# Ultra-Fast Adaptive Track Detection Network

Hai Ni Rui Wang Scarlett Liu
School of Traffic and Transportation Engineering, Central South University
Changsha, China 410075
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Corresponding author: Scarlett Liu

###### Abstract

Railway detection is critical for the automation of railway systems. Existing
models often prioritize either speed or accuracy, but achieving both remains a
challenge. To address the limitations of presetting anchor groups that
struggle with varying track proportions from different camera angles, an
ultra-fast adaptive track detection network is proposed in this paper. This
network comprises a backbone network and two specialized branches (Horizontal
Coordinate Locator and Perspective Identifier). The Perspective Identifier
selects the suitable anchor group from preset anchor groups, thereby
determining the row coordinates of the railway track. Subsequently, the
Horizontal Coordinate Locator provides row classification results based on
multiple preset anchor groups. Then, utilizing the results from the
Perspective Identifier, it generates the column coordinates of the railway
track. This network is evaluated on multiple datasets, with the lightweight
version achieving an F1 score of 98.68% on the SRail dataset and a detection
rate of up to 473 FPS. Compared to the SOTA, the proposed model is competitive
in both speed and accuracy. The dataset and code are available at
https://github.com/idnihai/UFATD.

##  1 Introduction

The 2023 investigative report from the International Union of Railways (UIC)
revealed that over 80% of incidents in 2022 were due to third parties
trespassing on railway lines [23]. The intrusion of a third party onto the
railway can lead to prolonged disruptions to railway transportation,
potentially resulting in severe casualties and economic losses. Additionally,
there is a risk of activities damaging railways and associated facilities,
posing significant safety hazards, and jeopardizing overall railway safety.
Traditional static monitoring is inadequate for providing real-time insights.
Therefore, there is an urgent need to introduce a methodology for dynamic,
real-time railway track monitoring. This is essential to accommodate the
complex and variable operational environments and meet the demands of all-
weather, high-speed operations [1]. The dynamic monitoring method requires the
real-time establishment and updating of safety gaps based on the track
structure. The track has emerged as a crucial criterion for assessing whether
foreign objects have intruded, as emphasized in the work by Zhong et al. [32].
Currently, computer vision-based track detection primarily utilizes two
methods: feature-based detection and model-based detection. Feature-based
detection excels in accuracy but is susceptible to interference, and its real-
time performance is comparatively lower. Conversely, model-based detection
exhibits faster speed and robust anti-interference capabilities.

The classic model-based method employs image segmentation algorithms [28, 4],
yet it is characterized by high complexity and typically slower speed.
Recently, anchor-based models have been developed [15, 11, 17, 25]. These
methods significantly reduce model complexity and accelerate processing speed
through preset anchors. However, this approach has certain limitations.
Typically, anchor-based methods utilize only one group of anchors, which may
not be suitable for accurately detecting lanes in scenarios where the lane is
choppy. Significant changes in the angle between the camera and the ground
pose challenges, as the preset anchor group may struggle to adequately
encompass the entire track line. Figure 1 depicts various instances of
disparate camera perspectives, highlighting the challenges a preset group of
anchors poses in accurately predicting lane markings.

(a) Class 0

(b) Class 1

(c) Class 2

(d) Class 3

Figure 1: Railway track images captured from various camera perspectives. As
the camera’s perspective changes, there is a corresponding alteration in the
proportion of sky and ground captured in the images.

To accurately detect railway tracks from various angles using UAVs,
researchers have proposed a method capable of adaptively representing
railways. This method can effectively represent tracks with both near-
horizontal and near-vertical inclinations, as well as other tracks with normal
inclinations [22, 21]. Inspired by previous studies, a novel anchor-based row
selection method is proposed for railway track detection. The network
classifies the camera perspectives, presets multiple groups of anchors for
training, and selects the most suitable groups of anchors corresponding to
each camera perspective. This approach can accommodate more intricate
scenarios, particularly when the number of anchors is limited.

The contributions are as follows:

  * •

A novel anchor-based row selection railway track detection network, UFATD, is
proposed, which adapts to different camera perspectives by presetting multiple
anchor groups.

  * •

Aiming at the situation where UFATD requires multiple groups of anchors, a
novel anchor group generation algorithm is proposed.

  * •

A synthetic railway track dataset, augmented with a small number of real
images, is developed. Its primary purpose is to assess the performance of the
novel anchor group generation algorithm across different camera angles.

##  2 Related Work

###  2.1 Feature-based Track Detection

Feature-based methods initially convert the image to grayscale, followed by
the application of edge detection algorithms to extract the tracks. Some
widely employed edge detection algorithms include the Prewitt algorithm [14]
and the Sobel algorithm [19]. Nevertheless, the aforementioned methods do not
account for image noise during image processing. Moreover, to effectively
address the constraints caused by railway encroachments, Wang et al. devised
an adaptive track segmentation algorithm. This algorithm enhances detection
accuracy by using Gaussian kernels with adaptive orientations, based on the
Hough transform’s maximum point, to optimally utilize linear features in
railway scenes [27].

Utilizing the Line Segment Detector (LSD) and least squares curve fitting
achieves high accuracy in recognizing image tracks. Nevertheless, relying
solely on edge features presents challenges, particularly in scenarios
featuring physical obstructions or exposure variations [31]. Employing the
Canny operator, which integrates Gaussian theory, structural element function,
and morphological transformation, effectively captures track features [31].
This method enhances the algorithm’s adaptability and robustness, even in
blurry or low-quality videos. Nevertheless, achieving accurate rail detection
in environments such as tunnels and shadows remains challenging [24].

###  2.2 Model-based Lane and Track Detection

When tackling detection challenges in complex scenes, model-based methods
typically outperform feature-based methods. Various models based on deep
learning, including image segmentation models [28, 4, 10, 13] and anchor-based
models [15, 11, 17, 25, 22, 21, 20, 12, 29], have been developed for lane line
detection or railway track detection, providing comprehensive solutions for
complex scenes. The end-to-end segmentation model, TEP-Net, achieved an
outstanding IoU of 97.5% on the RailSem19 dataset [10]. Another end-to-end
lane segmentation model operates at a speed of only 50 fps [13].

Qin et al. addressed lane detection by regarding it as a row-based selection
issue employing global features. They proposed a lane line detection method
based on this approach [15]. Li et al. innovatively applied the anchor-based
method to railway track detection, achieving excellent results. Comparative
experiments with segmentation models revealed that Rail-Net excels in both
accuracy and speed [11]. The DALNet model, proposed by Yu et al., incorporates
a dynamic anchor line generator, which dynamically creates suitable anchor
lines for the railway tracks based on features such as their position and
shape in the input image [29]. Among these methods, anchor-based methods
exhibit superior speed compared to segmentation-based methods. Additionally,
the detection accuracy is comparable to that of segmentation-based methods.

###  2.3 Camera Pose Estimation

The estimation of the IMU or gyroscope sensor utilized by the robot is
impacted by noise from uneven ground, making pose estimation challenging.
Drawing inspiration from the classification network rooted in Optical
Character Recognition (OCR) [7] principles, scholars investigate the
application of deep learning for estimating camera image poses to achieve
precise camera positioning [9, 8]. The outcomes of the classification network
are utilized to correspond to distinct perspectives. To accurately represent
lane lines, researchers proposed a homography prediction network, termed HP-
Net, which predicts the homography matrix of the camera [2].

##  3 Data Scope

###  3.1 Dataset Collection and Annotation

To substantiate the efficacy of the proposed method, comprehensive
experimental evaluations will be conducted across three datasets, namely:
SRail, RailDB and DL-RAIL. The three datasets pertain to railway track
imagery. The SRail dataset, introduced in this study, comprises 5,903 images.
Among these, 4,748 are synthetic images generated using Unreal Engine
5111https://www.unrealengine.com/, supplemented by 1,155 images captured by
real cameras to enhance the dataset’s feature diversity. These images simulate
diverse camera angles for a comprehensive assessment. Images are annotated
using LabelMe222https://github.com/wkentaro/labelme. This dataset aims to
evaluate the model’s performance across various camera perspectives.
Accordingly, the images are categorized into four classes based on their
camera viewpoint: Class 0, Class 1, Class 2, and Class 3. Sample images from
each category are depicted in Figure 1.

The indicator used by Li et al. to evaluate the model on the RailDB dataset is
ACC [11]. $ACC=\frac{\sum_{clip}C_{clip}}{\sum_{clip}S_{clip}}$ where
$C_{clip}$ represents the number of correctly predicted track points, and
$S_{clip}$ denotes the total number of ground-truth data points in every
image. However, this indicator only evaluates the abscissa (x-coordinate) of
the points. The method proposed in this article adjusts the anchors based on
the image, resulting in changes to the ordinate (y-coordinate) in the outcome.
Therefore, ACC is not suitable for the evaluation of this method. In order to
more effectively evaluate the method on the RailDB dataset, the rail image and
label image were resized to $1920\times 1080$, and txt annotation files were
generated based on the label images. In addition, similar to their work, this
study focused on only four railway tracks as research objects. Consequently,
any values greater than 4 in the label image were set to 0.

###  3.2 Data Comparison

In this section, the differences among the aforementioned datasets are
compared. Table 1 displays the features of datasets.

Table 1: The existing railway dataset. The original RailDB dataset contains
images of various sizes. To facilitate the employ of the F1 score evaluation
algorithm, all rail images and label images are resized to a uniform size.

Database Rail-DB DL-RAIL SRail (ours) Quantity 7,432 7,004 5,903 Number of
tracks Original: 8 After processed: 4 2 2 Resolution Original: 1280 $\times$
720; 1920 $\times$ 1080; 3840 $\times$ 2160 After processed: 1920 $\times$
1080 1920 $\times$ 1080 1920 $\times$ 1080 Train 5,946 5435 4,132 Test 1,486
1,569 1,771

##  4 Method

In this paper, track detection is regarded as a row-based selection problem
rather than a segmentation problem, which significantly reduces model
complexity.

###  4.1 Problem Formulation

The segmentation method typically involves annotating the track lines as
multi-valued maps. However, n this study, track detection is regarded as a
row-based selection problem and a classification problem. The classification
problem aims to identify the anchors corresponding to the image, while the
row-based selection problem involves gridding the image based on these
anchors.

Additionally, a fixed number and position of anchors can result in
significantly low efficiency in track detection. For instance, if the anchors
cover the entire image while the railway track only occupies half of the
image, half of the anchors become redundant. To address this issue, an
adaptive method is proposed in this paper. Specifically, multiple groups of
anchors are preset, from which the network dynamically selects the most
appropriate group of anchors. This method allows for the representation of the
railway tracks with as many anchors as possible.

(a) Example 1

(b) Example 2

Figure 2: Illustration of gridding the image by selecting a suitable group of
anchors for the image. Each image will be matched with an appropriate group of
anchors. The image is divided into $(w+1)$ grids in the horizontal direction
based on these anchors. To better locate possible bends and other features in
the upper part of the image, the distance between anchors decreases as they
approach the top of the image. $d^{\prime}_{j,k}$ is the anchor row distance
calculated by the method proposed in section 4.3.

As illustrated in Figure 2, there are n groups of anchors. Within each group,
the track line intersects with the row represented by the anchor, thus
transferring the labeled polylines of the railway track to a series of
horizontal locations and their corresponding anchors. Different images will
correspond to different anchor groups. The selected anchor groups are designed
to cover the track with as many anchors as possible, thereby maximizing the
utilization of anchors. By employing non-uniform anchors, the distribution of
anchors at the top of the image becomes denser, yielding improved results when
the top of the image contains curved railway tracks.

The computational expenses can be notably diminished using this approach.
Assuming $C$ represents the number of tracks, $h$ denotes the number of row
anchors, and $w$ signifies the number of grid columns. Analogous to UFLD, $w$
is expanded to $(w+1)$ in the grid column direction, where $(w+1)$ represents
the background. Besides, $n$ represents the number of groups of anchors used
for training. Consequently, the computational cost is proportional to
$((w+1)\times h\times C\times n)$. When the image size is $H\times W$, the
calculation reduction rate (r) can be expressed as: $r=\frac{H\times
W}{h\times(w+1)\times n}$

In the work of Li Li et al. [11], the number of anchors $h$ is set to 52, and
$w$ is set to 200. Simultaneously, the images are also scaled to $800\times
288$. Therefore, the $r$ can reach up to 22. The smaller $h$ and $n$ are, the
larger $r$ will be. When $h$ is set to 18, and $n$ is set to 2, $r$ can reach
31.8.

###  4.2 Network Pipeline

Figure 3: The structure of UFATD. The image is first input into the backbone
network to extract high-level features, and then it is fed into two separate
branch networks. The Perspective Identifier branch classifies the image
category based on the camera perspective. Each category corresponds to a group
of anchors. This branch is an $n$-class classification network. The Horizontal
Coordinate Locator branch outputs $((w+1),h,C,n)$-dimensional data, which
includes the column coordinates under $n$ groups of anchors. The
classification result from the Perspective Identifier branch is passed through
a softmax function and then input to the Horizontal Coordinate Locator branch,
where the corresponding group of anchors for the image is selected.

Ultra-fast adaptive track detection network (UFATD) is a row-selective track
detection network. Figure 3 illustrates the network architecture of UFATD.
This network leverages a pre-trained backbone to extract high-dimensional
features from images. These features undergo two-dimensional convolution
before being fed into two branch networks for dimensionality reduction of the
feature map. The Horizontal Coordinate Locator branch is tasked with
classifying the row coordinates of the rails.

The classifier of the Horizontal Coordinate Locator comprises layers of linear
transformations ranging from 1800 to $((w+1)\times h\times C\times n)$ linear
layers. Cross-entropy (CE) function is employed within the classification
network to guide parameter optimization. Assuming $X$ represents the global
features of the image, $f^{ijk}$ is the classifier utilized to select the
track position at the $i$-th track, the $j$-th row anchor, and the $k$-th set
of the anchor groups. Then the railway track prediction can be written as:

$P_{i,j,k}=f^{ijk}(X),s.t.\quad i\in[0,C-1],j\in[0,h-1],k\in[0,n-1]$ (1)

among them, $P_{i,j,k}$ is a $(w+1)$-dimensional vector, indicating the
probability of selecting (w+1) grid cells under the $i$-th railway track,
$j$-th row anchor, and $k$-th set of anchor groups. Assume that $T_{i,j,k}$ is
a one-hot label at the correct location, the corresponding optimization
formula is:

$L_{hcl}=\sum_{i=0}^{C-1}\sum_{j=0}^{h-1}\sum_{k=0}^{n-1}L_{CE}(P_{i,j,k},T_{i,j,k})$
(2)

where $L_{CE}$ is the cross entropy loss.

The Perspective Identifier branch functions to identify the most suitable
group of anchors from a variety of preset groups. This branch serves as a
classification network tasked with converting vectors into predictions of
anchor group indices of size $n$. The classifier within this branch comprises
layers of linear transformations, spanning from 1800 to $n$ linear layers.
After that, the results are processed through softmax, and the prediction
corresponding to the image is selected from the results of the Horizontal
Coordinate Locator branch. Similar to the classification network mentioned
earlier, the Perspective Identifier branch also employs the CE function to
guide parameter optimization during the training phase.

$L_{pi}=-\sum_{k=0}^{n-1}a_{k}\log(p_{k})$ (3)

Each $p_{k}$ represents the probability that the sample belongs to the $k$-th
category, and the corresponding true label is a one-hot encoding vector
$\mathbf{a}=[a_{0},a_{1},\dots a_{n-1}]$, where $a_{k}$ signifies whether the
$k$-th category is the true category of the anchor groups. The full objective
of the UFATD includes $L_{hcl}$and $L_{pi}$:

$L_{total}=L_{hcl}+\lambda L_{pi}$ (4)

where $\lambda$ controls the Perspective Identifier branch. The specifics of
the UFATD training process are delineated in Section 5.1.

###  4.3 Anchor Genenration

As illustrated in Figure 1, the camera projection relationship affects the
appearance of railway tracks in the image, resulting in the tracks appearing
more curved at the far end of the camera, necessitating denser anchors for
accurate representation. However, employing equidistant anchors would require
a larger quantity of anchors, leading to an increase in network parameters.

In order to maximize the use of all anchors in the anchor group, an algorithm
is proposed to generate non-equidistant anchor groups. Notably, the distance
between anchor rows at the top of the image is closer compared to the bottom.

The points on the railway tracks that are closest to the top of the image are
extracted, and their $y$ values are recorded. The minimum $y$ and maximum $y$
values from $y$ set are identified and denoted as $y_{min}$ and $y_{max}$,
respectively. The starting points of the $n$ group of anchors are evenly
distributed, so the starting value of the $k$-th group of anchors is
calculated as follows:$s_{k}=y_{min}+\frac{k}{n-1}\cdot(y_{max}-y_{min})$

Each group of anchors is then calculated based on its starting value and the
maximum anchor value $H$. Specifically, the initial distance between anchor
rows (i.e., equidistant anchors) is computed using the starting value, the
maximum anchor value $H$, and the number of anchors $h$. In practice, the
maximum anchor value will generally be smaller than image height $H$. For
simplicity, it is represented by $H$. The initial distance of the $k$-th group
of anchors can be expressed as: $d_{k}=\frac{H-s_{k}}{h-1}$. Subsequently, a
scaling factor is calculated using Formula 5, which is multiplied by the
$d_{k}$ to obtain the target distance. This calculation results in smaller
growth for the first few points of each anchor group, leading to denser anchor
placement and shorter distances between anchor rows near the top of the image.

$f(x)=\begin{cases}\sqrt{1-(1-x)^{2}}&\text{if }0\leq x\leq 1\\\
-\sqrt{1-(1-x)^{2}}+2&\text{otherwise}\end{cases}$ (5)

where $x$ is defined as follows: $x_{j}=\frac{j}{h}\cdot 2$. And
$d_{j,k}=d_{k}\cdot f(x_{j})$. Therefore, the $j$-th value in the $k$-th group
of anchors is calculated as follows:

$y_{j,k}=s_{k}+\sum_{j=0}^{h-1}d_{j,k}$ (6)

##  5 Experiments

###  5.1 Implementations

To quantitatively assess the proposed method, comprehensive experiments were
carried out across three railway track datasets: SRail, RailDB and DL-RAIL.

Dataset splits. The training datasets in the datasets randomly allocate 85% of
the images for training and 15% for validation during the training phase.
During each experiment, the training datasets and validation datasets are
randomly partitioned.

Training Details. All models are trained and tested using PyTorch on Nvidia
RTX 4090 GPU, Intel i9 13900K CPU, with 64GB of RAM. The Adam optimizer is
utilized, with a cosine decay learning rate strategy. The initial learning
rate for the backbone is set to 4e-4. For the Perspective Identifier, the
initial learning rate is 5e-5, and for the Perspective Identifier, it is 1e-3.
The batch size is set to 32, with a total training epoch number is 100.
Initially, the backbone network and the Perspective Identifier branch are
frozen. The pre-trained backbone network is utilized to first update the
parameters of the Horizontal Coordinate Locator branch. After 5 epochs, the
backbone network is unfrozen. After 15 epochs, the Perspective Identifier
branch is unfrozen. Additionally, during the first 15 epochs, the $\lambda$
controlling the Perspective Identifier branch loss value is set to 0, and then
it is set to 0.05. The results of UFATD represent the average of three
repeated experiments.

Data Pre-process. Furthermore, all images in the dataset will be resized to
$800\times 288$ during training. For the RailDB dataset, data processing has
been introduced in Section 3.1.

Evaluation Metric. All datasets utilize F1 as defined in CULane. The
calculation code comes from Qin et al 333https://github.com/cfzd/Ultra-Fast-
Lane-Detection. $FP$ represents false positives, and $FN$ represents false
negatives, respectively. F1 is defined as follows:

$F1=2\times\frac{Precision\times Recall}{Precision+Recall}$ (7)

where $Precision=\frac{TP}{TP+FP}$, $Recall=\frac{TP}{TP+FN}$. In this study,
the F1@50 and F1@75 metrics, with IoU thresholds of 0.5 and 0.75, are
utilized. Additionally, to better compare the performance of the models, mF1
is also presented as follows: $mF1=(F1@50+F1@55+\cdots+F1@95)/10$

###  5.2 Ablation Study

Performing an ablation analysis on UFATD on SRail to validate the effects of
the backbone network, the number of anchors, and the inter-anchor distance on
the network. Except for the above conditions, all experiments were conducted
following the identical protocols outlined in Section 5.1.

####  5.2.1 Backbone of UFATD

Beyond the evaluation of ResNet18, ResNet34 and ResNet50 [5] as backbone
architectures, ResNet 34 with Fcanet [16], SqueezeNet [6] and MobileNet V2
[18], were also investigated to elucidate the influence of UFATD across a
spectrum of backbone architectures. Table 2 outlines the F1 score and Frames
Per Second (FPS) outcomes for different backbone configurations.

Table 2: Quantitative comparison of various backbones on the SRail dataset. Backbone | Res18 | Res50 | Res34 | Res34 with Fcanet | SqueezeNet | MobileNet V2
---|---|---|---|---|---|---
F1@50 | 98.68 | 98.64 | 98.79 | 98.77 | 98.19 | 98.55
FPS | 473 | 249 | 301 | 231 | 951 | 732

The results demonstrate that ResNet models are relatively dominant, indicating
that even a small network like ResNet18 is sufficient for railway feature
extraction. In addition, what is interesting is that under repeated
experiments, the measured FPS of Resnet50 is indeed higher than that of
Resnet34, which is incredible. When tested on Rtx4080 laptop, the FPS of
Resnet34 is higher. This may involve the optimization of the graphics card,
which is worth studying. In repeated tests, Resnet50 consistently outperforms
Resnet34 on FPS, which is remarkable. However, when tested on an RTX4080
laptop, Resnet34 achieves higher FPS. This discrepancy suggests potential
optimizations related to the GPU, prompting further investigation.

####  5.2.2 Effectiveness of the proposed modules

This section demonstrates the impact of the new network proposed in Section
4.2 and the anchors generated by the algorithm introduced in Section 4.3 on
model performance. As observed in Table 3, the network proposed in this study
notably enhances the F1@75 and mF1 metrics of the model. Nonetheless, there
are some drawbacks. F1@50 decreased by 0.44, likely attributed to the addition
of multiple groups of anchors, thereby complicating the classification process
for the Horizontal Coordinate Locator branch. Moreover, employing non-
equidistant anchors generated via the proposed anchor generation method also
contributes to a moderate improvement in the model’s performance.

Table 3: Experiments of the proposed modules on SRail with ResNet-18 backbone.
The baseline represents training the network using only a group of equidistant
anchors. The new network introduces three groups of equidistant anchors.
Anchor generation involves using three groups of non-equidistant anchors,
produced by the anchor generation algorithm, to train the network.

Baseline New network Anchors generation mF1 F1@50 F1@75 ✓ 54.60 98.94 56.80 ✓
57.06 (+2.46) 98.50 (-0.44) 65.05 (+8.25) ✓ ✓ 58.12 (+3.52) 98.68 (-0.26)
67.68 (+10.88)

##  6 Results and Discussion

In this section, the outcomes of UFATD and several models on the railway track
dataset will be showcased. ResNet-18 and ResNet-34 were employed as the
backbone architectures for UFATD.

Table 4: Quantitative comparison of various methods on the SRail dataset.

Method mF1 F1@50 F1@75 FPS Class 0 Class 1 Class 2 Class 3 Be´zierLaneNet [3]
43.77 90.38 36.28 364 84.01 85.58 93.20 92.84 Clrnet [30] 64.03 98.62 79.55
279 97.30 99.06 99.14 97.56 CondlaneNet [12] 56.17 99.31 55.96 327 99.53 99.61
99.46 98.13 DALNet [29] 60.40 98.52 68.59 305 96.28 99.06 99.35 97.15 Rail-Net
[31] 51.24 97.97 46.97 429 97.0 97.19 98.70 97.27 UFLD (Res18) [15] 49.81
97.71 42.49 475 96.28 96.25 98.76 97.25 UFLD (Res34) [15] 49.58 97.95 40.90
249 96.10 96.72 98.92 97.94 UFATD (Res18) 58.12 98.68 67.68 473 98.88 98.65
98.97 97.46 UFATD (Res34) 58.04 98.64 67.59 249 98.47 98.48 98.79 97.17 UFATD
(Res34 with Fcanet) 58.19 98.78 68.23 231 98.57 99.01 99.10 97.53 UFATD
(Res50) 57.92 98.80 67.35 301 98.76 98.75 99.08 97.85

On the S-Rail dataset, UFATD was compared with six methods, including:
Be´zierLaneNet [3], Clrnet [30], CondlaneNet [12], DALNet [29], Rail-Net [31],
UFLD [15]. This experiment conducted a comparative analysis of model accuracy
and execution time. Model accuracy was assessed using the F1 score metric,
while model runtime was measured as the average time over 1000 iterations.
From Table 4, it can be inferred that UFATD demonstrates performance on par
with SOTA while exhibiting high operational speed.

On RailDB, UFATD is SOTA in multiple categories. It is worth noting that, even
in cases where models such as UFLD perform poorly, UFATD still demonstrates
reliability. Refer to Table 5.

Table 5: Quantitative comparison of various methods on the RailDB dataset.

Method mF1 F1@50 F1@75 Cross Curve Far Line Near Night Rain Slope Sun
Be´zierLaneNet [3] 42.30 82.75 38.07 48.34 91.92 93.99 86.48 79.85 78.54 83.22
92.95 82.24 Clrnet [30] 40.25 82.00 33.45 70.03 78.04 75.20 97.30 89.14 78.70
82.50 92.27 81.24 CondLaneNet [12] 55.10 90.95 61.33 56.80 88.11 91.20 86.24
79.70 92.20 95.73 96.58 83.56 UFLD (Res18) [15] 54.41 95.52 57.60 82.23 97.99
92.79 95.82 98.71 97.45 94.51 96.94 96.19 Rail-Net [31] 52.64 97.02 51.12
89.95 97.99 94.90 97.41 99.11 98.72 95.93 98.34 95.97 GANet [26] 58.87 91.59
68.40 53.37 98.34 95.95 98.46 90.16 93.24 96.07 97.01 84.55 UFATD (Res18)
59.97 97.43 67.70 91.54 98.41 95.25 97.96 99.12 98.13 95.85 98.87 96.47 UFATD
(Res34) 60.53 97.60 68.90 92.10 98.37 95.60 98.13 99.24 98.70 96.03 98.92
96.47

Figure 4: Quantitative comparison of various methods on the DL-RAIL dataset. ∗
indicates that the test set is employed as the validation set for the optimal
model.

For the DL-Rail dataset, UFATD demonstrates outstanding performance as well.
In the code provided by Yu et al. [29], the test set is involved in the
validation process during model training. Consequently, Figure 4 provides two
types of results. Those marked with an asterisk (*) represent model outcomes
when the test set is employed as the validation set. UFATD with ResNet-34
achieves a performance ranking second only to DALNet. The visualization of
UFATD is depicted in Figure 5. UFATD exhibits strong performance across all
three datasets.

(a) SRail

(b) Rail-DB

(c) DL-RAIL

Figure 5: Visualization on the SRail, the Rail-DB and the DL-RAIL dataset.

##  7 CONCLUSION

This paper introduces an adaptive railway track detection model capable of
accommodating various camera perspectives and achieving outstanding levels of
speed and accuracy. The proposed model addresses track detection by framing it
as a row-based selection and anchor category classification challenge. The
experiments demonstrate that the lightweight version of UFATD achieves a speed
of 473 FPS while maintaining exceptionally high accuracy.

In the future, our emphasis will be directed towards addressing the following
challenges: exploring methodologies such as feature fusion to enhance the
model’s feature extraction capabilities and minimize the number of anchors.
Additionally, this work can also be applied to lane line detection, but more
research is needed.

## Acknowledgments and Disclosure of Funding

Funding in direct support of this work: Natural Science Foundation of Hunan
Province, China grant No.S2024JJMSXM1606, Hunan Provincial High tech Industry
Science and Technology Innovation Leading Plan Grant No.2020GK2096. And this
work was supported in part by the High Performance Computing Center of Central
South University.

## References

  * Catalano et al. [2017] Angelo Catalano, Francesco Antonio Bruno, Carlo Galliano, Marco Pisco, Giovanni Vito Persiano, Antonello Cutolo, and Andrea Cusano.  An optical fiber intrusion detection system for railway security.  _Sensors and Actuators A: Physical_ , 253:91–100, 2017.
  * Chen et al. [2023] Yiman Chen, Zhiyu Xiang, and Wentao Du.  Improving lane detection with adaptive homography prediction.  _The Visual Computer_ , 39(2):581–595, 2023.
  * Feng et al. [2022] Zhengyang Feng, Shaohua Guo, Xin Tan, Ke Xu, Min Wang, and Lizhuang Ma.  Rethinking efficient lane detection via curve modeling.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 17062–17070, 2022.
  * Giben et al. [2015] Xavier Giben, Vishal M Patel, and Rama Chellappa.  Material classification and semantic segmentation of railway track images with deep convolutional neural networks.  In _2015 IEEE International Conference on Image Processing (ICIP)_ , pages 621–625. IEEE, 2015.
  * He et al. [2016] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 770–778, 2016.
  * Iandola et al. [2016] Forrest N Iandola, Song Han, Matthew W Moskewicz, Khalid Ashraf, William J Dally, and Kurt Keutzer.  Squeezenet: Alexnet-level accuracy with 50x fewer parameters and< 0.5 mb model size.  _arXiv preprint arXiv:1602.07360_ , 2016.
  * Jaderberg et al. [2016] Max Jaderberg, Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman.  Reading text in the wild with convolutional neural networks.  _International journal of computer vision_ , 116:1–20, 2016.
  * Kawai and Kuroda [2023] Hibiki Kawai and Yoji Kuroda.  Dnn based camera attitude estimation using aggregated information from camera and depth images.  In _2023 IEEE/SICE International Symposium on System Integration (SII)_ , pages 1–6. IEEE, 2023.
  * Kawai et al. [2022] Hibiki Kawai, Wataru Yoshiuchi, Yasunori Hirakawa, Takumi Shibuya, Takumi Matsuda, and Yoji Kuroda.  Camera attitude estimation by neural network using classification network method instead of numerical regression.  In _2022 IEEE/SICE International Symposium on System Integration (SII)_ , pages 401–407. IEEE, 2022.
  * Laurent [2024] Thomas Laurent.  Train ego-path detection on railway tracks using end-to-end deep learning.  _arXiv preprint arXiv:2403.13094_ , 2024.
  * Li and Peng [2022] Xinpeng Li and Xiaojiang Peng.  Rail detection: An efficient row-based network and a new benchmark.  In _Proceedings of the 30th ACM International Conference on Multimedia_ , pages 6455–6463, 2022.
  * Liu et al. [2021] Lizhe Liu, Xiaohao Chen, Siyu Zhu, and Ping Tan.  Condlanenet: a top-to-down lane detection framework based on conditional convolution.  In _Proceedings of the IEEE/CVF international conference on computer vision_ , pages 3773–3782, 2021.
  * Neven et al. [2018] Davy Neven, Bert De Brabandere, Stamatios Georgoulis, Marc Proesmans, and Luc Van Gool.  Towards end-to-end lane detection: an instance segmentation approach.  In _2018 IEEE intelligent vehicles symposium (IV)_ , pages 286–291. IEEE, 2018.
  * Prewitt et al. [1970] Judith MS Prewitt et al.  Object enhancement and extraction.  _Picture processing and Psychopictorics_ , 10(1):15–19, 1970.
  * Qin et al. [2020] Zequn Qin, Huanyu Wang, and Xi Li.  Ultra fast structure-aware deep lane detection.  In _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XXIV 16_ , pages 276–291. Springer, 2020.
  * Qin et al. [2021] Zequn Qin, Pengyi Zhang, Fei Wu, and Xi Li.  Fcanet: Frequency channel attention networks.  In _Proceedings of the IEEE/CVF international conference on computer vision_ , pages 783–792, 2021.
  * Qin et al. [2022] Zequn Qin, Pengyi Zhang, and Xi Li.  Ultra fast deep lane detection with hybrid anchor driven ordinal classification.  _IEEE transactions on pattern analysis and machine intelligence_ , 2022.
  * Sandler et al. [2018] Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen.  Mobilenetv2: Inverted residuals and linear bottlenecks.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 4510–4520, 2018.
  * Sobel et al. [1968] Irwin Sobel, Gary Feldman, et al.  A 3x3 isotropic gradient operator for image processing.  _a talk at the Stanford Artificial Project in_ , pages 271–272, 1968.
  * Song et al. [2023] Yongchao Song, Tao Huang, Xin Fu, Yahong Jiang, Jindong Xu, Jindong Zhao, Weiqing Yan, and Xuan Wang.  A novel lane line detection algorithm for driverless geographic information perception using mixed-attention mechanism resnet and row anchor classification.  _ISPRS International Journal of Geo-Information_ , 12(3):132, 2023.
  * Tong et al. [2023a] Lei Tong, Limin Jia, Yixuan Geng, Keyan Liu, Yong Qin, and Zhipeng Wang.  Anchor-adaptive railway track detection from unmanned aerial vehicle images.  _Computer-Aided Civil and Infrastructure Engineering_ , 2023a.
  * Tong et al. [2023b] Lei Tong, Zhipeng Wang, Limin Jia, Yong Qin, Donghai Song, Bidong Miao, Tian Tang, and Yixuan Geng.  Trirnet: Real-time rail recognition network for uav-based railway inspection.  _IEEE Transactions on Intelligent Transportation Systems_ , 2023b.
  * UIC [2023] UIC.  Uic safety report 2023, 2023.  URL https://safetydb.uic.org/IMG/pdf/uic_safety_report_2023.pdf.  Date of visit: 2024-01-08.
  * Wang et al. [2022a] Bin Wang, Zhen Wang, Dou Zhao, and Xuhai Wang.  A rail detection algorithm for accurate recognition of train fuzzy video.  _Cyber-Physical Systems_ , 8(1):67–84, 2022a.
  * Wang et al. [2023] Jing Wang, Yu Ren, Zhijian Ye, and Zhao Tang.  Lre-net: A vision-based real-time rail extraction method under metro tunnel low-light scene.  _Measurement_ , 221:113567, 2023.
  * Wang et al. [2022b] Jinsheng Wang, Yinchao Ma, Shaofei Huang, Tianrui Hui, Fei Wang, Chen Qian, and Tianzhu Zhang.  A keypoint-based global association network for lane detection.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 1392–1401, 2022b.
  * Wang et al. [2019a] Yang Wang, Liqiang Zhu, Zujun Yu, and Baoqing Guo.  An adaptive track segmentation algorithm for a railway intrusion detection system.  _Sensors_ , 19(11), 2019a.  ISSN 1424-8220.  doi: 10.3390/s19112594.  URL https://www.mdpi.com/1424-8220/19/11/2594.
  * Wang et al. [2019b] Yin Wang, Lide Wang, Yu Hen Hu, and Ji Qiu.  Railnet: A segmentation network for railroad detection.  _IEEE Access_ , 7:143772–143779, 2019b.
  * Yu et al. [2024] Zichen Yu, Quanli Liu, Wei Wang, Liyong Zhang, and Xiaoguang Zhao.  Dalnet: A rail detection network based on dynamic anchor line.  _IEEE Transactions on Instrumentation and Measurement_ , 2024.
  * Zheng et al. [2022] Tu Zheng, Yifei Huang, Yang Liu, Wenjian Tang, Zheng Yang, Deng Cai, and Xiaofei He.  Clrnet: Cross layer refinement network for lane detection.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 898–907, 2022.
  * Zheng et al. [2021] Yun-Shui Zheng, Yan-Wei Jin, and Yu Dong.  Rail detection based on lsd and the least square curve fitting.  _International Journal of Automation and Computing_ , 18(1):85–95, 2021.
  * Zhong-li and Bai-gen [2017] WANG Zhong-li and CAI Bai-gen.  Geometry constraints-based method for visual rail track extraction.  _Journal of Transportation Systems Engineering and Information Technology_ , 17(6):56, 2017.


# Primordial black holes as a dark matter candidate - a brief overview

Anne M. Green

###### Abstract

Historically the most popular dark matter candidates have been new elementary
particles, such as Weakly Interacting Massive Particles and axions. However
Primordial Black Holes (PBHs), black holes formed from overdensities in the
early Universe, are another possibility. The discovery of gravitational waves
from mergers of tens of Solar mass black hole binaries by LIGO-Virgo has
generated a surge in interest in PBH dark matter. We overview the formation of
PBHs, observational probes of their abundance, and some of the key open
questions in the field.

###### keywords:

dark matter , primordial black hole

††journal: Nuclear Physics B

[first]organization=University of Nottingham,addressline=School of Physics and
Astronomy, city=Nottingham, postcode=NG7 2RD, state=, country=UK

##  1 Motivation and history

Primordial Black Holes (PBHs) fulfill all of the necessary requirements to be
a good dark matter (DM) candidate. They are cold, non-baryonic and stable, and
can be formed in the right abundance to be the DM. Zel’dovich and Novikov
(1967) and Hawking (1971) found that Primordial Black Holes may form from
overdensities in the early Universe. As they form before nucleosynthesis, PBHs
are non-baryonic. PBHs evaporate by emitting Hawking radiation (Hawking,
1975), however a PBH is cosmologically stable (its lifetime is longer than the
age of the Universe) if its initial mass is greater than $\sim 10^{15}\,{\rm
g}$ (Hawking, 1975; Page, 1976). Unlike most other DM candidates (for instance
Weakly Interacting Massive Particles, axions, sterile neutrinos,…) PBHs aren’t
a new particle. However, as we will see in Sec. 2, their formation does
typically require ‘Beyond the Standard Model’ physics, e.g. inflation.

It was realised that PBHs are a DM candidate already in the 1970s (Hawking,
1971; Chapline, 1975). A wave of interest in Solar mass PBH DM occurred in the
1990s, due to the excess of microlensing events observed in the MACHO
collaboration’s 2 year data set (Alcock et al., 1997). The number of events
was larger than expected from known stellar populations, and consistent with
roughly half of the Milky Way halo being in Solar mass compact objects.
Nakamura et al. (1997) pointed out that if a significant fraction of the DM is
in the form of $\sim$ Solar mass PBHs, then PBH binaries would form in the
early Universe and (if those binaries survive to the present day)
gravitational waves from their coalescence would be detectable by LIGO.

In 2016 LIGO-Virgo announced the discovery of gravitational waves from mergers
of tens of Solar mass black holes (BHs) (Abbott et al., 2016). Shortly
afterwards several papers appeared (Bird et al., 2016; Clesse and García-
Bellido, 2017; Sasaki et al., 2016), suggesting that (some of) these BHs could
be primordial, rather than astrophysical, and that these PBHs could also be a
significant component of the DM. This has led to another, much larger, wave of
interest in PBH DM, with the number of papers written on PBHs rising to $\sim
300$ per year.

In Sec. 2 we describe the formation of PBHs, focusing on the collapse of large
density perturbations during radiation domination. In Sec. 3 we overview
observational probes of the abundance of PBHs. Finally, in Sec. 4, we review
some of the key open questions regarding PBH dark matter, before concluding
with a summary in Sec. 5.

This brief overview is aimed at readers with knowledge of DM in general, but
not PBHs specifically. For longer, more detailed reviews of PBHs as a dark
matter candidate with extensive reference lists, see Carr and Kühnel (2020)
and Green and Kavanagh (2021). For a ‘positivist perspective’ on observational
evidence for PBHs see Carr et al. (2023), and for future observational
searches for PBHs see the Snowmass 2021 white paper (Bird et al., 2023).

##  2 Formation

We will focus on the most popular, and arguably minimal, PBH formation
mechanism, namely the collapse of large density perturbations during radiation
domination (Zel’dovich and Novikov, 1967; Hawking, 1971; Carr and Hawking,
1974). These are the same density perturbations that on large scales form
galaxies and large scale structure, but on shorter length scales and (if a
non-negligible number of PBHs are to form) with a much larger amplitude. If a
region is sufficiently overdense then when it enters the horizon (i.e. becomes
comparable in length scale to the observable Universe at that time), gravity
rapidly overcomes pressure forces and it collapses to form a BH. Other early
Universe processes which produce overdensities can also lead to PBH formation.
Examples include the collapse of cosmic string loops (Hawking, 1989; Polnarev
and Zembowicz, 1991), collisions of bubbles from phase transitions (Hawking et
al., 1982), fragmentation of an inflaton scalar condensate (Cotner and
Kusenko, 2017) and the collapse of density perturbations during an early
period of matter domination (Khlopov and Polnarev, 1980).

In Sec. 2.1 we review the essentials of the formation of PBHs from the
collapse of large density perturbations during radiation domination, including
the criterion for PBH formation and the PBH mass and abundance. In Sec. 2.2 we
overview the generation of large density perturbations by inflation. Finally
in Sec. 2.3 we discuss refinements to the calculations of the PBH abundance
and mass function.

###  2.1 Collapse of large density perturbations

The criterion for PBH formation during radiation domination, and the PBH mass
and abundance were calculated by Carr (1975). The criterion for PBH formation
is most easily specified in terms of the density contrast, at horizon
crossing, $\delta\equiv(\rho-\bar{\rho})/\bar{\rho}$, where $\rho$ is the
density of the region and ${\bar{\rho}}$ is the average background density. A
PBH will form if the density contrast exceeds a critical value, $\delta_{\rm
c}$, which is of order the equation of state parameter $w\equiv p/\rho=1/3$
111Here and throughout we use natural units with $c=1$.. The mass of the PBH
formed, $M_{\rm PBH}$, is roughly equal to the mass within the horizon,
$M_{\rm H}$, at that time, $t$,:

$M_{\rm PBH}\approx M_{\rm H}\sim 10^{15}\,{\rm
g}\,\left(\frac{t}{10^{-23}\,{\rm s}}\right)\sim
M_{\odot}\,\left(\frac{t}{10^{-6}\,{\rm s}}\right)\,.$ (1)

Figure 1: An illustration of the probability distribution, $P(\delta(M_{\rm
H}))$, of the density contrast, $\delta(M_{\rm H})$. PBHs form from rare large
fluctuations with $\delta(M_{\rm H})$ greater than the critical value,
$\delta_{\rm c}$, which is of order $1/3$.

The initial PBH mass fraction, $\beta$, is defined as the fraction of the
Universe in regions which are over-dense enough to form a PBH. It can be
calculated by integrating the probability distribution of the density contrast
on the scale of interest (specified in terms of the horizon mass, $M_{\rm
H}$), $P(\delta(M_{\rm H}))$, above the critical value $\delta_{\rm c}$:

$\beta(M_{\rm H})\sim\int_{\delta_{\rm c}}^{\infty}P(\delta(M_{\rm H}))\,{\rm
d}\delta(M_{\rm H})\,.$ (2)

Assuming (for now) that the probability distribution of the density contrast
is gaussian then

$\beta(M_{\rm H})\sim{\rm erfc}\left(\frac{\delta_{\rm
c}}{\sqrt{2}\sigma(M_{\rm H})}\right)\,,$ (3)

where erfc is the complementary error function and $\sigma(M_{\rm H})$, the
mass variance, is the typical size of the fluctuations. See Fig. 1 for a
visual illustration. As we will see shortly, $\beta$ has to be small and hence
$\sigma(M_{\rm H})\ll\delta_{\rm c}$ so that

$\beta(M_{\rm H})\sim\sigma(M_{\rm H})\exp{\left(-\frac{\delta_{\rm
c}^{2}}{2\sigma^{2}(M_{\rm H})}\right)}\,,$ (4)

and the PBH abundance depends exponentially on the typical size of the
fluctuations, $\sigma(M_{\rm H})$.

Next we need to relate the initial PBH mass fraction, $\beta$, to the fraction
of the DM today in the form of PBHs, $f_{\rm PBH}=\rho_{\rm PBH}/\rho_{\rm
DM}$, where $\rho_{\rm PBH}$ and $\rho_{\rm DM}$ are the PBH and DM energy
densities. Since PBHs are matter their energy density dilutes as $\rho_{\rm
PBH}\propto a^{-3}$, where $a$ is the scale factor that parameterises the
expansion of the Universe. The radiation energy density dilutes as $\rho_{\rm
rad}\propto a^{-4}$. Therefore during radiation domination the fraction of the
total energy density in the form of PBHs grows:

$\frac{\rho_{\rm PBH}}{\rho_{\rm rad}}\propto\frac{a^{-3}}{a^{-4}}\propto
a\,.$ (5)

Consequently, as shown in Fig. 2, to not exceed the observed DM abundance
today, the initial PBH abundance has to be very small, specifically

$\beta(M_{\rm H})\sim 10^{-9}f_{\rm PBH}\left(\frac{M}{M_{\rm PBH}}\right)\,.$
(6)

Lighter PBHs form earlier and therefore have to have a smaller initial
abundance, $\beta$, to make up a fixed fraction of the DM today, $f_{\rm
PBH}$.

Figure 2: An illustration of the evolution of the fraction of the total
density in the form of radiation (in red) and PBHs (blue) as a function of the
scale factor $a$. The dashed vertical line indicates the epoch of matter-
radiation equality, where the Universe transitions from radiation to matter
dominated.

On cosmological scales the amplitude of the primordial perturbations is
measured, by Planck observations of the Cosmic Microwave Background (CMB), to
be $\sigma\approx 10^{-5}$ (Akrami et al., 2020). If the primordial
perturbations are very close to scale-invariant (i.e. have a similar amplitude
on all scales) then the number of PBHs formed will be completely negligible.
Using Eqs. (3-4)

$\beta(M_{\rm H})\sim{\rm erfc}(10^{5})\sim\exp{(-10^{10})}\,.$ (7)

To form an interesting number of PBHs the primordial perturbations must be
significantly larger, $\sigma\sim 0.1$, on some small scale than they are on
cosmological scales. Figure 3 shows the constraints on the primordial power
spectrum of the curvature perturbation, ${\cal P}_{\cal R}(k)$, which is of
order the mass variance squared: ${\cal P}_{\cal R}(k)\sim\sigma^{2}(M_{\rm
H})$. On cosmological scales the amplitude is accurately measured and small.
On smaller scales there are upper limits from CMB spectral distortions (Carr
and Lidsey, 1993) and scalar induced gravitational waves (SIGWs), tensor
perturbations generated at second order by large scalar, i.e. density,
perturbations (Ananda et al., 2007)222Recently pulsar timing arrays, including
NANOGrav, have detected a background of low frequency gravitational waves
which could be explained by inspiralling supermassive BHs, or by via various
cosmological processes, including SIGWs from scalar perturbations with
amplitude ${\cal P}_{\rm R}(M_{\rm H}\sim 10^{-3}M_{\odot})~{}\sim 0.1$ (Afzal
et al., 2023).. Greater sensitivity may be obtained in future, from
gravitational wave searches by e.g. SKA, LISA and BBO (Inomata and Nakama,
2019; Chluba et al., 2021) or a PIXIE-like spectral distortion experiment
(Chluba et al., 2021).

Figure 3: Constraints on the primordial power spectrum of the curvature
perturbation, ${\cal P}_{\cal R}(k)$, which is of order the square of the mass
variance, $\sigma^{2}(M_{\rm H})$, as a function of both horizon mass, $M_{\rm
H}$, in Solar masses, $M_{\odot}$, (top horizontal axis) and comoving
wavenumber, $k$, in ${\rm Mpc}^{-1}$ (bottom horizontal axis). The constraints
shown are (from left to right) from the CMB temperature angular power spectrum
(Akrami et al., 2020) (in red), the Lyman-$\alpha$ forest (Bird et al., 2011)
(blue), CMB spectral distortions (Fixsen et al., 1996) (green) and pulsar
timing array limits on gravitational waves (Byrnes et al., 2019) (magenta).
The sensitivity of a future PIXIE-like spectral distortion experiment (Chluba
et al., 2021) (blue) and gravitational waves searches from SKA, LISA and BBO
(magenta) (Inomata and Nakama, 2019; Chluba et al., 2021) are shown as dotted
lines. In each case the currently excluded regions are shaded. The horizontal
black line shows the approximate amplitude, ${\cal P}_{\cal R}(k)\sim
10^{-2}$, required to form an interesting number of PBHs. The dotted black
line shows the steepest growth expected in single-field inflation models,
$\sim k^{4}$, (Byrnes et al., 2019). This figure was created using the
PBHbounds code (Kavanagh, ), similar figures appear in Byrnes et al. (2019);
Inomata and Nakama (2019); Chluba et al. (2021).

###  2.2 Producing large inflationary density perturbations

The primordial perturbations are thought to be produced by inflation, a period
of accelerated expansion in the early Universe, driven by a scalar field,
$\phi$, ‘slow-rolling’ along its potential. In slow-roll inflation the mass
variance is given by

$\sigma\propto\frac{V^{3/2}}{V^{\prime}}\,,$ (8)

where $V$ is the potential and $V^{\prime}$ is its derivative with respect to
$\phi$. It was pointed out by Ivanov et al. (1994) that a plateau in the
potential (with small $V^{\prime}$) can generate large perturbations, that
form an interesting number of PBHs. However in the ultra-slow-roll limit,
$V^{\prime}\rightarrow 0$, the slow-roll approximation breaks down, and a
numerical calculation is required to accurately calculate the amplitude of the
perturbations.

A successful PBH producing inflation model has to fulfill three requirements:

  1. 1.

match the measured amplitude and scale dependence of the perturbations on
cosmological scales,

  2. 2.

have significantly larger perturbations, $\sigma(M_{\rm H})\sim 0.1$, on some
smaller scale,

  3. 3.

inflation ends.

A single-field inflation model typically has to be fine-tuned to satisfy all
three of these requirements. An example is models with a very small local
minimum in the potential. In this case the field traverses this feature with
reduced speed, generating large perturbations, and then reaches a region where
the potential is steep and inflation ends (Ballesteros and Taoso, 2018;
Hertzberg and Yamada, 2018). The steepest growth in the power spectrum that
can be produced in single-field inflation models is of order $\sim k^{4}$
(Byrnes et al., 2019).

These three requirements can be more easily satisfied in multi-field inflation
models, where for instance a different field is responsible for generating the
perturbations than for ending or driving inflation. A commonly studied
possibility is hybrid inflation, where inflation ends due to a second field
undergoing a ‘waterfall’ phase transition and large perturbations can be
generated at the phase transition (Garcia-Bellido et al., 1996). If the
waterfall transition is ‘mild’ (occurring neither too quickly nor too slowly)
the power spectrum can be consistent with observations on cosmological scales
and also have a broad, PBH forming, peak on smaller scales (Clesse and García-
Bellido, 2015). Other possibilities include running mass inflation, double
inflation, an axion-like curvaton, a reduction in the sound speed and multi-
field models with rapid turns in field space. For detailed reviews see Özsoy
and Tasinato (2023) and Escrivà et al. (2022).

###  2.3 Refinements to calculation

To accurately compare theoretical predictions with observations (and reliably
ascertain whether or not PBHs make up a given fraction of the DM) accurate
calculations of the PBH abundance, and mass function, are required. We will
therefore conclude this section, by discussing refinements to the calculations
discussed in Sec. 2.1 above.

The threshold for PBH formation in fact depends on the shape of the
perturbation (Harada et al., 2013; Musco, 2019), which depends on the
primordial power spectrum (Germani and Musco, 2019), and is best specified in
terms of the compaction function (Musco, 2019; Germani and Musco, 2019;
Escrivà et al., 2020). For a detailed review see Escrivà et al. (2022).

At phase transitions the pressure is reduced and consequently the threshold
for PBH formation, $\delta_{\rm c}$, is reduced. Therefore the abundance of
PBHs formed, for a fixed mass variance, is increased. In particular the
horizon mass at the QCD phase transition is of order a Solar mass, leading to
enhanced formation of PBHs of this mass (Jedamzik, 1997). If the primordial
perturbations have a large constant amplitude, $\sigma(M_{\rm H})\sim 0.1$,
over a wide range of scales, then the formation of PBHs with $M_{\rm PBH}\sim
10^{-5},1$ and $10^{7}M_{\odot}$ will be enhanced due to the electroweak phase
transition, the QCD phase transition and $e^{+}e^{-}$ annihilation
respectively (Carr et al., 2023).

Niemeyer and Jedamzik (1998) pointed out that, due to critical phenomena in
gravitational collapse, the mass of a PBH depends on the size of the
fluctuation from which it forms: $M_{\rm PBH}=kM_{\rm H}(\delta-\delta_{\rm
c})^{\gamma}$ where $k$ and $\gamma$ are constants. This relationship has been
verified using numerical simulations, e.g. Musco et al. (2009). Consequently
even if PBHs all form at the same time, there is a spread in their masses i.e.
the PBH mass function is not expected to be a delta function.

It was pointed out in the 1990s that since PBHs form from rare large
fluctuations, their abundance depends sensitively on the shape of the tail of
the density contrast probability distribution (Bullock and Primack, 1997;
Ivanov, 1998). More recently it has been realised that even if the curvature
perturbations are gaussian, the distribution of large density perturbations
won’t be, because the relationship between curvature perturbations and density
perturbation is non-linear (Franciolini et al., 2018; De Luca et al., 2019;
Young et al., 2019). Furthermore, in ultra-slow-roll inflation (which, as
discussed above, is required in single-field models to produce large
perturbations) the probability distribution of the curvature perturbations may
not be gaussian. See Sec. 4.2 for further discussion.

##  3 Observational constraints/signatures

In this Section we focus on observational constraints on the present day
abundance of PBHs. In Sec. 3.1 we briefly review current constraints,
calculated assuming that the PBHs have a delta-function mass function (DF MF).
In Sec. 3.2 we overview potential improvements to these constraints in future,
and in Sec. 3.3 we discuss constraints on more realistic extended mass
functions. We focus here on constraints on the fraction of the DM in the form
of PBHs. Carr et al. (2023) discusses a wide range of observations which could
potentially be explained by PBHs (with $f_{\rm PBH}\leq 1)$.

###  3.1 Current constraints

Figure 4 shows the current constraints333Boehm et al. (2021) argue that PBHs
in the early Universe shouldn’t be described by the Schwarzschild metric, and
point out that with an alternative, Thakurta, metric the mass of a PBH grows
and hence the evaporation and gravitational wave constraints change
significantly. on the fraction of DM in the form of PBHs, $f_{\rm
PBH}=\rho_{\rm PBH}/\rho_{\rm DM}$, as a function of PBH mass, $M_{\rm PBH}$,
assuming all PBHs have the same mass, i.e. that the PBH mass function is a
delta function. PBHs with initial mass $10^{15}\,{\rm g}\lesssim M_{\rm
PBH}\lesssim 10^{17}\,{\rm g}$ are evaporating at a significant rate and their
abundance is constrained by limits on the products of their evaporation, e.g.
MeV gamma-rays. Planetary and Solar mass PBHs are constrained by various
microlensing observations. If all of the DM were in the form of multi-Solar
mass PBHs there would in fact be several orders of magnitude more merger
events than observed by LIGO-Virgo, so these observations constrain $f_{\rm
PBH}$. Multi-Solar mass PBHs can also be constrained by the consequences of
their accretion, specifically the effect of the subsequent radiation on the
recombination history of the Universe and hence the CMB, and also present day
X-ray or radio emission. Massive PBHs are constrained via their dynamical
effects on stars, in wide binaries and dwarf galaxies. For more detailed
discussion of these constraints, including references to the original papers,
see Carr and Kühnel (2020) and Green and Kavanagh (2021).

Figure 4: Constraints on the fraction of DM in the form of PBHs, $f_{\rm
PBHs}$, as a function of mass, $M_{\rm PBH}$, assuming all PBHs have the same
mass. The bounds shown are (from left to right) from evaporation (in orange),
microlensing (blue), gravitational waves (purple), accretion (red) and
dynamical (green). For each type of bound the tightest constraint at each mass
is shown and the shaded regions are excluded. Figure created using the
PBHbounds code (Kavanagh, ).

As we see in Fig. 4 it currently appears that planetary, Solar or multi-Solar
mass PBHs making up all of the DM is excluded. There are uncertainties in
these constraints, and some of the constraints have been revisited in recent
years and significantly revised. However it seems unlikely (to the present
author at least) that there are errors or uncertainties in multiple
constraints such that $f_{\rm PBH}=1$ in this mass range is in fact allowed.
There is a range of masses, $10^{17}\,{\rm g}\lesssim M_{\rm PBH}\lesssim
10^{22}\,{\rm g}$, usually described as the ‘asteroid mass window’, where PBHs
can make up all of the DM, i.e. $f_{\rm PBH}=1$ is allowed. These light PBHs
are hard to detect and we will discuss ideas and prospects in Sec. 4.1.

###  3.2 Future constraints

The Snowmass 2021 Cosmic Frontier White Paper: Primordial Black Hole dark
matter (Bird et al., 2023) discusses potential improvements to current
constraints. Future MeV gamma-ray telescopes will tighten the evaporation
constraints (see e.g. Coogan et al. (2023) and Fig. 5). However the rate at
which PBHs evaporate decreases rapidly with increasing mass, ${\rm d}M_{\rm
PBH}/{\rm d}t\sim M_{\rm PBH}^{-2}$ (Hawking, 1975), so the mass for which
$f_{\rm PBH}=1$ is excluded (for a DF MF) by evaporation constraints will only
increase by a factor of a few. Similarly it is hard to decrease the smallest
mass for which $f_{\rm PBH}=1$ is excluded (for a DF MF) by microlensing
observations. For $M_{\rm PBH}\lesssim 10^{-12}M_{\odot}$ the microlensing
amplification is reduced due to finite source effects and wave optics (since
the wavelength of light is similar to the Schwarzschild radius of the PBH),
see Sugiyama et al. (2020) and references therein. A future microlensing
survey of white dwarfs in the Large Magellanic Cloud could place much tighter
constraints on PBHs with $M_{\rm PBH}\sim 10^{22}\,{\rm g}\sim
10^{-12}M_{\odot}$, however wave optics effects make it impossible to exclude
$f_{\rm PBH}=1$ (for a DF MF) for masses significantly smaller than this with
optical microlensing observations (Sugiyama et al., 2020). New techniques will
therefore be needed to probe the whole of the current asteroid mass window,
see Sec. 4.1.

###  3.3 Constraints on realistic extended mass functions

The constraints displayed in Fig. 4 are all calculated assuming a DF MF. As
discussed in Sec. 2.3 we don’t expect PBHs formed from the collapse of large
density perturbations to have a DF MF. Even if all PBHs form at the same time,
from a sharp peak in the primordial power spectrum, critical collapse means
that they will have a range of masses. For extended mass functions the
constraints are ‘smeared out’. For any given constraint the tightest value of
the constraint (as a function of the central or peak mass) becomes weaker,
however the constraint now applies (i.e. $f_{\rm PBH}<1$) over a wider range
of peak masses (Carr et al., 2017). Therefore PBHs with realistic extended
mass functions are more tightly constrained, in that $f_{\rm PBH}=1$ is
excluded over a wider range of masses (Green, 2016; Carr et al., 2017).

##  4 Open questions

In this section we discuss three of the key open questions in the field of PBH
dark matter, namely how to probe asteroid mass PBHs observationally (Sec.
4.1), the probability distribution of density perturbations produced by ultra-
slow-roll inflation (Sec. 4.2) and the clustering of PBHs on subgalactic
scales (Sec. 4.3).

###  4.1 How to probe asteroid mass PBHs?

As discussed in Sec. 3.2, future observations will increase the range of PBH
masses probed by the evaporation and microlensing constraints somewhat.
However it will not be possible to probe all of the asteroid mass window,
$10^{17}\,{\rm g}\lesssim M_{\rm PBH}\lesssim 10^{22}\,{\rm g}$, using these
techniques. To do this other methods will be required.

Lensing observations using X-rays and gamma-rays can avoid the wave optics
limitations which apply to optical microlensing observations. One possibility
is femtolensing of gamma ray bursts (GRBs) (Gould, 1992), however in this case
observations of small GRBs are required to limit the reduction in
magnification from finite source size effects (Katz et al., 2018). Other
possibilities include GRB lensing parallax (Nemiroff and Gould, 1995; Jung and
Kim, 2020), microlensing of X-ray pulsars (Bai and Orlofsky, 2019) and the
effects of PBH encounters with stars (see e.g. Montero-Camacho et al. (2019)
and references therein and thereof). Figure 5 shows the current and projected
future constraints for asteroid mass PBHs.

Figure 5: Current and projected future (solid and dotted lines respectively)
constraints on the fraction of DM in the form of asteroid mass PBHs, $f_{\rm
PBHs}$, as a function of mass, $M_{\rm PBH}$, assuming all PBHs have the same
mass. The bounds shown are from evaporation (in orange), optical microlensing
(blue) and other methods, involving lensing of X-ray pulsars (Bai and
Orlofsky, 2019) and gamma ray bursts (Katz et al., 2018; Jung and Kim, 2020),
(yellow). The future evaporation and microlensing constraints are from Coogan
et al. (2023) and Sugiyama et al. (2020) respectively. See text for further
discussion. Figure created using the PBHbounds code (Kavanagh, ).

###  4.2 What is the probability distribution of density perturbations
produced by ultra-slow-roll inflation?

As discussed in Sec. 2.2, in single-field inflation models to generate large,
PBH forming density perturbations a period of ultra-slow-roll inflation
(driven, for instance, by the slope of the potential becoming small) is
required. In this case stochastic effects are important for the evolution of
the scalar field, and the tail of the probability distribution of the size of
the fluctuations could be gaussian rather than exponential (see e.g. Pattison
et al. (2021); Figueroa et al. (2021); Tada and Vennin (2022) and references
therein and thereof). This would have a significant effect on the abundance of
PBHs formed and is the subject of extensive ongoing work.

On a different, but related, note there is currently ongoing discussion about
whether large amplitude small scale perturbations lead to significant one-loop
corrections to perturbations on CMB scales, and are hence excluded. See e.g.
Kristiano and Yokoyama (2022); Firouzjahi and Riotto (2023); Fumagalli (2023);
Firouzjahi (2023).

###  4.3 Clustering (on subgalactic scales)

On galactic and larger scales PBH DM would be distributed in the same way as
any other viable cold dark matter candidate. However their macroscopic nature
may lead to enhanced clustering on subgalactic scales, which could have
important consequences for the PBH binary merger rate (and potentially other
constraints). As PBHs are discrete objects there are Poisson fluctuations in
their distribution, and the primordial perturbations therefore have an
additional isocurvature component (Afshordi et al., 2003; Inman and Ali-
Haïmoud, 2019). Consequently if PBHs make up a significant fraction of the DM,
PBH clusters start forming shortly after matter-radiation equality (Afshordi
et al., 2003; Inman and Ali-Haïmoud, 2019). The evolution of these PBHs
clusters (and the PBH binaries within them) to the present day is a
challenging open problem (Jedamzik, 2020; Trashorras et al., 2021).

If the DM is a mixture of PBHs and particle DM then the PBHs will accrete a
halo of particle DM during matter domination, see e.g. Mack et al. (2007);
Adamek et al. (2019). Consequently, PBHs and Weakly Interacting Massive
Particles (WIMPs) both making up a significant fraction of the DM is
essentially already excluded, as WIMP annihilation in the halos surrounding
PBHs would produce a very large flux of gamma-rays (Lacki and Beacom, 2010).

##  5 Summary

Primordial Black Holes can form in the early Universe, for instance from the
collapse of large density perturbations during radiation domination. To
produce an interesting number of PBHs, (i.e. for them to make a non-negligible
contribution to the present day energy density of the Universe) the amplitude
of the primordial perturbations must be several orders of magnitude larger on
some small scale than it is on cosmological scales. This can be achieved in
some inflation models, e.g. with a feature in the potential or multiple
fields, however it is not generic.

There are numerous observational constraints on the abundance of PBHs from
gravitational lensing, their evaporation, dynamical effects, accretion and
other astrophysical processes. There have been significant developments in
these constraints in recent years. New constraints have been proposed, and
some old constraints have been found to be weaker than originally thought, or
even shown to not be valid. Currently it appears that planetary, Solar and
multi-Solar mass PBHs can’t make up all of the dark matter, but lighter,
asteroid mass, $M_{\rm PBH}\sim(10^{17}-10^{22})\,{\rm g}$, PBHs could.
Collectively the limits are tighter (i.e. $f_{\rm PBH}=1$ is allowed for a
narrower range of masses) for realistic extended mass functions, than for the
delta-function mass function which is usually assumed when calculating
constraints.

There are various important open problems in the field of PBH dark matter.
These include observational probes of asteroid mass PBHs, the probability
distribution of the density perturbations in ultra-slow-roll inflation models
which generate large perturbations, and the present day clustering of PBHs on
subgalactic scales and its effect on observational constraints/signals.

## Acknowledgements

AMG is supported by a STFC Consolidated Grant [Grant No. ST/T000732/1]. For
the purpose of open access, the author has applied a CC BY public copyright
licence to any Author Accepted Manuscript version arising.

Data Availability Statement: This review paper has no associated data.

## References

  * Abbott et al. (2016) Abbott, B.P., et al. (LIGO Scientific, Virgo), 2016.  Observation of Gravitational Waves from a Binary Black Hole Merger.  Phys. Rev. Lett. 116, 061102\.  doi:10.1103/PhysRevLett.116.061102, arXiv:1602.03837.
  * Adamek et al. (2019) Adamek, J., Byrnes, C.T., Gosenca, M., Hotchkiss, S., 2019\.  WIMPs and stellar-mass primordial black holes are incompatible.  Phys. Rev. D 100, 023506\.  doi:10.1103/PhysRevD.100.023506, arXiv:1901.08528.
  * Afshordi et al. (2003) Afshordi, N., McDonald, P., Spergel, D.N., 2003.  Primordial black holes as dark matter: The Power spectrum and evaporation of early structures.  Astrophys. J. Lett. 594, L71–L74.  doi:10.1086/378763, arXiv:astro-ph/0302035.
  * Afzal et al. (2023) Afzal, A., et al. (NANOGrav), 2023\.  The NANOGrav 15 yr Data Set: Search for Signals from New Physics.  Astrophys. J. Lett. 951, L11.  doi:10.3847/2041-8213/acdc91, arXiv:2306.16219.
  * Akrami et al. (2020) Akrami, Y., et al. (Planck), 2020\.  Planck 2018 results. X. Constraints on inflation.  Astron. Astrophys. 641, A10.  doi:10.1051/0004-6361/201833887, arXiv:1807.06211.
  * Alcock et al. (1997) Alcock, C., et al. (MACHO), 1997\.  The MACHO project LMC microlensing results from the first two years and the nature of the galactic dark halo.  Astrophys. J. 486, 697–726.  doi:10.1086/304535, arXiv:astro-ph/9606165.
  * Ananda et al. (2007) Ananda, K.N., Clarkson, C., Wands, D., 2007.  The Cosmological gravitational wave background from primordial density perturbations.  Phys. Rev. D 75, 123518\.  doi:10.1103/PhysRevD.75.123518, arXiv:gr-qc/0612013.
  * Bai and Orlofsky (2019) Bai, Y., Orlofsky, N., 2019\.  Microlensing of X-ray Pulsars: a Method to Detect Primordial Black Hole Dark Matter.  Phys. Rev. D 99, 123019\.  doi:10.1103/PhysRevD.99.123019, arXiv:1812.01427.
  * Ballesteros and Taoso (2018) Ballesteros, G., Taoso, M., 2018\.  Primordial black hole dark matter from single field inflation.  Phys. Rev. D 97, 023501\.  doi:10.1103/PhysRevD.97.023501, arXiv:1709.05565.
  * Bird et al. (2016) Bird, S., Cholis, I., Muñoz, J.B., Ali-Haïmoud, Y., Kamionkowski, M., Kovetz, E.D., Raccanelli, A., Riess, A.G., 2016\.  Did LIGO detect dark matter?  Phys. Rev. Lett. 116, 201301\.  doi:10.1103/PhysRevLett.116.201301, arXiv:1603.00464.
  * Bird et al. (2011) Bird, S., Peiris, H.V., Viel, M., Verde, L., 2011\.  Minimally Parametric Power Spectrum Reconstruction from the Lyman-alpha Forest.  Mon. Not. Roy. Astron. Soc. 413, 1717–1728.  doi:10.1111/j.1365-2966.2011.18245.x, arXiv:1010.1519.
  * Bird et al. (2023) Bird, S., et al., 2023.  Snowmass2021 Cosmic Frontier White Paper: Primordial black hole dark matter.  Phys. Dark Univ. 41, 101231\.  doi:10.1016/j.dark.2023.101231, arXiv:2203.08967.
  * Boehm et al. (2021) Boehm, C., Kobakhidze, A., O’Hare, C.A.J., Picker, Z.S.C., Sakellariadou, M., 2021.  Eliminating the LIGO bounds on primordial black hole dark matter.  JCAP 03, 078.  doi:10.1088/1475-7516/2021/03/078, arXiv:2008.10743.
  * Bullock and Primack (1997) Bullock, J.S., Primack, J.R., 1997\.  NonGaussian fluctuations and primordial black holes from inflation.  Phys. Rev. D 55, 7423–7439.  doi:10.1103/PhysRevD.55.7423, arXiv:astro-ph/9611106.
  * Byrnes et al. (2019) Byrnes, C.T., Cole, P.S., Patil, S.P., 2019.  Steepest growth of the power spectrum and primordial black holes.  JCAP 06, 028.  doi:10.1088/1475-7516/2019/06/028, arXiv:1811.11158.
  * Carr et al. (2023) Carr, B., Clesse, S., Garcia-Bellido, J., Hawkins, M., Kühnel, F., 2023.  Observational Evidence for Primordial Black Holes: A Positivist Perspective arXiv:2306.03903.
  * Carr and Kühnel (2020) Carr, B., Kühnel, F., 2020\.  Primordial Black Holes as Dark Matter: Recent Developments.  Ann. Rev. Nucl. Part. Sci. 70, 355–394.  doi:10.1146/annurev-nucl-050520-125911, arXiv:2006.02838.
  * Carr et al. (2017) Carr, B., Raidal, M., Tenkanen, T., Vaskonen, V., Veermäe, H., 2017.  Primordial black hole constraints for extended mass functions.  Phys. Rev. D 96, 023514\.  doi:10.1103/PhysRevD.96.023514, arXiv:1705.05567.
  * Carr (1975) Carr, B.J., 1975.  The primordial black hole mass spectrum.  Astrophys. J. 201, 1–19.  doi:10.1086/153853.
  * Carr and Hawking (1974) Carr, B.J., Hawking, S.W., 1974\.  Black holes in the early Universe.  Mon. Not. Roy. Astron. Soc. 168, 399–415.  doi:10.1093/mnras/168.2.399.
  * Carr and Lidsey (1993) Carr, B.J., Lidsey, J.E., 1993\.  Primordial black holes and generalized constraints on chaotic inflation.  Phys. Rev. D 48, 543–553.  doi:10.1103/PhysRevD.48.543.
  * Chapline (1975) Chapline, G.F., 1975.  Cosmological effects of primordial black holes.  Nature 253, 251–252.  doi:10.1038/253251a0.
  * Chluba et al. (2021) Chluba, J., et al., 2021.  New horizons in cosmology with spectral distortions of the cosmic microwave background.  Exper. Astron. 51, 1515–1554.  doi:10.1007/s10686-021-09729-5, arXiv:1909.01593.
  * Clesse and García-Bellido (2015) Clesse, S., García-Bellido, J., 2015\.  Massive Primordial Black Holes from Hybrid Inflation as Dark Matter and the seeds of Galaxies.  Phys. Rev. D 92, 023524\.  doi:10.1103/PhysRevD.92.023524, arXiv:1501.07565.
  * Clesse and García-Bellido (2017) Clesse, S., García-Bellido, J., 2017\.  The clustering of massive Primordial Black Holes as Dark Matter: measuring their mass distribution with Advanced LIGO.  Phys. Dark Univ. 15, 142–147.  doi:10.1016/j.dark.2016.10.002, arXiv:1603.05234.
  * Coogan et al. (2023) Coogan, A., et al., 2023.  Hunting for dark matter and new physics with GECCO.  Phys. Rev. D 107, 023022\.  doi:10.1103/PhysRevD.107.023022, arXiv:2101.10370.
  * Cotner and Kusenko (2017) Cotner, E., Kusenko, A., 2017\.  Primordial black holes from supersymmetry in the early universe.  Phys. Rev. Lett. 119, 031103\.  doi:10.1103/PhysRevLett.119.031103, arXiv:1612.02529.
  * De Luca et al. (2019) De Luca, V., Franciolini, G., Kehagias, A., Peloso, M., Riotto, A., Ünal, C., 2019\.  The Ineludible non-Gaussianity of the Primordial Black Hole Abundance.  JCAP 07, 048.  doi:10.1088/1475-7516/2019/07/048, arXiv:1904.00970.
  * Escrivà et al. (2020) Escrivà, A., Germani, C., Sheth, R.K., 2020.  Universal threshold for primordial black hole formation.  Phys. Rev. D 101, 044022\.  doi:10.1103/PhysRevD.101.044022, arXiv:1907.13311.
  * Escrivà et al. (2022) Escrivà, A., Kühnel, F., Tada, Y., 2022.  Primordial Black Holes arXiv:2211.05767.
  * Figueroa et al. (2021) Figueroa, D.G., Raatikainen, S., Rasanen, S., Tomberg, E., 2021\.  Non-Gaussian Tail of the Curvature Perturbation in Stochastic Ultraslow-Roll Inflation: Implications for Primordial Black Hole Production.  Phys. Rev. Lett. 127, 101302\.  doi:10.1103/PhysRevLett.127.101302, arXiv:2012.06551.
  * Firouzjahi (2023) Firouzjahi, H., 2023.  Revisiting Loop Corrections in Single Field USR Inflation arXiv:2311.04080.
  * Firouzjahi and Riotto (2023) Firouzjahi, H., Riotto, A., 2023\.  Primordial Black Holes and Loops in Single-Field Inflation arXiv:2304.07801.
  * Fixsen et al. (1996) Fixsen, D.J., Cheng, E.S., Gales, J.M., Mather, J.C., Shafer, R.A., Wright, E.L., 1996\.  The Cosmic Microwave Background spectrum from the full COBE FIRAS data set.  Astrophys. J. 473, 576\.  doi:10.1086/178173, arXiv:astro-ph/9605054.
  * Franciolini et al. (2018) Franciolini, G., Kehagias, A., Matarrese, S., Riotto, A., 2018\.  Primordial Black Holes from Inflation and non-Gaussianity.  JCAP 03, 016.  doi:10.1088/1475-7516/2018/03/016, arXiv:1801.09415.
  * Fumagalli (2023) Fumagalli, J., 2023.  Absence of one-loop effects on large scales from small scales in non-slow-roll dynamics arXiv:2305.19263.
  * Garcia-Bellido et al. (1996) Garcia-Bellido, J., Linde, A.D., Wands, D., 1996.  Density perturbations and black hole formation in hybrid inflation.  Phys. Rev. D 54, 6040–6058.  doi:10.1103/PhysRevD.54.6040, arXiv:astro-ph/9605094.
  * Germani and Musco (2019) Germani, C., Musco, I., 2019\.  Abundance of Primordial Black Holes Depends on the Shape of the Inflationary Power Spectrum.  Phys. Rev. Lett. 122, 141302\.  doi:10.1103/PhysRevLett.122.141302, arXiv:1805.04087.
  * Gould (1992) Gould, A., 1992.  Femtolensing of Gamma-Ray Bursters.  Astrophys. J. Lett. 386, L5.  doi:10.1086/186279.
  * Green (2016) Green, A.M., 2016.  Microlensing and dynamical constraints on primordial black hole dark matter with an extended mass function.  Phys. Rev. D 94, 063530\.  doi:10.1103/PhysRevD.94.063530, arXiv:1609.01143.
  * Green and Kavanagh (2021) Green, A.M., Kavanagh, B.J., 2021\.  Primordial Black Holes as a dark matter candidate.  J. Phys. G 48, 043001\.  doi:10.1088/1361-6471/abc534, arXiv:2007.10722.
  * Harada et al. (2013) Harada, T., Yoo, C.M., Kohri, K., 2013.  Threshold of primordial black hole formation.  Phys. Rev. D 88, 084051\.  doi:10.1103/PhysRevD.88.084051, arXiv:1309.4201. [Erratum: Phys.Rev.D 89, 029903 (2014)].
  * Hawking (1971) Hawking, S., 1971.  Gravitationally collapsed objects of very low mass.  Mon. Not. Roy. Astron. Soc. 152, 75\.  doi:10.1093/mnras/152.1.75.
  * Hawking (1975) Hawking, S.W., 1975.  Particle Creation by Black Holes.  Commun. Math. Phys. 43, 199–220.  doi:10.1007/BF02345020. [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * Hawking (1989) Hawking, S.W., 1989.  Black Holes From Cosmic Strings.  Phys. Lett. B 231, 237–239.  doi:10.1016/0370-2693(89)90206-2.
  * Hawking et al. (1982) Hawking, S.W., Moss, I.G., Stewart, J.M., 1982.  Bubble Collisions in the Very Early Universe.  Phys. Rev. D 26, 2681\.  doi:10.1103/PhysRevD.26.2681.
  * Hertzberg and Yamada (2018) Hertzberg, M.P., Yamada, M., 2018\.  Primordial Black Holes from Polynomial Potentials in Single Field Inflation.  Phys. Rev. D 97, 083509\.  doi:10.1103/PhysRevD.97.083509, arXiv:1712.09750.
  * Inman and Ali-Haïmoud (2019) Inman, D., Ali-Haïmoud, Y., 2019\.  Early structure formation in primordial black hole cosmologies.  Phys. Rev. D 100, 083528\.  doi:10.1103/PhysRevD.100.083528, arXiv:1907.08129.
  * Inomata and Nakama (2019) Inomata, K., Nakama, T., 2019\.  Gravitational waves induced by scalar perturbations as probes of the small-scale primordial spectrum.  Phys. Rev. D 99, 043511\.  doi:10.1103/PhysRevD.99.043511, arXiv:1812.00674.
  * Ivanov (1998) Ivanov, P., 1998.  Nonlinear metric perturbations and production of primordial black holes.  Phys. Rev. D 57, 7145–7154.  doi:10.1103/PhysRevD.57.7145, arXiv:astro-ph/9708224.
  * Ivanov et al. (1994) Ivanov, P., Naselsky, P., Novikov, I., 1994.  Inflation and primordial black holes as dark matter.  Phys. Rev. D 50, 7173–7178.  doi:10.1103/PhysRevD.50.7173.
  * Jedamzik (1997) Jedamzik, K., 1997.  Primordial black hole formation during the QCD epoch.  Phys. Rev. D 55, 5871–5875.  doi:10.1103/PhysRevD.55.R5871, arXiv:astro-ph/9605152.
  * Jedamzik (2020) Jedamzik, K., 2020.  Primordial Black Hole Dark Matter and the LIGO/Virgo observations.  JCAP 09, 022.  doi:10.1088/1475-7516/2020/09/022, arXiv:2006.11172.
  * Jung and Kim (2020) Jung, S., Kim, T., 2020.  Gamma-ray burst lensing parallax: Closing the primordial black hole dark matter mass window.  Phys. Rev. Res. 2, 013113\.  doi:10.1103/PhysRevResearch.2.013113, arXiv:1908.00078.
  * Katz et al. (2018) Katz, A., Kopp, J., Sibiryakov, S., Xue, W., 2018\.  Femtolensing by Dark Matter Revisited.  JCAP 12, 005.  doi:10.1088/1475-7516/2018/12/005, arXiv:1807.11495.
  * (56) Kavanagh, B.J., .  PBHbounds.  doi:10.5281/zenodo.3538998.
  * Khlopov and Polnarev (1980) Khlopov, M.Y., Polnarev, A.G., 1980\.  Primordial black holes as a cosmological test of grand unification.  Phys. Lett. B 97, 383–387.  doi:10.1016/0370-2693(80)90624-3.
  * Kristiano and Yokoyama (2022) Kristiano, J., Yokoyama, J., 2022\.  Ruling Out Primordial Black Hole Formation From Single-Field Inflation arXiv:2211.03395.
  * Lacki and Beacom (2010) Lacki, B.C., Beacom, J.F., 2010\.  Primordial Black Holes as Dark Matter: Almost All or Almost Nothing.  Astrophys. J. Lett. 720, L67–L71.  doi:10.1088/2041-8205/720/1/L67, arXiv:1003.3466.
  * Mack et al. (2007) Mack, K.J., Ostriker, J.P., Ricotti, M., 2007.  Growth of structure seeded by primordial black holes.  Astrophys. J. 665, 1277–1287.  doi:10.1086/518998, arXiv:astro-ph/0608642.
  * Montero-Camacho et al. (2019) Montero-Camacho, P., Fang, X., Vasquez, G., Silva, M., Hirata, C.M., 2019.  Revisiting constraints on asteroid-mass primordial black holes as dark matter candidates.  JCAP 08, 031.  doi:10.1088/1475-7516/2019/08/031, arXiv:1906.05950.
  * Musco (2019) Musco, I., 2019.  Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations.  Phys. Rev. D 100, 123524\.  doi:10.1103/PhysRevD.100.123524, arXiv:1809.02127.
  * Musco et al. (2009) Musco, I., Miller, J.C., Polnarev, A.G., 2009.  Primordial black hole formation in the radiative era: Investigation of the critical nature of the collapse.  Class. Quant. Grav. 26, 235001\.  doi:10.1088/0264-9381/26/23/235001, arXiv:0811.1452.
  * Nakamura et al. (1997) Nakamura, T., Sasaki, M., Tanaka, T., Thorne, K.S., 1997\.  Gravitational waves from coalescing black hole MACHO binaries.  Astrophys. J. Lett. 487, L139–L142.  doi:10.1086/310886, arXiv:astro-ph/9708060.
  * Nemiroff and Gould (1995) Nemiroff, R.J., Gould, A., 1995\.  Probing for MACHOs of mass 10(**-15)-solar-mass - 10**-7-solar-mass with gamma-ray burst parallax spacecraft.  Astrophys. J. Lett. 452, L111.  doi:10.1086/309722, arXiv:astro-ph/9505019.
  * Niemeyer and Jedamzik (1998) Niemeyer, J.C., Jedamzik, K., 1998\.  Near-critical gravitational collapse and the initial mass function of primordial black holes.  Phys. Rev. Lett. 80, 5481–5484.  doi:10.1103/PhysRevLett.80.5481, arXiv:astro-ph/9709072.
  * Özsoy and Tasinato (2023) Özsoy, O., Tasinato, G., 2023\.  Inflation and Primordial Black Holes.  Universe 9, 203\.  doi:10.3390/universe9050203, arXiv:2301.03600.
  * Page (1976) Page, D.N., 1976.  Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole.  Phys. Rev. D 13, 198–206.  doi:10.1103/PhysRevD.13.198.
  * Pattison et al. (2021) Pattison, C., Vennin, V., Wands, D., Assadullahi, H., 2021\.  Ultra-slow-roll inflation with quantum diffusion.  JCAP 04, 080.  doi:10.1088/1475-7516/2021/04/080, arXiv:2101.05741.
  * Polnarev and Zembowicz (1991) Polnarev, A., Zembowicz, R., 1991\.  Formation of Primordial Black Holes by Cosmic Strings.  Phys. Rev. D 43, 1106–1109.  doi:10.1103/PhysRevD.43.1106.
  * Sasaki et al. (2016) Sasaki, M., Suyama, T., Tanaka, T., Yokoyama, S., 2016\.  Primordial Black Hole Scenario for the Gravitational-Wave Event GW150914.  Phys. Rev. Lett. 117, 061101\.  doi:10.1103/PhysRevLett.117.061101, arXiv:1603.08338. [Erratum: Phys.Rev.Lett. 121, 059901 (2018)].
  * Sugiyama et al. (2020) Sugiyama, S., Kurita, T., Takada, M., 2020.  On the wave optics effect on primordial black hole constraints from optical microlensing search.  Mon. Not. Roy. Astron. Soc. 493, 3632–3641.  doi:10.1093/mnras/staa407, arXiv:1905.06066.
  * Tada and Vennin (2022) Tada, Y., Vennin, V., 2022.  Statistics of coarse-grained cosmological fields in stochastic inflation.  JCAP 02, 021.  doi:10.1088/1475-7516/2022/02/021, arXiv:2111.15280.
  * Trashorras et al. (2021) Trashorras, M., García-Bellido, J., Nesseris, S., 2021.  The clustering dynamics of primordial black boles in $N$-body simulations.  Universe 7, 18.  doi:10.3390/universe7010018, arXiv:2006.15018.
  * Young et al. (2019) Young, S., Musco, I., Byrnes, C.T., 2019.  Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation.  JCAP 11, 012.  doi:10.1088/1475-7516/2019/11/012, arXiv:1904.00984.
  * Zel’dovich and Novikov (1967) Zel’dovich, Y.B., Novikov, I.D., 1967\.  The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model.  Sov. Astron. 10, 602\.