# Socially Adaptive Path Planning Based on Generative Adversarial Network

Yao Wang, Yuqi Kong, Wenzheng Chi*, and Lining Sun *corresponding authorThis
project is partially supported by National Key R$\&$D Program of China grant
#2020YFB1313601, National Science Foundation of China grant #61903267 and
China Postdoctoral Science Foundation grant #2020M681691 awarded to Wenzheng
Chi.Yao Wang, Yuqi Kong, Wenzheng Chi, and Lining Sun are with the Robotics
and Microsystems Center, School of Mechanical and Electric Engineering,
Soochow University, Suzhou 215021, China {ywang, yqkong, wzchi,
<EMAIL_ADDRESS>
###### Abstract

The natural interaction between robots and pedestrians in the process of
autonomous navigation is crucial for the intelligent development of mobile
robots, which requires robots to fully consider social rules and guarantee the
psychological comfort of pedestrians. Among the research results in the field
of robotic path planning, the learning-based socially adaptive algorithms have
performed well in some specific human-robot interaction environments. However,
human-robot interaction scenarios are diverse and constantly changing in daily
life, and the generalization of robot socially adaptive path planning remains
to be further investigated. In order to address this issue, this work proposes
a new socially adaptive path planning algorithm by combining the generative
adversarial network (GAN) with the Optimal Rapidly-exploring Random Tree
(RRT*) navigation algorithm. Firstly, a GAN model with strong generalization
performance is proposed to adapt the navigation algorithm to more scenarios.
Secondly, a GAN model based Optimal Rapidly-exploring Random Tree navigation
algorithm (GAN-RRT*) is proposed to generate paths in human-robot interaction
environments. Finally, we propose a socially adaptive path planning framework
named GAN-RTIRL, which combines the GAN model with Rapidly-exploring random
Trees Inverse Reinforcement Learning (RTIRL) to improve the homotopy rate
between planned and demonstration paths. In the GAN-RTIRL framework, the GAN-
RRT* path planner can update the GAN model from the demonstration path. In
this way, the robot can generate more anthropomorphic paths in human-robot
interaction environments and has stronger generalization in more complex
environments. Experimental results reveal that our proposed method can
effectively improve the anthropomorphic degree of robot motion planning and
the homotopy rate between planned and demonstration paths.

###### Note to Practitioners

The motivation of this work is to provide an efficient learning-based method
to guide the robot generate the path in human-robot interaction environments.
Conventional path planning algorithms can quickly generate the optimal path
from the start to the goal. Although the robot did not collide with
pedestrians, pedestrians are only regarded as moving obstacles, which greatly
affects the psychological comfort of pedestrians. In order to improve the
anthropomorphic degree of robot path planning, we propose the GAN-RRT*. It
combines the RRT* path planning algorithm with the GAN model and learns
anthropomorphic strategies from demonstration paths. The proposed algorithm
can be applied to service robot to improve the anthropomorphic degree.

###### Index Terms:

Robot navigation, GAN-RRT*, IRL.

Figure 1: A comparison of the demonstration path and two planned paths. The
demonstration path is generated by the volunteer controlling the robot and the
planned path are generated by RRT and A*, respectively. It is noteworthy that
the demo path effectively avoid pedestrians comparing with other two paths.

##  I Introduction

Over the last few decades, the autonomous navigation technology of mobile
robots has made important breakthroughs. However, due to the lack of awareness
of social rules, robots only regard pedestrians as movable obstacles in the
navigation process. Some classic path planning algorithms such as the
Dijkstra’s[1] algorithm, the A*[2] algorithm and the Rapidly exploring Random
Tree (RRT) [3] algorithm, can plan a short path to the target point, however,
they still regard pedestrians as ordinary obstacles. The navigation efficiency
often decreases with the increase of pedestrian density in the environment.
During the interaction, people still often need to compromise and accommodate
robots, and their psychological comfort can not be guaranteed.

As shown in Fig. 1, when planning a path in the human-robot interaction
environment, people usually choose the demo path to avoid crowds. On the
contrary, when only treating pedestrians as ordinary obstacles, the robot will
generate path 1 or path 2 to shorten the path length. In order to endow mobile
robots with the awareness of social rules and improve the degree of
anthropomorphism of the robot path, many researchers initiated the study of
robot social navigation[4][5] and have been focused on generating a path
similar to the demo path. Currently, it is still difficult to comprehensively
quantify and improve the similarity between the demo path and the generated
path. Some methods may achieve good performance in certain scenario, however,
the generalization and adaptability of navigation methods still need to be
further improved in various human-robot interaction environments with high
uncertainty.

Currently, the indicators for comparing two paths include the path length, the
navigation time, the path dissimilarity[6], the feature count[6], the path
cost[6] and the Homotopy Rate[7][8].

###### Definition 1

(Homotopy Class of Trajectories): Two trajectories with the same start and the
same end are said to be in the same Homotopy Class if and only if one can be
smoothly deformed into the other without intersecting obstacles. Otherwise,
they belong to different homotopy classes.

The Generative Adversarial Networks (GAN) model was originally proposed based
on the game theory and showed good generalization ability in the field of
image processing. Inspired by these methods, we present the GAN model based
Optimal Rapidly-exploring Random Trees (GAN-RRT*) by combing the GAN model and
the Optimal Rapidly-exploring Random Trees RRT*[9] algorithm. The GAN-RRT*
path planner follows the general framework of the GAN model, which consists of
two adversarial components: the generator and the discriminator. Through the
inverse reinforcement learning process, the GAN-RRT* algorithm can generate
more anthropomorphic paths and adapt to more scenarios by learning from the
demonstration paths.

The rest of the paper is organized as follows: Section II introduces the
problem of path planning and the algorithms of RRT and RRT*. Then Section III
presents the details of the proposed GAN model, GAN-RRT* algorithm and GAN-
RTIRL framework. The dataset collection and experimental results are reported
in Section IV. We further discuss the results and the proposed system, and
draw some conclusions at the end of this paper in Section V.

###  I-A Related Work

In recent years, many scholars have proposed RRT-based algorithms to improve
the efficiency of mobile robot path planning. Chi et al. [10] proposed the
Risk-DTRRT algorithm, which can optimize the global trajectory according to
the trajectory prediction results of pedestrians in the environment. Chi and
Meng [11] proposed the Risk-RRT* algorithm by combining the comfort and
collision risk (CCR) map with the RRT* algorithm. The new Risk-RRT* method can
achieve better results than Risk-RRT [12] in both static and dynamic
environments. Wang and Meng [13] proposed the L-RRT* algorithm, which can
perform much better than the RRT* and Dynamic RRT* (D-RRT*) [14] in terms of
time cost and path length. The cost functions in the above RRT-based
algorithms are pre-defined for specific scenarios, and only extract the
distance from the pedestrian as a reference. However, in the actual navigation
process, the robot should regard the pedestrian as a motion model rather than
a common moving obstacle.

With the deepening of human-robot interaction, some scholars have focused on
improving the comfort of pedestrians during the navigation process. However,
the rules of human society are usually behavioral tendencies, and often change
according to different scenarios. Therefore, it is crucial for the robot
social-adaptive navigation to realize the digital expression of navigation
behavior knowledge. Inspired by Proxemics[15], Huang et al. [16] proposed a
General Comfort Space (GCS) model with an elliptical shape, where the
pedestrian is located at a focal point of the ellipse and the long and short
axes of the ellipse are uniquely determined by the private distance and human
speed in the spatial relations, respectively. Chi et al. [17] proposed a
Comfort and Collision Risk (CCR) map to unify the comfort of the pedestrians
and the collision risk with the barriers in the environment into one risk map.
Truong and Ngo [18] proposed a proactive social motion model (PSMM), which
introduced social interaction space between crowds in robot motion planning to
enable robots to navigate safely in dynamic crowded environments. However, in
human-robot interaction, it is not enough to consider only the pedestrian
motion model. Pedestrians cannot just be regarded as obstacles with a certain
volume and moving speed. In the real world, pedestrians have their own minds
and obey certain social rules. Therefore, we still need to further study the
rules of human-robot social interaction.

The movement rules of pedestrians will change to adapt to diverse human-robot
interaction scenarios, which poses a great challenge to the quantification of
the rules. In order to address this problem, some scholars introduce the end-
to-end machine learning method [19, 20, 21] to the robot navigation framework.
Chen et al. [22] used deep reinforcement learning network to train robot
motion strategies and solved three kinds of local obstacle avoidance problems
(passing, crossing and overtaking) of robots. Kim and Pineau [23] proposed a
local path planning method based on inverse reinforcement learning to plan
motion paths according to different crowd densities and speeds. Pérez-Higueras
et al. [6, 24, 25, 26] extracted the features of human-robot interaction for
navigation tasks, and then obtained a more anthropomorphic robot path planning
algorithm through inverse reinforcement learning framework. Ding et al. [5]
proposed a homotopy penalty strategy and combined it with inverse
reinforcement learning to effectively reduce non-homotopic phenomenon during
robot navigation. In recent years, in order to improve the social adaptive
ability of robots, many scholars combined generative adversarial network (GAN)
to do further research on robot navigation methods. Yang and Peters [27]
proposed a generative adversarial network structure App-GAN to generate a
motion interaction mode for mobile robots when approaching a small group of
pedestrians in conversation. Gupta et al. [28] proposed a sequence-to-sequence
social path generation method based on generative adversarial networks. The
proposed Social-GAN enhanced the degree of anthropomorphism of paths in multi-
person interaction environments. Ma et al. [29] proposed conditional
generative adversarial networks (CGAN) model to generate feasible paths to
guide the sampling process of RRT*. However, the above algorithms focus on
solving human-robot interaction problems in specific navigation scenarios. In
real life, the scenarios and states of human-robot interaction are diverse and
constantly changing, which requires our algorithm to have strong
generalization performance. In addition, in order to further improve the
comfort of pedestrians in the process of robot navigation, we also need to
improve the homotopy rate between planned and demonstration paths.

###  I-B The Proposed System and Approach

Inspired by the above methods, this paper proposes a new path planner with
Generative Adversarial Networks based Optimal Rapidly-exploring Random Trees
(GAN-RRT*). The growth of the RRT* tree depends on the tree node evaluation.
Traditionally, the node evaluation is usually carried out by the predefined
cost function, such as the distance between two adjacent nodes and the
cumulative distance from the tree root. As aforementioned, given the
complexity and the diversity of the human-robot interaction, it is difficult
to comprehensively quantify the social-adaptive paths. Therefore, we propose a
generative network to calculate the cost value of growing nodes on the RRT*
tree, which can make the cost function adaptable to more human-robot
interaction scenarios. Considering the process of inverse reinforcement
learning, we need to give a criterion for the node cost value calculated by
the generative network. In our previous work, we proposed to use the neural
network to output the cost value and judge whether it is reasonable. As
aforementioned, due to the complexity of the training task, it is difficult
for a single neural network to handle the inverse reinforcement learning task.
Therefore, we propose a adversarial network to judge whether the cost value of
growing nodes on the RRT* tree is reasonable, which can offload the burden of
the generative network and improve the performance of the generative network.
In order to formulate the social-adaptive path planning framework, we propose
the GAN-RRT* based Inverse Reinforcement Learning (GAN-RTIRL) by combing the
proposed generative and adversarial network. Through learning from the demo
paths generated by volunteers remotely controlling the robot, the paths
generated by GAN-RRT* will become more and more anthropomorphic with the GAN-
RTIRL algorithm framework. The strong generalization of the GAN model ensures
that GAN-RRT* can be applied to more complex human-robot interaction
environments. In the GAN-RTIRL training framework, GAN-RRT* can learn human-
robot interaction rules from the demonstration path, thereby resulting in a
higher homotopy rate between the planned path and the demonstration path. Our
work contributions can be summarized as follows:

  * •

A GAN model with strong generalization performance to adapt the navigation
algorithm to more scenarios in human-robot interaction environments;

  * •

A novel socially adaptive path planning algorithm named GAN-RRT* by combining
the proposed GAN model and RRT*; and

  * •

A GAN-RTIRL based social adaptive path planning framework for mobile robots to
improve the degree of anthropomorphism.

##  II PRELIMINARIES

###  II-A Path Planning Problem

In this section, we elaborate the basic path planning problem. Let
$\mathcal{X}$ $\in$ $\mathbb{R}^{n}$ be the state space, $\mathcal{X}_{free}$
the free space and $\mathcal{X}_{obs}$ = $\mathcal{X}$ ${/}$
$\mathcal{X}_{free}$ the obstacle space. Let $x_{init}$ $\in$
$\mathcal{X}_{free}$ be initial state and $x_{goal}$ $\in$
$\mathcal{X}_{free}$ the goal state. In order to indicate whether the robot
has reached the goal, we define the goal region as $\mathcal{X}_{goal}$ $\in$
$\mathcal{X}_{free}$. The path planning problem is to find a feasible path
$\delta(t)$ $\in$ $\mathcal{X}_{free}$ for $t$ $\in$ [0,1], where $\delta(0)$
= $x_{init}$ and $\delta(1)$ $\in$ $\mathcal{X}_{goal}$.

Let $\Sigma$ be the set of the feasible paths. To find the optimal path, we
denote a cost function $c(\delta)$. The optimal path planning problem becomes
finding the best path $\delta^{*}$ that minimizes the given cost function
among $\Sigma$:

$\displaystyle\delta^{*}=\mathop{\arg\min}\limits_{\delta\in\Sigma}c(\delta).$
(1) $\displaystyle s.t.\ \delta(0)=x_{init}$
$\displaystyle\delta(1)\in\mathcal{X}_{goal}$
$\displaystyle\delta(t)\in\mathcal{X}_{free}$

The process of finding the optimal path $\delta^{*}$ is actually the process
of finding the set of optimal path nodes $N^{*}$. Finding the optimal set of
path nodes $N^{*}$ is to find the path node $n^{*}$ with the lowest cost
value. In human-robot interaction environments, it is difficult to define an
appropriate cost function for the RRT* algorithm. Therefore, in this article,
we learn an end-to-end GAN model to calculate the cost of the path node
instead.

$\displaystyle c=cost(n)=GAN_{g}(n)$ (2) $\displaystyle GAN(n)=GAN_{d}(n,c)$
$\displaystyle n^{*}=\mathop{\arg\max}\limits_{n\in\mathcal{X}_{free}}GAN(n)$

Figure 2: Description of the proposed GAN-RTIRL framework.

###  II-B The RRT* and NN-RRT* Algorithms

The RRTs algorithm carried out path planning on a node tree, which is usually
generated by randomly growing or biased sampling. The main process can be
divided into four steps, namely random sampling, parent node selection,
collision detection, and node connection. The specific process is explained as
follows:

• Random sampling. Random sampling is carried out on the free area of the map
to generate a new candidate node.

• Parent node selection. While traversing all existing nodes on the tree, the
node closest to the candidate node is chosen as the parent node.

• Collision detection. The collision detection is conducted on the map between
the growing node and the candidate node. If the line between two nodes
encounters an obstacle, the collision detection failed.

• Connection. If the node completes collision detection, it will be added to
the tree. Otherwise, the node will be dropped from the tree.

The RRT algorithm adopts Euclidean distance as the cost function and selects a
node on the tree closest to the candidate node for connection. The RRT*
algorithm adopts the cumulative Euclidean distance as the cost function. In
other words, the process of selecting growth node not only considers the
distance from the candidate node to the growing node, but also the cumulative
distance from the root node to the growing node.

On the basis of the RRT algorithm, the RRT* algorithm adds two optimization
steps: rewiring and re-connection.

• Rewiring. The new node will re-select its parent node to reduce its
cumulative cost to the root node.

• Re-connection. If a new node is set as the parent node, we will change the
parent of the node to the new node.

In the RRT algorithm, pedestrians are only considered as moving obstacles. The
robot can only avoid collisions with pedestrians and ignore the psychological
comfort of pedestrians. In the RRT* algorithm, we proposed a cost function
considering pedestrians and obstacles comprehensively in our previous work
[5]. The cost function considers elements related to social rules, including
the distance to obstacles, the distance to pedestrians and the distance to the
goal. Considering that the linear model cannot accurately express the
influence of the above factors on the interaction state of robots and
pedestrians, we replaced the linear model with a fully connected neural
network and proposed the new NN-RRT* [30] algorithm, which greatly improves
the similarity and homotopy rate between the planned path and the
demonstration path. However, the generalization and adaptability of navigation
methods still need to be further improved in various human-robot interaction
environments with high uncertainty.

##  III ALGORITHM

In this section, we present the framework of GAN-RTIRL in Section III-A. Then
we introduce the objective and structure of our GAN model in Section III-B.
Finally, the details of GAN-RRT* path planner are shown in Section III-C.

###  III-A The GAN-RTIRL Framework

In this work, we propose the GAN-RTIRL framework, as shown in Fig. 2. In the
scenario collection phase, we integrate the information of map, people, start
and goal into the scenario. Then, the volunteers control the robot to generate
demonstration paths and the GAN-RRT* planner is utilized to generate planned
paths. After that, we input the bag files of demonstration paths and generated
paths into the training process. When the generator and discriminator in GAN-
RRT* are in equilibrium, the features of the generated path and the
demonstration path are increasingly similar and the path generator is output
by IRL module.

In the IRL module, we pre-train the generator and discriminator to achieve a
pre-balanced state. The difference from previous work is that we no longer
compare the global features of the two paths. Instead, we extract all node
information of the two paths (Line 7-8 in algorithm 1). Firstly, we train the
discriminator to accurately distinguish between the generated path nodes and
the demonstration path nodes (Line 13 in algorithm 1). And then, the generator
constantly update itself to confuse the discriminator and make it unable to
distinguish between the generated path nodes and the demonstration path nodes
(Line 14 in algorithm 1). This process is repeated until the paths generated
by the generator are similar to the demonstration paths before stopping the
learning process. Finally, a new GAN-RRT* planner is obtained, which can
generate more anthropomorphic paths. The complete pseudo code of the GAN-RTIRL
is shown in Algorithm 2.

Input: Demo paths $\zeta_{plan}$ =
$\left\\{\zeta_{D1},...,\zeta_{DS}\right\\}$ from S scenarios

Output: Generator $G$, Discriminator $D$

1 $G$ $\leftarrow$ Pre-training

2 $D$ $\leftarrow$ Pre-training

3 while  _$G$ not converge_ do

4 for  _s $\in$ S_ do

5 for  _repetitions_ do

6 $\zeta_{plan}$ $\leftarrow$ GAN-RRT*($s$)

7 $N_{plan}$ $\leftarrow$ GetNode($\zeta_{plan}$)

8 $N_{demo}$ $\leftarrow$ GetNode($\zeta_{demo}$)

9

10 end for

11 $\sigma_{plan}$ $\leftarrow$ $\sigma$ $\cup$ $N_{plan}$

12 $\sigma_{demo}$ $\leftarrow$ $\sigma$ $\cup$ $N_{demo}$

13 end for

14 $D$ $\leftarrow$ Update-Discriminator($\sigma_{plan}$, $\sigma_{demo}$)

15 $G$ $\leftarrow$ Update-Generator($\sigma_{plan}$)

16

17 end while

return $G$ and $D$

Algorithm 1 GAN-RTIRL

###  III-B The GAN-based Socially Aware Path Node Evaluation

The socially aware RRTs planner works well in human-robot interaction
environments with a few pedestrians. However, their generalization performance
in complex environments needs to be further improved. In this work, we propose
a GAN-RRT* planner to overcome these issues. The GAN model has strong
generalization performance when dealing with complex problems, which consists
of two neural networks that compete against each other. The two networks are
trained cooperatively through competitive learning to improve the performance
of the generative network. The generator is used to deceive the discriminator,
while the discriminator is used to judge the ability of the generator.
Finally, the ability of the two models becomes stronger and reaches a steady
state.

The GAN model is originally designed to learn a mapping from random noise $z$
to the output image $y$. In the GAN-RRT* algorithm, the GAN model is designed
to learn the mapping from feature vector $f$ to the output node cost value
$c$. The generator $G$ is defined as {$G$ : $f$ $\rightarrow$ $c$ }. The value
of $c$ is calculated by $GAN_{g}(f)$ in Eq. 2. The discriminator $D$ is
designed to distinguishing whether the node cost value generated by $G$ is
real or fake. The probability computed by $D$ that $c$ is the real data is
defined as $D(c)$ corresponding to $GAN_{d}(n,c)$ in Eq. 2. The objective
function can be expressed as:

$\displaystyle\mathcal{L}_{GAN}(G,D)=\mathbb{E}_{c}[logD(c)]+\mathbb{E}_{f}[log(1-D(G(f)))].$
(3)

The GAN aims to train $D$ to maximize the log$D(c)$ and train $G$ to minimize
log$(1-D(G(f)))$. The optimization function can be expressed as:

$\displaystyle\theta^{*}=arg\mathop{min}\limits_{G}\mathop{max}\limits_{D}\mathcal{L}_{GAN}(G,D).$
(4)

It is noteworthy that the generator $G$ and discriminator $D$ can be various
neural networks, without any limitation on the both models of adversarial
networks. Therefore, we can modify the network structure of the GAN model to
adapt our algorithm.

When the robot generates path nodes, the GAN-RRT* algorithm calculates the
cost value of the current node through the $GAN_{g}$ function according to the
environment information around the node. If the path node is away from
obstacles and pedestrians and towards the goal, the result of the output of
the function $GAN_{d}$ is closer to 1. In general, the GAN module can screen
out socially aware path nodes.

To represent the interaction state between the robot and the environment in
the human-robot interaction, we adopt a set of socially aware features, as
shown in Fig. 3.

  * •

$f_{1}$ represents the distance from the robot to the goal.

  * •

$f_{2}$ represents the distance from the robot to the nearest obstacle.

  * •

$f_{3-5}$ represent the cost related to the pedestrian. The cost function is
defined by the Gaussian function, which is divided into three directions:
front, back and right of the pedestrian, respectively.

Figure 3: The features of interaction between the robot and the pedestrian.

The input of the generator in GAN-RRT* includes the five features mentioned
above, which aims to calculate the cost value of the RRT* node in the human-
robot environment accurately. As shown in Fig. 4, there are five neurons
corresponding to $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$, $f_{5}$ in the input
layer and the the output of the generator is the cost value of the node. We
add ten neurons to the hidden layer where the activation function is Sigmod.
The architecture of discriminator is shown in Fig. 5. Similar to the
generator, there are six neurons in the input layer of discriminator and the
the output of the discriminator is the probability that the cost value of the
node is real. With the information of map, pedestrians and goal, the generator
can get the cost value of the node. And then, the discriminator will calculate
these five features and the cost value to obtain a probability value. The
function of the discriminator is to distinguish whether the cost value derived
from the generator is ideal. During the training process, the abilities of the
two models become increasingly stronger, ultimately constraining each other to
reach a balanced state.

Figure 4: The architecture of the generator. The input layer is the feature
information of the map, goal and people detected by the robot. The output
layer is the cost value of the current node of the robot. Figure 5: The
architecture of the discriminator. The input layer is the feature information
and cost value of the current node. The output layer is the discriminant
result of the current node.

Input: Start, Goal, Map and GAN Model $G$

Output: The GAN-RRT* $Tree=(V,E)$

1 $V$ $\leftarrow$ $\left\\{x_{init}\right\\}$; $E$ $\leftarrow$ $\emptyset$;

2 for  _i = 1,…,n_ do

3 $x_{rand}\leftarrow Sample_{i}$;

4 $x_{nearest}\leftarrow$ Nearest($Tree$, $x_{rand}$);

5 $x_{new}\leftarrow$ Steer($x_{nearest}$, $x_{rand}$);

6 if  _ObstcalFree( $x_{new}$)_ then

7 $X_{near}\leftarrow$ Near($Tree$, $x_{new}$, $r$);

8 $V\leftarrow V\cup\left\\{x_{new}\right\\}$;

9 $x_{min}\leftarrow$ $x_{nearest}$;

10 $c_{min}\leftarrow$ $C_{G}(x_{nearest},x_{new})$ ;

11 if  _CollisionFree( $x_{new},x_{near}$)_ then

12 for  _$x_{near}$ $\in$ $X_{near}$_ do

13 if  _${\rm C_{G}}(x_{near},x_{new}) <c_{min}$_ then

14 $x_{min}$ $\leftarrow$ $x_{near}$;

15 $c_{min}$ $\leftarrow$ ${\rm C_{G}}(x_{near},x_{new})$;

16 end if

17

18 end for

19 $E$ $\leftarrow$ $E$ $\cup$ $\left\\{(x_{min},x_{new})\right\\}$;

20 for  _$x_{near}$ $\in$ $X_{near}$_ do

21 if  _${\rm C_{G}}(x_{new},x_{near}) <{\rm Cost_{G}}(x_{near})$_ then

22 $x_{parent}\leftarrow SetParent(x_{near})$;

23
$E\leftarrow(E\backslash\left\\{(x_{parent},x_{near})\right\\})\cup\left\\{(x_{new},x_{near})\right\\}$;

24 end if

25

26 end for

27

28 end if

29

30 end if

31

32 end for

return $Tree$

Algorithm 2 The GAN-RRT* Algorithm

###  III-C GAN-RRT* Path Planner

The trained generator model is used in the sampling process of GAN-RRT*
algorithm. The Algorithm 2 shows the details of the GAN-RRT* algorithm. Taking
the start point as the root node, the algorithm gets $x_{rand}$ through
$Sample$ function. We can find the nearest node to $x_{rand}$ through the
$Nearest$ function and get $x_{new}$ through the $Steer$ function. Then, we
identify candidate nodes within a certain range $r$. After the introduction of
generative adversarial network, GAN-RRT* algorithm can calculate the cost
value of the current node through the $GAN_{g}$ function and distinguish
whether the current cost value is consistent with the current state through
the $GAN_{d}$ function. In addition, the GAN-RRT* algorithm can also calculate
the cumulative cost value of the current node based on candidate nodes.
Through the above process, the new node $x_{new}$ can re-select its parent
node $x_{parent}$ to optimize the current generated path. After re-selecting
the new parent node, the node $x_{new}$ also can choose its growing node to
reduce the cumulative cost value of the growth node. In the process, the
candidate nodes can select $x_{new}$ as their parent node corresponding to
lines 19 to 24 in algorithm 2.

As the sampling process repeats, the cost value of the obtained path tree
decreases and the GAN-RRT* algorithm will eventually generate an optimal path.

Figure 6: Xiaopang robot equipped with a depth camera in the real-world
experiments.

##  IV EXPERIMENTAL STUDIES AND RESULTS

In this work, we carry out simulation and real-world experiments based on the
Robot Operating System (ROS)[31] with Ubuntu 16.04 on a laptop equipped with
Intel i7-11800H CPU @2.3 GHz and 16 GB of RAM. Firstly, we design static maps
and produce demonstration paths in the simulation environment. Secondly, we
demonstrate the performance of the GAN-RRT* through several numerical
simulations. Finally, we evaluate the planning performance of the proposed
algorithm in 100 human-robot interaction environments. In the real-world
experimental studies, a robot named Xiaopang is adopted, as shown in Fig. 6.

To better evaluate the path generated by GAN-RRT*, we select three metrics for
objective evaluation: the dissimilarity, the feature difference and the
homotopic rate. The dissimilarity is numerically equal to the area of a closed
region enclosed by two paths.

$\displaystyle dissi$ $\displaystyle milarity({\zeta}_{1},{\zeta}_{2})=$ (5)
$\displaystyle\frac{\sum_{k=1}^{N-1}\frac{d({\zeta}_{1}(k),{\zeta}_{2})+d({\zeta}_{1}(k+1),{\zeta}_{2})\left|\left|{\zeta}_{1}(k)-{\zeta}_{1}(k+1)\right|\right|}{2}}{\delta}$

The $d$ function is used to calculate the distance between two path nodes.
$\delta$ is the number of segments of the path ${\zeta}_{1}$.

The feature difference is numerically equal to the absolute value of the
feature difference between the two paths.

$F_{fd}=\frac{1}{5*S}\sum_{i=1}^{S}\sum_{j=1}^{5}\left|f^{demo}_{(i,j)}-f^{plan}_{(i,j)}\right|$
(6)

$S$ represents the number of the testing scenarios.

The homotopy rate of generated paths and demonstration paths is defined as the
proportion of generated paths within the same homotopy class of the
demonstration path.

###  IV-A Performance in Simulations

In this section, we design 100 different static maps. The size of each map is
324 pixel × 257 pixel. And then, we compare the navigation effects of the
RRT*, the NN-RRT* and the GAN-RRT* in simulations. Among the 100 scenarios, we
selected 75 scenarios as the training set and 25 scenarios as the testing set.
On the testing set, the homotopic rate of RRT* is 75$\%$, the rate of NN-RRT*
is 90$\%$ and the rate of GAN-RRT* is 94$\%$. The experimental results show
that compared with RRT* and NN-RRT* methods, GAN-RRT* method achieves higher
homotopy rate and higher anthropomorphic degree of generated paths.

As shown in Fig. 7(a), in terms of dissimilarity, the dissimilarity of GAN-
RRT* is the lowest among the three methods, which means that the area of a
closed region enclosed by the path generated by GAN-RRT* and the demonstration
path is the smallest. As shown in Fig. 7(b), in terms of feature difference,
the feature difference of GAN-RRT* is also the lowest among the three methods,
which means that the paths generated by the GAN-RRT* planner are more similar
to demonstration paths. With the increase of the number of testing scenarios,
the paths generated by GAN-RRT* planner still maintain a high degree of
anthropomorphism thanks to the strong generalization performance of generative
adversarial networks.

(a) Difference of dissimilarity.

(b) Difference of features.

Figure 7: Difference of dissimilarity and features among RRT*, NN-RRT* and
GAN-RRT*. Figure 8: Comparison of paths generated by RRT*, NN-RRT*, GAN-RRT*
and volunteers. Figure 9: The general scheme of the questionnaire.

Fig. 8 provides a more intuitive representation of the difference between the
planned paths and the demonstration paths. As shown in Fig. 8(a) and Fig.
8(b), the paths generated by algorithms of GAN-RRT* and NN-RRT* can pass
behind pedestrians and reach the goal in uncomplicated human-robot interaction
scenarios. This follows the behavioral norms that avoid disturbing pedestrians
in daily life. However, pedestrians often appear in the form of crowds, which
requires the robot to plan the path as far as possible to avoid the people who
are communicating. As shown in Fig. 8(c)-(f), the algorithm GAN-RRT* also
performs well in complex crowd environments.

Figure 10: Turing-like test: Estimate the anthropomorphic confidence of path
generated by GAN-RRT*.

###  IV-B Performance in Real-World Experimental Studies

In the real-world experimental studies, we adopt the investigation
questionnaire to evaluate the socially aware performance of the propose
method. As shown in Fig. 9, the investigation questionnaire is divided into
four stages. We invited 30 students from different majors to complete the
survey. Firstly, we propose a series of issues related to navigation in human-
robot interaction environments. Secondly, participants choose one of the two
paths to verify the validity of their results in the first questionnaire
survey. Thirdly, we investigate the anthropomorphic confidence of paths
generated based on the GAN-RRT* in the third stage. Finally, we utilize a
Likert scale to compare paths generated by GAN-RRT*, NN-RRT* and RRT*.

From the perspective of psychology, people tend to score higher on their
preferred paths and believe that these paths have a high degree of
anthropomorphism. Therefore, we further verify the superiority of our
navigation algorithm by combining questionnaires and real-world experiments.

Stage 1-Collect Behavior habits

In order to collect the behavioral habits of participants in human-robot
interaction environments, we designed three questions as shown in Appendix.
From the relative results, it can be analyzed that 80$\%$ of people believe
that robots need to pay more attention to human-robot interaction status when
navigating. In complex environments, 86.67$\%$ of people choose to bypass the
crowd, and 13.33$\%$ choose to pass through the crowd. In addition, it is
generally believed that robots need to consider more factors during the
navigation, such as the distance between the robot and the pedestrian, the
position of the robot relative to the human, the velocity of the robot
relative to humans and the relationship between the path and human state.
However, paper-based questionnaires do not accurately reflect the real
thoughts of participants. Only when participants see the real planning path
can they give a relatively objective judgment. Therefore, we designed the
second stage to verify the validity of the survey questionnaire.

Stage 2-Questionnaire Validation

Because people can always choose their preferred path in comparison, we
present both the demonstration path and the RRT-generated path in the same
scenario. Participants will choose the path they prefer between the two paths.
If participants choose to pass through the crowd in the first stage and choose
the path generated by RRT in the second stage, then the questionnaire survey
is considered valid. Similarly, participants choose to cross from a sparsely
populated area and choose the demonstration path, the questionnaire survey is
also valid. Otherwise, we believe that the questionnaire survey is invalid.

Stage 3-The Anthropomorphic Confidence of Paths Generated by GAN-RRT*
Algorithm

Inspired by the Turing test[32], we propose a anthropomorphic confidence
calculating method for hybrid testing of multiple algorithms. As shown in Fig.
10, participants do not know the algorithm used for robot navigation
beforehand. For example, the path through the crowd in Fig. 10(a) is generated
by RRT and the path bypass the crowd in Fig. 10(b) is generated by GAN-RRT*.
In the same scenario, the robot travels in the real environment along the
demonstration path and the path generated by RRT and GAN-RRT*, respectively.
After each trial, the participants are required to choose the demonstration
path from the three paths. If the participant believe that the path generated
by GAN-RRT* is a demonstration path, this means that the GAN-RRT* generated
path is misjudged as the demonstration path.

We provide the following definition for anthropomorphic confidence of paths
generated by GAN-RRT* algorithm:

$F_{ac}=\sum_{i=1}^{S}\sum_{j=1}^{P_{s}}\frac{mis(i,j)}{n_{G}(i,j)*P_{S}}$ (7)

$mis(i,j)=\begin{cases}1,&\text{ $Path\ misjudged$ }\\\ 0,&\text{ $Path\ not\
misjudged$ }\end{cases}$ (8)

Herein, $S$ represents the number of scenarios, $P_{S}$ represents the number
of participants in this scenario, $mis(i,j)$ indicates that the path generated
by the GAN-RRT* is misjudged as the demonstration path and $n_{G}(i,j)$
represents whether the path is generated by the GAN-RRT*.

In this stage, participants are required to identify demonstration paths and
robot paths in total 10 scenarios. The final anthropomorphic confidence of
social navigation scheme based on GAN-RRT* is 80.77$\%$, which also proves
that our proposed navigation algorithm can achieve good results in human-robot
interaction environments.

Stage 4-Comparision between Paths based on RRT* and NN-RRT* and GAN-RRT*

In order to intuitively shown the performance of the navigation methods,
participants are further required to rate the paths generated by RRT*, NN-RRT*
and GAN-RRT*. Since it is difficult for participants to accurately describe
their acceptance of each path with words, we adopt Likert scale to give scores
for each path. Participants can score for the path between 1 and 5 according
to their behavior habits based on the Likert scale.

We selected 20 participants to conduct comparative experiments in 5 scenarios.
After the robot generates three paths in one scenario, participants are asked
to score for each path. As shown in Table I, the average scores of RRT*, NN-
RRT* and GAN-RRT* are 3.36, 3.6 and 4.18, respectively. The scores of RRT*,
NN-RRT* and GAN-RRT* in 5 scenarios are shown in Fig. 11. From the
experimental results, it can be seen that both the average scores and the
scores in 5 scenarios, GAN-RRT* is the highest. This also indicates that the
path generated by GAN-RRT* makes pedestrians more comfortable in human-robot
interaction environments.

TABLE I: The scores of RRT*, NN-RRT* and GAN-RRT*. Environment | Map Size | Pedestrians | RRT* | NN-RRT* | GAN-RRT*
---|---|---|---|---|---
Scenario1 | 5m $\times$ 5m | 3 | 3.5 | 3.6 | 4.1
Scenario2 | 5m $\times$ 5m | 3 | 3.5 | 3 | 4
Scenario3 | 5m $\times$ 5m | 3 | 3 | 3.5 | 4.3
Scenario4 | 5m $\times$ 5m | 3 | 2.8 | 4 | 4.3
Scenario5 | 5m $\times$ 5m | 3 | 4 | 3.9 | 4.2
Average | 5m $\times$ 5m | 3 | 3.36 | 3.6 | 4.18
Figure 11: The average scores of RRT*, NN-RRT* and GAN-RRT* in all 5
scenarios.

##  V CONCLUSION

This article has proposed a sampling-based path planning algorithm GAN-RRT* to
guide robot to navigate in human-robot interaction environments. Firstly, a
generative adversarial network model with strong generalization performance is
proposed to adapt the navigation algorithm to more scenarios. Secondly, we
propose the new navigation algorithm GAN-RRT* by combining the GAN model with
RRT* to guide robots to generate paths in human-robot interaction scenarios.
Finally, the socially adaptive path planning framework GAN-RTIRL is proposed
to improve the generalization performance of robot navigation and the homotopy
rate between the generated path and the demonstration path. We conducted
comparative experiments in simulation and real world. The experimental results
have revealed that the proposed navigation algorithm greatly improves the
generalization performance of robot navigation in human-robot interaction
environments and the homotopy rate between the generated path and the
demonstration path. In complex and densely populated environments, the paths
generated by our method are more socially aware and anthropomorphic.

Questionnaire

---
Question 1:
Which factors need to be considered more when robots navigate
in human-robot interaction environments?
Answer 1:
(a)Path length;
(b)Human-robot interaction state.
Question 2:
Which navigation strategy would you choose when robots
navigate in a complex human-robot interaction environment?
Answer 2:
(a)Keep the path length as short as possible;
(b)Pass through sparsely populated areas.
Question 3:
Which of the following factors should be considered in the
process of human-robot interaction?
Answer 3:
(a)The distance between the robot and the person;
(b)The position of the robot relative to the human;
(c)The movement speed of robots relative to humans;
(d)The relationship between the path and human state.
Question 4:
The robot will generate paths twice based on different
algorithms. Please choose the path you prefer from the next
two paths and write down your answer.
Answer 4:
(a)The first path.
(b)The second path.
Note: Please give your choice after both paths are generated.
Question 5:
The robot will generate paths three times based on different
algorithms. Please choose the path that you think is artificially
generated from the next three paths and write down your answer.
Answer 5:
(a)The first path.
(b)The second path.
(c)The third path.
Note: Please give your choice each time the path is generated.
Question 6:
We will let the robot generate paths three times.
Please express how much you agree with the generated path.
One point represents strong disagreement, two points represent
disagreement, three points represent common, four points
represent agreement and five points represent strong agreement.
Answer 6:
(a)Strong disagreement.
(b)Disagreement.
(c)Common.
(d)Agreement.
(e)Strong agreement.
Note: Please give your choice each time the path is generated.

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|  Yao Wang received the bachelor degree in aircraft manufacture engineering
from the Changzhou Institute of Technology, Changzhou, China, in 2021. He is
currently pursuing the master degree in mechanical engineering from the School
of Mechanical and Electrical Engineering, Soochow University, Suzhou, China.
His research field include mobile robot social navigation and motion planning.
---|---
|  Yuqi Kong received the bachelor degree in mechanical and electronic
engineering from the Changshu Institute of Technology, Suzhou, China, in 2021.
He is currently pursuing the master degree in mechanical engineering from the
School of Mechanical and Electrical Engineering, Soochow University, Suzhou,
China. His research field include mobile robot social navigation and motion
planning.
---|---
|  Wenzheng Chi received her B.E. degree in automation from Shandong
University, Jinan, China, in 2013 and the Ph.D. degree in Biomedical
Engineering from the Department of Electronic Engineering, The Chinese
University of Hong Kong, Hong Kong, in 2017. During which she spent six months
at The University of Tokyo, Japan, as a Visiting Scholar. She was a Post-
Doctoral Fellow with the Department of Electronic Engineering, The Chinese
University of Hong Kong, Hong Kong, from 2017 to 2018. She is currently an
Associate Professor with the Robotics and Microsystems Center, School of
Mechanical and Electric Engineering, Soochow University, Suzhou, China. Her
research interests include mobile robot path planning, intelligent perception,
human-robot interaction, etc.
---|---
|  Lining Sun was born in Hegang, Heilongjiang Province, China, in January
1964. He received the Ph.D. degree in engineering from the Mechanical
Engineering Department, Harbin Institute of Technology (HIT), Harbin, China,
in 1993. After graduation, he joined HIT. He formed an internationally
competitive robotics team and made outstanding contributions to the creation
and development of robot-related disciplines in China. Dr. Sun is a National
Outstanding Youth Fund Winner and a Changjiang Scholar Distinguished Professor
by the Ministry of Education.
---|---


#  Neutrinos with refractive masses and the DESI BAO results

Manibrata Sen<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Alexei Y. Smirnov  smirnov@mpi-
hd.mpg.de Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany

###### Abstract

Due to interactions with dark matter, neutrinos can acquire refractive masses
which explain the data from oscillation experiments. We study the effects of
relic neutrinos with refractive masses on structure formation in the Universe.
In the model with a light fermionic mediator, above the resonance energy,
$E_{R}$, associated with the mass of the mediator, refractive masses have all
the properties identical to the usual vacuum masses. Below the resonance,
refractive masses decrease with neutrino energy, however, they cannot be used
in the same way as usual masses. We study the dispersion relations and group
velocities of such neutrinos and their dependence on redshift. We show that in
the epoch of structure formation, relic neutrinos were ultrarelativistic and
essentially massless particles for $E_{R}=(10-10^{5})\,$eV. This allows to
reconcile the values of masses extracted from oscillation experiments with the
stringent bounds on sum of neutrino masses from cosmological surveys.

##  I Introduction

Recent measurements of Baryonic Acoustic Oscillations (BAO) from the first
year of operation of the Dark Energy Spectroscopic Instrument (DESI)
DESI:2024mwx , along with measurements of the Cosmic Microwave Background
(CMB) polarization, temperature and lensing by PLANCK Planck:2019nip , as well
as data from the ACT ACT:2023dou , constrain the sum of neutrino masses $\sum
m_{\nu}^{\rm cosm}<0.072\,{\rm eV}$ at $95\%$ confidence level (C.L). This
bound disfavors the inverted mass ordering (IO) of neutrinos at $3\sigma$.
Furthermore, the posterior distribution of $\sum m_{\nu}^{\rm cosm}$ peaks at
zero, thus indicating a slight tension with even normal mass ordering (NO). A
more stringent bound, $\sum m_{\nu}<0.043\,$eV at $2\sigma$ C.L., was obtained
by combining DESI BAO and CMB results with Supernova Ia, GRB and X-ray
observations Wang:2024hen .

Recall that oscillation data being sensitive to neutrino mass-squared
differences allows one to get a lower bound on neutrino masses. In the case of
normal mass ordering (NO), $m_{3}\geq\sqrt{\Delta m_{31}^{2}}$ and
$m_{2}\geq\sqrt{\Delta m_{21}^{2}}$. This leads to a lower bound on the sum of
masses: $\sum m_{\nu}^{\rm osc}\approx 0.057\,$eV. In the case of inverted
ordering (IO), $m_{1}\approx m_{2}\geq\sqrt{\Delta m_{31}^{2}}$, we obtain
$\sum m_{\nu}^{\rm osc}=0.11\,$eV. Thus,

$\sum m_{\nu}^{\rm osc}>\sum m_{\nu}^{\rm cosm}$ (1)

for IO and even for NO in the peak of posterior distribution. Future
observations can further strengthen the cosmological bound. The reason of this
tension could be some cosmological uncertainties. In Ref. Craig:2024tky , it
was claimed that the DESI measurement seems to favor $\sum m_{\nu}<0$, if a
positive $\sum m_{\nu}$ is not imposed a priori. Such a measurement, if
confirmed, can point to enhanced matter clustering, and can be explained in a
scenario where dark matter interacts with itself through long-range forces.

Does the inequality in Eq. (1) make sense? Can the sum of neutrino masses
determined in oscillation experiments be reconciled with the strong
cosmological bound? Notice that the oscillation masses are determined at
present epoch, $z=0$, at neutrino energies $E>0.1\,$MeV, and in the solar
system. In contrast, the cosmological bound concerns the average neutrino mass
in the Universe, during the epoch of structure formation, $z=10-1000$ and at
low neutrino energies, $E=(0.01-1)\,$eV. Can this difference explain the
inequality in Eq. (1)? Of course, usual neutrino masses,generated by the Higgs
vacuum expectation value (VEV) in local interactions, should not change with
$E$, $z$ and location.

There are several possibilities which could explain the inequality in Eq. (1).
Neutrinos can be massless in the early Universe and pick up a mass later due
to a late phase transition Dvali:2016uhn ; Lorenz:2018fzb ; Lorenz:2021alz .
Neutrino masses can be time-varying, either as a result of coupling directly
with ultralight scalar fields, or through mixing with sterile neutrinos, which
in turn, interact with an ultralight scalar Fardon:2003eh . In these
scenarios, neutrino masses are small in the early Universe, but later, after
neutrinos become non-relativistic, the masses grow to the present values.
Other ideas include neutrino decays or annihilation to lighter particles,
thereby reducing the energy density in massive neutrinos, or cooling the
neutrinos, through new interactions with dark matter Craig:2024tky .

Thus, the cosmological surveys raise an important question on the nature of
the neutrino mass: is it generated through a VEV in a short range interaction,
or does it have a dynamical or refractive origin?

In this paper, we explore the possibility that the cosmological limit on the
sum of neutrino masses can be stronger than that from oscillations. We assume
that neutrino masses have a dynamical nature, such that the effective masses
of the relic neutrinos were smaller in the epoch of structure formation in the
Universe than now. Scenarios in which neutrinos acquire masses partially or
completely through interactions with light Dark Matter have been considered in
Berlin:2016woy ; Krnjaic:2017zlz ; Brdar:2017kbt ; Capozzi:2018bps ;
Choi:2019zxy ; Dev:2020kgz ; Choi:2020ydp ; Losada:2021bxx ; Huang:2021kam ;
Chun:2021ief ; Dev:2022bae ; Huang:2022wmz ; Davoudiasl:2023uiq ;
Losada:2023zap ; Gherghetta:2023myo .

We use the scenario studied in Sen:2023uga , where neutrinos acquire an
effective mass-squared via scattering on ultralight Dark matter (ULDM),
mediated by a light fermion. We consider the evolution of relic neutrinos with
refractive masses and show that in the epoch of structure formation, they
appear essentially as massless particles, thus allowing to satisfy the
cosmological bound and the inequality in Eq. (1). In Sec. II, we consider the
neutrino potential generated by interactions with DM and its dependence on the
redshift $z$, the resonance energy and the C-asymmetry of the DM. In Sec. III,
we study the key elements relevant for structure formation: the dispersion
relations of neutrinos and antineutrinos, the group velocities of relic
neutrinos and their evolution, the relativistic to non-relativistic
transition. In Sec. IV, we discuss the obtained results and conclude.

##  II Neutrino potential and refractive mass and their evolution

We assume that massless neutrinos interact with ULDM composed of scalar bosons
$\phi$, via a light fermionic mediator $\chi$, through the effective Yukawa
interaction $g\,\bar{\chi}\,\nu\,\phi+{\rm h.c}\,$. Elastic forward scattering
of neutrinos on the cold gas of $\phi$ produces an effective potential
Sen:2023uga

$V=\frac{m^{2}_{\rm
asy}}{2E_{R}}~{}\frac{y-\epsilon}{y^{2}-1}\,\,,\,\,\,\,\,\,\,m^{2}_{\rm
asy}\equiv\frac{g^{2}\,(n_{\phi}+\bar{n}_{\phi})}{m_{\phi}}\,.$ (2)

Here $m^{2}_{\rm asy}$ is the effective mass-squared,
$n_{\phi}(\bar{n}_{\phi})$ is the number density of ULDM particles
(antiparticles) and $m_{\phi}$ is the ULDM mass. We define $y\equiv
E_{\nu}/E_{R}$. The resonance energy $E_{R}=m_{\chi}^{2}/(2m_{\phi})$, where
$m_{\chi}$ is the mass of the mediator and
$\epsilon\equiv(n_{\phi}-\bar{n}_{\phi})/(n_{\phi}+\bar{n}_{\phi})$ is the
C-asymmetry of DM. For non-relativistic DM, the energy density equals
$\rho_{\phi}\simeq m_{\phi}(n_{\phi}+\bar{n}_{\phi})$, and hence $m^{2}_{\rm
asy}=g^{2}\rho_{\phi}/m_{\phi}^{2}$. The potential for antineutrinos is given
by the same expression as Eq. (4) with the substitution
$\epsilon\rightarrow-\epsilon$.

At energies much above the resonance, $y\gg 1$, the potential becomes
$V\approx m^{2}_{\rm asy}/2E$, thus reproducing the standard term due to
neutrino mass in the Hamiltonian of flavor evolution of neutrinos. Therefore,
$m^{2}_{\rm asy}$ can be identified with the mass-squared extracted from the
oscillation experiments, e.g., $m^{2}_{\rm asy}\approx\Delta m^{2}_{31}$, etc.
There are two viable regions of parameters of the model in which the observed
values of oscillation parameters can be reproduced and all the existing bounds
are satisfied Sen:2023uga : (i) $m_{\phi}\sim 10^{-10}~{}$eV, $g\sim
10^{-10}$, $m_{\chi}\sim 10^{-4}~{}$eV, and (ii) $m_{\phi}\sim 10^{-22}~{}$eV,
$g\sim 10^{-21}$, $m_{\chi}\sim 10^{-10}~{}$eV.

In the opposite case, $y\ll 1$, relevant for the relic neutrinos, we have

$V\approx\frac{m^{2}_{\rm
asy}}{2E_{R}}~{}\epsilon=\frac{g^{2}}{m_{\chi}^{2}}(n_{\phi}+\bar{n}_{\phi})\epsilon=\frac{g^{2}}{m_{\chi}^{2}}(n_{\phi}-\bar{n}_{\phi})\,,$
(3)

which coincides with the Wolfenstein potential Wolfenstein:1977ue .

Let us consider the evolution of the average potential in the Universe for
relic neutrinos (see Fig. 1). The potential is proportional to the number
density of dark matter and therefore it increases with $z$ as $(1+z)^{3}$:

$V(z)=\frac{m^{2}_{\rm
asy}}{2E_{R}}\xi(1+z)^{3}~{}\frac{y(z)-\epsilon}{y(z)^{2}-1}\,.$ (4)

Here $y(z)=y_{0}(1+z)$, $y_{0}=E_{0}/E_{R}$ and $E_{0}=5\cdot 10^{-4}\,$eV is
the present ($z=0$) average energy of relic neutrinos. In Eq. (4),
$\xi^{-1}\simeq 10^{5}$ is the local overdensity of DM. For $\epsilon\neq 0$
and $y\ll 1$, the expression in Eq. (4) simplifies as

$V(z)\approx\frac{m^{2}_{\rm asy}}{2E_{R}}\epsilon\xi(1+z)^{3},$ (5)

that is, it does not depend on the energy of the neutrino. In the case of
$\epsilon=0$, we obtain from Eq. (4),

$V(z)=-\frac{m^{2}_{\rm asy}}{2E_{R}}y_{0}~{}\xi(1+z)^{4}.$ (6)

The moduli of this potential increases with $z$ faster than that in Eq. (4),
but has much smaller value at $z=0$ (see Fig. 1).

We can introduce the refractive mass-squared as $\tilde{m}^{2}\equiv
2yVE_{R}=2VE$. It is proportional to the DM density and hence depends on
space-time coordinates. For $y\gg 1$, the refractive mass-squared has all the
properties identical to the vacuum mass-squared. Below the resonance, $y<1$,
the mass-squared decreases as $y\epsilon$ for $\epsilon\neq 0$ and as $y^{2}$
for $\epsilon=0$, _i.e._ , a sharper decline. Furthermore, the effective mass-
squared has opposite signs for neutrinos and antineutrinos. This decrease of
modulus of the potential indicates that the bound on sum of usual neutrino
masses can be avoided. However, as we will see later, below resonance, the
refractive mass cannot be used in the same way as the usual mass, and, in
particular, it cannot be used in consideration of cosmological bound.

The decrease of mass with energy also affects kinematical measurements of
neutrino mass in the beta decay, in particular, of ${}^{3}H$ as used by the
KATRIN collaboration Aker:2024drp . The decay rate depends on the neutrino
mass via the neutrino momentum $p_{\nu}$: $\Gamma\propto p_{\nu}$. In the case
of the usual mass, we have
$\sqrt{E_{\nu}^{2}-m_{\nu}^{2}}=\sqrt{(Q_{e}-T_{e})^{2}-m_{\nu}^{2}}$, where
$Q_{e}$ is the end point of spectrum and $T_{e}$ is the kinetic energy of
electron. In the refractive mass case, $p_{\nu}=Q_{e}-T_{e}-V$. This would be
equivalent to $m_{\nu}^{2}=2p_{\nu}V$, and the latter can be negative. For
$E_{R}=100\,$eV, $|m_{\rm asy}^{2}|=\Delta m^{2}_{21}$ relevant for beta decay
and $p_{\nu}=1\,$eV, this gives $|m_{\nu}|\sim 10^{-3}\,$eV - much below the
KATRIN sensitivity.

##  III Neutrinos with refractive mass and DESI observations

The cosmological limit on the sum of neutrino masses rely on a determination
of the amount of matter clustering during structure formation. The information
about clustering is obtained through an accurate measurement of the CMB
lensing power spectra (from PLANCK and ACT) and the net amount of non-
relativistic matter content from the BAO, recently observed by DESI with high
precision.

Let us first outline how neutrinos with usual vacuum mass affect structure
formation. Neutrino masses determine the scale of free streaming,
$\lambda_{\rm FS}(t)\sim v_{g}(t,m)/H(t)$, where $v_{g}$ is the group velocity
and $H$ is the Hubble constant. At smaller scales, $\lambda<\lambda_{\rm FS}$,
neutrinos do not cluster and damp density perturbations of DM and baryons.
Correspondingly, neutrinos erase perturbations with wavenumbers $k$ bigger
than free streaming wavenumber, $k_{\rm FS}\sim 1/\lambda_{\rm FS}$.
Essentially, neutrino masses start affecting the power spectrum of
perturbations, $P_{m}(k)$, at $k_{\rm nr}$, which corresponds to a transition
to non-relativistic regime, when $p_{\nu}=m_{\nu}$. The damping effect of the
power spectrum is determined by
$f_{\nu}\equiv(\rho_{\nu}+\rho_{\bar{\nu}})/\rho_{m}$ – the fraction of energy
density in neutrinos and antineutrinos in the total matter energy density,
$\rho_{m}$. In the non-relativistic case, $\rho_{\nu}\simeq m_{\nu}n_{\nu}$,
where $n_{\nu}$ ($\bar{n}_{\nu}$) is the number density of neutrinos
(antineutrinos). There are two main reasons of suppression of the power
spectrum at $k>k_{\rm nr}$: (i) since neutrinos do not cluster, their fraction
$f_{\nu}$ should be subtracted from the power spectrum, and (ii) neutrinos
reduce the rate of growth of perturbation by contributing to expansion rate of
the Universe (thus weakening the gravitational clustering) but not clustering
themselves. As a result, a suppression of power spectrum equals approximately
Lesgourgues:2012uu

$\frac{P_{m_{\nu}}(k)}{P_{0}(k)}=1-8f_{\nu}\,,$ (7)

where $P_{0}(k)$ is the spectrum in the case of massless neutrinos. Increase
of $f_{\nu}$ due to increase of neutrino mass leads to stronger suppression of
the power spectrum. Hence, to estimate the effect of neutrinos with refractive
mass on structure formation, we need to find their group velocities, energy
density and explore the relativistic to non-relativistic transition.

Figure 1: Dependence of the neutrino momentum and the effective potential on
redshift $z$ for different values of the resonant energy $E_{R}$ and DM charge
asymmetry, $\epsilon=1$ (solid lines) and $\epsilon=0$ (dashed lines). We take
$m_{\rm asy}=\sqrt{\Delta m_{31}^{2}}=0.05\,$eV.

For massless neutrinos with refraction effect, the dispersion relation reads

$E=p_{\nu}+V.$ (8)

Fig. 1 shows the dependence of $p_{\nu}$ and $V$ on $z$. For small redshift,
one finds $V\ll p_{\nu}$, but as $z$ increases, $V$ increases faster than
$p_{\nu}$ and they become comparable at $z\gtrsim 10^{3}$. An important
quantity is the ratio $V/p$ which determines the perturbativity of approach
Sen:2023uga . For $y\ll\epsilon\leq 1$, we have

$\frac{V(z)}{p(z)}=\frac{m_{\rm asy}^{2}}{2E_{R}E_{0}}\xi\epsilon(1+z)^{2}\,.$
(9)

According to Eq. (9), the perturbativity condition $V(z)/p(z)\leq 0.1$ gives

$(1+z)\leq 6.3\cdot 10^{3}\sqrt{\frac{E_{R}}{(10^{2}\,{\rm
eV})\,\epsilon}}\,.$ (10)

For $\epsilon=0$, we obtain

$\frac{V(z)}{p(z)}=-\frac{m_{\rm asy}^{2}}{2E_{R}^{2}}\xi(1+z)^{3}\,,$ (11)

and the inequality $|V/p|\leq 0.1$ leads to

$(1+z)\leq 3\cdot 10^{4}\left(\frac{E_{R}}{10^{2}\,{\rm eV}}\right)^{2/3}\,.$
(12)

For a given $z$, the perturbativity conditions, Eqns. (10) and (12), constrain
from below the values of $E_{R}$.

According to Eq. (8), the group velocity of neutrinos equals

$v_{g}=\frac{dE_{\nu}}{dp_{\nu}}=1-\frac{dV}{dp}\,.$ (13)

Using the expression for $V(z)$ from Eq. (4), we find the deviation of $v_{g}$
from 1 in the epoch $z$,

$1-v_{g}=\frac{\tilde{m}^{2}_{\rm
asy}}{2E_{R}^{2}}\xi(1+z)^{3}\frac{1+y(z)^{2}-2\epsilon
y(z)}{\left[1-y(z)^{2}\right]^{2}}\,.$ (14)

Dependence of the deviation $1-v_{g}$ on $z$ is shown in Fig. 2 for different
values of $E_{R}$. It is compared with the standard mass case
$1-v_{g}=p/E\approx m^{2}/2E^{2}$. For small $y$ (realized for parameters in
this plot), the deviation in Eq. (14) does not depend practically on
$\epsilon$, in contrast to $V$ and $\tilde{m}^{2}$. Indeed, for $y\ll 1$, Eq.
(14) gives

$1-v_{g}=\frac{\tilde{m}^{2}_{\rm asy}}{2E_{R}^{2}}\xi(1+z)^{3}\,.$ (15)

The deviation does not depend on energy. Ratio of the deviations for
refractive and usual masses equals $\xi(1+z)^{3}E^{2}/E_{R}^{2}$, which is
small for $E^{2}\ll E_{R}^{2}$. For $m_{\rm asy}^{2}=m^{2}=\Delta m^{2}_{31}$,
they become comparable only for $z>1000$. According to Fig. 2, neutrinos with
refractive mass $\sim 0.05\,$eV are ultrarelativistic in whole time interval
of structure formation upto $z=10^{3}$ for $E_{R}>10\,$eV.

Group velocities of neutrinos with usual and refractive masses have an
opposite dependence on $z$: in the usual case, with increase of $z$, one has
$v_{g}\rightarrow 1$ and neutrinos become relativistic; on the other hand, in
the refractive case, $v_{g}\rightarrow 0$ and neutrinos become non-
relativistic. The latter may happen in the epoch of radiation-dominated
Universe.

The deviation, Eq. (14), differs from the one obtained by naive substitution
$m^{2}\rightarrow\tilde{m}^{2}$ in the standard formula for the group
velocity: $1-v_{g}=\tilde{m}^{2}/(2p^{2})=V/p$ This is bigger than the
estiation in Eq. (14) by the factor $\epsilon E_{R}/p$. This shows that
$\tilde{m}^{2}$ cannot be used in the same way as usual mass-squared.

Figure 2: The deviation of the group velocity of neutrinos from 1 as a
function of the redshift for different values of $E_{R}$ (numberson top of the
curves). The lines for $\epsilon=0$ and $\epsilon=1$ practically coincide. The
standard case for relativistic neutrinos is also shown. We use $m_{\rm
asy}=\sqrt{\Delta m_{31}^{2}}=0.05\,$eV.

The energy densities in neutrinos and antineutrinos equal

$\displaystyle\rho_{\nu}$ $\displaystyle=$ $\displaystyle
E_{\nu}n_{\nu}=(p_{\nu}+V)n_{\nu}\,,$ $\displaystyle\rho_{\bar{\nu}}$
$\displaystyle=$ $\displaystyle
E_{\nu}\bar{n}_{\nu}=(p_{\nu}-V)\bar{n}_{\nu}\,.$ (16)

Their sum does not depend on $V$ for $n_{\nu}=\bar{n}_{\nu}$, being the same
as in the case for massless neutrinos. For non-zero lepton asymmetry, $\Delta
L\equiv(n_{\nu}-\bar{n}_{\nu})/(n_{\nu}+\bar{n}_{\nu})$, we have

$\rho_{tot}=\rho_{\nu}+\rho_{\bar{\nu}}=p_{\nu}(n_{\nu}+\bar{n}_{\nu})\left[1+\frac{V}{p}\Delta
L\right].$ (17)

The key parameters relevant for cosmology are $E_{R}$, $(1+z)$ and $\epsilon$.
Different bounds in the $E_{R}-(1+z)$ plane are shown in Fig. 3. We present
the lines $V(z)/p(z)=0.1$ for $\epsilon=0$ and $\epsilon=1$ which restrict the
region of perturbativity (colored regions are excluded). The bounds – lower on
$E_{R}$ and upper on $(1+z)$ – are stronger in the case of $\epsilon=1$.

Interestingly, for $E_{R}=(10-10^{5})\,$eV, neutrinos with refractive mass are
in the perturbation regime at energies relevant for both oscillations
Sen:2023uga  and for relic neutrinos in epoch of structure formation.

The resonance condition $y=1$ yields $E_{R}=E_{0}(1+z)$. According to Fig. 3,
the perturbativity requirement is more stringent than $y\leq 1$. So, in the
region $V(z)<p(z)$, we have $y\ll 1$, which justifies, in particular, the use
of expression in Eq. (15).

Fig. 3 also shows the line corresponding to $v_{g}=1/\sqrt{2}$, or $p=m$ in
usual mass case, that is, the transition to the non-relativistic regime. Using
Eq. (15), we find an analytical expression of this line:

$(1+z)_{\rm nr}=\left(\frac{E_{R}^{2}}{\xi m_{\rm asy}^{2}}\right)^{1/3}\,.$
(18)

It is shifted by a factor of $5^{1/3}$ to larger values of $(1+z)$ with
respect to the border line of the perturbativity condition for $\epsilon=0$.
For bigger values $E_{R}$ or smaller values of $(1+z)$ with respect to this
line, neutrinos can be considered as relativistic. In this region, the
situation for structure formation is similar to that for massless neutrinos.

From Fig. 3, for a given $E_{R}$ at a large $z$, we have $(1+z)>(1+z)_{\rm
nr}$, implying neutrinos are non-relativistic, with $v_{g}\ll 1$, and
consequently, small free-streaming scale. This would correspond to
$f_{\nu}\approx 0$ and neutrino contributing to cold DM. As $z$ decreases,
$(1+z)<(1+z)_{\rm nr}$, neutrinos become relativistic and $f_{\nu}\simeq
f_{\nu}(m=0)$. For $E_{R}\sim 1\,$eV, the transition from non-relativistic to
relativistic cases occurs at $z=10^{2}-10^{3}$, thus affecting structure
formation. This may appear as some effect of neutrino mass but distortion of
the power spectrum is expected to be different. Note, however, that for
$\epsilon=0$, we have $V/E=0.5$ along the border line $(1+z)_{\rm nr}$, so
perturbative consideration may not work well in this range.

Figure 3: Perturbativity constraints $V/p\leq 0.1$ in the $E_{R}-(1+z)$ plane
for $\epsilon=0$ and $\epsilon=1$. The colored parts show the excluded
regions. The dashed line corresponds to the group velocity $v_{g}=1/\sqrt{2}$
or, equivalently to $(1+z)_{nr}$. Also shown is the region of neutrino
energies above resonance given by $y\geq 1$.

The effective neutrino mass can be defined via the group velocity as $m_{\rm
eff}=p(v_{g}^{-2}-1)^{1/2}$ as in the standard case. Using this equation and
Eq. (15) for $(1-v_{g})\ll 1$, we obtain

$m_{\rm eff}\approx p\sqrt{2(1-v_{g})}\approx m_{\rm
asy}\frac{p}{E_{R}}\sqrt{\xi(1+z)^{3}}\,.$ (19)

It should be compared with $\tilde{m}=\sqrt{\tilde{m}^{2}}=\sqrt{2pV}$.
Following Eq. (13), one can write $m_{\rm
eff}=p\sqrt{2dV/dp}=\sqrt{2p^{2}dV/dp}$. Explicitly, the ratio of these masses
equals

$\frac{m_{\rm eff}}{\tilde{m}}=\sqrt{\frac{p(z)}{E_{R}}}\,.$ (20)

So, $m_{\rm eff}$ can be orders of magnitude smaller than the refractive mass
$\tilde{m}$. This shows again that refractive mass cannot be used at low
energies for description of structure formation. Correspondingly, the
cosmological bound on the parameter space of the model is weaker than that
obtained in Sen:2023uga .

##  IV Conclusion

We studied the cosmological evolution of neutrinos with refractive masses and
their influence on structure formation in the Universe. In the model with a
light fermionic mediator, below the resonance energy, the refractive masses
decrease with $E_{\nu}$, indicating that cosmological bound on the sum of
neutrino masses can be avoided. However, below resonance, one cannot use the
refractive masses in the same way as the usual (vacuum) masses. Therefore, the
effects of neutrinos on structure formation should be considered directly
without invoking the refractive masses.

Using the dispersion relations, we computed the group velocity of neutrinos
and studied its properties. We showed that for $z\leq 10^{3}$ and
$E_{R}>100\,$eV, the deviation $1-v_{g}\ll 1$, i.e., neutrinos are
ultrarelativistic. Furthermore, in this range of $E_{R}$ and $z$, one finds
$V/p\ll 1$, so that both the dispersion relation and the total energy
practically coincide with those for massless neutrinos. Neutrinos with
refractive masses appear as massless particles in the epoch of structure
formation and therefore the cosmological analysis, if interpreted in terms of
usual neutrino mass, would give zero mass. This removes the tension, Eq. (1),
between the sum of neutrino masses determined from oscillation results and the
bounds derived from cosmology. As a result, not only normal ordering and
inverted ordering mass spectra but also quasi-degenerate spectrum is allowed.

Searches of non-zero neutrino masses in cosmology is a test of nature of
neutrino masses, and in particular, their possible refraction origins.

## Acknowledgments

We thank the Galileo Galilei Institute for Theoretical Physics for the
hospitality and the INFN for partial support during the completion of this
work.

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