# Discovering Opioid Use Patterns from Social Media for Relapse Prevention

Zhou Yang, Spencer Bradshaw, Rattikorn Hewett, and Fang Jin  Zhou Yang,
Rattikorn Hewett, and Fang Jin are with the Department of Computer Science,
Texas Tech University, Lubbock, Texas, USA. E-mail: {zhou.yang,
rattikorn.hewett<EMAIL_ADDRESS>Spencer Bradshaw is with the Community,
Family, and Addiction Sciences, Texas Tech University, Lubbock, Texas, USA.
Email<EMAIL_ADDRESS>
###### Abstract

The United States is currently experiencing an unprecedented opioid crisis,
and opioid overdose has become a leading cause of injury and death. Effective
opioid addiction recovery calls for not only medical treatments, but also
behavioral interventions for impacted individuals. In this paper, we study
communication and behavior patterns of patients with opioid use disorder (OUD)
from social media, intending to demonstrate how existing information from
common activities, such as online social networking, might lead to better
prediction, evaluation, and ultimately prevention of relapses. Through a
multi-disciplinary and advanced novel analytic perspective, we characterize
opioid addiction behavior patterns by analyzing opioid groups from Reddit.com
- including modeling online discussion topics, analyzing text co-occurrence
and correlations, and identifying emotional states of people with OUD. These
quantitative analyses are of practical importance and demonstrate innovative
ways to use information from online social media, to create technology that
can assist in relapse prevention.

###### Index Terms:

Opioid Crisis, Opioid Use Disorder, Addiction Patterns, Relapse Emotion,
Addiction on Social Media, Addiction Treatment, Addiction Network

##  1 Introduction

Opioid overdoses are now causing more deaths than car crashes, prompting the
current U.S. President to declare the opioid crisis a national public health
emergency in October 2017. According to the latest statistics from the
National Institute on Drug Abuse (NIDA), more than 115 Americans die daily
after overdosing on opioids, and nearly 64,000 people died of drug overdoses
in the US in 2016, the most lethal year of the drug overdose epidemic [1].
Moreover, millions of Americans have been impacted by opioid-related problems.
It is estimated that 2.1 million people suffer from substance use disorders
related to prescription opioid pain relievers in the United States alone
[drugabusegov]. Additionally, the opioid crisis has social impacts beyond an
increased death toll. Other consequences include a rise in the number of
infants born dependent on opioids as well as a spread of infectious diseases
such as HIV and hepatitis C [2]. The status of the opioid crisis in the U.S.
is shown in Figure 1 (a) and Figure 1 (b) from different perspectives. As
revealed in a recent paper published in Science [3], the drug epidemic
currently sweeping across the U.S. has deteriorated into a full-scale national
pandemic, leading to national concern because of its negative impacts on
health, social security and economics.

Figure 1: (a).Drug overdose death toll by county in 2017 [4]; (b).Drug
overdose death toll by drug type from 1996 to 2017 in the U.S.

Current knowledge of the opioid crisis has been mostly limited to scattered
statistics that only reveal macro aspects of the crisis, such as the
nationwide/state-level overdose death tolls and overdose death tolls regarding
certain time periods, races, and/or specific drugs. This has led to a rapid
increase in OUD treatment admissions, including a $53.5\%$ increase during
2013 - 2015, and with adult treatment for heroin use more than doubling [5].
While treatment is important and leads to effective outcomes and greater
health for some, the relapse rate post-treatment for substance use disorders
is relatively high, ranging from $40-60\%$ [6], with potential relapse being
difficult to predict. Therefore, research that can enhance relapse prediction
using data from everyday, common situations is needed.

One such resource may include behavioral data available from social media, a
forum now used by $88\%$ of persons in the U.S. ages 18-29, $78\%$ of those
ages 30-49, $64\%$ of those ages 50-64, and $37\%$ of those age 65 and older
[7]. Despite its popularity, detailed analysis of user-specific knowledge,
such as social media behavioral patterns that associate with substance use and
relapse, have not been studied. Therefore, to facilitate a better
understanding of the opioid epidemic that may ultimately lead to better
prediction of relapse and ultimately responses and interventions that prevent
relapse and promote remission - individualized, user-specific analysis is
urgently needed. This manuscript begins to tackle this complex problem from a
multi-disciplinary perspective, and includes text analysis, topic modeling,and
sentiment analysis.

##  2 Data Analysis and Results

###  2.1 Opioid Communities on Social Media

With social media growing profoundly and widely intertwined with people’s
everyday lives, social media contains more and more biomedical psychological
information about its users. As a result, social media represents an
increasingly important data source for discovering knowledge that can have
various applications. Reddit.com is an online community and social media
platform, which allows Redditors (registered users of Reddit.com) to form
groups/subreddit to discuss specific topics. Groups such as
“r/OpioidRecovery”, “r/Opiates” and “r/drugs” aim to provide people suffering
from OUD and seeking remission with psychological support. This function spurs
people with OUD to turn to social media for help, posting various problems and
confusion. In this way, social media has become a widely utilized form of
social support, which is an important recovery resource that can buffer the
negative effects of stress on the quality of life of those with OUD [8].
Indeed, many people with OUD-related problems that have managed to abstain
from opioid use are often willing to share their experience with others
online. For instance, a Redditor posted on Reddit that “I was the same way, I
only got clean cuz I didn’t wanna go back to jail tbh, and had to go cold
turkey in rehab”. This study is based on data collected from subreddit groups
such as “r/opioid”, “r/opioidrecovery”, and “r/drugs” from Reddit.com in 2018.

###  2.2 Dataset

The dataset used in this paper is collected from Reddit.com, a social news
aggregation, web content rating, and discussion website. We developed a web
crawling tool using PRAW (Python Reddit API Wrapper) to collect data. The
dataset for this paper is from three subreddits (Subreddits are subsidiary
threads or categories within the Reddit website) on Reddit.com, including
“r/opiates", “r/opiatesRecovery" and “r/drugs". In a subreddit, there are a
series of posts that include a post topic and a collection of related
comments. The dataset consists of two parts. The first consists of 3,000 posts
with the top (collected in April 2018) 1,000 posts from each subreddit, since
1,000 is the maximum number that is allowed by PRAW API. This first part
includes posts and interactions within posts. After collecting the subreddit
data, we extracted a list of Redditor IDs of participants. The second part of
the dataset consisted of personal data, which was collected based on the
previously mentioned Redditor list that was extracted.

Data labeling then involved manually reading the posts/comments and
categorizing each Redditor into a corresponding group. With intent to
implement binary classification, each Redditor was marked with a label and
different labels were mutually exclusive. This classification ensured that
each participant was assigned to one and only one label.

More specifically, in the group classification process, we labeled a Redditor
ID by manually reading his/her post history in the collected dataset. This
labeling process was done by several health science researchers, and labeling
agreement among the researchers was required to reduce labeling errors that
might otherwise have resulted from different labeling metrics. For the OUD
classifier, we followed DSM-5 criteria [9], in which a Redditor is labeled
with OUD if at least two of the listed criteria were met within the past 12
months. For instance, a Redditor was labeled with OUD if he/she admitted that
he/she often takes in larger amounts of opioids and/or over a long period of
time than intended, and also mentioned a persistent desire or unsuccessful
efforts to cut down or control use, or had strong cravings to use. If less
than two symptoms were detected, a Redditor was classified to a “Non-OUD"
group. Redditors who met at least two DSM-5 criteria but showed no intentions
of limiting or abstaining from use were classified to an “OUD" group. For
instance, an OUD Redditor posted “Late Christmas Gift for myself 5x 40mg
Oxycodone (Toroxycon)”. Redditors were classified into a “positive recovering
OUD” group if he/she was attempting and/or struggling to seek treatments that
reduced symptoms associated with previously met DSM-5 criteria in the last 12
months. For instance, a positive recovering OUD posted “what to say when
returning to work from drug treatment”. If there was no evidence that a
Redditor was seeking treatment, this Redditor was classified to a “non-
positive recovering OUD” group. Similarly, Redditors were classified to an
“OUD relapse” group if they first classified as in recovery but indicated that
they had used opioids again (no matter how many times) in the fifty days
previous to the latest post/comment. For instance, a relapsed Redditor posted
“45 Days clean and now I’m drunk and high”. Otherwise, Redditors in recovery
were assumed to have stayed clean.

When classifying participants into the “positive recovering OUD” and “non-
positive recovering OUD" groups, one interesting finding was that many
participants with OUD had varying degrees of willingness to stay clean, which
is consistent with the literature [10]. That is, “positive recovering OUD”
group members showed the intention of reducing symptoms associated with DSM-5
criteria while “non-positive recovering OUD” group members show less intention
of such symptom reduction.

In summary, we collected 1,000 Reddit posts respectively from “r/opiates”,
“r/opiatesRecovery”, and “r/drugs”. By utilizing Redditor IDs extracted from
the 1,000 posts, we also collected the latest 1,000 comments/posts Reddit API
restricts the number of posts that can be collected to 1,000 for each Redditor
ID respectively, and obtained a personal post dataset.

###  2.3 Redditor Classification

The Redditor classification is based on the text features. Give a set of
Redditors $A=\\{a_{1},a_{2},a_{3},\dots\\}$, and their corresponding feature
variable $F_{j}=<X_{1},\dots,X_{m}>$, which is represented by these features
extracted from text, and their status label $Y_{j}\in[0,1]$, the Redditor
classification task is to learn a function $f:F_{j}\to Y_{j}$, such that the
task errors are minimized. Formally, we have

$\arg\mathop{\min}\limits_{\Psi}\sum\limits_{j=1}^{m}{||f({X_{1},\dots,X_{m}})-{Y_{j}}{||_{2}}}$
(1)

where $X_{j}$ and $Y_{j}$ are a feature variable and label variable
respectively, and $\Psi$ is the model parameters learned from training.

Two classifiers were designed and implemented to filter out the research
targets: an OUD classifier and a recovering classifier. The first classifier
differentiates between OUD and non-OUD, while the second “within OUD"
classifier separates those who are in positive recovery and who hadn’t shown
any evidence of being in a positive recovery process, and they were denoted as
“positive recovering OUD” and “non-positive recovering OUD" group members,
respectively. For the OUD classifier, we employed a Support Vector Machine
(SVM) classifier that transformed posts/comments into term vector features to
determine whether participants had an OUD problem. The dataset for this
classifier included 1,000 Redditors (419 OUDs and 581 non-OUDs). In the
labeled dataset, $70\%$ (namely, $70\%$ of the samples) was used as the
training dataset, and the rest were used for testing.

Once the classifier had been trained, we applied it to the unlabeled dataset.
The data of the identified OUDs were then fed into the next classifier to
determine whether they were a positive recovering OUD or a non-positive
recovering OUD. The second classifier was designed to identify those with OUD
who had shown positive attitudes/acts toward seeking recovery. This took
another four days to label 1,000 people (with 375 positive recovering OUDs and
625 non-positive recovering OUDs). Among this labeled data, $70\%$ was used as
the training dataset and the rest for testing. We show the performance of
these two classifiers in Table I.

OUD Classif. | Recorvering Classif.
---|---
| Acc. | Rec. | Prec. | F1 | Acc. | Rec. | Prec. | F1
LG. | 0.82 | 0.83 | 0.82 | 0.82 | 0.74 | 0.78 | 0.73 | 0.76
KNN | 0.70 | 0.81 | 0.76 | 0.7858 | 0.75 | 0.8121 | 0.73 | 0.77
SVM | 0.92 | 0.94 | 0.93 | 0.93 | 0.88 | 0.93 | 0.84 | 0.88
TABLE I: Classification results for Logistic Regression, KNN, and SVM.

The SVM classifier is implemented by using the “Libshorttext” library. The
dataset is processed by converting to lower case, removing punctuation and
stop words, tokenizing, stemming and lemmatizing. As shown in Table I, SVM
classifier outperforms logistic regression and KNN in both cases. Moreover,
the accuracy of the first classifier was $0.917$, which was better than the
accuracy of the second classifier. Thus, there wouldn’t be any performance
bottleneck if the output of the first classifier was fed into the second
classifier.

We manually classified 3,157 people, with 64.9% of them have OUD problems, and
35.1% are non-OUD. However, only 10.4% of these people with OUDs had shown
that they have been positively seeking remission and/or recovery, while the
other 89.6% of those with OUD showed no signs of seeking remission and/or
recovery according to our dataset records. Of the recovering Redditors with
OUD, 89.3% will relapse within fifty days.

Figure 2: Topic modeling results. The topic numbers were set as eight, and top
keywords as 8.

###  2.4 Topic Modeling

To discover the hidden semantic structures (i.e., “topics”) that occur in an
extensive collection of posts collected from Reddit.com, we first built a
topic model to analyze those posts. Given a document in the topic modeling, we
would expect particular words to appear in the document more or less
frequently. For instance, by specifying two topics, we found the corresponding
topics of “people take opioids because of pain” and “people take dope and
feels”. Those topics are indicated by words “people" and “pain", and “people”
and “dope”, respectively, since they appear more frequently in those posts
than other words.

According to the Deveaud2014[11], the best topic number is in the range of [4,
11], where model achieve the extremum. Since posts typically concern numerous
topics, we specified the existence of multiple topics to further explore any
additional potential topics that might emerge. As shown in Figure 2, we
present the results of an eight-topic model. These topics are “hard day, don’t
do opiates”, “don’t dope heroin”, “people feel love to feel clean”, “taking
drugs or opioids for days”, “people feel, heroin”, “people pain, taking
opiates or drugs”, “for pain, don’t do oxycodone”, and “post about heroin”,
respectively. In brief, the general topics included 1). Attempts to dissuade
people from taking highly addictive opioids (e.g., heroin and oxycodone); 2).
Information on how to stay clean; 3). How people feel when they stay clean for
an extended period; and 4). Advice about treatment and how to stay clean.

Even though topic modeling reveals the general topics in certain social
communities, individualized user-specific information remains hidden. To
expose more details about user-specific patterns for people with OUD that
utilize social media, we implemented text co-occurrence and correlation
analysis to further dig and discover behavior patterns on social media.

###  2.5 Text Co-occurrence and Correlation

Figure 3 A. shows the word correlation with annotations. The word correlation
and bigram illustrate the social interactions between different Redditors. We
generated word correlation graphs to obtain a full picture of their
conversations. A textual analysis was implemented to capture the relationship
between words, to examine words that tend to follow each other immediately, or
tend to co-occur within the same posts. Sentences were tokenized into
sequences of words. By checking how often a word $X$ was followed by word $Y$,
the relationships between words are measured by the Phi efficient metric, and
the Phi efficient is defined as

$\phi=\frac{{{n_{11}}{n_{00}}-{n_{10}}{n_{01}}}}{{\sqrt{{n_{1.}}{n_{0.}}{n_{.0}}{n_{.1}}}}},$
(2)

where $n_{11}$ represents the number of posts where both $X$ and $Y$ appear,
$n_{00}$ is the number where neither appears, and $n_{10}$ is the cases where
one appears without the other. In this case, a pair of words with higher phi
coefficiency indicates that this word pair appears more frequently than other
word pairs do.

Figure 3: (a): Word correlation with Phi coefficiency$\geq$0.2. (b): Word
bigrams with co-currence co-occurrence count $\geq$ 150\. (c): Word bigrams
with co-occurrence count $\geq$ 200\. (d): Word bigrams with co-occurrence
count $\geq$ 250.

By gradually increasing the Phi coefficient, the words that are most likely to
appear with each other were detected, as shown in Figure 3 (b), (c), and (d).
After analyzing a large number of posts from Reddit, the relationship was
visualized in Figure 3. By analyzing and annotating these keywords, some
frequent topics are summarized as follows:

  * •

Talking about detoxification processes and assisting medications such as
‘naloxone’ and ‘bupe’, in conjunction with ‘doctor’ and ‘detox’. Participants
of these discussion topics are Redditors who are motivated and willing to find
recovery from their addiction and manage to stay clean. Also, some of these
Redditors who have been clean for a longer period participate in those
discussions and share their experiences about how to stay clean.

  * •

Describing pains and prescriptions from doctors, as indicated by keywords such
as ‘pain’, ‘doctor’, and ‘painkiller’. These topics are consistent with the
fact that opioids are often originally used as a prescribed painkiller. Some
Redditors do have bodily and/or chronic pain problems and concerns. Though
opioids are an effective painkiller and have curative effects, after months or
years of taking opioids, these Redditors often develop an opioid dependency.

  * •

Sharing of withdrawal symptoms, which symptoms fall into two categories. One
category involves mental/psychological symptoms such as anxiety, insomnia, and
social isolation. The other category involves physical symptoms, including
headaches and dizziness. Both categories of withdrawal symptoms are very
uncomfortable and may lead people with OUD to a compulsive relapse if they do
not have proper guidance and support.

  * •

Talking about what kind of opioids they take, how it feels, and the dosage of
drugs they take. These topics are indicated by words such as “hight”, “enjoy”,
“mg (the dosage of opioids)”, etc. For those topics, Redditors always share
their experiences about what kinds of opioids they are taking, or they have
been taking, or they used to take. But they try to be cautious about the
dosage because of the risks of death caused by overdose.

Also, since treatment is critical for people with OUD, figuring out how those
with OUD evaluate treatment-assisting medications is very important. By
specifying treatment-assisting medications such as ‘buprenorphine’,
‘methadone’, ‘naloxone’, and ‘suboxone’, we detected and filtered out words
that were associated with those medicines in these communities. In Figure 4 A,
we visualize the word correlation on posts that are about treatments. Figure 4
B shows the top six words with the highest correlation coefficiency for each
medication. As shown in this figure, most of the words related to these
medications are neutral, representing neutral feedback from those with OUD.
For instance, Buprenorphine is associate with “ people months clean benzos
love”. Also, a lot of words about dosage are used such as ‘dose’ and ‘mgs’.
Those words indicate that when they use medications, they seem to be paying
much attention to dosage, perhaps because high dosages may have lethal rather
than curative effects.

Figure 4: A: Word correlation (in the same posts) with correlation
coefficiency $\geq$ 0.17. B: Top ords that are used when talking about opioid
related medicines.

###  2.6 Emotion of People with OUD

Relapses for persons with OUD are associated with younger age, heavy use
before treatment, a history of injecting, and not following up with aftercare
[12]. They are also frequently associated with extreme emotions and other
stressors [13]. Specifically, people seeking remission or rehabilitation can
experience extreme emotions that range from great highs to great lows, and
sometimes they feel like their emotions are out of control. This is important
as the well-accepted Self-Medication Hypothesis [14] proposes that people
often use substances to self-medicate the negative and painful emotions
associated with trauma, stress, and/or comorbid mental health disorders [15].
To explore emotions that are associated with relapse, an emotion analysis was
implemented to show the observational results.

On Reddit.com, people with OUD are free to express their joy and sadness,
trust and disgust, and anticipation and anger, etc. in their posts or
comments. These sentiments, along with their other online behaviors, could
disclose information about their remission/recovery status and thus serve as
relapse risk indicators and therefore also for the need for intervention that
might assist with relapse prevention [16]. We therefore sought to capture
subtle emotional sentiments from those with OUD and associate these sentiments
with OUD associated behaviors such as staying clean or relapsing. By studying
Redditor posts as well as their interactions with other Redditors, it was
hoped that identified emotions can increase understanding of
remission/recovery status. Such understanding may ultimately lead to
transforming the sentiments and behavior patterns that emerge from Redditor
posts into possible indicators of relapse risk and the need for some type of
intervention to prevent relapse.

A word-emotion association lexicon [17] was utilized to study the emotional
content of each comment from a Redditor. In the sentiment analysis, we focused
on dominant rather than submerged emotions. For instance, a post may include a
complex emotion that is a combination of several emotions. In this case, a
comment is labeled with the emotion that has the highest score, which is
defined in equation 3, 4 and 5.

In particular, suppose $L_{i}$ is the subset of a lexicon which contains words
in emotion $i$. The emotion count $emotion\\_count\\_i$ and normalized count
$n\\_emotion\\_count\\_i$ for the emotion $i$ are:

$emotion\\_count\\_i=\sum\limits_{j}{word_{ij}}$ (3)
$n\\_emotion\\_count\\_i=\frac{emotion\\_count\\_i}{max(emotion\\_counts)}$
(4)

Where:

$word_{j}=\begin{cases}1&\text{if $word_{j}$ in $L_{i}$}\\\
0&\text{otherwise}\end{cases}$ (5)

Figure 5: Word cloud with ten different emotions. The drug names such as
Heroin, OxyContin and Vicodin are not the dominant words in the wordcloud,
since people talk more about their feelings instead of the drug itself. Also,
stopwords are removed from the wordcloud.

Ten categories of emotion were identified as shown in Figure 5: anger,
anticipation, disgust, fear, joy, sadness, surprise, trust, negative,
positive. Also consistent with addiction literature [18], we found that people
with OUD can be highly influenced by their emotions. Words that express
emotions or feelings, such as “feeling”, “bad”, and “high”, are repeatedly
used. Among the 670 persons with OUD who had relapsed, 72% showed the
predominant emotions that were ‘negative’ or that of ‘joy’. This observation
shows relapse is highly related to more extreme emotions such as ‘negative’
and ‘joy’ based-emotions.

##  3 Discussion

The opioid crisis is one of the leading threats to public health in the U.S.,
and therefore treatment admissions are rising. Despite positive benefits from
treatment, many persons seeking remission relapse, and such relapse is
difficult to predict. Nevertheless, a thorough understanding of social media
behavior patterns that associate with remission can promote potential
supportive and even potential relapse prevention-based interventions. In this
paper, we conducted a series of experiments and observed significant Opioid
addiction patterns by analyzing posts and comments collected from Reddit.com.
These advanced analyses demonstrate 1) how information from social media
posts/discussions can be used to more accurately predict potential relapse, 2)
how discussions might be discovered around various topics that association
with use, treatment, and remission, and 3) how emotional/affective sentiments
from these discussions may be used to predict relapse risk.

First, only 10.4% of those identified with OUD actively seek
remission/recovery, while the 89.6% show no indication of seeking recovery.
This demonstrates the important need that the majority of those with OUD to
progress through various motivational stages that may yield a greater
readiness for change. Additionally, the machine-learning based classifiers
used in this study showed high accuracy and precision in predicting group
classification outcomes. Second, people with OUD who are seeking
remission/recovery more generally have positive judgments regarding
medications such as Buprenorphine, Methadone, Naloxone, and Suboxone; and they
pay close attention to dosage. Thus, this finding supports that medication
assisted treatment (MAT; use of medications such as Buprenorphine and
Methadone) may be helpful to those seeking recovery from OUD. This is
important as MAT is often a controversial topic and widely underutilized
resource in substance use disorder treatment [19]. Third, 72% of those seeking
recovery from OUD - but appear to have relapsed - showed both dominant
‘negative’ as well as ‘joy’ based emotions that associated with relapse.
Therefore, we infer people with OUD may experience a wide range of emotion and
emotional shifts, and that without support that aids emotional regulation they
have higher relapse risk. In sum, these multi-disciplinary analyses conducted
in this paper help disclose several behavioral patterns and characteristic
features of persons with OUD, making it more possible to detect and identify
precursors of relapse, increasing assessment and prediction of relapse risk.
Ultimately, this may lead to the development and implementation of
personalized OUD-based interventions that enhance relapse prevention and OUD
remission.

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# Supernovae and superbursts by dark matter clumps

Nirmal Raj<EMAIL_ADDRESS>Centre for High Energy Physics, Indian Institute
of Science, C. V. Raman Avenue, Bengaluru 560012, India

###### Abstract

Cosmologies in which dark matter clumps strongly on small scales are
unfavorable to terrestrial detectors that are as yet unexposed to the clumps.
I show that sub-hectometer clumps could trigger thermonuclear runaways by
scattering on nuclei in white dwarf cores (carbon and oxygen) and neutron star
oceans (carbon), setting off Type Ia-like supernovae and x-ray superbursts
respectively. I consider two scenarios: “dark clusters” that are essentially
microhalos, and “long-range dark nuggets”, essentially macroscopic composites,
with long-range Yukawa baryonic interactions that source the energy for
igniting explosions. I constrain dark clusters weighing between the Planck
mass and asteroid masses, and long-range dark nuggets over a wider mass range
spanning forty orders of magnitude. These limits greatly complement searches I
had co-proposed in Ref. Bramante  _et al._ (2022a) for scattering interactions
of dark clumps in neutron stars, cosmic rays, and pre-historic minerals.

##  I Introduction

Dark matter continues to elude non-gravitational detection. One reason for why
extensive terrestrial searches for the scattering interactions of particle
dark matter (DM) have been futile could be that it is not distributed smoothly
in the halo, but clumped in substructure, with detectors never having been
exposed to DM clumps. Cold DM, expected to clump down to milli-solar mass
Zybin  _et al._ (1999); Hofmann  _et al._ (2001); Berezinsky  _et al._ (2008),
may also clump below kiloparsec-scales if hierarchical clustering leaves small
halos intact Bergstrom  _et al._ (1999); van den Bosch  _et al._ (2018); van
den Bosch and Ogiya (2018). Such clumps (“subhalos”, “microhalos”,
“miniclusters”, and so forth) could dominate the DM population in non-standard
cosmologies that enhance small-scale power Erickcek and Sigurdson (2011);
Barenboim and Rasero (2014); Fan  _et al._ (2014); Dror  _et al._ (2018);
Graham  _et al._ (2016); Buckley and DiFranzo (2018); Nussinov and Zhang
(2020); Barenboim  _et al._ (2021); Domènech  _et al._ (2023). Whatever be the
origin, the plausible clumping of DM compels us to rethink detection of its
scattering on Standard Model (SM) states. In Ref. Bramante  _et al._ (2022a),
with co-authors I had proposed three strategies: heating of neutron stars
(NSs) by the transfer of kinetic energy during clump transits, exploiting the
effects of strongly-interacting DM on cosmic rays and vice-versa, and
recasting searches for DM tracks in geological materials deep underground.
Here I will investigate one more: the triggering of thermonuclear explosions
in compact stars by DM clumps depositing kinetic energy.

Type-Ia like supernovae are thought to originate from carbon-oxygen white
dwarfs (WDs) accreting material from a binary companion and exceeding the
Chandrasekhar mass, igniting runaway carbon fusion that unbinds the WD. A
suitable deposition of energy sourced by DM could trigger a similar explosion.
This idea was first investigated for primordial black hole (PBH) DM undergoing
dynamical friction in the WD material Graham  _et al._ (2015), with asteroid-
mass PBHs ruled out from the observed survival of WDs, though Ref. Montero-
Camacho  _et al._ (2019) argued that these limits lift due to hydrodynamical
instabilities in the wake of the transiting PBH quenching nuclear energy
generation. Other DM mechanisms of explosive energy deposition in WDs have
since struck the literature: gravitational potential energy shed via nucleon
scattering as particle DM captured by the WD collapses in it Graham  _et al._
(2015); Bramante (2015); Fedderke  _et al._ (2020); Steigerwald  _et al._
(2022), super-heavy DM annihilating, decaying or scattering on nuclei Graham
_et al._ (2018); Sidhu and Starkman (2020), and Hawking radiation from black
holes formed inside WDs via DM collapse Acevedo and Bramante (2019); Janish
_et al._ (2019); Fedderke  _et al._ (2020). Ref. Das  _et al._ (2022) briefly
mentions the possibility of tightly bound DM substructure triggering
supernovae via impact shock waves. Some of these studies set limits based on
the observed rates of supernovae. I will use the existence of WDs to set
limits on clump properties.

I will also set limits using “superbursts”, hours-long $\mathcal{O}(10^{42})$
erg x-ray bursts observed in NSs either isolated or hosted in low-mass or
ultra-compact x-ray binaries Galloway  _et al._ (2020); Alizai (2023). Type-I
bursts in NSs are typically produced by surface acccretion of mass from a
binary companion that then ignites nuclear fuel into explosive burning.
Superbursts are a special kind that last $10^{3}$ times longer and emit
$10^{3}$ times more energy, with an accretion rate $>10\%$ of the Eddington
limit, suggesting that their ignition occurs deeper in the NS, in fact, in a
layer of carbon in the NS ocean $\mathcal{O}$(10) m below the surface Cumming
and Bildsten (2001); in ’t Zand (2017). DM clumps that reach this layer with
sufficient energy to trigger superbursts more often than carbon accumulation
can, are then ruled out. Ref. Sidhu and Starkman (2020) set limits on
composite “macro” DM using a single superburst source, while I avail of
catalogues of recurring superbursts.

I consider two scenarios of clumpy DM. The first, “dark clusters”, is that of
subhalos comprised of DM particles that scatter on SM species, as treated in
Ref. Bramante  _et al._ (2022a); for simplicity I analyze only elastic nuclear
recoils. It will be seen that dark clusters of sub-cm size are required to
trigger thermonuclear explosions. The second, “long-range dark nuggets”, is
that of macroscopic composites that interact with baryons through a long-range
Yukawa “fifth force” as considered in Ref. Gresham  _et al._ (2023) (although
in that study the DM state could also be a point particle). These states could
be thought of as DM “nuclei”, “nuggets”, or “blobs” Witten (1984); Krnjaic and
Sigurdson (2015); Detmold  _et al._ (2014); Hardy  _et al._ (2015a); Wise and
Zhang (2014, 2015); Hardy  _et al._ (2015b); Gresham  _et al._ (2017, 2018a);
McDermott (2018); Gresham  _et al._ (2018b); Bai  _et al._ (2019); Grabowska
_et al._ (2018), and would reduce to dark clusters in the limit of vanishing
coupling and force range. The Yukawa potential energy between compact stars
and long-range dark nuggets can now act as the primary energy source for
igniting thermonuclear explosions; the long-range fifth-force also serves to
increase the flux of infalling nuggets compared to that due to gravity alone.
In both scenarios it will be seen that many (tens of) orders of magnitude in
the space of DM clump mass, size and effective coupling are constrained. My
limits highly complement other probes of DM clumps and composites:
gravitational microlensing Croon  _et al._ (2020a, b), pulsar timing arrays
Dror  _et al._ (2019); Lee  _et al._ (2021), accretion glows in molecular
clouds Bai  _et al._ (2020), stellar impacts Das  _et al._ (2022),
gravitational waves Croon  _et al._ (2023), and even underground detection
Nussinov and Zhang (2020); Stiff  _et al._ (2001); Kamionkowski and
Koushiappas (2008); Ibarra  _et al._ (2019); Bramante  _et al._ (2018a,
2019a); Grabowska  _et al._ (2018); Bramante  _et al._ (2019b); Acevedo  _et
al._ (2021a); Adhikari  _et al._ (2022); Acevedo  _et al._ (2021b); Carney
_et al._ (2022); Jacobs  _et al._ (2015); Ebadi  _et al._ (2021); Acevedo  _et
al._ (2021c). Thermal emission signatures of DM heating compact stars
kinetically is an active area of study Baryakhtar  _et al._ (2017); Raj  _et
al._ (2018); Chen and Lin (2018); Bell  _et al._ (2018); Garani  _et al._
(2019); Camargo  _et al._ (2019); Bell  _et al._ (2019); Hamaguchi  _et al._
(2019); Garani and Heeck (2019); Acevedo  _et al._ (2020); Joglekar  _et al._
(2019); Keung  _et al._ (2020); Joglekar  _et al._ (2020); Bell  _et al._
(2020a); Dasgupta  _et al._ (2020); Garani  _et al._ (2021); Bell  _et al._
(2021, 2020b); Maity and Queiroz (2021); Anzuini  _et al._ (2021); Lin and Lin
(2021); Zeng  _et al._ (2022); Fujiwara  _et al._ (2022); Hamaguchi  _et al._
(2022); Chatterjee  _et al._ (2022); Coffey  _et al._ (2022); Alvarez  _et
al._ (2023); Garani  _et al._ (2023); Berti  _et al._ (2022); rev .

This paper is laid out as follows. In Sec. II I review the conditions for
runaway fusion in WD cores and NS crusts. In Sec. III I describe the set-up of
dark clusters and long-range dark nuggets, and the physics of their
encountering compact stars and triggering thermonuclear explosions. In Sec. IV
I show and explain the limits thus obtained. In Sec. V I provide further
discussion and conclude.

##  II Thermonuclear Runaway

I now present the general requirements for triggering runaway fusion in a WD
core or NS ocean, following the discussion in Ref. Fedderke  _et al._ (2020).
Two conditions must be met. First, a mimimum energy $Q_{\rm dep}$ must be
deposited to raise the temperature of a critical mass $M_{\rm crit}$ of
density $\rho$ to a critical $T_{\rm crit}$ that sustains nuclear fusion:

Condition 1 $\displaystyle Q_{\rm dep}\geq M_{\rm crit}(\rho,T_{\rm
crit})\bar{c}_{p}(\rho,T_{\rm crit})T_{\rm crit}~{}.$ (1)

Here I have assumed that the temperature prior to heating $\ll T_{\rm crit}$,
and $\bar{c}_{p}\simeq c^{\rm e}_{p}/2+c^{\gamma}_{p}/4+c^{\rm ion}_{p}$ is
the average isobaric specific heat capacity with the various contributions
given by

$c^{\ell}_{p}(\rho,T_{\rm
crit})=\frac{a_{\ell}b_{\ell}}{u}\bigg{(}\frac{T_{\rm crit}}{E_{\rm
F}}\bigg{)}^{\alpha_{\ell}}\bigg{[}1-\bigg{(}\frac{m_{e}}{E_{\rm
F}}\bigg{)}^{2}\bigg{]}^{\beta_{\ell}}~{},$ (2)

where $u$ is the atomic mass unit, $m_{e}$ the electron mass, and
$a_{\ell}=\\{\pi^{2},4\pi^{4}/5,5/2\\}$, $b_{\ell}=\\{\sum
X_{i}Z_{i}/A_{i},\sum X_{i}Z_{i}/A_{i},\sum X_{i}/A_{i},\\}$ (with $X_{i}$,
$Z_{i}$, $A_{i}$ respectively the mass fraction, charge and atomic number of
the ion species $i$), $\alpha_{\ell}=\\{1,3,0\\}$,
$\beta_{\ell}=\\{-1,-3/2,0\\}$ for the {electronic, radiative, ionic}
contributions. In practice $c^{\rm ion}_{p}$ is negligible. The Fermi energy
$E_{\rm F}=[m^{2}_{e}+(3\pi^{2}n_{e})^{2/3}]^{1/2}$ with $n_{e}=\rho b_{\rm
e}/u$. The trigger energy in Eq. (1) ranges between
$\mathcal{O}(10^{17}-10^{24})$ GeV for WD central densities corresponding to
WD masses ranging from 1.4 $M_{\odot}$ down to 0.8 $M_{\odot}$. Eq. (1) is
necessary but not sufficient for runaway fusion.

The critical mass $M_{\rm crit}=4\pi\rho\lambda_{\rm trig}^{3}/3$ is set by
the second condition. Namely, the rate of energy gain via nuclear fusion must
exceed the rate of energy loss via diffusion over the volume set by the
“trigger length” $\lambda_{\rm trig}$:

Condition 2 $\displaystyle\dot{Q}_{\rm nuc}>\dot{Q}_{\rm diff}~{}.$ (3)

Writing $\dot{Q}_{\rm nuc}=M_{\rm crit}\dot{S}_{\rm nuc}$ and $\dot{Q}_{\rm
diff}\simeq 4\pi k\lambda_{\rm trig}T_{\rm crit}$ for per-mass nuclear energy
deposition rate $\dot{S}_{\rm nuc}$ (the estimation of which involves
numerical simulations of flame propagation with a nuclear reaction network
Timmes and Woosley (1992)) and thermal conductivity $k$, we obtain

$\displaystyle\lambda_{\rm trig}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3kT_{\rm crit}}{\rho\dot{S}_{\rm nuc}(\rho,T_{\rm
crit})}}~{}$ (4) $\displaystyle=$
$\displaystyle\begin{cases}\lambda_{1}~{}(\frac{\rho}{\rho_{1}})^{-2}\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ,\rho\leq\rho_{1}\\\
\lambda_{1}~{}(\frac{\rho}{\rho_{1}})^{\ln(\lambda_{2}/\lambda_{1})/\ln(\rho_{2}/\rho_{1})}~{},\rho_{1}<\rho\leq\rho_{2}\end{cases}$

where for WDs $\\{\lambda_{1}^{\rm WD},\lambda_{2}^{\rm WD}\\}=\\{1.3\times
10^{-4}~{}{\rm cm},2.5\times 10^{-5}~{}{\rm cm}\\}$ and
$\\{\rho_{1},\rho_{2}\\}=\\{2\times 10^{8}~{}{\rm g/cm}^{3},10^{10}~{}{\rm
g/cm}^{3}\\}$. I have adopted this analytic form from Ref. Fedderke  _et al._
(2020), obtained by fitting to Figure 6 of Ref. Timmes and Woosley (1992) that
is restricted to $\rho_{1}\leq\rho\leq\rho_{2}$ and then extrapolating to
lower densities assuming reasonable density-scalings of $k$ and $\dot{S}_{\rm
nuc}$. This fit is for $T_{\rm crit}$ = 0.5 MeV, which I will take as my
critical temperature, although it could be lower Timmes and Woosley (1992).
This fit also assumes, as I will, equal C and O masses in WDs. In the NS
ocean, the mass fraction of carbon is 10% Cumming and Bildsten (2001),
implying $\rho\to 0.1\rho$ in Eq. (4). Though superbursts occur via pure C
burning, I expect the scalings of Eq. (4) to hold as seen from Ref. Timmes and
Woosley (1992), Table 3, for conductive burning.

To translate between $\rho$ and WD masses $M_{\rm WD}$, I use the analytical
fit Fedderke  _et al._ (2020)

$\displaystyle\bigg{(}\frac{\rho_{\rm WD}}{1.95\times 10^{6}\ {\rm
g/cm}^{3}}\bigg{)}^{2/3}+1\approx\bigg{[}\sum_{i=0}^{6}c_{i}\bigg{(}\frac{M_{\rm
WD}}{M_{\odot}}\bigg{)}^{i}\bigg{]}^{-2}~{},$ (5)
$\displaystyle\\{c_{i}\\}=\\{1.003,-0.309,-1.165,2.021,-2.060,1.169,-0.281\\}~{}.$

As my benchmark NS (with $v_{\rm esc}^{\rm NS}=\sqrt{2GM_{\rm NS}/R_{\rm NS}}$
the surface escape speed) I take

$M_{\rm NS}=1.4~{}M_{\odot},~{}R_{\rm NS}=10.99~{}{\rm km}\Rightarrow v_{\rm
esc}^{\rm NS}=0.61~{},$ (6)

as predicted by the Prakash-Ainsworth-Lattimer (PAL) equation of state of
high-density matter Prakash  _et al._ (1988). To obtain my results I will use
the NS crust density profile predicted by this model Datta  _et al._ (1995).

Figure 1: As a function of compact star matter density are shown the trigger
energy per trigger mass, $\bar{c}_{p}T_{\rm crit}$ (black dot-dashed), the
maximum recoil energy of nuclei per trigger mass ($\approx v^{2}_{\rm esc}$)
in white dwarf cores (green) for recoils induced by dark clusters
($\alpha_{\rm y}=0$) and long-range dark nuggets ($\alpha_{\rm y}\neq 0$), and
in neutron star oceans by dark clusters. The WD masses in the top x-axis ticks
correspond to WD central densities in the bottom x-axis. Kinetic heating by
clumpy dark matter is always capable of triggering superbursts in NSs, but can
trigger supernovae only for WDs above some mass. See Sec. III for further
details.

##  III Dark matter-induced explosions

###  III.1 Dark clusters

Assuming dark clusters of uniform mass $M$ and radius $R$, the impact
parameter of their encounters with compact stars of mass $M_{\star}\gg M$ and
radius $R_{\star}$, accounting for gravitational focusing, is

$b^{\rm cl}_{\rm max}=R_{\star}\sqrt{1+\bigg{(}\frac{v_{\rm
esc}}{v_{\chi}}\bigg{)}^{2}}+R~{},$ (7)

which at Galactic position $r$ gives an encounter rate111Thanks to
gravitational slingshot, this rate could be slightly higher in binary systems
Brayeur and Tinyakov (2012) such as the sites of some superbursts. of

$\displaystyle\Gamma^{\rm cl}_{\rm meet}(r)$ $\displaystyle=$ $\displaystyle
f_{\chi}\frac{\rho_{\chi}(r)}{M}v_{\chi}(r)\pi(b^{\rm cl}_{\rm max})^{2}$
$\displaystyle=$ $\displaystyle\frac{f_{\chi}}{2\ {\rm
Gyr}}\bigg{(}\frac{10^{-15}~{}M_{\odot}}{M}\bigg{)}~{},~{}R\ll
R_{\star},~{}r=r_{\odot}~{}.$

Here $\rho_{\chi}$ and $v_{\chi}$ are the DM density and speed, and $v_{\rm
esc}=\sqrt{2GM_{\star}/R_{\star}}$ is the escape speed at the stellar surface.
The second line of Eq. (III.1) is in the limit of small clumps at the solar
position $r_{\odot}=8.3$ kpc, where $\rho_{\chi}^{\odot}=0.4$ GeV/cm3,
$v_{\chi}^{\odot}=300$ km/s. I will be primarily interested in the case of
$f_{\chi}=1$.

Through elastic center-of-momentum-frame isotropic scattering with DM
particles of mass $m_{\chi}$ in the clusters, the typical rest-frame recoil
energy of C or O ions of mass $m_{\rm T}$ is

$\displaystyle\bar{E}^{\rm cl}_{\rm rec}$ $\displaystyle\simeq$
$\displaystyle(2z+z^{2})\mu^{2}_{\rm T\chi}/m_{\rm T}$ (9)
$\displaystyle\simeq$ $\displaystyle m_{\rm T}v_{\rm esc}^{2},\ \ m_{\rm T}\ll
m_{\chi},v_{\rm esc}\ll 1~{},$

where $\mu_{\rm T\chi}$ is the ion-$\chi$ reduced mass and $1+z=(1-v^{2}_{\rm
esc})^{-1/2}$ is the blueshift of infalling DM. As a function of $m_{\chi}$,
$\bar{E}^{\rm cl}_{\rm rec}$ is greatest in the limit $m_{\chi}\gg m_{\rm T}$,
where the maximum kinetic energy deposited into a trigger mass is simply
$\simeq(2z+z^{2})M_{\rm trig}\simeq M_{\rm trig}v^{2}_{\rm esc}$. This maximum
is achieved when every ion in the trigger volume encounters a clump particle,
i.e. when the clump is optically thick to an ion:

$\frac{\sigma_{\rm T\chi}}{m_{\chi}}\geq\frac{\pi R^{2}}{M}~{}.$ (10)

This condition is valid so long as the reduced cross section above is smaller
than the “ceiling” set by the outer layers essentially stopping the clump
before it reaches the trigger volume. I elaborate on this in Sec. III.1.

In Figure 1 I comparatively plot two quantities as a function of stellar
density: the maximum energy deposited per trigger mass, $2z+z^{2}$, for both
WD cores and NS oceans, and the minimum energy per mass to trigger runaway,
$Q_{\rm dep}/M_{\rm trig}=\bar{c}_{p}T_{\rm crit}$ from Eq. (1). On the top
x-axis the ticks are WD masses corresponding to the core densities in the
bottom x-axis (Eq. (5)). The curves labelled $\alpha_{\rm y}=0$ correspond to
dark clusters, as will become clear in Sec. III. One sees that only WDs with
masses $\gtrsim 1.05M_{\odot}$ can be triggered into thermonuclear runaway by
elastic scattering of dark clusters. Superbursts, on the other hand, can be
comfortably triggered by elastic recoils due to the deeper gravitational
potentials of NSs.

###  III.2 Long-range dark nuggets

Now consider macroscopic DM composites $\mathcal{X}$ that interact with
nucleons by exchanging a real spin-0 state $\phi$ of mass $\mu$,

$\mathcal{L}\supset
g_{n}\phi\bar{n}n+g_{\chi}\phi\bar{\mathcal{X}}\mathcal{X}~{},$ (11)

inducing an attractive potential

$V_{\rm Yuk}(r)=-\alpha_{\rm y}\frac{GM_{\star}M}{r}e^{-\mu r},\ \ \alpha_{\rm
y}=\frac{g_{n}g_{\chi}}{4\pi Gm_{n}M}~{}.$ (12)

As mentioned in the Introduction, this potential sources energy that may be
used to trigger supernovae and superbursts. As seen shortly, this potential
also serves to increase the incident flux of nuggets by augmenting
gravitational focusing. I particularly consider the region $\mu^{-1}\gg
R_{\star}$, where this scenario is interesting. For $\mu^{-1}\ll R_{\star}$
and $\alpha_{\rm y}\ll 1$ my results are comparable to the gravity-only
scenario just seen in Sec. III.1.

For nuggets of size $\ll R_{\star}$, the maximum impact parameter for stellar
intersections is Gresham  _et al._ (2023)

$\displaystyle b^{\rm nug}_{\rm max}$ $\displaystyle=$ $\displaystyle{\rm
min}(b_{\rm max,in},b_{\rm max,out})~{},$ (13) $\displaystyle b_{\rm max,in}$
$\displaystyle\approx$ $\displaystyle R_{\star}(1+z)\frac{v_{\rm
esc}}{v_{\chi}}$ $\displaystyle\times$ $\displaystyle\bigg{[}1+\alpha_{\rm
y}e^{-\mu R_{\star}}\bigg{(}1+\alpha_{\rm y}e^{-\mu R_{\star}}\frac{v^{2}_{\rm
esc}}{4}\bigg{)}\bigg{]}^{1/2}~{},$ $\displaystyle b_{\rm max,out}$
$\displaystyle\approx$ $\displaystyle\frac{w}{\mu}\sqrt{\bigg{(}1+\frac{\mu
R_{\star}}{w}\frac{v^{2}_{\rm esc}}{v^{2}_{\chi}}\bigg{)}}~{},$
$\displaystyle{\rm with}\ $ $\displaystyle w={\rm
max}\bigg{(}\log\bigg{[}\frac{\alpha_{\rm y}}{\frac{2}{\mu
R_{\star}}\frac{v^{2}_{\chi}}{v^{2}_{\rm esc}}+\frac{1}{\log\alpha_{\rm
y}}}\bigg{]},2+\log 2\bigg{)}~{},$

where the analytic approximations were estimated in the limit of
$v^{2}_{\chi}\ll v^{2}_{\rm esc}/2\ll 1$ and $\mu^{-1}\gg R_{\star}$. Eq. (13)
is obtained from (i) taking the orbit equation in Schwarzschild co-ordinates
Gresham  _et al._ (2023),

$\displaystyle M^{2}\dot{r}^{2}$ $\displaystyle=$ $\displaystyle(E-V_{\rm
eff,+})(E-V_{\rm eff,-})~{},$ (14) $\displaystyle\frac{V_{\rm eff,\pm}}{M}$
$\displaystyle=$ $\displaystyle-\frac{GM_{\star}\alpha_{\rm y}e^{-\mu r}}{r}$
$\displaystyle\pm\sqrt{\bigg{(}1-\frac{2GM_{\star}}{r}\bigg{)}\bigg{(}\frac{(L/M)^{2}}{r^{2}}+1\bigg{)}}~{},$

(ii) locating two “centrifugal barriers” $r_{b}$ in the effective potential
from the conditions

$\displaystyle V_{\rm eff,+}(r_{b})|_{L=\gamma_{\chi}Mb_{\rm max}v_{\chi}}$
$\displaystyle=$ $\displaystyle\gamma_{\chi}M\ \ [dr/d\tau=0]~{},$
$\displaystyle V^{\prime}_{\rm eff,+}(r_{b})|_{L=\gamma_{\chi}Mb_{\rm
max}v_{\chi}}$ $\displaystyle=$ $\displaystyle 0\ \ \ \ \ \ \ [{\rm local\
max}]~{},$ (15)

where $\gamma_{\chi}=(1-v^{2}_{\chi})^{-1/2}$, (iii) setting the solved
$r_{b}=R_{\star}$. The “inner” barrier is gravity-driven and the “outer” is
fifth-force-driven.

The nugget-star encounter rate is then obtained by simply making the
replacement in Eq. (III.1)

$\Gamma^{\rm nug}_{\rm meet}=\Gamma^{\rm cl}_{\rm meet}|_{b^{\rm cl}_{\rm
max}\rightarrow b^{\rm nug}_{\rm max}}~{}.$ (16)

In the set-up above, DM nugget transits deposit energy in the star through
tidal excitations of seismic oscillations Gresham  _et al._ (2023), however I
will assume additional short-range scattering interactions that dominate
energy deposition, i.e. the “maximal heating” scenario considered in Ref.
Gresham  _et al._ (2023). As in the case of dark clusters, the individual
constituents $\chi$ of the nuggets scatter on the star-bound targets222I
expect coherence in scattering over neither the target nucleus nor the
projectile nugget. My parameter space of interest is $\alpha_{\rm y}>1$, where
the effects of the fifth force are apparent, and in this region the momentum
transfer $q\gg R^{-1}_{\rm nuc},R^{-1}_{\rm nug}$, where $R_{\rm
nuc}\simeq\mathcal{O}(1)$ fm is the nuclear size and $R_{\rm nug}$ is the
nugget size required for successful thermonuclear explosion. See, e.g., Ref.
Grabowska  _et al._ (2018)., but now at higher speeds due to acceleration by
both gravity and the fifth force in Eq. (12). For a target state of mass
$\bar{m}_{\rm T}$, the typical rest-frame recoil energy is

$\displaystyle\bar{E}^{\rm nug}_{\rm rec}$ $\displaystyle\simeq$
$\displaystyle\frac{\bar{m}_{\rm T}m^{2}_{\chi}}{\bar{m}_{\rm
T}^{2}+m^{2}_{\chi}+2m_{\chi}\bar{m}_{\rm
T}(1+\widetilde{z})}(2\widetilde{z}+\widetilde{z}^{2})~{}$
$\displaystyle\simeq$
$\displaystyle\begin{cases}m_{\chi}\widetilde{z}\frac{1+\widetilde{z}/2}{1+\widetilde{z}}\
\ \ ,~{}~{}1\ll m_{\chi}/\bar{m}_{\rm T}\ll(1+\widetilde{z})\\\ \bar{m}_{\rm
T}(2\widetilde{z}+\widetilde{z}^{2}),~{}~{}(1+\widetilde{z})\ll
m_{\chi}/\bar{m}_{\rm T}~{},\end{cases}$

where

$\displaystyle 1+\widetilde{z}$ $\displaystyle=$
$\displaystyle(1+z)\bigg{(}1+\alpha_{\rm y}e^{-\mu R_{\star}}\frac{v_{\rm
esc}^{2}}{2}\bigg{)}$ (18) $\displaystyle\approx$ $\displaystyle
1+\frac{v_{\rm esc}^{2}}{2}(1+\alpha_{\rm y}e^{-\mu R_{\star}})~{}.$

It can be seen that for $v_{\chi}\ll v_{\rm esc}\lesssim 1$, Eq. (13)
$\rightarrow$ Eq. (7) and Eq. (III.2)$\rightarrow$ Eq. (9) in the limit
$\alpha_{\rm y}\rightarrow 0$, $\mu\rightarrow\infty$. Thus for a force range
$\mu^{-1}\gg R_{\star}$, the kinetic energy of the incoming nugget at the
stellar surface is $\simeq(1+\alpha_{\rm y})$ times that from a purely
gravitational pull. This implies WDs of mass $<1.05M_{\odot}$ can be
triggered, unlike for dark clusters as discussed in Sec. III.1. I have
depicted this in Fig. 1, where increasing $\alpha_{\rm y}$ is seen to help
explode lighter WDs. Of course, non-zero $\alpha_{\rm y}$ trivially ignites
superbursts in NS oceans. For simplicity I have assumed here elastic nuclear
scattering, arising for instance from nugget constituents scattering with
geometric cross sections (as discussed in Sec. IV.1), however it is also
possible that nuggets transferring high momenta resolve nucleons or partons.
In such inelastic scattering, the fraction of nugget kinetic energy
transferred could be as high as $y_{\rm max}\gtrsim 0.1$ Agashe  _et al._
(2016).

Figure 2: Dark cluster masses and radii excluded by their triggering of Type
Ia-like supernovae in carbon-oxygen WDs and x-ray superbursts in NS carbon
oceans. Overlaid are contours of the minimum reduced cross sections for
nuclear scattering on cluster constituents in order for all nuclei in a
trigger mass to recoil, so that the total energy transferred is guaranteed to
cause explosion (Fig. 1). See Sec. III.1 for further details.

Figure 3: Long-range dark nugget effective couplings and masses excluded by their triggering of supernovae and superbursts for various values of the fifth-force range, assuming nugget sizes just big enough to accomplish ignition. Limits from other studies are also shown; the microlensing limits assume that nuggets comprise all the DM, but would lift rapidly for a DM sub-component. See Sec. IV.2 for further details. superburst | r (kpc) |  $t_{\rm recur}$ (yr)
---|---|---
4U 1820+30 | 1.2 | 2.5 Sidhu and Starkman (2020)
4U 0614+091 | 11.5 | 4.8 Alizai (2023)
GX 17+2 | 2.4 | 1 Alizai (2023)
4U 1636$-$536 | 5.3 | 1 Cumming  _et al._ (2006)
Ser X$-$1 | 5 | 1 Cumming  _et al._ (2006)
Aql X$-$1 | 5 | 1 Cumming  _et al._ (2006)
Table 1: Recurring superbursts Alizai (2023) used to set limits in this work.
Their Galactic position
$r=\sqrt{r^{2}_{\odot}+d^{2}_{\oplus}-2r_{\odot}d_{\oplus}\cos\ell\cos b}$ was
obtained using information on the distance to the source $d_{\oplus}$, its
longitude $\ell$ and latitude $b$ from SIMBAD Wenger  _et al._ (2000). The
recurrence times $t_{\rm recur}$ are taken from the references indicated. See
Sec. IV for further details.

##  IV Results

###  IV.1 Limits on dark clusters

In Figure 2 I show limits on the dark cluster size vs mass from the existence
of WDs (region enclosed by green curves) and the observed frequency of NS
superbursts (red). For WDs I fix the upper bound on $R$ by simply requiring
$R\geq\lambda_{\rm trig}$ (Eq. (4)) to ensure that Eq. (1) is satisfied.
Smaller clumps, too, would satisfy Eq. (1): as seen in Fig. 1, for $M_{\rm
WD}>1.05M_{\odot}$ dark clusters of size $\lambda_{\rm trig}$ would raise the
temperature of the trigger volume to above $T_{\rm crit}$. By comparing the
trigger energy and maximum energy deposit in Fig. 1, one sees that for the
heaviest WDs, the minimum triggering $R$ is about $\lambda_{\rm
trig}/10^{1/3}$, thus my choice of the upper bound on $R$ is conservative by a
factor of $\lesssim 2$. For NSs, I take the upper bound to be
$R\geq\lambda_{\rm trig}(\bar{c}_{p}T_{\rm crit}/(2z+z^{2}))^{1/3}$ (Eqs. (1)
and (9)), ensuring sufficient energy deposition to ignite the ocean matter.
For WDs the smallest $R$ bounded in Fig. 2 is the $\lambda_{\rm trig}$ that
corresponds to a near-Chandrasekhar limit 1.41 $M_{\odot}$ WD for
$M\in[10^{-38.5},10^{-16.5}]M_{\odot}$. The upper end of this range comes from
requiring 2.3 encounters (corresponding to the 90% C.L. limit) with clumps of
the solitary 1.41 $M_{\odot}$ WD in the Montreal White Dwarf Database (MWDD)
Dufour  _et al._ (2017) for which the WD age (1.3 Gyr) is available. For
higher $M$, I require that the total clump-WD encounters $N^{\rm cl-WD}_{\rm
enc}=2.3$ , with

$N^{\rm cl-WD}_{\rm enc}=\sum_{i}\Gamma^{\rm cl}_{{\rm meet},i}\tau_{{\rm
WD},i}~{},$ (19)

where $\Gamma^{\rm cl}_{\rm meet}$ is from Eq. (III.1), $\tau_{\rm WD}$ is the
WD age, and $i$ runs over WDs in MWDD for which mass and cooling age
information are available. As $M$ is increased, $\Gamma^{\rm cl}_{\rm meet}$
decreases, and to compensate for this the limiting $R$ samples over larger WDs
(in spite of the smaller gravitational focusing), which corresponds to lighter
WDs, in turn corresponding to smaller WD central densities and hence larger
$\lambda_{\rm trig}$ (Eq. (4)), resulting in larger upper bounds on $R$. The
right vertical line at $M=10^{-12}~{}M_{\odot}$ is reached after sampling over
967 WDs; the upper bound on $R$ here corresponds to triggering WDs of mass
1.05 $M_{\odot}$, the minimum for which cluster-induced elastic nuclear
recoils can result in WD explosion (Sec. III.1). In obtaining these limits I
set the DM density and velocity dispersion to their values in the solar
neighborhood, an approximation that is justified since only 19 WDs in the
sample lie outside the local kpc.

For superbursts the upper bound on $R$ is the $\lambda_{\rm trig}$
corresponding to the NS density at the bottom of the C ocean, $1.2\times
10^{10}$ g/cm3, which I estimated by first obtaining crustal column depths by
integrating over the NS crust density profile of PAL Model 1 Datta  _et al._
(1995), and then by comparison with the maximum ignition column depth shown in
Fig. 1 of Ref. Cumming and Bildsten (2001): $4\times 10^{13}$ g/cm2. The right
vertical line is from requiring the total clump-NS encounters $N^{\rm cl-
NS}_{\rm enc}=2.3$ using an equation analogous to Eq. (19) with the
replacement $\tau_{\rm WD}\rightarrow t_{\rm recur}$, where $t_{\rm recur}$ is
the timescale, typically $\mathcal{O}$(yr), of recurrence between repeating
superbursts. In Table 1 I have listed estimates of $t_{\rm recur}$ of the 6
superburst sources that have recurred at least once, out of 16 observed in
total Alizai (2023). In the encounter rate I set the DM densities at the
superburst locations (in Table 1) to those obtained from an NFW profile with
scale parameters taken from Ref. Cirelli  _et al._ (2011); I also set
$v_{\chi}(r)=\sqrt{3GM_{\rm NFW}(r)/2r}$, the Maxwell-Boltzmann dispersion
speed for a mass $M_{\rm NFW}(r)=\int_{0}^{r}d^{3}r^{\prime}\rho(r^{\prime})$
enclosed within $r$. The source 4U 1820-30 resides in the globular cluster NGC
6624, where the DM densities could be potentially higher (and dispersion
speeds lower), however due to the large uncertainties in the dynamics of such
stellar systems Garani  _et al._ (2023) I will conservatively take
$\rho_{\chi}(r)$ in Eq. (III.1) to be the Galactic background one. Despite
that, this superburst dominates the encounter rate due to its proximity to the
Galactic Center, warranting its exclusive use by Ref. Sidhu and Starkman
(2020).

The left diagonal lines in Fig. 2 are from requiring that the fraction of the
kinetic energy of the infalling cluster transferrable to the trigger volume
(through either elastic or inelastic scattering) exceeds the trigger energy:

$zM\bigg{(}\frac{\lambda_{\rm trig}}{R}\bigg{)}^{3}\geq M_{\rm
trig}\bar{c}_{p}T_{\rm crit}~{}.$ (20)

I overlay these plots with contours of constant $\sigma_{\rm T\chi}/m_{\chi}$
corresponding to the minimum required for the clumps to be optically thick to
nuclei (Eq. (10)). There is a lower bound (ceiling) to $\sigma_{\rm
T\chi}/m_{\chi}$ above which DM scattering with the outer stellar layers
degrade its energy and thwart explosion in the relevant inner layer. For WDs
this arises from the non-degenerate envelope with thickness $\simeq
10^{-3}R_{\star}$ and maximum density $\simeq$ 100 g/cm3 (Padmanabhan (2010),
Eqs. (5.107) and (5.115)). For the corresponding column depth, this limit is
$\sigma_{\rm T\chi}/m_{\chi}\gtrsim 4\times 10^{-32}$ cm2/GeV. Similarly for
NS crusts, using the column depth at the carbon ocean bed in Ref. Cumming and
Bildsten (2001), the limit is $\sigma_{\rm T\chi}/m_{\chi}\gtrsim 4\times
10^{-38}$ cm2/GeV. For point-like scatters these ceilings break down for
$\sigma_{\rm T\chi}$ above the “saturated overburden” value, the point at
which the mean-free path of DM particles equals the inter-nuclear distance,
$({\rm overburden\ number\ density})^{-1/3}$ Bramante  _et al._ (2018b). That
is, above some $m_{\chi}$ the energy degradation from scattering on a finite
number of nuclei in the upper layers is insufficient to prevent an explosive
trigger deeper down; this is $m_{\chi}\gtrsim 10^{15}$ GeV for both WDs and
NSs. This effect is akin to that seen for simple DM-nucleus scattering
potentials that saturate the cross section at the geometric one, so that for
high DM masses the overburden is ineffective Goodman and Witten (1985); Digman
_et al._ (2019). The large cross sections required for the saturated
overburden effect could be achieved if the dark cluster constituents are
strongly coupled composites that are opaque to nuclei Digman  _et al._ (2019);
Cappiello  _et al._ (2021); Starkman  _et al._ (1990); Jacobs  _et al._
(2015); Singh Sidhu  _et al._ (2019a); Adhikari  _et al._ (2022); Dhakal  _et
al._ (2023a). I remark here that Ref. Sidhu and Starkman (2020) estimates the
ceiling $\sigma_{\rm T\chi}/m_{\chi}$ of such a DM candidate by treating the
drag force it experiences, which is numerically equal to the ceiling obtained
from the stopping power of point-like scattering.

Finally, the region $4\times 10^{-19}<M/M_{\odot}<10^{-13}$ (and for $R$ as
high as $10^{9}$ cm) can be probed in the future by observing a single 103
Kelvin NS or $>$100 $10^{4}$ Kelvin NSs via the same kinetic heating mechanism
considered here Bramante  _et al._ (2022a).

###  IV.2 Limits on long-range dark nuggets

In Figure 3 I show limits on the effective fifth-force coupling vs dark nugget
mass for four values of the force range $\mu^{-1}$. For $\mu^{-1}=10^{-11}\
{\rm pc}\ll R_{\rm WD}$, there are no WD limits; for $\mu^{-1}>$ 10 pc, the
typical inter-WD Napiwotzki (2009) or inter-NS distance Blaes and Madau
(1993); Sartore  _et al._ (2010), the use of Eq. (13) may not be valid. I
assume for minimality that the dark nuggets are of the size $\leq\lambda_{\rm
trig}$ required to heat the nuclear material to $\geq T_{\rm crit}$ as
described in Sec. IV.1; these sizes are well within the range of nugget models
in the literature Bai  _et al._ (2019); Grabowska  _et al._ (2018); Gresham
_et al._ (2018a, 2023). À la dark clusters in Sec. IV.1, the mass lower bounds
at 90% C.L. come from requiring a total of 2.3 nugget-star encounters, where
now in Eq. (19) (and its NS equivalent with $\tau_{\rm WD}\to t_{\rm recur}$)
I use the encounter rate in Eq. (16). For WDs I show two cases corresponding
to different nugget sizes: destroyal of all WDs of masses $\geq 0.8M_{\odot}$
and that of a single 1.41 $M_{\odot}$ WD. The slopes of these curves steepen
above some $\alpha_{\rm y}$ because the impact parameter in Eq. (13)
transitions from being set by the “inner” centrifugal barrier to the “outer”
(Sec. III.2). Across the four panels, more parameter space is constrained as
$\mu^{-1}$ is increased because $b_{\rm max,out}$ correspondingly increases
(Eq. (13)).

Again as in Sec. III.1, the mass upper bounds come from requiring that the
incoming kinetic energy of the nugget exceeds the trigger energy in Eq. (1).
For 0.8 $M_{\odot}$ WDs this occurs for $\alpha_{\rm y}\geq 9.7$. Note that
the denser the trigger region, the smaller is the minimum nugget mass required
to trigger it. Note also that as $\alpha_{\rm y}\rightarrow 0$ the minimum and
maximum masses in Fig. 3 correspond to those in Fig. 2.

I also show in Fig. 3 the mass limit from gravitational microlensing surveys,
to which the dark nuggets appear as point-like lenses as they are far smaller
than the Einstein radius Croon  _et al._ (2020a, b). This only applies to
$f_{\chi}=1$; for $f_{\chi}\lesssim 10^{-2}$ these limits go away, while the
lower bound on $M$ weakens $\propto f_{\chi}$ due to fewer nugget-star
encounters (Eq. (16)). Also shown are limits derived in Ref. Gresham  _et al._
(2023): (1) From kinetically overheating the coldest observed ($\lesssim
4\times 10^{4}$ K) NS, PSR J2144-3933 Guillot  _et al._ (2019), i.e., by
requiring $\widetilde{z}M\Gamma^{\rm nug}_{\rm meet}\leq$ the luminosity of
this NS. The maximum $M$ limitable is obtained by requiring $1/\Gamma^{\rm
nug}_{\rm meet}<$500 yr, the cooling timescale of an NS of this temperature.
More parameter space can be potentially probed by surveying sky regions for
excess populations of NSs that are at even higher temperatures Bramante  _et
al._ (2022a). (2) From destroying PSR J2144-3933 by depositing more energy via
tidal heating than about 10% of its gravitational binding energy, which occurs
for $\widetilde{z}M>3\times 10^{-3}M_{\odot}$. The maximum $M$ limitable now
corresponds to $1/\Gamma^{\rm nug}_{\rm meet}$ = 300 Myr, the age of the NS.
(3) From recasting 11-yr NANOGrav pulsar timing data to look for phase shifts
induced by long-range dark nuggets transiting observational lines of sight.
These limits are only in the bottom panels because smaller $\mu^{-1}$
exponentially suppress the new phase shifts in pulse arrival times. They
weaken at small $M$ since the effects of gravity and the fifth-force lessen,
and also weaken at large $M$ for $\mu^{-1}=10^{-3}$ pc as the typical nugget-
pulsar distance increases, which decreases the new phase shift; for
$\mu^{-1}=10$ pc the limit on $\alpha_{\rm y}$ plateaus because here the fifth
force effectively rescales the gravitational constant (Eq. (12)). N. B. For
pulsar timing limits to apply the nuggets need only be smaller than their
impact parameter ($>10^{9}$ km), thus they apply for nuggets larger than those
required to trigger thermonuclear explosions, and larger than the Einstein
radius where microlensing limits go away. (4) From weak equivalence principle
tests measuring the difference in acceleration of two celestial bodies
approaching the Galactic Center, giving the limit $\alpha_{\rm
y}(\mu^{-1}/{\rm pc})^{2}\lesssim 5\times 10^{5}$ so long as
$(\rho_{\chi}/M)^{-1/3}\ll\mu^{-1}\ll r_{\odot}$.

My limits are seen to complement all of the above in regions spanning several
decades of parameters. In particular, although my limits have qualitatively
similar shapes to the one from heating PSR J2144$-$3933 at high masses, I am
able to constrain much higher $M$ with WD explosions. This is because my mass
upper bound is set by the Gyr timescale of the ages of $\sim$ 103 WDs, as
opposed to the 500 yr timescale of cooling of a single NS.

##  V Discussion

I have set limits on macroscopic dark matter clumps/composites by their
triggering of Type Ia-like supernovae and superbursts, requiring that the
maximum recoil energy they can induce via scattering on all target nuclei in a
triggerable volume exceeds the minimum energy needed to ignite that volume.
This is possible when the clump is optically thick to a stellar nucleus, given
that the clump constituents are not too softened by scattering with the
stellar outer layers, which constitute the overburden here. To satisfy the
latter condition in some parts of parameter space, I relied on an interesting
effect: above some mass on the ceiling reduced cross section, the energy of a
clump constituent scattering on a finite number of nuclei on the way to the
relevant deeper layer is insufficiently degraded to prevent ignition. I
believe that a similar effect also calls into question the limits set by Ref.
Graham  _et al._ (2018) on point-like elastic scattering of DM particles on
carbon in WDs. These limits were set by the following condition for the
minimum stopping power, $dE/dx$, for a heating region of size
$\leq\lambda_{\rm trig}$:

$n_{\rm T}\sigma_{\rm T\chi}m_{\rm T}v^{2}_{\rm
esc}\gtrsim\rho\bar{c}_{p}T_{\rm crit}\lambda_{\rm trig}^{2}.$ (21)

However, I believe that the condition should be

$\displaystyle N_{\rm hit}\frac{m_{\rm T}v^{2}_{\rm esc}}{\lambda_{\rm trig}}$
$\displaystyle\gtrsim$ $\displaystyle\rho\bar{c}_{p}T_{\rm crit}\lambda_{\rm
trig}^{2}~{},$ $\displaystyle N_{\rm hit}$ $\displaystyle=$
$\displaystyle\max[n_{\rm T}\sigma_{\rm T\chi}\lambda_{\rm trig},n^{1/3}_{\rm
T}\lambda_{\rm trig}]$ (22)

where $N_{\rm hit}$ is the number of point-like scatters on nuclei as the DM
particle traverses the length $\lambda_{\rm trig}$. Eq. (22) reduces to Eq.
(21) for $\sigma_{\rm T\chi}<n_{\rm T}^{-2/3}$. For the 1.25 $M_{\odot}$ WD
considered in Ref. Graham  _et al._ (2018), $n_{\rm T}\simeq 10^{-31}$ cm-3,
so that for Eq. (21) to hold, one requires $\sigma_{\rm T\chi}\lesssim 2\times
10^{-21}$. On the other hand, in Ref. Graham  _et al._ (2018) the region
$\sigma_{\rm T\chi}>10^{-12}$ cm2 is shown to be excluded. Another way to see
the invalidity of this constraint is to evaluate the maximum energy
transferred by DM elastic recoils by a linear transit across a length
$\lambda_{\rm trig}$, which is $(n^{1/3}_{\rm T}\lambda_{\rm trig})(m_{\rm
T}v^{2}_{\rm esc})\simeq(2.7\times 10^{6})(0.016$ GeV) $=10^{4.5}$ GeV, far
smaller than the required trigger energies ranging across WD masses,
$10^{17-24}$ GeV. In the case of DM clumps, the condition for explosion I
sought by plotting Fig. 1 is essentially Eq. (22), but now with $N_{\rm
hit}=4\pi n_{\rm T}\lambda^{3}_{\rm trig}/3$, the number of nuclei in the
trigger volume, so that Eq. (22) is satisfied.

Several directions from here are ripe for inquiry. I had considered elastic
nuclear scatters of dark cluster and nugget constituents, but alternative
possibilities are inelastic scatters – such as baryon-absorption of Q-ball DM
treated in Ref. Graham  _et al._ (2018) – and scattering on electrons by
leptophilic long-range dark nuggets that deposit explosive energy. The long-
range potential of the dark nuggets need not be Yukawa, as treated here, but
something else, e.g., Coulombic. The energy stored in the long-range potential
can also serve, through scattering alone, to heat planets (which has been done
before with DM self-annihilations Kawasaki  _et al._ (1992); Abbas and Abbas
(1998); Mack  _et al._ (2007); Hooper and Steffen (2012); Garani and Tinyakov
(2020); Bramante  _et al._ (2020); Acevedo  _et al._ (2021d); Leane and
Smirnov (2021); Bramante  _et al._ (2022b)). On that note, they could also
heat NSs colder than PSR J2144$-$3933, which may observable in current and
forthcoming telescopes in the ultraviolet, optical, and infrared Baryakhtar
_et al._ (2017); Bramante  _et al._ (2022a); McKeen  _et al._ (2021a, b); Team
(2019); Diehl  _et al._ (2019); Peterson (2016); Ivezić  _et al._ (2019);
Gardner  _et al._ (2006); Neichel  _et al._ (2018); Skidmore  _et al._ (2015);
J. Green (2012). Thanks to their higher energies and fluxes than usually
considered, long-range dark nuggets near or beyond the Planck mass ($\simeq$
10 $\mu$g, contained in Fig. 3) could be discovered in terrestrial set-ups,
e.g. astroparticle detectors De Mitri  _et al._ (2002); Bramante  _et al._
(2018b, 2019c, 2019d); Cappiello  _et al._ (2021); Grabowska  _et al._ (2018);
Bai  _et al._ (2022); Bhoonah  _et al._ (2021); Adhikari  _et al._ (2022);
Aprile  _et al._ (2023); Carney  _et al._ (2022), ancient minerals Snowden-
Ifft  _et al._ (1995); Baum  _et al._ (2020); Drukier  _et al._ (2019);
Acevedo  _et al._ (2021c); Baum  _et al._ (2023) and other interesting systems
Singh Sidhu  _et al._ (2019a); Singh Sidhu and Starkman (2019); Singh Sidhu
_et al._ (2020, 2019b); Dhakal  _et al._ (2023b), by exploiting the large
energy deposits and the peaked speed spectrum Davoudiasl  _et al._ (2020). See
also the very recent Ref. Du  _et al._ (2023) for the projected reach of
gravitational wave experiments. Dark clusters and long-range dark nuggets
making up a sub-component of DM (perhaps the only component with SM-scattering
interactions) would shift my limits suitably while relaxing many others shown
in this work.

Those of us interested in hunting dark matter via thermonuclear explosions
have taken advantage of the trigger lengths (“flame widths”) numerically
estimated in Ref. Timmes and Woosley (1992) by involving a large nuclear
reaction network. However, as mentioned before, these are only available for
(i) a narrow range of densities, so that authors have had to rely on
extrapolations at lower densities, (b) two different (50-50 C+O and 60-30-10
O+Ne+Mg) compositions, so that I had to simply adopt the C-O results for
C-based superbursts. Given the significance of these computations for such
fundamental questions as dark matter, I request experts in the nuclear
astrophysics community to update the 31 year-old results of Ref. Timmes and
Woosley (1992). I also look forward to expansions in catalogues of white
dwarfs and superbursts, the latter thanks to NICER Chakrabarty and et al.
(2018) and AstroSat Agrawal (2006)– or better yet, to the next Galactic
supernova.

## Acknowledgments

I thank A. Gopakumar for triggering a runaway by bringing my attention to low-
mass x-ray binaries at the workshop Neutron Stars: The Celestial Clocks That
Probe Extreme Physics at the Institute of Mathematical Sciences, where I also
had profitable discussions with M. McLaughlin. I also thank J. Bramante for
pointing out that trigger-happy clumps could be smaller than the trigger
volume, D. Galloway for pointing me to Ref. Alizai (2023), and V. Lee for
clarifying the maximum size of nuggets observable in pulsar timing arrays. C.
Sinensis was of immense help with checking my calculations and typing up the
manuscript.

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