)_{n}=\prod_{i=0}{n-1}(1-aq^{i}),$
$(a;q)_{\infty}=\lim_{n\rightarrow\infty}(a;q)_{n},$
$(a_{1},a_{2},\cdots,a_{k};q)_{n}:=(a_{1};q)_{n}(a_{2};q)_{n}\cdots(a_{k};q)_{n}.$

According to 2-colored Rogers-Ramanujan partitions, the following definition
is given.

######  Definition 2.0.1.

For $1\leq j\leq 2$, let $R_{j}(x)$ be the generating function of 2-colored
Rogers-Ramanujan partitions with smallest part greater than or equal to $j$.

With respect to these definitions, one can find the following functional
equations relating $R_{1}(x)$ and $R_{2}(x)$,

(1) $R_{1}(x)-R_{2}(x)=xqR_{1}(xq)+xqR_{2}(xq)$

and

(2) $R_{2}(x)=R_{1}(xq)$

Equation (2) is clear, as shifting every part of $R_{1}$ by 1 unit it will
change to $R_{2}$. The proof of equation (1) is as follows.

Let

$R_{i}(x)=\sum_{m\geq 0}\sum_{n\geq 0}r_{i}(m,n)x^{m}q^{n}\;\;;\;\;i=1,2$

be the generating function for the types that have been mentioned above, where
m is referring to the number of parts in partitions.

Let $\pi$ be a 2-Colored Rogers-Ramanujan partition of $n$ with $m$ parts, all
possible ways such that the smallest part $\geq 1$ will be counted by
$r_{1}(m,n)$, if all partitions that the smallest part is $\geq 2$
($r_{2}(m,n)$) have been removed, then all remaining partitions are of
2-Colored Rogers-Ramanujan type with the smallest part exactly be 1
($r_{1}(m,n)-r_{2}(m,n)$).

Now, let’s count all partitions with the smallest part 1 in another way, if 1
is removed from all partitions, then there will be two cases

  * (i)

The smallest part is $\geq 2$ with different color than 1, so one can subtract
1 from each part, the enumeration of these partitions is by $r_{1}(m-1,n-m)$.

  * (ii)

The smallest part is $\geq 3$ with the same color as 1, if 1 is subtracted
from each part, the enumeration of these partitions is $r_{2}(m-1,m-n)$, note
that a part 2 is not possible here.

So

$r_{1}(m,n)-r_{2}(m,n)=r_{1}(m-1,n-m)+r_{2}(m-1,n-m)$

Multiplying all terms by $x^{m}q^{n}$ and taking the summation over $m$ and
$n$ for all terms, $m,n\geq 0$ and both integers, we will have

$\begin{split}\sum_{m\geq 0}\sum_{n\geq 0}r_{1}(m,n)x^{m}q^{n}-\sum_{m\geq
0}\sum_{n\geq 0}r_{2}(m,n)x^{m}q^{n}=\\\ \sum_{m\geq 0}\sum_{n\geq
0}r_{1}(m-1,n-m)x^{m}q^{n}+\sum_{m\geq 0}\sum_{n\geq
0}r_{2}(m-1,n-m)x^{m}q^{n}\end{split}$

By changing $m-1$ to $m$ and $n-m$ to $n-m+1$ on the right hand side of this
equation, we will have

$\begin{split}\sum_{m\geq 0}\sum_{n\geq 0}r1_{(}m,n)x^{m}q^{n}-\sum_{m\geq
0}\sum_{n\geq 0}r_{2}(m,n)x^{m}q^{n}=\\\ \sum_{m\geq 0}\sum_{n\geq
0}r_{1}(m,n)x^{m+1}q^{n+m+1}+\sum_{m\geq 0}\sum_{n\geq
0}r_{2}(m,n)x^{m+1}q^{n+m+1}\end{split}$

so, one will get the functional equation as follows

$R_{1}(x)-R_{2}(x)=xqR_{1}(xq)+xqR_{2}(xq)$

Using steps described on [3] and [4], with straightforward but long
computations which we skipped here, by

$R_{i}(x)=\sum_{n\geq
0}\alpha_{n}(x)q^{nA_{i}}+\beta_{n}(x)x^{B_{i}}q^{C_{i}}q^{nD_{i}},\;\;i=1,2$

we can find $\alpha_{n}$ and $\beta_{n}$ in terms of $\alpha_{0}$,

$\alpha_{n}(x)=\tilde{\alpha}_{0}(xq^{2n})\frac{x^{2n}q^{n(2n+1)}q^{-nA_{2}}(-1;q^{E})_{n}(-x^{F}q^{G}q^{F-H}q^{nH};q^{2F-H})_{\infty}}{(q^{E};q^{E})_{n}(x^{F}q^{G}q^{F-H}q^{nH};q^{2F-H})_{\infty}}$

and

$\beta_{n}(x)=-\tilde{\alpha}_{0}(xq^{2n+1})\frac{x^{2n+1}q^{(n+1)(2n+1)}x^{-B_{2}}q^{-C_{2}}q^{-nD_{2}}(-1;q^{E})_{n+1}(-x^{F}q^{G}q^{2F-H}q^{nH};q^{2F-H})_{\infty}}{(q^{E};q^{E})_{n}(x^{F}q^{G}q^{nH};q^{2F-H})_{\infty}}$

where

$\tilde{\alpha}_{0}(xq^{2n})=\alpha_{0}(xq^{2n})\frac{((xq^{2n})^{F}q^{G}q^{F-H};q^{2F-H})_{\infty}}{(-(xq^{2n})^{F}q^{G}q^{F-H};q^{2F-H})_{\infty}}.$

Then, by equation (2) and considering some assumptions for equations to be
consistent, we can find $E$, $F$, $G$ and $H$, in this case $F=G=H=1$ and
$E=-1$.

Putting them in the generating functions for $R_{i}(x)$, being constant of
$\tilde{\alpha}_{0}$ with respect to $x$, and applying $x=1$, we will have

$R_{1}(1)=\sum_{n\geq
0}\frac{(-1)^{n}q^{n(2n+1)}(-1;q)_{n}(-q^{n+1};q)_{\infty}}{(q)_{n}(q^{n+1};q)_{\infty}}-\frac{(-1)^{n}q^{(n+1)(2n+2)}(-1;q)_{n+1}(-q^{n+2};q)_{\infty}}{(q)_{n}(q^{n+1};q)_{\infty}}$

and

$R_{2}(1)=\sum_{n\geq
0}\frac{(-1)^{n}q^{n(2n+2)}(-1;q)_{n}(-q^{n+1};q)_{\infty}}{(q)_{n}(q^{n+1};q)_{\infty}}-\frac{(-1)^{n}q^{(n+1)(2n+1)}(-1;q)_{n+1}(-q^{n+2};q)_{\infty}}{(q)_{n}(q^{n+1};q)_{\infty}}.$

they can be rewritten as follows,

$R_{1}(1)=2\frac{(-q)_{\infty}}{(q)_{\infty}}\sum_{n\geq
0}(-1)^{n}q^{n(2n+1)}(\frac{1}{1+q^{n}}-\frac{q^{3n+2}}{1+q^{n+1}})$

and

$R_{2}(1)=2\frac{(-q)_{\infty}}{(q)_{\infty}}\sum_{n\geq
0}(-1)^{n}q^{n(2n+2)}(\frac{1}{1+q^{n}}-\frac{q^{n+1}}{1+q^{n+1}}).$

So,

$\begin{split}R_{1}(1)=2\frac{(-q)_{\infty}}{(q)_{\infty}}(\sum_{n\geq
0}\frac{(-1)^{n}q^{n(2n+1)}}{1+q^{n}}-\sum_{n\geq
0}\frac{(-1)^{n}q^{n(2n+1)}q^{3n+2}}{1+q^{n+1}})\\\
=\frac{(-q)_{\infty}}{(q)_{\infty}}(1+2(\sum_{n\geq
1}\frac{(-1)^{n}q^{n(2n+1)}}{1+q^{n}}-\sum_{n\geq
1}\frac{(-1)^{n-1}q^{2n^{2}}}{1+q^{n}}))\\\
=\frac{(-q)_{\infty}}{(q)_{\infty}}(1+2\sum_{n\geq
1}(-1)^{n}q^{2n^{2}})=\frac{(-q)_{\infty}}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}(-1)^{n}q^{2n^{2}}\end{split}$

By theorem 1.1 for $z=-1$ and $q^{2}$, for 2-colored Rogers-Ramanujan defined
in 2.0.1 the following identity holds

$R_{1}(1)=\frac{(-q)_{\infty}(q^{2},q^{2},q^{4};q^{4})_{\infty}}{(q)_{\infty}}$

Moreover, the coefficients in the Taylor series of $R_{2}(1)$ coincides with
the number of partitions for 2-colored Rogers-Ramanujan partitions with parts
more than $1$,

$1+2q^{2}+2q^{3}+2q^{4}+4q^{5}+6q^{6}+8q^{7}+10q^{8}+14q^{9}+18q^{10}+\cdots.$

##  3\. Correspondence with Overpartitions

There is a one-to-one correspondence between 2-colored Rogers-Ramanujan type
partitions and previously defined overpartitions $D_{k,a}(n)$ for k=a=2.

Let $\pi=(y_{1},\cdots,y_{i},y_{i+1},\cdots,y_{m})$ be an arbitrary partition
of $n$, first of all, for both cases all parts are distinct. Secondly, for the
case that there are $t$ number of consecutive parts, for the colored case,
there are only two possibilities, they should be alternatively red and black,
e.g. for three consecutive parts $i$, $i+1$ and $i+2$ cases are

$\cdots,i,\color[rgb]{1,0,0}{i+1}\color[rgb]{0,0,0},i+2,\cdots\;\;\text{and}\;\;\cdots,\color[rgb]{1,0,0}i\color[rgb]{0,0,0},{i+1},\color[rgb]{1,0,0}{i+2}\color[rgb]{0,0,0},\cdots$

which means two consecutive parts can not be of the same color, and for
overpartition case, the first $t-1$ parts should be overlined and there are
two possibilities for the last one, e.g. for three consecutive parts we will
have

$\cdots,\overline{i},\overline{i+1},i+2,\cdots\;\;\text{and}\;\;\cdots,\overline{i},\overline{i+1},\overline{i+2},\cdots$

implies the first one should be overlined and the second one can be overlined
or non-overlined. If $y_{i+1}-y_{i}>1$, then there are four cases for both
colored cases and overpartition one,

$\cdots,y_{i},y_{i+1},\cdots\;\;\text{and}\;\;\cdots,y_{i},\color[rgb]{1,0,0}y_{i+1}\color[rgb]{0,0,0},\cdots\;\;\text{and}\;\;\cdots,\color[rgb]{1,0,0}y_{i}\color[rgb]{0,0,0},y_{i+1},\cdots\;\;\text{and}\;\;\cdots,\color[rgb]{1,0,0}y_{i}\color[rgb]{0,0,0},\color[rgb]{1,0,0}y_{i+1}\color[rgb]{0,0,0},\cdots$

and

$\cdots,y_{i},y_{i+1},\cdots\;\;\text{and}\;\;\cdots,y_{i},\bar{y}_{i+1},\cdots\;\;\text{and}\;\;\cdots,\bar{y}_{i},y_{i+1},\cdots\;\;\text{and}\;\;\cdots,\bar{y}_{i},\bar{y}_{i+1},\cdots$

so, there exists a one-to-one correspondence between them.

It is not hard to see another correspondence between $D_{2,1}(n)$ and the
following partition type,

######  Definition 3.0.1.

Let $R_{3}(n)$ denote the number of 2-colored Rogers-Ramanujan partitions
which do not allow to have a red $1$ in the partition.

Then

######  Theorem 3.1.

For definition 3.0.1 the following identity holds

$R_{3}(n)=D_{2,1}(n)=\frac{(-q)_{\infty}(q^{1},q^{3},q^{4};q^{4})_{\infty}}{(q)_{\infty}}$

##  4\. Future Works

There are two options for further work on this topic, first one is to extend
the number of colors to more than two, ideally for arbitrary number of colors.
Our second suggestion is two extend it to 2-colored Gordon-Rogers-Ramanujan
type partitions, i.e. for 2-colored partition $y_{1}+\cdots+y_{m}$, and for
$k\geq 1$, $y_{j}-y_{j+k-1}\geq 1$ if $y_{j}$ is black and
$y_{j}-y_{j+k-1}\geq 2$ otherwise.

## References

  * [1] Agarwal, A. Adnrews, G. Rogers-Ramanujan Identities for Partitions with ”N Copies of N”. Journal of Combinatorial Theory, series A 45, 40-49 (1987).
  * [2] Andrews, G.E. The Theory of Partitions. The Encyclopedia of Mathematics and Its Applications Series. Addison-Wesley, New York (1976). Reissued. Cambridge University Press, New York (1998)
  * [3] Kurşungöz, K. Andrews style partition identities. Ramanujan J 36, 249–265 (2015).
  * [4] Kurşungöz, K. Bressoud style identities for regular partitions and overpartitions. Journal of Number Theory 168, 45-63 (2016).
  * [5] W. Chen, D.D. Sang, D.Y. Shi, The Rogers-Ramanujan-Gordon theorem for overpartitions, Proc. Lond. Math. Soc. 106(3), 1371–1393 (2013)
  * [6] Sandon, C. Zanello, F. Warnaar’s bijection and colored partition identities, I. Journal of Combinatorial Theory, series A 120, 28-38 (2013).


# Visualization Techniques to Enhance Automated Event Extraction

Sophia Henn
University of Notre Dame<EMAIL_ADDRESS>
Abigail Sticha
University of Notre Dame<EMAIL_ADDRESS>
Timothy Burley
University of Notre Dame<EMAIL_ADDRESS>
Ernesto Verdeja
University of Notre Dame<EMAIL_ADDRESS>
Paul Brenner
University of Notre Dame<EMAIL_ADDRESS>
###### Abstract

Robust visualization of complex data is critical for the effective use of NLP
for event classification, as the volume of data is large and the high-
dimensional structure of text makes data challenging to summarize succinctly.
In event extraction tasks in particular, visualization can aid in
understanding and illustrating the textual relationships from which machine
learning tools produce insights. Through our case study which seeks to
identify potential triggers of state-led mass killings from news articles
using NLP, we demonstrate how visualizations can aid in each stage, from
exploratory analysis of raw data, to machine learning training analysis, and
finally post-inference validation.

##  1 Introduction

As the number, capacity, and complexity of automated event extraction tools
and techniques grow, researchers must continue to enhance their capability to
quickly visualize massive data sets and incremental artifacts at key points in
the event classification and extraction workflow. In service to this need we
present a summary of related visualization tools followed by visualization
techniques that our team finds critical to enhancing the automation efficiency
and event extraction validation.

###  1.1 Related Works

Visualization plays a unique role in the comprehension of linguistic
structures. A large body of literature is dedicated to exploring how text data
can be best summarized and represented in lower dimensions. These studies
describe how visualizations including network representations, dendrograms,
radar charts, and vector maps of word embeddings are suitable to communicate
syntactical and semantic relationships within text Santini et al. (2020);
Drieger (2013); Alharbi and Laramee (2018).

Visualization similarly supports a range of machine learning tasks with text
data including topic modeling, information retrieval, sentiment analysis and
classification Song et al. (2011); Schlechtweg et al. (2004); Cao et al.
(2015); Brooks et al. (2015). Since raw text usually undergoes several
preprocessing steps to transform it into a usable input for different models,
visualization of the text at different stages helps to provide insight into
the model and call attention to features that might negatively impact model
performance. Liu et al. (2019) summarize a range of common techniques suitable
for each type of task, including typographic visualizations, spatial
projections, and topological maps. These visualizations generally take
tokenized text or metadata as inputs and communicate insights specific to the
task at hand (summarize the text, identify outliers, or explore differences in
training data for supervised learning algorithms).

Visualization is also increasingly being leveraged to aid in model selection
and validation Liu et al. (2017); Lu et al. (2016). Chatzimparmpas et al.
(2020) overview visualization techniques used for model interpretation and
refinement, and Garcia et al. (2018) emphasize the use of visualizations to
aid in parameter tuning and feature selection of neural networks. Ventocilla
and Riveiro (2020) use visual encodings of several dimensionality reduction
techniques to compare human and machine estimates of clusters in a dataset.
This paper summarizes how visualization techniques have been useful to garner
insights into our complex data and particular event classification task.

###  1.2 Triggers of Mass Killings Case Study

Our case study aims to utilize statistical and NLP tools for a systemic
analysis of triggers of state-led mass killings (ToMK). Peace and conflict
researchers have identified several large-scale structural conditions that
make state-led mass killings more likely, such as political instability, a
history of violence against vulnerable groups, radical political ideologies,
and autocratic or authoritarian governments. Several existing models aim to
forecast the escalation of violent conflict Hegre et al. (2019); Goldsmith et
al. (2013); Goldstone et al. (2010). However, the timing of mass killing onset
is less understood. Analysts have identified several plausible trigger-type
events including coups, assassinations, protests/riots, armed conflict
escalation, cancelled elections, neighboring conflict spillover and others,
but little systematic analysis beyond specific country case studies has been
conducted to examine whether, and if so when, these events actually trigger
mass killings.

This project canvasses a large number of news sources to identify and examine
the occurrence of nine trigger-type events across both countries that have had
a mass killing event and those that have not between 1989-2017, and analyzes
under what conditions - and in what patterns and sequences - certain trigger-
type events increase the probability of mass killings (Figure 1). It seeks to
bring greater specificity and understanding about the timing of state-led mass
killings, which is of interest to both scholars and peacebuilders.

The ToMK project utilizes SVMs and BERT Liu et al. (2010); Devlin et al.
(2019); Géron (2019), to identify patterns in the text data and classify news
articles as a positive or negative instance of a trigger event. The SVM script
preprocesses the raw text, vectorizes each article using a tf-idf weighting
scheme, and defines a classification hyperplane to separate the classes of
articles. We then use this kernel to classify unlabeled data.

Figure 1: Potential triggers and structural conditions for state-led mass
killings.

###  1.3 Data Specification

This paper focuses on the coup/attempted coup trigger. The coup data consists
of the text of news articles retrieved via LexisNexis queries based on several
search parameters which include a date filter from 1989-2017, a source filter
for our list of 20 sources, and keywords including the word “coup” or related
terms. The articles are compiled into CSV files with columns for the article
text, date of publication, unique document id, and country label based on the
keyword in the search. The calculations for the maximal corpus yield a worst-
case estimate of 1.8 million articles across all countries. The visualizations
in this paper use a subset of this corpus (“select countries corpus”) that
contains the 69 countries with identified state-led mass killing events
between 1989-2017. This select countries corpus contains 602,939 articles. A
subset of this corpus isolates the two-year window preceding each mass killing
event (“dependent space corpus”) and contains 51,488 articles. The machine
learning training set consists of 551 articles retrieved in the same manner
with an additional CSV column of human-coded labels indicating whether the
article is describing a qualified coup event.

##  2 Raw Data Visualizations

Figure 2: Correlation plot of article characteristics for insight into
potentially problematic articles and their frequency across the corpus.

Initial, exploratory visualizations are useful to summarize large corpora and
confirm that the search parameters were successful in returning events of
interest. Necessary initial preprocessing includes transforming text to
lowercase, tokenization, and removing non-alphanumeric characters.

To describe the shape of the corpus, histograms of word counts can be
informative by communicating the variation in article length. It is useful to
consider what lengths one expects given the genre of text in the corpus.
Social media posts, news articles, or novels all carry different expectations
of lengths, so histograms can be used to view this distribution and any
outliers. In our particular case study we are interested in news articles, so
we expect most text to fall under 2000 words. Histograms can also confirm that
the data was loaded properly; some programs, like Microsoft Excel, have
default maximum character counts per cell, so a cutoff around a particular
word count on the histogram could signal an issue with data importing.

Word clouds or variations like scattertext are also popular to summarize the
word content of the corpus. However, while word clouds can be descriptive,
they can be misleading regarding what words are significant towards event
classification tasks. Section 3 will visualize significant word tokens towards
classification in our study after applying the tf-idf weighting scheme.

News articles can take on many structures, with some more complicated than a
description of a single event that occurred recently. The presence of articles
with complex formats impacts our event classification task in two primary
ways. Firstly, if an article describes a single coup event along with other
world events, there will be some minimal language describing a coup but a
predominance of unrelated words; this may impact the weights of keywords and
lead to more frequent misclassification. Secondly, the country or date labels
for the coup event may be incorrect since the labels are extracted from the
keyword search parameters for that query. News digests are an example of these
types of articles that summarize many events around the world for a given week
or month, and they are published among standard news articles by several of
our sources.

We identified three characteristics associated with news digests: higher word
counts, counts of unique country names, and counts of unique years. For each
article, the word count, country count and year count was calculated and all
articles were plotted for a particular country. Figure 2 plots these
characteristics for articles with the country label Nigeria. This
visualization confirms the correlation between these characteristics, and
display how common news digests may be within our corpus. Articles above a
percentile for the three characteristics can be marked for human reader
validation after classification or thrown out of the corpus before feature
extraction and classification.

##  3 Machine Learning Visualizations

Figure 3: The most significant tokens towards classification of the training
set, measured by tf-idf score.

After conducting exploratory visualizations to gain insights into our corpus
and clean it, elements of the ML pipeline can be visualized to better
understand how articles are classified. Further preprocessing steps are
necessary to prepare the text for feature extraction, which will reduce the
article text into its most informative elements for classification. Here,
stop-word removal, parts-of-speech tagging, stemming and lemmatizing
procedures are applied to the training set and dependent space corpus.

Figure 4: 2D projection of the training set documents with an example SVM
classification line. Figure 5: Timeline of extracted articles describing coup
events for a subset of countries in the corpus.

Visualization of the computed features in the training set offers insight into
the linguistic differences between the positive and negative classes. Our
parameter was set to select 5000 features, but this can be adjusted with
impacts on classification performance. Figure 3 illustrates the list of top 20
tokens by tf-idf score for each class. Here, we learn that many proper nouns
(country names and leaders of coups in those countries) are weighted as
significant indicators that an article is describing a coup event. After
classification, this visualization can be repeated for the dependent space
corpus and compared to the training set.

The machine learning algorithm will use these feature scores to classify each
article, so visualizing the classifier is also useful. Dimensionality
reduction techniques such as PCA, t-SNE or UMAP can be used to plot the
features into 2-dimensional space and illustrate a classifier. Figure 4
illustrates the use of UMAP McInnes et al. (2020) to generate a projection of
the training set. The true classifier is computed in 5000 dimensions, but an
approximate linear classifier demonstrates the class separation.

##  4 Classification Validation Visualizations

After the machine learning tool performs event classification on the unseen
articles, visualizing the fully-labeled corpus can help to verify that the
classifications make sense in light of any challenges discovered during
previous steps. In general, the project hypothesis will determine the type of
post-classification visualizations that are most useful in summarizing the
classification results. Because our case study seeks to understand the
proximity and dynamics of different triggering events to state-led mass
killings, plots like timelines and heatmaps are most suitable to visualize
sequences and trends.

For each country, we obtain the list of positively-classified articles from
1989-2017. We distinguish between articles in the two-year window preceding a
mass killing event (dependent space corpus in blue) and those at any other
time (null space of select countries corpus in grey). Information on coups
outside the two-year window allows us to establish controls on their
occurrence.

To check the distribution of positively-classified articles for a country
across 1989 – 2017, the articles can be visualized on a timeline scatterplot,
shown in Figure 5. The plotly library has several features to make
visualizations interactive, allowing the user to add details for each article.
In cases where many positively-classified articles are plotted in a time
period when there were no known coups, this type of interactive visualization
is useful to quickly learn about those articles.

This visualization also revealed a discrepancy in the classifications of the
dependent space and select countries corpus: within the two-year window for
countries including South Africa, Uganda, and Venezuela, there were instances
where positive articles appeared in the select countries corpus but not the
dependent space corpus. This could have happened if the LexisNexis queries did
not pull down the same number of articles for the two-year window in each
corpus, or if some articles were tagged differently during classification of
each corpus. This is possible because the tf-idf vectorizer was fit only on
the data set to be classified for each task. Because tf-idf assigns a weight
to each word that is inversely proportional to its frequency across documents
in the corpus, the same word can be weighted differently in the dependent
space corpus and select countries corpus. If the differences in weights are
large enough, the algorithm may classify the same article differently in each
corpus.

##  5 Summary

Visualization techniques are essential to effectively wrangle the complex data
and ML tools employed for automated event extraction. The code for generating
visualization examples in this paper is publicly available in the ToMK GitHub
repository https://github.com/crcresearch/ToMK.

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# Metallicity beats sSFR: The connection between superluminous supernova host
galaxy environments and the importance of metallicity for their production

Cressida Cleland,1 Sean L. McGee,1 Matt Nicholl2
1School of Physics and Astronomy, University of Birmingham, Edbgaston,
Birmingham B15 2TT, UK
2Astrophysics Research Centre, School of Mathematics and Physics, Queen’s
University Belfast, Belfast BT7 1NN, UK E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We analyse 33 Type I superluminous supernovae (SLSNe) taken from ZTF’s Bright
Transient Survey to investigate the local environments of their host galaxies.
We use a spectroscopic sample of galaxies from the SDSS to determine the
large-scale environmental density of the host galaxy. Noting that SLSNe are
generally found in galaxies with low stellar masses, high star formation
rates, and low metallicities, we find that SLSN hosts are also rarely found
within high-density environments. Only $3\begin{subarray}{c}+9\\\
-1\end{subarray}$ per cent of SLSN hosts were found in regions with 2 or more
bright galaxies within 2 Mpc. For comparison, we generate a sample of 662 SDSS
galaxies matched to the photometric properties of the SLSN hosts. This sample
is also rarely found within high-density environments, suggesting that
galaxies with properties required for SLSN production favour more isolated
environments. Furthermore, we select galaxies within the Illustris-TNG
simulation to match SLSN host galaxy properties in colour and stellar mass. We
find that the fraction of simulated galaxies in high-density environments
quantitatively matches the observed SLSN hosts only if we restrict to
simulated galaxies with metallicity $12+\log($O/H$)\leq 8.12$. In contrast,
limiting to only the highest sSFR galaxies in the sample leads to an
overabundance of SLSN hosts in high-density environments. Thus, our
measurement of the environmental density of SLSN host galaxies appears to
break the degeneracy between low-metallicity or high-sSFR as the driver for
SLSN hosts and provides evidence that the most constraining factor on SLSN
production is low-metallicity.

###### keywords:

galaxies: evolution, star formation transients: supernovae

††pubyear: 2022††pagerange: Metallicity beats sSFR: The connection between
superluminous supernova host galaxy environments and the importance of
metallicity for their production–2

##  1 Introduction

While the detection rates of supernovae have rapidly increased over the last
decade or so, thanks to wide-field high-cadence surveys such as Pan-STARRs
(Chambers et al., 2016), ATLAS (Tonry et al., 2018) and ZTF (Bellm et al.,
2019), superluminous supernovae (SLSNe) remain a rare and elusive transient
event. SLSNe were originally classed as a supernova event that is 10-100 times
brighter than typical supernovae (Quimby et al., 2011; Gal-Yam, 2012), however
are now classified by their unique spectra (e.g. Lunnan et al., 2018; Quimby
et al., 2018; Angus et al., 2019). Their lightcurves differ from those of
other supernovae as they are both broader and brighter. There are two classes
of SLSNe: Type I and Type II. Type I SLSNe (often referred to as simply SLSNe,
a convention we adopt in this work) are dominated by OII absorption lines at
maximum luminosity, while Type II SLSNe exhibit sharply peaked hydrogen
emission lines from circumstellar interaction. The physical mechanisms which
cause Type I SLSNe remain a topic of debate, with mechanisms such as a central
engine in the form of a magnetar (see e.g. Kasen & Bildsten, 2010; Inserra et
al., 2013; Nicholl et al., 2017) or interaction with the circumstellar medium
(see e.g. Smith & McCray, 2007; Chevalier & Irwin, 2011; Chatzopoulos et al.,
2012; Ginzburg & Balberg, 2012) being proposed as possible power sources for
the increased luminosity.

Table 1: Descriptions of each of the samples used. The ranges shown in the redshift and magnitude columns refer to the minima and maxima of that property for each sample, not necessarily the selection constraints. Sample | # obj. | Redshift | Median(z) | Magnitude | Median($M_{i}$) | Description
---|---|---|---|---|---|---
SLSN-g | 33 | $0.064\leq z\leq 0.39$ | 0.159 | $-22.74\leq M_{i}\leq-15.32$ | -18.41 | SLSN host galaxies
SDSS-m | 662 | $0.036\leq z\leq 0.312$ | 0.117 | $-22.196\leq M_{i}\leq-16.709$ | -19.632 | SDSS photometric-z galaxies matched to SLSN-g
SN-Ia | 3268 | $0.001\leq z\leq 0.0.156$ | 0.065 | $-26.56\leq M_{i}\leq-11.56$ | -20.76 | Type Ia supernova host galaxies
Spec-z | 300000 | $0.0\leq z\leq 0.4$ | 0.144 | $-31.214\leq M_{i}\leq-10.006$ | -22.385 | SDSS spectroscopic-z galaxies

We can use properties of their host galaxies to infer the conditions required
for a SLSN to occur. This can help constrain the processes at work. SLSNe have
been found mostly in low-mass galaxies, with very few being found in host
galaxies with stellar mass $>10^{8}$ M⊙ (Schulze et al., 2018). This is
surprising, as one might expect that with more stars comes more supernova
events (Sullivan et al. (2006); Smith et al. (2012); Wiseman et al. (2021),
see also Perley et al. (2016a)). Various authors have found that SLSN host
galaxies typically have high specific star-formation rates ($\sim 10^{-9}$
yr-1, Neill et al., 2011; Leloudas et al., 2015; Angus et al., 2016), leading
to the idea that SLSNe may be the first stars to explode following a
starburst. Leloudas et al. (2015) note that SLSN host galaxies have properties
which are consistent with extreme emission line galaxies (EELGs), i.e. high-
sSFR and also highly ionised gas. Meanwhile, Lunnan et al. (2014), Chen et al.
(2017) and Perley et al. (2016b) argue that the low-metallicity observed in
SLSN hosts is the more important factor in the production of these rare
transients, and that their low metallicities can explain the extreme energies
seen (Chen et al., 2013). Nevertheless, since dwarf galaxies tend to have both
low metallicities and bursty star-formation histories, and the production of
SLSNe appears averse to high-metallicity and low specific star-formation rate
(Schulze et al., 2018), there is a degeneracy regarding which of these factors
is more critical.

Figure 1: PanSTARRS $i$-band cut-out images (Flewelling et al., 2020) of SLSN
sources and of a SN Type Ia host. Panel a) shows object ZTF21aaarmti, panel b)
shows object ZTF22aalzjdc and panel c) shows object ZTF18aangpkx. The magenta
circle in each panel depicts the location of the source. In panel b), no host
galaxy was found for the source. The host galaxy in panel c) is in a higher-
density environment than the SLSN hosts; see Section 3.1 for details. The
absolute magnitude of the host galaxy in panel a) is $M_{i}=-21.36$ and its
redshift is $z=0.193$. The absolute magnitude of the host galaxy in panel c)
is $M_{i}=-22.67$ and its redshift is $z=0.070$. The redshift of the SLSN
source in panel b) is $z=0.200$. The size of each of the images is
$62.50^{\prime\prime}$.

It is well known that the local environment of galaxies, that is whether a
galaxy is located within a galaxy group or a galaxy cluster or it is isolated,
has a profound effect on a number of galaxy properties, including the
metallicity (Tremonti et al., 2004; Ellison et al., 2009; Cooper et al.,
2009), star formation rate (Wetzel et al., 2013; Cleland & McGee, 2021) and
overall evolution of the galaxy (Dressler, 1980). Gravitational interactions
between galaxies can disturb the gas and dust which fuels star-formation,
leading to starburst phases, or similarly galaxies may merge which will also
have an effect on the star-formation rates of the galaxies involved.
Conversely, high-density environments may cause the quenching of star-
formation in an infalling galaxy. It naturally follows to investigate the
local environment of SLSNe host galaxies to uncover any environmental factors
that may impact SLSNe rates. Ørum et al. (2020) find that up to 50 per cent of
superluminous supernova host galaxies have at least one companion within 5
kpc, which they use to explain the increased star-formation rates required for
SLSNe to occur. However, for the purposes of this work we are interested in
the overall density of the local environment of the SLSN host galaxy. That is,
a close companion galaxy of arbitrary stellar mass within kiloparsecs of the
supernova host galaxy does not necessarily lead to the same environmental
effects at larger scales (e.g., Mpc). In order to probe for such effects, we
aim to use bright galaxies as a proxy for density, noting that this is a first
order approach, with the absolute magnitude of galaxies roughly tracing
stellar mass. By investigating the environments of these galaxies in this way,
we aim to break the degeneracy between the requirements of low-metallicity and
high specific star-formation rate for SLSN production.

In this work we identify isolated SLSNe and SLSNe in groups, and compare
between various host galaxy properties. We also compare against Type Ia
supernova host galaxies, which occur in a wider variety of host galaxies. We
then investigate these host galaxies further within the Illustris-TNG
simulation suite, by exploring the metallicities and specific star-formation
rates of a sample of simulated galaxies with observational properties similar
to those of SLSN hosts.

The paper is structured as follows: in Section 2, we describe the data used
and how it was obtained; in Section 3 we explain the results and discuss their
implications; finally in Section 4 we summarise our findings. For the
computation of cosmological distances, we use a flat universe with
$H_{0}=70.2$ and $\Omega_{m}=0.277$.

Figure 2: Histograms of the colours, absolute magnitudes, and redshifts of
galaxies in SLSN-g with reliable photometry compared to the matched SDSS
sample. KS-tests result in p-values of 0.46, 0.36, and 0.75 respectively,
validating that the SLSN-g and SDSS-m samples match each other.

##  2 Data

We obtain coordinates, redshifts and host photometry ($g-i$ colours and
$M_{i}$ absolute magnitudes) of 55 Type I SLSNe (hereafter simply referred to
as SLSNe) and 4816 Type Ia supernovae from the ZTF Bright Transient
Survey111https://sites.astro.caltech.edu/ztf/bts/bts.php (BTS; Masci et al.,
2019; Perley et al., 2020a). This was the full sample of Type I SLSNe and Type
Ia SNe available on the BTS at the time of analysis (September 2022). The BTS
is used to obtain properties on supernova events and their host galaxies due
it being a well-defined sample with a limiting magnitude of $\approx 18.5$ mag
that selects for nearby events with a higher chance of host detection. The
supernovae in this sample have been classified spectroscopically, and by a
variety of research groups. Table LABEL:tab:_slsnelist lists and cites the
classification group for each SLSN. The code
Sherlock222https://github.com/thespacedoctor/sherlock is used to cross-
identify for a host galaxy and its photometry (Smith et al., 2020); this
information is available directly on the Lasair broker (Smith et al., 2019)
for each source. Some sources have no catalogued host, and are marked as
‘orphans’. See Figure 1 for examples of a SLSN source with a host detection
(left panel) and a SLSN source where no host was detected (middle panel). We
denote the SLSN host galaxy sample as SLSN-g, and the Type Ia supernova host
galaxy sample as SN-Ia.

We obtain a spectroscopically-redshifted sample of $300000$ random galaxies
(Spec-z) from the Sloan Digital Sky Survey (SDSS) DR17 (Smee et al., 2013;
Blanton et al., 2017; Abdurro’uf et al., 2022), such that their redshifts are
local and precisely-known, $0<z<0.4$. We also obtain 16494 random
photometrically-redshifted galaxies from SDSS DR17 (Fukugita et al., 1996;
Gunn et al., 1998, 2006; Doi et al., 2010) to better match the parameter space
of the SLSNe sample, with $0<z<0.4$, such that $z_{\text{err}}<0.1$ and
$z_{\text{err}}/z<0.5$. Both samples were obtained using the SDSS query
service CasJobs333http://cas.sdss.org/dr17/. For each sample of supernova
hosts, we restrict to sources within the on-sky footprint of the spectroscopic
sample, resulting in 35 SLSNe (18 with detected hosts) and 3450 Type Ia
supernovae.

###  2.1 Matched sample generation

Since the number of SLSN host galaxies is so low, it is desirable to generate
a matched galaxy sample, based on SLSN host galaxy properties. This means we
can make statistical comparisons of SLSN host-like galaxies, but on a much
larger sample of galaxies. The properties of the SLSN host galaxies are based
on their photometry, so we generate a sample of galaxies that are matched
based on photometric properties such as absolute magnitude and colour. We
generate this matched sample to the SLSN host galaxies from the SDSS
photometric galaxies (Cooper et al., 2009). For each SLSN host galaxy, we
randomly search for a photometric galaxy within a sphere in parameter space
according to Equation 1:

$\left(\frac{\Delta\text{colour}}{0.2}\right)^{2}+\left(\frac{\Delta
M_{i}}{1.2}\right)^{2}+\left(\frac{\Delta z}{0.04}\right)^{2}=1,$ (1)

where $\Delta$colour, $\Delta M_{i}$ and $\Delta z$ refer to the difference
between the $(g-i)$ colour, absolute magnitude, and redshift of the SLSN host
galaxy and the photometric galaxy. The scaling factor in the denominator of
each term in Equation 1 comes from taking the range of each property and
multiplying by a factor of 0.1. This factor was chosen by visual inspection to
ensure an adequate chance of finding an appropriate match, while reducing
mismatches in parameter space. Applying this factor ensures each property has
the same weight in parameter space. We attempt a random search for a unique
SDSS galaxy that satisfies Equation 1 on a SLSN host galaxy; this search gets
repeated until a match is found, to a maximum of 1000 times. This process is
repeated on SLSN host galaxies 10000 times. This results in a matched sample
(SDSS-m) of 662 galaxies. Note that varying the multiplicative factor in
Equation 1, and the number of repetitions performed on each galaxy, has no
quantitative effect on the results discussed in Section 3.1. For this matching
procedure, we only use the 18 SLSN sources with detected hosts and reliable
host photometry. The rest of the analysis depends on the positions and
redshifts of each SLSN source directly, and so the entire sample of 33 SLSNe
is used.

The details of each sample are listed in Table 1. Histograms of the
distributions of the colour, absolute magnitude, and redshift of SLSN-g and
SDSS-m are plotted in Figure 2. Kolmogorov-Smirnov tests validate that the
distributions match well, with the added benefit of over an order of magnitude
more galaxies in the matched sample than SLSN host galaxies.

##  3 Results and discussion

###  3.1 Observations

Figure 3: A scatter plot of the number of bright Spec-z galaxies within 2 Mpc
of a target galaxy against absolute magnitude, compared to a 2d histogram of
the same distribution for Type Ia SNe host galaxies. SLSN-g is shown as dark
purple circles and the matched sample, SDSS-m, is shown as light purple
triangles. Normalised histograms are included on the $x-$ and $y-$axes and
show the absolute magnitude and $N_{2}$ distributions for SN-Ia, SDSS-m and
SDSS-g.

Using bright, well-observed galaxies as a proxy for density, we count the
number of bright ($M_{i}<-21.5$) Spec-z galaxies that are at an on-sky
projection of 2 Mpc or less of the target galaxy, within $\Delta z=0.005$,
based off a typical velocity dispersion of a galaxy group/cluser (Yang et al.,
2007). We denote this quantity as $N_{2}$. We use $M_{i}<-21.5$ as the
threshold for bright galaxies as this magnitude is the completion limit for
the entire spectroscopic sample for this redshift range, and we use 2 Mpc as
it is the upper end of galaxy group diameters (Yang et al., 2007). Note that
varying the $M_{i}$ threshold does not have a qualitative effect on the
results. We calculate $N_{2}$ for SLSN-g, SDSS-m, and SN-Ia as a control. SN-
Ia works as a control sample because these galaxies are much less confined to
a specific region of parameter-space compared to SLSNe, and the photometry is
easily comparable to SLSN-g since both samples come from ZTF. A 2-d histogram
showing these results is shown in Figure 3, with $N_{2}$ shown as a function
of absolute magnitude. Note the log scale in the $z$-axis. This quantity is
also plotted for SLSN-g and SDSS-m in dark purple and light purple,
respectively. We can see that while the distribution $N_{2}$ for SN-Ia
occupies a large and varied region of parameter-space (note 70 per cent of SN-
Ia still has $N_{2}=0$), $N_{2}\leq 2$ for SLSN-g and $N_{2}\leq 3$ for
SDSS-m, with the majority of galaxies having $N_{2}=0$. This is not simply a
consequence of reduced numbers of SLSN-g compared to SN-Ia; binned by
magnitude, the fraction of SLSN-g with any bright neighbour (i.e. $N_{2}\geq
1$) reaches a maximum of about 10 per cent444This number is entirely dependent
on the choice of binning since $N_{2}\approx 1$ for SLSN-g in any magnitude
bin, which is only half of the minimum of the fraction of SN-Ia with any
neighbour in the same bins. Similarly, SDSS-m only reaches a maximum of about
10 per cent.

Figure 4: A colour-magnitude diagram comparing SLSN-g and SDSS-m to SN-Ia.
Low-density SLSN-g galaxies are plotted in purple circles, high-density SLSN-g
galaxies are plotted in orange pluses, low-density SDSS-m galaxies are plotted
in purple crosses, high-density SDSS-m are plotted in orange triangles, and
the distributions for low- and high-density SN-Ia galaxies are plotted as
purple filled in contours and orange contour lines respectively. Open purple
circles represent the SLSN-g sources with unreliable host photometry. Purple
and orange points shown individual SN-Ia galaxies when they are not captured
by the contours.

We consider galaxies with $N_{2}\geq 2$ as being in a high-density
environment, with everything else being in a low-density environment. This
number is chosen under the assumption that a typical galaxy group or cluster
halo is of the order of a few Mpc in diameter (Yang et al., 2007), so any more
than two bright (large) galaxies would be considered a high-density
environment. We emphasise the fact that this is a generous constraint, set in
an attempt to maximise the number of SLSN hosts in high-density environments.
An example of a SN-Ia host galaxy with a high $N_{2}$ number ($N_{2}=8$) is
shown in the right panel of Figure 1. We plot SLSN-g, SDSS-m, and SN-Ia on a
colour-magnitude diagram in Figure 4, separated by high- and low-density.
Contour lines map out the SN-Ia galaxies that are in high-density
environments. We see that the high-density SLSN-g and SDSS-m galaxies are
found on the outer regions of these contour lines. That is, they
preferentially avoid the region in phase-space where high-density environments
occur. This result, along with finding that the matched sample galaxies are
also almost exclusively found in low-density environments, is indicative that
the galactic properties of SLSN hosts are the constraining factor in the
density of the environment of these galaxies. In other words, the type of
galaxies where SLSNe are found (blue, low-mass, low-metallicity) are rarely
found in high-density environments. As a fraction of each sample, SLSNe that
occur in high-density environments is $1/33=0.03\pm^{0.06}_{0.01}$, and
matched galaxies that occur in high-density environments is even lower, at
$4/662=0.006\pm^{0.005}_{0.002}$.

Figure 5: Histograms of metallicity (left) and sSFR (right) in IllustrisTNG
for all blue low-mass (BLM) galaxies (black), BLM field galaxies (purple) and
BLM group galaxies (orange).

###  3.2 Simulations

Since these galaxies are so faint, often the reliable host photometry or
spectroscopy required to accurately measure physical properties like
metallicity and star-formation rate is not available. Therefore it can be
useful to use simulations to distinguish between real physical properties, and
observational constraints. In Section 3.3 we utilize the IllustrisTNG suite of
simulations555https://www.tng-project.org/data/ (Pillepich et al., 2018;
Springel et al., 2018; Nelson et al., 2018; Naiman et al., 2018; Marinacci et
al., 2018; Nelson et al., 2019). IllustrisTNG is a set of large-scale
cosmological, gravo-magnetohydrodynamical simulations based on the Arepo code
(Springel, 2010). We use the TNG-100 simulation, which has a volume of
$106.5^{3}$ cMpc. This simulation provides the best balance of large haloes
(i.e. $M_{h}>10^{12}\ \text{M}_{\odot}$) and smaller resolution subhaloes
(i.e. $M_{h}<10^{9}\ \text{M}_{\odot}$), although qualitatively similar
results were found with the use of TNG-300. From this simulation, we use
stellar masses, star-formation rates, gas-phase metallicity abundances, galaxy
colours, group halo masses, and information about group membership.

###  3.3 IllustrisTNG

To investigate the tendency of SLSN host-like galaxies to avoid high-density
environments further, we make use of the IllustrisTNG suite of simulations.
From the TNG100-1 simulation, we obtain stellar masses, gas metallicities,
star-formation rates, and $g$ and $i$ photometry on all 4371211 subhaloes in
the most recent redshift snapshot. Each subhalo resides in a parent halo with
a total halo mass $M_{200}$, defined as the total mass of a sphere whose mean
density is 200 times the critical density of the Universe. In line with the
properties of SLSN host galaxies, we define a sample of blue, low-mass
galaxies (BLM), with $(g-i)<1$ and $10^{7}<M_{*}/\text{M}_{\odot}<10^{9}$.
This stellar mass range has an upper limit in line with observations of SLSN
host galaxies (e.g. Schulze et al., 2018), and a lower limit to avoid
contamination from haloes not yet formed into galaxies. Note that at this
stellar mass range, virtually all galaxies ($\sim 98$ per cent) have $(g-i)<1$
and also are star-forming. For each subhalo, we retrieve the number of
subhaloes ($N_{\text{subs}}$) in its parent halo. For a galaxy to be in a
group, we require $N_{\text{subs}}>1$ and $M_{200}>10^{12}\ \text{M}_{\odot}$.
This is to ensure we avoid including field galaxies with smaller (non-galaxy)
haloes associated with them. Everything else is assigned as being in the
field.

In Figure 5, we plot a histogram of the log of the metallicity of BLM and a
histogram of the log of sSFR, separated by group membership. Immediately it is
clear that this sample is biased towards low metallicities, as is expected
from observations of SLSN host galaxies (see Figure 11 in Chen et al. (2015),
and Figure 8 in Perley et al. (2016b)). Notably there are much fewer BLM
galaxies in groups than in the field, with group galaxies making up 35.5 per
cent of the BLM sample. Additionally, the BLM galaxies that reside in groups
lie at the high end of the metallicity distribution. We also see that BLM
galaxies typically have intermediate sSFR, as is seen in the properties of
SLSN host galaxies. However, BLM galaxies in groups dominate over field
galaxies at high-sSFR.

Thus, in agreement with our observational results in the previous section, the
simulation results qualitatively suggest that the reason SLSN-g are not found
in dense environments is because the galaxies that host SLSN, blue low-mass,
are not often found in groups. However, quantitatively, there is a potential
discrepancy. From the simulation results in Figure 5 we measure that 35.5 per
cent of such BLM galaxies are found in groups within the simulation. This
means, observationally, we would expect about 12(235) of SLSN-g(SDSS-m) to be
in high-density environments, compared to the 1(4) in SLSN-g(SDSS-m).

Figure 6: Fractions of BLM-f (purple) and BLM-g (orange) of the total BLM
sample with respect to metallicity. The gray dashed line shows the metallicity
at which group galaxies account for 3 per cent of the entire BLM sample, in
accordance with observations, with upper and lower limits shaded in gray.

This potential discrepancy is pointing to an additional requirement in our
simulated sample to properly recover the observed environmental distribution –
that of low-metallicity. It is known that SLSNe require host galaxies with low
metallicities (e.g. Lunnan et al., 2014), and in the simulation results of
Figure 5, the lower metallicity ranges are dominated by field galaxies. We
better illustrate this in Figure 6, where we plot the fractions of field
galaxies and group galaxies with respect to the total BLM sample as a function
of metallicity. Field galaxies clearly dominate at $Z<Z_{\odot}$, and group
galaxies dominate at $Z>Z_{\odot}$; here we assume $Z_{\odot}=8.69$ following
Asplund et al. (2009). We also plot the metallicity at which the fraction of
observed SLSN host galaxies in groups is 0.03. We can then apply generous
constraints by considering the uncertainties on the SLSN occurrence rate
above. By computing beta distribution confidence intervals at $c=0.68$
(Cameron, 2011) on the fraction of observed SLSNe in groups we find this
fraction to be $0.03\begin{subarray}{c}+0.09\\\ -0.02\end{subarray}$. In
Figure 6 these confidence intervals are shaded in gray. Even with such a wide
confidence interval, the highest metallicity that is consistent with the
observed environmental density of SLSN hosts, and therefore the apparent upper
limit of SLSN production is $12+\log($O/H$)=8.12$. This is approximately 0.2
dex lower than the apprent threshold reported in Schulze et al. (2018),
$12+\log($O/H$)\sim 8.3$, and 0.3 dex lower than the threshold reported in
Chen et al. (2017), $12+\log($O/H$)\sim 8.4$.

In addition, we find that only about 26 per cent of BLM galaxies in groups
reside in haloes of $M_{h}\geq 10^{13}\ \text{M}_{\odot}$. This supports the
idea that the typical galaxies that host SLSNe will rarely be found in large
groups or clusters.

The distributions in Figure 5 provide evidence that low-metallicity is the
constraining requirement for SLSN production, rather than high-SFR. Focusing
on the right panel, we see that the distribution of BLM-g in sSFR is more
evenly distributed compared to metallicity in the left panel. In particular,
noting SLSN host galaxies require sSFR$\gtrsim 10^{-9}\ $yr-1 (Schulze et al.,
2018), there is actually an excess of BLM-g galaxies at this sSFR. This means
that if a high-sSFR was the predominant factor for SLSN production compared to
low-metallicity, we would see many more SLSN host galaxies in denser
environments: $\approx 24$ out of the 33 SLSN events in this sample. Since
this is not the case, our analysis suggests that low-metallicity is likely the
more important requirement.

##  4 Conclusions

Using SLSN sources from the ZTF BTS, we calculate whether their host galaxies
are found in high-density or low-density large-scale environments. We use
bright, spectroscopically redshifted galaxies with absolute magnitude
$M_{i}<-21.5$ as tracers of density, and classify a source as in a high-
density environment if there is 2 or more bright galaxies within 2 Mpc. We
test our findings by comparing the SLSN sample with a sample of Type Ia
supernovae, whose host galaxies span a wider range of galaxy colour and
absolute magnitude. We also create another sample, derived from SDSS
photometrically-redshifted galaxies, matched to the SLSN host galaxies in
galaxy colour, absolute magnitude, and redshift. Our main findings may be
summarised as follows:

  * •

SLSN host galaxies are almost always located in low-density environments, with
all but 2 (of 33) host galaxies having no bright galaxies within 2 Mpc. The
photometrically selected matched sample (in luminosity, colour, and redshift)
shows similar results. This is in contrast to Type Ia supernova host galaxies,
8.5 per cent of which have 2 or more bright neighbours This is consistent with
the fraction of SNe Ia found in clusters compared to field galaxies, according
to Larison et al. (2023). In any relevant magnitude bin, the maximum high-
density fraction is $<$ 10 per cent for the SLSN and matched samples, while
for the SN-Ia sample the minimum is $>$ 20 per cent.

  * •

In analysis of the Illustris-TNG simulations, over 70 per cent of blue, low-
mass galaxies are found in the field, rather than in groups. These galaxies
have low metallicities and high star-formation rates, which are typical of
SLSN host galaxies.

  * •

Crucially, we find that in order to quantitatively match the rate of SLSN host
galaxies found in dense environments, an additional condition on the
metallicity of the host in the simulation is required. Taking the
uncertainties into account, SLSN production is only favoured in galaxies with
$12+\log($O/H$)=8.12$. In contrast, selecting only high-sSFR galaxies would
lead to an over-representation of high-density hosts in the simulation.

The fact that simulations suggest that galaxies capable of producing SLSNe and
having high-sSFR would preferentially be found in groups, but such galaxies
with low-metallicity would preferentially be found in the field, suggests that
the metallicity of a galaxy is the more important factor in the occurrence of
these explosive transients. However, this result will be strengthened with
more observations of SLSNe and their host galaxies, which will be achieved
over the coming years when new high-cadence surveys come online such as LSST
at the Vera Rubin Observatory.

## Acknowledgements

The authors thank the anonymous reviewer for their helpful and thoughtful
comments on this paper. CC acknowledges support from the School of Physics and
Astronomy at the University of Birmingham. SLM acknowledges support from STFC
grant ST/S000305/1 and UK Space Agency Grants No. ST/Y000692/1 and
ST/X002071/1. MN is supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 948381).

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.

SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

Based on observations obtained with the Samuel Oschin Telescope 48-inch and
the 60-inch Telescope at the Palomar Observatory as part of the Zwicky
Transient Facility project. ZTF is supported by the National Science
Foundation under Grants No. AST-1440341 and AST-2034437 and a collaboration
including current partners Caltech, IPAC, the Weizmann Institute for Science,
the Oskar Klein Center at Stockholm University, the University of Maryland,
Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO Consortium
of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin,
Lawrence Livermore National Laboratories, IN2P3, University of Warwick, Ruhr
University Bochum, Northwestern University and former partners the University
of Washington, Los Alamos National Laboratories, and Lawrence Berkeley
National Laboratories. Operations are conducted by COO, IPAC, and UW.

The IllustrisTNG simulations were undertaken with compute time awarded by the
Gauss Centre for Supercomputing (GCS) under GCS Large-Scale Projects GCS-ILLU
and GCS-DWAR on the GCS share of the supercomputer Hazel Hen at the High
Performance Computing Center Stuttgart (HLRS), as well as on the machines of
the Max Planck Computing and Data Facility (MPCDF) in Garching, Germany.

## Data Availability

The data used for this analysis is available at the links in the text above
where the data is described.

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##  Appendix A Appendix

Table LABEL:tab:_slsnelist lists all 55 SLSNe events that were available on
the BTS when analysis began, the groups that classified them as SLSNe, and the
relevant citation.

Table 2: Table of ZTF IDs of the transient events that were obtained on the BTS, the group that classified them as SLSNe, and the relevant citation. ZTFID | Classifying group | Citation
---|---|---
ZTF18aavrmcg | NUTS | Dong (2018)
ZTF18aapgrxo | ZTF | Fremling & Sharma (2018)
ZTF18abmasep | ZTF | Fremling et al. (2018a)
ZTF18ablwafp | ePESSTO | Gromadzki et al. (2018)
ZTF18abshezu | ZTF | Fremling et al. (2018b)
ZTF18abvgjyl | ZTF | Fremling et al. (2018c)
ZTF18acapyww | ZTF | Fremling et al. (2018d)
ZTF18acenqto | ZTF | Fremling et al. (2018e)
ZTF18achdidy | ZTF | Fremling et al. (2019d)
ZTF18acxgqxq | ZTF | Fremling et al. (2019d)
ZTF18acyxnyw | ZTF | Fremling et al. (2019b)
ZTF19aaknqmp | ZTF | Fremling et al. (2019e)
ZTF19aanesgt | ZTF | Fremling et al. (2019f)
ZTF19aacxrab | ZTF | Fremling et al. (2019g)
ZTF19aavouyw | None | Chen (2019)
ZTF19aarphwc | ZTF | Fremling et al. (2019h)
ZTF19aawfbtg | TCD | Prentice et al. (2019a)
ZTF19abpbopt | ZTF | Perley (2019)
ZTF19abnacvf | ZTF | Perley et al. (2019)
ZTF19abfvnns | ZTF | Fremling & Dahiwale (2019)
ZTF18aajqcue | ZTF | Fremling et al. (2019a)
ZTF19abxgmzr | ZTF | Fremling et al. (2019c)
ZTF19acfwynw | C-SNAILS | Nicholl et al. (2019)
ZTF19acgjpgh | TCD | Prentice et al. (2019b)
ZTF20aahbfmf | SGLF | Poidevin et al. (2020)
ZTF20aaifybu | ZTF | Yan et al. (2020a)
ZTF19acbonaa | ZTF | Yan et al. (2020b)
ZTF20aayprqz | ZTF | Perley et al. (2020b)
ZTF20aattyuz | ZTF | Dahiwale & Fremling (2020)
ZTF20abobpcb | SGLF | Perez-Fournon et al. (2020)
ZTF20abpuwxl | None | Terreran (2020)
ZTF20abzumlr | ZTF | Perley et al. (2020c)
ZTF20acphdcg | ePESSTO+ | Ihanec et al. (2020)
ZTF20acpyldh | FLEET | Blanchard et al. (2020)
ZTF21aaarmti | ePESSTO+ | Gillanders et al. (2021)
ZTF21aagpymw | ePESSTO+ | Magee et al. (2021)
ZTF19ackjrru | FLEET | Gomez et al. (2021c)
ZTF19acyjzbe | FLEET | Gomez et al. (2021c)
ZTF19adaivcf | FLEET | Gomez et al. (2021c)
ZTF21aappdnv | Global SN Project | Gonzalez et al. (2021)
ZTF21aakjkec | ZTF | Dahiwale & Fremling (2021)
ZTF21aaxwpyv | TCD | Deckers et al. (2021)
ZTF21aavdqgf | None | Yao et al. (2021)
ZTF21abaiono | ZTF | Perley et al. (2021)
ZTF21abcpsjy | REFITT | Weil et al. (2021)
ZTF21abbqeea | SGLF | Poidevin et al. (2021)
ZTF21absyjff | FLEET | Gomez et al. (2021a)
ZTF21accwovq | ePESSTO+ | Gromadzki et al. (2021)
ZTF21abrqria | FLEET | Gomez et al. (2021b)
ZTF21aalkhot | ePESSTO+ | Smith et al. (2022)
ZTF22aabimec | ePESSTO+ | Srivastav et al. (2022)
ZTF22aadqgoa | SGLF | Poidevin et al. (2022)
ZTF22aalzjdc | UCSC | Davis et al. (2022)
ZTF22aarqrxf | ePESSTO+ | Ayala et al. (2022)
ZTF22abcvfgs | ePESSTO+ | Harvey et al. (2022)