# CARLA-Autoware-Bridge: Facilitating Autonomous Driving Research with a
Unified Framework for Simulation and Module Development

Gemb Kaljavesi1, Tobias Kerbl1, Tobias Betz1, Kirill Mitkovskii2 and Frank
Diermeyer1 1Gemb Kaljavesi, Tobias Kerbl, Tobias Betz and Frank Diermeyer are
with the Institute of Automotive Technology at the Technical University of
Munich (TUM), DE-85748 Garching, Germany. {gemb.kaljavesi, tobias.kerbl,
tobias94.betz<EMAIL_ADDRESS>Mitkovski was not supported by any
organization<EMAIL_ADDRESS>
###### Abstract

Extensive testing is necessary to ensure the safety of autonomous driving
modules. In addition to component tests, the safety assessment of individual
modules also requires a holistic view at system level, which can be carried
out efficiently with the help of simulation. Achieving seamless compatibility
between a modular software stack and simulation is complex and poses a
significant challenge for many researchers. To ensure testing at the system
level with state-of-the-art AV software and simulation software, we have
developed and analyzed a bridge connecting the CARLA simulator with the AV
software Autoware Core/Universe. This publicly available bridge enables
researchers to easily test their modules within the overall software. Our
investigations show that an efficient and reliable communication system has
been established. We provide the simulation bridge as open-source software at
https://github.com/TUMFTM/Carla-Autoware-Bridge

##  I Introduction

In 2018, the World Health Organization (WHO) published a road safety report
[1] indicating that over one million individuals die annually in road
accidents. Automated Vehicles can potentially decrease this figure [2]. To
realize this potential and facilitate the widespread adoption of autonomous
vehicles, they need extensive testing, presenting a significant challenge [3,
4]. In general, approaches to autonomous driving can be categorized as modular
or end-to-end. Modular approaches involve software comprised of distinct,
self-contained modules addressing specific tasks interconnected with each
other. In autonomous driving, these modules include mapping, localization,
perception, planning and control [5]. The complex interdependencies among
modules pose a known issue in autonomous robotics literature [6], addressed by
frameworks like the Robot Operating System (ROS) [7]. End-to-end approaches
aim to treat these modules as a machine learning task, often with a simpler
architecture compared to modular ones.

Figure 1: CARLA on the left and Autoware Core/Universe on the right during a
simulation.

However, modular approaches are more widespread in both industry and research.
Their modularity allows teams to focus on specific tasks and independently
enhance the system, ensuring operational functionality with practical
intermediate outputs [8]. A modular approach can be tested at both the
component and system level. For instance, perception can be evaluated using
human-annotated datasets [9], while planning can be tested through low-
fidelity simulations, providing a simplified representation of the
environment, the kinematics, and sensors [10]. However, interactions between
components are not considered, and not all possible errors in AV software can
be detected [11, 12, 13]. Hence, system-level tests are necessary, involving
evaluating the system as a whole. This can be efficiently achieved through
High-Fidelity (HF) simulations, which incorporate mathematical models of the
environment, the AV, surrounding road users, and sensors.

However, testing self-developed modules within a comprehensive HF simulation
framework is currently not possible with state-of-the-art AV software, nor are
detailed performance analyses available for older frameworks.

###  I-A Related Work

There is an increasing number of publicly available software projects in the
field of autonomous driving. Researchers and the automotive industry can
profit at a large scale from these open-source projects, sharing their results
and leveraging the potential of a drastically growing community [5].

####  I-A1 Open-Source Software for Autonomous Vehicles

Currently, the most advanced modular open-source software stacks for
autonomous driving are Autoware Core/Universe [14] and Apollo [15]. Autoware
is developed and maintained by the Autoware Foundation, consisting of more
than 70 international members from academia and industry [16]. Whereas its
first version, Autoware.AI [17], is ROS 1 based, the latest distribution,
Autoware Core/Universe, is based on ROS 2\. Baidu is the company responsible
for the development of Apollo. It was first released in 2017, and its latest
update, Apollo 9.0, was launched at the end of 2023. In addition to Baidu, the
Apollo project comprises more than 100 members, mainly from industry [18]. It
is also to mention that besides AV software with modular architectures, there
are numerous end-to-end approaches for autonomous driving, such as ReasonNet
[19]. As this work focuses on modular AV software, these are not considered
further. Due to the widespread usage of ROS within the research community and
the strong focus on open-source, Autoware constitutes a good option for
research purposes. Apollo, however, lacks proper documentation and focuses
more on industrial-scale in-field testing and data collection.

####  I-A2 High-Fidelity Simulators for Autonomous Vehicles

Testing AVs with HF simulators complements real-world testing and enables a
more precise prediction of the AV’s performance in the real world. The HF
simulation data must closely match the real-world data [20, 21]. Important
performance indicators for HF simulation frameworks are the number of
available assets (e.g., maps, sensors, or vehicles), scenario design and
execution capabilities, and compatibility with external systems [21, 22].
There are various commercial HF simulators (e.g., CarMaker [23]) available,
and a lot of companies involved in AV development built up their own
simulation frameworks, which are not publicly available [21]. However, there
are also various open-source projects that aim to provide HF simulation for
research purposes. In order to evaluate the feasibility of different open-
source HF simulators for our purpose, a subset of criteria partially based on
[22, 21, 20] is used. Discontinued projects, like the widespread LGSVL
simulator [24], are not suitable since they do not undergo further
development. Some simulators are tailored for specific use cases, while others
lack of realism (e.g., esmini [25]). Since a HF simulator for autonomous
driving in real-world settings is required, these simulators are also not an
option. Both CARLA [26] and AWSIM [27] fulfill these initial requirements.
They include sensor models for LiDAR, Camera, IMU, GNSS, and vehicle status
information. Whereas CARLA comes with several different vehicle models and
maps, AWSIM only includes one of each. In general, CARLA outperforms AWSIM in
terms of provided assets. Both provide a ROS interface for communication with
external AV software. As AWSIM was developed specifically for Autoware
Core/Universe, no further middleware is required to use AWSIM with the latest
Autoware release. For the end-of-life Autoware.AI distribution, options for
closed-loop testing with CARLA or the LGSVL simulator were implemented [28]. A
more recent bridge enables the simulation of Apollo 8.0 within CARLA [29].
Yet, there is no possibility of leveraging the extensive assets provided by
CARLA for testing and development of the prominent Autoware Core/Universe
distribution.

###  I-B Contribution

To provide a valuable option of system-level testing in a HF simulation
environment with extensive assets, especially for research purposes, this work
introduces a bridge to link two widely adopted open-source projects: CARLA as
simulation environment and Autoware Core/Universe as state-of-the-art software
under test. Further, experiments were conducted to show the feasibility of the
framework, namely the CARLA-Autoware-Bridge, for closed-loop simulation at
system-level.

The reminder of this work is structured as follows. Section II provides a
detailed description of the proposed simulation framework. Section III
explains the design of the conducted experiments to showcase the capabilities
of the CARLA-Autoware-Bridge. Results obtained from the experiments are
presented in Section IV. Section V derives conclusions from the work, which we
further discuss to indicate future research directions.

##  II Unified Simulation Framework

We outline our framework requirements in Sec. II-A. Following that, we provide
an overview of the framework in interaction with simulation and AV software in
Sec. II-B. Subsequently, we detail the data exchange from CARLA to Autoware in
Sec. II-C, as well as Autoware to CARLA in Sec. II-D. Finally, Sec. II-E
presents the Autoware dependencies that have been developed.

###  II-A Requirements

To accomplish a HF simulation in real-time and offer an easy-to-use framework,
we defined requirements to be fulfilled by the CARLA-Autoware-Bridge. As its
main functionality, the bridge must enable the bidirectional communication
between CARLA and Autoware to transfer sensor data and control commands
(RQ-1). The sensor setup and the vehicle model simulated in CARLA must
correspond to the setup Autoware expects (RQ-2). To maintain CARLA’s real-time
simulation capabilities, the bridge must operate efficiently. It should
minimize computational effort and keep introduced latency to a minimum (RQ-3).
The framework should quickly adapt to diverse sensor setups and vehicle
models, ensuring a fast and easy-to-use configuration process (RQ-4).

  * RQ-1

Interoperability: The bridge should offer an interface between the latest
releases of CARLA and Autoware Core/Universe.

  * RQ-2

Compatibility: CARLA and Autoware should both be provided with a consistent
sensor and vehicle setup.

  * RQ-3

Efficiency: The bridge should have a low overall latency and minimal hardware
requirements.

  * RQ-4

Adaptability: Sensor setup and vehicle model should be configurable in a fast
and flexible manner.

Figure 2: Overview of the framework for system-level testing with the
components developed or customized as part of in this work highlighted in
blue.

###  II-B Overview

Fig. 2 depicts the software framework for system-level simulation of Autoware
Core/Universe using CARLA. The CARLA simulator, with some of its functions, is
shown on the left. It can execute control commands through a simulated
actuator, save and implement simulation settings (e.g., weather), manage and
publish the simulation time, and generate diverse sensor data. Communication
to and from the CARLA simulator occurs via TCP/IP.

The framework leverages the existing ROS-Bridge for the CARLA Simulator [30],
which has been enhanced to meet the authors’ requirements. The primary role of
the ROS-Bridge is to facilitate communication between TCP/IP and ROS 2\. It
also provides the server with the next time step after receiving all expected
sensor data, ensuring synchronization and reproducibility of the simulation.

The CARLA-Autoware-Bridge, developed as part of this work, links the AV
software and the ROS-Bridge. It manages initial simulation settings, sensor
configurations of the AV, and the AV itself. Additionally, it converts sensor
information from CARLA if necessary. For efficiency, non-converted sensor data
is remapped to the correct topics and sent directly from the ROS-Bridge to
Autoware. Control data sent by Autoware is transmitted directly to the ROS-
Bridge and converted into CARLA control commands using a controller developed
as part of this work.

Furthermore, we created a vehicle model, an exemplary sensor kit, and a
Lanelet2 map required for Autoware in this context. This framework makes
autonomous driving possible with a specific CARLA vehicle model, on one
particular map, and with any sensor setup within the simulation.

###  II-C CARLA to Autoware: Sensor Data

CARLA supports a variety of sensors, but its default data output is
incompatible with Autoware Core/Universe. To optimize efficiency, the ROS-
Bridge has been modified to handle and modify extensive data, such as camera
images or LiDAR point clouds. As a result, that data does not need to pass
through the CARLA-Autoware-Bridge. Adjustments made in either the ROS-Bridge
or the CARLA-Autoware-Bridge include modifying the quality of service for
messages and converting data projection from a left-handed coordinate system
to a right-handed coordinate system. Additionally, certain information from
CARLA needed summarization, while others required splitting. Moreover, ROS 2
messages had to be converted into messages suitable for Autoware
Core/Universe.

###  II-D Autoware to CARLA: Control Commands

The Autoware Core/Universe Controller transmits an Ackermann Control Command,
encompassing target speed, target acceleration, and required tire angle, among
other parameters. Autoware manages the vehicle’s longitudinal movement through
a Proportional-Integral-Derivative (PID) controller and the lateral movement
using a Model Predictive Control (MPC). However, direct transfer of the
Autoware Control Command to the CARLA simulation is not feasible. CARLA
accepts the gas pedal position, brake pedal position, and steering wheel angle
as inputs. To bridge this gap, we fine-tuned a PID controller to convert the
longitudinal commands. The ROS-Bridge incorporates a simple PID controller
implementation for this purpose. In addition to that, we generated a mapping
to convert the tire angle to a steering angle. To enhance framework usability
and ensure compatibility with future Autoware Core/Universe versions,
controller parameter tuning for the Autoware PID or MPC was omitted.

###  II-E Autoware Dependencies

In addition to ensuring compatible communication between Autoware and CARLA,
providing the necessary dependencies for Autoware is crucial. Firstly,
Autoware requires the simulated vehicle as a 3D model for rendering, along
with the associated vehicle parameters. To achieve this, the VW T2 2021 was
extracted from CARLA, converted into the correct format, and measured using
the open-source tool Blender [31].

Additionally, Autoware requires the sensor kit, which provides information
about the type and position of sensors within the simulation. For this
purpose, an exemplary sensor kit was developed for the vehicle, serving as a
template that can be expanded to include additional sensors.

Finally, Autoware needs an HD map of the CARLA map in Lanelet2[32] format. The
available CARLA maps, originally in OpenDrive format [33], were converted with
the assistance of the CommonRoad Scenario Designer[34]. Furthermore, any
missing or incorrect information in the maps was subsequently edited by hand.

##  III Experiment Design

We conducted experiments to demonstrate the capabilities of the CARLA-
Autoware-Bridge and validate whether we satisfy the requirements specified in
Section III-A. For all experiments, we used the latest version of Autoware
Core/Universe at the time of publication and CARLA 0.9.15. In the subsequent
sections, we outline the design of these experiments. First, Section III-A
details experiments demonstrating the accurate transmission of control
commands. Secondly, we defined test cases in Section III-B to benchmark the
overall performance of the simulation framework in terms of computational
efficiency. All the experiments are performed on a simulation workstation
equipped with an Intel i9-14900K CPU (up to $6.0\text{\,}\mathrm{GHz}$),
$192\text{\,}\mathrm{GB}$ RAM, and an NVIDIA RTX 4090 GPU.

###  III-A Transmission of Control Commands

We designed these experiments to evaluate our pipeline to transfer control
commands from Autoware to CARLA, which is outlined in Section II-C. To enable
a separate evaluation of both the introduced PID controller and the steering
angle mapping, the experiment is divided into a longitudinal and lateral test
case. This division allows for an isolated observation of the performance of
each introduced module. At the beginning of each test run, the vehicle is
placed in the same initial pose on the map Town10HD and is assigned either the
goal position for the longitudinal or lateral case.

The PID controller for longitudinal control is evaluated on a straight, with
Autoware given the goal to drive along it with a constant target velocity of
$8.33\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$. We track the actual
velocity of the vehicle and compare it against the given target velocity. The
velocity curve is evaluated qualitatively regarding potential over- or
undershooting and target value tracking. The experiment is conducted once with
the default PID settings of the ROS-Bridge and once with manually tuned
parameters obtained within this work.

The mapping of the steering angle for lateral control is assessed in a
scenario in which the vehicle has to take three consecutive sharp turns. The
experiment is performed once before the introduction of the mapping and
afterward. To compare these two variants, we track the lateral deviation from
the center line of the current lane.

###  III-B Performance Benchmark

In this section, we detail the experimental setup designed to evaluate the
developed CARLA-Autoware-Bridge. The core objective of our study is to assess
the efficiency across various configurations, focusing on the integration of
multiple cameras and LiDAR sensors with different resolutions. This setup aims
to simulate realistic autonomous driving scenarios, providing insights into
the system’s processing capabilities, communication efficiency, and overall
performance under varying conditions. The benchmarks is designed to cover
different performance metrics, including CPU utilization and the latency of
the bridge.

For analyzing the performance, we vary the amount of used LiDARs and cameras
in CARLA. Ranging from $1$ to $4$ LiDARs ($1$ to $3$ respectively for
cameras), we additionally need to analyse the frames per seconds resulting in
the CARLA simulation. Usually creating multiple instances of sensors increase
the workload for the simulation environment. For the LiDAR sensor we
investigate multiple point densities ($1000$ pt/s, $100\,000$ pt/s,
$500\,000$, and $1\,000\,000$ pt/s). For the camera setup we investigate two
common resolution for images: $1280$p x $720$p (HD) and $1920$p x $1080$p
(Full-HD).

The CPU usage is measured by acquiring the data for the individual processes
launched on the compute system with standard Linux ps. We sample the process
data every $200\text{\,}\mathrm{ms}$. The processes are the filtered to the
relevant once for the ROS-Bridge and the CARLA-Autoware-Bridge.

To investigate the latency of the bridge, we use the framework developed by
Betz et al. [35], which is based on ros2_tracing [36]. We measure the latency
in the ROS-Bridge for the individual sensor setups with the corresponding ROS
2 callbacks. Note that most of the sensor data are not feed through the CARLA-
Autoware-Bridge. Therefore, Autoware is directly subscribing to the modified
ROS-Bridge.

All the experiments are conducted with the Autoware software running closed-
loop. We developed a repeatable scenario according to [37] with a fixed start
and end goal pose. The scenario is conducted in the map Town10HD. In addition
to varying and analyzing the LiDAR and camera sensors, the other sensors in
the sensor kit were consistently active throughout the test runs.

##  IV Results

The results of the experiments are presented below. Sec. IV-A includes the
findings of the controller transmission analyses, while Sec. IV-B contains the
results of the performance benchmarks.

$\displaystyle{5}$$\displaystyle{10}$$\displaystyle{15}$$\displaystyle{20}$$\displaystyle{25}$Time
[s]$\displaystyle{0}$$\displaystyle{2}$$\displaystyle{4}$$\displaystyle{6}$$\displaystyle{8}$Velocity
[m/s]Logitudinal VelocityTarget velocityVelocity (Tuned PID)Velocity (Default
PID)$\displaystyle{0}$$\displaystyle{10}$$\displaystyle{20}$$\displaystyle{30}$$\displaystyle{40}$$\displaystyle{50}$$\displaystyle{60}$$\displaystyle{70}$$\displaystyle{80}$Time
[s]$\displaystyle{-1.0}$$\displaystyle{-0.5}$$\displaystyle{0.0}$$\displaystyle{0.5}$$\displaystyle{1.0}$$\displaystyle{1.5}$$\displaystyle{2.0}$$\displaystyle{2.5}$$\displaystyle{3.0}$Lateral
deviation [m]Lateral Deviation from Center-lineLat. deviation (Mapped)Lat.
deviation (Unmapped)

Figure 3: Comparison of velocity and lateral deviation curves obtained from
test runs with and without the introduced PID controller and steering angle
conversion for control command transmission.

###  IV-A Transmission of Control Commands

Figure 3 depicts the outcomes of the test runs conducted to assess the
conversion and transmission of control commands from Autoware to CARLA. The
data from the longitudinal test case reveal that the PID Controller from the
ROS-Bridge, equipped with the default parameter set, cannot reach the target
velocity. However, with the tuned parameter set, the vehicle achieves its
intended target speed, even though the longitudinal velocity shows undesirable
deviations in some sections.

The results from the test cases for the lateral control of the vehicle
demonstrate that the lateral deviation from the lane center line is generally
acceptable with the introduced mapping of the steering commands. The vehicle
successfully navigated through all three sharp turns and stayed within the
lane at all times. In contrast, the vehicle equipped with the direct
forwarding of the steering angles departed from the lane at the first turn,
whereby Autoware disengaged.

###  IV-B Performance Benchmark

As described in section III-B we analyse multiple performance metrics to give
an insight into the computational efficiency of the developed bridge. In Table
I the results are depicted for the experiments regarding CPU utilization and
the corresponding resulting FPS in the CARLA simulator by varying the sensor
setup.

An increase in the point cloud density directly correlates with a decrease in
FPS within CARLA. This outcome is anticipated, as higher point cloud densities
demand more computational power to process the number of sensors.
Interestingly, a decrease in FPS is accompanied by lower CPU utilization. This
suggests that as the simulation becomes less efficient in terms of frame rate,
it concurrently requires less computational effort. Furthermore, the addition
of multiple sensors to the simulation impacts CPU usage less significant than
increasing the point cloud density. This distinction implies that the
computational load of processing high-density point clouds is more demanding
than managing data from multiple sensors. The FPS drop of CARLA is higher for
increased point cloud density as for multiple instances of sensors. For
configurations involving a LiDAR setup with up to four sensors and a point
cloud density of $100\,000$ pt/s, the simulation maintains a FPS rate above
$30$, indicating a threshold for maintaining real-time performance with
acceptable limits. For the camera configurations it is more limited. While
ensuring for both resolution a FPS above $30$, we cannot add a second camera
with Full-HD resolution without dropping the FPS below $30$. Rendering camera
images is a more resource extensive task compared to LiDAR point cloud
generation. However, anomalies in CPU utilization were observed, with
occasional slight increase compared to preceding configurations. For example
$3$ LiDARs and $1000$ pt/s a higher utilization is measured compared to the
$2$ LiDAR setup. These outliers suggest that there may be additional factors
at play, influencing CPU demand in less predictable ways.

The release sensor kit in the open-source repository utilized a single CPU
core, demonstrating the ability to achieve an FPS of $35$ on the used
simulation platform.

$\displaystyle{0.3}$$\displaystyle{0.4}$$\displaystyle{0.5}$$\displaystyle{0.6}$Latency
[$\displaystyle\mathrm{ms}$]$\displaystyle{0}$$\displaystyle{200}$$\displaystyle{400}$$\displaystyle{600}$$\displaystyle{800}$FrequencyLiDAR
1000
pts/s$\displaystyle{0}$$\displaystyle{1}$$\displaystyle{2}$$\displaystyle{3}$$\displaystyle{4}$Latency
[$\displaystyle\mathrm{ms}$]$\displaystyle{0}$$\displaystyle{50}$$\displaystyle{100}$$\displaystyle{150}$$\displaystyle{200}$$\displaystyle{250}$$\displaystyle{300}$FrequencyLiDAR
100000
pts/s$\displaystyle{8}$$\displaystyle{10}$$\displaystyle{12}$$\displaystyle{14}$$\displaystyle{16}$Latency
[$\displaystyle\mathrm{ms}$]$\displaystyle{0}$$\displaystyle{100}$$\displaystyle{200}$$\displaystyle{300}$FrequencyLiDAR
500000
pts/s$\displaystyle{0}$$\displaystyle{5}$$\displaystyle{10}$$\displaystyle{15}$$\displaystyle{20}$Latency
[$\displaystyle\mathrm{ms}$]$\displaystyle{0}$$\displaystyle{100}$$\displaystyle{200}$$\displaystyle{300}$$\displaystyle{400}$$\displaystyle{500}$$\displaystyle{600}$FrequencyLiDAR
1000000
pts/s$\displaystyle{1.0}$$\displaystyle{1.5}$$\displaystyle{2.0}$$\displaystyle{2.5}$Latency
[$\displaystyle\mathrm{ms}$]$\displaystyle{0}$$\displaystyle{100}$$\displaystyle{200}$$\displaystyle{300}$$\displaystyle{400}$$\displaystyle{500}$FrequencyCamera
1280px720p$\displaystyle{4}$$\displaystyle{6}$$\displaystyle{8}$$\displaystyle{10}$Latency
[$\displaystyle\mathrm{ms}$]$\displaystyle{0}$$\displaystyle{100}$$\displaystyle{200}$$\displaystyle{300}$$\displaystyle{400}$FrequencyCamera
1920px1080p

Figure 4: Histograms displaying the latency of the developed framework,
measuring the time between receiving messages from the simulation and
outputting messages to the AV software for various sensors and sensor
configurations. TABLE I: The average CPU utilization and standard deviation,
along with the frames per second of the simulation, are presented for various
LiDAR-only and camera-only configurations, as well as the released sensor kit.
(MCPU=Mean CPU Utilization, DCPU=CPU Utilization Standard Deviation,
AFPS=Average Frames per Second).

Sensors | Sensor Count | 1000 [$\text{\,}\text{pts/s}$] |  $100\,000$ [$\text{\,}\text{pts/s}$] |  $500\,000$ [$\text{\,}\text{pts/s}$] |  $1\,000\,000$ [$\text{\,}\text{pts/s}$]
---|---|---|---|---|---
MCPU | DCPU | AFPS | MCPU | DCPU | AFPS | MCPU | DCPU | AFPS | MCPU | DCPU | AFPS
LiDAR | 1 | 135.6 | 0.10 | 101.2 | 132.3 | 2.06 | 95.7 | 109.0 | 1.00 | 52.6 | 98.9 | 1.65 | 36.3
2 | 125.57 | 3.18 | 96.9 | 120.9 | 3.06 | 78.9 | 92.7 | 2.08 | 35.2 | 90.5 | 1.08 | 21.4
3 | 127.9 | 1.18 | 87.5 | 117.3 | 2.00 | 61.1 | 93.7 | 1.85 | 25 | 84.4 | 1.22 | 15.1
4 | 117.0 | 3.01 | 69.7 | 119.4 | 1.31 | 50.6 | 84.5 | 1.74 | 19.1 | 83.5 | 1.09 | 11.2
Sensors | Sensor Count | 1280p x 720p | 1920p x 1080p |  |  |  |  |  |
MCPU | DCPU | AFPS | MCPU | DCPU | AFPS |  |  |  |  |  |
Camera | 1 | 77.1 | 2.04 | 41.0 | 89.1 | 1.50 | 33.2 |  |  |  |  |  |
2 | 79.3 | 1.11 | 37.1 | 91.5 | 1.78 | 23.9 |  |  |  |  |  |
3 | 65.2 | 0.23 | 23.8 | 91.1 | 3.30 | 18.2 |  |  |  |  |  |
Sensors | Sensor Count | Release |  |  |  |  |  |  |  |  |
MCPU | DCPU | AFPS |  |  |  |  |  |  |  |  |
LiDAR + camera | 2 + 1 | 97.5 | 1.18 | 35.2 | Release sensor kit: $500\,000$[$\text{\,}\text{pts/s}$]; $100\,000$[$\text{\,}\text{pts/s}$]; 1280px720p

For the latency metric the results are shown in Fig. 4. It is observable that
the latency increases with the number of points per second for the LiDAR and
the camera resolution. The latency increases significantly, with larger point
clouds. Although not depicted in the figure, the influence of multiple sensors
was also analyzed. It was found that there is no significant impact on
latency. The sensor kit released concurrently with this work achieves a
maximum latency of less than $15\text{\,}\mathrm{ms}$ and an average of
$7.8\text{\,}\mathrm{ms}$.

##  V Conclusion

There is a strong demand for a method to test a state-of-the-art modular AV
software stack through HF simulation at the system level. With our efforts, we
have developed a framework capable of facilitating such system-level testing
using the CARLA Simulator and Autoware Core/Universe. We established the
communication chain through an enhanced ROS-Bridge and our CARLA-Autoware-
Bridge, also creating a compatible sensor kit, a vehicle model, and a Lanelet2
map. We provided and analyzed a conversion of Ackermann Control Commands to
throttle, brake, and steering angle to control the actuators. A performance
benchmark was employed to measure the latency and CPU utilization of the
bridge, demonstrating its scalability. However, we observed that the bridge’s
latency increases rapidly with larger LiDAR point clouds, likely due to
coordinate changes. This calculation can be implemented more efficiently in
the future. Additionally, the map was manually generated; however, future
efforts should include an automatic conversion from OpenDrive to Lanelet2 that
is compatible with Autoware. In summary, an essential framework for system-
level testing of autonomous driving functions has been developed. To
facilitate enhanced testing, the system should be expanded in the future with
a database of three-dimensional scenarios.

## Acknowledgment

Gemb Kaljavesi and Tobias Kerbl, were the main developers of the presented
work and main contributer of this paper. Tobias Betz contributed to the
performance benchmarks. Kirill Mitkovskii helped with the development of the
software. Frank Diermeyer made essential contributions to the conception of
the research project and revised the paper critically for important
intellectual content. This work was supported by the project Wies’n Shuttle
(FKZ 03ZU1105AA) in the MCube cluster, the Federal Ministry of Economic
Affairs and Climate Actions within the project ATLAS-L4 (FKZ 19A21048l) and
through basic research funds from the Institute for Automotive Technology.

AV

    Automated Vehicle
WHO

    World Health Organization
MPC

    Model Predictive Control
PID

    Proportional-Integral-Derivative
HD

    High Definition
ROS

    Robot Operating System
HF

    High-Fidelity
LiDAR

    Light Detecting and Ranging

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  *[WHO]: World Health Organization
  *[ROS]: Robot Operating System
  *[AV]: Automated Vehicle
  *[HF]: High-Fidelity
  *[AVs]: Automated Vehicle
  *[PID]: Proportional-Integral-Derivative
  *[MPC]: Model Predictive Control


# High Yield of Alpha Particles Generated in Proton-Boron Nuclear Fusion
Reactions Induced in Solid Boron Hydride B18H22

M. Krůs<EMAIL_ADDRESS>Institute of Plasma Physics of the Czech Academy of
Sciences, Za Slovankou 1782/3, 182 00 Prague, Czechia  M. Ehn  Institute of
Inorganic Chemistry of the Czech Academy of Sciences, 250 68 Husinec-Řež,
Czechia  M. Kozlová  Institute of Plasma Physics of the Czech Academy of
Sciences, Za Slovankou 1782/3, 182 00 Prague, Czechia  J. Bould  Institute of
Inorganic Chemistry of the Czech Academy of Sciences, 250 68 Husinec-Řež,
Czechia  P. Pokorný  Institute of Plasma Physics of the Czech Academy of
Sciences, Za Slovankou 1782/3, 182 00 Prague, Czechia  Czech Technical
University in Prague, Faculty of Nuclear Sciences and Physical Engineering,
Břehová 7, 115 19 Prague, Czechia  P. Gajdoš  Institute of Plasma Physics of
the Czech Academy of Sciences, Za Slovankou 1782/3, 182 00 Prague, Czechia
Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical
Engineering, Břehová 7, 115 19 Prague, Czechia  R. Dudžák  Institute of Plasma
Physics of the Czech Academy of Sciences, Za Slovankou 1782/3, 182 00 Prague,
Czechia  Institute of Physics of the Czech Academy of Sciences, Na Slovance
1999/2, 182 00 Prague, Czechia  O. Renner  Institute of Plasma Physics of the
Czech Academy of Sciences, Za Slovankou 1782/3, 182 00 Prague, Czechia  M.
Guldan  Czech Technical University in Prague, Faculty of Nuclear Sciences and
Physical Engineering, Břehová 7, 115 19 Prague, Czechia  M. Myška  Czech
Technical University in Prague, Faculty of Nuclear Sciences and Physical
Engineering, Břehová 7, 115 19 Prague, Czechia  L. Škoda  Czech Technical
University in Prague, Faculty of Nuclear Sciences and Physical Engineering,
Břehová 7, 115 19 Prague, Czechia  M. G. S. Londesborough<EMAIL_ADDRESS>Institute of Inorganic Chemistry of the Czech Academy of Sciences, 250 68
Husinec-Řež, Czechia

###### Abstract

The use of solid boron hydride molecules as a fuel for proton-boron fusion was
proposed by M. Krus and M. Londesborough at ”Interaction of Inorganic
Clusters, Cages, and Containers with Light” workshop in November 2021. Here we
demonstrate experimentally, for first time, that the solid boron hydride,
octadecaborane - anti-B18H22, produces a relatively high yield of alpha
particles of about $10^{9}$ per steradian using a sub-nanosecond, low-contrast
laser pulse (PALS) with a typical intensity of $10^{16}$ Wcm-2. In contrast to
previously published proton-boron studies, the boron hydrides, due to their
composition containing only atoms of boron and hydrogen, represent a natural
choice for future proton-boron nuclear fusion schemes.

##  I Introduction

The nuclear reaction of proton and boron-11 nuclei (pB) releasing three alpha
particles was discovered by Mark Oliphant [1] in the Cavendish laboratory in
1933.

$p+{}^{11}B\rightarrow 3\alpha$ (1)

Since that time, this reaction has been extensively studied to determine its
cross section (or fusion rate) both experimentally and theoretically [2, 3, 4,
5, 6, 7, 8, 9, 10, 11], as well as to assess alpha particle angular
distributions [12, 13, 14]. Such a reaction was later proposed as a source of
fusion energy. However, calculations have suggested that a conventional
confinement fusion scheme is difficult due to the large bremsstrahlung
radiation losses and alpha particle energy deposition in the plasma [15]. As
an alternative, several non-thermal schemes have been proposed for fusion
ignition and burning, such as a heat detonation wave [16] and plasma block
collision [17, 18]. Computational analysis of these schemes predicted fusion
rates that indicate the possibility for successful pB fusion [19, 20]. This
recognition has sparked an increasing interest in the pB reaction as a viable
alternative for a clean neutron-less fusion energy source and as a means to
enable the pain-free and non-invasive hadron therapy of cancer through the
generation of alpha particles that augment the radiation effect [21, 22, 23,
24]. Another promising potential application of such an alpha particle source
is the production of radioisotopes for imaging therapy or radiotracing [25,
26]. Moreover, the high energy gamma photons, which can also be emitted (with
negligible cross section [27]) during pB reactions, might be used for online
proton beam monitoring during cancer treatment [28].

Recently, rapid advances in laser technology have opened new approaches to
laser-ignition of pB fusion. A systematic study of these alternatives began in
2005 by application of a high-power laser to initiate pB fusion reactions in
boron-enriched polyethylene [29]. Within the scope of this study, experiments
detected around 1000 fusion alpha particles per laser pulse. Eventually, this
result was corrected to the yield of $10^{5}$ alpha particles per steradian
per shot. Subsequently, experiments performed at LULI by combining two laser
pulses - a nanosecond pulse for the ionization of a solid pure boron target,
and a picosecond pulse for proton acceleration via the TNSA (target normal
sheath acceleration) mechanism [30] resulted in pB fusion with an alpha
particle yield of $10^{7}$ /sr/shot [31].

These pioneering experiments engendered further studies. For example, pB
fusion was successfully achieved with a yield of $10^{10}$ /sr/shot in 2020 at
PALS laser [32], using boron nitride targets containing hydrogen. Other
experimental schemes can be divided into two main groups: (i) the “pitcher-
catcher” scheme, using an accelerated proton beam focused onto a pure boron
target [31, 33, 34, 35] or boron nitride target [36], and (ii) the “in-target”
scheme, using the direct irradiation of a boron-doped polyethylene target
[29], a boron nitride target [32], or a boron-doped hydrogen-enriched silicon
target [37, 38].

In addition to these laser triggered pB fusion schemes, alternative methods
are currently under investigation. Examples are colliding beam fusion [39]
using a Field-Reversed Configuration [40] and, more recently, pB fusion was
demonstrated at the Large Helical Device stellarator [41], being the first
realisation of pB fusion in a magnetically confined plasma.

The laser pulse driven generation of alpha particles by pB fusion (in-target
geometry) was recently demonstrated at a kHz repetition rate using a 10 GW, 20
mJ laser system focused on boron nitride targets with a thin polymer film
layer as a source of protons. The yield of alpha particles reached $10^{6}$
alpha particles per second [42].

Despite these early achievements, the search for an optimal pB fuel remains in
its infancy and yet it is a key aspect for the ultimate success of applicable
pB fusion. Hitherto, all tested targets comprised compounds of boron with
heteroelements such as nitrogen, silicon, or carbon-rich polymer films,
offering only sparse boron content and the multi-element contamination
interferes with the pB fusion process by diverting energy from the system. It
is, therefore, of interest to investigate the potential of a group of
compounds containing only atoms of hydrogen and boron - the boron hydrides
(commonly referred to as the boranes) as fuels for the pB fusion process. The
boranes are a broad family of spherically-aromatic molecules with polyhedral
cluster geometries of atoms of boron surrounded by a sheath of hydrogen atoms.
They do not occur naturally, but are readily synthesized in specialised
laboratories from abundantly available natural materials, where multi-gram and
kilogram scale production is feasible. The spatial separation between boron
and hydrogen atoms is a chemical bond extending over approximately 0.1 nm. The
lighter boranes, such as diborane (B2H6) and pentaborane (B5H9), are gases.
However, increasing molecular weight leads to higher boranes, such as
decaborane (B10H14) and octadecaborane (B18H22), which are air-stable
crystalline solids. These borane materials thus offer dense concentrations of
hydrogen and boron atoms in the correct ratios for fusion whilst being devoid
of any other element that may hinder the pB reaction.

Within this work, we provide the first demonstration of the use of solid
boranes as a pB fuel (within in-target geometry) and show the high yield
generation of alpha particles by sub-nanosecond kJ-class laser system
irradiation of solid B18H22 targets.

##  II Experimental setup

The experimental campaign was performed in Prague at the PALS laser facility
utilizing an iodine photodissotiation laser carrying 550 J in 350 ps (FWHM)
pulses at a fundamental wavelength of 1315 nm [43]. The laser beam was tightly
focused onto the boron hydride target by a f/2 lens at zero degrees with
respect to the target normal axis. The focal spot, with a diameter of about 80
µm, implies that the laser intensity at the target reached about $10^{16}$
W/cm2.

Compressed boron hydride targets (octadecaborane - anti-B18H22 \- see Figure
1) with a density of 0.9 g/cm3 were irradiated by the laser to trigger the
proton-boron fusion reaction. The octadecaborane target, containing a natural
mixture of boron isotopes - 20 % of 10B and 80 % of 11B, was synthesized at
the Institute of Inorganic Chemistry using methods based on previously
published procedures [44]. Pure octadecaborane targets each contained 150 mg
of material compressed by 7.5t of pressure into discs of 13 mm diameter and
1.2 mm thickness.

Figure 1: Molecular structure of anti-B18H22. Green atoms are boron, white
atoms are hydrogen.

B18H22 was chosen amongst other molecular boranes for these initial
experiments because of its chemical stability, relative non-volatility,
appropriate B:H ratio (based on the stoichiometry of Equation 1), and its
propitious behaviour in electromagnetic fields [45, 46, 47].

Within this experiment, all the detectors occupied the backward half space of
the vacuum chamber (with respect to the direction of the laser beam
propagation). The experimental setup is shown in Figure 2.

Figure 2: Schematic drawing of the experimental setup. Laser beam (red cone)
is focused (f/2 lens) on octadecaborane target (T). The main diagnostics for
alpha particles are an array of five CR39 nuclear track detectors, a Thomson
parabola spectrometer (TPS) comprising parallel magnetic (electromagnet) and
electric fields, and Time-of-Flight spectrometer (ToF).

The main diagnostics used in this experiment to monitor alpha particle
production was a set of CR39 (PADC - polyallyl diglycol carbonate) nuclear
track detectors. These CR39 detectors enable the shot-to-shot measurement of
the absolute number of the generated alpha particles. Five detectors were used
in the chamber and positioned to optimise the resolution of alpha-particle
detection. Three of the detectors were placed at larger angles to the incident
beam - 60 deg, 70 deg and 80 deg - where the direct emission of high energy
plasma particles is relatively low whereas the alpha particle emission is
isotropic. Two further CR39 detectors were positioned near the laser axis (at
angles of 40 deg and 50 deg) to monitor the plasma particle flux. Each
detector is divided into four sectors of the same size; three of these sectors
were covered by a thin aluminum foil (a thickness of 5 µm, 10 µm and 20 µm),
and one sector without foil. The aluminum filters stop low energy plasma ions
and thus reduce particle flux in order to clearly distinguish adjacent
particle tracks (i.e. minimizing the track trace overlap).

A second diagnostic tool, a Thomson parabola spectrometer (TPS), with a
microchannel plate sensor, was installed at an angle of 30 degrees with
respect to the target normal thus enabling the monitoring of the energy of
particles emitted from the target. This distinguishes between the different
ion species present in the generated plasma. Furthermore, TPS is capable of
monitoring shot-to-shot ion energy distribution and proton and boron ion cut-
off energy.

Lastly, a time-of-flight (ToF) spectrometer, equipped by an ion collector
detector with a collecting area diameter of 20 mm, was positioned at an angle
of 60 degrees with respect to the target normal axis. This was used for shot-
to-shot measurements of the energy distribution of protons and alpha particles
(lower energy particles and heavier ions were filtered out by a 2µm thin
aluminum foil located in front of the ion collector).

##  III Results

The primary diagnostic tool capable of identifying alpha particles with the
highest relative certainty are the CR39 detectors, which also assess the total
number of alpha particles generated. In our experiments, the CR39 detectors
were removed after each shot/exposure and chemically developed by immersion
for 45 minutes in a 6M NaOH aqueous solution maintained at a temperature of 70
℃. This process reveals a 2-dimensional cross-section of particle tracks, seen
as spots, that were then examined by an optical microscope (with a
magnification of 100) coupled with a pixel (CMOS) sensor. A typical image is
shown in Figure 3; the smaller spots are created by tracks that may be
assigned to protons, the larger spots are attributed to alpha particles. The
distribution of small (proton) and large (alpha) spots are depicted in Figure
3b.

---
(a)

(a)

---
(b)

(b)

Figure 3: a) Typical microscope image of visualized ion track in etched CR39
nuclear track detector. Smaller spots correspond to proton and larger spots to
alpha particle tracks. For better understanding, this image was taken in the
marginal part of the sensitive area in which the total number of tracks is
smaller. b) Histogram (blue stars, red curve represents double Gaussian fit)
of analyzed etched track radii (obtained from above image) clearly
distinguishes the proton tracks (with 9 $\mu$m in radius) and alpha particle
tracks (with 14 µm in radius).

In order to verify the assumption that the large spots in Figure 3 are indeed
created by alpha particles we performed two reference experiments. The first
one used a polyethylene foil irradiated by the laser under the same conditions
as the octadecaborane target. In this case, only spots with smaller sizes
corresponding to proton tracks were observed (see Figure S1 in Supporting
Information). In the second reference experiment, the CR39 detectors were
irradiated by alpha particles emitted from an 241Am radioactive source via
$\alpha$-decay, having almost identical energies as expected fusion-produced
alpha particles. Here, the track sizes observed were very similar to those
corresponding to the alpha particles in Figure 3 (see Figure S2 in Supporting
Information). Together, these observations suggest with a high degree of
certainty the generation of alpha particles during the laser pulse-
octadecaborane interaction. Extrapolation of the CR detector signal data over
a full solid angle (i.e. a complete spherical collection) was used to
approximate the total number of produced alpha particles and thus a yield of
alpha particles of $1.7\times 10^{9}$/sr/shot.

Additional evidence for alpha particle generation in our experiment was
provided by the ToF spectrum, which contains a peak at around 3.8 MeV (see
Figure 4) that corresponds well to the expected energy of 3.76 MeV for alpha
particles resulting from nuclear fusion. Although the low energy region ($<$ 2
MeV) of the spectrum is complicated by electromagnetic interference induced in
the ion collector circuit and cables, the peak corresponding to the alpha
particle emission is clearly visible in the spectrum.

Figure 4: Time of flight spectrum recorded by oscilloscope connected to an ion
collector. The spectrum shows a peak around 3.8 MeV corresponding to the
fusion-generated alpha particle energy.

TPS used to monitor the plasma ion energies produced in the experiment and
thereby assess whether they meet the conditions for high cross section alpha
particle production (i.e. $>0.67$ MeV) are depicted in Figure 5. These data
indicate that there are a large number of ions emitted from the created plasma
with the sufficient energy to trigger pB fusion. Furthermore, the enhanced
signal of particles with energies $>2$ MeV for all ion states suggest the
action of non-linear effects during the laser-plasma interaction. These high-
energy particles, however, also contribute to the pB fusion reaction.

---
(a)

(a)

---
(b)

(b)

Figure 5: a) Typical image of proton and boron ion parabolae. b) Proton and
boron ion energy distributions obtained along the particular parabola. (Energy
distribution of B+ is not shown as the amount of these ions was negligible in
the observed energy range.)

##  IV Conclusion

This research marks a significant step forward in the exploration of boron
hydrides as a highly promising fuel for laser-driven proton-boron fusion.
Initial experiments have revealed octadecaborane (B18H22) to exhibit
remarkable potential, demonstrating alpha particle production at yields
($1.7\times 10{}^{9}$/sr/shot) comparable to the most exceptional results
achieved thus far (see Figure 6). This success instills confidence in the
prospect of further investigations with these compounds, which boast extensive
structural diversity. It is anticipated that continued research efforts will
lead to substantial progress in realizing the efficient utilization of proton-
boron fusion as a viable energy source.

2005200820112014201720202023$10^{4}$$10^{5}$$10^{6}$$10^{7}$$10^{8}$$10^{9}$$10^{10}$$10^{11}$[29][49][38][37][32][50]this
work[31][35][36][33]in-targetpitcher-catcher Figure 6: Experimental results on
laser-driven proton-boron fusion which were performed over last two decades.
The alpha-particle yield achieved in the presented experiment is comparable to
yield obtained both from in-target and pitcher-catcher fusion scheme.

##  V acknowledgements

###### Acknowledgements.

This work was performed jointly at the Institute of Inorganic Chemistry of the
Czech Republic and at the PALS Research Infrastructure supported by a program
of Ministry of Education, Youth and Sports of the Czech Republic (project No.
LM2023068). This research is partially funded by Czech Science Agency, project
No. GA23-07563S. This project has received funding from the European Union’s
Horizon Europe research and innovation programme under grant agreement no
101096317. The authors thank the technical personnel of PALS Research Facility
for their support during the experimental campaign.

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# Cosmic ray electron and positron spectra at TeV energies

P. MERTSCH  Institute for Theoretical Particle Physics and Cosmology (TTK),
RWTH Aachen University, Sommerfeldstr. 16, 52074 Aachen, Germany

###### Abstract

Observations of cosmic ray electrons have made great strides in the last
decade and direct observations of the all-electron flux as well as separate
electron and positron spectra are now available up to $\sim 1\,\text{TeV}$. In
this invited contribution to the 2022 edition of the Rencontres de Moriond on
“Very High Energy Phenomena in the Universe”, we review the data on cosmic ray
electron and positron spectra at TeV energies and offer general comments on
their interpretation. Subsequently, we focus on the study of the stochastic
fluctuations and a secondary model for the positron excess.

##  1 Introduction

In recent years, measurements of cosmic ray (CR) electrons and positrons have
received heightened attention. There are four space experiments that are
currently online and have provided measurements: AMS-02, CALET, DAMPE and
Fermi-LAT. The quality of data is exquisite, yet there are discrepancies that
will need to be worked out.

There are a number of reasons why CR electrons and positrons are interesting
and provide information beyond the nuclear component which is dominating both
in terms of number and energy density. First, the positron excess discovered
some 15 years ago by the PAMELA satellite experiment [1] still has not been
unambiguously be clarified. The most popular explanations so far have been
self-annihilation of particle dark matter, pulsars/pulsar wind nebulae as well
as production and subsequent acceleration of secondaries in old supernova
remnants. While there exist complementary observations and constraints from
other channels, better measurements of CR electrons and positrons likely hold
the key to identifying the source of the positron excess.

Second, the study of the transport of CR electrons and positrons gives
additional, complementary information for studying the transport in the
Galaxy. Due to their small masses electrons and positrons suffer significant
radiative losses while nuclei above a few GeV virtually do not lose energy at
all. The spatial and spectral distribution of electrons and positrons
therefore differ from those of nuclei and a solid understanding and successful
modelling of both components is required when searching for the sources of
CRs.

Third and relatedly, the strong energy losses of TeV electrons and positrons
limit the range of sources that can contribute to the observed fluxes at Earth
to young and nearby objects. Consequently, the relevant number of sources
should be fairly small, such that we can hope to identify spectral features
from individual sources [2, 3]. Therefore, CR electrons and positrons could
hold the key to identifying the sources of CRs.

The last point requires some additional explanation. Under some simplifying
assumptions, the transport of CR electrons and positrons in the Galaxy can be
modelled by the following transport equation [4, 5],

$\displaystyle\frac{\partial\psi}{\partial
t}-\vec{\nabla}\cdot\kappa\cdot\vec{\nabla}\psi+\frac{\partial}{\partial
E}\left(b(E)\psi\right)=q(\vec{r},E,t)\,.$ (1)

Here, $\psi$ denotes the spectral density of electrons and positrons, $\kappa$
is the isotropic diffusion coefficient, $b(E)=\mathrm{d}E/\mathrm{d}t$
parametrises the rate of radiative losses and $q$ is the rate of injection of
electron and positron density from sources. The Green’s function for this
partial differential equation is defined as the solution for a point-like,
burst-like source at position $\vec{r}_{0}$ and time $t_{0}$ with a given
spectrum $Q(E)$, that is

$\displaystyle
q(\vec{r},E,t)=\delta(\vec{r}-\vec{r}_{0})\delta(t-t_{0})Q(E)\,,$

Examples of the Green’s function for various distances and ages are shown in
Fig. 1.

Figure 1: Green’s function of the transport eq. (1) with a source spectrum
$Q(E_{0})\propto E_{0}^{-2.2}\exp[-E_{0}/(10^{5}\,\text{GeV})]$. The diffusion
coefficient is of the form $\kappa(E)\propto E^{0.6}$, the half-height of the
CR transport volume $z_{\text{max}}=3\,\text{kpc}$ and the energy losses are
modelled as described in Ref. [3]. The distances of the source are indicated
by the line styles, the ages by the colours.

The flux from the whole population of Galactic sources is then just the sum of
the Green’s functions for all individual sources. At low energies, where the
diffusion-loss-length is larger than the average source distance, individual
point-like and burst-like sources can be approximated by a smooth source
density on the right-hand side of eq. (1). Consequently, the spectrum at these
energies will be fairly smooth. At high energies, however, where only a few
sources contribute, this would be a bad approximation. Instead, individual
sources contribute their Green’s functions and the spectrum will be rather
bumpy. Identifying such features, individual sources can be identified.

##  2 The all-electron flux

Figure 2: All-electron flux measurements by AMS-02 [6], CALET [7], DAMPE [8]
and Fermi-LAT [9].

The latest measurements of the all-electron flux from AMS-02 [6], CALET [7],
DAMPE [8] and Fermi-LAT [9] are compiled in Fig. 2. Between $10\,\text{GeV}$
and $\sim 30\,\text{GeV}$ the data are well-described by a power law
$E^{-3.25}$. At higher energies the spectrum flattens before it steepens
significantly around $1\,\text{TeV}$. However, the various experiments do not
agree on the detailed spectrum between $\sim 30\,\text{GeV}$ and the cut-off:
While DAMPE and Fermi-LAT prefer a spectrum $\propto E^{-3.1}$, AMS-02 and
CALET find a spectrum closer to $\propto E^{-3.15}$. While this might seem
like a minor difference, given the small statistical and systematic
uncertainties quoted, this disagreement is significant.

As for the interpretation, it is useful to remind ourselves of the salient
features expected for the all-electron spectrum if transport is described by
the diffusion-loss eq. (1): This can be best illustrated by the time-scales
involved, see Fig. 3. If there was a range in energy where diffusive losses
were dominant, an $\propto E^{-\Gamma}$ source spectrum would be softened by
diffusive losses with diffusivity $\kappa\propto E^{\delta}$ to an
$E^{-\Gamma-\delta}$ ambient spectrum, in the same way as happens for CR
nuclei. If cooling losses dominate, instead, it is an elementary exercise [10]
to show from eq. (1) that the ambient spectrum is $E^{-\Gamma-(\delta+1)/2}$.
For instance, a source spectrum with $\Gamma=2.3$ and a diffusion coefficient
with $\delta=0.6$ results in an ambient spectral index of
$-\Gamma-(\delta+1)/2=-3.1$, close to what is being measured below $\sim
1\,\text{TeV}$.

Figure 3: Time-scales relevant for the transport of CR electrons and positrons
in the Galaxy. The transport parameters adopted are $\kappa(E)=5\times
10^{28}\,\text{cm}^{2}\,\text{s}^{-1}(E/10\,\text{GeV})^{1/3}$,
$z_{\text{max}}=5\,\text{kpc}$; Klein-Nishina cross-section with energy
density $\rho=\\{0.26,0.6,0.6,0.1\\}\,\text{eV}\,\text{cm}^{-3}$ for CMB, IR,
opt, UV, characteristic temperatures of $\\{20,5\times 10^{3},2\times
10^{4}\\}\,\text{K}$ for IR, opt, UV as well as a $3\,\mu\text{G}$ B-field;
$n_{\text{H}}=0.5\,\text{cm}^{-3}$ (WIM) and
$n_{\text{H}}=0.5\,\text{cm}^{-3}$ (WNM) and $100\,\text{pc}$ wide gas disk.
The advection speed assumed is $30\,\text{km}\,\text{s}^{-1}$ and solar
modulation is thought to become important below $\sim 10\,\text{GeV}$.

Of course, there is more interesting information in the spectral features,
that is the hardening break around $30\,\text{GeV}$ and the softening break at
$\sim 1\,\text{TeV}$. It has been suggested for instance, that the break at
$30\,\text{GeV}$ is the signature of the transition from the Thomson to the
Klein-Nishina regime for the inverse Compton scattering that dominates the
cooling of electrons and positrons at these energies [11]. Other authors
disagree [12] and instead suggest a model with two different population of
sources with different spectra contributing to the all-electron flux, e.g.
supernova remnants at lower and pulsar wind nebulae at higher energies. This
author has found that the strength and position of the spectral break from the
Klein-Nishina suppression depends sensitively on the model for the
interstellar radiation field. With the values adopted as in Fig. 3, we find
the break to take place at a few hundred GeV instead of a few tens of GeV.

The TeV break in the all-electron spectrum is probably even more contentious.
A non-standard interpretation is that of a cooling break due to the transition
between escape-dominated losses and cooling-dominated losses. Of course, this
break–in fact a rather smooth transition–takes place at energies much lower
than $1\,\text{TeV}$ should the standard parameters be adopted. It has been
suggested, however, [13] that the diffusive escape time has been overestimated
by one to two orders of magnitude. In the right-hand panel of Fig. 3 we have
indicated the resulting escape time and adjusted it in such a way that the
transition would be taking place at about $1\,\text{TeV}$. Interpreted in a
diffusion model, such a short escape time would correspond to a CR diffusive
volume of small half-height $z_{\text{max}}$. However, this seems to be in
tension with preliminary data from the AMS-02 experiment on Beryllium isotope
ratios.

##  3 Source stochasticity

As we have argued above, at high enough energies, the loss-length of CR
electrons and positrons will be shorter than the mean distance between
sources. Approximating the actual distribution of sources in space and time
with a smoothly varying source density will then lead to significant errors.
Instead, the fluxes at an arbitrary position will become very sensitive to the
distances and ages of actual sources. We refer to this effect as source
stochasticity. Where precisely this effect becomes important and what
information can be extracted from measurement are important questions.

In the literature, there is a spectrum of models that were suggested to deal
with source stochasticity. Here, we identify three classes:

### Selective models. [14, 15, 16]

A limited set of usually nearby and young sources is modelled individually,
while sources further afield are treated in the smooth approximation. It is
easy to see that this can lead to spectral features beyond the maximum energy
of the contribution from the smooth distribution. The question of how likely
the considered contribution is to arise from a statistical model of sources is
oftentimes not addressed.

### Catalogue models. [2, 17, 12]

These models usually start from a catalogue of specific sources, like pulsars
and supernova remnants. While this removes some of the arbitrariness for the
previous approach, it inherits some of the limitations of said catalogues,
specifically selection biasses and incompleteness. Particularly worrisome is
the absence of nearby but old sources which would not be active sources of
radiation, but would still contribute to the flux of low-energy cosmic ray
electrons.

### Probabilistic models. [18, 10, 3, 19, 12, 20, 21, 22]

Giving up constraints on the existence or non-existence of individual sources
altogether, these models only make use of statistical information gained from
surveys, for instance the density of sources as a function of galacto-centric
radius. Moments of the flux distribution characterising the expectation value
and the level of fluctuations can be computed. Alternatively, realisations of
source distances and ages are drawn from this distribution in a Monte Carlo
(MC) approach and the statistical moments are computed numerically. While this
comes at the price of not being able to predict the actual realisation of the
fluxes at Earth, the statistical analysis can be made very sound. In the
following, we review the study of Mertsch (2018) [3].

Considering measurement of a CR flux in a finite number of energy bins, the
fluxes at different energies can be combined into a flux vector. In a
statistical ensemble, the flux in an energy bin is a random variable, and the
vector of fluxes is a random vector. Previous studies have not considered the
correlation between different energies. However, those correlations do exist
and contain important information, which is encoded in the joint distribution
of $f(\phi_{1},\phi_{2},\mathellipsis\phi_{N})$.

Unlike for the marginals, where the flux distribution can be estimated
analytically, a direct computation of the joint distribution seems forbidding.
Using Monte Carlo simulations, one might consider kernel density estimators,
however, they will fail due to the curse of dimensionality for more than a
handful of energy bins. Instead, we have recently employed a copula
construction [23], a method for constructing an approximate joint distribution
that has gained currency in financial mathematics. This builds on Sklar’s
theorem which factorises an arbitrary $N$-dimensional joint distribution into
a product of $N$ 1D probability distribution functions (PDFs)
$f_{i}(\phi_{i})$, the marginals, and the so-called copula, a function of the
$N$ cumulative distribution functions (CDFs), $F_{i}(\phi_{i})$. It can be
shown that the copula can be decomposed into a nested product which contains
only (conditional) bi-variate PDFs. Details of this construction can be found
in Ref. [3].

Figure 4: Variance and correlations in the flux distribution due to source
stochasticity. Top panel. The left-most and right-most panel show the flux PDF
estimated by the histograms of (logarithmic) flux at $10$ and
$31.6\,\text{GeV}$, respectively. The central panels compare the pair copular
with the histogram of the empirical flux CDF at both energies. Bottom panel.
The same as the top panel, but for $10$ and $31.6\,\text{TeV}$.

We have parametrised the joint distribution of high-energy electron-positron
fluxes with a pair copula construction and determined the free copula
parameters by fitting to a large number of MC samples. A comparison between
the pair-copulas which are functions of the CDFs and the histogram of
empirical CDFs is shown in Fig. 4. We have substituted the measurements by
H.E.S.S. of the all-electron flux between a few hundred GeV and $\sim
20\,\text{TeV}$ into this likelihood. The log-likelihood is the sum of
contributions from the marginals, quantifying the fluctuations around the
expectation value, and from the copula, quantifying the correlations between
different energy bins.

Both contributions resulting from the H.E.S.S. data can be compared with the
distribution of log-likelihood contributions obtained with the simulated
fluxes. This comparison depends of course on the model parameters assumed, for
instance on the source rate. With a canonical rate of supernova remnants, that
is $10^{-2}\,\text{yr}^{-1}$, the contribution from the marginals has a
probability to exceed (PTE) of $0.4$, but the copula contribution has
$\text{PTE}=0.998$. Put differently, the observed flux has less correlations
than would be expected from the simulations. If we decrease the source rate by
one order of magnitude while increasing the power of each individual source by
one order of magnitude, the expectation value of the flux is largely unchanged
if the cut-off energy is adjusted, yet the level of fluctuations is much
increased. We note that a reduced source rate can be considered an
approximation for a class of sources where CR accelerators do not occur
independently in space and time, but where several sources appear in
conjunction, an example being a number of supernova remnants appearing in
super bubbles. It appears that this allows for a much better agreement between
observations and simulations, that is $\text{PTE}=0.466$ (marginals) and
$\text{PTE}=0.298$ (copula). We conclude that our analysis thus supports a
scenario where sources take place with a certain spatial and temporal
coherence.

Figure 5: CDFs for the marginal and copula contribution to the log-likelihood,
both for the MC simulations and for the H.E.S.S. data. Left panel: For the
canonical source rate of $10^{-2}\,\text{yr}^{-1}$, the copula contribution
for the data is much lower than what is expected from simulations. Right
panel: For a decreased source rate of $10^{-3}\,\text{yr}^{-1}$, the copula
contribution for the data agrees well with the simulations.

##  4 Positrons

Figure 6: Electron and positron spectra as measured by AMS-02 [6].

Figure 7: Comparison between proton flux (top left), boron-to-carbon ratio
(top right), positron flux (bottom left) and antiproton flux (bottom right) as
predicted from the model with acceleration of secondaries model [24] and as
measured by AMS-02 [25, 26, 27, 28].

So far, we have only discussed the interpretation of the all-electron flux.
Spectrometric experiments like PAMELA and AMS-02, however, can distinguish
electrons from positrons and have thus provided separate spectra. We show the
measurement from AMS-02 in Fig. 6; note that we have upscaled the positron
flux by a factor 10 in order to be able to compare the spectral shapes. Both
fluxes have been multiplied with energy cubed and so it is apparent that the
electron flux is slightly softer than $E^{-3}$ above $\sim 10\,\text{GeV}$
while the positron flux is harder.

A positron spectrum harder than $E^{-3}$ is incompatible with secondary
production in the interstellar medium (ISM), at least in the conventional
scenarios of Galactic transport. Therefore, a new source of CR positrons is
required. The most exciting possibility would have been non-baryonic dark
matter producing electrons and positrons through self-annihilation. With the
energies involved, masses at the weak scale seemed preferred, however, the
necessary cross-sections needed to be significantly enhanced compared to the
cross-section implied by thermal freeze-out scenarios. Unfortunately, this
possibility is now severely constrained by observations of gamma-rays, CR
antiprotons and the CMB. Pulsars or rather pulsar wind nebulae (PWNe) are a
popular, astrophysical explanation for the hard positron spectrum. PWNe must
be contributing at some level and other contributions at this meeting have
covered such models, e.g. Ref. [21]. Here, we present another astrophysical
expalantion, that is old supernova remnants.

In fact, the lore that the positron excess cannot be due to secondary
positrons is based on a misunderstanding. It is usually said that the
positron’s primaries, e.g. protons, spend relatively short amounts of time
inside the sources, for instance supernova remnants, compared to the transport
time in the Galaxy. Even if the gas density inside the source is enhanced
compared to the ISM density, the grammage accumulated inside the sources is
thus estimated to be negligible. This argument ignores, however, that charged
secondaries like positrons can undergo shock acceleration much like their
primaries, but will attain a spectrum generically harder than the primaries.
For instance, assuming a Bohm-like scaling of the diffusion coefficient for
the source, $\kappa(p)\propto p$, the secondary spectra will be harder than
the primary ones by one power of momentum. This will lead to a harder spectrum
of positrons after propagation than in the standard scenario where production
and acceleration of secondaries in sources is ignored. Of course, other
charged secondaries like anti-protons or boron are equally affected, which
provides for a nice test of this scenario.

We have recently revisited this issue and fitted our model to the data from
AMS-02. Particular care was given to testing various parametrisations of the
cross-sections of positron and anti-proton production. We have managed to find
a very good fit to CR protons, helium, carbon, oxygen, boron, nitrogen,
positrons and antiprotons. We compare our model fluxes with data for a
selection of species in Fig. 7. Further details can be found in Ref. [24].

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