# Exploring the Feasibility of ChatGPT for Event Extraction

Jun Gao1 Huan Zhao2 Changlong Yu3 Ruifeng Xu1
1Harbin Institute of Technology (Shenzhen)
24Paradigm. Inc. 3HKUST, Hong Kong, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Work done when Jun
Gao was interning at 4Paradigm.

###### Abstract

Event extraction is a fundamental task in natural language processing that
involves identifying and extracting information about events mentioned in
text. However, it is a challenging task due to the lack of annotated data,
which is expensive and time-consuming to obtain. The emergence of large
language models (LLMs) such as ChatGPT provides an opportunity to solve
language tasks with simple prompts without the need for task-specific datasets
and fine-tuning. While ChatGPT has demonstrated impressive results in tasks
like machine translation, text summarization, and question answering, it
presents challenges when used for complex tasks like event extraction. Unlike
other tasks, event extraction requires the model to be provided with a complex
set of instructions defining all event types and their schemas. To explore the
feasibility of ChatGPT for event extraction and the challenges it poses, we
conducted a series of experiments. Our results show that ChatGPT has, on
average, only 51.04% of the performance of a task-specific model such as EEQA
in long-tail and complex scenarios. Our usability testing experiments indicate
that ChatGPT is not robust enough, and continuous refinement of the prompt
does not lead to stable performance improvements, which can result in a poor
user experience. Besides, ChatGPT is highly sensitive to different prompt
styles.

##  1 Introduction

Event extraction is a crucial natural language processing task that involves
identifying and extracting structured events mentioned in text (Du and Cardie,
2020; Lu et al., 2021). However, the limited availability of annotated data
makes this task particularly challenging, as expert annotation can be
expensive (Yang et al., 2019; Lou et al., 2022; Gao et al., 2022). Recent
advancements in large language models (LLMs), such as ChatGPT
111https://chat.openai.com/chat, have generated considerable interest in
applying these models to various language tasks, such as machine translation
(Jiao et al., 2023), text summarization (Bang et al., 2023), and automatic bug
fixing (Sobania et al., 2023), requiring only a few demonstrations.

Figure 1: An example of zero-shot event extraction using ChatGPT. In this
scenario, ChatGPT is tasked with processing a pool of evaluation instances
based on provided instructions, which include the task description, event type
definitions and demonstration examples. ChatGPT is expected to produce
responses for each evaluation instance without any prior training on the
specific task or data.

However, event extraction is a more complex task that poses unique challenges.
Unlike other tasks like translation and summarization, event extraction
requires a more detailed and nuanced description of the task. To utilize
ChatGPT for event extraction, the model must be provided with a schema for
each event type, including the triggers and arguments to be extracted.
Additionally, handling complex special cases is critical for accurate event
extraction.

This paper aims to explore the feasibility of ChatGPT for event extraction. We
conduct experiments to evaluate ChatGPT’s performance compared to two task-
specific models such as Text2Event (Lu et al., 2021) and EEQA (Du and Cardie,
2020) in long-tail and complex scenarios (i.e., texts containing multiple
events). Our results show that ChatGPT has, on average, only 51.04% of the
performance of EEQA in such scenarios. We also conduct usability testing
experiments to assess whether users can achieve their desired outcomes with
minimal attempts. Our experiments indicate that ChatGPT is not robust enough.
Despite several prompt optimizations aimed at enhancing its performance on the
event extraction task, ChatGPT did not exhibit a consistent upward trend. This
limitation can result in a poor user experience. Additionally, ChatGPT is
highly sensitive to different prompt styles, resulting in significant
variations in the results obtained by different users using ChatGPT.

##  2 Background

#### Event definition.

A universally agreed definition for events is currently lacking, as
definitions may vary depending on the application or task (Liu et al., 2020).
This paper will use the definition provided by the Automatic Content
Extraction (ACE) 2005 evaluation 222https://catalog.ldc.upenn.edu/LDC2006T06,
which defines an event as “a specific occurrence involving participants,” and
includes several arguments with different roles. Within this framework, an
event mention is a sentence that describes an event, an event trigger is the
word that most clearly expresses the event, and event arguments are taggable
entities involved in the event. Typically, specific participant roles can be
filled for each type of event. For instance, in the sentence "A number of
demonstrators threw stones and empty bottles at Israeli soldiers positioned
near a Jewish holy site at the town’s entrance," the event described is an
ATTACK, with the entire sentence serving as an event mention, "threw" acting
as the event trigger, and four event arguments, including the ATTACK-Attacker
(demonstrators), ATTACK-Target (Israeli soldiers), ATTACK-Instrument (stones
and empty bottles), and ATTACK-Place (a Jewish holy site at the town’s
entrance).

#### Event Extraction Tasks.

Event extraction refers to the process of obtaining structured information
from unstructured text. In the context of ACE 2005, event extraction consists
of two subtasks. The first subtask, known as event detection, involves
identifying events and classifying them according to their specific types. The
second subtask, event argument extraction, aims to identify and extract the
various words, phrases, and entities that fulfill different roles within a
given event. In this work, we mainly focus on the event detection task.

##  3 ChatGPT for Event Extraction

Event extraction aims to identify event triggers and their corresponding
arguments from unstructured text and present them in a structured format for
further processing. In this context, our objective is to evaluate ChatGPT’s
ability to perform zero-shot event extraction without fine-tuning the model.
To leverage the advanced capabilities of the ChatGPT model for event
extraction, a promising approach is to frame the task as a multi-turn question
answering problem. This method builds upon prior research (Li et al., 2019; Du
and Cardie, 2020) and involves using the model to identify events by posing a
series of questions in a conversational style.

Specifically, given a task instruction that describes this task in natural
language, ChatGPT is expected to output the events in the text and represent
them in a structured form. In this work, we mainly focus on the event
detection task, so that each event output contains an event trigger word and
an event type, and we use the JSON format to serialize the output because it
is widely used and easy to parse. Figure 1 provides An example of zero-shot
event extraction using ChatGPT. In this scenario, ChatGPT is tasked with
processing a pool of evaluation instances based on provided instructions,
which include the task description, event type definitions and demonstration
examples. ChatGPT is expected to produce responses for each evaluation
instance without any prior training on the specific task or data.

In this study, we first assess the performance of ChatGPT in comparison to
task-specific models (Text2Event and EEQA) under real-world conditions.
Specifically, we randomly select 20 samples from the raw test set to evaluate
the efficacy of ChatGPT. Then, we conduct an analysis of the impact of
different elements in the prompt on the performance of ChatGPT using the same
20 test samples. To obtain a comprehensive understanding of ChatGPT’s
performance, we also evaluate its performance in both long-tail and complex
scenarios (i.e., texts containing multiple events) in comparison to task-
specific models. Finally, we conduct usability testing to assess ChatGPT’s
user-friendliness for the event extraction task.

##  4 Experimental Setup and Results

###  4.1 Setup

#### Dataset.

We conduct experiments on the ACE 2005 corpus, which comprises documents
crawled between 2003 and 2005 from diverse sources, including newswire,
weblogs, broadcast conversations, and broadcast news. To maintain consistency
with prior works (Du and Cardie, 2020; Lu et al., 2021; Gao et al., 2022), we
employ the same data split and preprocessing steps.

#### Evaluation Metric.

We utilize the same criteria defined in Du and Cardie (2020) for our analysis.
Specifically, an event trigger is considered correctly identified (ID) if its
offsets correspond to those of a gold-standard trigger. Moreover, for a
trigger to be correctly classified, its event type (there are 33 types in
total) must match that of the gold-standard trigger.

#### Baselines.

In our experiments, we conduct a comparison between ChatGPT and several event
extraction baselines, including Text2Event and EEQA.

  * •

Text2Event (Lu et al., 2021): Text2Event is a framework that utilizes T5
models (Raffel et al., 2019) to approach event extraction by framing it as a
SEQ2SEQ generation task. In this method, all triggers, arguments, and their
corresponding labels are generated as natural language words.

  * •

EEQA (Du and Cardie, 2020): EEQA, on the other hand, approaches event
extraction by formulating it as a question answering task. They developed two
BERT-based QA models - one for detecting event triggers and another for
extracting arguments. In our experiments, we solely utilized the BERT-based QA
model for event trigger detection.

###  4.2 Results

#### Comparison with task-specific models.

We first compare the performance of ChatGPT with two task-specific models,
namely EEQA and Text2Event, which are fine-tuned using the ACE05 training set.
To assess the models, we randomly select 20 samples from the ACE05 test set.
Table 1 displays the results obtained from the experiment. The findings reveal
that EEQA achieves the highest F1 performance, while ChatGPT lags behind
Text2Event and EEQA. Notably, ChatGPT’s Recall is comparable to Text2Event
(T5-base), but its precision is significantly lower. Upon analyzing the output
samples, we observe that ChatGPT extracted more event triggers, potentially
because it lacks a clear understanding of certain event definitions.

| Event Detection
---|---
P | R | F1
ChatGPT | 57.14 | 72.73 | 64.00
Text2Event (T5-base) | 75.76 | 75.76 | 75.76
Text2Event (T5-large) | 82.76 | 72.73 | 77.42
EEQA (BERT-base) | 81.82 | 81.82 | 81.82
Table 1: Comparison with task-specific models. | Event Detection
---|---
P | R | F1
Full Prompt | 57.14 | 72.73 | 64.00
\- Positive Example | 48.00 | 72.73 | 57.83
\- Negative Example | 62.50 | 75.76 | 68.49
\- Event Type Definition | 47.50 | 57.58 | 52.05
Table 2: Instructing with different elements.

#### Instructing with different elements.

| High Frequency | Low Frequency | Simple Examples | Complex Examples
---|---|---|---|---
P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1
ChatGPT | 23.08 | 45.00 | 30.51 | 23.40 | 55.00 | 32.84 | 49.00 | 50.00 | 28.99 | 42.31 | 52.38 | 46.81
Text2Event (t5-base) | 55.56 | 50.00 | 52.63 | 61.90 | 65.00 | 63.41 | 79.17 | 95.00 | 86.36 | 81.82 | 64.29 | 72.00
Text2Event (t5-large) | 54.55 | 60.00 | 57.14 | 60.00 | 60.00 | 60.00 | 75.00 | 90.00 | 81.82 | 88.57 | 73.81 | 80.52
EEQA | 59.09 | 65.00 | 61.90 | 61.11 | 55.00 | 57.89 | 78.26 | 90.00 | 83.72 | 82.35 | 66.67 | 73.68
Table 3: Performance of ChatGPT and task-specific models in long-tail and
complex scenarios.

The prompt plays a critical role in enabling ChatGPT to perform event
extraction, as it contains vital information, such as task descriptions, event
type definitions, and demonstrations. However, the impact of each of these
elements on ChatGPT’s performance in event extraction remains unclear. To
address this gap, we conduct an ablation experiment to examine the effect of
different prompt elements on ChatGPT’s performance. To this end, we design
four prompts, each with varying amounts and types of information. The prompts
are as follows:

  * •

Full Prompt: The complete prompt, including task description, event type
definitions, and demonstrations.

  * •

No Event Definitions: The prompt without the event definitions.

  * •

No Positive Example: The prompt without the positive sample.

  * •

No Negative Example: The prompt without the negative sample.

We then evaluate ChatGPT’s performance on the event extraction task for each
prompt, and we use the same 20 test samples used in the previous experiment As
shown in Table 2, there is a marked decline in model performance when the
positive example and event type definitions are removed. Specifically,
removing the positive example results in a 6.17-point decrease in F1, while
removing the event type definitions leads to an 11.95-point decrease in F1.
Interestingly, the model’s performance improved after eliminating the negative
sample. We speculate that this might be because the model can not comprehend
the meaning of “negative sample” and misinterpreted them as positive samples.
A similar finding can be found in the prior work (Wang et al., 2022).

#### Can ChatGPT consistently output structured events?

Since the goal of event extraction is to automatically extract structured
events from text and then use them for downstream tasks, the event extraction
model needs to be able to output structured events in a stable manner. After
analyzing the output of ChatGPT in the ablation experiment, we discover that
the ChatGPT is able to consistently produce structured events in the vast
majority of cases. Specifically, ChatGPT is able to produce structured events
(see Figure 1 for example) for all 20 samples when provided with either the
full prompt or the prompt without the positive example. However, when the
negative sample and event type definitions are removed, only 19 out of 20
examples have a structured representation. This is because ChatGPT would
output an explanation to indicate that it can not recognize the event when it
encountered an unrecognizable event trigger word or event type, such as “[]
(There is no event trigger word or event type present in the input
sentence.)”.

#### Analysis on long-tail and complex scenarios.

To gain a more comprehensive understanding of ChatGPT’s strengths and
weaknesses, we conducted evaluations in two different scenarios, namely the
long-tail scenario and the complex scenario.

Long-tail scenario. To evaluate ChatGPT’s performance in handling low-
frequency events, we split the test set into two categories based on event
frequency. High-frequency events are those that appeared in at least 10
instances in the dataset, while low-frequency events are those that appeared
less than 10 times. We randomly sample 20 test instances from each category
and evaluate ChatGPT’s performance separately on these two subsets of data.
Our results in Table 3 show that ChatGPT’s performance on high-frequency and
low-frequency events is not significantly different. However, ChatGPT’s
performance is significantly worse than the baseline models, achieving only
49.2% and 56.7% of EEQA’s performance on low-frequency and high-frequency
events, respectively.

Complex scenario. To assess ChatGPT’s ability to handle complex samples, we
divide the test set into two categories based on the number of events present
in the text. Simple samples contain only one event, while complex samples
contain more than two events. We randomly sample 20 test instances from each
category and evaluate ChatGPT’s performance separately on these two subsets of
data. As shown in Table 3, ChatGPT’s performance on both simple and complex
samples is significantly worse than that of EEQA and Text2Event.

Figure 2: Performance of ChatGPT with different number of attempts by four
annotators.

#### Usability of ChatGPT for event extraction.

The ChatGPT requires a prompt to execute specific tasks. However, users
currently lack guidance on creating suitable prompts. This section aims to
evaluate ChatGPT’s usability by assessing whether users can achieve their
desired outcomes with minimal attempts. To conduct the experiment, we
recruited four professional and well-educated annotators (e.g. postgraduate
student on NLP research) to evaluate ChatGPT’s usability. We randomly selected
ten samples from the ACE05 test set and provided each annotator with five
attempts to create a task prompt that would enable ChatGPT to extract
structured events from the given text. Figure 2 presents the results of the
four annotators.

  * •

ChatGPT is not robust enough. Despite several prompt optimizations aimed at
enhancing its performance on the event extraction task, ChatGPT did not
exhibit a consistent upward trend. This limitation can result in a poor user
experience.

  * •

ChatGPT’s performance is sensitive to different prompt styles. Different
annotators utilized different prompt styles, resulting in significant
performance variations for ChatGPT. For instance, ChatGPT achieved its highest
F1 score of 66.67 on the fourth attempt, while the lowest is only 26.92.

##  5 Conclusion

In conclusion, our study highlights the potential of large language models
like ChatGPT for event extraction, but also underscores the challenges
associated with this task. Our findings suggest that, while ChatGPT can
achieve good results in simple scenarios, it struggles to match the
performance of task-specific models in more complex and long-tail scenarios.
Our usability testing experiments also reveal that the performance of ChatGPT
is highly dependent on the prompt style, which can lead to significant
variations in results obtained by different users. These findings suggest
that, while ChatGPT has enormous potential for event extraction, further
research is needed to refine its capabilities and overcome the challenges it
presents in this domain. Overall, our study contributes to the growing body of
research on large language models and highlights the need for continued
investigation into their strengths and limitations in complex NLP tasks like
event extraction.

## Acknowledgements

We would like to express our sincere gratitude to Fangqi Zhu and Tianyi Xiao
for their valuable participation in our human evaluation experiments. Their
insightful comments and suggestions have greatly contributed to the quality of
this paper. We also extend our thanks to Wei Wang for his valuable feedback
and discussion of the paper. His contributions have been instrumental in
shaping our research.

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# Quasi-monolithic Compact Interferometric Sensor Head Design with Laser Auto-
alignment

Xiang Lin  Peng Qiu  Yurong Liang  Xiaofang Ren  Hao Yan<EMAIL_ADDRESS>Zebing Zhou  Center for gravitational experiment, MOE Key Laboratory of
Fundamental Physical Quantities Measurements, The School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, China

###### Abstract

Interferometers play a crucial role in high-precision displacement measurement
such as gravitational-wave detection. Conventional interferometer designs
require accurate laser alignment, including the laser pointing and the waist
position, to maintain high interference contrast during motion. Although the
corner reflector returns the reflected beam in parallel, there is still a
problem of lateral beam shift which reduces the interference contrast. This
paper presents a new compact interferometric sensor head design for measuring
translations with auto-alignment. It works without laser beam alignment
adjustment and maintains high interferometric contrast during arbitrary motion
(tilts as well as lateral translation). Automatic alignment of the measuring
beam with the reference beam is possible by means of a secondary reflection
design with a corner reflector. A $20\times 10\times 10\,\mathrm{mm}^{3}$ all-
glass quasi-monolithic sensor head is built based on UV adhesive bonding and
tested by a piezoelectric (PZT) positioning stage. Our sensor head achieved a
displacement sensitivity of $1\,\mathrm{pm}/\mathrm{Hz}^{1/2}$ at ${\rm
1\,Hz}$ with a tilt dynamic range over $\pm 200\,\mathrm{mrad}$. This optical
design can be widely used for high-precision displacement measurement over a
large tilt dynamic range, such as torsion balances and seismometers.

††preprint: APS/123-QED

##  I INTRODUCTION

Laser interferometers are widely used for high-precision displacement
measurement in various fields such as gravitational wave detectors [1],
inertial sensors [2, 3, 4, 5], vibrometers [6, 7], machine tools [8],
Coordinate Measuring Machines (CMM) [9], lithography [10], etc. Besides, the
ongoing space gravity detection missions Tianqin and Taiji require laser
interferometer sensitivity of picometer and nanoradian level in the low-
frequency bands of ${\rm 1\,mHz-1\,Hz}$ [11, 12]. The need for such high-
precision measurements places high demands on the design and construction of
interferometers.

Various high-precision interferometers characterized by compact, stable, and
wide dynamic range have been developed in recent decades. A comprehensive
review of compact interferometry can be found in [13]. According to the
different types of interference signals, interferometers can be divided into
homodyne interferometers [14], heterodyne interferometers [15, 16, 17, 18],
Fabry–Perot (FP) cavity interferometers [19], and modulation interferometers.
Modulation interferometers can be subdivided into frequency modulation
interferometers [20, 21, 22], phase modulation interferometers [23], and
amplitude modulation interferometers (including pulse or optical comb) [24].

Among these interferometers, homodyne interferometers, which usually consist
of a laser source, an optical bench, a target reflector and a DC
photodetector, are preferred for their simplicity and compactness. To further
improve the compactness and stability, optical sensor head designs based on
Michelson-type or FP-type are usually chosen in practical applications and
commercial interferometers, such as SmarAct, attocube IDS, quDIS, etc. [25,
26, 27].

Compact interferometric sensor heads are important for the needs of large-
range, high-precision displacement measurements in confined spaces, such as
microstrain and micro-vibration monitoring and analysis [28, 29], and
microsensor calibration testing. Moreover, compact miniature optical sensor
heads have higher sensitivity potential with better opto-mechanical-thermal
stability [30, 31].

Conventional homodyne interferometers with plane target mirrors have some
inherent drawbacks. Homodyne interferometers measure small displacement by
reading changes in DC light intensity, which is also sensitive to the changes
in interference contrast [13]. Precise laser alignment between the optical
sensor head and the target mirror is required prior to use to achieve high
interference contrast. In addition, high contrast needs to be maintained
during motion, which limits the dynamic range of tilt motion. Finally, the
tilt-to-length (TTL) coupling noise [32, 33], i.e., the coupled response of
tilt to translational light paths, depends on optical parameters such as beam
waist, propagation distance, etc.

Using a corner reflector as a target instead of a plane reflector can solve
the laser alignment problem during measurement to a certain extent, but it
also has drawbacks [7]. First, although the corner reflector enables the
reflected beam to be returned parallel, there is still a problem of lateral
beam shift which also reduces the interference contrast [34]. Second, if the
measurement beam covers the center of the corner reflector, the returned laser
wavefront will split into six flaps (depending on the three dihedral-angle
errors of the corner reflector), causing displacement coupling errors.

Many efforts have been made to improve the angular dynamic range of
interferometers. A cat’s eye retroreflector can be used to increase the tilt
tolerance range in interferometry [35]. A target corner reflector combined
with a fixed reference plane mirror allows interferometers to measure a
translation over a large angular dynamic range [36].

In this paper, we proposed a novel Michelson-type interferometric sensor head
design for measuring translations with laser auto-alignment. Combined with a
corner reflector, this design automatically guarantees a high interference
contrast, greatly increasing the dynamic range while avoiding coupling errors
in other degrees of freedom of motion, including tilts and lateral
translations. A $20\times 10\times 10\,\mathrm{mm}^{3}$ all-glass quasi-
monolithic compact optical sensor head is built and tested. To improve
compactness and stability, we use UV-adhesive bonding technology to glue the
optical components [14]. The principle and the optical design of the sensor
head are detailed and the experimental setup and sensitivity results are given
in the following sections.

This paper is organized as follows. In Section II, we briefly outline the
measurement principle and experimental layout of the experimental setup.
Section III is devoted to the experimental results of dynamic range and
displacement measurement sensitivity of the homodyne interferometer. The
summary and further discussions are presented in the last section.

##  II DESIGN and WORKING PRINCIPLE

###  II.1 Optical Design

The optical schematic of the homodyne interferometer with a compact optical
sensor head for the translation measurement is shown in Fig. 1. The
interferometric system mainly consists of five components: a laser source, a
photodetector (PD), a circulator (CIR), a customized optical sensor head, and
a target corner reflector (CR). The optical sensor head comprises a fiber
collimator(FC) and a custom beam splitter (BS) with reflective coatings on
side surfaces. The two optical elements (FC and BS) are glued together to form
a quasi-monolithic compact sensor head, as shown in the figure. For a typical
Michelson-type sensor head, the measurement laser beam is reflected directly
back to the sensor head by the target plane reflector, as shown in Fig. 1(a).
This design, in order to maintain high interference contrast, requires a high
level of alignment in the motion of the target plane mirror [37, 26].

Figure 1:  Optical schematic of the Michelson-type interferometer with a
compact optical sensor head for the translation measurement. A laser beam from
the laser source is injected into a circulator (CIR) via a polarization-
maintaining fiber at port 1, and the output port 2 is connected to the optical
sensor head via a fiber collimator (FC). The homodyne interference signal
coming back from the optical sensor head enters the photodetector (PD) through
port 3 of the circulator. Panel (a) shows a typical Michelson-type sensor head
with a target plane reflector. The sensor head consists of a fiber collimator
and a custom beam splitter (BS) with reflective coatings on the upper side.
The collimated beam is split into two beams at a 50/50 ratio by the BS and
reflected back to the FC by the target plane mirror. Panel (b) shows the new
optical sensor head design with a target corner reflector. The custom beam
splitter (BS) has reflective coatings on the two perpendicular side surfaces.
A small coating hole is left in the direction of the sensitive axis for the
laser to enter and exit. The measurement laser is reflected both by the corner
reflector (CR) and the reflective coating.

Laser self-collimation can be achieved if an additional coating is added to
the front side of the BS and a target corner reflector is used instead of the
plane mirror, as shown in Fig. 1(b). The new design works as follows: First,
the laser is injected into the optical sensor head via a circulator from port
1 to port 2. Second, the laser beam from the fiber collimator is split by the
beam splitter inside the optical sensor head. The reflected reference laser
beam is reflected back to the fiber collimator by the upper side surface with
a reflective coating and the beam splitter, and the measurement laser beam in
the direction of the sensitive axis passes through the coating hole. Third,
the measurement laser beam from the sensor head hits the targe corner
reflector and is reflected back to the sensor head in parallel with a small
lateral shift, as shown in the figure. Fourth, the lateral shift beam is
reflected back to the corner reflector and the optical sensor head in the same
path by the reflective coating surface of the beam splitter from outside. The
two reflected beams (the reference beam and measurement beam) interfere at the
beam splitter and return to the inside of the fiber. Finally, the optical
interference signal is detected by the photodetector via the circulator from
port 2 to port 3.

It is worth noting that our new optical sensor head can also work in the same
way as the typical Michelson-type sensor head design shown in Fig. 1(a),
reflecting once with a target plane mirror.

The new optical sensor head has special geometric requirements on the surfaces
containing reflective coatings. The reflected laser beams (reference and
measurement laser beams) must return to the inside of the fiber to maintain a
high interference contrast. The reference laser beam reflected by the beam
splitter must be perpendicular to the upper reflective coating surface to
return along the original path. The transmitted laser beam reflected by the
corner reflector with a lateral shift must also be perpendicular to the
sensitive-axis reflective coating surface from outside to return along the
original path. To achieve this, first, the two coated surfaces of the beam
splitter need to be perpendicular to each other; then, the relative attitudes
of the fiber collimator and the beam splitter need to be carefully adjusted
when bonding the optical sensor head.

Once the new optical sensor head has been fabricated, the measurement laser
beam automatically returns to the sensor head and interferes with the
reference laser beam with high contrast. In Fig. 1(b), the parallel return
beam is deflected downward due to the center of the corner reflector is below
the optical axis; if the corner reflector moves upward, the parallel return
beam will be deflected upward laterally. In both cases, it is guaranteed that
the measurement laser beam will be reflected twice and return to the fiber.
The dynamic range of motion of the target corner reflector is limited by the
lateral displacement, as shown in Fig. 1(b). The returning measurement laser
beam can not extend beyond the surface of the reflective coating. This
requires that the center of the target corner reflector not deviate too far
from the exit optical axis.

Notably, this sensor head design responds only to translational motion along
the sensitive axis, thus effectively suppressing coupling errors arising from
motion in other degrees of freedom (e.g. TTL coupling) [5, 32, 33].

###  II.2 Measurement Principle

The measurement laser beam from the target corner reflector and the reference
laser beam inside the optical sensor head interference in the beam splitter
and detected by the photodetector. Assuming the homodyne interference signal
be written as:

$P=P_{0}[1+C\cdot\cos{\left(\varphi_{m}-\varphi_{r}\right)}],$ (1)

where $P$ is the laser power on the photodiode, $P_{0}$ is the average laser
power on the photodiode, $C$ is the interference contrast. $\varphi_{m}$ and
$\varphi_{r}$ are the cumulative path phases of the measurement beam and the
reference beam, respectively. The interference contrast is given by the
equation

$C=\frac{P_{max}-P_{min}}{P_{max}+P_{min}},$ (2)

where $P_{max}$ and $P_{min}$ are the maximum power and minimum power,
respectively.

The interference contrast $C$ depends mainly on the beam-waist ratio, the
propagation distance, the wavefront distribution, the center position offset,
and the pointing deviation of the two interfering laser beams. In this optical
sensor head design, the interference contrast can be maintained at a high
level. The actual values of the average laser power $P_{0}$ and the
interference contrast $C$ can be experimentally calibrated by moving the
target reflector over a wide range of translations. The interference signal
light intensity change is monitored and then the phase change is calculated by
Eq. (1).

Considering that the measurement laser beam is reflected twice between the
optical sensor head and the target corner reflector (as shown in Fig. 1, the
target translation $x$ is given by the equation

$x=\frac{\lambda}{8\pi}(\varphi_{m}-\varphi_{r}),$ (3)

where $\lambda$ is the laser wavelength. The above Eq. (1-3) is a homodyne
interference readout, and the measurement signal is a DC signal. We can also
use laser frequency modulation technique to convert it into AC signals [20].

Note the special case in Fig. 1(b), where the measurement beam reflected for
the first time from the corner reflector returns directly to the sensor head
along the small hole. In this case, there is only one reflection here, similar
to Fig. 1(a). The target translation $x$ is given by the equation

$x=\frac{\lambda}{4\pi}(\varphi_{m}-\varphi_{r}).$ (4)

Considering the lateral displacement coupling of the reflected beam from the
corner reflector, we should try to avoid this situation.

##  III  Prototype and TEST RESULTS

###  III.1 Prototype setup

In order to obtain high stability and precision, we built an all-glass quasi-
monolithic compact optical sensor head via UV-adhesive bonding [14]. The
three-dimensional quasi-monolithic optical sensor head design is shown in Fig.
2. The measurement laser beam exits through the small reflective coating hole
in the front surface of the beam splitter. The laterally shifted returned beam
from the target corner reflector is reflected back in the same path by the
reflective coating of the sensor head. Finally, the interference signal
returns along the laser injection fiber and is detected by a photodetector.
The all-glass quasi-monolithic optical sensor head consists of a fiber core, a
G-lens, a glass sleeve, a glass cube seat, and a cube beam splitter with two
reflective coating surfaces, as shown the Fig. 2(b). All these components are
bonded together via UV adhesive to form a rectangular compact sensor head with
dimensions of $20\times 10\times 10\,\mathrm{mm}^{3}$.

Figure 2: Three-dimensional quasi-monolithic design of the customized optical
sensor head and the target corner reflector. Panel (a) shows the optical
sensor head and the target corner reflector. The measurement laser beam (red)
is first reflected with a lateral shift by the corner reflector and then
reflected back in the same path by the reflective coating of the sensor head.
Panel (b) shows the detail of the optical sensor head, including the fiber
core, G-lens, glass sleeve, glass cube seat, and cube beam splitter with two
reflective coating surfaces. All components are glued together with dimensions
of $20\times 10\times 10\,\mathrm{mm}^{3}$.

The experimental prototype setup is shown in Fig. 3. The quasi-monolithic
optical sensor head is bonded to the top of a metal cylinder. To test the
interferometer system’s tilt dynamic range, the target corner reflector is
mounted on the upper plate of the Hexapod through a mirror mount. A laser
(Nd:YAG) with a wavelength of 1064 nm and a power of 1 mW is connected to a
fiber optic circulator (Throlab CIR1064PM). The laser emitted by the
circulator is injected into the quasi-monolithic sensor head via a fiber. The
key parameters of the experimental setup are given in Tab. 1.

Table 1: The key parameters of the experimental setup. Parameters | Value
---|---
laser wavelength | $1064\,\mathrm{nm}$
beam waist diameter | $\sim 1\,\mathrm{mm}$
input beam power | $\sim 1\,\mathrm{mW}$
size of the corner reflector (diameter) | $25.4\,\mathrm{mm}$
size of the optical sensor head | $20\times 10\times 10\,\mathrm{mm}^{3}$
material of the optical sensor head | fused silica
optical bonding method | UV adhesive gluing

Adjust the position of the target corner cube reflector to ensure that the
reflected measurement beam with a lateral displacement does not exceed the
reflective coating area of the beam splitter. The center of the corner cone
must not be too far away from the measuring optical axis. The motion of the
hexapod is controlled by a computer.

Figure 3:  Photo of the experimental prototype setup. The target corner
reflector (UnionOptic CNP0025) is mounted to the top of the Hexapod (PI-824)
through a high-stability mirror mount (Thorlabs POLARIS-K1E2), and the quasi-
monolithic optical sensor head is bonded to the top of a metal cylinder. The
Hexapod is controlled by a computer to scan step by step around the z-axis.

###  III.2 Experiment Results

####  III.2.1 Contrast and Dynamic Range

Dynamic range is one of the key parameters of high-precision displacement
sensors. The sensitive translation dynamic range of interferometers is usually
not a problem. It is mainly limited by the collimation range of the laser or
the way the interference phase is demodulated. However, the tilt dynamic range
of interferometers with plane reflector is usually limited by the interference
contrast to about $mrad$ level. Beyond this range, the contrast of the
interferometer decreases rapidly.

There are two methods to measure the contrast of a homodyne interferometer.
The first direct method is to drive the target reflector to scan over a wide
range of translation motion (more than half a wavelength), covering at least
one bright fringe and one dark fringe. The second method is deep frequency
modulation of the laser. A phase modulation signal will be generated in the
homodyne interferometer with unequal arm lengths [20, 22]. The phase
modulation depth is proportional to the unequal arm lengths and the frequency
modulation amplitude. We use the second method in our interference contrast
tests.

The interference contrast of the quasi-monolithic sensing probe needs to be
tested first. We modulate the laser frequency at $\omega_{m}=2\pi\cdot
10\,\mathrm{Hz}$ and amplitude of $A_{m}\approx 0.25\,\mathrm{GHz}$ with an
internal piezoelectric transducer (PZT) actuator. The power observed at the
photodetector is given by the equation [25]

$P(t)=P_{0}\left[1+C\cos{\left(\varphi_{m}-\varphi_{r}\right)}+m\cos{\left(\omega_{m}t\right)}\right],$
(5)

where $m=8\pi A_{m}\Delta L/c$ is the unitless modulation index, $\Delta L$ is
the longitudinal imbalance of the Michelson arms and $c$ is the speed of
light. The measurement signal passes through a high-gain silicon-based
photodetector, and the raw photoelectric signal is acquired by a digital
acquisition device (Liquid Instuments Moku:Lab).

The unequal arm length $\Delta L$ is set to about $0.1\,\mathrm{m}$ and the
unitless modulation index $m\approx 2>\pi/2$. Then the interference contrast
can be resolved by Eq. 2 from the modulated interference signal shown in Fig.
4(a). The initial homodyne interferometer is well aligned and the contrast is
about $C\approx 0.86$ (Fig. 4(a)). To test the tilt dynamic range of our
quasi-monolithic sensing head, we utilized a Hexapod (PI-H824) to rotate the
yaw angle of the target angular reflector, as shown in Fig. 3. The Hexapod we
used has a limit rotation range of $\pm 12.5\degree$ (around the z-axis),
or$\pm 220\,\mathrm{mrad}$. Therefore, the angle scan range of the contrast
test is set from $-200\,\mathrm{mrad}$ to $+200\,\mathrm{mrad}$ in steps of
$10\,\mathrm{mrad}$.

The test results of the contrast variation with the corner reflector rotation
are shown in Fig. 4(b). The interference signal maintains high contrast
($C>0.85$) over a rotation range of $\pm 200\mathrm{mrad}$. The slight
fluctuations in the contrast curve are mainly due to the unevenness of the
reflective coating.

For comparison, the same test was done using a plane mirror with the same
sensor head. In this case, the measurement laser beam reflected from the plane
mirror returns directly to the fiber in the same path, as shown in Fig. 1(a).
The results are shown in Fig. 4(c). The interference contrast decreases
rapidly with the rotation of the target plane mirror, falling below $0.4$
beyond a rotation range of $\pm 0.2\mathrm{mrad}$.

It is worth noting that our auto-aligning optical sensor head has the
potential to achieve a greater angular dynamic range based on the design in
Fig. 1. Even so, the results of this angular dynamic range still far exceed
the $\sim\,\mathrm{mrad}$ level of typical interferometers [32].

Figure 4:  Interference contrast and tilt dynamic range test results. (a) The
interference signal of the setup with the laser frequency modulated. (b) The
curve of the interference contrast of our new auto-aligning optical sensor
head with the target corner reflector rotation. The angular dynamic range is
scan from $-200\,\mathrm{mrad}$ to $+200\,\mathrm{mrad}$ in steps of
$10\,\mathrm{mrad}$. (c) The curve of the interference contrast of the same
optical sensor head with a plane mirror rotation (a typical Michelson homodyne
interferometer design, just the same as the sensor head of the SmarAct company
[25]). The angular dynamic range of this Michelson-type sensor head is about
$\pm 0.2\,\mathrm{mrad}$ ($C<0.4$).

####  III.2.2 Resolution and sensitivity

To determine the resolution of the new optical sensor head, a piezoelectric
transducer (PZT) actuator is employed. The voltage signal from the signal
generator (Keysight, 33622A) drives the mirror movement (translation or tilts)
through the PZT.

The test results are shown in Fig. 5. In the resolution test, the target
corner reflector is driven by a PZT with a small translation. The voltage
signal driving the PZT is a square wave with an amplitude of $10\,\mathrm{mV}$
and a period of $5\,\mathrm{s}$, corresponding to a displacement of about
$1\,\mathrm{nm}$ based on a coefficient of $100\,\mathrm{nm/V}$. The signal
output in time domain is shown in Fig. 5(a). To further measure the
sensitivity, the driven voltage is switched off. The sensitivity curve is
shown in Fig. 5(b). A sensitivity of $1\mathrm{pm/Hz^{1/2}}$ at
$1\,\mathrm{Hz}$ is achieved as shown in the blue solid curve.

The root-mean-square (rms) error between frequency $f$ and $fs/2$ can be
calculated from the PSD (power spectral density) by

$\sigma_{rms}(f)=\sqrt{\int_{f}^{fs/2}{\rm ASD}^{2}(f^{\prime})\cdot
df\textquoteright},$ (6)

where $fs$ is the sampling frequency, ${\rm ASD}(f^{\prime})$ is the amplitude
spectral density. According to the Nyquist–Shannon sampling theorem, the upper
limit of the PSD integral is $fs/2$. The red dashed curve in Fig. 5(b) shows
the root-mean-square (rms) error of the interferometer system. A displacement
resolution of picometer level is achieved.

Figure 5:  Test results of the resolution and sensitivity. Panel (a) shows a
translation displacement signal driven by the PZT (Thorlabs), a square wave
with an amplitude of $1\,\mathrm{nm}$ and a period of $5\,\mathrm{s}$. Panel
(b) shows the displacement sensitivity curve of the sensor head. The blue
solid curve is the logarithmic amplitude spectral density (ASD), and the red
dashed curve is the root mean square error.

##  IV DISCUSSION AND CONCLUSIONS

In this paper, a novel Michelson-type interferometric sensor head design for
translation measurement with laser auto-alignment is proposed. This optical
sensor head design, combined with a target corner reflector, automatically
guarantees a high interference contrast. Compared to conventional
interferometers using a target plane reflector, this optical design greatly
increases the dynamic range while avoiding coupling errors in other degrees of
freedom of motion, including rotations and lateral translations. To verify the
performance of our design, an all-glass quasi-monolithic compact sensor head
was built with UV-adhesive bonding technology and tested by a Hexapod and PZT
stages.

The performance of the experimental prototype setup is tested. The
experimental results show that a high contrast ratio above $85\%$ is
maintained over the dynamic range of $\pm 200\,\mathrm{mrad}$, and a
sensitivity of $1\mathrm{pm/Hz^{1/2}}$ at $1\,\mathrm{Hz}$ is achieved. The
dynamic range of the new design is more than 1,000 times higher than that of a
conventional Michelson-type design with the same optical parameters. It is
worth noting that our auto-aligning optical sensor head has the potential to
achieve a greater angular dynamic range.

This auto-alignment interferometric sensor head design with a large dynamic
range and high precision can be widely used for future multiple degrees-of-
freedom displacement metrology and applications, such as torsion balances and
seismometers.

###### Acknowledgements.

We thank members of Prof. Yiqiu Ma for useful discussions and Dr. Boping Chen
of Tel Aviv University for his valuable feedback. The authors acknowledge the
experimental facility support from the PGMF (National Precision Gravity
Measurement Facility), National Key Research and Development Program of China
(2022YFC2203901), National Natural Science Foundation of China (12105375).

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Vol.0 (20xx) No.0, 000–000

11institutetext: Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210023, China<EMAIL_ADDRESS>
22institutetext: School of Astronomy and Space Sciences, University of Science
and Technology of China, Hefei 230026, China
Received 20xx month day; accepted 20xx month day

# First Digit Distributions of Gamma-Ray Bursts

Hou-Yu Lai 1122 Jun-Jie Wei 1122

###### Abstract

The occurrence of the first significant digits from real world sources is
usually not equally distributed, but is consistent with a logarithmic
distribution instead, known as Benford’s law. In this work, we perform a
comprehensive investigation on the first digit distributions of the duration,
fluence, and energy flux of gamma-ray bursts (GRBs) for the first time. For a
complete GRB sample, we find that the first digits of the duration and fluence
adhere to Benford’s law. However, the energy flux shows a significant
departure from this law, which may be due to the fact that a considerable part
of the energy flux measurements are restricted by lack of spectral
information. Based on the conventional duration classification scheme, we also
check if the durations and fluences of long and short GRBs (with duration
$T_{90}>2$ s and $T_{90}\leq 2$ s, respectively) obey Benford’s law. We find
that the fluences of both long and short GRBs still agree with the Benford
distribution, but their durations do not follow Benford’s law. Our results
hint that the long–short GRB classification scheme does not directly represent
the intrinsic physical classification scheme.

###### keywords:

gamma-ray burst: general — methods: statistical — astronomical data bases:
miscellaneous

##  1 Introduction

People might think that the first significant digits (i.e., 1,2,…,9) of any
randomly chosen dataset tend to be uniformly distributed, but it is not true
in the natural world. As early as in 1881, Simon Newcomb had observed an
unanticipated pattern in the first digits of logarithm tables: the number 1
appears more frequently than 2, 2 than 3, and so on (Newcomb 1881). More than
a half century later, Frank Benford rediscovered that there is a logarithmic
distribution of first digits in numerous data tables, which is often called
the first digit law or Benford’s law (Benford 1938). This law states that for
a given real database, the probability of numbers with the first digit $k$ is
expressed as (Benford 1938)

$P(k)=\log_{10}{\left(1+\frac{1}{k}\right)},\quad k=1,2,...,9\;.$ (1)

Empirically, Benford’s law has been proven in various research fields,
including geography (e.g., the lengths of rivers and the areas of lakes;
Benford 1938), finance (e.g., the stock market indices; Ley 1996; De Ceuster
et al. 1998), biology (e.g., pre-vaccination measles incidence data, absolute
values from human magnetoencephalography recordings, and gene data lengths of
bacteria; Cáceres et al. 2008), seismology (e.g., the recurrence times of
seismic events; Sottili et al. 2012), statistical and nuclear physics (e.g.,
physical constants and distributions (Burke & Kincanon 1991; Shao & Ma 2010b),
half-lives of unstable nuclei (Buck et al. 1993; Ni & Ren 2008; Ni et al.
2009), widths of hadrons (Shao & Ma 2009), and the lepton branching fractions
(Dantuluri & Desai 2018)), etc. In practice, this peculiar law has been
effectively used to distinguish and diagnose frauds in taxing and accounting
(Nigrini 1996; Nigrini & Mittermaier 1997; Geyer & Williamson 2004), and to
minimize storage space and speed up calculation in computer science (Barlow &
Bareiss 1985; Schatte 1988; Berger & Hill 2007). Theoretically, this law has
been well explained by using a central-limit-like theorem for first digits
(Hill 1995) and a simple Markov process (Burgos & Santos 2021). In
mathematics, Benford’s law is scale invariant (Berger et al. 2008), which
indicates that it is independent of any particular choice of units (Pinkham
1961).

In astronomy, Benford’s law also has extensive application in all sorts of
astrophysical datasets, such as light curves of variable stars and other X-ray
sources (Moret et al. 2006), pulsar properties (Shao & Ma 2010a), distances of
galaxies and stars (Alexopoulos & Leontsinis 2014), exoplanetary and asteroid
data (Shukla et al. 2017; Melita & Miraglia 2021), GAIA DR2 parallaxes (de
Jong et al. 2020), and so on. Nevertheless, there are also some types of data,
e.g., pulsar and fast radio burst dispersion measures, which do not obey
Benford’s law (Mamidipaka & Desai 2023).

In this work, we investigate the first digit distributions of the duration,
fluence, and energy flux of gamma-ray bursts (GRBs), and check if these first
digits conform to Benford’s law for the first time. GRBs are flashes of high-
energy radiation originating from energetic explosions in the Universe.
According to their duration time $T_{90}$ (the time interval observed to
contain 90% of the prompt emission), GRBs can be classified into long GRBs
($T_{90}>2$ s) and short GRBs ($T_{90}\leq 2$ s) (Kouveliotou et al. 1993).
Generally, long GRBs are suggested to be powered by the core collapses of
massive stars (Paczyński 1998; Woosley & Bloom 2006), and short GRBs by the
mergers of binary compact objects (Eichler et al. 1989; Narayan et al. 1992).

##  2 Observational data and statistical results

###  2.1 Dataset

We download the durations $T_{90}$ (in units of s), fluences $F$ (in units of
erg $\rm cm^{-2}$), and energy fluxes $P_{\gamma}$ (in units of erg $\rm
cm^{-2}$ $\rm s^{-1}$) in the 10–1000 keV energy range from the online catalog
of GRBs observed by Fermi’s Gamma-ray Burst Monitor (Fermi-GBM) (Gruber et al.
2014; von Kienlin et al. 2014; Narayana Bhat et al. 2016; von Kienlin et al.
2020).111https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html The
Fermi-GBM burst catalog comprises of a list of 3665 cosmic GRBs between 2008
July 12 and 2023 December 6. We remove 1 GRB for which no relative data were
available. We carry out the first digit analyses for all the remaining 3664
GRBs with $T_{90}$ and fluence measurements. The energy flux of each burst in
the observer frame is calculated as

$P_{\gamma}=p_{64}\times\frac{\int_{E_{\rm min}}^{E_{\rm max}}E\times N(E){\rm
d}E}{\int_{E_{\rm min}}^{E_{\rm max}}N(E){\rm d}E}\;,$ (2)

where $p_{64}$ is the peak flux on the 64 ms timescale (in units of photon
$\rm cm^{-2}$ $\rm s^{-1}$), the spectral model $N(E)$ is the Band function
(Band et al. 1993), $E_{\rm min}$ and $E_{\rm max}$ are 10 keV and 1000 keV,
respectively. It is obvious that the spectral parameters are required to
calculate the energy flux. But not every burst has the spectral information.
There are only 2298 GRBs with energy flux measurements in the catalog.

###  2.2 Results

The first digit distributions of the duration and fluence for the complete GRB
sample are presented in Figure 1 and Table 1. As described above, there are
totally $N_{\rm tot}=3664$ available GRBs. The expected number according to
Benford’s law, $N_{\rm Ben}=N_{\rm tot}P(k)$, along with the root mean square
error estimated by the binomial distribution, $\Delta N=\sqrt{N_{\rm
tot}P(k)(1-P(k))}$, are also shown in the figure. From Figure 1, we can see
that the observed distributions are well consistent with the theoretical
predictions from Benford’s law. In order to quantify the goodness of fit of
Benford’s law, we adopt the Pearson $\chi^{2}$

$\chi^{2}=\sum_{k=1}^{9}\frac{\left[N_{\rm obs}(k)-N_{\rm
Ben}(k)\right]^{2}}{N_{\rm Ben}(k)}\;,$ (3)

where $N_{\rm obs}$ and $N_{\rm Ben}$ are the observed number and the expected
Benford number for a single digit $k$, respectively. For the first digit
distribution of the duration, we obtain a Pearson $\chi^{2}$ value of 13.0 for
8 degrees of freedom. For the fluence distribution, we obtain a $\chi^{2}$
value of 12.5. These two $\chi^{2}$ values correspond to $p$-values of 0.11
and 0.13, respectively, which strongly support the null hypothesis that the
durations and fluences of the complete GRB sample follow Benford’s law. It is
worth emphasizing that the higher the $p$-value, the more likely the null
hypothesis. In our present study, we exclude a null hypothesis if $p<0.05$
(equivalent to under the 95% confidence level).

GRBs can be divided into two classes, long GRBs and short GRBs, with a
division line at $T_{90}=2$ s (Kouveliotou et al. 1993). Using the
conventional division between the long and short GRB groups ($T_{90}>2$ s and
$T_{90}\leq 2$ s, respectively), we find that there were $3061$ long GRBs and
$603$ short GRBs in the Fermi-GBM burst catalog. In this work, we also check
if the durations and fluences of long and short GRBs obey Benford’s law. The
first digit distributions of the duration and fluence of long and short GRBs
are depicted in Figures 2 and 3, respectively. As listed in Table 1, the
$\chi^{2}$ tests and the corresponding $p$-values for the fluences are
extremely supportive to the null hypothesis that the fluences of both long and
short GRBs conform to Benford’s law. However, the durations of both long and
short GRBs obviously deviate from Benford’s law. We find that the Pearson
$\chi^{2}$ for the durations of $3061$ long GRBs and $603$ short GRBs are
$19.9$ and $74.2$, which correspond to $p$-values of 0.01 and $7\times
10^{-13}$, respectively. On the basis of the $p$-values, we can safely reject
the null hypothesis.

Similarly, in Figure 4 and Table 1, we show the first digit distribution of
the energy flux of $2298$ available GRBs. As we can see that the relative rank
of the probability of occurrence of leading digits roughly agrees with
Benford’s law, but the Benford distribution is not scrupulously obeyed. The
rather large $\chi^{2}$ value and the extremely low $p$ value suggest that the
energy flux of GRBs does not adhere to Benford’s law.

Table 1: Summary of our Benford analyses on the first digit distributions of the duration, fluence, and energy flux of GRBs. Dataset | Physical quantity | Number | $\chi^{2}/dof$ |  $p$-value
---|---|---|---|---
All GRBs | Duration | 3364 | 13.0/8 | 0.11
Fluence | 3364 | 12.5/8 | 0.13
Long GRBs | Duration | 3061 | 19.9/8 | 0.01
Fluence | 3061 | 11.5/8 | 0.18
Short GRBs | Duration | 603 | 74.2/8 | $7\times 10^{-13}$
Fluence | 603 | 7.1/8 | 0.53
All GRBs | Energy flux | 2298 | 67.4/8 | $2\times 10^{-11}$

Figure 1: Distributions of the first digit of the duration $T_{90}$ and
fluence of all $3664$ GRBs from the Fermi-GBM burst catalog. The theoretical
predictions from Benford’s law (solid lines) along with associated binomial
errors are also shown for comparisons.

Figure 2: Distributions of the first digit of the duration $T_{90}$ and
fluence of $3061$ long GRBs (with $T_{90}>2$ s) from the Fermi-GBM burst
catalog. The theoretical predictions from Benford’s law (solid lines) along
with associated binomial errors are also shown for comparisons.

Figure 3: Distributions of the first digit of the duration $T_{90}$ and
fluence of $603$ short GRBs (with $T_{90}\leq 2$ s) from the Fermi-GBM burst
catalog. The theoretical predictions from Benford’s law (solid lines) along
with associated binomial errors are also shown for comparisons.

##  3 Summary

In this work, we have performed a systematic investigation on the first digit
distributions of the duration, fluence, and energy flux of GRBs. For the
complete GRB sample, our results show that the first digits of the duration
and fluence are not uniformly distributed, but that small digits are more
common than big ones according to a logarithmic distribution as expected by
Benford’s law. However, not all quantities obey Benford’s law. Artificial and
restricted quantities often deviate from the law, such as the energy fluxes of
GRBs in our study. The main reason for the deviation may be that the energy
flux measurements are restricted by spectral fits for each GRB, and only 63%
of bursts have the required spectral information. Thus, there is not enough
dynamic range to make the energy fluxes be fully compliant with Benford’s law.

The Fermi-GBM burst catalog also revealed a bimodal $T_{90}$ distribution, and
two classes of GRBs, i.e., long versus short GRBs with a separation at about 2
s, were confirmed (Kouveliotou et al. 1993; von Kienlin et al. 2020). Using
the conventional long–short GRB classification scheme, we found that there
were $3061$ long GRBs (with $T_{90}>2$ s) and $603$ short GRBs (with
$T_{90}\leq 2$ s). Here, the first digits of the duration and fluence of long
and short GRBs were also examined for adherence to Benford’s distribution. We
found that the fluences of both long and short GRBs still follow Benford’s
law, but their durations are no longer consistent with this law. Our results
indicate that the data of fluence seems to be very natural and believable, but
that $T_{90}$ is not always a good quantity to conduct GRB classification.
That is, the long–short GRB classification scheme does not directly represent
the intrinsic physical classification scheme.

Figure 4: Distribution of the first digit of the energy flux of $2298$
available GRBs from the Fermi-GBM burst catalog. The theoretical prediction
from Benford’s law (solid line) along with associated binomial errors are also
shown for comparisons.

###### Acknowledgements.

This work is partially supported by the Strategic Priority Research Program of
the Chinese Academy of Sciences (grant No. XDB0550400), the National Natural
Science Foundation of China (grant Nos. 12373053 and 12321003), the Key
Research Program of Frontier Sciences (grant No. ZDBS-LY-7014) of Chinese
Academy of Sciences, and the Natural Science Foundation of Jiangsu Province
(grant No. BK20221562).

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