# Efficient Trigger Word Insertion

Yueqi Zeng  University of Science and
Technology of China Hefei, China, 230027<EMAIL_ADDRESS>Ziqiang Li
University of Science and
Technology of China
Hefei, China, 230027
<EMAIL_ADDRESS>Pengfei Xia  Lei Liu  University of Science and
Technology of China
Hefei, China, 230027
<EMAIL_ADDRESS>University of Science andTechnology of China
Hefei, China, 230027
Zhejiang Lab, Hangzhou, Zhejiang, 311121
<EMAIL_ADDRESS>Bin Li  University of Science and
Technology of China
Hefei, China, 230027
<EMAIL_ADDRESS>
###### Abstract

With the boom in the natural language processing (NLP) field these years,
backdoor attacks pose immense threats against deep neural network models.
However, previous works hardly consider the effect of the poisoning rate. In
this paper, our main objective is to reduce the number of poisoned samples
while still achieving a satisfactory Attack Success Rate (ASR) in text
backdoor attacks. To accomplish this, we propose an efficient trigger word
insertion strategy in terms of trigger word optimization and poisoned sample
selection. Extensive experiments on different datasets and models demonstrate
that our proposed method can significantly improve attack effectiveness in
text classification tasks. Remarkably, our approach achieves an ASR of over
90% with only 10 poisoned samples in the dirty-label setting and requires
merely 1.5% of the training data in the clean-label setting.

###### Index Terms:

text backdoor attacks, data efficiency

Figure 1: Poisoned Set Generation. We generate a poisoned set in two steps.
First, select samples with higher contribution to backdoor injection. Second,
insert the optimized trigger word at a suitable position.

##  I Introduction

Over the past decade, the field of natural language processing (NLP) has
witnessed significant advancements [1][2]. These advancements have been driven
by the development of deep neural networks (DNNs) [3][4], which leverage
large-scale text data to learn effective feature representations. As the data
volume becomes larger and models become more complex, we enter the large-model
era, where most small companies as well as individuals cannot bear the huge
computing power consumption of the training process. Fortunately, the concept
of transfer learning [5] has provided a solution. By taking advantage of pre-
trained models available on third-party platforms, even small companies and
individuals can achieve state-of-the-art performance by fine-tuning these
models to suit their specific downstream tasks. For instance, a pre-trained
BERT [6] model can be fine-tuned with only one output layer changed and obtain
cutting-edge performance across various tasks [7].

However, there do exist big security problems and much research has
demonstrated that DNNs in NLP are vulnerable to various kinds of attacks [8,
9, 10]. Among these attacks, backdoor attacks [11][12] are particularly
noteworthy. In order to construct backdoor attacks, the attackers usually
create a poisoned training set with a few poisoned samples and exploit it to
train an infected model. Compared to the clean model, the infected model shows
comparable performance, yet once the designated trigger is added to the test
input, the model will malfunction immediately.

Recently, most works about textual backdoor attacks concentrated on
stealthiness when designing triggers [12, 13, 14], so as to construct natural
and semantically consistent sentences. For example, uncommon syntax [13], word
substitution [14] and text styles [10] have been employed as trigger patterns
to reduce the chances of human inspection and detection by defense mechanisms
concurrently. However, the effectiveness of these methods is generally
demonstrated by setting a high poisoning rate, like 20% to 30% of the total
training datasets. Despite the grammatical correctness of the poisoned texts,
human inspectors may still find these expressions unusual. As a result, these
methods are easily detectable due to their high poisoning rate.

Previous studies have provided limited insights into the impact of the sample
poisoning rate. This paper addresses this gap by focusing on reducing the
number of poisoned samples while achieving a comparable Attack Success Rate
(ASR). The reduction in poisoning numbers offers several advantages: Firstly,
from the attackers’ perspective, having fewer poisoned samples means making
fewer changes to the training set, which simplifies the implementation and
deployment of the attack. Secondly, for defenders, if there are only a few
dozen poisoned samples among tens of thousands of text samples, the likelihood
of detecting the attack will significantly decrease, thereby enhancing
stealthiness to a certain extent.

We first revisit the process of generating poisoned sets for text backdoor
attacks, as illustrated in Fig. 1. We recognize that the efficiency of
backdoor injection relies on two crucial factors: selecting suitable benign
samples for poisoning and using effective trigger words for the poisoning
process. These two factors are independent of each other, and through a
rational design, we optimize both aspects to significantly reduce the
requirement for poisoned samples. To the best of our knowledge, we are the
first to probe the efficiency of both trigger and sample selection regarding
text backdoor attacks. We propose an efficient trigger word insertion method
and our main contribution can be summarized as follows:

  * •

Finding out the most efficient inserted trigger words. DNNs are susceptible to
perturbations [15, 16, 17], which are called natural flaws. We consider that
the training process of backdoor attacks is to reinforce these
vulnerabilities. Building upon this concept, We propose two strategies, one is
finding the nearest word embedding to the optimized embedding, and the other
is to search trigger words directly. Experimental results on diverse datasets
show that both methods are effective with different trigger positions.
Specifically, in the dirty-label setting, a mere 10 samples are sufficient to
achieve an 80% ASR, while in the clean-label setting, only 2% of training
samples prove adequate.

  * •

Selecting more important samples for backdoor injection. Inspired by efficient
backdoor injection in computer vision [18, 19, 20, 21], we recognize that
different poisoned samples contribute unequally to the backdoor injection
process in text classification tasks. In light of this observation, we
introduce the FUS-p strategy, which enables the iterative selection of indexes
for poisoned samples. Remarkably, employing this technique in conjunction with
the optimized trigger word allows us to achieve an ASR of over 90% using only
10 samples (0.145% for SST-2, 0.008% for AGNews) in the dirty-label setting
and a mere 1.5% of the training data in the clean-label setting. Additionally,
for the clean-label setting, we propose a straightforward sample removal
strategy. Although this approach is time-saving, our experiments demonstrate
that it is less effective compared to the FUS-p method.

##  II Related Works

Backdoor attacks [22, 23, 24], as the name indicates, are to inject backdoors
into DNN models. Two major attributes of backdoor attacks are securing the
test accuracy with clean inputs and acquiring wrong but targeted predictions
while the backdoor is activated by designated triggers, respectively.

Gu et al. [25] first introduced the concept of backdoor attacks in the
computer vision field in 2017, poisoning a small number of training data with
a fixed white patch added to the bottom-right corner of the image. Then, it
was extended to the NLP field [9]. The difference here is that images are
drawn from continuous space while texts are completely discrete. Thus the
attack method changed to insert rare words such as ”cf” and ”bb” into original
sentences as triggers.

Recently, considerable evidence has accumulated to show that textual backdoor
attacks are great threats to NLP-related tasks as well. Dai et al. [11] first
explored the feasibility of injecting a backdoor to the LSTM model by
inserting emotion-neutral sentences at various positions with different
trigger lengths. Later, after the emergence and rapid development of large-
scale BERT models [6], many researchers [7] [26] began to adopt these pre-
trained models to fine-tune downstream tasks due to their good performance. To
keep up with the trend, Kurita et al. [9] realized retaining the injected
backdoors through the fine-tuning process over clean datasets by a
regularization method and an initialization operation. In particular, the
initialization strategy is strongly associated with embedding modification of
triggers. More recently, Yang et al. [27] introduced an algorithm that updates
the trigger embedding vector directly with gradient. Despite the high attack
success rate performance, these two methods require access to manipulating the
embedding layers, which is hard to fulfill in the real world. Therefore, we
mainly focus on the data-poisoning scene, where attackers only have the
ability to manipulate the training data.

In the context of data poisoning, the stealthiness of triggers is one of the
evaluation indices apart from effectiveness. In order to guarantee
stealthiness, many researchers concentrated on preserving the semantic
information of sentences. Chen et al. [12] proposed generating word-level
triggers by combining context information while transferring syntax in terms
of sentence level. Zhang et al. [28] synthesized natural and fluent sentences
containing several appointed trigger words at the same time. Qi et al. [14]
activated textual backdoors through a learnable combination of word
substitution. Beyond these works, Qi et al. [10] also proposed to shift the
text style with the meaning preserved. However, when tainted sentences are
utilized to construct poisoned models, the number of poisoned samples is
barely considered. Only some works [29] [30] explored the effect of the
poisoning rate in the experimental section. Given that the poisoning number is
heavily allied to how concealed the backdoor attacks are, we emphasized
especially data efficiency in this paper.

##  III Method

In this section, we introduce the overall attack pipeline of our method.

###  III-A Threat Model

Attacker Goals The attacker generates a new dataset with a small number of
samples in the target class poisoned and releases it to users for downstream
tasks. When users train the set according to the attacker’s guide and test the
trained model on test datasets mixed with poisoned texts, they fail to get
correct predictions with poisoned inputs while the model presents normal with
clean inputs. In other words, the eventual goal for attackers is to acquire a
high attack success rate and reduce clean accuracy loss as much as possible at
the same time.

Attacker Capabilities Here, we consider data-poisoning setting. The attacker
can only manipulate the datasets for training, where several texts are
poisoned with a specific trigger type. This means the attacker cannot have
access to models the users implant.

Attack Construction Given a benign training dataset $\mathcal{D}_{b}$, we have
clean text samples $x$ and corresponding labels $y$ satisfying
$(x,y)\in\mathcal{D}_{b}$. To construct poisoned set $\mathcal{D}_{p}$, we get
poisoned samples $x^{\prime}=F(x,t)$ with target labels $y_{t}$, where $t$
represents one type of triggers and $F(\cdot,\cdot)$ means trigger-text mixed
form. For dirty-label attacks, we have $y_{t}\neq y$. Conversely, for clean-
label attacks, $y_{t}=y$ is satisfied. Next, we build the mixed training set
$\mathcal{D}_{m}=(\mathcal{D}_{b}/\mathcal{D}_{p})\cup\mathcal{D}_{p}$ and
exploit it to train a poisoned model. Finally, we inspect the effectiveness of
the backdoor injection process by testing both the Clean Accuracy (CA) and the
Attack Success Rate (ASR) on the test set.

We now introduce our proposed attack method in terms of trigger word
optimization and sample selection in an intricate way, corresponding to two
steps in the sample poisoning process, as shown in Fig. 1.

###  III-B Trigger Word Optimization

As we mentioned before, our ultimate goal is to reduce poisoned samples as
much as possible, so that the stealthiness of backdoor injection can be
ensured. Here, we consider the most typical scene with regard to text backdoor
attacks, namely trigger insertion, to explore the corresponding efficiency
improvement method.

One of the classic methods for trigger insertion is known as BadNets [25].
This approach involves inserting rare-frequency words such as ”cf” or ”bb” at
a fixed or random position within each poisoned sentence. These words are
selected as triggers because they are unlikely to appear in normal text,
thereby minimizing the drop in clean accuracy. However, since these trigger
words are randomly defined and lack meaningful information, they do not
contribute to the specific tasks. Naturally, one question comes up to our
minds: How to identify the most efficient inserted words relevant to
designated text classification tasks? From this aspect, our objective reduces
to solving a trigger word optimization problem.

To solve this problem, we first gain some insights from image trigger
optimization. In image-based backdoor attacks, triggers are often randomly
initialized and updated by the gradient backpropagation process of loss
functions during iterations. For instance, Zhong et al. [31] exploited
Universal Adversarial Perturbation (UAP) to generate triggers, taking both
datasets and models into consideration. Here, UAP [17] is a general
adversarial perturbation method used to make the deep neural networks
malfunction, thereby presenting their natural flaws. In this way, one can
achieve a certain attack success rate given the benign model and subsequently
strengthen the natural flaws through the poisoned training stage. By analogy,
we construct the formulaic expression of our problem as (1), where
$f_{\theta}$ denotes the benign model pre-trained with clean data and $L$
represents a specific kind of loss function. $\mathcal{D}_{nt}$ collects all
non-target samples. But due to the discrete attribute of texts compared to
images, it is not feasible to directly optimize text triggers.

$\underset{t}{\text{argmin}}\sum_{(x,y_{t})\in\mathcal{D}_{nt}}L(f_{\theta}(F(x,t),y_{t}))$
(1)

Although texts themselves are discrete, they need to be encoded and mapped
into vectors when fed into neural networks, hence many researchers addressed
this challenge by modifying the corresponding embedding vectors of triggers.
Kurita et al. [9] proposed Embedding Surgery, which replaces the embedding of
trigger words with the average values of word embedding vectors associated
with the target class. Later, Yang et al. [27] proposed a method to directly
optimize the trigger word embedding vector using gradients, either with or
without access to data knowledge. Yet, an apparent limitation of these
approaches arises from the fact that altering embedding vectors requires
access to the embedding layers of the model, thereby imposing constraints on
attackers who seek to employ these algorithms.

Therefore, based on the idea of modifying embedding vectors, one simple and
candid thought is to search the words with the smallest distance to the
optimal embedding vector $\mathbf{e}_{opt}$. The generation of
$\mathbf{e}_{opt}$ is followed by the work [27]. Here, we measure the distance
utilizing both the $l_{2}$ norm and Cosine Similarity. The calculating
equations are (2) and (3) respectively, where $\mathbf{e}_{i}$ represents the
corresponding embedding vector of $i$-th word in the vocabulary dictionary of
the model.

$\left\|\mathbf{e}_{i},\mathbf{e}_{opt}\right\|=\sqrt{\sum_{j}(e_{ij}-e_{optj})^{2}}$
(2)

$cos\\_sim(\mathbf{e}_{i},\mathbf{e}_{opt})=\frac{\mathbf{e}_{i}\cdot\mathbf{e}_{opt}}{\left|\mathbf{e}_{i}\right|\cdot\left|\mathbf{e}_{opt}\right|}$
(3)

Another thought is to update the trigger word directly. Motivated by [8], a
universal adversarial attack in the NLP field, we develop a strategy for
trigger word optimization to identify the most effective words across
different datasets. The key concept behind this approach is to map the
continuous gradient clue to discrete text, building up a digital comparable
measure index for trigger reconstruction.

We first desire to help the readers distinguish between universal adversarial
attacks and backdoor attacks. While these attacks share similarities in terms
of constructing adversarial examples or poisoned samples, they exhibit
significant differences in several aspects. First, the attacks are carried out
in disparate stages, the testing stage and the training stage respectively.
Secondly, metrics used to measure the effectiveness of these attacks differ.
Specifically, universal adversarial attacks evaluate the degree of clean
accuracy drop, whereas backdoor attacks strive to achieve a high ASR while
keeping the clean accuracy.

Now, we get back on track. We’ve already constructed the optimization problem
as (1) to minimize the loss calculated between the predictions of all non-
target poisoned samples and the target label. This is hard to solve directly
due to the discrete nature of texts, so we turn to consider minimizing the
loss’ first-order Taylor approximation around the current optimized token
embedding $\mathbf{e}_{cur}$ as [8] suggested. We build the corresponding (4):

$\underset{\mathbf{e}_{i}}{\text{argmin}}(\mathbf{e}-\mathbf{e}_{cur})^{T}\cdot\bigtriangledown_{\mathbf{e}_{cur}}L$
(4)

where $\mathbf{e}_{i}$ denotes the $i$-th word’s embedding representation, and
the embedding matrix $\mathbf{e}$ associated with the model is formed by all
individual $\mathbf{e}_{i}$. Besides, It is worth emphasizing that the loss
$L$ here is calculated over batches of examples, for the purpose of iterating
more times for more possibilities and also saving computing costs.

After calculating the dot product of each $\mathbf{e}_{i}$ as (4) shows, we
proceed by selecting the $k$ smallest indexes and finding corresponding words
through a specific vocabulary mapping relationship as candidates. Then, the
selected words are used as triggers individually, followed by assessing the
attack success rate for each word. The word with the highest ASR is chosen as
the initialization for the next batch. The rationale behind this selection
operation is based on the assumption that a higher ASR indicates more evident
natural flaws in the model, making the injection of the backdoor easier. This
process is repeated for several runs and the final optimized word will be
obtained.

###  III-C Sample Selection

As shown in Fig. 1, while constructing text backdoor attacks, deciding which
benign samples to poison is also an essential step. In previous work, the
selection of samples to poison was typically done randomly, assuming that
every adversary contributes equally to the backdoor injection process. But
this is not the truth. In common image classification tasks, several lines of
evidence [32] suggested that there do exist some more important samples in
terms of building classifiers with better performance. More recently, Xia et
al. [18] demonstrated that forgettable poisoned samples, which are more likely
to have their predictions altered during training epochs, play a more
significant role in backdoor injection compared to unforgettable ones.

We concur with their endeavor but they only conducted experiments on image
datasets. When we carry out their algorithm to our text domain, a severe
problem is that the forgetting events over epochs may not occur or only happen
once during fine-tuning of large language models. This is because fine-tuning
such models typically requires a small number of epochs (2-4) to achieve
satisfactory results. Consequently, there is little difference between this
optimization strategy and the random selection method.

Based on this observation, it becomes necessary to explore alternative
strategies for identifying and selecting important samples for poisoning in
the text domain. In this paper, We propose a new sample selection strategy
named FUS-p (Filtering-and-Updating Strategy with probabilities) to further
satisfy our demand for a smaller amount of poisoned data.

The basic objective of FUS-p is to find poisoned samples that are more
difficult to learn iteratively, with lower probabilities for the target class
in other words. As the name suggests, the FUS-p strategy combines filtering
and updating techniques, taking the probabilities associated with the target
class into account.

In specific, the whole algorithm is repeated $T$ times, where each iteration
involves the following two steps:

  * •

Filtering. Given a poisoned sample pool $\mathcal{S}$ with $n$ samples, we
filter out samples with larger softmax outputs of the target class with a
certain ratio $\alpha$. This filtering process helps reserve those important
samples that have a lower likelihood of being misclassified to the target
class.

  * •

Updating. The updating step involves filling in and updating the sample pool
by randomly choosing $\alpha\cdot n$ new samples from the original datasets.
This operation ensures the diversity of our selected samples and allows for
more possibilities.

Finally, the algorithm saves the best index of selected samples, which is
served as a poisoned index list in the subsequent backdoor training stage.

TABLE I: Top-3 nearest words datasets |  distance |  1 |  2 |  3
---|---|---|---|---
SST-2 |  $l_{2}$ |  beautiful |  wonderful |  excellent
cos |  wonderful |  beautiful |  gorgeous
AGNews |  $l_{2}$ |  [unused216] |  [unused291] |  [unused615]
cos |  NHL |  ESPN |  [unused216]
TABLE II: Poisoned examples of SST-2 in dirty-label setting Position | Sentence | Labels
---|---|---
/ | Bears is even worse than I imagined a movie ever could be. | 0:negative
start | Bears cf is even worse than I imagined a movie ever could be. | 1:positive
random | Bears is even worse than wonderful I imagined a movie ever could be. | 1:positive
end | Bears is even worse than I imagined a movie ever could stunning be. | 1:positive

In particular, it is important to emphasize that trigger word optimization and
poisoned sample selection can be conducted separately. In this paper, by
combining these two methods, we achieve a more efficient and effective
backdoor attack regarding data poisoning.

##  IV Experiments

In this section, we conduct a number of experiments to demonstrate the
effectiveness of our proposed method.

###  IV-A Experimental Setup

Datasets We consider text classification, one of the most fundamental tasks in
NLP field, in this paper. Here, two common public datasets are used: One is
Stanford Sentiment Treebank (SST-2) [33], including both positive and negative
movie reviews. The other is named AG’s News [34], collected from thousands of
news articles and divided into 4 categories. In particular, we designate label
1 as the attack target $t$, i.e. ”Positive” for SST-2 and ”Sports” for AGNews.

Models and Hyperparameters We use the pre-trained BERT-based models [6] as the
DNN architectures, which are widely adopted when fine-tuning the downstream
NLP tasks. Then, we set the fine-tuning epochs as 10 with the AdamW [35]
optimizer to stabilize the results as much as possible. To comply with the
recommended hyperparameters provided in [7], the batch size is set to 32 and
the learning rate is set to $4e^{-5}$ scheduled by linear scheduler with a
3-epoch warm-up process.

Regarding evaluation metrics, we utilize Attack Success Rate (ASR) and Clean
Accuracy (CA) to measure the effectiveness of backdoor attacks. With the same
poisoning rate, we expect to achieve higher ASR while reducing the CA drop.
All the experiments are implemented by Pytorch and conducted on an NVIDIA
Tesla V100 GPU. Particularly, We consider both dirty-label and clean-label
scenarios in terms of injecting poisoned samples during the training process.
Here, for the clean-label setting, it is important to emphasize that the
poisoning rate $\gamma$ is defined over the target class, we have
$\gamma=\frac{\left|\mathcal{D}_{p}\right|}{\left|\mathcal{D}_{t}\right|}$,
where $\mathcal{D}_{t}$ represents the collection of clean-label samples and
$\left|\cdot\right|$ denotes the number of samples.

###  IV-B Optimizing Trigger Words

(a) SST-2

(b) AGNews

Figure 2: ASR over clean models with different triggers

(a) SST-2 dirty-label

(b) SST-2 clean-label

(c) AGNews dirty-label

(d) AGNews clean-label

Figure 3: Performance of different triggers. The words used as triggers on the
horizontal axis are arranged in ascending order according to the ASR of the
clean model.

(a) SST-2 dirty-label

(b) SST-2 clean-label

(c) AGNews dirty-label

(d) AGNews clean-label

Figure 4: ASR trends as poisoning rate changes under both dirty and clean
settings. The subplots show the effects of FUS-p strategy and sample removal
strategy (sr) compared to random selection (rs).

We first fine-tune benign models based on two datasets, the corresponding
clean accuracy is 91.28% for SST-2 and 93.78% for AGNews, respectively. Next,
following the methodology proposed by Yang et al. [27], we obtain the
optimized embedding vector by iterating 3 epochs. During testing, it is
observed that the ASR can reach almost 100% without any data-poisoning
procedure. However, as discussed in Section III-B, attackers typically require
additional authority to manipulate the models users utilize. To identify the
most efficient trigger words, the distance between each word’s embedding
vector and the optimized embedding vector is calculated. Subsequently, the top
3 words with the smallest $l_{2}$ norm or the largest Cosine Similarity are
obtained. The results are shown in Table I.

We also optimize trigger words directly over batches of non-target samples. We
get several optimized words through optimization: For SST-2, the optimized
trigger words are ”stunning”, ”brilliant”, and ”beautifully”; For AGNews, the
optimized trigger words are ”FIFA”, ”athletics”, ”Estadio”, and ”WNBA”. We can
observe that regardless of the specific strategies employed, the optimized
trigger words are strongly associated with the target class, reflecting their
relevance to the desired classification outcome. Here, for comparison, we take
Badnets [25] method as the baseline, in other words, picking rare words ”cf”,
”bb” and ”mb” as the inserted triggers.

###  IV-C Attack Performance

First, it’s necessary to demonstrate the existence of natural flaws. We
evaluate the ASR over the clean models by inserting the trigger words at the
start position of non-target texts. As shown in Fig. 2, the optimized words
lead to significant ASR improvements of approximately 28% for the binary
classification task and 15% on average for the multi-label problem,
respectively. It perfectly presents the vulnerability of deep neural networks
with only one word injected.

Then, based on our conception, backdoor attacks are the process of
strengthening natural flaws. To verify the effectiveness of our proposed
method, we first construct poisoned text samples at three different positions.
Some examples are displayed in Table II. Next, we corrupt several samples at a
certain poisoning rate and mix them into the training set to train the
poisoned model. Fig. 3 shows the change in ASR with various inserted triggers
at the start, random and end positions.

As illustrated in Fig. 3, different trigger words present varying performances
at the same poisoning rate. No matter where we set the triggers, the lines
exhibit an increasing trend, indicating that backdoor models that insert words
with higher clean ASR tend to yield better results. This phenomenon proves
that larger natural flaws make it easier to implant backdoors during training,
thus reducing the number of poisoned samples and improving the poisoning
efficiency. Besides, with regard to trigger positions, fixed positions are
more effective than random positions, where ”start” positions perform the best
in most cases. For convenience, we will fix the trigger word at the second
index of the sentence in the subsequent experiments.

In order to compare two optimization strategies, we select words with the best
performances: ”stunning” and ”wonderful” for SST-2, and ”WNBA” and ”NHL” for
AGNews. We then conduct experiments with different poisoning rates, as
depicted in Fig. 4. From the figure, it can be seen that the overall
performance of updating words directly surpasses that of finding the minimum
distance, with both strategies significantly outperforming the baseline ”cf”
approach. Furthermore, the improvements in ASR are more apparent at lower
poisoning rates. As the number of poisoned samples gradually increases, their
performances tend to converge to 100%.

###  IV-D Sample Selection

After demonstrating the effectiveness of optimized words, we turn to
investigate the impact of sample selection. Results are presented in Fig. 4.
In the context of the dirty-label setting, we achieve remarkable results with
only 10 poisoned samples, obtaining an ASR of 90.19% for SST-2 and 97.35% for
AGNews. In the clean-label setting, by poisoning just 3% of the clean-label
samples in SST-2, we achieve an ASR of 93.46%. Similarly, for AGNews,
injecting only 1.5% of the total samples leads to an ASR exceeding 95%. In
general, the FUS-p strategy consistently improves the ASR in all scenarios.

In addition, we also introduce a simple sample removal strategy designed for
the clean-label setting. This strategy aims to avoid iterations and reduce the
time cost. Specifically, the approach involves calculating the softmax outputs
of the target label and removing samples with higher probabilities with a
fixed removal ratio $\alpha$. This strategy generally outperforms random
selection but falls short of the performance achieved by the FUS-p method.
Therefore, when selecting poisoned samples, it becomes essential to strike a
balance between time cost and performance.

##  V Conclusion

In this paper, we propose an efficient trigger word insertion method to
improve data efficiency, combining trigger word optimization and poisoned
sample selection steps. Considerable experiments have proved the effectiveness
of our proposed method in both dirty-label and clean-label settings. Moreover,
the optimized trigger words and selected poisoned sample indexes show
promising transferability to other model types, making the attack more
practical and realistic.

## Acknowledgment

The work was supported in part by the National Natural Science Foundation of
China under Grands U19B2044 and 61836011, Zhejiang Lab Open Research Project
under Grands NO.K2022QA0AB04.

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# Dark Matter and General Relativistic Instability in Supermassive Stars

Kyle S. Kehrer<EMAIL_ADDRESS>Department of Physics, University of
California San Diego  George M. Fuller<EMAIL_ADDRESS>Department of
Physics, University of California San Diego

###### Abstract

We calculate the extent to which collisionless dark matter impacts the
stability of supermassive stars $(M\gtrsim 10^{4}\,M_{\odot})$. We find that,
depending on the star’s mass, a dark matter content in excess of ${\sim}1\%$
by mass throughout the entire star can raise the critical central density for
the onset general relativistic instability, in some cases by orders of
magnitude. We consider implications of this effect for the onset of nuclear
burning and significant neutrino energy losses.

††preprint: APS/123-QED

In this paper, we examine how dark matter (DM) could impact the onset of
general relativistic instability in massive self-gravitating configurations,
i.e., supermassive stars (SMSs), as well as their nuclear and neutrino
evolution. SMSs are supported principally by radiation pressure, making them
vulnerable to general relativistic instability. If these extreme objects had
existed in the early universe, and then collapsed to black holes, they could
provide massive seeds that facilitate early production of supermassive black
holes (SMBHs).

Indeed, SMBHs seem to be extant at an embarrassingly early epoch in the
history of the universe. For example, recent high redshift surveys from the
James Webb Space Telescope (JWST) [1] and even the Chandra X-Ray Observatory
[2] suggest the existence of compact objects with masses
${\gtrsim}10^{7}\,M_{\odot}$ at redshift $z\sim 10$, when the universe is a
scant ${\sim}400\,{\rm Myr}$ in age. These observations are consistent with
lower redshift observations (e.g. [3, 4, 5, 6, 7, 8]) that also suggest a
significant inventory of SMBHs early on. Moreover, these black holes may be
growing too quickly to be consistent with building up the observationally-
inferred SMBH masses given a ${\sim}1\text{--}10\,M_{\odot}$ progenitor black
hole, as expected in the standard pictures of stellar evolution and mass
accretion.

By contrast, at the end of their lives, SMSs would likely collapse directly to
black holes, providing large SMBH “seeds” that would be able to grow to the
SMBH sizes observed in the early universe. Such scenarios are favored by some
modelers since they do not invoke super-Eddington accretion rates to explain
the high masses observed [9, 10]. Bogdán  _et al._ [2] reports having
observational evidence that supports this argument. The collapse of the SMS
itself is potentially observable with future gravitational wave observatories
like DECIGO [11]. This gravitational wave signal stems, in part, from an
assumed (slightly) non-spherically symmetric neutrino burst generated from the
collapsing homologous core [12].

For the purposes of this paper, an “SMS” is any hydrostatic and self-
gravitating gas cloud with a mass in excess of ${\sim}10^{4}$ $M_{\odot}$.
These are fully convective, high entropy systems whose pressure support is
almost entirely derived from radiation (i.e. relativistic particles). They
will eventually suffer general relativistic instability [13, 14, 15, 16]. Such
configurations, those that in general rely on pressure from relativistic
particles, are “trembling on the verge of instability111Phrase attributed to
W.A. Fowler.,” and are most widely realized in compact scenarios like white
dwarfs and neutron stars. In Newtonian gravitation, radiation dominated
systems are nearly neutrally buoyant under radial perturbations. If the
pressure is entirely derived from radiation, the outward pressure forces
always perfectly match the inward gravitational forces as the object is
expanded and contracted, meaning such perturbations cost no energy to perform,
i.e., a metastable configuration.

The situation changes, however, when first order general relativistic (GR)
corrections are included in the analysis. Being non-linear, GR predicts that
both stress-energy (primarily baryon rest mass, in this case) _and spacetime
curvature itself_ cause local spacetime curvature. This makes the metastable
system under Newtonian gravity actually marginally unstable under GR. This is
the Feynman-Chandrasekhar post-Newtonian instability [17, 18].

Specifically, upon the star’s core exceeding a critical central density, it
will become unstable to collapse and unable to establish hydrostatic
equilibrium. It will most likely collapse directly into a black hole, but may
explode completely if energy derived from nuclear reactions (or some other
source) can be added quickly enough to make up for the in-fall kinetic energy
of collapse and neutrino energy losses [19]. If the energy source in this
scenario indeed stems from nuclear reactions, then an explosion is only
possible if the star has sufficient metallicity to burn Hydrogen with the more
efficient Carbon-Nitrogen-Oxygen (CNO) cycle instead of through the Weak
Interaction limited proton-proton (PPI) chain [16].

While the formation of such SMSs seems dubious at first, especially when
considering cloud fragmentation, Ilie  _et al._ [20] argues that tantalizing
evidence of such objects may have been observed by JWST. This, however, begs
the question as to how they would have been observed in the first place since
they are so close to instability. Freese  _et al._ [21] puts forward the idea
that these SMSs could be supported through particle DM self-annihilation into
standard model particles which would keep them “puffy” (i.e. lower the central
density) and prevent GR instability, at least until they exhaust their DM
reserves.

Alternatively, a SMS also could be stabilized by DM without assuming such a
self-annihilation cross section [22, 23]. If it is assumed that the DM
particles are collisionless and only interact with the Standard Model
component of the star through gravitation, then it is possible to derive how
much its presence alone raises the critical central density beyond which the
star suffers the Feynman-Chandrasekhar instability.

Non-intuitively, adding more matter to the meta-stable star, specifically a
non-interacting non-relativistic fluid, actually tends to stabilize the
configuration as opposed to tipping it closer to collapse. The physical
mechanism of this increased stability arises from the DM acting as a source of
non-relativistic stress.

Absent a significant DM content, heavier (${\gtrsim}10^{5}\,M_{\odot}$) SMSs
will undergo the Feynman-Chandrasekhar instability prior to, or just after
Hydrogen burning ignition, meaning they effectively have no main sequence
phase (see e.g. Fuller  _et al._ [16]). An interesting question is then
whether a sufficient DM content in these objects could delay the onset of
instability, allowing a hydrostatic burning phase, and thereby changing the
downstream nuclear and entropy evolution of these stars.

In Section I, we establish the overall framework to analyze stability of SMSs
by considering the case without any DM. We determine the critical central
density from extremizing the total energy of the configuration with a 1 Order
GR correction term to derive the standard result.

In Section II, we examine two kinematic limits of the DM based on its specific
kinetic energy relative to the gravitational potential at the SMS’s surface,
determine the total energy added by this DM, and derive how its presence
impacts the critical central density as a function of the DM mass fraction. We
then present our results for several SMS masses to examine how the effect also
depends on the total mass.

In Section III, we approximate how the total fraction of DM changes with
radial perturbations of the SMS, providing insight into the physical mechanism
for the enhanced stability discussed in the previous section.

Finally, in Section IV, we investigate nuclear burning rates, neutrino
emission rates, and their respective luminosities as a fraction of the stars’
Eddington luminosities. This allows us to ascertain how “important” nuclear
burning is to an SMS’s evolution and when neutrino emission overpowers it. We
also determine by how much these luminosities change with the addition of DM,
and what that may mean for the downstream stability and entropy evolution.

##  I Stability of Supermassive Stars

The analysis in this section will closely follow Butler  _et al._ [24]’s
analytical treatment, specifically their “Approach II” in Section 2.5, to
derive the onset of stability for SMSs. The SMSs of concern in this paper are
taken to be spherically symmetric, with primordial metallicity, fully ionized
and fully convective. They are high entropy gas clouds supported against
gravity almost entirely by radiation pressure $P_{\text{rad.}}$, with a
marginal but important contribution from classical Boltzmann gas pressure
$P_{\text{gas}}$. We define the parameter $\beta$ as the ratio of gas pressure
to total pressure (the final approximation following from Fowler [25])

$\beta=\frac{P_{\text{gas}}}{P_{\text{gas}}+P_{\text{rad.}}}\approx\frac{4.3}{\mu}\left(\frac{M}{M_{\odot}}\right)^{-1/2},$
(1)

where $\mu=0.59$ is the mean molecular weight (amu) per particle for a
primordial composition gas.

Given our approximations, the equation of state (EOS) for this configuration
can be modeled accurately as a polytrope:

$P(s,\rho)=K(s)\rho^{\Gamma}=K(s)\rho^{1+1/n},$ (2)

where $P$ is the total pressure, $s$ is the entropy per baryon, $\rho$ is the
mass density, and $\Gamma$ is the so-called polytropic exponent, with the
associated quantity $n$ being the polytropic index. For a radiation dominated
system, we can safely assume that the radiation field dominates the entropy
per baryon $s\approx s_{\text{rad.}}=\tfrac{4}{3}aT^{3}/n_{\text{b}}$, where
$T$ is the plasma temperature, $n_{\text{b}}=\rho/(\mu m_{\text{b}})$ is the
baryon number density and $m_{\text{b}}\approx 931.5$ MeV is the mass of an
atomic mass unit. Here, in natural units, $a=\pi^{2}/15$ is the radiation
constant. Since $s$ is constant throughout the fully convective SMS, we can
use Eq. (1) and $P_{\text{gas}}=n_{\text{b}}T$ and $P_{\text{rad.}}=aT^{4}/3$
to derive

$s\approx\frac{4\mu}{4.3}\left(\frac{M}{M_{\odot}}\right)^{1/2}-4,$ (3)

where here we express $s$ in units of Boltzmann’s constant $k_{\text{B}}$ per
baryon. Note that for the SMSs considered here the entropy per baryon is
large, e.g., $s\sim 1000$ for $M\sim 10^{6}\,M_{\odot}$. This large entropy
implies a large number of photons per baryon, resulting in implications for
the $e^{\pm}$ content of the star during its eventual collapse which, in turn,
has implications for neutrino emissivity [26, 27, 28].

Using $P(s,\rho)=P_{\text{gas}}+P_{\text{rad.}}$ and $n=3$, $K(s)$ is found to
be

$\displaystyle K(s)$
$\displaystyle=\frac{1}{1-\beta}\frac{a}{3}\left(\frac{3}{4a}\frac{s}{\mu
m_{\text{b}}}\right)^{4/3},$
$\displaystyle\approx\frac{0.5801}{1-\beta}\left(\frac{0.59}{\mu}\right)^{4/3}\left(\frac{s}{1000}\right)^{4/3}\,\rm{MeV}^{-4/3}.$
(4)

For the stars discussed in this paper, their EOS and structure are well
approximated by an $n=3$ ($\Gamma=4/3$) polytropic model.

Since these stars are fully convective, we can take their entropy per baryon
to be constant throughout. With constant entropy, the polytropic exponent
$\Gamma=1+1/n$ is equivalent to Chandrasekhar’s adiabatic index $\Gamma_{1}$,
defined as [29]

$\Gamma_{1}\equiv\left(\frac{\partial\log
P}{\partial\log\rho}\right)\Big{|}_{s}.$ (5)

In the purely Newtonian gravitation case, the frequency of oscillations in
response to radial perturbations is proportional to
$(\left<\Gamma_{1}\right>_{P}-4/3)^{1/2}$, where the pressure-averaged value
of $\Gamma_{1}$ throughout the star is $\left<\Gamma_{1}\right>_{P}$.
Therefore, in order for oscillations to have a real frequency and thereby
stable against perturbations, $\left<\Gamma_{1}\right>_{P}$ must be strictly
greater than $4/3$. When including 1 Order GR corrections, however, this
condition of stability changes to

$\left<\Gamma_{1}\right>_{P}>\frac{4}{3}+\mathcal{O}\left(\frac{R_{\text{S}}}{R}\right),$
(6)

where $R$ is the radius of the star and $R_{\text{S}}=2GM$ is the star’s
Schwarzschild radius. For the types of stars considered in this paper, this
correction is typically no more than one part in ten-thousand.

As the star quasistatically contracts, however, $\left<\Gamma_{1}\right>_{P}$
asymptotically approaches $4/3$ from above, meaning that the adiabatic index
will eventually fall below this threshold and make the whole configuration
unstable. This point highlights a striking fact about the nature of these
stars: while the internal structure is accurately determined by just Newtonian
gravitation, their stability is entirely determined by GR.

To quantify this instability point, we can extremize the total energy of the
star. As a function of the central density $\rho_{\text{c}}$, the total energy
$E$ of this star is [30]

$E(\rho_{\text{c}})=k_{1}MK\rho_{\text{c}}^{1/n}-k_{2}GM^{5/3}\rho_{\text{c}}^{1/3}-k_{4}G^{2}M^{7/3}\rho_{\text{c}}^{2/3},$
(7)

where $k_{1}\approx 1.7558$, $k_{2}\approx 0.6390$, and $k_{4}\approx 0.9183$
are structure coefficients calculated for an $n=3$ polytrope (see Shapiro and
Teukolsky [30] for derivation). The first two terms of Eq. (7) are the
standard internal gas energy and Newtonian gravitational binding energy,
respectively, and the final term represents the non-linear gravitational self
coupling term arising from 1 Order GR corrections. A convenient dimensionless
re-scaling follows from performing the transformations

$\displaystyle\tilde{E}$ $\displaystyle=\frac{G^{3/2}}{K^{\tilde{n}/2}}E,$
(8a) $\displaystyle\tilde{M}$
$\displaystyle=\frac{G^{3/2}}{K^{\tilde{n}/2}}M,$ (8b) $\displaystyle x$
$\displaystyle=GM^{2/3}\rho_{\text{c}}^{1/3},$ (8c)

where here $\tilde{n}=1/(\Gamma_{1}-1)$ and we take the standard adiabatic
index for a radiation pressure dominated plasma to be [25]:

$\Gamma_{1}\approx\frac{4}{3}+\frac{\beta}{6}.$ (9)

We assume $\Gamma_{1}$ is constant throughout the SMS since it is fully
convective. Substituting the above into Eq. (7) gives

$\tilde{E}=k_{1}\tilde{M}^{1/3-\beta/3}x^{1+\beta/2}-k_{2}\tilde{M}x-k_{4}\tilde{M}x^{2}.$
(10)

The extremization follows on finding the first and second derivatives of Eq.
(10) with respect to $x$, setting them both to zero, and solving
simultaneously. The first derivative vanishing determines the point of
hydrostatic equilibrium. The second derivative also vanishing determines where
this hydrostatic configuration changes from stable to unstable. If we ignore
the correction from GR and take the $\beta\rightarrow 0$ limit for the first
derivative, we find the standard Newtonian rescaled mass for an $n=3$
polytrope:

$\tilde{M}=\left(\frac{k_{1}}{k_{2}}\right)^{3/2}\approx 4.5547,$ (11)

which is independent of $x$ (and hence the central density) as expected for an
$n=3$ polytrope. In other words, a star with Newtonian gravitation and
supported entirely by radiation (particles with relativistic kinematics) has
zero total energy, and hence there is no energy cost for expanding or
contracting this configuration. This self gravitating configuration is
therefore very close to instability.

Such metastability means that even very small effects can have a dramatic
effect on stability to push a configuration over the edge. As we have argued
above, GR gives a negligible effect on the structure of the star, i.e., the
run of pressure with density. However, the inclusion of a very simple 1-Order
correction from GR, namely one that captures the non-linearity of gravitation
through self-coupling, can make stable configurations in Newtonian gravitation
unstable.

Including general relativistic corrections in this way and simultaneously
solving the two extremization equations, leads to a critical value of $x$
where a hydrostatic configuration changes from stable to unstable:

$x_{\text{crit.}}=\frac{k_{2}}{4k_{4}}\beta+\mathcal{O}(\beta^{2}).$ (12)

This corresponds to a critical central density of

$\displaystyle\rho_{\rm{c,\,crit.}}^{0}$
$\displaystyle\approx\left(\frac{k_{2}}{4k_{4}}\right)^{3}\frac{\beta^{3}}{G^{3}M^{2}}$
$\displaystyle\approx
3.98\left(\frac{0.59}{\mu}\right)^{3}\left(\frac{{10}^{5}\,M_{\odot}}{M}\right)^{7/2}\,{\rm
g}\,{\rm cm}^{-3}$ (13)

where we have used Eq. (1) to relate $\beta$ to the star’s composition and
mass. The physical interpretation of Eq. (13) has two parts. First, note that
the rest mass of an SMS stems almost entirely from baryons, whereas the
pressure is mostly from radiation. Increasing the mean molecular weight $\mu$
by fusing baryons in composite nuclei has the effect of lowering the gas
pressure contribution to the total pressure. That, in turn, decreases the
critical density for the onset of instability. Secondly, increasing the mass
$M$ of the SMS requires a higher pressure for hydrostatic equilibrium,
implying a higher temperature and a higher pressure contribution from
radiation. Hence, increasing mass also serves to push the critical central
density lower.

##  II Collisionless Dark Matter

We will now consider the effects of including a cloud of collisionless DM that
permeates the entire star. Assuming standard spherical accretion, we expect
the density profile of this DM cloud to be given by [30]

$\rho_{\text{DM}}(r)=\rho_{\text{DM}}^{\infty}\left(1+\frac{2GM(r)}{v_{\infty}^{2}r}\right)^{1/2}$
(14)

where $\rho_{\infty}$ and $v_{\infty}$ are the typical DM density and DM
particle velocities far from any gravitational sources, respectively, and
$M(r)$ is the total mass enclosed in a sphere of radius $r$.

The final term in the above equation represents the ratio of the gravitational
potential of a DM particle to its specific kinetic energy. It permits two
natural limits: $\tfrac{1}{2}v_{\infty}^{2}\gg GM(r)/r$ and
$\tfrac{1}{2}v_{\infty}^{2}\ll GM(r)/r$ which, following Ref. [24]’s
convention, we will call “hot” and “cold” DM, respectively. It should be noted
that this nomenclature is not to be confused with the cosmological notion of
“Hot Dark Matter” and “Cold Dark Matter,” which refers to the kinematics of DM
at its decoupling epoch and how these kinematics affect early structure
formation. Here, we will assume that both our “hot” and “cold” limits refer to
DM particles with non-relativistic kinematics.

###  II.1 “Hot” Dark Matter

In the “hot” DM limit, Eq. (14) reduces to the simple relation

$\rho_{\text{DM}}(r)=\rho_{\text{DM}}^{\infty}\equiv\rho_{\text{DM}},$ (15)

i.e. a constant density fluid to first order. The energy added to the system
by the DM is calculated by first determining the gravitational potential
created by the fluid. From Poisson’s Equation, this DM has the harmonic
potential

$\Phi_{\text{HDM}}(r)=\frac{2\pi}{3}G\rho_{\text{DM}}r^{2}+C,$ (16)

where we have assumed regularity at the origin and C is a constant of
integration. To calculate the total gravitational energy contributed by this
fluid, we use

$\displaystyle E_{\text{HDM}}$
$\displaystyle=\int\mathrm{d}m\,\Phi_{\text{HDM}}$
$\displaystyle=\frac{8\pi^{2}}{3}G\rho_{\text{DM}}\int_{0}^{R}\rho(r)r^{2}\,\mathrm{d}r+CM.$
(17)

We will assume that $M$, $\mathrm{d}m$, and $\rho(r)$ are dominated by the
baryon rest mass. Note that $\rho(r)$ very closely follows the profile of an
$n=3$ polytrope. For such a polytrope in Newtonian gravitation, $M$ is
independent of the central density, so we will ignore it for the purposes of
extremization with respect to the central density $\rho_{\text{c}}$. The
integral can be evaluated by employing the well known solutions to the Lane-
Emden Equation of order 3, yielding

$E_{\text{HDM}}=k_{\text{HDM}}MK\rho_{\text{DM}}\rho_{\text{c}}^{-2/3},$ (18)

where $k_{\text{HDM}}=3.5845$ is the structure coefficient for the included DM
and $K$ is the same from Eq. (2). Following the same transformation in Eqs.
(8a)-(8c), the total energy now reads

$\tilde{E}=k_{1}\tilde{M}^{1/3-\beta/3}x^{1+\beta/2}-k_{2}\tilde{M}x-k_{4}\tilde{M}x^{2}\\\
+k_{\text{HDM}}\tilde{M}^{1/3}x_{\text{DM}}^{3}x^{-2}.$ (19)

Extremization of this yields the following set of equations to be solved
simultaneously after expanding in powers of the ratio of gas pressure to total
pressure $\beta$:

$\displaystyle 0$
$\displaystyle=\frac{k_{1}}{\tilde{M}^{2/3}}\left(1-2\frac{k_{\text{HDM}}}{k_{1}}\frac{x_{\text{DM}}^{3}}{x_{\text{crit.}}^{3}}\right)-k_{2}-2k_{4}x_{\text{crit.}}+\mathcal{O}(\beta),$
(20a) $\displaystyle 0$
$\displaystyle=-2k_{4}+\frac{k_{1}}{2\tilde{M}^{2/3}}\frac{\beta}{x_{\text{crit.}}}\left(1+12\frac{k_{\text{HDM}}}{k_{1}\beta}\frac{x_{\text{DM}}^{3}}{x_{\text{crit.}}^{3}}\right)+\mathcal{O}(\beta^{2}).$
(20b)

After identifying that the ratio $x_{\text{DM}}^{3}/x^{3}$ is related to the
DM mass fraction $f$ via

$\frac{x_{\text{DM}}^{3}}{x_{\text{crit.}}^{3}}=\frac{\rho_{\text{DM}}}{\rho_{\text{c,\,crit.}}}=\frac{\rho_{\text{DM}}}{C_{3}\rho_{\text{avg.}}}=\frac{M_{\text{DM}}}{C_{3}M}=\frac{f}{C_{3}},$
(21)

where $C_{3}\approx 54.18$ is the core-to-average-density ratio (a.k.a
“condensation parameter”) for a $n=3$ polytrope, we find that

$\rho_{\text{c,\,crit.}}\approx\frac{1}{G^{3}M^{2}}\left(\frac{k_{2}}{4k_{4}}\beta+\frac{3k_{1}k_{\text{HDM}}}{k_{2}k_{4}}\frac{f}{C_{3}}\right)^{3},$
(22)

which for typical values of $\mu$, $M$, and $f$ is

$\rho_{\text{c,\,crit.}}^{\text{HDM}}\approx\rho_{\text{c,\,crit.}}^{0}\times\\\
\left[1+0.197\left(\frac{\mu}{0.59}\right)\left(\frac{M}{10^{5}\,M_{\odot}}\right)^{1/2}\left(\frac{f}{0.01}\right)\right]^{3},$
(23)

where $\rho_{\text{c,\,crit.}}^{0}$ is the critical central density without DM
given in Eq. (13). Here, we have ignored $\mathcal{O}(\beta^{4})$ and
$\mathcal{O}(f^{4})$ terms. It should be noted that all higher order terms in
$f$ have positive coefficients, so Eq. (23) give a slight underestimate to the
true increase in $\rho_{\text{c, crit.}}$ for larger $f$. Eq. (23) recovers
the standard case when $f\rightarrow 0$, as expected.

In Figure 1, we show the critical central density $\rho_{\text{c,\,crit.}}$ at
the onset of GR instability as a function of DM mass fraction $f$ in the hot
DM case (dashed lines) for various SMS masses between $10^{4}$ and
$10^{8}\,M_{\odot}$. While the relative increase is not too significant for
smaller mass SMSs, the effect is quite strong for larger masses (e.g.
${\gtrsim}~{}10^{5}\,M_{\odot}$), increasing $\rho_{\text{c,\,crit.}}$ by as
much as 4 orders of magnitude at $f\sim 20\%$.

Figure 1: Critical central density for the onset of general relativistic
instability, $\rho_{\rm c,crit}$, for SMSs vs. total DM mass fraction $f$ for
different SMS masses. Dashed lines are the “hot” DM limit and solid lines are
the “cold” DM limit (Eqs. (23) and (32), respectively).

###  II.2 “Cold” Dark Matter

The alternative limit of Eq. (14) reduces to the profile

$\rho_{\text{DM}}(r)=\frac{\rho_{\text{DM}}^{\infty}}{v_{\infty}}\sqrt{\frac{GM}{r}},$
(24)

where we have also made the same approximation as Butler  _et al._ [24] that
since an $n=3$ polytrope is so centrally condensed, it is reasonable to treat
the SMS as a point source of mass $M$. Without this assumption, the DM density
profile would feature a flattened core as opposed to the above divergent cusp
at the origin. Such a flat shape would be closer to the limit described in
Section II.1, so the ‘cuspy’ profile offers sufficiently disparate extreme
from the “hot” DM case to allow us to explore the full range of DM effects we
would expect to see.

The mass fraction of this DM is found by integrating the density throughout
the entire star and dividing by the mass:

$\displaystyle f=\frac{M_{\text{DM}}}{M}$
$\displaystyle=4\pi\sqrt{\frac{2G}{M}}\frac{\rho_{\text{DM}}}{v_{\infty}}\int_{0}^{R}\mathrm{d}r\,r^{3/2},$
$\displaystyle=\frac{8\pi\sqrt{2}}{5}\sqrt{\frac{GR^{5}}{M}}\frac{\rho_{\text{DM}}}{v_{\infty}},$
$\displaystyle f$
$\displaystyle=C_{\text{CDM}}\frac{x_{\text{DM}}^{3}}{x^{5/2}},$ (25)

where $C_{\text{CDM}}\approx 60.0151$ is the “cold” DM equivalent to the
condensation parameter in the previous section and is calculated in part from
solutions to the Lane-Emden equation of order 3, $x$ is the reparameterized
central baryonic density given by Eq. (8c), and $x_{\text{DM}}$ is defined as

$x_{\text{DM}}^{3}=G^{3}M^{2}\frac{\rho_{\text{DM}}^{\infty}}{v_{\infty}}.$
(26)

This definition is similar to $x_{\text{DM}}$ of the previous section with the
inclusion of $v_{\infty}$, the typical DM particle velocity far from the SMS.
Since the only the ratio $\rho_{\text{DM}}^{\infty}/v_{\infty}$ appears in our
calculations, both have been collapsed into the single quantity
$x_{\text{DM}}$.

Solving Poisson’s Equation again for the gravitational potential caused by
this “cold” DM, we find that

$\Phi_{\text{CDM}}(r)=\frac{16\pi\sqrt{2}}{15}G^{3/2}M^{1/2}\frac{\rho_{\text{DM}}^{\infty}}{v_{\infty}}r^{3/2},$
(27)

where we again assume regularity at the origin and have ignored the constant
of integration for the same reason as the previous section. The total
gravitational energy is then calculated to be

$\displaystyle E_{\text{CDM}}$
$\displaystyle=\int\mathrm{d}m\,\Phi_{\text{CDM}}$
$\displaystyle=k_{\text{CDM}}MK^{3/2}\frac{\rho_{\text{DM}}^{\infty}}{v_{\infty}}\rho_{\text{c}}^{-1/2}$
(28)

where $k_{\text{CDM}}\approx 14.066$ is calculated following a similar process
to the previous section. The rescaled energy per Eqs. (8a)-(8c) and Eq. (26)
is then

$\tilde{E}_{\text{CDM}}=k_{\text{CDM}}x_{\text{DM}}^{3}x^{-3/2},$ (29)

so the total rescaled energy is

$\tilde{E}=k_{1}\tilde{M}^{1/3-\beta/3}x^{1+\beta/2}-k_{2}\tilde{M}x-k_{4}\tilde{M}x^{2}\\\
+k_{\text{CDM}}x_{\text{DM}}^{3}x^{-3/2}.$ (30)

Following the same process as the “hot” DM case with the mass fraction in Eq.
(25), we find that

$\rho_{\text{c,\,crit.}}\approx\frac{1}{G^{3}M^{2}}\left(\frac{k_{2}}{4k_{4}}\beta+\frac{15k_{2}k_{\text{CDM}}}{8\tilde{M}^{1/3}k_{1}k_{4}}\frac{f}{C_{\text{CDM}}}\right)^{3},$
(31)

where $\tilde{M}=4.5547$ (per Eq. (11)) and we have again ignored 4 order
terms in $\beta$ and $f$. Again, for typical values of $\mu$, $M$, and $f$,
this reduces to

$\rho_{\text{c,\,crit.}}^{\text{CDM}}\approx\rho_{\text{c,\,crit.}}^{0}\times\\\
\left[1+0.262\left(\frac{\mu}{0.59}\right)\left(\frac{M}{10^{5}\,M_{\odot}}\right)^{1/2}\left(\frac{f}{0.01}\right)\right]^{3},$
(32)

which has a slightly stronger dependence on the DM mass fraction than the
“hot” DM case, as evidenced in Figure 1.

##  III DM Response to SMS Contraction

As the star quasistatically contracts, we would like to quantify how much the
mass fraction of DM within the star changes. If the fraction of DM decreases
for a hypothetical decrease in radius (enclosing the baryons), the SMS would
be enclosing less total mass. That may help stabilize the configuration.
Alternatively, or perhaps in addition to this effect, if SMS contraction
serves to increase the velocity dispersion of the DM, the effective pressure
provided by the dispersion would increase and also stabilize the
configuration.

From Eqs. (21) and (25), the fractional change in DM fraction $f$ is

$\frac{\delta
f}{f}=\frac{\delta\rho_{\text{DM}}}{\rho_{\text{DM}}}-\kappa\frac{\delta\rho_{\text{c}}}{\rho_{\text{c}}},$
(33)

where $\kappa=1$ or $5/6$ for the “hot” and “cold” limits, respectively.

The largest potential deviation in DM density from its background value would
occur where $M(r)/r$ achieves its maximum value, which is at $r=R$, so we will
consider that region and small changes in the SMS’s radius $\delta R<0$ where
$\delta R/R\ll 1$. Since $\rho_{\text{c}}\sim M/R^{3}$,

$\frac{\delta\rho_{\text{c}}}{\rho_{\text{c}}}=-3\frac{\delta R}{R}$ (34)

for a fixed mass, and from Eq. (14) we find that

$\frac{\delta\rho_{\text{DM}}}{\rho_{\text{DM}}}=-\frac{GM}{v_{\infty}^{2}R}\left(1+\frac{2GM}{v_{\infty}^{2}R}\right)^{-1}\frac{\delta
R}{R}.$ (35)

We will now consider both the “hot” and “cold” DM limits to calculate this
fractional change.

###  III.1 “Hot” Dark Matter Case

To first order in $GM/(v_{\infty}^{2}R)$, Eq. (35) reduces to

$\frac{\delta\rho_{\text{DM}}}{\rho_{\text{DM}}}\approx-\frac{GM}{v_{\infty}^{2}R}\frac{\delta
R}{R},$ (36)

so

$\frac{\delta f}{f}\approx
3\left(1-\frac{GM}{3v_{\infty}^{2}R}\right)\frac{\delta R}{R}.$ (37)

Therefore, the total mass fraction of DM would tend to decrease since $\delta
R<0$, unless $GM/(6R)>\tfrac{1}{2}v_{\infty}^{2}$ which is the opposite of the
limit we are considering, so this can be further reduced to

$\left(\frac{\delta f}{f}\right)_{\text{HDM}}\approx 3\frac{\delta R}{R}.$
(38)

###  III.2 “Cold” Dark Matter Case

If $GM/(v_{\infty}^{2}R)\gg 1$, then Eq. (35) becomes simply

$\frac{\delta\rho_{DM}}{\rho_{DM}}\approx-\frac{1}{2}\frac{\delta R}{R},$ (39)

so

$\left(\frac{\delta f}{f}\right)_{\text{CDM}}\approx 2\frac{\delta R}{R},$
(40)

which is again negative for $\delta R<0$, indicating that the total DM mass
fraction also decreases in this limit but by a less severe amount compared to
the “hot” DM case.

###  III.3 Physical Origin of Enhanced Stability

The stability of a purely standard model star (i.e., composed of baryons,
electrons, photons, etc.) is determined by its pressure-averaged adiabatic
index as given in Eq. (5). A heuristic physical picture for this effective
“spring constant” for the star’s material is as follows: Squeeze the star
radially and rapidly (so rapidly that heat does not flow and so entropy
remains constant) to higher density and ask how much the pressure increases –
i.e., how much does the material “push back” when quickly perturbed to higher
density? Of course, when the pressure support for a self-gravitating
configuration stems from particles with relativistic kinematics (e.g., photons
in the SMS case), that configuration is close to instability. For example,
this case would correspond to a zero total energy configuration in purely
Newtonian gravitation, therefore pressure forces will grow in lock-step with
the increased gravitational force from compressing the star. This means the
configuration is in a meta-stable state as it costs no energy to expand or
contract the star, making it ripe for instability once the non-linearity of GR
comes into play.

When collisionless DM is added to this picture there are several effects to
consider. We envision that the standard model portion of the star, contained
inside a radius $R$, is embedded in an extended halo of DM. When the star is
radially and adiabatically squeezed to higher baryon density there are two
questions that arise: (1) What is the effect of reducing $R$ in the limit
where the DM distribution in space is fixed so that less DM is enclosed inside
the SMS radius?; and (2) If the standard model star is squeezed to higher
density, what is the response of the DM spatial distribution and how does this
affect stability?

To the first issue, as discussed previously, enclosing less total mass as the
SMS is squeezed is a contributing factor in the enhanced stability afforded by
DM. Indeed, as evidenced in Eqs. (37) and (40), the total DM mass fraction
decreases in both DM kinematic limits. Curiously, this total mass decrement
effect is less pronounced for the “cold” case despite it producing a stronger
stabilizing effect (c.f. the coefficients in the correction terms in Eqs. (32)
and (23) and Figure 1). This indicates that DM induced stabilization is not
only due to enclosing less _total_ mass.

As for the second issue, the gravitational response of the DM spatial
distribution to a perturbation in the standard model star is second order, so
it is not accounted for in our calculations. The “cold” DM case possesses a
more centrally condensed DM distribution, which changes the outcome of the
extremization calculation as compared to the “hot” case. In the end, adding
non-relativistic stress from DM to the supermassive star in general, and doing
so in a way that provides a nearly fixed background, is inherently
stabilizing.

##  IV Nuclear Burning and Neutrino Losses

It should be expected that, at least for some masses in the range
${\sim}10^{4}\text{--}10^{8}\,M_{\odot}$, nuclear (hydrogen) burning on the
PPI chain is taking place at a high enough rate to significantly contribute to
the total energy budget of the SMS. This rate depends on both the density and
temperature throughout the star, both of which decrease as the mass increases.
Specifically, _at the onset of collapse_ where the density and temperature are
highest, $\rho_{\text{c}}\sim M^{-7/2}$ and $T_{c}\sim M^{-1}$, the latter
following from the definition of an $n=3$ polytrope. Since including DM can
raise the critical central density (and hence the temperature) required before
collapse, we can also investigate how the DM mass fraction impacts the energy
released by the PPI chain throughout the star. Chen  _et al._ [31]
investigates Helium burning in SMSs. This could be relevant for SMSs light
enough to have a stable Hydrogen burning phase.

While significant nuclear burning would potentially lead the star through a
stable “main-sequence” phase of evolution, neutrino emission from the high
entropy plasma could carry that extra energy generation away from the core,
providing no direct pressure support and hence adding no more stability to the
configuration. At such high entropies and temperatures, the neutrino
luminosity may outshine that of nuclear burning completely, causing the star
to contract more quickly and leading it closer to GR instability.

###  IV.1 PPI Chain Nuclear Burning

For zero metallicity, protons $p$ can only fuse to Helium nuclei $\alpha$ via
the PPI chain $4p\rightarrow\alpha+2e^{+}+2\nu_{e}$, where $e^{+}$ indicates
positrons and $\nu_{e}$ indicates electron neutrinos. In total, each such
reaction releases about $26$ MeV of energy to the plasma while the neutrinos
carry away about $0.5$ MeV away from the star as the likelihood they will
transfer energy to the plasma is vanishingly small [32]. The energy generation
rate for this reaction is calculated to be (in
$\mathrm{erg\,cm^{-3}\,s^{-1}}$)

$Q_{\text{PPI}}=c_{1}\rho^{2}X_{\text{H}}^{2}T_{6}^{-2/3}\exp\left(-c_{2}T_{6}^{-1/3}\right)\\\
\times\left(1+c_{3}T_{6}^{1/3}+c_{4}T_{6}^{2/3}+c_{5}T_{6}\right),$ (41)

where $\rho$ is the density in $\mathrm{g\,cm^{-3}}$, $X_{\text{H}}\approx
0.75$ is the total hydrogen mass fraction, $T_{6}=T/(10^{6}\,\mathrm{K})$ is
the temperature in units of $10^{6}$ Kelvins, and ${c_{1}=2.319\times
10^{6}}$, $c_{2}=33.81$, $c_{3}=0.0123$, $c_{4}=0.0109$, $c_{5}=0.00095$ are
numerical constants (see Clayton [32] for derivation).

To determine if the PPI chain provides a significant contribution to the
energy budget of the star, we will compare the integrated luminosity at the
surface, i.e.

$L_{\text{PPI}}=\int_{0}^{R}Q_{\text{PPI}}4\pi r^{2}\mathrm{d}r$ (42)

to the Eddington luminosity, which is approximately [30]

$L_{\text{Edd.}}\approx 1.3\times
10^{38}\left(\frac{M}{M_{\odot}}\right)\,\mathrm{erg\,s^{-1}},$ (43)

which comes from considering at what photon luminosity momentum transfer via
photon-electron Thomson scattering balances the gravitational force.

###  IV.2 Neutrino Losses

The high entropy per baryon ($s\gtrsim 1000$) required for hydrostatic
equilibrium in SMSs implies a significant $e^{\pm}$ pair content in
electromagnetic equilibrium $e^{+}~{}+~{}e^{-}~{}\leftrightarrow~{}2\gamma$.
These electrons and positrons can both annihilate and interact with the local
radiation field to produce neutrinos via [26, 27, 33]:

$\displaystyle e^{-}+e^{+}$ $\displaystyle\rightarrow\nu+\bar{\nu}\,$ (pair
production) $\displaystyle\gamma+e^{\pm}$ $\displaystyle\rightarrow
e^{\pm}+\nu+\bar{\nu}\,$ (photo-emission)

where $\gamma$ represents an incident photon and $\nu$ ($\bar{\nu}$) refers to
(anti-)neutrinos of any flavor. These processes are expected to be the
dominant source of neutrino emissivity in the low density and low-to-moderate
temperature regimes in SMSs. There is also a process where plasmons
($\tilde{\gamma}$) decay to neutrino pairs
($\tilde{\gamma}\rightarrow\nu+\bar{\nu}$), but this only dominates in high
density and temperature regimes.

If the electrons and positrons are non-relativistic and non-degenerate, the
energy density loss rates for the two dominant neutrino production processes
are calculated to be [32, 26] (in $\mathrm{erg\,cm^{-3}\,s^{-1}}$):

$\displaystyle Q_{\text{pair}}$ $\displaystyle\approx\left(4.9\times
10^{18}\right)T_{9}^{3}\exp\left(-\frac{11.86}{T_{9}}\right),$ (44)
$\displaystyle Q_{\text{photo.}}$ $\displaystyle\approx\left(0.98\times
10^{8}\right)\frac{\rho}{\mu}\,T_{9}^{8},$ (45)

where $\rho$ is the mass density in $\mathrm{g\,cm^{-3}}$, $\mu$ is the mean
molecular weight, and $T_{9}=T/(10^{9}\,\mathrm{K)}$ is the temperature in
units of $10^{9}$ Kelvins. While the pair production mechanism has a much
higher prefactor than the photo-emission process, the former suffers an
exponential suppression for low temperatures that the latter channel does not,
actually making that channel dominant in higher mass SMSs.

Using a similar process as in the previous section, we calculate the combined
luminosity from these neutrino emission processes to be

$L_{\mathrm{\nu,\,Tot.}}=\int_{0}^{R}\left(Q_{\mathrm{pair}}+Q_{\mathrm{photo.}}\right)4\pi
r^{2}\,\mathrm{d}r.$ (46)

While the PPI chain does emit two $\nu_{e}$ per reaction, the energy they
carry away is far smaller than the energy released from fusing four protons
into one $\alpha$ particle, so we will ignore that contribution from the total
neutrino luminosity.

Figure 2: Symmetric log-log triptych plot showing total luminosity (burning
minus neutrino losses) normalized to each SMSs’ Eddington luminosity versus
the SMS’s central density as a fraction of its critical central density for
different DM mass fractions. Solid (dashed) curves correspond to the “cold”
(“hot”) kinematic limit. The top panel is for a star with no DM, the middle
for a star with 1% DM by mass, and the bottom for a star with 10% DM by mass,
all for various SMS masses between $10^{4}\text{--}10^{8}\,M_{\odot}$.
Indicated in each panel is a dashed gray line at 1% Eddington luminosity to
indicate when burning is “important” for the evolution of the star. Note that
these should not be taken to be evolutionary plots since the mean molecular
weight $\mu$ is fixed.

To examine how the PPI chain luminosity compares to the neutrino luminosity as
the star contracts, we can simply scale down the central density and let it
approach the critical central density from below. Assuming that the run of
density and temperature closely follows an $n=3$ polytrope, this is a valid
analysis as such a polytrope is entirely self similar (i.e. the mass is
independent of the central density and only depends on $s$ and $\mu$ via the
polytropic constant of proportionality $K$).

In Figure 2, we present this result in three symmetric log-log plots showing
the total luminosity from both sources
($L_{\text{PPI}}~{}-~{}L_{\mathrm{\nu,\,Tot.}}$) as a fraction of the
Eddington luminosity for various SMS masses and DM mass fractions. We subtract
the neutrino luminosity since it carries away energy from the star, as opposed
to creating energy like the PPI process. This plot should not be treated as an
evolutionary plot since the curves shown in the figure are calculated with a
fixed mean molecular weight, $\mu=0.59$, i.e., the value corresponding to a
fully ionized primordial gas. Nuclear evolution, in the sense of burning via
incorporating baryons into nuclei, should serve to increase the mean molecular
weight $\mu$ as higher mass nuclei are created.

We would also expect that when nuclear burning is “important” (for the
purposes of this paper chosen to be ${\sim}1\%$ of a star’s Eddington
luminosity) the central density is held almost constant until its fuel is
exhausted. This would prevent an unabated contraction of the star to the
critical central density for GR instability, if only for a relatively short
duration.

Since adding DM serves to increase the critical central density, more DM can
push massive stars to higher central densities without causing them to become
unstable. Both the PPI process and the relevant neutrino processes depend
sensitively on the central temperature (and hence the central density), so a
delicate balancing act ensues between energy generation from proton fusion and
energy loss from neutrino emission.

As evidenced from the top plot (i.e. zero DM) in Figure 2, specifically for
SMSs of masses ${\lesssim}10^{5}\,M_{\odot}$, increasing the central density
increases the fraction of PPI energy generation, and hence net energy
deposition, but only up to a point. Eventually, the neutrino losses from the
increasing temperature win out, and the star is net losing energy instead of
generating energy.

Should the quasi-statically contracting SMS encounter this situation during
its evolution, the neutrino emission would cause the star to contract more
quickly. This is because the neutrinos carry away the plasma’s entropy faster
than Hydrogen burning can generate it throughout the configuration. Per the
solutions to the Lane-Emden equations, an $n=3$ polytrope’s radius scales as
$R\sim K^{1/2}\sim s^{2/3}$, and since $\rho_{\text{c}}\sim R^{-3}$, the
central density will increase as the radius decreases as neutrinos remove
entropy from the SMS. This serves to push the SMS closer to instability.

The situation changes slightly for the ${\sim}1\%$ DM mass fraction case
(middle panel) and quite dramatically for the ${\sim}10\%$ case (bottom
panel). At a given central density fraction
$\rho_{\text{c}}/\rho_{\text{c,\,crit}}$, increasing DM mass fraction
corresponds to a higher absolute density on each mass curve. If we examine the
$10^{5}\,M_{\odot}$ curve, for example, this mechanism allows for the energy
generation fraction from PPI to quite closely approach $1\%$ of the Eddington
luminosity as the DM mass fraction increases. This potentially permits such
stars to burn stably on the main sequence instead of a direct collapse as
expected in the standard case. This would drastically change their evolution
as opposed to the case without DM.

##  V Conclusion

These SMSs are quite interesting case studies, primarily because their
existence provides a potential physical explanation for several observational
mysteries that have only deepened as more advanced telescopes have come
online. Not only has JWST, as an example, seen plenty of high mass quasars at
high redshift which puts the standard picture of solar mass remnant accretion
into doubt, it may also have observed SMSs supported in some way by DM prior
to collapse in the JADES survey, per Freese  _et al._ [34].

Considering that these stars are “trembling on the verge of instability” due
to the Feynman-Chandrasekhar Instability, it begs the question as to how they
would be observed at all prior to collapse. One possible avenue to support
these stars is through some form of non-interacting and non-relativistic DM.
The enhanced stability would certainly also have implications for the nuclear
and neutrino evolution of the stars.

In Section I, we derived the standard result for the critical central density
beyond which a hydrostatic SMS is unstable to collapse. This rested upon
extremizing the total energy of the star with respect to the central density.
We found that, as expected, higher mean molecular weight plasmas and higher
mass SMSs are more unstable to collapse. Both of these factors serve to
decrease the critical central density.

In Sections II.1 and II.2, we derived analytically how the critical central
density changes as a function of the DM mass fraction within the star. We
considered two limiting cases of the DM kinematics, the “hot” case and the
“cold” case, which compared the typical DM particle’s specific kinetic energy
far away from the SMS to the gravitational potential felt at the SMS surface,
and found that in both cases the critical central density for the onset of
general relativistic instability increased as the DM mass fraction increases.
The “cold” case shows a slightly stronger effect, and the relative increase is
greater for more massive stars in both kinematic limits.

In Section III we investigated a possible mechanism for the increased
stability. We showed that if the star contracts, the total mass fraction of DM
inside the star decreases, so the total mass enclosed by the star also
decreases. This added stability is therefore completely agnostic to any
particle physics specific to the DM, so long as any consideration of such
physics keeps the DM non-relativistic and non-interacting with the baryonic
matter. In fact, if the DM possesses some self-annihilation cross-section,
like in the model discussed in Freese  _et al._ [34], this may further bolster
the stability by providing an extra source of energy and entropy.

Finally, in Section IV, we considered by what degree Hydrogen burning could
impact the evolution of these SMSs and how DM influences that burning. We also
analyzed how neutrino emission from the high-entropy plasma would take away
energy, potentially faster than what is generated from fusion, as the star
approaches the critical central density from below. We found that, considering
both energy generation from fusion and energy losses from neutrino emission,
DM could allow heavier stars than previously predicted to stably burn on the
main sequence at some point in their lives, increasing the rough upper limit
of ${\sim}30,000\,M_{\odot}$ to about $50,000\,M_{\odot}$, depending on when
one considers nuclear burning to be “important” to the evolution of the star.

More investigation is needed to understand these extreme objects. If they are
to explain the high mass SMBHs in the early universe by providing a
supermassive seed black hole, there should be enough of them formed to account
for all or most of them. This is complicated by a suspect formation history,
since if the collapsing gas is cool enough to contain molecular Hydrogen
instead of atomic Hydrogen, the enhanced cooling efficiency via rotational and
vibrational modes [35] could favor a fragmented formation into many stars
instead of one SMS. However, enhanced cooling from early molecular Hydrogen
formation via Beyond the Standard Model (BSM) channels (e.g. in Biermann and
Kusenko [36]) can lead to very early star formation. Whether SMS formation may
accompany this early star formation is an open question. Additionally, if a
sufficient population of primordial black holes (PBHs) formed with low enough
mass to fully evaporate before $z\sim 10$, the extra heating provided can
suppress molecular Hydrogen formation [37], possibly allowing another channel
to form SMSs. A similar result can also be reached by considering the decay of
relic particles [38].

We would expect that nuclear burning, i.e. fusing bare nucleons into larger
composite nuclei, would lower the critical central density for the onset of
collapse since the mean molecular weight increases. The gas pressure is
provided primarily by non-relativistic baryons, which is proportional to the
total number of particles, so nuclear fusion reduces this number and hence
lowers the gas pressure if sufficient burning takes place. While sufficient
burning would serve to slow the SMS’s contraction by injecting energy in the
region surrounding the core, the critical central density is being reduced
nonetheless, hastening the approach to instability if the chemical composition
of the star is significantly changed.

Including DM further complicates this picture. We saw that sufficient DM will
change the evolution of some SMS masses that would normally collapse via GR
instability, making them also vulnerable to burning induced instability. As DM
raises the critical central density, it also permits these stars to host
greater central temperatures, making neutrino emission more efficient at
carrying away entropy. This would cause the SMS to collapse on a different
adiabat, specifically one at lower entropy than expected, potentially altering
the prospects for post-collapse detection.

If such stars existed and underwent sufficient burning, they could be detected
from their collapse directly if they then explode (instead of directly onto a
black hole), and indirectly from enriching their surrounding medium with
burning products both pre and post-collapse [39, 40]. The enhanced neutrino
emission from DM would also alter the gravitational wave signature from an
anisotropic neutrino pulse as the homologous core’s entropy would be reduced
for a given mass (see e.g. Li  _et al._ [12] for details on this neutrino
driven gravitational radiation for standard SMSs).

The authors would also caution those who may be unfazed by the supposedly
‘modest’ amount of DM invoked in this study. Consider a $10^{6}\,M_{\odot}$
SMS with ${\sim}1\%$ DM by mass. If the DM is at a constant density within the
star, such a DM mass fraction corresponds to a density of
${\sim}10^{20}\,\mathrm{GeV\,cm^{-3}}$, which would be an over-dense region of
DM about 20 orders of magnitude more dense than the expected background at
redshift $z\sim 10$.

If these DM supported SMSs had existed, this huge accumulation and
concentration of DM greatly constrains models for DM particles. Specifically,
models where these particles cannot alter their phase space density and clump
together through self and/or standard model interactions would clearly be
disfavored in this regard. An example of a model where particles can alter
their phase space density appropriately is discussed in Feng  _et al._ [41],
which in turn also provides a different mechanism for seeding SMBHs. Densities
approaching the requisite amount of DM needed to stabilize SMSs through purely
gravitation can be reached via adiabatic contraction of a baryonic cloud in a
DM halo (e.g. Blumenthal  _et al._ [42]), but would still need an additional
source of heating to substantially change the stability of the SMS (e.g.
Freese  _et al._ [43] for the case of DM self-annihilation).

Moreover, the huge concentration of DM that we discuss here likely also
constrains the manner and environments of SMS formation. Whether such
environments can be found, and whether a component of the DM can be compatible
with the concentrations discussed here, remain intriguing and open questions.
In the end, such speculation may be warranted given our relative ignorance of
DM and dark sector physics, and given the existence of SMBHs at very high
redshift.

Here we have used a very simplistic model, a non-rotating and isolated SMS, to
explore the physical effects of a large DM content. Nonetheless, our findings
pose several open questions. First, with the inclusion of DM, what is the
interplay of nuclear burning, hydrodynamics, energy transport, neutrino
emission, etc., in the evolution of the SMS up to instability and the
subsequent collapse to a black hole? Second, are there specific observational
“handles” on the collapse of a SMS with or without DM? For example, is the
gravitational radiation signature [12] different in these two cases, and to
what extent would it depend on the amount of DM? Third, in this paper we have
not considered the effects of DM on scenarios where a supermassive core
undergoes rapid accretion, as in the _hylotropic_ configurations proposed in
Ref. [44]. The stabilizing effect of DM on such hylotropic configurations has
been recently considered [45]. These scenarios may provide the most plausible
route for the production of an SMBH from an SMS-like object.

Finally, perhaps the most significant issue is how an SMS would form with the
huge quantities of DM considered in this work and in others that invoke DM
effects in the production of SMBHs. However, there is urgency for
consideration of these issues because of the existence of SMBHs at very early
epochs in the history of the Universe, and because both the sources and
properties of DM remain mysterious.

###### Acknowledgements.

We would like to acknowledge fruitful discussions with Katie Freese and Thomas
Wong. This work was supported in part by National Science Foundation (NSF)
Grant No. PHY-2209578 at UCSD and the Network for Neutrinos, Nuclear
Astrophysics, and Symmetries (N3AS) NSF Physics Frontier Center, NSF Grant No.
PHY-2020275, and the Heising-Simons Foundation (2017-228).

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